Table of Contents
 
 
 
Contents Overview viii List of Illustrations xiv Words of Thanks xix Part I: GEB  
 
 
 
Contents  II  
 
 
Part II EGB  
 
Prelude ... 275 Chapter X: Levels of Description, and Computer Systems 285 Ant Fugue 311 Chapter XI: Brains and Thoughts 337 English French German Suit 366 Chapter XII: Minds and Thoughts 369 Aria with Diverse Variations 391 Chapter XIII: BlooP and FlooP and GlooP 406 Air on G's String 431 Chapter XIV: On Formally Undecidable Propositions of TNT and Related Systems 438 Birthday Cantatatata ... 461 Chapter XV: Jumping out of the System 465 Edifying Thoughts of a Tobacco Smoker 480 Chapter XVI: SelfRef and SelfRep 495 The Magn fierab, Indeed 549 Chapter XVII: Church, Turing, Tarski, and Others 559 SHRDLU, Toy of Man's Designing 586 Chapter XVIII: Artificial Intelligence: Retrospects 594 Contrafactus 633 Chapter XIX: Artificial Intelligence: Prospects 641 Sloth Canon 681 Chapter XX: Strange Loops, Or Tangled Hierarchies 684 SixPart Ricercar 720  
 
Notes 743 Bibliography 746 Credits 757 Index 759  
 
Contents  III  
 
 
Overview Part I: GEB Introduction: A MusicoLogical Offering. The book opens with the story of Bach's Musical Offering. Bach made an impromptu visit to King Frederick the Great of Prussia, and was requested to improvise upon a theme presented by the King. His improvisations formed the basis of that great work. The Musical Offering and its story form a theme upon which I "improvise" throughout the book, thus making a sort of "Metamusical Offering". Selfreference and the interplay between different levels in Bach are discussed: this leads to a discussion of parallel ideas in Escher's drawings and then Gödel’s Theorem. A brief presentation of the history of logic and paradoxes is given as background for Gödel’s Theorem. This leads to mechanical reasoning and computers, and the debate about whether Artificial Intelligence is possible. I close with an explanation of the origins of the bookparticularly the why and wherefore of the Dialogues. ThreePart Invention. Bach wrote fifteen threepart inventions. In this threepart Dialogue, the Tortoise and Achillesthe main fictional protagonists in the Dialoguesare "invented" by Zeno (as in fact they were, to illustrate Zeno's paradoxes of motion). Very short, it simply gives the flavor of the Dialogues to come. Chapter I: The MUpuzzle. A simple formal system (the MIL'system) is presented, and the reader is urged to work out a puzzle to gain familiarity with formal systems in general. A number of fundamental notions are introduced: string, theorem, axiom, rule of inference, derivation, formal system, decision procedure, working inside/outside the system. TwoPart Invention. Bach also wrote fifteen twopart inventions. This twopart Dialogue was written not by me, but by Lewis Carroll in 1895. Carroll borrowed Achilles and the Tortoise from Zeno, and I in turn borrowed them from Carroll. The topic is the relation between reasoning, reasoning about reasoning, reasoning about reasoning about reasoning, and so on. It parallels, in a way, Zeno's paradoxes about the impossibility of motion, seeming to show, by using infinite regress, that reasoning is impossible. It is a beautiful paradox, and is referred to several times later in the book. Chapter II: Meaning and Form in Mathematics. A new formal system (the pqsystem) is presented, even simpler than the MIUsystem of Chapter I. Apparently meaningless at first, its symbols are suddenly revealed to possess meaning by virtue of the form of the theorems they appear in. This revelation is the first important insight into meaning: its deep connection to isomorphism. Various issues related to meaning are then discussed, such as truth, proof, symbol manipulation, and the elusive concept, "form". Sonata for Unaccompanied Achilles. A Dialogue which imitates the Bach Sonatas for unaccompanied violin. In particular, Achilles is the only speaker, since it is a transcript of one end of a telephone call, at the far end of which is the Tortoise. Their conversation concerns the concepts of "figure" and "ground" in various  
 
Overview  IV  
 
 
contexts e.g., Escher's art. The Dialogue itself forms an example of the distinction, since Achilles' lines form a "figure", and the Tortoise's linesimplicit in Achilles' linesform a "ground". Chapter III: Figure and Ground. The distinction between figure and ground in art is compared to the distinction between theorems and nontheorems in formal systems. The question "Does a figure necessarily contain the same information as its ground%" leads to the distinction between recursively enumerable sets and recursive sets. Contracrostipunctus. This Dialogue is central to the book, for it contains a set of paraphrases of Gödel’s selfreferential construction and of his Incompleteness Theorem. One of the paraphrases of the Theorem says, "For each record player there is a record which it cannot play." The Dialogue's title is a cross between the word "acrostic" and the word "contrapunctus", a Latin word which Bach used to denote the many fugues and canons making up his Art of the Fugue. Some explicit references to the Art of the Fugue are made. The Dialogue itself conceals some acrostic tricks. Chapter IV: Consistency, Completeness, and Geometry. The preceding Dialogue is explicated to the extent it is possible at this stage. This leads back to the question of how and when symbols in a formal system acquire meaning. The history of Euclidean and nonEuclidean geometry is given, as an illustration of the elusive notion of "undefined terms". This leads to ideas about the consistency of different and possibly "rival" geometries. Through this discussion the notion of undefined terms is clarified, and the relation of undefined terms to perception and thought processes is considered. Little Harmonic Labyrinth. This is based on the Bach organ piece by the same name. It is a playful introduction to the notion of recursivei.e., nested structures. It contains stories within stories. The frame story, instead of finishing as expected, is left open, so the reader is left dangling without resolution. One nested story concerns modulation in musicparticularly an organ piece which ends in the wrong key, leaving the listener dangling without resolution. Chapter V: Recursive Structures and Processes. The idea of recursion is presented in many different contexts: musical patterns, linguistic patterns, geometric structures, mathematical functions, physical theories, computer programs, and others. Canon by Intervallic Augmentation. Achilles and the Tortoise try to resolve the question, "Which contains more informationa record, or the phonograph which plays it This odd question arises when the Tortoise describes a single record which, when played on a set of different phonographs, produces two quite different melodies: BACH and CAGE. It turns out, however, that these melodies are "the same", in a peculiar sense. Chapter VI: The Location of Meaning. A broad discussion of how meaning is split among coded message, decoder, and receiver. Examples presented include strands of DNA, undeciphered inscriptions on ancient tablets, and phonograph records sailing out in space. The relationship of intelligence to "absolute" meaning is postulated. Chromatic Fantasy, And Feud. A short Dialogue bearing hardly any resemblance, except in title, to Bach's Chromatic Fantasy and Fugue. It concerns the proper way to manipulate sentences so as to preserve truthand in particular the question  
 
Overview  V  
 
 
of whether there exist rules for the usage of the word "arid". This Dialogue has much in common with the Dialogue by Lewis Carroll. Chapter VII: The Propositional Calculus. It is suggested how words such as .,and" can be governed by formal rules. Once again, the ideas of isomorphism and automatic acquisition of meaning by symbols in such a system are brought up. All the examples in this Chapter, incidentally, are "Zentences"sentences taken from Zen koans. This is purposefully done, somewhat tongueincheek, since Zen koans are deliberately illogical stories. Crab Canon. A Dialogue based on a piece by the same name from the Musical Offering. Both are so named because crabs (supposedly) walk backwards. The Crab makes his first appearance in this Dialogue. It is perhaps the densest Dialogue in the book in terms of formal trickery and levelplay. Gödel, Escher, and Bach are deeply intertwined in this very short Dialogue. Chapter VIII: Typographical Number Theory. An extension of the Propositional Calculus called "TNT" is presented. In TNT, numbertheoretical reasoning can be done by rigid symbol manipulation. Differences between formal reasoning and human thought are considered. A Mu Offering. This Dialogue foreshadows several new topics in the book. Ostensibly concerned with Zen Buddhism and koans, it is actually a thinly veiled discussion of theoremhood and nontheoremhood, truth and falsity, of strings in number theory. There are fleeting references to molecular biologyparticular) the Genetic Code. There is no close affinity to the Musical Offering, other than in the title and the playing of selfreferential games. Chapter IX: Mumon and Gödel. An attempt is made to talk about the strange ideas of Zen Buddhism. The Zen monk Mumon, who gave well known commentaries on many koans, is a central figure. In a way, Zen ideas bear a metaphorical resemblance to some contemporary ideas in the philosophy of mathematics. After this "Zennery", Gödel’s fundamental idea of Gödelnumbering is introduced, and a first pass through Gödel’s Theorem is made. Part II: EGB Prelude ... This Dialogue attaches to the next one. They are based on preludes and fugues from Bach's WellTempered Clavier. Achilles and the Tortoise bring a present to the Crab, who has a guest: the Anteater. The present turns out to be a recording of the W.T.C.; it is immediately put on. As they listen to a prelude, they discuss the structure of preludes and fugues, which leads Achilles to ask how to hear a fugue: as a whole, or as a sum of parts? This is the debate between holism and reductionism, which is soon taken up in the Ant Fugue. Chapter X: Levels of Description, and Computer Systems. Various levels of seeing pictures, chessboards, and computer systems are discussed. The last of these is then examined in detail. This involves describing machine languages, assembly languages, compiler languages, operating systems, and so forth. Then the discussion turns to composite systems of other types, such as sports teams, nuclei, atoms, the weather, and so forth. The question arises as to how man intermediate levels existor indeed whether any exist.  
 
Overview  VI  
 
 
…Ant Fugue. An imitation of a musical fugue: each voice enters with the same statement. The themeholism versus reductionismis introduced in a recursive picture composed of words composed of smaller words. etc. The words which appear on the four levels of this strange picture are "HOLISM", "REDLCTIONIsM", and "ML". The discussion veers off to a friend of the Anteater's Aunt Hillary, a conscious ant colony. The various levels of her thought processes are the topic of discussion. Many fugal tricks are ensconced in the Dialogue. As a hint to the reader, references are made to parallel tricks occurring in the fugue on the record to which the foursome is listening. At the end of the Ant Fugue, themes from the Prelude return. transformed considerably. Chapter XI: Brains and Thoughts. "How can thoughts he supported by the hardware of the brain is the topic of the Chapter. An overview of the large scale and smallscale structure of the brain is first given. Then the relation between concepts and neural activity is speculatively discussed in some detail. English French German Suite. An interlude consisting of Lewis Carroll's nonsense poem "Jabberwocky`' together with two translations: one into French and one into German, both done last century. Chapter XII: Minds and Thoughts. The preceding poems bring up in a forceful way the question of whether languages, or indeed minds, can be "mapped" onto each other. How is communication possible between two separate physical brains: What do all human brains have in common? A geographical analogy is used to suggest an answer. The question arises, "Can a brain be understood, in some objective sense, by an outsider?" Aria with Diverse Variations. A Dialogue whose form is based on Bach's Goldberg Variations, and whose content is related to numbertheoretical problems such as the Goldbach conjecture. This hybrid has as its main purpose to show how number theory's subtlety stems from the fact that there are many diverse variations on the theme of searching through an infinite space. Some of them lead to infinite searches, some of them lead to finite searches, while some others hover in between. Chapter XIII: BlooP and FlooP and GlooP. These are the names of three computer languages. BlooP programs can carry out only predictably finite searches, while FlooP programs can carry out unpredictable or even infinite searches. The purpose of this Chapter is to give an intuition for the notions of primitive recursive and general recursive functions in number theory, for they are essential in Gödel’s proof. Air on G's String. A Dialogue in which Gödel’s selfreferential construction is mirrored in words. The idea is due to W. V. O. Quine. This Dialogue serves as a prototype for the next Chapter. Chapter XIV: On Formally Undecidable Propositions of TNT and Related Systems. This Chapter's title is an adaptation of the title of Gödel’s 1931 article, in which his Incompleteness Theorem was first published. The two major parts of Gödel’s proof are gone through carefully. It is shown how the assumption of consistency of TNT forces one to conclude that TNT (or any similar system) is incomplete. Relations to Euclidean and nonEuclidean geometry are discussed. Implications for the philosophy of mathematics are gone into with some care.  
 
Overview  VII  
 
 
Birthday Cantatatata ... In which Achilles cannot convince the wily and skeptical Tortoise that today is his (Achilles') birthday. His repeated but unsuccessful tries to do so foreshadow the repeatability of the Gödel argument. Chapter XV: Jumping out of the System. The repeatability of Gödel’s argument is shown, with the implication that TNT is not only incomplete, but "essentially incomplete The fairly notorious argument by J. R. Lucas, to the effect that Gödel’s Theorem demonstrates that human thought cannot in any sense be "mechanical", is analyzed and found to be wanting. Edifying Thoughts of a Tobacco Smoker. A Dialogue treating of many topics, with the thrust being problems connected with selfreplication and selfreference. Television cameras filming television screens, and viruses and other subcellular entities which assemble themselves, are among the examples used. The title comes from a poem by J. S. Bach himself, which enters in a peculiar way. Chapter XVI: SelfRef and SelfRep. This Chapter is about the connection between selfreference in its various guises, and selfreproducing entities e.g., computer programs or DNA molecules). The relations between a selfreproducing entity and the mechanisms external to it which aid it in reproducing itself (e.g., a computer or proteins) are discussedparticularly the fuzziness of the distinction. How information travels between various levels of such systems is the central topic of this Chapter. The Magnificrab, Indeed. The title is a pun on Bach's Magnifacat in D. The tale is about the Crab, who gives the appearance of having a magical power of distinguishing between true and false statements of number theory by reading them as musical pieces, playing them on his flute, and determining whether they are "beautiful" or not. Chapter XVII: Church, Turing, Tarski, and Others. The fictional Crab of the preceding Dialogue is replaced by various real people with amazing mathematical abilities. The ChurchTuring Thesis, which relates mental activity to computation, is presented in several versions of differing strengths. All are analyzed, particularly in terms of their implications for simulating human thought mechanically, or programming into a machine an ability to sense or create beauty. The connection between brain activity and computation brings up some other topics: the halting problem of Turing, and Tarski's Truth Theorem. SHRDLU, Toy of Man's Designing. This Dialogue is lifted out of an article by Terry Winograd on his program SHRDLU: only a few names have been changed. In it. a program communicates with a person about the socalled "blocks world" in rather impressive English. The computer program appears to exhibit some real understandingin its limited world. The Dialogue's title is based on Jesu, joy of Mans Desiring, one movement of Bach's Cantata 147. Chapter XVIII: Artificial Intelligence: Retrospects, This Chapter opens with a discussion of the famous "Turing test"a proposal by the computer pioneer Alan Turing for a way to detect the presence or absence of "thought" in a machine. From there, we go on to an abridged history of Artificial Intelligence. This covers programs that canto some degreeplay games, prove theorems, solve problems, compose music, do mathematics, and use "natural language" (e.g., English).  
 
Overview  VIII  
 
 
Contrafactus. About how we unconsciously organize our thoughts so that we can imagine hypothetical variants on the real world all the time. Also about aberrant variants of this abilitysuch as possessed by the new character, the Sloth, an avid lover of French fries, and rabid hater of counterfactuals. Chapter XIX: Artificial Intelligence: Prospects. The preceding Dialogue triggers a discussion of how knowledge is represented in layers of contexts. This leads to the modern Al idea of "frames". A framelike way of handling a set of visual pattern puzzles is presented, for the purpose of concreteness. Then the deep issue of the interaction of concepts in general is discussed, which leads into some speculations on creativity. The Chapter concludes with a set of personal "Questions and Speculations" on Al and minds in general. Sloth Canon. A canon which imitates a Bach canon in which one voice plays the same melody as another, only upside down and twice as slowly, while a third voice is free. Here, the Sloth utters the same lines as the Tortoise does, only negated (in a liberal sense of the term) and twice as slowly, while Achilles is free. Chapter XX: Strange Loops, Or Tangled Hierarchies. A grand windup of many of the ideas about hierarchical systems and selfreference. It is concerned with the snarls which arise when systems turn back on themselvesfor example, science probing science, government investigating governmental wrongdoing, art violating the rules of art, and finally, humans thinking about their own brains and minds. Does Gödel’s Theorem have anything to say about this last "snarl"? Are free will and the sensation of consciousness connected to Gödel’s Theorem? The Chapter ends by tying Gödel, Escher, and Bach together once again. SixPart Ricercar. This Dialogue is an exuberant game played with many of the ideas which have permeated the book. It is a reenactment of the story of the Musical Offering, which began the book; it is simultaneously a "translation" into words of the most complex piece in the Musical Offering: the SixPart Ricercar. This duality imbues the Dialogue with more levels of meaning than any other in the book. Frederick the Great is replaced by the Crab, pianos by computers, and so on. Many surprises arise. The Dialogue's content concerns problems of mind, consciousness, free will, Artificial Intelligence, the Turing test, and so forth, which have been introduced earlier. It concludes with an implicit reference to the beginning of the book, thus making the book into one big selfreferential loop, symbolizing at once Bach's music, Escher's drawings, and Gödel’s Theorem.  
 
Overview  IX  
 
 
 
FIGURE 1. Johann Sebastian Bach, in 1748. From a painting by Elias Gottlieb Hanssmann.  
 
Introduction: A MusicoLogical Offering  10  
 
 
Introduction:  
 
A MusicoLogical Offering Author: FREDERICK THE GREAT, King of Prussia, came to power in 1740. Although he is remembered in history books mostly for his military astuteness, he was also devoted to the life of the mind and the spirit. His court in Potsdam was one of the great centers of intellectual activity in Europe in the eighteenth century. The celebrated mathematician Leonhard Euler spent twentyfive years there. Many other mathematicians and scientists came, as well as philosophersincluding Voltaire and La Mettrie, who wrote some of their most influential works while there. But music was Frederick's real love. He was an avid flutist and composer. Some of his compositions are occasionally performed even to this day. Frederick was one of the first patrons of the arts to recognize the virtues of the newly developed "pianoforte" ("softloud"). The piano had been developed in the first half of the eighteenth century as a modification of the harpsichord. The problem with the harpsichord was that pieces could only be played at a rather uniform loudnessthere was no way to strike one note more loudly than its neighbors. The "softloud", as its name implies, provided a remedy to this problem. From Italy, where Bartolommeo Cristofori had made the first one, the softloud idea had spread widely. Gottfried Silbermann, the foremost German organ builder of the day, was endeavoring to make a "perfect" pianoforte. Undoubtedly King Frederick was the greatest supporter of his effortsit is said that the King owned as many as fifteen Silbermann pianos! Bach Frederick was an admirer not only of pianos, but also of an organist and composer by the name of J. S. Bach. This Bach's compositions were somewhat notorious. Some called them "turgid and confused", while others claimed they were incomparable masterpieces. But no one disputed Bach's ability to improvise on the organ. In those days, being an organist not only meant being able to play, but also to extemporize, and Bach was known far and wide for his remarkable extemporizations. (For some delightful anecdotes about Bach's extemporization, see The Bach Reader, by H. T. David and A. Mendel.) In 1747, Bach was sixtytwo, and his fame, as well as one of his sons, had reached Potsdam: in fact, Carl Philipp Emanuel Bach was the Capellmeister (choirmaster) at the court of King Frederick. For years the King had let it be known, through gentle hints to Philipp Emanuel, how  
 
Introduction: A MusicoLogical Offering  11  
 
 
pleased he would be to have the elder Bach come and pay him a visit; but this wish had never been realized. Frederick was particularly eager for Bach to try out his new Silbermann pianos, which lie (Frederick) correctly foresaw as the great new wave in music. It was Frederick's custom to have evening concerts of chamber music in his court. Often he himself would be the soloist in a concerto for flute Here we have reproduced a painting of such an evening by the German painter Adolph von Menzel, who, in the 1800's, made a series of paintings illustrating the life of Frederick the Great. At the cembalo is C. P. E. Bach, and the figure furthest to the right is Joachim Quantz, the King's flute masterand the only person allowed to find fault with the King's flute playing. One May evening in 1747, an unexpected guest showed up. Johann Nikolaus Forkel, one of Bach's earliest biographers, tells the story as follows: One evening, just as lie was getting his flute ready, and his musicians were ssembled, an officer brought him a list of the strangers who had arrived. With his flute in his hand he ran ever the list, but immediately turned to the assembled musicians, and said, with a kind of agitation, "Gentlemen, old Bach is come." The Hute was now laid aside, and old Bach, who had alighted at his son's lodgings, was immediately summoned to the Palace. Wilhelm Friedemann, who accompanied his father, told me this story, and I must say that 1 still think with pleasure on the manner in which lie related it. At that time it was the fashion to make rather prolix compliments. The first appearance of J. S. Bach before se great a King, who did not even give him time to change his traveling dress for a black chanter's gown, must necessarily be attended with many apologies. I will net here dwell en these apologies, but merely observe, that in Wilhelm Friedemann's mouth they made a formal Dialogue between the King and the Apologist. But what is mere important than this is that the King gave up his Concert for this evening, and invited Bach, then already called the Old Bach, to try his fortepianos, made by Silbermann, which steed in several rooms of the palace. [Forkel here inserts this footnote: "The pianofortes manufactured by Silbermann, of Frevberg, pleased the King se much, that he resolved to buy them all up. He collected fifteen. I hear that they all now stand unfit for use in various corners of the Royal Palace."] The musicians went with him from room to room, and Bach was invited everywhere to try them and to play unpremeditated compositions. After he had gene en for some time, he asked the King to give him a subject for a Fugue, in order to execute it immediately without any preparation. The King admired the learned manner in which his subject was thus executed extempore: and, probably to see hew far such art t could be carried, expressed a wish to hear a Fugue with six Obligato parts. But as it is not every subject that is fit for such full harmony, Bach chose one himself, and immediately executed it to the astonishment of all present in the same magnificent and learned manner as he had done that of the King. His Majesty desired also to hear his performance en the organ. The next day therefore Bach was taken to all the organs in Potsdam, as lie had before been to Silbermann's fortepianos. After his return to Leipzig, he composed the subject, which he had received from the King, in three and six parts. added several artificial passages in strict canon to it, and had it engraved, under the title of "Musikalisches Opfer" [Musical Offering], and dedicated it to the Inventor.'  
 
Introduction: A MusicoLogical Offering  12  
 
 
 
Introduction: A MusicoLogical Offering  13  
 
 
 
FIGURE 3. The Royal Theme.  
 
When Bach sent a copy of his Musical Offering to the King, he included a dedicatory letter, which is of interest for its prose style if nothing else rather submissive and flattersome. From a modern perspective it seems comical. Also, it probably gives something of the flavor of Bach's apology for his appearance.2 MOST GRACIOUS KING! In deepest humility I dedicate herewith to Your Majesty a musical offering, the noblest part of which derives from Your Majesty's own august hand. With awesome pleasure I still remember the very special Royal grace when, some time ago, during my visit in Potsdam, Your Majesty's Self deigned to play to me a theme for a fugue upon the clavier, and at the same time charged me most graciously to carry it out in Your Majesty's most august presence. To obey Your Majesty's command was my most humble dim. I noticed very soon, however, that, for lack of necessary preparation, the execution of the task did not fare as well as such an excellent theme demanded. I resoled therefore and promptly pledged myself to work out this right Royal theme more fully, and then make it known to the world. This resolve has now been carried out as well as possible, and it has none other than this irreproachable intent, to glorify, if only in a small point, the fame of a monarch whose greatness and power, as in all the sciences of war and peace, so especially in music, everyone must admire and revere. I make bold to add this most humble request: may Your Majesty deign to dignify the present modest labor with a gracious acceptance, and continue to grant Your Majesty's most august Royal grace to Your Majesty's most humble and obedient servant, THE AUTHOR Leipzig, July 7 1747 Some twentyseven years later, when Bach had been dead for twentyfour years, a Baron named Gottfried van Swietento whom, incidentally, Forkel dedicated his biography of Bach, and Beethoven dedicated his First Symphonyhad a conversation with King Frederick, which he reported as follows: He [Frederick] spoke to me, among other things, of music, and of a great organist named Bach, who has been for a while in Berlin. This artist [Wilhelm Friedemann Bach] is endowed with a talent superior, in depth of harmonic knowledge and power of execution, to any 1 have heard or can imagine, while those who knew his father claim that he, in turn, was even greater. The King  
 
Introduction: A MusicoLogical Offering  14  
 
 
is of this opinion, and to prove it to me he sang aloud a chromatic fugue subject which he had given this old Bach, who on the spot had made of it a fugue in four parts, then in five parts, and finally in eight parts.' Of course there is no way of knowing whether it was King Frederick or Baron van Swieten who magnified the story into largerthanlife proportions. But it shows how powerful Bach's legend had become by that time. To give an idea of how extraordinary a sixpart fugue is, in the entire WellTempered Clavier by Bach, containing fortyeight Preludes and Fugues, only two have as many as five parts, and nowhere is there a sixpart fugue! One could probably liken the task of improvising a sixpart fugue to the playing of sixty simultaneous blindfold games of chess, and winning them all. To improvise an eightpart fugue is really beyond human capability. In the copy which Bach sent to King Frederick, on the page preceding the first sheet of music, was the following inscription:  
 
FIG URE 4. ("At the King's Command, the Song and the Remainder Resolved with Canonic Art.") Here Bach is punning on the word "canonic", since it means not only "with canons" but also "in the best possible way". The initials of this inscription are R I C E R C A R an Italian word, meaning "to seek". And certainly there is a great deal to seek in the Musical Offering. It consists of one threepart fugue, one sixpart fugue, ten canons, and a trio sonata. Musical scholars have concluded that the threepart fugue must be, in essence, identical with the one which Bach improvised for King Frederick. The sixpart fugue is one of Bach's most complex creations, and its theme is, of course, the Royal Theme. That theme, shown in Figure 3, is a very complex one, rhythmically irregular and highly chromatic (that is, filled with tones which do not belong to the key it is in). To write a decent fugue of even two voices based on it would not be easy for the average musician! Both of the fugues are inscribed "Ricercar", rather than "Fuga". This is another meaning of the word; "ricercar" was, in fact, the original name for the musical form now known as "fugue". By Bach's time, the word "fugue" (or fuga, in Latin and Italian) had become standard, but the term "ricercar" had survived, and now designated an erudite kind of fugue, perhaps too austerely intellectual for the common ear. A similar usage survives in English today: the word "recherche" means, literally, "sought out", but carries the same kind of implication, namely of esoteric or highbrow cleverness. The trio sonata forms a delightful relief from the austerity of the fugues and canons, because it is very melodious and sweet, almost dance  
 
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able. Nevertheless, it too is based largely on the King's theme, chromatic and austere as it is. It is rather miraculous that Bach could use such a theme to make so pleasing an interlude. The ten canons in the Musical Offering are among the most sophisticated canons Bach ever wrote. However, curiously enough, Bach himself never wrote them out in full. This was deliberate. They were posed as puzzles to King Frederick. It was a familiar musical game of the day to give a single theme, together with some more or less tricky hints, and to let the canon based on that theme be "discovered" by someone else. In order to know how this is possible, you must understand a few facts about canons. Canons and Fugues The idea of a canon is that one single theme is played against itself. This is done by having "copies" of the theme played by the various participating voices. But there are means' ways to do this. The most straightforward of all canons is the round, such as "Three Blind Mice", "Row, Row, Row Your Boat", or " Frere Jacques". Here, the theme enters in the first voice and, after a fixed timedelay, a "copy" of it enters, in precisely the same key. After the same fixed timedelay in the second voice, the third voice enters carrying the theme, and so on. Most themes will not harmonize with themselves in this way. In order for a theme to work as a canon theme, each of its notes must be able to serve in a dual (or triple, or quadruple) role: it must firstly be part of a melody, and secondly it must be part of a harmonization of the same melody. When there are three canonical voices, for instance, each note of the theme must act in two distinct harmonic ways, as well as melodically. Thus, each note in a canon has more than one musical meaning; the listener's ear and brain automatically figure out the appropriate meaning, by referring to context. There are more complicated sorts of canons, of course. The first escalation in complexity comes when the "copies" of the theme are staggered not only in time, but also in pitch; thus, the first voice might sing the theme starting on C, and the second voice, overlapping with the first voice, might sing the identical theme starting five notes higher, on G. A third voice, starting on the D yet five notes higher, might overlap with the first two, and so on. The next escalation in complexity comes when the speeds of' the different voices are not equal; thus, the second voice might sing twice as quickly, or twice as slowly, as the first voice. The former is called diminution, the latter augmentation (since the theme seems to shrink or to expand). We are not yet done! The next stage of complexity in canon construction is to invert the theme, which means to make a melody which jumps down wherever the original theme jumps up, and by exactly the same number of semitones. This is a rather weird melodic transformation, but when one has heard many themes inverted, it begins to seem quite natural. Bach was especially fond of inversions, and they show up often in his workand the Musical Offering is no exception. (For a simple example of  
 
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inversion, try the tune "Good King Wenceslas". When the original and its inversion are sung together, starting an octave apart and staggered with a timedelay of two beats, a pleasing canon results.) Finally, the most esoteric of "copies" is the retrograde copywhere the theme is played backwards in time. A canon which uses this trick is affectionately known as a crab canon, because of the peculiarities of crab locomotion. Bach included a crab canon in the Musical Offering, needless to say. Notice that every type of "copy" preserves all the information in the original theme, in the sense that the theme is fully recoverable from any of the copies. Such an information preserving transformation is often called an isomorphism, and we will have much traffic with isomorphisms in this book. Sometimes it is desirable to relax the tightness of the canon form. One way is to allow slight departures from perfect copying, for the sake of more fluid harmony. Also, some canons have "free" voicesvoices which do not employ the canon's theme, but which simply harmonize agreeably with the voices that are in canon with each other. Each of the canons in the Musical Offering has for its theme a different variant of the King's Theme, and all the devices described above for making canons intricate are exploited to the hilt; in fact, they are occasionally combined. Thus, one threevoice canon is labeled "Canon per Augmentationem, contrario Motu"; its middle voice is free (in fact, it sings the Royal Theme), while the other two dance canonically above and below it, using the devices of augmentation and inversion. Another bears simply the cryptic label "Quaerendo invenietis" ("By seeking, you will discover"). All of the canon puzzles have been solved. The canonical solutions were given by one of Bach's pupils, Johann Philipp Kirnberger. But one might still wonder whether there are more solutions to seek! I should also explain briefly what a fugue is. A fugue is like a canon, in that it is usually based on one theme which gets played in different voices and different keys, and occasionally at different speeds or upside down or backwards. However, the notion of fugue is much less rigid than that of canon, and consequently it allows for more emotional and artistic expression. The telltale sign of a fugue is the way it begins: with a single voice singing its theme. When it is done, then a second voice enters, either five scalenotes up, or four down. Meanwhile the first voice goes on, singing the "countersubject": a secondary theme, chosen to provide rhythmic, harmonic, and melodic contrasts to the subject. Each of the voices enters in turn, singing the theme, often to the accompaniment of the countersubject in some other voice, with the remaining voices doing whatever fanciful things entered the composer's mind. When all the voices have "arrived", then there are no rules. There are, to be sure, standard kinds of things to dobut not so standard that one can merely compose a fugue by formula. The two fugues in the Musical Offering are outstanding examples of fugues that could never have been "composed by formula". There is something much deeper in them than mere fugality. All in all, the Musical Offering represents one of Bach's supreme accomplishments in counterpoint. It is itself one large intellectual fugue, in  
 
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which many ideas and forms have been woven together, and in which playful double meanings and subtle allusions are commonplace. And it is a very beautiful creation of the human intellect which we can appreciate forever. (The entire work is wonderfully described in the book f. S. Bach's Musical Offering, by H. T. David.) An Endlessly Rising Canon There is one canon in the Musical Offering which is particularly unusual. Labeled simply "Canon per Tonos", it has three voices. The uppermost voice sings a variant of the Royal Theme, while underneath it, two voices provide a canonic harmonization based on a second theme. The lower of this pair sings its theme in C minor (which is the key of the canon as a whole), and the upper of the pair sings the same theme displaced upwards in pitch by an interval of a fifth. What makes this canon different from any other, however, is that when it concludesor, rather, seems to concludeit is no longer in the key of C minor, but now is in D minor. Somehow Bach has contrived to modulate (change keys) right under the listener's nose. And it is so constructed that this "ending" ties smoothly onto the beginning again; thus one can repeat the process and return in the key of E, only to join again to the beginning. These successive modulations lead the ear to increasingly remote provinces of tonality, so that after several of them, one would expect to be hopelessly far away from the starting key. And yet magically, after exactly six such modulations, the original key of C minor has been restored! All the voices are exactly one octave higher than they were at the beginning, and here the piece may be broken off in a musically agreeable way. Such, one imagines, was Bach's intention; but Bach indubitably also relished the implication that this process could go on ad infinitum, which is perhaps why he wrote in the margin "As the modulation rises, so may the King's Glory." To emphasize its potentially infinite aspect, I like to call this the "Endlessly Rising Canon". In this canon, Bach has given us our first example of the notion of Strange Loops. The "Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started. (Here, the system is that of musical keys.) Sometimes I use the term Tangled Hierarchy to describe a system in which a Strange Loop occurs. As we go on, the theme of Strange Loops will recur again and again. Sometimes it will be hidden, other times it will be out in the open; sometimes it will be right side up, other times it will be upside down, or backwards. "Quaerendo invenietis" is my advice to the reader.  
 
Escher To my mind, the most beautiful and powerful visual realizations of this notion of Strange Loops exist in the work of the Dutch graphic artist M. C. Escher, who lived from 1902 to 1972. Escher was the creator of some of the  
 
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FIGURE 5. Waterfall, by M. C. Escher (lithograph, 1961).  
 
most intellectually stimulating drawings of all time. Many of them have their origin in paradox, illusion, or doublemeaning. Mathematicians were among the first admirers of Escher's drawings, and this is understandable because they often are based on mathematical principles of symmetry or pattern ... But there is much more to a typical Escher drawing than just symmetry or pattern; there is often an underlying idea, realized in artistic form. And in particular, the Strange Loop is one of the most recurrent themes in Escher's work. Look, for example, at the lithograph Waterfall (Fig. 5), and compare its sixstep endlessly falling loop with the sixstep endlessly rising loop of the "Canon per Tonos". The similarity of vision is  
 
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FIGURE 6. Ascending and Descending, by M. C. Escher (lithograph, 1960).  
 
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remarkable. Bach and Escher are playing one single theme in two different "keys": music and art. Escher realized Strange Loops in several different ways, and they can be arranged according to the tightness of the loop. The lithograph Ascending and Descending (Fig. 6), in which monks trudge forever in loops, is the loosest version, since it involves so many steps before the starting point is regained. A tighter loop is contained in Waterfall, which, as we already observed, involves only six discrete steps. You may be thinking that there is some ambiguity in the notion of a single "step"for instance, couldn't Ascending and Descending be seen just as easily as having four levels (staircases) as fortyfive levels (stairs)% It is indeed true that there is an inherent  
 
 
FIGURE 7. Hand with Reflecting Globe. Selfportrait In, M. C. Escher (lithograph, 1935).  
 
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haziness in levelcounting, not only in Escher pictures, but in hierarchical, manylevel systems. We will sharpen our understanding of this haziness later on. But let us not get too distracted now' As we tighten our loop, we come to the remarkable Drawing Hands (Fig. 135), in which each of two hands draws the other: a twostep Strange Loop. And finally, the tightest of all Strange Loops is realized in Print Gallery (Fig. 142): a picture of a picture which contains itself. Or is it a picture of a gallery which contains itself? Or of a town which contains itself? Or a young man who contains himself'? (Incidentally, the illusion underlying Ascending and Descending and Waterfall was not invented by Escher, but by Roger Penrose, a British mathematician, in 1958. However, the theme of the Strange Loop was already present in Escher's work in 1948, the year he drew Drawing Hands. Print Gallery dates from 1956.) Implicit in the concept of Strange Loops is the concept of infinity, since what else is a loop but a way of representing an endless process in a finite way? And infinity plays a large role n many of Escher's drawings. Copies of one single theme often fit into each' other, forming visual analogues to the canons of Bach. Several such patterns can be seen in Escher's famous print Metamorphosis (Fig. 8). It is a little like the "Endlessly Rising Canon": wandering further and further from its starting point, it suddenly is back. In the tiled planes of Metamorphosis and other pictures, there are already suggestions of infinity. But wilder visions of infinity appear in other drawings by Escher. In some of his drawings, one single theme can appear on different levels of reality. For instance, one level in a drawing might clearly be recognizable as representing fantasy or imagination; another level would be recognizable as reality. These two levels might be the only explicitly portrayed levels. But the mere presence of these two levels invites the viewer to look upon himself as part of yet another level; and by taking that step, the viewer cannot help getting caught up in Escher's implied chain of levels, in which, for any one level, there is always another level above it of greater "reality", and likewise, there is always a level below, "more imaginary" than it is. This can be mindboggling in itself. However, what happens if the chain of levels is not linear, but forms a loop? What is real, then, and what is fantasy? The genius of Escher was that he could not only concoct, but actually portray, dozens of halfreal, halfmythical worlds, worlds filled with Strange Loops, which he seems to be inviting his viewers to enter. Gödel In the examples we have seen of Strange Loops by Bach and Escher, there is a conflict between the finite and the infinite, and hence a strong sense of paradox. Intuition senses that there is something mathematical involved here. And indeed in our own century a mathematical counterpart was discovered, with the most enormous repercussions. And, just as the Bach and Escher loops appeal to very simple and ancient intuitionsa musical scale, a staircaseso this discovery, by K. Gödel, of a Strange Loop in  
 
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FIGURE 9. Kurt Godel.  
 
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mathematical systems has its origins in simple and ancient intuitions. In its absolutely barest form, Godel's discovery involves the translation of an ancient paradox in philosophy into mathematical terms. That paradox is the socalled Epimenides paradox, or liar paradox. Epimenides was a Cretan who made one immortal statement: "All Cretans are liars." A sharper version of the statement is simply "I am lying"; or, "This statement is false". It is that last version which I will usually mean when I speak of the Epimenides paradox. It is a statement which rudely violates the usually assumed dichotomy of statements into true and false, because if you tentatively think it is true, then it immediately backfires on you and makes you think it is false. But once you've decided it is false, a similar backfiring returns you to the idea that it must be true. Try it! The Epimenides paradox is a onestep Strange Loop, like Escher's Print Gallery. But how does it have to do with mathematics? That is what Godel discovered. His idea was to use mathematical reasoning in exploring mathematical reasoning itself. This notion of making mathematics "introspective" proved to be enormously powerful, and perhaps its richest implication was the one Godel found: Godel's Incompleteness Theorem. What the Theorem states and how it is proved are two different things. We shall discuss both in quite some detail in this book. The Theorem can De likened to a pearl, and the method of proof to an oyster. The pearl is prized for its luster and simplicity; the oyster is a complex living beast whose innards give rise to this mysteriously simple gem. Godel's Theorem appears as Proposition VI in his 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I." It states: To every wconsistent recursive class K of formulae there correspond recursive classsigns r, such that neither v Gen r nor Neg (v Gen r) belongs to Fig (K) (where v is the free variable of r). Actually, it was in German, and perhaps you feel that it might as well be in German anyway. So here is a paraphrase in more normal English: All consistent axiomatic formulations of number theory include undecidable propositions. This is the pearl. In this pearl it is hard to see a Strange Loop. That is because the Strange Loop is buried in the oysterthe proof. The proof of Godel's Incompleteness Theorem hinges upon the writing of a selfreferential mathematical statement, in the same way as the Epimenides paradox is a selfreferential statement of language. But whereas it is very simple to talk about language in language, it is not at all easy to see how a statement about numbers can talk about itself. In fact, it took genius merely to connect the idea of selfreferential statements with number theory. Once Godel had the intuition that such a statement could be created, he was over the major hurdle. The actual creation of the statement was the working out of this one beautiful spark of intuition.  
 
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We shall examine the Godel construction quite carefully in Chapters to come, but so that you are not left completely in the dark, I will sketch here, in a few strokes, the core of the idea, hoping that what you see will trigger ideas in your mind. First of all, the difficulty should be made absolutely clear. Mathematical statementslet us concentrate on numbertheoretical onesare about properties of whole numbers. Whole numbers are not statements, nor are their properties. A statement of number theory is not about a. statement of number theory; it just is a statement of number theory. This is the problem; but Godel realized that there was more here than meets the eye. Godel had the insight that a statement of number theory could be about a statement of number theory (possibly even itself), if only numbers could somehow stand for statements. The idea of a code, in other words, is at the heart of his construction. In the Godel Code, usually called "Godelnumbering", numbers are made to stand for symbols and sequences of symbols. That way, each statement of number theory, being a sequence of specialized symbols, acquires a Godel number, something like a telephone number or a license plate, by which it can be referred to. And this coding trick enables statements of number theory to be understood on two different levels: as statements of number theory, and also as statements about statements of number theory. Once Godel had invented this coding scheme, he had to work out in detail a way of transporting the Epimenides paradox into a numbertheoretical formalism. His final transplant of Epimenides did not say, "This statement of number theory is false", but rather, "This statement of number theory does not have any proof". A great deal of confusion can be caused by this, because people generally understand the notion of "proof" rather vaguely. In fact, Godel's work was just part of a long attempt by mathematicians to explicate for themselves what proofs are. The important thing to keep in mind is that proofs are demonstrations within fixed systems of propositions. In the case of Godel's work, the fixed system of numbertheoretical reasoning to which the word "proof" refers is that of Principia Mathematica (P.M.), a giant opus by Bertrand Russell and Alfred North Whitehead, published between 1910 and 1913. Therefore, the Godel sentence G should more properly be written in English as: This statement of number theory does not have any proof in the system of Principia Mathematica. Incidentally, this Godel sentence G is not Godel's Theoremno more than the Epimenides sentence is the observation that "The Epimenides sentence is a paradox." We can now state what the effect of discovering G is. Whereas the Epimenides statement creates a paradox since it is neither true nor false, the Godel sentence G is unprovable (inside P.M.) but true. The grand conclusion% That the system of Principia Mathematica is "incomplete"there are true statements of number theory which its methods of proof are too weak to demonstrate.  
 
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But if Principia Mathematica was the first victim of this stroke, it was certainly not the last! The phrase "and Related Systems" in the title of Godel's article is a telling one: for if Godel's result had merely pointed out a defect in the work of Russell and Whitehead, then others could have been inspired to improve upon P.M. and to outwit Godel's Theorem. But this was not possible: Godel's proof pertained to any axiomatic system which purported to achieve the aims which Whitehead and Russell had set for themselves. And for each different system, one basic method did the trick. In short, Godel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved. Therefore Godel's Theorem had an electrifying effect upon logicians, mathematicians, and philosophers interested in the foundations of mathematics, for it showed that no fixed system, no matter how complicated, could represent the complexity of the whole numbers: 0, 1, 2, 3, ... Modern readers may not be as nonplussed by this as readers of 1931 were, since in the interim our culture has absorbed Godel's Theorem, along with the conceptual revolutions of relativity and quantum mechanics, and their philosophically disorienting messages have reached the public, even if cushioned by several layers of translation (and usually obfuscation). There is a general mood of expectation, these days, of "limitative" resultsbut back in 1931, this came as a bolt from the blue. Mathematical Logic: A Synopsis A proper appreciation of Godel's Theorem requires a setting of context. Therefore, I will now attempt to summarize in a short space the history of mathematical logic prior to 1931an impossible task. (See DeLong, Kneebone, or Nagel and Newman, for good presentations of history.) It all began with the attempts to mechanize the thought processes of reasoning. Now our ability to reason has often been claimed to be what distinguishes us from other species; so it seems somewhat paradoxical, on first thought, to mechanize that which is most human. Yet even the ancient Greeks knew that reasoning is a patterned process, and is at least partially governed by statable laws. Aristotle codified syllogisms, and Euclid codified geometry; but thereafter, many centuries had to pass before progress in the study of axiomatic reasoning would take place again. One of the significant discoveries of nineteenthcentury mathematics was that there are different, and equally valid, geometrieswhere by "a geometry" is meant a theory of properties of abstract points and lines. It had long been assumed that geometry was what Euclid had codified, and that, although there might be small flaws in Euclid's presentation, they were unimportant and any real progress in geometry would be achieved by extending Euclid. This idea was shattered by the roughly simultaneous discovery of nonEuclidean geometry by several peoplea discovery that shocked the mathematics community, because it deeply challenged the idea that mathematics studies the real world. How could there be many differ  
 
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ent kinds of "points" and "lines" in one single reality? Today, the solution to the dilemma may be apparent, even to some nonmathematiciansbut at the time, the dilemma created havoc in mathematical circles. Later in the nineteenth century, the English logicians George Boole and Augustus De Morgan went considerably further than Aristotle in codifying strictly deductive reasoning patterns. Boole even called his book "The Laws of Thought"surely an exaggeration, but it was an important contribution. Lewis Carroll was fascinated by these mechanized reasoning methods, and invented many puzzles which could be solved with them. Gottlob Frege in Jena and Giuseppe Peano in Turin worked on combining formal reasoning with the study of sets and numbers. David Hilbert in Gottingen worked on stricter formalizations of geometry than Euclid's. All of these efforts were directed towards clarifying what one means by "proof". In the meantime, interesting developments were taking place in classical mathematics. A theory of different types of infinities, known as the theory of sets, was developed by Georg Cantor in the 1880's. The theory was powerful and beautiful, but intuitiondefying. Before long, a variety of settheoretical paradoxes had been unearthed. The situation was very disturbing, because just as mathematics seemed to be recovering from one set of paradoxesthose related to the theory of limits, in the calculusalong came a whole new set, which looked worse! The most famous is Russell's paradox. Most sets, it would seem, are not members of themselvesfor example, the set of walruses is not a walrus, the set containing only Joan of Arc is not Joan of Arc (a set is not a person)and so on. In this respect, most sets are rather "runofthemill". However, some "selfswallowing" sets do contain themselves as members, such as the set of all sets, or the set of all things except Joan of Arc, and so on. Clearly, every set is either runofthemill or selfswallowing, and no set can be both. Now nothing prevents us from inventing R: the set of all runo,themill sets. At first, R might seem a rather runofthemill inventionbut that opinion must be revised when you ask yourself, "Is R itself "a runofthemill set or a selfswallowing set?" You will find that the answer is: "R is neither runofthemill nor selfswallowing, for either choice leads to paradox." Try it! But if R is neither runofthemill nor selfswallowing, then what is it? At the very least, pathological. But no one was satisfied with evasive answers of that sort. And so people began to dig more deeply into the foundations of set theory. The crucial questions seemed to be: "What is wrong with our intuitive concept of 'set'? Can we make a rigorous theory of sets which corresponds closely with our intuitions, but which skirts the paradoxes?" Here, as in number theory and geometry, the problem is in trying to line up intuition with formalized, or axiomatized, reasoning systems. A startling variant of Russell's paradox, called "Grelling's paradox", can be made using adjectives instead of sets. Divide the adjectives in English into two categories: those which are selfdescriptive, such as "pentasyllabic", "awkwardnessful", and "recherche", and those which are not, such  
 
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as "edible", "incomplete", and "bisyllabic". Now if we admit "nonselfdescriptive" as an adjective, to which class does it belong? If it seems questionable to include hyphenated words, we can use two terms invented specially for this paradox: autological (= "selfdescriptive"), and heterological (= "nonselfdescriptive"). The question then becomes: "Is 'heterological' heterological?" Try it! There seems to he one common culprit in these paradoxes, namely selfreference, or "Strange Loopiness". So if the goal is to ban all paradoxes, why not try banning selfreference and anything that allows it to arise? This is not so easy as it might seem, because it can be hard to figure out just where selfreference is occurring. It may be spread out over a whole Strange Loop with several steps, as in this "expanded" version of Epimenides, reminiscent of Drawing Hands: The following sentence is false. The preceding sentence is true. Taken together, these sentences have the same effect as the original Epimenides paradox: yet separately, they are harmless and even potentially useful sentences. The "blame" for this Strange Loop can't he pinned on either sentenceonly on the way they "point" at each other. In the same way, each local region of Ascending and Descending is quite legitimate; it is only the way they are globally put together that creates an impossibility. Since there are indirect as well as direct ways of achieving selfreference, one must figure out how to ban both types at onceif one sees selfreference as the root of all evil. Banishing Strange Loops Russell and Whitehead did subscribe to this view, and accordingly, Principia Mathematica was a mammoth exercise in exorcising Strange Loops from logic, set theory, and number theory. The idea of their system was basically this. A set of the lowest "type" could contain only "objects" as membersnot sets. A set of the next type up could only contain objects, or sets of the lowest type. In general, a set of a given type could only contain sets of lower type, or objects. Every set would belong to a specific type. Clearly, no set could contain itself because it would have to belong to a type higher than its own type. Only "runof'themill" sets exist in such a system; furthermore, old Rthe set of all runofthemill setsno longer is considered a set at all, because it does not belong to any finite type. To all appearances, then, this theory of types, which we might also call the "theory of the abolition of Strange Loops", successfully rids set theory of its paradoxes, but only at the cost of introducing an artificialseeming hierarchy, and of disallowing the formation of certain kinds of setssuch as the set of all runofthemill sets. Intuitively, this is not the way we imagine sets. The theory of types handled Russell's paradox, but it did nothing about the Epimenides paradox or Grelling's paradox. For people whose  
 
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interest went no further than set theory, this was quite adequatebut for people interested in the elimination of paradoxes generally, some similar "hierarchization" seemed necessary, to forbid looping back inside language. At the bottom of such a hierarchy would be an object language. Here, reference could be made only to a specific domainnot to aspects of the object language itself (such as its grammatical rules, or specific sentences in it). For that purpose there would be a metalanguage. This experience of two linguistic levels is familiar to all learners of foreign languages. Then there would be a metametalanguage for discussing the metalanguage, and so on. It would be required that every sentence should belong to some precise level of the hierarchy. Therefore, if one could find no level in which a given utterance fit, then the utterance would be deemed meaningless, and forgotten.
An analysis can be attempted on the twostep Epimenides loop given above. The first sentence, since it speaks of the second, must be on a higher level than the second. But by the same token, the second sentence must be on a higher level than the first. Since this is impossible, the two sentences are "meaningless". More precisely, such sentences simply cannot be formulated at all in a system based on a strict hierarchy of languages. This prevents all versions of the Epimenides paradox as well as Grelling's paradox. (To what language level could "heterological" belong?)
Now in set theory, which deals with abstractions that we don't use all the time, a stratification like the theory of types seems acceptable, even if a little strangebut when it comes to language, an allpervading part of life, such stratification appears absurd. We don't think of ourselves as jumping up and down a hierarchy of languages when we speak about various things. A rather matteroffact sentence such as, "In this book, I criticize the theory of types" would be doubly forbidden in the system we are discussing. Firstly, it mentions "this book", which should only be mentionable in a
metabook"and secondly, it mentions mea person whom I should not be allowed to speak of at all! This example points out how silly the theory of types seems, when you import it into a familiar context. The remedy it adopts for paradoxestotal banishment of selfreference in any formis a real case of overkill, branding many perfectly good constructions as meaningless. The adjective "meaningless", by the way, would have to apply to all discussions of the theory of linguistic types (such as that of this very paragraph) for they clearly could not occur on any of the levelsneither object language, nor metalanguage, nor metametalanguage, etc. So the very act of discussing the theory would be the most blatant possible violation of it!
Now one could defend such theories by saying that they were only intended to deal with formal languagesnot with ordinary, informal language. This may be so, but then it shows that such theories are extremely academic and have little to say about paradoxes except when they crop up in special tailormade systems. Besides, the drive to eliminate paradoxes at any cost, especially when it requires the creation of highly artificial formalisms, puts too much stress on bland consistency, and too little on the
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quirky and bizarre, which make life and mathematics interesting. It is of course important to try to maintain consistency, but when this effort forces you into a stupendously ugly theory, you know something is wrong. These types of issues in the foundations of mathematics were responsible for the high interest in codifying human reasoning methods which was present in the early part of this century. Mathematicians and philosophers had begun to have serious doubts about whether even the most concrete of theories, such as the study of whole numbers (number theory), were built on solid foundations. If paradoxes could pop up so easily in set theorya theory whose basic concept, that of a set, is surely very intuitively appealingthen might they not also exist in other branches of mathematics? Another related worry was that the paradoxes of logic, such as the Epimenides paradox, might turn out to be internal to mathematics, and thereby cast in doubt all of mathematics. This was especially worrisome to thoseand there were a good numberwho firmly believed that mathematics is simply a branch of logic (or conversely, that logic is simply a branch of mathematics). In fact, this very question"Are mathematics and logic distinct, or separate%"was the source of much controversy. This study of mathematics itself became known as metamathematicsor occasionally, metalogic, since mathematics and logic are so intertwined. The most urgent priority of metamathematicians was to determine the true nature of mathematical reasoning. What is a legal method of procedure, and what is an illegal one? Since mathematical reasoning had always been done in "natural language" (e.g., French or Latin or some language for normal communication), there was always a lot of possible ambiguity. Words had different meanings to different people, conjured up different images, and so forth. It seemed reasonable and even important to establish a single uniform notation in which all mathematical work could be done, and with the aid of which any two mathematicians could resolve disputes over whether a suggested proof was valid or not. This would require a complete codification of the universally acceptable modes of human reasoning, at least as far as they applied to mathematics. Consistency, Completeness, Hilbert's Program This was the goal of Principia Mathematica, which purported to derive all of mathematics from logic, and, to be sure, without contradictions! It was widely admired, but no one was sure if (1) all of mathematics really was contained in the methods delineated by Russell and Whitehead, or (2) the methods given were even selfconsistent. Was it absolutely clear that contradictory results could never be derived, by any mathematicians whatsoever, following the methods of Russell and Whitehead? This question particularly bothered the distinguished German mathematician (and metamathematician) David Hilbert, who set before the world community of mathematicians (and metamathematicians) this chal  
 
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lenge: to demonstrate rigorouslyperhaps following the very methods outlined by Russell and Whiteheadthat the system defined in Principia Mathematica was both consistent (contradictionfree), and complete (i.e., that every true statement of, number theory could be derived within the framework drawn up in P.M.). This was a tall order, and one could criticize it on the grounds that it was somewhat circular: how can you justify your methods of reasoning on the basis of those same methods of reasoning? It is like lifting yourself up by your own bootstraps. (We just don't seem to be able to get away from these Strange Loops!) Hilbert was fully aware of this dilemma, of course, and therefore expressed the hope that a demonstration of consistency or completeness could be found which depended only on "finitistic" modes of reasoning. "these were a small set of reasoning methods usually accepted by mathematicians. In this way, Hilbert hoped that mathematicians could partially lift themselves by their own bootstraps: the sum total of mathematical methods might be proved sound, by invoking only a smaller set of methods. This goal may sound rather esoteric, but it occupied the minds of many of the greatest mathematicians in the world during the first thirty years of this century. In the thirtyfirst year, however, Godel published his paper, which in some ways utterly demolished Hilbert's program. This paper revealed not only that there were irreparable "holes" in the axiomatic system proposed by Russell and Whitehead, but more generally, that no axiomatic system whatsoever could produce all numbertheoretical truths, unless it were an inconsistent system! And finally, the hope of proving the consistency of a system such as that presented in P.M. was shown to be vain: if such a proof could be found using only methods inside P.M., thenand this is one of the most mystifying consequences of Godel's workP.M. itself would be inconsistent! The final irony of it all is that the proof of Gi del's Incompleteness Theorem involved importing the Epimenides paradox right into the heart ofPrincipia Mathematica, a bastion supposedly invulnerable to the attacks of Strange Loops! Although Godel's Strange Loop did not destroy Principia Mathematica, it made it far less interesting to mathematicians, for it showed that Russell and Whitehead's original aims were illusory. Babbage, Computers, Artificial Intelligence ... When Godel's paper came out, the world was on the brink of developing electronic digital computers. Now the idea of mechanical calculating engines had been around for a while. In the seventeenth century, Pascal and Leibniz designed machines to perform fixed operations (addition and multiplication). These machines had no memory, however, and were not, in modern parlance, programmable. The first human to conceive of the immense computing potential of machinery was the Londoner Charles Babbage (17921871). A character who could almost have stepped out of the pages of the Pickwick Papers,  
 
Introduction: A MusicoLogical Offering  32  
 
 
Babbage was most famous during his lifetime for his vigorous campaign to rid London of "street nuisances"organ grinders above all. These pests, loving to get his goat, would come and serenade him at any time of day or night, and he would furiously chase them down the street. Today, we recognize in Babbage a man a hundred years ahead of his time: not only inventor of the basic principles of modern computers, he was also one of the first to battle noise pollution. His first machine, the "Difference Engine", could generate mathematical tables of many kinds by the "method of differences". But before any model of the "D.E." had been built, Babbage became obsessed with a much more revolutionary idea: his "Analytical Engine". Rather immodestly, he wrote, "The course through which I arrived at it was the most entangled and perplexed which probably ever occupied the human mind."' Unlike any previously designed machine, the A.E. was to possess both a "store" (memory) and a "mill" (calculating and decisionmaking unit). These units were to be built of thousands of intricate geared cylinders interlocked in incredibly complex ways. Babbage had a vision of numbers swirling in and out of the mill tinder control of a program contained in punched cardsan idea inspired by the jacquard loom, a cardcontrolled loom that wove amazingly complex patterns. Babbage's brilliant but illfated Countess friend, Lady Ada Lovelace (daughter of Lord Byron), poetically commented that "the Analytical Engine weaves algebraic patterns just as the Jacquardloom weaves flowers and leaves." Unfortunately, her use of the present tense was misleading, for no A.E. was ever built, and Babbage died a bitterly disappointed man. Lady Lovelace, no less than Babbage, was profoundly aware that with the invention of the Analytical Engine, mankind was flirting with mechanized intelligenceparticularly if the Engine were capable of "eating its own tail" (the way Babbage described the Strange Loop created when a machine reaches in and alters its own stored program). In an 1842 memoir,5 she wrote that the A.E. "might act upon other things besides number". While Babbage dreamt of creating_ a chess or tictactoe automaton, she suggested that his Engine, with pitches and harmonies coded into its spinning cylinders, "might compose elaborate and scientific pieces of music of any degree of complexity or extent." In nearly the same breath, however, she cautions that "The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform." Though she well understood the power of artificial computation, Lady Lovelace was skeptical about the artificial creation of intelligence. However, could her keen insight allow her to dream of the potential that would be opened up with the taming of electricity? In our century the time was ripe for computerscomputers beyond the wildest dreams of Pascal, Leibniz, Babbage, or Lady Lovelace. In the 1930's and 1940's, the first "giant electronic brains" were designed and built. They catalyzed the convergence of three previously disparate areas: the theory of axiomatic reasoning, the study of mechanical computation, and the psychology of intelligence. These same years saw the theory of computers develop by leaps and  
 
Introduction: A MusicoLogical Offering  33  
 
 
bounds. This theory was tightly linked to metamathematics. In fact, Godel's Theorem has a counterpart in the theory of computation, discovered by Alan Turing, which reveals the existence of inelucPable "holes" in even the most powerful computer imaginable. Ironically, just as these somewhat eerie limits were being mapped out, real computers were being built whose powers seemed to grow and grow beyond their makers' power of prophecy. Babbage, who once declared he would gladly give up the rest of his life if he could come back in five hundred years and have a threeday guided scientific tour of the new age, would probably have been thrilled speechless a mere century after his deathboth by the new machines, and by their unexpected limitations. By the early 1950's, mechanized intelligence seemed a mere stone's throw away; and yet, for each barrier crossed, there always cropped up some new barrier to the actual creation of a genuine thinking machine. Was there some deep reason for this goal's mysterious recession? No one knows where the borderline between nonintelligent behavior and intelligent behavior lies; in fact, to suggest that a sharp borderline exists is probably silly. But essential abilities for intelligence are certainly: to respond to situations very flexibly; to take advantage of fortuitous circumstances; to make sense out of ambiguous or contradictory messages; to recognize the relative importance of different elements of a situation; to find similarities between situations despite differences which may separate them; to draw distinctions between situations despite similarities may link them; to synthesize new concepts by taking old them together in new ways; to come up with ideas which are novel. Here one runs up against a seeming paradox. Computers by their very nature are the most inflexible, desireless, rulefollowing of beasts. Fast though they may be, they are nonetheless the epitome of unconsciousness. How, then, can intelligent behavior be programmed? Isn't this the most blatant of contradictions in terms? One of the major theses of this book is that it is not a contradiction at all. One of the major purposes of this book is to urge each reader to confront the apparent contradiction head on, to savor it, to turn it over, to take it apart, to wallow in it, so that in the end the reader might emerge with new insights into the seemingly unbreathable gulf between the formal and the informal, the animate and the inanimate, the flexible and the inflexible. This is what Artificial Intelligence (A1) research is all about. And the strange flavor of AI work is that people try to put together long sets of rules in strict formalisms which tell inflexible machines how to be flexible. What sorts of "rules" could possibly capture all of what we think of as intelligent behavior, however? Certainly there must be rules on all sorts of  
 
Introduction: A MusicoLogical Offering  34  
 
 
different levels. There must be many "just plain" rules. There must be "metarules" to modify the "just plain" rules; then "metametarules" to modify the metarules, and so on. The flexibility of intelligence comes from the enormous number of different rules, and levels of rules. The reason that so many rules on so many different levels must exist is that in life, a creature is faced with millions of situations of completely different types. In some situations, there are stereotyped responses which require "just plain" rules. Some situations are mixtures of stereotyped situationsthus they require rules for deciding which of the 'just plain" rules to apply. Some situations cannot be classifiedthus there must exist rules for inventing new rules ... and on and on. Without doubt, Strange Loops involving rules that change themselves, directly or indirectly, are at the core of intelligence. Sometimes the complexity of our minds seems so overwhelming that one feels that there can be no solution to the problem of understanding intelligencethat it is wrong to think that rules of any sort govern a creature's behavior, even if one takes "rule" in the multilevel sense described above. ...and Bach In the year 1754, four years after the death of J. S. Bach, the Leipzig theologian Johann Michael Schmidt wrote, in a treatise on music and the soul, the following noteworthy passage: Not many years ago it was reported from France that a man had made a statue that could play various pieces on the Fleuttraversiere, placed the flute to its lips and took it down again, rolled its eyes, etc. But no one has yet invented an image that thinks, or wills, or composes, or even does anything at all similar. Let anyone who wishes to be convinced look carefully at the last fugal work of the abovepraised Bach, which has appeared in copper engraving, but which was left unfinished because his blindness intervened, and let him observe the art that is contained therein; or what must strike him as even more wonderful, the Chorale which he dictated in his blindness to the pen of another: Wenn wir in hochsten Nothen seen. I am sure that he will soon need his soul if he wishes to observe all the beauties contained therein, let alone wishes to play it to himself or to form a judgment of the author. Everything that the champions of Materialism put forward must fall to the ground in view of this single example.6 Quite likely, the foremost of the "champions of Materialism" here alluded to was none other than Julien Offroy de la Mettriephilosopher at the court of Frederick the Great, author of L'homme machine ("Man, the Machine"), and Materialist Par Excellence. It is now more than 200 years later, and the battle is still raging between those who agree with Johann Michael Schmidt, and those who agree with Julien Offroy de la Mettrie. I hope in this book to give some perspective on the battle. "Godel, Escher, Bach" The book is structured in an unusual way: as a counterpoint between Dialogues and Chapters. The purpose of this structure is to allow me to  
 
Introduction: A MusicoLogical Offering  35  
 
 
present new concepts twice: almost every new concept is first presented metaphorically in a Dialogue, yielding a set of concrete, visual images; then these serve, during the reading of the following`Chapter, as an intuitive background for a more serious and abstract presentation of the same concept. In many of the Dialogues I appear to be talking about one idea on the surface, but in reality I am talking about some other idea, in a thinly disguised way. Originally, the only characters in my Dialogues were Achilles and the Tortoise, who came to me from Zeno of Elea, by way of Lewis Carroll. Zeno of Elea, inventor of paradoxes, lived in the fifth century B.C. One of his paradoxes was an allegory, with Achilles and the Tortoise as protagonists. Zeno's invention of the happy pair is told in my first Dialogue, ThreePart Invention. In 1895, Lewis Carroll reincarnated Achilles and the Tortoise for the purpose of illustrating his own new paradox of infinity. Carroll's paradox, which deserves to be far better known than it is, plays a significant role in this book. Originally titled "What the Tortoise Said to Achilles", it is reprinted here as TwoPart Invention. When I began writing Dialogues, somehow I connected them up with musical forms. I don't remember the moment it happened; I just remember one day writing "Fugue" above an early Dialogue, and from then on the idea stuck. Eventually I decided to pattern each Dialogue in one way or another on a different piece by Bach. This was not so inappropriate. Old Bach himself used to remind his pupils that the separate parts in their compositions should behave like "persons who conversed together as if in a select company". I have taken that suggestion perhaps rather more literally than Bach intended it; nevertheless I hope the result is faithful to the meaning. I have been particularly inspired by aspects of Bach's compositions which have struck me over and over, and which are so well described by David and Mendel in The Bach Reader: His form in general was based on relations between separate sections. These relations ranged from complete identity of passages on the one hand to the return of a single principle of elaboration or a mere thematic allusion on the other. The resulting patterns were often symmetrical, but by no means necessarily so. Sometimes the relations between the various sections make up a maze of interwoven threads that only detailed analysis can unravel. Usually, however, a few dominant features afford proper orientation at first sight or hearing, and while in the course of study one may discover unending sub tleties, one is never at a loss to grasp the unity that holds together every single creation by Bach.' I have sought to weave an Eternal Golden Braid out of these three strands: Godel, Escher, Bach. I began, intending to write an essay at the core of which would be Godel's Theorem. I imagined it would be a mere pamphlet. But my ideas expanded like a sphere, and soon touched Bach and Escher. It took some time for me to think of making this connection explicit, instead of just letting it be a private motivating force. But finally 1 realized that to me, Godel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book.  
 
Introduction: A MusicoLogical Offering  36  
 
 
ThreePart Invention  
 
Achilles (a Greek warrior, the fleetest of foot of all mortals) and a Tortoise are standing together on a dusty runway in the hot sun. Far down the runway, on a tall flagpole, there hangs a large rectangular flag. The flag is sold red, except where a thin ringshaped holes has been cut out of it, through which one can see the sky. ACHILLES: What is that strange flag down at the other end of the track? It reminds me somehow of a print by my favourite artists M.C. Escher. TORTOISE: That is Zeno’ s flag ACHILLES: Could it be that the hole in it resembles the holes in a Mobian strip Escher once drew? Something is wrong about the flag, I can tell. TORTOISE: The ring which has been cut from it has the shape of the numeral for zero, which is Zeno´s favourite number. ACHILLES: The ring which hasn´t been invented yet! It will only be invented by a Hindu mathematician some millennia hence. And thus, Mr. T, mt argument proves that such a flag is impossible. TORTOISE: Your argument is persuasive, Achilles, and I must agree that such a flag is indeed impossible. But it is beautiful anyway, is it not? ACHILLES: Oh, yes, there is no doubt of its beauty. TORTOISE: I wonder if it´s beauty is related to it´s impossibility. I don´t know, I´ve never had the time to analyze Beauty. It´s a Capitalized Essence, and I never seem to have time for Capitalized Essences. ACHILLES: Speaking of Capitalized Essences, Mr. T, have you ever wondered about the Purpose of Life? TORTOISE: Oh, heavens, no; ACHILLES: Haven’t you ever wondered why we are here, or who invented us? TORTOISE: Oh, that is quite another matter. We are inventions of Zeno (as you will shortly see) and the reason we are here is to have a footrace. ACHILLES::: A footrace? How outrageous! Me, the fleetest of foot of all mortals, versus you, the ploddingest of the plodders! There can be no point to such a race. TORTOISE: You might give me a head start. ACHILLES: It would have to be a huge one. TORTOISE: I don’t object. ACHILLES: But I will catch you, sooner or later – most likely sooner. TORTOISE: Not if things go according to Zeno´s paradox, you won’t. Zeno is hoping to use our footrace to show that motion is impossible, you see. It is only in the mind that motion seems possible, according to Zeno. In truth, Motion Is Inherently Impossible. He proves it quite elegantly.  
 
ThreePart Invention  37  
 
 
 
Figure 10. Mobius strip by M.C.Escher (woodengraving printed from four blocks, 1961) ACHILLES: Oh, yes, it comes back to me now: the famous Zen koan about Zen Master Zeno. As you say it is very simple indeed. TORTOISE: Zen Koan? Zen Master? What do you mean? ACHILLES: It goes like this: Two monks were arguing about a flag. One said, “The flag is moving.” The other said, “The wind is moving.” The sixth patriarch, Zeno, happened to be passing by. He told them, “Not the wind, not the flag, mind is moving.” TORTOISE: I am afraid you are a little befuddled, Achilles. Zeno is no Zen master, far from it. He is in fact, a Greek philosopher from the town of Elea (which lies halfway between points A and B). Centuries hence, he will be celebrated for his paradoxes of motion. In one of those paradoxes, this very footrace between you and me will play a central role. ACHILLES: I’m all confused. I remember vividly how I used to repeat over and over the names of the six patriarchs of Zen, and I always said, “The sixth patriarch is Zeno, The sixth patriarch is Zeno…” (Suddenly a soft warm breeze picks up.) Oh, look Mr. Tortoise – look at the flag waving! How I love to watch the ripples shimmer through it’s soft fabric. And the ring cut out of it is waving, too!  
 
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TORTOISE: Don´t be silly. The flag is impossible, hence it can’t be waving. The wind is waving. (At this moment, Zeno happens by.) Zeno: Hallo! Hulloo! What’s up? What’s new? ACHILLES: The flag is moving. TORTOISE: The wind is moving. Zeno: O friends, Friends! Cease your argumentation! Arrest your vitriolics! Abandon your discord! For I shall resolve the issue for you forthwith. Ho! And on such a fine day. ACHILLES: This fellow must be playing the fool. TORTOISE: No, wait, Achilles. Let us hear what he has to say. Oh Unknown Sir, do impart to us your thoughts on this matter. Zeno: Most willingly. Not thw ind, not the flag – neither one is moving, nor is anything moving at all. For I have discovered a great Theorem, which states; “Motion Is Inherently Impossible.” And from this Theorem follows an even greater Theorem – Zeno’s Theorem: “Motion Unexists.” ACHILLES: “Zeno’s Theorem”? Are you, sir, by any chance, the philosopher Zeno of Elea? Zeno: I am indeed, Achilles. ACHILLES: (scratching his head in puzzlement). Now how did he know my name? Zeno: Could I possibly persuade you two to hear me out as to why this is the case? I’ve come all the way to Elea from point A this afternoon, just trying to find someone who’ll pay some attention to my closely honed argument. But they’re all hurrying hither and thither, and they don’t have time. You’ve no idea how disheartening it is to meet with refusal after refusal. Oh, I’m sorry to burden you with my troubles, I’d just like to ask you one thing: Would the two of you humour a sill old philosopher for a few moments – only a few, I promise you – in his eccentric theories. ACHILLES: Oh, by all means! Please do illuminate us! I know I speak for both of us, since my companion, Mr. Tortoise, was only moments ago speaking of you with great veneration – and he mentioned especially your paradoxes. Zeno: Thank you. You see, my Master, the fifth patriarch, taught me that reality is one, immutable, and unchanging, all plurality, change, and motion are mere illusions of the sense. Some have mocked his views; but I will show the absurdity of their mockery. My argument is quite simple. I will illustrate it with two characters of my own Invention: Achilles )a Greek warrior, the fleetest of foot of all mortals), and a Tortoise. In my tale, they are persuaded by a passerby to run a footrace down a runway towards a distant flag waving in the breeze. Let us assume that, since the Tortoise is a much slowerrunner, he gets a head start of, say, ten rods. Now the race begins. In a few bounds Achilles has reached the spot where the Tortoise started.  
 
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ACHILLES: Hah! Zeno: And now the Tortoise is but a single rod ahead of Achilles. Within only a moment, Achilles has attained that spot. ACHILLES: Ho ho! Zeno: Yet, in that short moment, the Tortoise has managed to advance a slight amount. In a flash, Achilles covers that distance too. ACHILLES: Hee hee hee! Zeno: But in that very short flash, the Tortoise has managed to inch ahead by ever so little, and so Achilles is still behind. Now you see that in order for Achilles to catch the Tortoise, this game of “trytocatchme” will have to be played an INFINITE number of times – and therefore Achilles can NEVER catch up with the Tortoise. TORTOISE: Heh heh heh heh! ACHILLES: Hmm… Hmm… Hmm… Hmm… Hmm…That argument sounds wrong to me. And yes, I can’t quite make out what’s wrong with it Zeno: Isn’t it a teaser? It’s my favourite paradox. TORTOISE: Excuse me, Zeno, but I believe your tale illustrates the wrong principle, doe sit not? You have just told us what will come to known, centuries hence, as Zeno’s “Achilles paradox” , which shows (ahem!) that Achilles will never catch the Tortoise; but the proof that Motion Is Inherently Impossible (and thence that Motion Unexists) is your “dichotomy paradox”, isn’t that so? Zeno: Oh, shame on me. Of course, you’re right. That’s the new one about how, in going from A to B, one has to go halfway first – and of that stretch one also has to go halfway, and so on and so forth. But you see, both those paradoxes really have the same flavour. Frankly, I’ve only had one Great Idea – I just exploit it in different ways. ACHILLES: I swear, these arguments contain a flaw. I don’t quite see where, but they cannot be correct. Zeno: You doubt the validity of my paradox? Why not just try it out? You see that red flag waving down here, at the far end of the runway? ACHILLES: The impossible one, based on an Escher print? Zeno: Exactly. What do you say to you and Mr. Tortoise racing for it, allowing Mr. T a fair head start of, well, I don’t know – TORTOISE: How about ten rods? Zeno: Very good – ten rods. ACHILLES: Any time. Zeno: Excellent! How exciting! An empirical test of my rigorously proven Theorem! Mr. Tortoise, will you position yourself ten rods upwind? (The Tortoise moves ten rods closer to the flag) Tortoise and Achlles: Ready! Zeno: On your mark! Get set! Go!  
 
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Chapter 1  
 
The MUpuzzle  
 
Formal Systems ONE OF THE most central notions in this book is that of a formal system. The type of formal system I use was invented by the American logician Emil Post in the 1920's, and is often called a "Post production system". This Chapter introduces you to a formal system and moreover, it is my hope that you will want to explore this formal system at least a little; so to provoke your curiosity, I have posed a little puzzle. "Can you produce MU?" is the puzzle. To begin with, you will be supplied with a string (which means a string of letters).* Not to keep you in suspense, that string will be MI. Then you will be told some rules, with which you can change one string into another. If one of those rules is applicable at some point, and you want to use it, you may, butthere is nothing that will dictate which rule you should use, in case there are several applicable rules. That is left up to youand of course, that is where playing the game of any formal system can become something of an art. The major point, which almost doesn't need stating, is that you must not do anything which is outside the rules. We might call this restriction the "Requirement of Formality". In the present Chapter, it probably won't need to be stressed at all. Strange though it may sound, though, I predict that when you play around with some of the formal systems of Chapters to come, you will find yourself violating the Requirement of Formality over and over again, unless you have worked with formal systems before. The first thing to say about our formal systemthe MIUsystemis that it utilizes only three letters of the alphabet: M, I, U. That means that the only strings of the MIUsystem are strings which are composed of those three letters. Below are some strings of the MIUsystem: MU UIM MUUMUU UIIUMIUUIMUIIUMIUUIMUIIU * In this book, we shall employ the following conventions when we refer to strings. When the string is in the same typeface as the text, then it will be enclosed in single or double quotes. Punctuation which belongs to the sentence and not to the string under discussion will go outside of the quotes, as logic dictates. For example, the first letter of this sentence is 'F', while the first letter of 'this ‘sentence’.is 't'. When the string is in Quadrata Roman, however, quotes will usually be left off, unless clarity demands them. For example, the first letter of Quadrata is Q.  
 
The MUpuzzle  41  
 
 
But although all of these are legitimate strings, they are not strings which are "in your possession". In fact, the only string in your possession so far is MI. Only by using the rules, about to be introduced, can you enlarge your private collection. Here is the first rule: RULE I: If you possess a string whose last letter is I, you can add on a U at the end. By the way, if up to this point you had not guessed it, a fact about the meaning of "string" is that the letters are in a fixed order. For example, MI and IM are two different strings. A string of symbols is not just a "bag" of symbols, in which the order doesn't make any difference. Here is the second rule: RULE II: Suppose you have Mx. Then you may add Mxx to your collection. What I mean by this is shown below, in a few examples. From MIU, you may get MIUIU. From MUM, you may get MUMUM. From MU, you may get MUU. So the letter `x' in the rule simply stands for any string; but once you have decided which string it stands for, you have to stick with your choice (until you use the rule again, at which point you may make a new choice). Notice the third example above. It shows how, once you possess MU, you can add another string to your collection; but you have to get MU first! I want to add one last comment about the letter `x': it is not part of the formal system in the same way as the three letters `M', `I', and `U' are. It is useful for us, though, to have some way to talk in general about strings of the system, symbolicallyand that is the function of the `x': to stand for an arbitrary string. If you ever add a string containing an 'x' to your "collection", you have done something wrong, because strings of the MIUsystem never contain "x" “s”! Here is the third rule: RULE III: If III occurs in one of the strings in your collection, you may make a new string with U in place of III. Examples: From UMIIIMU, you could make UMUMU. From MII11, you could make MIU (also MUI). From IIMII, you can't get anywhere using this rule. (The three I's have to be consecutive.) From MIII, make MU. Don't, under any circumstances, think you can run this rule backwards, as in the following example:  
 
The MUpuzzle  42  
 
 
From MU, make MIII < This is wrong. Rules are oneway. Here is the final rule. RULE IV: If UU occurs inside one of your strings, you can drop it. From UUU, get U. From MUUUIII, get MUIII. There you have it. Now you may begin trying to make MU. Don't worry you don't get it. Just try it out a bitthe main thing is for you to get the flavor of this MUpuzzle. Have fun. Theorems, Axioms, Rules The answer to the MUpuzzle appears later in the book. For now, what important is not finding the answer, but looking for it. You probably hay made some attempts to produce MU. In so doing, you have built up your own private collection of strings. Such strings, producible by the rules, are called theorems. The term "theorem" has, of course, a common usage mathematics which is quite different from this one. It means some statement in ordinary language which has been proven to be true by a rigorous argument, such as Zeno's Theorem about the "unexistence" of motion, c Euclid's Theorem about the infinitude of primes. But in formal system theorems need not be thought of as statementsthey are merely strings c symbols. And instead of being proven, theorems are merely produced, as if F machine, according to certain typographical rules. To emphasize this important distinction in meanings for the word "theorem", I will adopt the following convention in this book: when "theorem" is capitalized, its meaning will be the everyday onea Theorem is a statement in ordinary language which somebody once proved to be true by some sort of logic argument. When uncapitalized, "theorem" will have its technical meaning a string producible in some formal system. In these terms, the MUpuzzle asks whether MU is a theorem of the MIUsystem. I gave you a theorem for free at the beginning, namely MI. Such "free" theorem is called an axiomthe technical meaning again being qui different from the usual meaning. A formal system may have zero, or several, or even infinitely many axioms. Examples of all these types v appear in the book. Every formal system has symbolshunting rules, such as the four rules of the MIUsystem. These rules are called either rules of production or rules of inference. I will use both terms. The last term which I wish to introduce at this point is derivation. Shown below is a derivation of the theorem MUIIU: (1) MI axiom (2) MII from (1) by rule II  
 
The MUpuzzle  43  
 
 
(3) MIII from (2) by rule II (4) MIIIIU from (3) by rule I (5) MUIU from (4) by rule III (6) MUIUUIU from (5) by rule II (7) MUIIU from (6) by rule IV A derivation of a theorem is an explicit, linebyline demonstration of how to produce that theorem according to the rules of the formal system. The concept of derivation is modeled on that of proof, but a derivation is an austere cousin of a proof. It would sound strange to say that you had proven MUIIU, but it does not sound so strange to say you have derived MUIIU. Inside and Outside the System Most people go about the MUpuzzle by deriving a number of theorems, quite at random, just to see what kind of thing turns up. Pretty soon, they begin to notice some properties of the theorems they have made; that is where human intelligence enters the picture. For instance, it was probably not obvious to you that all theorems would begin with M, until you had tried a few. Then, the pattern emerged, and not only could you see the pattern, but you could understand it by looking at the rules, which have the property that they make each new theorem inherit its first letter from an earlier theorem; ultimately, then, all theorems' first letters can be traced back to the first letter of the sole axiom MIand that is a proof that theorems of the MIUsystem must all begin with M. There is something very significant about what has happened here. It shows one difference between people and machines. It would certainly be possiblein fact it would be very easyto program a computer to generate theorem after theorem of the MIUsystem; and we could include in the program a command to stop only upon generating U. You now know that a computer so programmed would never stop. And this does not amaze you. But what if you asked a friend to try to generate U? It would not surprise you if he came back after a while, complaining that he can't get rid of the initial M, and therefore it is a wild goose chase. Even if a person is not very bright, he still cannot help making some observations about what he is doing, and these observations give him good insight into the taskinsight which the computer program, as we have described it, lacks. Now let me be very explicit about what I meant by saying this shows a difference between people and machines. I meant that it is possible to program a machine to do a routine task in such a way that the machine will never notice even the most obvious facts about what it is doing; but it is inherent in human consciousness to notice some facts about the things one is doing. But you knew this all along. If you punch "1" into an adding machine, and then add 1 to it, and then add 1 again, and again, and again, and continue doing so for hours and hours, the machine will never learn to anticipate you, and do it itself, although any person would pick up the  
 
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pick up the idea, no matter how much or how well it is driven, that it i supposed to avoid other cars and obstacles on the road; and it will never learn even the most frequently traveled routes of its owner. The difference, then, is that it is possible for a machine to act unobservant; it is impossible for a human to act unobservant. Notice I am not saying that all machines are necessarily incapable of making sophisticated observations; just that some machines are. Nor am I saying that all people are always making sophisticated observations; people, in fact, are often very unobservant. But machines can be made to be totally unobservant; any people cannot. And in fact, most machines made so far are pretty close ti being totally unobservant. Probably for this reason, the property of being; unobservant seems to be the characteristic feature of machines, to most people. For example, if somebody says that some task is "mechanical", i does not mean that people are incapable of doing the task; it implies though, that only a machine could do it over and over without eve complaining, or feeling bored. Jumping out of the System It is an inherent property of intelligence that it can jump out of the tas which it is performing, and survey what it has done; it is always looking for and often finding, patterns. Now I said that an intelligence can jump out o its task, but that does not mean that it always will. However, a little prompting will often suffice. For example, a human being who is reading a boo may grow sleepy. Instead of continuing to read until the book is finished he is just as likely to put the book aside and turn off the light. He ha stepped "out of the system" and yet it seems the most natural thing in the world to us. Or, suppose person A is watching television when person B comes in the room, and shows evident displeasure with the situation Person A may think he understands the problem, and try to remedy it b exiting the present system (that television program), and flipping the channel knob, looking for a better show. Person B may have a more radio concept of what it is to "exit the system"namely to turn the television oft Of course, there are cases where only a rare individual will have the vision to perceive a system which governs many peoples lives, a system which ha never before even been recognized as a system; then such people often devote their lives to convincing other people that the system really is there and that it ought to be exited from! How well have computers been taught to jump out of the system? I w cite one example which surprised some observers. In a computer chess: tournament not long ago in Canada, one programthe weakest of all the competing oneshad the unusual feature of quitting long before the game was over. It was not a very good chess player, but it at least had the redeeming quality of being able to spot a hopeless position, and to resign then and there, instead of waiting for the other program to go through the  
 
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boring ritual of checkmating. Although it lost every game it played, it did it in style. A lot of local chess experts were impressed. Thus, if you define "the system" as "making moves in a chess game", it is clear that this program had a sophisticated, preprogrammed ability to exit from the system. On the other hand, if you think of "the system" as being "whatever the computer had been programmed to do", then there is no doubt that the computer had no ability whatsoever to exit from that system. It is very important when studying formal systems to distinguish working within the system from making statements or observations about the system. I assume that you began the MUpuzzle, as do most people, by working within the system; and that you then gradually started getting anxious, and this anxiety finally built up to the point where without any need for further consideration, you exited from the system, trying to take stock of what you had produced, and wondering why it was that you had not succeeded in producing MU. Perhaps you found a reason why you could not produce MU; that is thinking about the system. Perhaps you produced MIU somewhere along the way; that is working within the system. Now I do not want to make it sound as if the two modes are entirely incompatible; I am sure that every human being is capable to some extent of working inside a system and simultaneously thinking about what he is doing. Actually, in human affairs, it is often next to impossible to break things neatly up into "inside the system" and "outside the system"; life is composed of so many interlocking and interwoven and often inconsistent "systems" that it may seem simplistic to think of things in those terms. But it is often important to formulate simple ideas very clearly so that one can use them as models in thinking about more complex ideas. And that is why I am showing you formal systems; and it is about time we went back to discussing the MIUsystem.  
 
MMode, IMode, UMode The MUpuzzle was stated in such a way that it encouraged some amount of exploration within the MIUsystemderiving theorems. But it was also stated in a way so as not to imply that staying inside the system would necessarily yield fruit. Therefore it encouraged some oscillation between the two modes of work. One way to separate these two modes would be to have two sheets of paper; on one sheet, you work "in your capacity as a machine", thus filling it with nothing but M's, I's, and U's; on the second sheet, you work "in your capacity as a thinking being", and are allowed to do whatever your intelligence suggestswhich might involve using English, sketching ideas, working backwards, using shorthand (such as the letter `x'), compressing several steps into one, modifying the rules of the system to see what that gives, or whatever else you might dream up. One thing you might do is notice that the numbers 3 and 2 play an important role, since I's are gotten rid of in three's, and U's in two'sand doubling of length (except for the M) is allowed by rule II. So the second sheet might  
 
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also have some figuring on it. We will occasionally refer back to these two modes of dealing with a formal system, and we will call them the Mechanic mode (Mmode) and the Intelligent mode (Imode). To round out our mode with one for each letter of the MIUsystem, I will also mention a fin modethe Unmode (Umode), which is the Zen way of approaching thing. More about this in a few Chapters.  
 
Decision Procedures An observation about this puzzle is that it involves rules of two opposite tendenciesthe lengthening rules and the shortening rules. Two rules (I and II) allow you to increase the size of strings (but only in very rigid, pr scribed ways, of course); and two others allow you to shrink strings somewhat (again in very rigid ways). There seems to be an endless variety to the order in which these different types of rules might be applied, and this gives hope that one way or another, MU could be produced. It might involve lengthening the string to some gigantic size, and then extracting piece after piece until only two symbols are left; or, worse yet, it might involve successive stages of lengthening and then shortening and then lengthening and then shortening, and so on. But there is no guarantee it. As a matter of fact, we already observed that U cannot be produced at all and it will make no difference if you lengthen and shorten till kingdom come. Still, the case of U and the case of MU seem quite different. It is by very superficial feature of U that we recognize the impossibility of producing it: it doesn't begin with an M (whereas all theorems must). It is very convenient to have such a simple way to detect nontheorems. However who says that that test will detect all nontheorems? There may be lots strings which begin with M but are not producible. Maybe MU is one of them. That would mean that the "firstletter test" is of limited usefulness able only to detect a portion of the nontheorems, but missing others. B there remains the possibility of some more elaborate test which discriminates perfectly between those strings which can be produced by the rules and those which cannot. Here we have to face the question, "What do mean by a test?" It may not be obvious why that question makes sense, of important, in this context. But I will give an example of a "test" which somehow seems to violate the spirit of the word. Imagine a genie who has all the time in the world, and who enjoys using it to produce theorems of the MIUsystem, in a rather methodical way. Here, for instance, is a possible way the genie might go about it Step 1: Apply every applicable rule to the axiom MI. This yields two new theorems MIU, MII. Step 2: Apply every applicable rule to the theorems produced in step 1. This yields three new theorems: MIIU, MIUIU, MIIII.  
 
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Step 3: Apply every applicable rule to the theorems produced in step 2. This yields five new theorems: MIIIIU, MIIUIIU, MIUIUIUIU, MIIIIIIII, MUI.  
 
This method produces every single theorem sooner or later, because the rules are applied in every conceivable order. (See Fig. 11.) All of the lengtheningshortening alternations which we mentioned above eventually get carried out. However, it is not clear how long to wait for a given string  
 
 
FIGURE 11. A systematically constructed "tree" of all the theorems of the MIUsystem. The N th level down contains those theorems whose derivations contain exactly N steps. The encircled numbers tell which rule was employed. Is MU anywhere in this tree?  
 
to appear on this list, since theorems are listed according to the shortness of their derivations. This is not a very useful order, if you are interested in a specific string (such as MU), and you don't even know if it has any derivation, much less how long that derivation might be. Now we state the proposed "theoremhoodtest": Wait until the string in question is produced; when that happens, you know it is a theoremand if it never happens, you know that it is not a theorem. This seems ridiculous, because it presupposes that we don't mind waiting around literally an infinite length of time for our answer. This gets to the crux of the matter of what should count as a "test". Of prime importance is a guarantee that we will get our answer in a finite length of time. If there is a test for theoremhood, a test which does always terminate in a finite  
 
The MUpuzzle  48  
 
 
amount of time, then that test is called a decision procedure for the given formal system. When you have a decision procedure, then you have a very concrete characterization of the nature of all theorems in the system. Offhand, it might seem that the rules and axioms of the formal system provide no less complete a characterization of the theorems of the system than a decision procedure would. The tricky word here is "characterization". Certainly the rules of inference and the axioms of the MIUsystem do characterize, implicitly, those strings that are theorems. Even more implicitly, they characterize those strings that are not theorems. But implicit characterization is not enough, for many purposes. If someone claims to have a characterization of all theorems, but it takes him infinitely long to deduce that some particular string is not a theorem, you would probably tend to say that there is something lacking in that characterizationit is not quite concrete enough. And that is why discovering that a decision procedure exists is a very important step. What the discovery means, in effect, is that you can perform a test for theoremhood of a string, and that, even if the test is complicated, it is guaranteed to terminate. In principle, the test is just as easy, just as mechanical, just as finite, just as full of certitude, as checking whether the first letter of the string is M. A decision procedure is a "litmus test" for theoremhood! Incidentally, one requirement on formal systems is that the set of axioms must be characterized by a decision procedurethere must be a litmus test for axiomhood. This ensures that there is no problem in getting off the ground at the beginning, at least. That is the difference between the set of axioms and the set of theorems: the former always has a decision procedure, but the latter may not. I am sure you will agree that when you looked at the MIUsystem for the first time, you had to face this problem exactly. The lone axiom was known, the rules of inference were simple, so the theorems had been implicitly characterizedand yet it was still quite unclear what the consequences of that characterization were. In particular, it was still totally unclear whether MU is, or is not, a theorem.  
 
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FIGURE 12. Sky Castle, by M. C.: Escher (woodcut, 1928).  
 
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TwoPart Invention  
 
or, What the Tortoise Said to Achilles by Lewis Carroll' Achilles had overtaken the Tortoise, and had seated himself comfortably on its back. "So you've got to the end of our racecourse?" said the Tortoise. "Even though it DOES consist of an infinite series of distances? I thought some wiseacre or other had proved that the thing couldn't be done?" "It CAN be done," said Achilles. "It HAS been done! Solvitur ambulando. You see the distances were constantly DIMINISHING; and so" "But if they had been constantly INCREASING?" the Tortoise interrupted. "How then?" "Then I shouldn't be here," Achilles modestly replied; "and You would have got several times round the world, by this time!" "You flatter meFLATTEN, I mean," said the Tortoise; "for you ARE a heavy weight, and NO mistake! Well now, would you like to hear of a racecourse, that most people fancy they can get to the end of in two or three steps, while it REALLY consists of an infinite number of distances, each one longer than the previous one?" "Very much indeed!" said the Grecian warrior, as he drew from his helmet (few Grecian warriors possessed POCKETS in those days) an enormous notebook and pencil. "Proceed! And speak SLOWLY, please! SHORTHAND isn't invented yet!" "That beautiful First Proposition by Euclid!" the Tortoise murmured dreamily. "You admire Euclid?" "Passionately! So far, at least, as one CAN admire a treatise that won't be published for some centuries to come!" "Well, now, let's take a little bit of the argument in that First Proposition just TWO steps, and the conclusion drawn from them. Kindly enter them in your notebook. And in order to refer to them conveniently, let's call them A, B, and Z: (A) Things that are equal to the same are equal to each other. (B) The two sides of this Triangle are things that are equal to the same. (Z) The two sides of this Triangle are equal to each other. Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, MUST accept Z as true?" "Undoubtedly! The youngest child in a High Schoolas soon as High  
 
TwoPart Invention  51  
 
 
Schools are invented, which will not be till some two thousand years laterwill grant THAT." "And if some reader had NOT yet accepted A and B as true, he might still accept the SEQUENCE as a VALID one, I suppose?" "No doubt such a reader might exist. He might say, `I accept as true the Hypothetical Proposition that, IF A and B be true, Z must be true; but I DON'T accept A and B as true.' Such a reader would do wisely in abandoning Euclid, and taking to football." "And might there not ALSO be some reader who would say `I accept A and B as true, but I DON'T accept the Hypothetical'?" "Certainly there might. HE, also, had better take to football." "And NEITHER of these readers," the Tortoise continued, "is AS YET under any logical necessity to accept Z as true?" "Quite so," Achilles assented. "Well, now, I want you to consider ME as a reader of the SECOND kind, and to force me, logically, to accept Z as true." "A tortoise playing football would be" Achilles was beginning. `an anomaly, of course," the Tortoise hastily interrupted. "Don't wander from the point. Let's have Z first, and football afterwards!" "I'm to force you to accept Z, am I?" Achilles said musingly. "And your present position is that you accept A and B, but you DON'T accept the Hypothetical" "Let's call it C," said the Tortoise. "but you DON'T accept (C) If A and B are true, Z must be true." "That is my present position," said the Tortoise. "Then I must ask you to accept C." "I'll do so," said the Tortoise, "as soon as you've entered it in that notebook of yours. What else have you got in it?" "Only a few memoranda," said Achilles, nervously fluttering the leaves: "a few memoranda ofof the battles in which I have distinguished myself!" "Plenty of blank leaves, I see!" the Tortoise cheerily remarked. "We shall need them ALL!" (Achilles shuddered.) "Now write as I dictate: (A) Things that are equal to the same are equal to each other. (B) The two sides of this Triangle are things that are equal to the same. (C) If A and B are true, Z must be true. (Z) The two sides of this Triangle are equal to each other." "You should call it D, not Z," said Achilles. "It comes NEXT to the other three. If you accept A and B and C, you MUST accept Z.  
 
TwoPart Invention  52  
 
 
“And why must I?” "Because it follows LOGICALLY from them. If A and B and C are true, Z MUST be true. You can't dispute THAT, I imagine?" "If A and B and C are true, Z MUST be true," the Tortoise thoughtfully repeated. "That's ANOTHER Hypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and STILL not accept Z, mightn't I?" "You might," the candid hero admitted; "though such obtuseness would certainly be phenomenal. Still, the event is POSSIBLE. So I must ask you to grant ONE more Hypothetical." "Very good, I'm quite willing to grant it, as soon as you've written it down. We will call it (D) If A and B and C are true, Z must be true. Have you entered that in your notebook?" "I HAVE!" Achilles joyfully exclaimed, as he ran the pencil into its sheath. "And at last we've got to the end of this ideal racecourse! Now that you accept A and B and C and D, OF COURSE you accept Z." "Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I STILL refused to accept Z?" "Then Logic would take you by the throat, and FORCE you to do it!" Achilles triumphantly replied. "Logic would tell you, `You can't help yourself. Now that you've accepted A and B and C and D, you MUST accept Z!' So you've no choice, you see.", "Whatever LOGIC is good enough to tell me is worth WRITING DOWN," said the Tortoise. "So enter it in your book, please. We will call it (E) If A and B and C and D are true, Z must be true. Until I've granted THAT, of course I needn't grant Z. So it's quite a NECESSARY step, you see?" "I see," said Achilles; and there was a touch of sadness in his tone. Here the narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the muchenduring Tortoise, and was writing in his notebook, which appeared to be nearly full. The Tortoise was saying, "Have you got that last step written down? Unless I've lost count, that makes a thousand and one. There are several millions more to come. And WOULD you mind, as a personal favour, considering what a lot of instruction this colloquy of ours will provide for the Logicians of the Nineteenth CenturyWOULD you mind adopting a pun that my cousin the MockTurtle will then make, and allowing yourself to be renamed TAUGHTUS?" "As you please," replied the weary warrior, in the hollow tones of despair, as he buried his face in his hands. "Provided that YOU, for YOUR part, will adopt a pun the MockTurtle never made, and allow yourself to be renamed A KILLEASE!"  
 
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CHAPTER 11 Meaning and Form in Mathematics. THIS TwoPart Invention was the inspiration for my two characters. Just as Lewis Carroll took liberties with Zeno's Tortoise and Achilles, so have I taken liberties with Lewis Carroll's Tortoise and Achilles. In Carroll's dialogue, the same events take place over and over again, only each time on a higher and higher level; it is a wonderful analogue to Bach's EverRising Canon. The Carrollian Dialogue, with its wit subtracted out, still leaves a deep philosophical problem: Do words and thoughts follow formal rules, or do they not? That problem is the problem of this book. In this Chapter and the next, we will look at several new formal systems. This will give us a much wider perspective on the concept of formal system. By the end of these two Chapters, you should have quite a good idea of the power of formal systems, and why they are of interest to mathematicians and logicians. The pqSystem The formal system of this Chapter is called the pqsystem. It is not important to mathematicians or logiciansin fact, it is just a simple invention of mine. Its importance lies only in the fact that it provides an excellent example of many ideas that play a large role in this book. There are three distinct symbols of the pqsystem: p q  the letters p, q, and the hyphen. The pqsystem has an infinite number of axioms. Since we can't write them all down, we have to have some other way of describing what they are. Actually, we want more than just a description of the axioms; we want a way to tell whether some given string is an axiom or not. A mere description of axioms might characterize them fully and yet weaklywhich was the problem with the way theorems in the MIUsystem were characterized. We don't want to have to struggle for an indeterminatepossibly infinite length of time, just to find out if some string is an axiom or not. Therefore, we will define axioms in such a way that there is an obvious decision procedure for axiomhood of a string composed of p's, q's, and hyphens.  
 
Meaning and Form in Mathematics  54  
 
 
DEFINITION: xpqx is an axiom, whenever x is composed of hyphens only. Note that 'x' must stand for the same string of hyphens in both occurrences For example, pqis an axiom. The literal expression `xpqx' i,, not an axiom, of course (because `x' does not belong to the pqsystem); it is more like a mold in which all axioms are castand it is called an axiom schema. The pqsystem has only one rule of production: RULE: Suppose x, y, and z all stand for particular strings containing only hyphens. And suppose that x py qz is known to be a theorem. The` xpyqz is a theorem. For example, take x to be'', y to be'', and z to be''. The rule tells us: If pq turns out to be a theorem, then so will pq.  
 
As is typical of rules of production, the statement establishes a causal connection between the theoremhood of two strings, but without asserting theoremhood for either one on its own. A most useful exercise for you is to find a decision procedure for the theorems of the pqsystem. It is not hard; if you play around for a while you will probably pick it up. Try it.  
 
The Decision Procedure I presume you have tried it. First of all, though it may seem too obvious to mention, I would like to point out that every theorem of the pqsystem has three separate groups of hyphens, and the separating elements are one p, and one q, in that order. (This can be shown by an argument based on "heredity", just the way one could show that all MIUsystem theorems had to begin with M.) This means that we can rule out, from its form alone, o string such as pppq . Now, stressing the phrase "from its form alone" may seem silly; what else is there to a string except its form? What else could possibly play a roll in determining its properties? Clearly nothing could. But bear this in mint as the discussion of formal systems goes on; the notion of "form" will star to get rather more complicated and abstract, and we will have to think more about the meaning of the word "form". In any case, let us give the name well formed string to any string which begins with a hyphengroup, then ha one p, then has a second hyphengroup, then a q, and then a final hyphengroup. Back to the decision procedure ... The criterion for theoremhood is that the first two hyphengroups should add up, in length, to the third  
 
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hyphengroup. for instance, pq  is a theorem, since 2 plus 2 equals 4, whereas pqis not, since 2 plus 2 is not 1. To see why this is the proper criterion, look first at the axiom schema. Obviously, it only manufactures axioms which satisfy the addition criterion. Second, look at the rule of production. If the first string satisfies the addition criterion, so must the second oneand conversely, if the first string does not satisfy the addition criterion, then neither does the second string. The rule makes the addition criterion into a hereditary property of theorems: any theorem passes the property on to its offspring. This shows why the addition criterion is correct. There is, incidentally, a fact about the pqsystem which would enable us to say with confidence that it has a decision procedure, even before finding the addition criterion. That fact is that the pqsystem is not complicated by the opposing currents of lengthening and shortening rules; it has only lengthening rules. Any formal system which tells you how to make longer theorems from shorter ones, but never the reverse, has got to have a decision procedure for its theorems. For suppose you are given a string. First check whether it's an axiom or not (I am assuming that there is a decision procedure for axiomhoodotherwise, things are hopeless). If it is an axiom, then it is by definition a theorem, and the test is over. So suppose instead that it's not an axiom. Then, to be a theorem, it must have come from a shorter string, via one of the rules. By going over the various rules one by one, you can pinpoint not only the rules that could conceivably produce that string, but also exactly which shorter strings could be its forebears on the "family tree". In this way, you "reduce" the problem to determining whether any of several new but shorter strings is a theorem. Each of them can in turn be subjected to the same test. The worst that can happen is a proliferation of more and more, but shorter and shorter, strings to test. As you continue inching your way backwards in this fashion, you must be getting closer to the source of all theoremsthe axiom schemata. You just can't get shorter and shorter indefinitely; therefore, eventually either you will find that one of your short strings is an axiom, or you'll come to a point where you're stuck, in that none of your short strings is an axiom, and none of them can be further shortened by running some rule or other backwards. This points out that there really is not much deep interest in formal systems with lengthening rules only; it is the interplay of lengthening and shortening rules that gives formal systems a certain fascination.. Bottomup vs. Topdown The method above might be called a topdown decision procedure, to be contrasted with a bottomup decision procedure, which I give now. It is very reminiscent of the genie's systematic theoremgenerating method for the MIUsystem, but is complicated by the presence of an axiom schema. We are going to form a "bucket" into which we throw theorems as they are generated. Here is how it is done:  
 
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(1a) Throw the simplest possible axiom (pq) into the bucket. (I b) Apply the rule of inference to the item in the bucket, and put the result into the bucket. (2a) Throw the secondsimplest axiom into the bucket. (2b) Apply the rule to each item in the bucket, and throw all results into the bucket. (3a) Throw the thirdsimplest axiom into the bucket. (3b) Apply the rule to each item in the bucket, and throw all results into the bucket. etc., etc. A moment's reflection will show that you can't fail to produce every theorem of the pqsystem this way. Moreover, the bucket is getting filled with longer and longer theorems, as time goes on. It is again a consequence of that lack of shortening rules. So if you have a particular string, such as pq , which you want to test for theoremhood, just follow the numbered steps, checking all the while for the string in question. If it turns uptheorem! If at some point everything that goes into the bucket is longer than the string in question, forget itit is not a theorem. This decision procedure is bottom=up because it is working its way up from the basics, which is to say the axioms. The previous decision procedure is topdown because it does precisely the reverse: it works its way back down towards the basics. Isomorphisms Induce Meaning Now we come to a central issue of this Chapterindeed of the book. Perhaps you have already thought to yourself that the pqtheorems are like additions. The string pq is a theorem because 2 plus 3 equals 5. It could even occur to you that the theorem pqis a statement, written in an odd notation, whose meaning is that 2 plus 3 is 5. Is this a reasonable way to look at things? Well, I deliberately chose 'p' to remind you of 'plus', and 'q' to remind you of 'equals' . . . So, does the string pq actually mean "2 plus 3 equals 5"? What would make us feel that way? My answer would be that we have perceived an isomorphism between pqtheorems and additions. In the Introduction, the word "isomorphism" was defined as an information preserving transformation. We can now go into that notion a little more deeply, and see it from another perspective. The word "isomorphism' applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two part play similar roles in their respective structures. This usage of the word "isomorphism" is derived from a more precise notion in mathematics.  
 
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It is cause for joy when a mathematician discovers an isomorphism between two structures which he knows. It is often a "bolt from the blue", and a source of wonderment. The perception of an isomorphism between two known structures is a significant advance in knowledgeand I claim that it is such perceptions of isomorphism which create meanings in the minds of people. A final word on the perception of isomorphisms: since they come in many shapes and sizes, figuratively speaking, it is not always totally clear when you really have found an isomorphism. Thus, "isomorphism" is a word with all the usual vagueness of wordswhich is a defect but an advantage as well. In this case, we have an excellent prototype for the concept of isomorphism. There is a "lower level" of our isomorphismthat is, a mapping between the parts of the two structures: p <= => plus q <= => equals  <= => one  <= => two  <= => three etc. This symbolword correspondence has a name: interpretation. Secondly, on a higher level, there is the correspondence between true statements and theorems. Butnote carefullythis higherlevel correspondence could not be perceived without the prior choice of an interpretation for the symbols. Thus it would be more accurate to describe it as a correspondence between true statements and interpreted theorems. In any case we have displayed a twotiered correspondence, which is typical of all isomorphisms. When you confront a formal system you know nothing of, and if you hope to discover some hidden meaning in it, your problem is how to assign interpretations to its symbols in a meaningful waythat is, in such a way that a higherlevel correspondence emerges between true statements and theorems. You may make several tentative stabs in the dark before finding a good set of words to associate with the symbols. It is very similar to attempts to crack a code, or to decipher inscriptions in an unknown language like Linear B of Crete: the only way to proceed is by trial and error, based on educated guesses. When you hit a good choice, a "meaningful" choice, all of a sudden things just feel right, and work speeds up enormously. Pretty soon everything falls into place. The excitement of such an experience is captured in The Decipherment of Linear B by John Chadwick. But it is uncommon, to say the least, for someone to be in the position of "decoding" a formal system turned up in the excavations of a ruined civilization! Mathematicians (and more recently, linguists, philosophers, and some others) are the only users of formal systems, and they invariably have an interpretation in mind for the formal systems which they use and publish. The idea of these people is to set up a formal system whose  
 
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Theorems reflect some portion of reality isomorphically. In such a case, the choice of symbols is a highly motivated one, as is the choice of typographical rules of production. When I devised the pqsystem, I was in position. You see why I chose the symbols I chose. It is no accident theorems are isomorphic to additions; it happened because I deliberately sought out a way to reflect additions typographically. Meaningless and Meaningful Interpretations You can choose interpretations other than the one I chose. You need make every theorem come out true. But there would be very little reason make an interpretation in which, say, all theorems came out false, certainly even less reason to make an interpretation under which there is no correlation at all, positive or negative, between theoremhood and tri Let us therefore make a distinction between two types of interpretations a formal system. First, we can have a meaningless interpretation, one un which we fail to see any isomorphic connection between theorems of system, and reality. Such interpretations aboundany random choice a will do. For instance, take this one: p <= => horse q <= => happy  <= => apple  
 
Now pq acquires a new interpretation: "apple horse apple hat apple apple"a poetic sentiment, which might appeal to horses, and mi! even lead them to favor this mode of interpreting pqstrings! However, t interpretation has very little "meaningfulness"; under interpretative, theorems don't sound any truer, or any better, than nontheorems. A ho might enjoy "happy happy happy apple horse" (mapped onto q q q) just as much as any interpreted theorem. The other kind of interpretation will be called meaningful. Under si an interpretation, theorems and truths correspondthat is, an isomorphism exists between theorems and some portion of reality. That is why it is good to distinguish between interpretations and meanings. Any old word can be used as an interpretation for `p', but `plus' is the only meaningful choice we've come up with. In summary, the meaning of `p' seems to be 'plus’ though it can have a million different interpretations. Active vs. Passive Meanings Probably the most significant fact of this Chapter, if understood deeply this: the pqsystem seems to force us into recognizing that symbols of a formal system, though initially without meaning, cannot avoid taking on "meaning" of sorts at least if an isomorphism is found. The difference between meaning it formal system and in a language is a very important one, however. It is this:  
 
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in a language, when we have learned a meaning for a word, we then marc new statements based on the meaning of the word. In a sense the meaning becomes active, since it brings into being a new rule for creating sentences. This means that our command of language is not like a finished product: the rules for making sentences increase when we learn new meanings. On the other hand, in a formal system, the theorems are predefined, by the rules of production. We can choose "meanings" based on an isomorphism (if we can find one) between theorems and true statements. But this does not give us the license to go out and add new theorems to the established theorems. That is what the Requirement of Formality in Chapter I was warning you of. In the MIUsystem, of course, there was no temptation to go beyond the four rules, because no interpretation was sought or found. But here, in our new system, one might be seduced by the newly found "meaning" of each symbol into thinking that the string pppq is a theorem. At least, one might wish that this string were a theorem. But wishing doesn't change the fact that it isn't. And it would be a serious mistake to think that it "must" be a theorem, just because 2 plus 2 plus 2 plus 2 equals 8. It would even be misleading to attribute it any meaning at all, since it is not wellformed, and our meaningful interpretation is entirely derived from looking at wellformed strings. In a formal system, the meaning must remain passive; we can read each string according to the meanings of its constituent symbols, but we do not have the right to create new theorems purely on the basis of the meanings we've assigned the symbols. Interpreted formal systems straddle the line between systems without meaning, and systems with meaning. Their strings can be thought of as "expressing" things, but this must come only as a consequence of the formal properties of the system. DoubleEntendre! And now, I want to destroy any illusion about having found the meanings for the symbols of the pqsystem. Consider the following association: p <= => equals q <= => taken from  <= => one  <= => two etc. Now, pq has a new interpretation: "2 equals 3 taken from 5". Of course it is a true statement. All theorems will come out true under this new interpretation. It is just as meaningful as the old one. Obviously, it is silly to ask, "But which one is the meaning of the string?" An interpreta  
 
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tion will me meaningful to the extent that it accurately reflects some isomorphism to the real world. When different aspects of the real world a isomorphic to each other (in this case, additions and subtractions), or single formal system can be isomorphic to both, and therefore can take ( two passive meanings. This kind of doublevaluedness of symbols at strings is an extremely important phenomenon. Here it seems trivial curious, annoying. But it will come back in deeper contexts and bring with it a great richness of ideas. Here is a summary of our observations about the pqsystem. Und either of the two meaningful interpretations given, every wellform( string has a grammatical assertion for its counterpartsome are true, son false. The idea of well formed strings in any formal system is that they a those strings which, when interpreted symbol for symbol, yield grammatical sentences. (Of course, it depends on the interpretation, but usually, there one in mind.) Among the wellformed strings occur the theorems. The: are defined by an axiom schema, and a rule of production. My goal in inventing the pqsystem was to imitate additions: I wanted every theorem] to express a true addition under interpretation; conversely, I wanted every true addition of precisely two positive integers to be translatable into a string, which would be a theorem. That goal was achieved. Notice, then fore, that all false additions, such as "2 plus 3 equals 6", are mapped into strings which are wellformed, but which are not theorems.  
 
Formal Systems and Reality This is our first example of 'a case where a formal system is based upon portion of reality, and seems to mimic it perfectly, in that its theorems a] isomorphic to truths about that part of reality. However, reality and tt formal system are independent. Nobody need be aware that there is a isomorphism between the two. Each side stands by itselfone plus or equals two, whether or not we know that pq is a theorem; and pq is still a theorem whether or not we connect it with addition. You might wonder whether making this formal system, or any form system, sheds new light on truths in the domain of its interpretation. Hat we learned any new additions by producing pqtheorems? Certainly not but we have learned something about the nature of addition as processnamely, that it is easily mimicked by a typographical rule governing meaningless symbols. This still should not be a big surprise sing addition is such a simple concept. It is a commonplace that addition can I captured in the spinning gears of a device like a cash register. But it is clear that we have hardly scratched the surface, as far formal systems go; it is natural to wonder about what portion of reality co be imitated in its behavior by a set of meaningless symbols governed I formal rules. Can all of reality be turned into a formal system? In a very broad sense, the answer might appear to be yes. One could suggest, for instance, that reality is itself nothing but one very complicated formal  
 
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system. Its symbols do not move around on paper, but rather in a threedimensional vacuum (space); they are the elementary particles of which everything is composed. (Tacit assumption: that there is an end to the descending chain of matter, so that the expression "elementary particles" makes sense.) The "typographical rules" are the laws of physics, which tell how, given the positions and velocities of all particles at a given instant, to modify them, resulting in a new set of positions and velocities belonging to the "next" instant. So the theorems of this grand formal system are the possible configurations of particles at different times in the history of the universe. The sole axiom is (or perhaps, was) the original configuration of all the particles at the "beginning of time". This is so grandiose a conception, however, that it has only the most theoretical interest; and besides, quantum mechanics (and other parts of physics) casts at least some doubt on even the theoretical worth of this idea. Basically, we are asking if the universe operates deterministically, which is an open question. Mathematics and Symbol Manipulation Instead of dealing with such a big picture, let's limit ourselves to mathematics as our "real world". Here, a serious question arises: How can we be sure, if we've tried to model a formal system on some part of mathematics, that we've done the job accuratelyespecially if we're not one hundred per cent familiar with that portion of mathematics already? Suppose the goal of the formal system is to bring us new knowledge in that discipline. How will we know that the interpretation of every theorem is true, unless we've proven that the isomorphism is perfect? And how will we prove that the isomorphism is perfect, if we don't already know all about the truths in the discipline to begin with? Suppose that in an excavation somewhere, we actually did discover some mysterious formal system. We would try out various interpretations and perhaps eventually hit upon one which seemed to make every theorem come out true, and every nontheorem come out false. But this is something which we could only check directly in a finite number of cases. The set of theorems is most likely infinite. How will we know that all theorems express truths under this interpretation, unless we know everything there is to know about both the formal system and the corresponding domain of interpretation? It is in somewhat this odd position that we will find ourselves when we attempt to match the reality of natural numbers (i.e., the nonnegative integers: 0, 1, 2, ...) with the typographical symbols of a formal system. We will try to understand the relationship between what we call "truth" in number theory and what we can get at by symbol manipulation. So let us briefly look at the basis for calling some statements of number theory true, and others false. How much is 12 times 12? Everyone knows it is 144. But how many of the people who give that answer have actually at  
 
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any time in their lives drawn a 12 by 12 rectangle, and then counted the little squares in it? Most people would regard the drawing and counting unnecessary. They would instead offer as proof a few marks on paper, such as are shown below: 12 X 12  24 12  144  
 
And that would be the "proof". Nearly everyone believes that if you counted the squares, you would get 144 of them; few people feel that outcome is in doubt. The conflict between the two points of view comes into sharper focus when you consider the problem of determining the value 987654321 x 123456789. First of all, it is virtually impossible to construct the appropriate rectangle; and what is worse, even if it were constructed and huge armies of people spent centuries counting the little squares, o a very gullible person would be willing to believe their final answer. It is just too likely that somewhere, somehow, somebody bobbled just a little bit. So is it ever possible to know what the answer is? If you trust the symbolic process which involves manipulating digits according to certain simple rules, yes. That process is presented to children as a device which gets right answer; lost in the shuffle, for many children, are the rhyme reason of that process. The digitshunting laws for multiplication are based mostly on a few properties of addition and multiplication which are assumed to hold for all numbers. The Basic Laws of Arithmetic The kind of assumption I mean is illustrated below. Suppose that you down a few sticks: / // // // / / Now you count them. At the same time, somebody else counts them, starting from the other end. Is it clear that the two of you will get the s: answer? The result of a counting process is independent of the way in which it is done. This is really an assumption about what counting i would be senseless to try to prove it, because it is so basic; either you s or you don'tbut in the latter case, a proof won't help you a bit. From this kind of assumption, one can get to the commutativity and associativity of addition (i.e., first that b + c = c + b always, and second that b + (c + d) = (b + c) + d always). The same assumption can also you to the commutativity and associativity of multiplication; just think of  
 
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many cubes assembled to form a large rectangular solid. Multiplicative commutativity and associativity are just the assumptions that when you rotate the solid in various ways, the number of cubes will not change. Now these assumptions are not verifiable in all possible cases, because the number of such cases is infinite. We take them for granted; we believe them (if we ever think about them) as deeply as we could believe anything. The amount of money in our pocket will not change as we walk down the street, jostling it up and down; the number of books we have will not change if we pack them up in a box, load them into our car, drive one hundred miles, unload the box, unpack it, and place the books in a new shelf. All of this is part of what we mean by number. There are certain types of people who, as soon as some undeniable fact is written down, find it amusing to show why that "fact" is false after all. I am such a person, and as soon as I had written down the examples above involving sticks, money, and books, I invented situations in which they were wrong. You may have done the same. It goes to show that numbers as abstractions are really quite different from the everyday numbers which we use. People enjoy inventing slogans which violate basic arithmetic but which illustrate "deeper" truths, such as "1 and 1 make 1" (for lovers), or "1 plus 1 plus 1 equals 1" (the Trinity). You can easily pick holes in those slogans, showing why, for instance, using the plussign is inappropriate in both cases. But such cases proliferate. Two raindrops running down a windowpane merge; does one plus one make one? A cloud breaks up into two cloudsmore evidence for the same? It is not at all easy to draw a sharp line between cases where what is happening could be called "addition", and where some other word is wanted. If you think about the question, you will probably come up with some criterion involving separation of the objects in space, and making sure each one is clearly distinguishable from all the others. But then how could one count ideas? Or the number of gases comprising the atmosphere? Somewhere, if you try to look it up, you can probably find a statement such as, "There are 17 languages in India, and 462 dialects." There is something strange about precise statements like that, when the concepts "language" and "dialect" are themselves fuzzy. Ideal Numbers Numbers as realities misbehave. However, there is an ancient and innate sense in people that numbers ought not to misbehave. There is something clean and pure in the abstract notion of number, removed from counting beads, dialects, or clouds; and there ought to be a way of talking about numbers without always having the silliness of reality come in and intrude. The hardedged rules that govern "ideal" numbers constitute arithmetic, and their more advanced consequences constitute number theory. There is only one relevant question to be asked, in making the transition from numbers as practical things to numbers as formal things. Once you have  
 
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FIGURE 13. Liberation, by M.C. Escher (lithograph, 1955).  
 
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decided to try to capsulize all of number theory in an ideal system, is it really possible to do the job completely? Are numbers so clean and crystalline and regular that their nature can be completely captured in the rules of a formal system? The picture Liberation (Fig. 13), one of Escher's most beautiful, is a marvelous contrast between the formal and the informal, with a fascinating transition region. Are numbers really as free as birds? Do they suffer as much from being crystallized into a ruleobeying system? Is there a magical transition region between numbers in reality and numbers on paper? When I speak of the properties of natural numbers, I don't just mean properties such as the sum of a particular pair of integers. That can be found out by counting, and anybody who has grown up in this century cannot doubt the mechanizability of such processes as counting, adding, multiplying, and so on. I mean the kinds of properties which mathematicians are interested in exploring, questions for which no countingprocess is sufficient to provide the answernot even theoretically sufficient. Let us take a classic example of such a property of natural numbers. The statement is: "There are infinitely many prime numbers." First of all, there is no counting process which will ever be able to confirm, or refute, this assertion. The best we could do would be to count primes for a while and concede that there are "a lot". But no amount of counting alone would ever resolve the question of whether the number of primes is finite or infinite. There could always be more. The statementand it is called "Euclid's Theorem" (notice the capital "T")is quite unobvious. It may seem reasonable, or appealing, but it is not obvious. However, mathematicians since Euclid have always called it true. What is the reason? Euclid's Proof The reason is that reasoning tells them it is so. Let us follow the reasoning involved. We will look at a variant of Euclid's proof. This proof works by showing that whatever number you pick, there is a prime larger than it. Pick a numberN. Multiply all the positive integers starting with 1 and ending with N; in other words, form the factorial of N, written "N!". What you get is divisible by every number up to N. When you add 1 to N!, the result can't be a multiple of 2 (because it leaves 1 over, when you divide by 2); can't be a multiple of 3 (because it leaves I over, when you divide by 3); can't be a multiple of 4 (because it leaves 1 over, when you divide by 4);  
 
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can't be a multiple of N (because it leaves 1 over, when you divide by N); In other words, N! + 1, if it is divisible at all (other than by 1 and itself only is divisible by numbers greater than N. So either it is itself prime, or prime divisors are greater than N. But in either case we've shown the must exist a prime above N. The process holds no matter what number is. Whatever N is, there is a prime greater than N. And thus ends the demonstration of the infinitude of the primes. This last step, incidentally, is called generalization, and we will meet again later in a more formal context. It is where we phrase an argument terms of a single number (N), and then point out that N was unspecified and therefore the argument is a general one. Euclid's proof is typical of what constitutes "real mathematics". It simple, compelling, and beautiful. It illustrates that by taking several rash short steps one can get a long way from one's starting point. In our case, t starting points are basic ideas about multiplication and division and forth. The short steps are the steps of reasoning. And though eve individual step of the reasoning seems obvious, the end result is not obvious. We can never check directly whether the statement is true or not; } we believe it, because we believe in reasoning. If you accept reasoning there seems to be no escape route; once you agree to hear Euclid out, you’ll have to agree with his conclusion. That's most fortunatebecause it mea that mathematicians will always agree on what statements to label "true and what statements to label "false". This proof exemplifies an orderly thought process. Each statement related to previous ones in an irresistible way. This is why it is called "proof'' rather than just "good evidence". In mathematics the goal always to give an ironclad proof for some unobvious statement. The very fact of the steps being linked together in an ironclad way suggests ti there may be a patterned structure binding these statements together. TI structure can best be exposed by finding a new vocabularya stylized vocabulary, consisting of symbolssuitable only for expressing statements about numbers. Then we can look at the proof as it exists in its translated version. It will be a set of statements which are related, line by line, in some detectable way. But the statements, since they're represented by means a small and stylized set of symbols, take on the aspect of patterns. In other words, though when read aloud, they seem to be statements about numb and their properties, still when looked at on paper, they seem to be abstract patternsand the linebyline structure of the proof may start to look like slow transformation of patterns according to some few typographical rules. Getting Around Infinity Although Euclid's proof is a proof that all numbers have a certain property it avoids treating each of the infinitely many cases separately. It gets around  
 
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it by using phrases like "whatever N is", or "no matter what number N is". We could also phrasethe proof over again, so that it uses the phrase "all N". By knowing the appropriate context and correct ways of using such phrases, we never have to deal with infinitely many statements. We deal with just two or three concepts, such as the word "all"which, though themselves finite, embody an infinitude; and by using them, we sidestep the apparent problem that there are an infinite number of facts we want to prove. We use the word "all" in a few ways which are defined by the thought processes of reasoning. That is, there are rules which our usage of "all" obeys. We may be unconscious of them, and tend to claim we operate on the basis of the meaning of the word; but that, after all, is only a circumlocution for saying that we are guided by rules which we never make explicit. We have used words all our lives in certain patterns, and instead of calling the patterns "rules", we attribute the courses of our thought processes to the "meanings" of words. That discovery was a crucial recognition in the long path towards the formalization of number theory. If we were to delve into Euclid's proof more and more carefully, we would see that it is composed of many, many smallalmost infinitesimal steps. If all those steps were written out line after line, the proof would appear incredibly complicated. To our minds it is clearest when several steps are telescoped together, to form one single sentence. If we tried to look at the proof in slow motion, we would begin to discern individual frames. In other words, the dissection can go only so far, and then we hit the "atomic" nature of reasoning processes. A proof can be broken down into a series of tiny but discontinuous jumps which seem to flow smoothly when perceived from a higher vantage point. In Chapter VIII, I will show one way of breaking the proof into atomic units, and you will see how incredibly many steps are involved. Perhaps it should not surprise you, though. The operations in Euclid's brain when he invented the proof must have involved millions of neurons (nerve cells), many of which fired several hundred times in a single second. The mere utterance of a sentence involves hundreds of thousands of neurons. If Euclid's thoughts were that complicated, it makes sense for his proof to contain a huge number of steps! (There may be little direct connection between the neural actions in his brain, and a proof in our formal system, but the complexities of the two are comparable. It is as if nature wants the complexity of the proof of the infinitude of primes to be conserved, even when the systems involved are very different from each other.) In Chapters to come, we will lay out a formal system that (1) includes a stylized vocabulary in which all statements about natural numbers can be expressed, and (2) has rules corresponding to all the types of reasoning which seem necessary. A very important question will be whether the rules for symbol manipulation which we have then formulated are really of equal power (as far as number theory is concerned) to our usual mental reasoning abilitiesor, more generally, whether it is theoretically possible to attain the level of our thinking abilities, by using some formal system.  
 
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Sonata for Unaccompanied Achilles The telephone rings; Achilles picks it up.  
 
Achilles: Hello, this is Achilles. Achilles: Oh, hello, Mr. T. How are you? Achilles: A torticollis? Oh, I'm sorry to hear it. Do you have any idea what caused it? Achilles: How long did you hold it in that position? Achilles: Well, no wonder it's stiff, then. What on earth induced you keep your neck twisted that way for so long? Achilles: Wondrous many of them, eh? What kinds, for example? Achilles: What do you mean, "phantasmagorical beasts"? FIGURE 14. Mosaic II, by M. C. Escher (lithograph, 1957).  
 
 
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Achilles: Wasn't it terrifying to see so many of them at the same time? Achilles: A guitar!? Of all things to be in the midst of all those weird creatures. Say, don't you play the guitar? Achilles: Oh, well, it's all the same to me. Achilles: You're right; I wonder why I never noticed that difference between fiddles and guitars before. Speaking of fiddling, how would you like to come over and listen to one of the sonatas for unaccompanied violin by your favorite composer, J. S. Bach? I just bought a marvelous recording of them. I still can't get over the way Bach uses a single violin to create a piece with such interest. Achilles: A headache too? That's a shame. Perhaps you should just go to bed. Achilles: I see. Have you tried counting sheep? Achilles: Oh, oh, I see. Yes, I fully know what you mean. Well, if it's THAT distracting, perhaps you'd better tell it to me, and let me try to work on it, too. Achilles: A word with the letters `A', `D', `A', `C' consecutively inside it ... Hmm ... What about "abracadabra"? Achilles: True, "ADAC" occurs backwards, not forwards, in that word. Achilles: Hours and hours? It sounds like I'm in for a long puzzle, then. Where did you hear this infernal riddle? Achilles: You mean he looked like he was meditating on esoteric Buddhist matters, but in reality he was just trying to think up complex word puzzles? Achilles: Aha!the snail knew what this fellow was up to. But how did you come to talk to the snail? Achilles: Say, I once heard a word puzzle a little bit like this one. Do you want to hear it? Or would it just drive you further into distraction? Achilles: I agreecan't do any harm. Here it is: What's a word that begins with the letters "HE" and also ends with "HE"? Achilles: Very ingeniousbut that's almost cheating. It's certainly not what I meant! Achilles: Of course you're rightit fulfills the conditions, but it's a sort of "degenerate" solution. There's another solution which I had in mind. Achilles: That's exactly it! How did you come up with it so fast? Achilles: So here's a case where having a headache actually might have helped you, rather than hindering you. Excellent! But I'm still in the dark on your "ADAC" puzzle. Achilles: Congratulations! Now maybe you'll be able to get to sleep! So tell me, what is the solution? Achilles: Well, normally I don't like hints, but all right. What's your hint? Achilles: I don't know what you mean by "figure" and "ground" in this case. Achilles: Certainly I know Mosaic II! I know ALL of Escher's works. After all, he's my favorite artist. In any case, I've got a print of Mosaic II hanging on my wall, in plain view from here.  
 
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Achilles:: Yes, t see all the black animals. Achilles: Yes, I also see how their "negative space"  what's left out defines the white animals. Achilles: So THAT'S what you mean by "figure" and "ground". But what does that have to do with the "ADAC" puzzle? Achilles: Oh, this is too tricky for me. I think I'M starting to get a headache Achilles: You want to come over now? But I thoughtAchilles: Very well. Perhaps by then I'll have thought of the right answer to YOUR puzzle, using your figureground hint, relating it to MY puzzle Achilles: I'd love to play them for you. Achilles: You've invented a theory about them? Achilles: Accompanied by what instrument? Achilles: Well, if that's the case, it seems a little strange that he would have written out the harpsichord part, then, and had it published a s well. Achilles: I see  sort of an optional feature. One could listen to them either way  with or without accompaniment. But how would one know what the accompaniment is supposed to sound like? Achilles: Ah, yes, I guess that it is best, after all, to leave it to the listener’s imagination. And perhaps, as you said, Bach never even had accompaniment in mind at all. Those sonatas seem to work very indeed as they are. Achilles: Right. Well, I'll see you shortly. Achilles: Goodbye, Mr. T.  
 
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CHAPTER III  
 
Figure and Ground Primes vs. Composites THERE IS A strangeness to the idea that concepts can be captured by simple typographical manipulations. The one concept so far captured is that of addition, and it may not have appeared very strange. But suppose the goal were to create a formal system with theorems of the form Px, the letter `x' standing for a hyphenstring, and where the only such theorems would be ones in which the hyphenstring contained exactly a prime number of hyphens. Thus, P would be a theorem, but P would not. How could this be done typographically? First, it is important to specify clearly what is meant by typographical operations. The complete repertoire has been presented in the MIUsystem and the pqsystem, so we really only need to make a list of the kinds of things we have permitted: (1) reading and recognizing any of a finite set of symbols; (2) writing down any symbol belonging to that set; (3) copying any of those symbols from one place to another; (4) erasing any of those symbols; (5) checking to see whether one symbol is the same as another; (6) keeping and using a list of previously generated theorems. The list is a little redundant, but no matter. What is important is that it clearly involves only trivial abilities, each of them far less than the ability to distinguish primes from nonprimes. How, then, could we compound some of these operations to make a formal system in which primes are distinguished from composite numbers? The tqSystem A first step might be to try to solve a simpler, but related, problem. We could try to make a system similar to the pqsystem, except that it represents multiplication, instead of addition. Let's call it the tqsystem, `t' for times'. More specifically, suppose X, Y, and Z are, respectively, the numbers of hyphens in the hyphenstrings x, y, and z. (Notice I am taking special pains to distinguish between a string and the number of hyphens it contains.) Then we wish the string x ty q z to be a theorem if and only if X times Y equals Z. For instance, tq  should be a theorem because 2 times 3 equals 6, but  tq should not be a theorem. The tqsystem can be characterized just about as easily as the pqsystem namely, by using just one axiom schema and one rule of inference:  
 
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AXIOM SCHEMA: xtqx is an axiom, whenever x is a hyphen string. . RULE OF INFERENCE: Suppose that x, y, and z are all hyphenstrings. An suppose that x ty qz is an old theorem. Then, xtyqzx is a ne' theorem. Below is the derivation of the theorem tq  : (1) tq (axiom) (2) t—q (by rule of inference, using line (1) as the old theorem) (3) tq  (by rule of inference, using line (2) as the old theorem) Notice how the middle hyphenstring grows by one hyphen each time the rule of inference is applied; so it is predictable that if you want a theorem with ten hyphens in the middle, you apply the rule of inference nine times in a row. Capturing Compositeness Multiplication, a slightly trickier concept than addition, has now bee] "captured" typographically, like the birds in Escher's Liberation. What about primeness? Here's a plan that might seem smart: using the tqsystem define a new set of theorems of the form Cx, which characterize compost. numbers, as follows: RULE: Suppose x, y, and z are hyphenstrings. If xtyqz is a theorem then C z is a theorem. This works by saying that Z (the number of hyphens in z) is composite a long as it is the product of two numbers greater than 1namely, X + (the number of hyphens in x), and Y + 1 (the number of hyphens in y I am defending this new rule by giving you some "Intelligent mode justifications for it. That is because you are a human being, and want t, know why there is such a rule. If you were operating exclusively in the "Mechanical mode", you would not need any justification, since Mmod. workers just follow the rules mechanically and happily, never questioning; them! Because you work in the Imode, you will tend to blur in your mind the distinction between strings and their interpretations. You see, things Cal become quite confusing as soon as you perceive "meaning" in the symbol which you are manipulating. You have to fight your own self to keep from thinking that the string'' is the number 3. The Requirement of Formality, which in Chapter I probably seemed puzzling (because it seemed so obvious), here becomes tricky, and crucial. It is the essential thing which keeps you from mixing up the Imode with the Mmode; or said another way, it keeps you from mixing up arithmetical facts with typographical theorems.  
 
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Illegally Characterizing Primes It is very tempting to jump from the Ctype theorems directly to Ptype theorems, by proposing a rule of the following kind: PROPOSED RULE: Suppose x is a hyphenstring. If Cx is not a theorem, then Px is a theorem. The fatal flaw here is that checking whether Cx is not a theorem is not an explicitly typographical operation. To know for sure that MU is not a theorem of the MIUsystem, you have to go outside of the system ... and so it is with this Proposed Rule. It is a rule which violates the whole idea of formal systems, in that it asks you to operate informallythat is, outside the system. Typographical operation (6) allows you to look into the stockpile of previously found theorems, but this Proposed Rule is asking you to look into a hypothetical "Table of Nontheorems". But in order to generate such a table, you would have to do some reasoning outside the systemreasoning which shows why various strings cannot be generated inside the system. Now it may well be that there is another formal system which can generate the "Table of Nontheorems", by purely typographical means. In fact, our aim is to find just such a system. But the Proposed Rule is not a typographical rule, and must be dropped. This is such an important point that we might dwell on it a bit more. In our Csystem (which includes the tqsystem and the rule which defines Ctype theorems), we have theorems of the form Cx, with `x' standing, as usual, for a hyphenstring. There are also nontheorems of the form Cx. (These are what I mean when I refer to "nontheorems", although of course ttCqq and other illformed messes are also nontheorems.) The difference is that theorems have a composite number of hyphens, nontheorems have a prime number of hyphens. Now the theorems all have a common "form", that is, originate from a common set of typographical rules. Do all nontheorems also have a common "form", in the same sense? Below is a list of Ctype theorems, shown without their derivations. The parenthesized numbers following them simply count the hyphens in them. C (4) C  (6) C  (8) C  (9) C  (10) C  (12) C  (14) C  (15) C  (16) C  (18)  
 
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I he "holes" in this list are the nontheorems. I o repeat the earlier quest Do the holes also have some "form" in common? Would it be reasonable say that merely by virtue of being the holes in this list, they share a common form? Yes and no. That they share some typographical quality is and able, but whether we want to call it "form" is unclear. The reason hesitating is that the holes are only negatively definedthey are the things that are left out of a list which is positively defined. Figure and Ground This recalls the famous artistic distinction between figure and ground. When a figure or "positive space" (e.g., a human form, or a letter, or a still life is drawn inside a frame, an unavoidable consequence is that its complementary shapealso called the "ground", or "background", or "negative space"has also been drawn. In most drawings, however, this fig ground relationship plays little role. The artist is much less interested in ground than in the figure. But sometimes, an artist will take interest in ground as well. There are beautiful alphabets which play with this figureground distinction. A message written in such an alphabet is shown below. At fir looks like a collection of somewhat random blobs, but if you step back ways and stare at it for a while, all of a sudden, you will see seven letters appear in this ..  
 
 
FIGURE 15.  
 
For a similar effect, take a look at my drawing Smoke Signal (Fig. 139). Along these lines, you might consider this puzzle: can you somehow create a drawing containing words in both the figure and the ground? Let us now officially distinguish between two kinds of figures: cursively drawable ones, and recursive ones (by the way, these are my own terms are not in common usage). A cursively drawable figure is one whose ground is merely an accidental byproduct of the drawing act. A recursive figure is one whose ground can be seen as a figure in its own right. Usually this is quite deliberate on the part of the artist. The "re" in "recursive" represents the fact that both foreground and background are cursively drawable – the figure is "twicecursive". Each figureground boundary in a recursive figure is a doubleedged sword. M. C. Escher was a master at drawing recursive figuressee, for instance, his beautiful recursive drawing of birds (Fig. 16).  
 
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FIGURE 16. Tiling of the plane using birds, by M. C. Escher (from a 1942 notebook). Our distinction is not as rigorous as one in mathematics, for who can definitively say that a particular ground is not a figure? Once pointed out, almost any ground has interest of its own. In that sense, every figure is recursive. But that is not what I intended by the term. There is a natural and intuitive notion of recognizable forms. Are both the foreground and background recognizable forms? If so, then the drawing is recursive. If you look at the grounds of most line drawings, you will find them rather unrecognizable. This demonstrates that There exist recognizable forms whose negative space is not any recognizable form. In more "technical" terminology, this becomes: There exist cursively drawable figures which are not recursive. Scott Kim's solution to the above puzzle, which I call his "FIGUREFIGURE Figure", is shown in Figure 17. If you read both black and white,  
 
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FIGURE 17. FIGUREFIGURE Figure, by Scott E. Kim (1975).  
 
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you will see "FIGURE" everywhere, but "GROUND" nowhere! It is a paragon of recursive figures. In this clever drawing, there are two nonequivalent ways of characterizing the black regions: (1) as the negative space to the white regions; (2) as altered copies of the white regions (produced by coloring and shifting each white region). (In the special case of the FIGUREFIGURE Figure, the two characterizations are equivalentbut in most blackandwhite pictures, they would not be.) Now in Chapter VIII, when we create our Typographical Number Theory (TNT), it will be our hope that the set of all false statements of number theory can be characterized in two analogous ways: (1) as the negative space to the set of all TNTtheorems; (2) as altered copies of the set of all TNTtheorems (produced by negating each TNTtheorem). But this hope will be dashed, because: (1) inside the set of all nontheorems are found some truths (2) outside the set of all negated theorems are found some falsehoods . You will see why and how this happens, in Chapter XIV. Meanwhile, ponder over a pictorial representation of the situation (Fig. 18). Figure and Ground in Music One may also look for figures and grounds in music. One analogue is the distinction between melody and accompanimentfor the melody is always in the forefront of our attention, and the accompaniment is subsidiary, in some sense. Therefore it is surprising when we find, in the lower lines of a piece of music, recognizable melodies. This does not happen too often in postbaroque music. Usually the harmonies are not thought of as foreground. But in baroque musicin Bach above allthe distinct lines, whether high or low or in between, all act as "figures". In this sense, pieces by Bach can be called "recursive". Another figureground distinction exists in music: that between onbeat and offbeat. If you count notes in a measure "oneand, twoand, threeand, fourand", most melodynotes will come on numbers, not on "and"'s. But sometimes, a melody will be deliberately pushed onto the "and" 's, for the sheer effect of it. This occurs in several etudes for the piano by Chopin, for instance. It also occurs in Bachparticularly in his Sonatas and Partitas for unaccompanied violin, and his Suites for unaccompanied cello. There, Bach manages to get two or more musical lines going simultaneously. Sometimes he does this by having the solo instrument play "double stops"two notes at once. Other times, however, he  
 
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FIGURE 18. Considerable visual symbolism is featured in this diagram of the relation between various classes of TNT strings. The biggest box represents the set of all TNT strings The nextbiggest box represents the set of all wellformed TNT strings. Within it is found~ set of all sentences of TNT. Now things begin to get interesting. The set of theorems pictured as a tree growing out of a trunk (representing the set of axioms). The treesymbol chosen because of the recursive growth pattern which it exhibits: new branches (theorems constantly sprouting from old ones. The fingerlike branches probe into the corners of constraining region (the set of truths), yet can never fully occupy it. The boundary beta the set of truths and the set of falsities is meant to suggest a randomly meandering coastline which, no matter how closely you examine it, always has finer levels of structure, an consequently impossible to describe exactly in any finite way. (See B. Mandelbrot's book Fractals.) The reflected tree represents the set of negations of theorems: all of them false yet unable collectively to span the space of false statements. [Drawing by the author.] puts one voice on the onbeats, and the other voice on the offbeats, so ear separates them and hears two distinct melodies weaving in and out,  harmonizing with each other. Needless to say, Bach didn't stop at this level of complexity... Recursively Enumerable Sets vs. Recursive Sets Now let us carry back the notions of figure and ground to the domain formal systems. In our example, the role of positive space is played by Ctype theorems, and the role of negative space is played by strings with a  
 
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prime number of hyphens. So far, the only way we have found to represent prime numbers typographically is as a negative space. Is there, however, some wayI don't care how complicatedof representing the primes as a positive spacethat is, as a set of theorems of some formal system? Different people's intuitions give different answers here. I remember quite vividly how puzzled and intrigued I was upon realizing the difference between a positive characterization and a negative characterization. I was quite convinced that not only the primes, but any set of numbers which could be represented negatively, could also be represented positively. The intuition underlying my belief is represented by the question: "How could a figure and its ground not carry exactly the same information?" They seemed to me to embody the same information, just coded in two complementary ways. What seems right to you? It turns out I was right about the primes, but wrong in general. This astonished me, and continues to astonish me even today. It is a fact that: There exist formal systems whose negative space (set of nontheorems) is not the positive space (set of theorems) of any formal system. This result, it turns out, is of depth equal to Gödel’s Theoremso it is not surprising that my intuition was upset. I, just like the mathematicians of the early twentieth century, expected the world of formal systems and natural numbers to be more predictable than it is. In more technical terminology, this becomes: There exist recursively enumerable sets which are not recursive. The phrase recursively enumerable (often abbreviated "r.e.") is the mathematical counterpart to our artistic notion of "cursively drawable"and recursive is the counterpart of "recursive". For a set of strings to be "r.e." means that it can be generated according to typographical rulesfor example, the set of Ctype theorems, the set of theorems of the MIUsystemindeed, the set of theorems of any formal system. This could be compared with the conception of a "figure" as "a set of lines which can be generated according to artistic rules" (whatever that might mean!). And a "recursive set" is like a figure whose ground is also a figurenot only is it r.e., but its complement is also r.e. It follows from the above result that: There exist formal systems for which there is no typographical decision procedure. How does this follow? Very simply. A typographical decision procedure is a method which tells theorems from nontheorems. The existence of such a test allows us to generate all nontheorems systematically, simply by going down a list of all strings and performing the test on them one at a time, discarding illformed strings and theorems along the way. This amounts to  
 
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a typographical method for generating the set of nontheorems. But according to the earlier statement (which we here accept on faith), for some systems this is not possible. So we must conclude that typographical decision procedures do not exist for all formal systems. Suppose we found a set F of natural numbers (`F' for `Figure') whi4 we could generate in some formal waylike the composite numbers. Suppose its complement is the set G (for 'Ground')like the primes. Together F and G make up all the natural numbers, and we know a rule for making all the numbers in set F, but we know no such rule for making all tl numbers in set G. It is important to understand that if the members of were always generated in order of increasing size, then we could always characterize G. The problem is that many r.e. sets are generated I methods which throw in elements in an arbitrary order, so you never know if a number which has been skipped over for a long time will get included you just wait a little longer. We answered no to the artistic question, "Are all figures recursive We have now seen that we must likewise answer no to the analogous question in mathematics: "Are all sets recursive?" With this perspective, 1 us now come back to the elusive word "form". Let us take our figureset and our groundset G again. We can agree that all the numbers in set have some common "form"but can the same be said about numbers in s G? It is a strange question. When we are dealing with an infinite set to sta withthe natural numbersthe holes created by removing some subs may be very hard to define in any explicit way. And so it may be that th< are not connected by any common attribute or "form". In the last analysis it is a matter of taste whether you want to use the word "form"but just thinking about it is provocative. Perhaps it is best not to define "form", bi to leave it with some intuitive fluidity. Here is a puzzle to think about in connection with the above matter Can you characterize the following set of integers (or its negative space) 1 3 7 12 18 26 35 45 56 69... How is this sequence like the FIGUREFIGURE Figure?  
 
Primes as Figure Rather than Ground Finally, what about a formal system for generating primes? How is it don< The trick is to skip right over multiplication, and to go directly to nondivisibility as the thing to represent positively. Here are an axiom schema and rule for producing theorems which represent the notion that one number does not divide (D N D) another number exactly: AXIOM SCHEMA: xy D N Dx where x and y are hyphenstrings. For example D N D, where x has been replaced by''and y by ‘“.  
 
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RULE: If x D N Dy is a theorem, then so is x D N Dx y. If you use the rule twice, you can generate this theorem:  D N D  which is interpreted as "5 does not divide 12". But D N D  is not a theorem. What goes wrong if you try to produce it? Now in order to determine that a given number is prime, we have to build up some knowledge about its nondivisibility properties. In particular, we want to know that it is not divisible by 2 or 3 or 4, etc., all the way up to 1 less than the number itself. But we can't be so vague in formal systems as to say "et cetera". We must spell things out. We would like to have a way of saying, in the language of the system, "the number Z is divisor free up to X", meaning that no number between 2 and X divides Z. This can be done, but there is a trick to it. Think about it if you want. Here is the solution: RULE: If D N D z is a theorem, so is z D F. RULE: If z D Fx is a theorem and also xD N Dz is a theorem, z D Fx is a theorem. These two rules capture the notion of divisor freeness. All we need to do is to say that primes are numbers which are divisorfree up to 1 less than themselves: RULE: If zDFz is a theorem, then Pz is a theorem. Ohlet's not forget that 2 is prime! Axiom: P. And there you have it. The principle of representing primality formally is that there is a test for divisibility which can be done without any backtracking. You march steadily upward, testing first for divisibility by 2, then by 3, and so on. It is this "monotonicity" or unidirectionalitythis absence of crossplay between lengthening and shortening, increasing and decreasingthat allows primality to be captured. And it is this potential complexity of formal systems to involve arbitrary amounts of backwardsforwards interference that is responsible for such limitative results as Gödel’s Theorem, Turing's Halting Problem, and the fact that not all recursively enumerable sets are recursive.  
 
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Contracrostipunctus Achilles has come to visit his friend and jogging companion, the Tortoise, at his home Achilles: Heavens, you certainly have an admirable boomerang collection Tortoise: Oh, pshaw. No better than that of any other Tortoise. And now would you like to step into the parlor? Achilles: Fine. (Walks to the corner of the room.) I see you also have a large collection of records. What sort of music do you enjoy? Tortoise: Sebastian Bach isn't so bad, in my opinion. But these days, I must say, I am developing more and more of an interest in a rather specialized sort of music. Achilles: Tell me, what kind of music is that? Tortoise: A type of music which you are most unlikely to have heard of. call it "music to break phonographs by". Achilles: Did you say "to break phonographs by"? That is a curious concept. I can just see you, sledgehammer in hand, whacking on phonograph after another to pieces, to the strains of Beethoven's heroic masterpiece Wellington's Victory. Tortoise: That's not quite what this music is about. However, you might find its true nature just as intriguing. Perhaps I should give you a brief description of it? Achilles: Exactly what I was thinking. Tortoise: Relatively few people are acquainted with it. It all began whet my friend the Crabhave you met him, by the way?paid m• a visit. Achilles: ' twould be a pleasure to make his acquaintance, I'm sure Though I've heard so much about him, I've never met him Tortoise: Sooner or later I'll get the two of you together. You'd hit it of splendidly. Perhaps we could meet at random in the park on day ... Achilles: Capital suggestion! I'll be looking forward to it. But you were going to tell me about your weird "music to smash phone graphs by", weren't you? Tortoise: Oh, yes. Well, you see, the Crab came over to visit one day. You must understand that he's always had a weakness for fang gadgets, and at that time he was quite an aficionado for, of al things, record players. He had just bought his first record player, and being somewhat gullible, believed every word the salesman had told him about itin particular, that it was capable of reproducing any and all sounds. In short, he was convinced that it was a Perfect phonograph.  
 
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Achilles: Naturally, I suppose you disagreed. Tortoise: True, but he would hear nothing of my arguments. He staunchly maintained that any sound whatever was reproducible on his machine. Since I couldn't convince him of the contrary, I left it at that. But not long after that, I returned the visit, taking with me a record of a song which I had myself composed. The song was called "I Cannot Be Played on Record Player 1". Achilles: Rather unusual. Was it a present for the Crab? Tortoise: Absolutely. I suggested that we listen to it on his new phonograph, and he was very glad to oblige me. So he put it on. But unfortunately, after only a few notes, the record player began vibrating rather severely, and then with a loud "pop", broke into a large number of fairly small pieces, scattered all about the room. The record was utterly destroyed also, needless to say. Achilles: Calamitous blow for the poor fellow, I'd say. What was the matter with his record player? Tortoise: Really, there was nothing the matter, nothing at all. It simply couldn't reproduce the sounds on the record which I had brought him, because they were sounds that would make it vibrate and break. Achilles: Odd, isn't it? I mean, I thought it was a Perfect phonograph. That's what the salesman had told him, after all. Tortoise: Surely, Achilles, you don't believe everything that salesmen tell you! Are you as naive as the Crab was? Achilles: The Crab was naiver by far! I know that salesmen are notorious prevaricators. I wasn't born yesterday! Tortoise: In that case, maybe you can imagine that this particular salesman had somewhat exaggerated the quality of the Crab's piece of equipment ... perhaps it was indeed less than Perfect, and could not reproduce every possible sound. Achilles: Perhaps that is an explanation. But there's no explanation for the amazing coincidence that your record had those very sounds on it ... Tortoise: Unless they got put there deliberately. You see, before returning the Crab's visit, I went to the store where the Crab had bought his machine, and inquired as to the make. Having ascertained that, I sent off to the manufacturers for a description of its design. After receiving that by return mail, I analyzed the entire construction of the phonograph and discovered a certain set of sounds which, if they were produced anywhere in the vicinity, would set the device to shaking and eventually to falling apart. Achilles: Nasty fellow! You needn't spell out for me the last details: that you recorded those sounds yourself, and offered the dastardly item as a gift ...  
 
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Tortoise: Clever devil! You jumped ahead of the story! But that wasn't t end of the adventure, by any means, for the Crab did r believe that his record player was at fault. He was quite stubborn. So he went out and bought a new record player, this o even more expensive, and this time the salesman promised give him double his money back in case the Crab found a soul which it could not reproduce exactly. So the Crab told r excitedly about his new model, and I promised to come over and see it. Achilles: Tell me if I'm wrongI bet that before you did so, you on again wrote the manufacturer, and composed and recorded new song called "I Cannot Be Played on Record Player based on the construction of the new model. Tortoise: Utterly brilliant deduction, Achilles. You've quite got the spirit. Achilles: So what happened this time? Tortoise: As you might expect, precisely the same thing. The phonograph fell into innumerable pieces, and the record was shattered. Achilles: Consequently, the Crab finally became convinced that there could be no such thing as a Perfect record player. Tortoise: Rather surprisingly, that's not quite what happened. He was sure that the next model up would fill the bill, and having twice the money, h e Achilles: OhoI have an idea! He could have easily outwitted you, I obtaining a LOWfidelity phonographone that was not capable of reproducing the sounds which would destroy it. In that way, he would avoid your trick. Tortoise: Surely, but that would defeat theoriginal purposenamely, to have a phonograph which could reproduce any sound whatsoever, even its own selfbreaking sound, which is of coup impossible. Achilles: That's true. I see the dilemma now. If any record playersi Record Player Xis sufficiently highfidelity, then when attempts to play the song "I Cannot Be Played on Record Player X", it will create just those vibrations which will cause to break. .. So it fails to be Perfect. And yet, the only way to g, around that trickery, namely for Record Player X to be c lower fidelity, even more directly ensures that it is not Perfect It seems that every record player is vulnerable to one or the other of these frailties, and hence all record players are defective. Tortoise: I don't see why you call them "defective". It is simply an inherent fact about record players that they can't do all that you might wish them to be able to do. But if there is a defect anywhere, is not in THEM, but in your expectations of what they should b able to do! And the Crab was just full of such unrealistic expectations.  
 
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Achilles: Compassion for the Crab overwhelms me. High fidelity or low fidelity, he loses either way. Tortoise: And so, our little game went on like *_his for a few more rounds, and eventually our friend tried to become very smart. He got wind of the principle upon which I was basing my own records, and decided to try to outfox me. He wrote to the phonograph makers, and described a device of his own invention, which they built to specification. He called it "Record Player Omega". It was considerably more sophisticated than an ordinary record player. Achilles: Let me guess how: Did it have no of cotton? Or Tortoise: Let me tell you, instead. That will save some time. In the first place, Record Player Omega incorporated a television camera whose purpose it was to scan any record before playing it. This camera was hooked up to a small builtin computer, which would determine exactly the nature of the sounds, by looking at the groovepatterns. Achilles: Yes, so far so good. But what could Record Player Omega do with this information? Tortoise: By elaborate calculations, its little computer figured out what effects the sounds would have upon its phonograph. If it deduced that the sounds were such that they would cause the machine in its present configuration to break, then it did something very clever. Old Omega contained a device which could disassemble large parts of its phonograph subunit, and rebuild them in new ways, so that it could, in effect, change its own structure. If the sounds were "dangerous", a new configuration was chosen, one to which the sounds would pose no threat, and this new configuration would then be built by the rebuilding subunit, under direction of the little computer. Only after this rebuilding operation would Record Player Omega attempt to play the record. Achilles: Aha! That must have spelled the end of your tricks. I bet you were a little disappointed. Tortoise: Curious that you should think so ... I don't suppose that you know Godel's Incompleteness Theorem backwards and forwards, do you? Achilles: Know WHOSE Theorem backwards and forwards? I've heard of anything that sounds like that. I'm sure it's fascinating, but I'd rather hear more about "music to break records by". It's an amusing little story. Actually, I guess I can fill in the end. Obviously, there was no point in going on, and so you sheepishly admitted defeat, and that was that. Isn't that exactly it? Tortoise: What! It's almost midnight! I'm afraid it's my bedtime. I'd love to talk some more, but really I am growing quite sleepy.  
 
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Achilles: As am 1. Well, 1 u be on my way. (As he reaches the door, he suddenly stops, and turns around.) Oh, how silly of me! I almost forgo brought you a little present. Here. (Hands the Tortoise a small neatly wrapped package.) Tortoise: Really, you shouldn't have! Why, thank you very much indeed think I'll open it now. (Eagerly tears open the package, and ins discovers a glass goblet.) Oh, what an exquisite goblet! Did y know that I am quite an aficionado for, of all things, gl goblets? Achilles: Didn't have the foggiest. What an agreeable coincidence! Tortoise: Say, if you can keep a secret, I'll let you in on something: I trying to find a Perfect goblet: one having no defects of a sort in its shape. Wouldn't it be something if this gobleth call it "G"were the one? Tell me, where did you come across Goblet G? Achilles: Sorry, but that's MY little secret. But you might like to know w its maker is. Tortoise: Pray tell, who is it? Achilles: Ever hear of the famous glassblower Johann Sebastian Bach? Well, he wasn't exactly famous for glassblowingbut he dabbled at the art as a hobby, though hardly a soul knows ita: this goblet is the last piece he blew. Tortoise: Literally his last one? My gracious. If it truly was made by Bach its value is inestimable. But how are you sure of its maker Achilles: Look at the inscription on the insidedo you see where tletters `B', `A', `C', `H' have been etched? Tortoise: Sure enough! What an extraordinary thing. (Gently sets Goblet G down on a shelf.) By the way, did you know that each of the four letters in\Bach's name is the name of a musical note? Achilles:' tisn't possible, is it? After all, musical notes only go from ‘A’ through `G'. Tortoise: Just so; in most countries, that's the case. But in Germany, Bach’s own homeland, the convention has always been similar, except that what we call `B', they call `H', and what we call `Bflat', they call `B'. For instance, we talk about Bach's "Mass in B Minor whereas they talk about his "Hmoll Messe". Is that clear? Achilles: ... hmm ... I guess so. It's a little confusing: H is B, and B Bflat. I suppose his name actually constitutes a melody, then Tortoise: Strange but true. In fact, he worked that melody subtly into or of his most elaborate musical piecesnamely, the final Contrapunctus in his Art of the Fugue. It was the last fugue Bach ever wrote. When I heard it for the first time, I had no idea how would end. Suddenly, without warning, it broke off. And the ... dead silence. I realized immediately that was where Bach died. It is an indescribably sad moment, and the effect it had o me wasshattering. In any case, BACH is the last theme c that fugue. It is hidden inside the piece. Bach didn't point it out  
 
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FIGURE 19. The last page of Bach's Art of the Fugue. In the original manuscript, in the handwriting of Bach's son Carl Philipp Emanuel, is written: "N.B. In the course of this fugue, at the point where the name B.A.C.H. was brought in as countersubject, the composer died." (BACH in box.) I have let this final page of Bach's last fugue serve as an epitaph. [Music Printed by Donald Byrd's program "SMUT", developed at Indiana University]  
 
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Explicitly, but if you know about it, you can find it without much trouble. Ah, methere are so many clever ways of hiding things in music .. . Achilles: . . or in poems. Poets used to do very similar things, you know (though it's rather out of style these days). For instance, Lewis Carroll often hid words and names in the first letters (or characters) of the successive lines in poems he wrote. Poems which conceal messages that way are called "acrostics". Tortoise: Bach, too, occasionally wrote acrostics, which isn't surprising. After all, counterpoint and acrostics, with their levels of hidden meaning, have quite a bit in common. Most acrostics, however, have only one hidden levelbut there is no reason that one couldn't make a doubledeckeran acrostic on top of an acrostic. Or one could make a "contracrostic"where the initial letters, taken in reverse order, form a message. Heavens! There's no end to the possibilities inherent in the form. Moreover, it's not limited to poets; anyone could write acrosticseven a dialogician. Achilles: A dialalogician? That's a new one on me. Tortoise: Correction: I said "dialogician", by which I meant a writer of dialogues. Hmm ... something just occurred to me. In the unlikely event that a dialogician should write a contrapuntal acrostic in homage to J. S. Bach, do you suppose it would be more proper for him to acrostically embed his OWN nameor that of Bach? Oh, well, why worry about such frivolous matters? Anybody who wanted to write such a piece could make up his own mind. Now getting back to Bach's melodic name, did you know that the melody BACH, if played upside down and backwards, is exactly the same as the original? Achilles: How can anything be played upside down? Backwards, I can seeyou get HCABbut upside down? You must be pulling my leg. Tortoise: ' pon my word, you're quite a skeptic, aren't you? Well, I guess I'll have to give you a demonstration. Let me just go and fetch my fiddle (Walks into the next room, and returns in a jiffy with an ancientlooking violin.) and play it for you forwards and backwards and every which way. Let's see, now ... (Places his copy of the Art of the Fugue on his music stand and opens it to the last page.) ... here's the last Contrapunctus, and here's the last theme ... The Tortoise begins to play: BAC  but as he bows the final H, suddenly, without warning, a shattering sound rudely interrupts his performance. Both he and Achilles spin around, just in time to catch a glimpse of myriad fragments of glass tinkling to the floor from the shelf where Goblet G had stood, only moments before. And then ... dead silence.  
 
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Chapter IV Consistency, Completeness, and Geometry Implicit and Explicit Meaning IN CHAPTER II, we saw how meaningat least in the relatively simple context of formal systemsarises when there is an isomorphism between rulegoverned symbols, and things in the real world. The more complex the isomorphism, in general, the more "equipment"both hardware and softwareis required to extract the meaning from the symbols. If an isomorphism is very simple (or very familiar), we are tempted to say that the meaning which it allows us to see is explicit. We see the meaning without seeing the isomorphism. The most blatant example is human language, where people often attribute meaning to words in themselves, without being in the slightest aware of the very complex "isomorphism" that imbues them with meanings. This is an easy enough error to make. It attributes all the meaning to the object (the word), rather than to the link between that object and the real world. You might compare it to the naive belief that noise is a necessary side effect of any collision of two objects. This is a false belief; if two objects collide in a vacuum, there will be no noise at all. Here again, the error stems from attributing the noise exclusively to the collision, and not recognizing the role of the medium, which carries it from the objects to the ear. Above, I used the word "isomorphism" in quotes to indicate that it must be taken with a grain of salt. The symbolic processes which underlie the understanding of human language are so much more complex than the symbolic processes in typical formal systems, that, if we want to continue thinking of meaning as mediated by isomorphisms, we shall have to adopt a far more flexible conception of what isomorphisms can be than we have up till now. In my opinion, in fact, the key element in answering the question "What is consciousness?" will be the unraveling of the nature of the "isomorphism" which underlies meaning. Explicit Meaning of the Contracrostipunctus All this is by way of preparation for a discussion of the Contracrostipunctusa study in levels of meaning. The Dialogue has both explicit and implicit meanings. Its most explicit meaning is simply the story  
 
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Which was related. This “explicit meaning is, strictly speaking extremely implicit, in the sense that the brain processes required to understand the events in the story, given only the black marks on paper, are incredibly complex. Nevertheless, we shall consider the events in the story to be the explicit meaning of the Dialogue, and assume that every reader of English uses more or less the same "isomorphism" in sucking that meaning from the marks on the paper. Even so, I'd like to be a little more explicit about the explicit meaning of the story. First I'll talk about the record players and the records. The main point is that there are two levels of meaning for the grooves in the records. Level One is that of music. Now what is "music"a sequence of vibrations in the air, or a succession of emotional responses in a brain? It is both. But before there can be emotional responses, there have to be vibrations. Now the vibrations get "pulled" out of the grooves by a record player, a relatively straightforward device; in fact you can do it with a pin, just pulling it down the grooves. After this stage, the ear converts the vibrations into firings of auditory neurons in the brain. Then ensue a number of stages in the brain, which gradually transform the linear sequence of vibrations into a complex pattern of interacting emotional responsesfar too complex for us to go into here, much though I would like to. Let us therefore content ourselves with thinking of the sounds in the air as the "Level One" meaning of the grooves. What is the Level Two meaning of the grooves? It is the sequence of vibrations induced in the record player. This meaning can only arise after the Level One meaning has been pulled out of the grooves, since the vibrations in the air cause the vibrations in the phonograph. Therefore, the Level Two meaning depends upon a chain of two isomorphisms: (1) Isomorphism between arbitrary groove patterns and air vibrations; (2) Isomorphism between graph vibrations. arbitrary air vibrations and phonograph vibrations This chain of two isomorphisms is depicted in Figure 20. Notice that isomorphism I is the one which gives rise to the Level One meaning. The Level Two meaning is more implicit than the Level One meaning, because it is mediated by the chain of two isomorphisms. It is the Level Two meaning which "backfires", causing the record player to break apart. What is of interest is that the production of the Level One meaning forces the production of the Level Two meaning simultaneouslythere is no way to have Level One without Level Two. So it was the implicit meaning of the record which turned back on it, and destroyed it. Similar comments apply to the goblet. One difference is that the mapping from letters of the alphabet to musical notes is one more level of isomorphism, which we could call "transcription". That is followed by "translation"conversion of musical notes into musical sounds. Thereafter, the vibrations act back on the goblet just as they did on the escalating series of phonographs.  
 
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FIGURE 20. Visual rendition of the principle underlying Gödel’s Theorem: two backtoback mappings which have an unexpected boomeranging effect. The first is from groove patterns to sounds, carried out by a phonograph. The secondfamiliar, but usually ignored  is from sounds to vibrations of the phonograph. Note that the second mapping exists independently of the first one, for any sound in the vicinity, not just ones produced by the phonograph itself, will cause such vibrations. The paraphrase of Gödel’s Theorem says that for any record player, there are records which it cannot play because they will cause its indirect selfdestruction. [Drawing by the author.  
 
Implicit Meanings of the Contracrostipunctus What about implicit meanings of the Dialogue? (Yes, it has more than one of these.) The simplest of these has already been pointed out in the paragraphs abovenamely, that the events in the two halves of the dialogue are roughly isomorphic to each other: the phonograph becomes a violin, the Tortoise becomes Achilles, the Crab becomes the Tortoise, the grooves become the etched autograph, etc. Once you notice this simple isomorphism, you can go a little further. Observe that in the first half of the story, the Tortoise is the perpetrator of all the mischief, while in the second half, he is the victim. What do you know, but his own method has turned around and backfired on him! Reminiscent of the backfiring of the records' muusicor the goblet's inscriptionor perhaps of the Tortoise's boomerang collection? Yes, indeed. The story is about backfiring on two levels, as follows ... Level One: Goblets and records which backfire; Level Two: The Tortoise's devilish method of exploiting implicit meaning to cause backfireswhich backfires. Therefore we can even make an isomorphism between the two levels of the story, in which we equate the way in which the records and goblet boomerang back to destroy themselves, with the way in which the Tortoise's own fiendish method boomerangs back to get him in the end. Seen this  
 
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way, the story itself is an example of the backfirings which it discusses. So we can think of the Contracrostipunctus as referring to itself indirectly that its own structure is isomorphic to the events it portrays. (Exactly goblet and records refer implicitly to themselves via the backtoback morphisms of playing and vibrationcausing.) One may read the Dialogue without perceiving this fact, of coursebut it is there all the time. Mapping Between the Contracrostipunctus and Gödel’s Theorem Now you may feel a little dizzybut the best is yet to come. (Actually, levels of implicit meaning will not even be discussed herethey will 1 for you to ferret out.) The deepest reason for writing this Dialogue illustrate Gödel’s Theorem, which, as I said in the Introduction, heavily on two different levels of meaning of statements of number t1 Each of the two halves of the Contracrostipunctus is an "isomorphic co Gödel’s Theorem. Because this mapping is the central idea of the Dialogue and is rather elaborate, I have carefully charted it out below. Phonograph <= =>axiomatic system for number theory lowfidelity phonograph <= =>"weak" axiomatic system highfidelity phonograph <= =>"strong" axiomatic system "Perfect" phonograph" <= => complete system for number theory' Blueprint" of phonograph <= => axioms and rules of formal system record <= => string of the formal system playable record<= => theorem of the axiomatic system unplayable record <= =>nontheorem of the axiomatic system sound <= =>true statement of number theory reproducible sound <= => 'interpreted theorem of the system unreproducible sound <= => true statement which isn't a theorem: song title <= =>implicit meaning of Gödel’s string: "I Cannot Be Played "I Cannot Be Derived on Record Player X" in Formal System X" This is not the full extent of the isomorphism between Gödel’s theorem and the Contracrostipunctus, but it is the core of it. You need not if you don't fully grasp Gödel’s Theorem by nowthere are still Chapters to go before we reach it! Nevertheless, having read this Dialogue you have already tasted some of the flavor of Gödel’s Theorem without necessarily being aware of it. I now leave you to look for any other types of implicit meaning in the Contracrostipunctus. "Quaerendo invenietis!"  
 
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The Art of the Fugue A few words on the Art of the Fugue ... Composed in the last year of Bach's life, it is a collection of eighteen fugues all based on one theme. Apparently, writing the Musical Offering was an inspiration to Bach. He decided to compose another set of fugues on a much simpler theme, to demonstrate the full range of possibilities inherent in the form. In the Art of the Fugue, Bach uses a very simple theme in the most complex possible ways. The whole work is in a single key. Most of the fugues have four voices, and they gradually increase in complexity and depth of expression. Toward the end, they soar to such heights of intricacy that one suspects he can no longer maintain them. Yet he does . . . until the last Contrapunctus. The circumstances which caused the breakoff of the Art of the Fugue (which is to say, of Bach's life) are these: his eyesight having troubled him for years, Bach wished to have an operation. It was done; however, it came out quite poorly, and as a consequence, he lost his sight for the better part of the last year of his life. This did not keep him from vigorous work on his monumental project, however. His aim was to construct a complete exposition of fugal writing, and usage of multiple themes was one important facet of it. In what he planned as the nexttolast fugue, he inserted his own name coded into notes as the third theme. However, upon this very act, his health became so precarious that he was forced to abandon work on his cherished project. In his illness, he managed to dictate to his soninlaw a final chorale prelude, of which Bach's biographer Forkel wrote, "The expression of pious resignation and devotion in it has always affected me whenever I have played it; so that I can hardly say which I would rather missthis Chorale, or the end of the last fugue." One day, without warning, Bach regained his vision. But a few hours later, he suffered a stroke; and ten days later, he died, leaving it for others to speculate on the incompleteness of the Art of the Fugue. Could it have been caused by Bach's attainment of selfreference? Problems Caused by Gödel’s Result The Tortoise says that no sufficiently powerful record player can be perfect, in the sense of being able to reproduce every possible sound from a record. Godel says that no sufficiently powerful formal system can be perfect, in the sense of reproducing every single true statement as a theorem. But as the Tortoise pointed out with respect to phonographs, this fact only seems like a defect if you have unrealistic expectations of what formal systems should be able to do. Nevertheless, mathematicians began this century with just such unrealistic expectations, thinking that axiomatic reasoning was the cure to all ills. They found out otherwise in 1931. The fact that truth transcends theoremhood, in any given formal system, is called "incompleteness" of that system. A most puzzling fact about Gödel’s method of proof is that he uses  
 
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reasoning methods which seemingly cannot be "encapsulated"they re being incorporated into any formal system. Thus, at first sight, it seems that Gödel has unearthed a hitherto unknown, but deeply significant, difference between human reasoning and mechanical reasoning. This mysterious discrepancy in the power of living and nonliving systems is mirrored in the discrepancy between the notion of truth, and that of theoremhood or at least that is a "romantic" way to view the situation. The Modified pqSystem and Inconsistency In order to see the situation more realistically, it is necessary to see in, depth why and how meaning is mediated, in formal systems, by isomorphisms. And I believe that this leads to a more romantic way to view i situation. So we now will proceed to investigate some further aspects of 1 relation between meaning and form. Our first step is to make a new formal system by modifying our old friend, the pqsystem, very slightly. We a one more axiom schema (retaining the original one, as well as the sin rule of inference): Axiom SCHEMA II: If x is a hyphenstring, then xpqx is an axiom. Clearly, then, pq is a theorem in the new system, and so pq. And yet, their interpretations are, respectively, "2 plus; equals 2", and "2 plus 2 equals 3". It can be seen that our new system contain a lot of false statements (if you consider strings to be statement Thus, our new system is inconsistent with the external world. As if this weren't bad enough, we also have internal problems with < new system, since it contains statements which disagree with one another such as pq (an old axiom) and pq (a new axiom). So our system is inconsistent in a second sense: internally. Would, therefore, the only reasonable thing to do at this point be drop the new system entirely? Hardly. I have deliberately presented the "inconsistencies" in a woolpulling manner: that is, I have tried to press fuzzyheaded arguments as strongly as possible, with the purpose of n leading. In fact, you may well have detected the fallacies in what I hi said. The crucial fallacy came when I unquestioningly adopted the very same interpreting words for the new system as I had for the old of Remember that there was only one reason for adopting those words in I last Chapter, and that reason was that the symbols acted isomorphically to concepts which they were matched with, by the interpretation. But when y modify the rules governing the system, you are bound to damage t isomorphism. It just cannot be helped. Thus all the problems which we lamented over in preceding paragraphs were bogus problems; they can made to vanish in no time, by suitably reinterpreting some of the symbols of system. Notice that I said "some"; not necessarily all symbols will have to mapped onto new notions. Some may very well retain their "meaning while others change.  
 
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Suppose, for instance, that we reinterpret just the symbol q, leaving all the others constant; in particular, interpret q by the phrase "is greater than or equal to". Now, our "contradictory" theorems pqand pqcome out harmlessly as: "1 plus 1 is greater than or equal to 1", and "1 plus 1 is greater than or equal to 2". We have simultaneously gotten rid of (1) the inconsistency with the external world, and (2) the internal inconsistency. And our new interpretation is a meaningful interpretation; of course the original one is meaningless. That is, it is meaningless for the new system; for the original pqsystem, it is fine. But it now seems as pointless and arbitrary to apply it to the new pqsystem as it was to apply the "horseapplehappy" interpretation to the old pqsystem. The History of Euclidean Geometry Although I have tried to catch you off guard and surprise you a little, this lesson about how to interpret symbols by words may not seem terribly difficult once you have the hang of it. In fact, it is not. And yet it is one of the deepest lessons of all of nineteenth century mathematics! It all begins with Euclid, who, around 300 B.C., compiled and systematized all of what was known about plane and solid geometry in his day. The resulting work, Euclid's Elements, was so solid that it was virtually a bible of geometry for over two thousand yearsone of the most enduring works of all time. Why was this so? The principal reason was that Euclid was the founder of rigor in mathematics. The Elements began with very simple concepts, definitions, and so forth, and gradually built up a vast body of results organized in such a way that any given result depended only on foregoing results. Thus, there was a definite plan to the work, an architecture which made it strong and sturdy. Nevertheless, the architecture was of a different type from that of, say, a skyscraper. (See Fig. 21.) In the latter, that it is standing is proof enough that its structural elements are holding it up. But in a book on geometry, when each proposition is claimed to follow logically from earlier propositions, there will be no visible crash if one of the proofs is invalid. The girders and struts are not physical, but abstract. In fact, in Euclid's Elements, the stuff out of which proofs were constructed was human languagethat elusive, tricky medium of communication with so many hidden pitfalls. What, then, of the architectural strength of the Elements? Is it certain that it is held up by solid structural elements, or could it have structural weaknesses? Every word which we use has a meaning to us, which guides us in our use of it. The more common the word, the more associations we have with it, and the more deeply rooted is its meaning. Therefore, when someone gives a definition for a common word in the hopes that we will abide by that  
 
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FIGURE 21. Tower of Babel, by M. C. Escher (woodcut, 1928).  
 
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definition, it is a foregone conclusion that we will not do so but will instead be guided, largely unconsciously, by what our minds find in their associative stores. I mention this because it is the sort of problem which Euclid created in his Elements, by attempting to give definitions of ordinary, common words such as "point", "straight line", "circle", and so forth. How can you define something of which everyone already has a clear concept? The only way is if you can make it clear that your word is supposed to be a technical term, and is not to be confused with the everyday word with the same spelling. You have to stress that the connection with the everyday word is only suggestive. Well, Euclid did not do this, because he felt that the points and lines of his Elements were indeed the points and lines of the real world. So by not making sure that all associations were dispelled, Euclid was inviting readers to let their powers of association run free ... This sounds almost anarchic, and is a little unfair to Euclid. He did set down axioms, or postulates, which were supposed to be used in the proofs of propositions. In fact, nothing other than those axioms and postulates was supposed to be used. But this is where he slipped up, for an inevitable consequence of his using ordinary words was that some of the images conjured up by those words crept into the proofs which he created. However, if you read proofs in the Elements, do not by any means expect to find glaring "jumps" in the reasoning. On the contrary, they are very subtle, for Euclid was a penetrating thinker, and would not have made any simpleminded errors. Nonetheless, gaps are there, creating slight imperfections in a classic work. But this is not to be complained about. One should merely gain an appreciation for the difference between absolute rigor and relative rigor. In the long run, Euclid's lack of absolute rigor was the cause of some of the most fertile pathbreaking in mathematics, over two thousand years after he wrote his work. Euclid gave five postulates to be used as the "ground story" of the infinite skyscraper of geometry, of which his Elements constituted only the first several hundred stories. The first four postulates are rather terse and elegant: (1) A straight line segment can be drawn joining any two points. (2) Any straight line segment can be extended indefinitely in a straight line. (3) Given any straight line segment, a circle can be drawn having the segment as radius and one end point as center. (4) All right angles are congruent. The fifth, however, did not share their grace: (5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough  
 
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Though he never explicitly said so, Euclid considered this postulate to be somehow inferior to the others, since he managed to avoid using it in t proofs of the first twentyeight propositions. Thus, the first twentyeight propositions belong to what might be called "fourpostulate geometry" that part of geometry which can be derived on the basis of the first to postulates of the Elements, without the help of the fifth postulate. (It is al often called absolute geometry.) Certainly Euclid would have found it 1 preferable to prove this ugly duckling, rather than to have to assume it. B he found no proof, and therefore adopted it. But the disciples of Euclid were no happier about having to assume this fifth postulate. Over the centuries, untold numbers of people ga untold years of their lives in attempting to prove that the fifth postulate s itself part of fourpostulate geometry. By 1763, at least twentyeight deficient proofs had been publishedall erroneous! (They were all criticized the dissertation of one G. S. Klugel.) All of these erroneous proofs involve a confusion between everyday intuition and strictly formal properties. It safe to say that today, hardly any of these "proofs" holds any mathematic or historical interestbut there are certain exceptions. The Many Faces of Noneuclid Girolamo Saccheri (16671733) lived around Bach's time. He had t ambition to free Euclid of every flaw. Based on some earlier work he h; done in logic, he decided to try a novel approach to the proof of the famous fifth: suppose you assume its opposite; then work with that as your fif postulate ... Surely after a while you will create a contradiction. Since i mathematical system can support a contradiction, you will have shown t unsoundness of your own fifth postulate, and therefore the soundness Euclid's fifth postulate. We need not go into details here. Suffice it to s that with great skill, Saccheri worked out proposition after proposition "Saccherian geometry" and eventually became tired of it. At one point, decided he had reached a proposition which was "repugnant to the nature of the straight line". That was what he had been hoping forto his mind was the longsought contradiction. At that point, he published his work under the title Euclid Freed of Every Flaw, and then expired. But in so doing, he robbed himself of much posthumous glory, sir he had unwittingly discovered what came later to be known as "hyperbolic geometry". Fifty years after Saccheri, J. H. Lambert repeated the "near miss", this time coming even closer, if possible. Finally, forty years after Lambert, and ninety years after Saccheri, nonEuclidean geometry was recognized for what it wasan authentic new brand of geometry, a bifurcation the hitherto single stream of mathematics. In 1823, nonEuclidean geometry was discovered simultaneously, in one of those inexplicable coincidences, by a Hungarian mathematician, Janos (or Johann) Bolyai, age twentyone, and a Russian mathematician, Nikolay Lobachevskiy, ag thirty. And, ironically, in that same year, the great French mathematician  
 
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AdrienMarie Legendre came up with what he was sure was a proof of Euclid's fifth postulate, very much along the lines of Saccheri. Incidentally, Bolyai's father, Farkas (or Wolfgang) Bolyai, a close friend of the great Gauss, invested much effort in trying to prove Euclid's fifth postulate. In a letter to his son Janos, he tried to dissuade him from thinking about such matters: You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone.... I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. For here it is true that si paullum a summo discessit, vergit ad imum. I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind.... I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness sut Caesar aut nihil.' But later, when convinced his son really "had something", he urged him to publish it, anticipating correctly the simultaneity which is so frequent in scientific discovery: When the time is ripe for certain things, these things appear in different places in the manner of violets coming to light in early spring. How true this was in the case of nonEuclidean geometry! In Germany, Gauss himself and a few others had more or less independently hit upon nonEuclidean ideas. These included a lawyer, F. K. Schweikart, who in 1818 sent a page describing a new "astral" geometry to Gauss; Schweikart's nephew, F. A. Taurinus, who did nonEuclidean trigonometry; and F. L. Wachter, a student of Gauss, who died in 1817, aged twentyfive, having found several deep results in nonEuclidean geometry. The clue to nonEuclidean geometry was "thinking straight" about the propositions which emerge in geometries like Saccheri's and Lambert's. The Saccherian propositions are only "repugnant to the nature of the straight line" if you cannot free yourself of preconceived notions of what "straight line" must mean. If, however, you can divest yourself of those preconceived images, and merely let a "straight line" be something which satisfies the new propositions, then you have achieved a radically new viewpoint. Undefined Terms This should begin to sound familiar. In particular, it harks back to the pqsystem, and its variant, in which the symbols acquired passive meanings by virtue of their roles in theorems. The symbol q is especially interesting,  
 
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since its "meaning" changed when a new axiom schema was added. In the very same way, one can let the meanings of "point", "line", and so on I determined by the set of theorems (or propositions) in which they occur. This was th great realization of the discoverers of nonEuclidean geometry. The found different sorts of nonEuclidean geometries by denying Euclid's fifth postulate in different ways and following out the consequences. Strict] speaking, they (and Saccheri) did not deny the fifth postulate directly, but rather, they denied an equivalent postulate, called the parallel postulate, which runs as follows: Given any straight line, and a point not on it, there exists one, and only one, straight line which passes through that point and never intersects the first line, no matter how far they are extended. The second straight line is then said to be parallel to the first. If you assert that no such line exists, then you reach elliptical geometry; if you assert that, at east two such lines exist, you reach hyperbolic geometry. Incidentally, tf reason that such variations are still called "geometries" is that the cot elementabsolute, or fourpostulate, geometryis embedded in them. is the presence of this minimal core which makes it sensible to think of the] as describing properties of some sort of geometrical space, even if the spa( is not as intuitive as ordinary space. Actually, elliptical geometry is easily visualized. All "points", "lines and so forth are to be parts of the surface of an ordinary sphere. Let t write "POINT" when the technical term is meant, and "point" when t1 everyday sense is desired. Then, we can say that a POINT consists of a pa of diametrically opposed points of the sphere's surface. A LINE is a great circle on the sphere (a circle which, like the equator, has its center at tI center of the sphere). Under these interpretations, the propositions ( elliptical geometry, though they contain words like "POINT" and "LINE speak of the goingson on a sphere, not a plane. Notice that two LINT always intersect in exactly two antipodal points of the sphere's surface that is, in exactly one single POINT! And just as two LINES determine POINT, so two POINTS determine a LINE. By treating words such as "POINT" and "LINE" as if they had only tt meaning instilled in them by the propositions in which they occur, we take step towards complete formalization of geometry. This semiformal version still uses a lot of words in English with their usual meanings (words such "the", ` if ", "and", "join", "have"), although the everyday meaning has bee drained out of special words like "POINT" and "LINE", which are consequently called undefined terms. Undefined terms, like the p and q of th pqsystem, do get defined in a sense: implicitlyby the totality of all propos dons in which they occur, rather than explicitly, in a definition. One could maintain that a full definition of the undefined tern resides in the postulates alone, since the propositions which follow from them are implicit in the postulates already. This view would say that the postulates are implicit definitions of all the undefined terms, all of the undefined terms being defined in terms of the others.  
 
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The Possibility of Multiple Interpretations A full formalization of geometry would take the drastic step of making every term undefinedthat is, turning every term into a "meaningless" symbol of a formal system. I put quotes around "meaningless" because, as you know, the symbols automatically pick up passive meanings in accordance with the theorems they occur in. It is another question, though, whether people discover those meanings, for to do so requires finding a set of concepts which can be linked by an isomorphism to the symbols in the formal system. If one begins with the aim of formalizing geometry, presumably one has an intended interpretation for each symbol, so that the passive meanings are built into the system. That is what I did for p and q when I first created the pqsystem. But there may be other passive meanings which are potentially perceptible, which no one has yet noticed. For instance, there were the surprise interpretations of p as "equals" and q as "taken from", in the original pqsystem. Although this is rather a trivial example, it contains the essence of the idea that symbols may have many meaningful interpretationsit is up to the observer to look for them. We can summarize our observations so far in terms of the word "consistency". We began our discussion by manufacturing what appeared to be an inconsistent formal systemone which was internally inconsistent, as well as inconsistent with the external world. But a moment later we took it all back, when we realized our error: that we had chosen unfortunate interpretations for the symbols. By changing the interpretations, we regained consistency! It now becomes clear that consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system. Varieties of Consistency We have been speaking of "consistency" and "inconsistency" all along, without defining them. We have just relied on good old everyday notions. But now let us say exactly what is meant by consistency of a formal system (together with an interpretation): that every theorem, when interpreted, becomes a true statement. And we will say that inconsistency occurs when there is at least one false statement among the interpreted theorems. This definition appears to be talking about inconsistency with the external worldwhat about internal inconsistencies? Presumably, a system would be internally inconsistent if it contained two or more theorems whose interpretations were incompatible with one another, and internally consistent if all interpreted theorems were compatible with one another. Consider, for example, a formal system which has only the following three theorems: TbZ, ZbE, and EbT. If T is interpreted as "the Tortoise", Z as "Zeno", E as "Egbert", and x by as "x beats y in chess always", then we have the following interpreted theorems:  
 
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The Tortoise always beats Zeno at chess Zeno always beats Egbert at chess. Egbert always beats the Tortoise at chess. The statements are not incompatible, although they describe a rather bizarre circle of chess players. Hence, under this interpretation, the form; system in which those three strings are theorems is internally consistent although, in point of fact, none of the three statements is true! Intern< consistency does not require all theorems to come out true, but merely that they come out compatible with one another. Now suppose instead that x by is to be interpreted as "x was invented by y". Then we would have: The Tortoise was invented by Zeno. Zeno was invented by Egbert. Egbert was invented by the Tortoise. In this case, it doesn't matter whether the individual statements are true c falseand perhaps there is no way to know which ones are true, and which are not. What is nevertheless certain is that not all three can be true at one Thus, the interpretation makes the system internally inconsistent. The internal inconsistency depends not on the interpretations of the three capital letters, but only on that of b, and on the fact that the three capita are cyclically permuted around the occurrences of b. Thus, one can have internal inconsistency without having interpreted all of the symbols of the formal system. (In this case it sufficed to interpret a single symbol.) By tl time sufficiently many symbols have been given interpretations, it may t clear that there is no way that the rest of them can be interpreted so that a theorems will come out true. But it is not just a question of truthit is question of possibility. All three theorems would come out false if the capitals were interpreted as the names of real peoplebut that is not why we would call the system internally inconsistent; our grounds for doing s would be the circularity, combined with the interpretation of the letter I (By the way, you'll find more on this "authorship triangle" in Chapter XX.; Hypothetical Worlds and Consistency We have given two ways of looking at consistency: the first says that systemplusinterpretation is consistent with the external world if every theorem comes out true when interpreted; the second says that a systemplus: interpretation is internally consistent if all theorems come out mutually compatible when interpreted. Now there is a close relationship between these two types of consistency. In order to determine whether several statements at mutually compatible, you try to imagine a world in which all of them could be simultaneously true. Therefore, internal consistency depends upon consistency with the external worldonly now, "the external world" allowed to be any imaginable world, instead of the one we live in. But this is  
 
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an extremely vague, unsatisfactory conclusion. What constitutes an “imaginable" world? After all, it is possible to imagine a world in which three characters invent each other cyclically. Or is it? Is it possible to imagine a world in which there are square circles? Is a world imaginable in which Newton's laws, and not relativity, hold? Is it possible to imagine a world in which something can be simultaneously green and not green? Or a world in which animals exist which are not made of cells? In which Bach improvised an eightpart fugue on a theme of King Frederick the Great? In which mosquitoes are more intelligent than people? In which tortoises can play footballor talk? A tortoise talking football would be an anomaly, of course. Some of these worlds seem more imaginable than others, since some seem to embody logical contradictionsfor example, green and not greenwhile some of them seem, for want of a better word, "plausible"  such as Bach improvising an eightpart fugue, or animals which are not made of cells. Or even, come to think of it, a world in which the laws of physics are different ... Roughly, then, it should be possible to establish different brands of consistency. For instance, the most lenient would be "logical consistency", putting no restraints on things at all, except those of logic. More specifically, a systemplusinterpretation would be logically consistent just as long as no two of its theorems, when interpreted as statements, directly contradict each other; and mathematically consistent just as long as interpreted theorems do not violate mathematics; and physically consistent just as long as all its interpreted theorems are compatible with physical law; then comes biological consistency, and so on. In a biologically consistent system, there could be a theorem whose interpretation is the statement "Shakespeare wrote an opera", but no theorem whose interpretation is the statement "Cellless animals exist". Generally speaking, these fancier kinds of inconsistency are not studied, for the reason that they are very hard to disentangle from one another. What kind of inconsistency, for example, should one say is involved in the problem of the three characters who invent each other cyclically? Logical? Physical? Biological? Literary? Usually, the borderline between uninteresting and interesting is drawn between physical consistency and mathematical consistency. (Of course, it is the mathematicians and logicians who do the drawinghardly an impartial crew . . .) This means that the kinds of inconsistency which "count", for formal systems, are just the logical and mathematical kinds. According to this convention, then, we haven't yet found an interpretation which makes the trio of theorems TbZ, ZbE, EbT inconsistent. We can do so by interpreting b as "is bigger than". What about T and Z and E? They can be interpreted as natural numbersfor example, Z as 0, T as 2, and E as 11. Notice that two theorems come out true this way, one false. If, instead, we had interpreted Z as 3, there would have been two falsehoods and only one truth. But either way, we'd have had inconsistency. In fact, the values assigned to T, Z, and E are irrelevant, as long as it is understood that they are restricted to natural numbers. Once again we see a case where only some of the interpretation is needed, in order to recognize internal inconsistency.  
 
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Embedding of One Formal System In Another The preceding example, in which some symbols could have interpretations while others didn't, is reminiscent of doing geometry in natural languag4 using some words as undefined terms. In such a case, words are divide into two classes: those whose meaning is fixed and immutable, and, those whose meaning is to be adjusted until the system is consistent (these are th undefined terms). Doing geometry in this way requires that meanings have already been established for words in the first class, somewhere outside c geometry. Those words form a rigid skeleton, giving an underlying structure to the system; filling in that skeleton comes other material, which ca vary (Euclidean or nonEuclidean geometry). Formal systems are often built up in just this type of sequential, c hierarchical, manner. For example, Formal System I may be devised, wit rules and axioms that give certain intended passive meanings to its symbol Then Formal System I is incorporated fully into a larger system with more symbolsFormal System II. Since Formal System I's axioms and rules at part of Formal System II, the passive meanings of Formal System I symbols remain valid; they form an immutable skeleton which then plays large role in the determination of the passive meanings of the new symbols of Formal System II. The second system may in turn play the role of skeleton with respect to a third system, and so on. It is also possiblean geometry is a good example of thisto have a system (e.g., absolute geometry) which partly pins down the passive meanings of its undefined terms, and which can be supplemented by extra rules or axioms, which then further restrict the passive meanings of the undefined terms. This the case with Euclidean versus nonEuclidean geometry. Layers of Stability in Visual Perception In a similar, hierarchical way, we acquire new knowledge, new vocabulary or perceive unfamiliar objects. It is particularly interesting in the case understanding drawings by Escher, such as Relativity (Fig. 22), in which there occur blatantly impossible images. You might think that we won seek to reinterpret the picture over and over again until we came to interpretation of its parts which was free of contradictionsbut we dot do that at all. We sit there amused and puzzled by staircases which go eve which way, and by people going in inconsistent directions on a sing staircase. Those staircases are "islands of certainty" upon which we base of interpretation of the overall picture. Having once identified them, we try extend our understanding, by seeking to establish the relationship which they bear to one another. At that stage, we encounter trouble. But if i attempted to backtrackthat is, to question the "islands of certainty"s would also encounter trouble, of another sort. There's no way of backtracking and "undeciding" that they are staircases. They are not fishes, or whip or handsthey are just staircases. (There is, actually, one other on ti leave all the lines of the picture totally uninterpreted, like the "meaningless  
 
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FIGURE 22. Relativity, by M. C. Escher (lithograph, 1953). symbols" of a formal system. This ultimate escape route is an example of a "Umode" responsea Zen attitude towards symbolism.) So we are forced, by the hierarchical nature of our perceptive processes, to see either a crazy world or just a bunch of pointless lines. A similar analysis could be made of dozens of Escher pictures, which rely heavily upon the recognition of certain basic forms, which are then put together in nonstandard ways; and by the time the observer sees the paradox on a high level, it is too latehe can't go back and change his mind about how to interpret the lowerlevel objects. The difference between an Escher drawing and nonEuclidean geometry is that in the latter, comprehensible interpretations can be found for the undefined terms, resulting in a com  
 
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prehensible total system, whereas for the former, the end result is not reconcilable with one's conception of the world, no matter how long or stares at the pictures. Of course, one can still manufacture hypothetic worlds, in which Escherian events can happen ... but in such worlds, t1 laws of biology, physics, mathematics, or even logic will be violated on or level, while simultaneously being obeyed on another, which makes the: extremely weird worlds. (An example of this is in Waterfall (Fig. 5), whet normal gravitation applies to the moving water, but where the nature space violates the laws of physics.) Is Mathematics the Same in Every Conceivable World? We have stressed the fact, above, that internal consistency of a form; system (together with an interpretation) requires that there be some imaginable worldthat is, a world whose only restriction is that in it, mathematics and logic should be the same as in our worldin which all the interpreted theorems come out true. External consistency, however consistency with the external worldrequires that all theorems come of true in the real world. Now in the special case where one wishes to create consistent formal system whose theorems are to be interpreted as statements of mathematics, it would seem that the difference between the two types of consistency should fade away, since, according to what we sat above, all imaginable worlds have the same mathematics as the real world. Thus, i every conceivable world, 1 plus 1 would have to be 2; likewise, there would have to be infinitely many prime numbers; furthermore, in every conceivable world, all right angles would have to be congruent; and of cours4 through any point not on a given line there would have to be exactly on parallel line ... But wait a minute! That's the parallel postulateand to assert i universality would be a mistake, in light of what's just been said. If in all conceivable worlds the parallel postulateis obeyed, then we are asserting that nonEuclidean geometry is inconceivable, which puts us back in the same mental state as Saccheri and Lambertsurely an unwise move. But what, then, if not all of mathematics, must all conceivable worlds share? Could it I as little as logic itself? Or is even logic suspect? Could there be worlds where contradictions are normal parts of existenceworlds where contradictious are not contradictions? Well, in some sense, by merely inventing the concept, we have shoe that such worlds are indeed conceivable; but in a deeper sense, they are al: quite inconceivable. (This in itself is a little contradiction.) Quite serious] however, it seems that if we want to be able to communicate at all, we ha, to adopt some common base, and it pretty well has to include logic. (The are belief systems which reject this point of viewit is too logical. particular, Zen embraces contradictions and noncontradictions with equ eagerness. This may seem inconsistent, but then being inconsistent is pa of Zen, and so ... what can one say?)  
 
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Is Number Theory the Same In All Conceivable Worlds? If we assume that logic is part of every conceivable world (and note that we have not defined logic, but we will in Chapters to come), is that all? Is it really conceivable that, in some worlds, there are not infinitely many primes? Would it not seem necessary that numbers should obey the same laws in all conceivable worlds? Or ... is the concept "natural number" better thought of as an undefined term, like "POINT" or "LINE"? In that case, number theory would be a bifurcated theory, like geometry: there would be standard and nonstandard number theories. But there would have to be some counterpart to absolute geometry: a "core" theory, an invariant ingredient of all number theories which identified them as number theories rather than, say, theories about cocoa or rubber or bananas. It seems to be the consensus of most modern mathematicians and philosophers that there is such a core number theory, which ought to be included, along with logic, in what we consider to be "conceivable worlds". This core of number theory, the counterpart to absolute geometryis called Peano arithmetic, and we shall formalize it in Chapter VIII. Also, it is now well establishedas a matter of fact as a direct consequence of Gödel’s Theoremthat number theory is a bifurcated theory, with standard and nonstandard versions. Unlike the situation in geometry, however, the number of "brands" of number theory is infinite, which makes the situation of number theory considerably more complex. For practical purposes, all number theories are the same. In other words, if bridge building depended on number theory (which in a sense it does), the fact that there are different number theories would not matter, since in the aspects relevant to the real world, all number theories overlap. The same cannot be said of different geometries; for example, the sum of the angles in a triangle is 180 degrees only in Euclidean geometry; it is greater in elliptic geometry, less in hyperbolic. There is a story that Gauss once attempted to measure the sum of the angles in a large triangle defined by three mountain peaks, in order to determine, once and for all, which kind of geometry really rules our universe. It was a hundred years later that Einstein gave a theory (general relativity) which said that the geometry of the universe is determined by its content of matter, so that no one geometry is intrinsic to space itself. Thus to the question, "Which geometry is true?" nature gives an ambiguous answer not only in mathematics, but also in physics. As for the corresponding question, "Which number theory is true?", we shall have more to say on it after going through Gödel’s Theorem in detail. Completenes If consistency is the minimal condition under which symbols acquire passive meanings, then its complementary notion, completeness, is the maximal confirmation of those passive meanings. Where consistency is the property  
 
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way round: "Every true statement is produced by the system". Now I refine the notion slightly. We can't mean every true statement in th worldwe mean only those which belong to the domain which we at attempting to represent in the system. Therefore, completeness mean! "Every true statement which can be expressed in the notation of the system is a theorem." Consistency: when every theorem, upon interpretation, comes out true (in some imaginable world). Completeness: when all statements which are true (in some imaginable world), and which can be expressed as wellformed strings of the system, are theorems. An example of a formal system which is complete on its own mode level is the original pqsystem, with the original interpretation. All true additions of two positive integers are represented by theorems of th system. We might say this another way: "All true additions of two positive integers are provable within the system." (Warning: When we start using th term "provable statements" instead of "theorems", it shows that we at beginning to blur the distinction between formal systems and their interpretations. This is all right, provided we are very conscious of th blurring that is taking place, and provided that we remember that multiple interpretations are sometimes possible.) The pqsystem with the origin interpretation is complete; it is also consistent, since no false statement is, use our new phraseprovable within the system. Someone might argue that the system is incomplete, on the grounds that additions of three positive integers (such as 2 + 3 + 4 =9) are not represented by theorems of the pqsystem, despite being translatable into the notation of the system (e.g., ppq  ). However, this string is not wellformed, and hence should be considered to I just as devoid of meaning as is p q pq p q. Triple additions are simply not expressible in the notation of the systemso the completeness of the system is preserved. Despite the completeness of the pqsystem under this interpretation, certainly falls far short of capturing the full notion of truth in numb theory. For example, there is no way that the pqsystem tells us how mat prime numbers there are. Gödel’s Incompleteness Theorem says that any system which is "sufficiently powerful" is, by virtue of its power, incomplete, in the sense that there are wellformed strings which express tr statements of number theory, but which are not theorems. (There a truths belonging to number theory which are not provable within the system.) Systems like the pqsystem, which are complete but not very powerful, are more like lowfidelity phonographs; they are so poor to beg with that it is obvious that they cannot do what we would wish them donamely tell us everything about number theory.  
 
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How an Interpretation May Make or Break Completeness What does it mean to say, as I did above, that "completeness is the maximal confirmation of passive meanings"? It means that if a system is consistent but incomplete, there is a mismatch between the symbols and their interpretations. The system does not have the power to justify being interpreted that way. Sometimes, if the interpretations are "trimmed" a little, the system can become complete. To illustrate this idea, let's look at the modified pqsystem (including Axiom Schema II) and the interpretation we used for it. After modifying the pqsystem, we modified the interpretation for q from "equals" to "is greater than or equal to". We saw that the modified pqsystem was consistent under this interpretation; yet something about the new interpretation is not very satisfying. The problem is simple: there are now many expressible truths which are not theorems. For instance, "2 plus 3 is greater than or equal to 1" is expressed by the nontheorem pq. The interpretation is just too sloppy! It doesn't accurately reflect what the theorems in the system do. Under this sloppy interpretation, the pqsystem is not complete. We could repair the situation either by (1) adding new rules to the system, making it more powerful, or by (2) tightening up the interpretation. In this case, the sensible alternative seems to be to tighten the interpretation. Instead of interpreting q as "is greater than or equal to", we should say "equals or exceeds by 1". Now the modified pqsystem becomes both consistent and complete. And the completeness confirms the appropriateness of the interpretation. Incompleteness of Formalized Number Theory In number theory, we will encounter incompleteness again; but there, to remedy the situation, we will be pulled in the other directiontowards adding new rules, to make the system more powerful. The irony is that we think, each time we add a new rule, that we surely have made the system complete now! The nature of the dilemma can be illustrated' by the following allegory ... We have a record player, and we also have a record tentatively labeled "Canon on BACH". However, when we play the record on the record player, the feedbackinduced vibrations (as caused by the Tortoise's records) interfere so much that we do not even recognize the tune. We conclude that something is defectiveeither our record, or our record player. In order to test our record, we would have to play it on friends' record players, and listen to its quality. In order to test our phonograph, we would have to play friends' records on it, and see if the music we hear agrees with the labels. If our record player passes its test, then we will say the record was defective; contrariwise, if the record passes its test, then we will say our record player was defective. What, however, can we conclude when we find out that both pass their respective tests? That is the moment to remember the chain of two isomorphisms (Fig. 20), and think carefully!  
 
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Little Harmonic Labyrinth  
 
The Tortoise and Achilles are spending a day at Coney Island After buying a couple of cotton candies, they decide to take a ride on the Ferris wheel. Tortoise: This is my favorite ride. One seems to move so far, and reality one gets nowhere. Achilles: I can see why it would appeal to you. Are you all strapped in? Tortoise: Yes, I think I've got this buckle done. Well, here we go. Achilles: You certainly are exuberant today. Tortoise: I have good reason to be. My aunt, who is a fortuneteller me that a stroke of Good Fortune would befall me today. So I am tingling with anticipation. Achilles: Don't tell me you believe in fortunetelling! Tortoise: No ... but they say it works even if you don't believe ii Achilles: Well, that's fortunate indeed. Tortoise: Ah, what a view of the beach, the crowd, the ocean, the city. . . Achilles: Yes, it certainly is splendid. Say, look at that helicopter there. It seems to be flying our way. In fact it's almost directly above us now. Tortoise: Strangethere's a cable dangling down from it, which is very close to us. It's coming so close we could practically grab it Achilles: Look! At the end of the line there's a giant hook, with a note (He reaches out and snatches the note. They pass by and are on their z down.) Tortoise: Can you make out what the note says? Achilles: Yesit reads, "Howdy, friends. Grab a hold of the hook time around, for an Unexpected Surprise." Tortoise: The note's a little corny but who knows where it might lead, Perhaps it's got something to do with that bit of Good Fortune due me. By all means, let's try it! Achilles: Let's! (On the trip up they unbuckle their buckles, and at the crest of the ride, grab for the giant hook. All of a sudden they are whooshed up by the ca which quickly reels them skyward into the hovering helicopter. A It strong hand helps them in.) Voice: Welcome aboardSuckers. Achilles: Whwho are you?  
 
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Voce: Allow me to introduce myself. I am Hexachlorophene J. Goodforttune, Kidnapper AtLarge, and Devourer of Tortoises par Excellence, at your service. Tortoise: Gulp! Achilles (whispering to his friend): UhohI think that this "Goodfortune" is not exactly what we'd anticipated. (To Goodfortune) Ahif I may be so boldwhere are you spiriting us off to? Goodfortune: Ho ho! To my allelectric kitcheninthesky, where I will prepare THIS tasty morsel(leering at the Tortoise as he says this)in a delicious pieinthesky! And make no mistakeit's all just for my gobbling pleasure! Ho ho ho! Achilles: All I can say is you've got a pretty fiendish laugh. Goodfortune (laughing fiendishly): Ho ho ho! For that remark, my friend, you will pay dearly. Ho ho! Achilles: Good griefI wonder what he means by that! Goodfortune: Very simpleI've got a Sinister Fate in store for both of you! Just you wait! Ho ho ho! Ho ho ho! Achilles: Yikes! Goodfortune: Well, we have arrived. Disembark, my friends, into my fabulous allelectric kitcheninthesky. (They walk inside.) Let me show you around, before I prepare your fates. Here is my bedroom. Here is my study. Please wait here for me for a moment. I've got to go sharpen my knives. While you're waiting, help yourselves to some popcorn. Ho ho ho! Tortoise pie! Tortoise pie! My favorite kind of pie! (Exit.) Achilles: Oh, boypopcorn! I'm going to munch my head off! Tortoise: Achilles! You just stuffed yourself with cotton candy! Besides, how can you think about food at a time like this? Achilles: Good gravyoh, pardon meI shouldn't use that turn of phrase, should I? I mean in these dire circumstances ... Tortoise: I'm afraid our goose is cooked. Achilles: Saytake a gander at all these books old Goodfortune has in his study. Quite a collection of esoterica: Birdbrains I Have Known; Chess and UmbrellaTwirling Made Easy; Concerto for Tapdancer and Orchestra ... Hmmm. Tortoise: What's that small volume lying open over there on the desk, next to the dodecahedron and the open drawing pad? Achilles: This one? Why, its title is Provocative Adventures of Achilles and the Tortoise Taking Place in Sundry Spots of the Globe. Tortoise: A moderately provocative title. Achilles: Indeedand the adventure it's opened to looks provocative. It's called "Djinn and Tonic". Tortoise: Hmm ... I wonder why. Shall we try reading it? I could take the Tortoise's part, and you could take that of Achilles.  
 
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Achilles: I’m game. Here goes nothing . . . (They begin reading "Djinn and Tonic".) (Achilles has invited the Tortoise over to see his collection of prints by his favorite artist, M. C. Escher.) Tortoise: These are wonderful prints, Achilles. Achilles: I knew you would enjoy seeing them. Do you have any particular favorite? Tortoise: One of my favorites is Convex and Concave, where two internally consistent worlds, when juxtaposed, make a completely inconsistent composite world. Inconsistent worlds are always fun places to visit, but I wouldn't want to live there. Achilles: What do you mean, "fun to visit"? Inconsistent worlds don't EXIST, so how can you visit one? Tortoise: I beg your pardon, but weren't we just agreeing that in this Escher picture, an inconsistent world is portrayed? Achilles: Yes, but that's just a twodimensional worlda fictitious worlda picture. You can't visit that world. Tortoise: I have my ways ... Achilles: How could you propel yourself into a flat pictureuniverse? Tortoise: By drinking a little glass of PUSHINGPOTION. That does the trick. Achilles: What on earth is pushingpotion? Tortoise: It's a liquid that comes in small ceramic phials, and which, when drunk by someone looking at a picture, "pushes'' him right into the world of that picture. People who aren't aware of the powers of pushingpotion often are pretty surprised by the situations they wind up in. Achilles: Is there no antidote? Once pushed, is one irretrievably lost? Tortoise: In certain cases, that's not so bad a fate. But there is, in fact, another potionwell, not a potion, actually, but an elixirno, not an elixir, but aa Tortoise: He probably means "tonic". Achilles: Tonic? Tortoise: That's the word I was looking for! "POPPINGTONIC" iu what it's called, and if you remember to carry a bottle of it in your right hand as you swallow the pushingpotion, it too will be pushed into the picture; then, whenever you get a hanker ing to "pop" back out into real life, you need only take a swallow of poppingtonic, and presto! You're back in the rea. world, exactly where you were before you pushed yourself in. Achilles: That sounds very interesting. What would happen it you took some poppingtonic without having previously pushed yourself into a picture?  
 
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Tortoise: I don’t precisely know, Achilles, but I would be rather wary of horsing around with these strange pushing and popping liquids. Once I had a friend, a Weasel, who did precisely what you suggestedand no one has heard from him since. Achilles: That's unfortunate. Can you also carry along the bottle of pushingpotion with you? Tortoise: Oh, certainly. Just hold it in your left hand, and it too will get pushed right along with you into the picture you're looking at. Achilles: What happens if you then find a picture inside the picture which you have already entered, and take another swig of pushingpotion? Tortoise: Just what you would expect: you wind up inside that pictureinapicture. Achilles: I suppose that you have to pop twice, then, in order to extricate yourself from the nested pictures, and reemerge back in real life. Tortoise: That's right. You have to pop once for each push, since a push takes you down inside a picture, and a pop undoes that. Achilles: You know, this all sounds pretty fishy to me . . . Are you sure you're not just testing the limits of my gullibility? Tortoise: I swear! Lookhere are two phials, right here in my pocket. (Reaches into his lapel pocket, and pulls out two rather large unlabeled phials, in one of which one can hear a red liquid sloshing around, and in the other of which one can hear a blue liquid sloshing around.) If you're willing, we can try them. What do you say? Achilles: Well, I guess, ahm, maybe, ahm ... Tortoise: Good! I knew you'd want to try it out. Shall we push ourselves into the world of Escher's Convex and Concave? Achilles: Well, ah, .. . Tortoise: Then it's decided. Now we've got to remember to take along this flask of tonic, so that we can pop back out. Do you want to take that heavy responsibility, Achilles? Achilles: If it's all the same to you, I'm a little nervous, and I'd prefer letting you, with your experience, manage the operation. Tortoise: Very well, then. (So saying, the Tortoise pours two small portions of pushingpotion. Then he picks up the flask of tonic and grasps it firmly in his right hand, and both he and Achilles lift their glasses to their lips.) Tortoise: Bottoms up! (They swallow.)  
 
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FIGURE 23. Convex and Concave, by M. C. Escher (lithograph, 1955).  
 
Achilles: That's an exceedingly strange taste. Tortoise: One gets used to it. Achilles: Does taking the tonic feel this strange? Tortoise: Oh, that's quite another sensation. Whenever you taste the tonic, you feel a deep sense of satisfaction, as if you'd been waiting to taste it all your life. Achilles: Oh, I'm looking forward to that. Tortoise: Well, Achilles, where are we? Achilles (taking cognizance of his surroundings): We're in a little gondola, gliding down a canal! I want to get out. Mr.Gondolier, please let us out here. (The gondolier pays no attention to this request.) 
Tortoise: He doesn't speak English. If we want to get out here, we'd better just clamber out quickly before he
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Enters the sinister “Tunnel of Love”; just ahead of us. (Achilles, his face a little pale scrambles out in a split second and then pulls his slower friend out.) Achilles: I didn't like the sound of that place, somehow. I'm glad we got out here. Say, how do you know so much about this place, anyway? Have you been here before? Tortoise: Many times, although I always came in from other Escher pictures. They're all connected behind the frames, you know. Once you're in one, you can get to any other one. Achilles: Amazing! Were I not here, seeing these things with my own eyes, I'm not sure I'd believe you. (They wander out through a little arch.) Oh, look at those two cute lizards! Tortoise: Cute? They aren't cuteit makes me shudder just to think of them! They are the vicious guardians of that magic copper lamp hanging from the ceiling over there. A mere touch of their tongues, and any mortal turns to a pickle. Achilles: Dill, or sweet? Tortoise: Dill. Achilles: Oh, what a sour fate! But if the lamp has magical powers, I would like to try for it. Tortoise: It's a foolhardy venture, my friend. I wouldn't risk it. Achilles: I'm going to try just once. (He stealthily approaches the lamp, making sure not to awaken the sleeping lad nearby. But suddenly, he slips on a strange shelllike indentation in the floor, and lunges out into space. Lurching crazily, he reaches for anything, and manages somehow to grab onto the lamp with one hand. Swinging wildly, with both lizards hissing and thrusting their tongues violently out at him, he is left dangling helplessly out in the middle of space.) Achilles: Heeeelp! (His cry attracts the attention of a woman who rushes downstairs and awakens the sleeping boy. He takes stock of the situation, and, with a kindly smile on his face, gestures to Achilles that all will be well. He shouts something in a strange guttural tongue to a pair of trumpeters high up in windows, and immediately,  
 
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Weird tones begin ringing out and making beats each other. The sleepy young lad points at the lizards, and Achilles sees that the music is having a strong soporific effect on them. Soon, they are completely unconscious. Then the helpful lad shouts to two companions climbing up ladders. They both pull their ladders up and then extend them out into space just underneath the stranded Achilles, forming a sort of bridge. Their gestures make it clear that Achilles should hurry and climb on. But before he does so, Achilles carefully unlinks the top link of the chain holding the lamp, and detaches the lamp. Then he climbs onto the ladderbridge and the three young lads pull him in to safety. Achilles throws his arms around them and hugs them gratefully.) Achilles: Oh, Mr. T, how can I repay them? Tortoise: I happen to know that these valiant lads just love coffee, and down in the town below, there's a place where they make an incomparable cup of espresso. Invite them for a cup of espresso! Achilles: That would hit the spot. (And so, by a rather comical series of gestures, smiles, and words, Achilles manages to convey his invitation to the young lads, and the party of five walks out and down a steep staircase descending into the town. They reach a charming small cafe, sit down outside, and order five espressos. As they sip their drinks, Achilles remembers he has the lamp with him.) Achilles: I forgot, Mr. TortoiseI've got this ma; lamp with me! Butwhat's magic about it? Tortoise: Oh, you know, just the usuala genie. Achilles: What? You mean a genie comes out when you rub it, and grants you wishes? Tortoise: Right. What did you expect? Pennies fry heaven? Achilles: Well, this is fantastic! I can have any wish want, eh? I've always wished this would happen to me ... (And so Achilles gently rubs the large letter `L' which is etched on the lamp's copper surface ... Suddenly a huge puff of smoke appears, and in the forms of the smoke the five friends can make out a weird, ghostly figure towering above them.)  
 
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I Genie: Hello, my friends – and thanks ever so much for rescuing my Lamp from the evil LizardDuo. (And so saying, the Genie picks up the Lamp, and stuffs it into a pocket concealed among the folds of his long ghostly robe which swirls out of the Lamp.) As a sign of gratitude for your heroic deed, I would like to offer you, on the part of my Lamp, the opportunity to have any three of your wishes realized. Achilles: How stupefying! Don't you think so, Mr. T? Tortoise: I surely do. Go ahead, Achilles, take the first wish. Achilles: Wow! But what should I wish? Oh, I know! It's what I thought of the first time I read the Arabian Nights (that collection of silly (and nested) tales)I wish that I had a HUNDRED wishes, instead of just three! Pretty clever, eh, Mr. T? I bet YOU never would have thought of that trick. I always wondered why those dopey people in the stories never tried it themselves. Tortoise: Maybe now you'll find out the answer. Genie: I am sorry, Achilles, but I don't grant metawishes. Achilles: I wish you'd tell me what a "metawish" is! Genie: But THAT is a metametawish, Achillesand I don't grant them, either. Achilles: Whaaat? I don't follow you at all. Tortoise: Why don't you rephrase your last request, Achilles? Achilles: What do you mean? Why should I? Tortoise: Well, you began by saying "I wish". Since you're just asking for information, why don't you just ask a question? Achilles: All right, though I don't see why. Tell me, Mr. Geniewhat is a metawish? Genie: It is simply a wish about wishes. I am not allowed to grant metawishes. It is only within my purview to grant plain ordinary wishes, such as wishing for ten bottles of beer, to have Helen of Troy on a blanket, or to have an allexpensespaid weekend for two at the Copacabana. You knowsimple things like that. But metawishes I cannot grant. GOD won't permit me to. Achilles: GOD? Who is GOD? And why won't he let you grant metawishes? That seems like such a puny thing compared to the others you mentioned.  
 
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Genie: Well, it’s a complicated matter, you see. Why don’t you just go ahead and make your three wishes? Or at least make one of them. I don't have all I time in the world, you know ... Achilles: Oh, I feel so rotten. I was REALLY HOPING wish for a hundred wishes ... Genie: Gee, I hate to see anybody so disappointed that. And besides, metawishes are my favorite k of wish. Let me just see if there isn't anything I do about this. This'll just take one moment (The Genie removes from the wispy folds of his robe an object which looks just like the copper Lamp he had put away, except that this one is made of silver; and where the previous one had 'L' etched on it, this one has 'ML' in smaller letters, so as to cover the same area.) I Achilles: And what is that? Genie: This is my MetaLamp ... (He rubs the MetaLamp, and a huge puff of smoke appears. In the billows of smoke, they can all make out a ghostly form towering above them.) MetaGenie: I am the MetaGenie. You summoned me, 0 Genie? What is your wish? Genie: I have a special wish to make of you, 0 Djinn and of GOD. I wish for permission for tempos suspension of all typerestrictions on wishes, for duration of one Typeless Wish. Could you ph grant this wish for me? MetaGenie: I'll have to send it through Channels, of course. One half a moment, please (And, twice as quickly as the Genie did, this MetaGenie removes from the wispy folds of her robe an object which looks just like the silver MetaLamp, except that it is made of gold; and where the previous one had 'ML' etched on it, this one has 'MML' in smaller letters, so as to cover the same area.) Achilles (his voice an octave higher than before): And what is that? MetaGenie: This is my MetaMetaLamp. . . (She rubs the MetaMetaLamp, and a hugs puff of smoke appears. In the billows o smoke, they can all make out a ghostly fore towering above them.)  
 
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MetaMetaGenie: I am the MetaMetaGenie. You summoned me, 0 MetaGenie? What is your wish? MetaGenie: I have a special wish to make of you, 0 Djinn, and of GOD. I wish for permission for temporary suspension of all typerestrictions on wishes, for the duration of one Typeless Wish. Could you please grant this wish for me? MetaMetaGenie: I'll have to send it through Channels, of course. One quarter of a moment, please. (And, twice as quickly as the MetaGenie did, this MetaMetaGenie removes from the folds of his robe an object which looks just like the gold MetaLamp, except that it is made of ...) . . . . . . .{GOD} . . . . ( ... swirls back into the MetaMetaMetaLamp, which the MetaMetaGenie then folds back into his robe, half as quickly as the MetaMetaMetaGenie did.) Your wish is granted, 0 MetaGenie. MetaGenie: Thank you, 0 Djinn, and GOD. (And the MetaMetaGenie, as all the higher ones before him, swirls back into the MetaMetaLamp, which the MetaGenie then folds back into her robe, half as quickly as the MetaMetaGenie did.) Your wish is granted, 0 Genie. Genie: Thank you, 0 Djinn, and GOD. (And the MetaGenie, as all the higher ones before her,  
 
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swirls back into the MetaLamp, which the Genie folds back into his robe, half as quickly as the M Genie did.) Your wish is granted, Achilles. (And one precise moment has elapsed since he "This will just take one moment.") Achilles: Thank you, 0 Djinn, and GOD. Genie: I am pleased to report, Achilles, that you r have exactly one (1) Typeless Wishthat is to sa wish, or a metawish, or a metametawish, as many "meta"'s as you wisheven infinitely many (if wish). Achilles: Oh, thank you so very much, Genie. But curiosity is provoked. Before I make my wish, would you mind telling me whoor whatGOD is? Genie: Not at all. "GOD" is an acronym which stands "GOD Over Djinn". The word "Djinn" is used designate Genies, MetaGenies, MetaMetaGen etc. It is a Typeless word. Achilles: Butbuthow can "GOD" be a word in own acronym? That doesn't make any sense! Genie: Oh, aren't you acquainted with recursive acronyms? I thought everybody knew about them. \ see, "GOD" stands for "GOD Over Djinn"which can be expanded as "GOD Over Djinn, O, Djinn"and that can, in turn, be expanded to "G( Over Djinn, Over Djinn, Over Djinn"which can its turn, be further expanded ... You can go as as you like. Achilles: But I'll never finish! Genie: Of course not. You can never totally expand GOD. Achilles: Hmm ... That's puzzling. What did you me when you said to the MetaGenie, "I have a sped wish to make of you, 0 Djinn, and of GOD"? Genie: I wanted not only to make a request of MetaGenie, but also of all the Djinns over her. 'I recursive acronym method accomplishes this qL naturally. You see, when the MetaGenie received my request, she then had to pass it upwards to I GOD. So she forwarded a similar message to I MetaMetaGenie, who then did likewise to t MetaMetaMetaGenie ... Ascending the chain this way transmits the message to GOD.  
 
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Achilles: I see. You mean GOD sits up at the top of the ladder of djinns? Genie: No, no, no! There is nothing "at the top", for there is no top. That is why GOD is a recursive acronym. GOD is not some ultimate djinn; GOD is the tower of djinns above any given djinn. Tortoise: It seems to me that each and every djinn would have a different concept of what GOD is, then, since to any djinn, GOD is the set of djinns above him or her, and no two djinns share that set. Genie: You're absolutely rightand since I am the lowest djinn of all, my notion of GOD is the most exalted one. I pity the higher djinns, who fancy themselves somehow closer to GOD. What blasphemy! Achilles: By gum, it must have taken genies to invent GOD. Tortoise: Do you really believe all this stuff about GOD, Achilles? Achilles: Why certainly, I do. Are you atheistic, Mr. T? Or are you agnostic? Tortoise: I don't think I'm agnostic. Maybe I'm metaagnostic. Achilles: Whaaat? I don't follow you at all. Tortoise: Let's see . . . If I were metaagnostic, I'd be confused over whether I'm agnostic or notbut I'm not quite sure if I feel THAT way; hence I must be metametaagnostic (I guess). Oh, well. Tell me, Genie, does any djinn ever make a mistake, and garble up a message moving up or down the chain? Genie: This does happen; it is the most common cause for Typeless Wishes not being granted. You see, the chances are infinitesimal, that a garbling will occur at any PARTICULAR link in the chainbut when you put an infinite number of them in a row, it becomes virtually certain that a garbling will occur SOMEWHERE. In fact, strange as it seems, an infinite number of garblings usually occur, although they are very sparsely distributed in the chain. Achilles: Then it seems a miracle that any Typeless Wish ever gets carried out. Genie: Not really. Most garblings are inconsequential, and many garblings tend to cancel each other out. But occasionallyin fact, rather seldomthe nonfulfillment of a Typeless Wish can be traced back to a single unfortunate djinn's garbling. When this happens, the guilty djinn is forced to run an infinite  
 
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Gauntlet and get paddled on his or her rump, by GOD. It's good fun for the paddlers, and q harmless for the paddlee. You might be amused by the sight. Achilles: I would love to see that! But it only happens when a Typeless Wish goes ungranted? Genie: That's right. Achilles: Hmm ... That gives me an idea for my w Tortoise: Oh, really? What is it? Achilles: I wish my wish would not be granted! (At that moment, an eventor is "event" the word for it? takes place which cannot be described, and hence no attempt will be made to describe it.) Achilles: What on earth does that cryptic comment mean? Tortoise: It refers to the Typeless Wish Achilles made. Achilles: But he hadn't yet made it. Tortoise: Yes, he had. He said, "I wish my wish would not be granted", and the Genie took THAT to be his wish. (At that moment, some footsteps are heard coming down the hallway in their direction.) Achilles: Oh, my! That sounds ominous. (The footsteps stop; then they turn around and fade away.) Tortoise: Whew! Achilles: But does the story go on, let's see. or is that the end? Turn the page and let’s see. (The Tortoise turns the page of "Djinn and Tonic", where they find that the story goes on ...) Achilles: Hey! What happened? Where is my Genie: lamp? My cup of espresso? What happened to young friends from the Convex and Concave worlds? What are all those little lizards doing hi Tortoise: I'm afraid our context got restored incorrectly Achilles. Achilles: What on earth does that cryptic comment mean? Tortoise: I refer to the Typeless Wish you made. Achilles: But I hadn't yet made it. Tortoise: Yes, you had. You said, "I wish my wish would not be granted", and the Genie took THAT to be your wish. Achilles: Oh, my! That sounds ominous. Tortoise; It spells PARADOX. For that Typeless wish to be  
 
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granted, it had to be denied – yet not to grant it would be to grant it. Achilles: So what happened? Did the earth come to a standstill? Did the universe cave in? Tortoise: No. The System crashed. Achilles: What does that mean? Tortoise: It means that you and I, Achilles, were suddenly and instantaneously transported to Tumbolia. Achilles: To where? Tortoise: Tumbolia: the land of dead hiccups and extinguished light bulbs. It's a sort of waiting room, where dormant software waits for its host hardware to come back up. No telling how long the System was down, and we were in Tumbolia. It could have been moments, hours, dayseven years. Achilles: I don't know what software is, and I don't know what hardware is. But I do know that I didn't get to make my wishes! I want my Genie back! Tortoise: I'm sorry, Achillesyou blew it. You crashed the System, and you should thank your lucky stars that we're back at all. Things could have come out a lot worse. But I have no idea where we are. Achilles: I recognize it nowwe're inside another of Escher's pictures. This time it's Reptiles. Tortoise: Aha! The System tried to save as much of our context as it could before it crashed, and it got as far as recording that it was an Escher picture with lizards before it went down. That's commendable. Achilles: And lookisn't that our phial of poppingtonic over there on the table, next to the cycle of lizards? Tortoise: It certainly is, Achilles. I must say, we are very lucky indeed. The System was very kind to us, in giving us back our poppingtonicit's precious stuff! Achilles: I'll say! Now we can pop back out of the Escher world, into my house. Tortoise: There are a couple of books on the desk, next to the tonic. I wonder what they are. (He picks up the smaller one, which is open to a random page.) This looks like a moderately provocative book. Achilles: Oh, really? What is its title? Tortoise: Provocative Adventures of the Tortoise and Achilles Taking Place in Sundry Parts of the Globe. It sounds like an interesting book to read out of.  
 
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FIGURE 24. Reptiles, by M. C. Escher (lithograph, 1943). Achilles: Well, You can read it if you want, but as for I'm not going to take any chances with t poppingtonicone of the lizards might knock it off the table, so I'm going to get it right now! (He dashes over to the table and reaches for the poppingtonic, but in his haste he somehow bumps the flask of tonic, and it tumbles off the desk and begins rolling.) Oh, no! Mr. Tlook! I accidentally knocked tonic onto the floor, and it's rolling toward towardsthe stairwell! Quickbefore it falls!  
 
(The Tortoise, however, is completely wrapped up in the thin volume which he has in his hands.) Achilles: Well, You can read it if you want, but as for I'm not going to take any chances with t poppingtonicone of the lizards might knock it off the table, so I'm going to get it right  
 
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Tortoise (muttering): Eh? This story looks fascinating. Achilles: Mr. T, Mr. T, help! Help catch the tonicflask! Tortoise: What's all the fuss about? Achilles: The tonicflaskI knocked it down from the desk, and now it's rolling and (At that instant it reaches the brink of the stairwell, and plummets over ... ) Oh no! What can we do? Mr. Tortoisearen't you alarmed? We're losing our tonic! It's just fallen down the stairwell! There's only one thing to do! We'll have to go down one story! Tortoise: Go down one story? My pleasure. Won't you join me? (He begins to read aloud, and Achilles, pulled in two directions at once, finally stays, taking the role of the Tortoise.) Achilles: It's very dark here, Mr. T. I can't see a thing. Oof! I bumped into a wall. Watch out! Tortoise: HereI have a couple of walking sticks. Why don't you take one of them? You can hold it out in front of you so that you don't bang into things. Achilles: Good idea. (He takes the stick.) Do you get the sense that this path is curving gently to the left as we walk? Tortoise: Very slightly, yes. Achilles: I wonder where we are. And whether we'll ever see the light of day again. I wish I'd never listened to you, when you suggested I swallow some of that "DRINK ME" stuff. Tortoise: I assure you, it's quite harmless. I've done it scads of times, and not a once have I ever regretted it. Relax and enjoy being small. Achilles: Being small? What is it you've done to me, Mr. T? Tortoise: Now don't go blaming me. You did it of your own free will. Achilles: Have you made me shrink? So that this labyrinth we're in is actually some teeny thing that someone could STEP on?  
 
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FIGURE 25. Cretan Labyrinth (Italian engraving; School of Finiguerra). [From N Matthews, Mazes and Labyrinths: Their History and Development (New York: Dover Publications, 1970).  
 
Tortoise: Labyrinth? Labyrinth? Could it Are we in the notorious Little Harmonic Labyrinth of the dreaded Majotaur? Achilles: Yiikes! What is that? Tortoise: They sayalthough I person never believed it myselfthat an I Majotaur has created a tiny labyrinth sits in a pit in the middle of it, waiting innocent victims to get lost in its fears complexity. Then, when they wander and dazed into the center, he laughs and laughs at themso hard, that he laughs them to death! Achilles: Oh, no! Tortoise: But it's only a myth. Courage, Achilles. (And the dauntless pair trudge on.) Achilles: Feel these walls. They're like o gated tin sheets, or something. But the corrugations have different sizes.  
 
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(To emphasize his point, he sticks out his walking stick against the wall surface as he walks. As the stick bounces back and forth against the corrugations, strange noises echo up and down the long curved corridor they are in.) Tortoise (alarmed): What was THAT? Achilles: Oh, just me, rubbing my walking stick against the wall. Tortoise: Whew! I thought for a moment it was the bellowing of the ferocious Majotaur! Achilles: I thought you said it was all a myth. Tortoise: Of course it is. Nothing to be afraid of. (Achilles puts his walking stick back against the wall, and continues walking. As he does so, some musical sounds are heard, coming from the point where his stick is scraping the wall.) Tortoise: Uhoh. I have a bad feeling, Achilles. That Labyrinth may not be a myth, after all. Achilles: Wait a minute. What makes you change your mind all of a sudden? Tortoise: Do you hear that music? (To hear more clearly, Achilles lowers the stick, and the strains of melody cease.) Hey! Put that back! I want to hear the end of this piece! (Confused, Achilles obeys, and the music resumes.) Thank you. Now as I was about to say, I have just figured out where we are. Achilles: Really? Where are we? Tortoise: We are walking down a spiral groove of a record in its jacket. Your stick scraping against the strange shapes in the wall acts like a needle running down the groove, allowing us to hear the music. Achilles: Oh, no, oh, no ... Tortoise: What? Aren't you overjoyed? Have you ever had the chance to be in such intimate contact with music before?  
 
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Achzltes: How am I ever going to win footraces against fullsized people when I am smaller than a flea, Mr. Tortoise? Tortoise: Oh, is that all that's bothering you That's nothing to fret abopt, Achilles. Achilles: The way you talk, I get the impression that you never worry at all. Tortoise: I don't know. But one thing for certain is that I don't worry about being small. Especially not when faced with the awful danger of the dreaded Majotaur! Achilles: Horrors! Are you telling me Tortoise: I'm afraid so, Achilles. The music gave it away. Achilles: How could it do that? Tortoise: Very simple. When I heard melody BACH in the top voice, I immediately realized that the grooves we're walking through could only be Little Harmonic Labyrinth, one of Bach's er known organ pieces. It is so named cause of its dizzyingly frequent modulations. Achilles: Whwhat are they? Tortoise: Well, you know that most music pieces are written in a key, or tonality, as C major, which is the key of this o; Achilles: I had heard the term before. Do that mean that C is the note you want to on? Tortoise: Yes, C acts like a home base, in a Actually, the usual word is "tonic". Achilles: Does one then stray away from tonic with the aim of eventually returning Tortoise: That's right. As the piece develops ambiguous chords and melodies are t which lead away from the tonic. Little by little, tension builds upyou feel at creasing desire to return home, to hear the tonic. Achilles: Is that why, at the end of a pie always feel so satisfied, as if I had waiting my whole life to hear the ton Tortoise: Exactly. The composer has uses knowledge of harmonic progressions to  
 
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manipulate your emotions, and to build up hopes in you to hear that tonic. Achilles: But you were going to tell me about modulations. Tortoise: Oh, yes. One very important thing a composer can do is to "modulate" partway through a piece, which means that he sets up a temporary goal other than resolution into the tonic. Achilles: I see ... I think. Do you mean that some sequence of chords shifts the harmonic tension somehow so that I actually desire to resolve in a new key? Tortoise: Right. This makes the situation more complex, for although in the short term you want to resolve in the new key, all the while at the back of your mind you retain the longing to hit that original goalin this case, C major. And when the subsidiary goal is reached, there is Achilles (suddenly gesturing enthusiastically): Oh, listen to the gorgeous upwardswooping chords which mark the end of this Little Harmonic Labyrinth! Tortoise: No, Achilles, this isn't the end. It's merely Achilles: Sure it is! Wow! What a powerful, strong ending! What a sense of relief! That's some resolution! Gee! (And sure enough, at that moment the music stops, as they emerge into an open area with no walls.) You see, it Is over. What did I tell you? Tortoise: Something is very wrong. This record is a disgrace to the world of music. Achilles: What do you mean? Tortoise: It was exactly what I was telling you about. Here Bach had modulated from C into G, setting up a secondary goal of hearing G. This means that you experience two tensions at oncewaiting for resolution into G, but also keeping in mind that ultimate desireto resolve triumphantly into C Major. Achilles: Why should you have to keep any  
 
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thing in mind when listening to a piece of music? Is music only an intellectual exercise? Tortoise: No, of course not. Some music is highly intellectual, but most music is not. And most of the time your ear or br the "calculation" for you, and lets your emotions know what they want to hear, don't have to think about it consciously in this piece, Bach was playing tricks hoping to lead you astray. And in your case Achilles, he succeeded. Achilles: Are you telling me that I responded to a resolution in a subsidiary key? Tortoise: That's right. Achilles: It still sounded like an ending to me Tortoise: Bach intentionally made it sot way. You just fell into his trap. It was deliberately contrived to sound like an ending but if you follow the harmonic progression carefully, you will see that it is in the wrong key. Apparently not just you but this miserable record company fell for the same trickand they truncated the piece early. Achilles: What a dirty trick Bach played Tortoise: That is his whole gameto m lose your way in his Labyrinth! 'l Majotaur is in cahoots with Bach, And if you don't watch out, he i laugh you to deathand perhaps n with you! Achilles: Oh, let us hurry up and get here! Quick! Let's run backwards grooves, and escape on the outside record before the Evil Majotaur finds us. Tortoise: Heavens, no! My sensibility is delicate to handle the bizarre the gressions which occur when time versed. Achilles: Oh, Mr. T, how will we ever get out of here, if we can't just retrace our steps Tortoise: That's a very good question. (A little desperately, Achilles starts runt about aimlessly in the dark. Suddenly t is a slight gasp, and then a "thud".)  
 
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Achillesare you all right? Achilles: Just a bit shaken up but otherwise fine. I fell into some big hole. Tortoise: You've fallen into the pit of the Evil Majotaur! Here, I'll come help you out. We've got to move fast! Achilles: Careful, Mr. TI don't want You to fall in here, too ... Tortoise: Don't fret, Achilles. Everything will be all  (Suddenly, there is a slight gasp, and then a "thud".) Achilles: Mr. Tyou fell in, too! Are you all right? Tortoise: Only my pride is hurtotherwise I'm fine. Achilles: Now we're in a pretty pickle, aren't we? (Suddenly, a giant, booming laugh is heard, alarmingly close to them.) Tortoise: Watch out, Achilles! This is no laughing matter. Majotaur: Hee hee hee! Ho ho! Haw haw haw! Achilles: I'm starting to feel weak, Mr. T ... Tortoise: Try to pay no attention to his laugh, Achilles. That's your only hope. Achilles: I'll do my best. If only my stomach weren't empty! Tortoise: Say, am I smelling things, or is there a bowl of hot buttered popcorn around here? Achilles: I smell it, too. Where is it coming from? Tortoise: Over here, I think. Oh! I just ran into a big bowl of the stuff. Yes, indeedit seems to be a bowl of popcorn! Achilles: Oh, boypopcorn! I'm going to munch my head off! Tortoise: Let's just hope it isn't pushcorn! Pushcorn and popcorn are extraordinarily difficult to tell apart. Achilles: What's this about Pushkin? Tortoise: I didn't say a thing. You must be hearing things. Achilles: Gogolly! I hope not. Well, let's dig in!  
 
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(And the two Jriends begin muncnai popcorn (or pushcorn?)and t once POP! I guess it was popcorn; all.) Tortoise: What an amusing story. Did you en Achilles: Mildly. Only I wonder whether the' out of that Evil Majotaur's pit or r Achilleshe wanted to be fullsized again Tortoise: Don't worrythey're out, and he is again. That's what the "POP" was all abo Achilles: Oh, I couldn't tell. Well, now I REAL: find that bottle of tonic. For some reason, burning. And nothing would taste bett drink of poppingtonic. Tortoise: That stuff is renowned for its thirst powers. Why, in some places people very crazy over it. At the turn of the century the Schonberg food factory stopped ma] and started making cereal instead. You cai the uproar that caused. Achilles: I have an inkling. But let's go look fo Hey just a moment. Those lizards on the you see anything funny about them? Tortoise: Umm ... not particularly. What do you see of such great interest? Achilles: Don't you see it? They're emerging flat picture without drinking any pop] How are they able to do that? Tortoise: Oh, didn't I tell you? You can ge picture by moving perpendicularly to it you have no poppingtonic. The little li learned to climb UP when they want to ge twodimensional sketchbook world. Achilles: Could we do the same thing to get Escher picture we're in? Tortoise: Of course! We just need to go UP one story. you want to try it? Achilles: Anything to get back to my house! I all these provocative adventures. Tortoise: Follow me, then, up this way. (And they go up one story.) Achilles: It's good to be back. But something seems wrong. This isn't my house! This is YOUR house, Mr. Tortoise Tortoise: Well, so it isand am I glad for that! I wasn’t looking  
 
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forward one whit to the long walk back from your house. I am bushed, and doubt if I could have made it. Achilles: I don't mind walking home, so I guess it's lucky we ended up here, after all. Tortoise: I'll say! This certainly is a piece of Good Fortune!  
 
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Recursive Structures and Processes What Is Recursion?  
 
WHAT IS RECURSION? It is what was illustrated in the Dialogue Little Harmonic Labyrinth: nesting, and variations on nesting. The concept is very general. (Stories inside stories, movies inside movies, paintings inside paintings, Russian dolls inside Russian dolls (even parenthetical comments in. side parenthetical comments!)these are just a few of the charms of recursion.) However, you should he aware that the meaning of "recursive' in this Chapter is only faintly related to its meaning in Chapter 111. The relation should be clear by the end of this Chapter. Sometimes recursion seems to brush paradox very closely. For example, there are recursive definitions. Such a definition may give the casual viewer the impression that something is being defined in terms of itself. That would be circular and lead to infinite regress, if not to paradox proper. Actually, a recursive definition (when properly formulated) never leads to infinite regress or paradox. This is because a recursive definition never defines something in terms of itself, but always in terms of simpler versions of itself. What I mean by this will become clearer shortly, when ' show some examples of recursive definitions. One of the most common ways in which recursion appears in daily life is when you postpone completing a task in favor of a simpler task, often o the same type. Here is a good example. An executive has a fancy telephone and receives many calls on it. He is talking to A when B calls. To A he say,, "Would you mind holding for a moment?" Of course he doesn't really car if A minds; he just pushes a button, and switches to B. Now C calls. The same deferment happens to B. This could go on indefinitely, but let us not get too bogged down in our enthusiasm. So let's say the call with C terminates. Then our executive "pops" back up to B, and continues. Meanwhile A is sitting at the other end of the line, drumming his fingernails again some table, and listening to some horrible Muzak piped through the phone lines to placate him ... Now the easiest case is if the call with B simply terminates, and the executive returns to A finally. But it could happen that after the conversation with B is resumed, a new callerDcalls. B is once again pushed onto the stack of waiting callers, and D is taken care of. Aft D is done, back to B, then back to A. This executive is hopelessly mechanical, to be surebut we are illustrating recursion in its most precise form  
 
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Pushing, Popping, and Stacks In the preceding example, I have introduced some basic terminology of recursionat least as seen through the eyes of computer scientists. The terms are push, pop, and stack (or pushdown stack, to be precise) and they are all related. They were introduced in the late 1950's as part of IPL, one of the first languages for Artificial Intelligence. You have already encountered "push" and "pop" in the Dialogue. But I will spell things out anyway. To push means to suspend operations on the task you're currently working on, without forgetting where you areand to take up a new task. The new task is usually said to be "on a lower level" than the earlier task. To pop is the reverseit means to close operations on one level, and to resume operations exactly where you left off, one level higher. But how do you remember exactly where you were on each different level? The answer is, you store the relevant information in a stack. So a stack is just a table telling you such things as (1) where you were in each unfinished task (jargon: the "return address"), (2) what the relevant facts to know were at the points of interruption (jargon: the "variable bindings"). When you pop back up to resume some task, it is the stack which restores your context, so you don't feel lost. In the telephonecall example, the stack tells you who is waiting on each different level, and where you were in the conversation when it was interrupted. By the way, the terms "push", "pop", and "stack" all come from the visual image of cafeteria trays in a stack. There is usually some sort of spring underneath which tends to keep the topmost tray at a constant height, more or less. So when you push a tray onto the stack, it sinks a littleand when you remove a tray from the stack, the stack pops up a little. One more example from daily life. When you listen to a news report on the radio, oftentimes it happens that they switch you to some foreign correspondent. "We now switch you to Sally Swumpley in Peafog, England." Now Sally has got a tape of some local reporter interviewing someone, so after giving a bit of background, she plays it. "I'm Nigel Cadwallader, here on scene just outside of Peafog, where the great robbery took place, and I'm talking with ..." Now you are three levels down. It may turn out that the interviewee also plays a tape of some conversation. It is not too uncommon to go down three levels in real news reports, and surprisingly enough, we scarcely have any awareness of the suspension. It is all kept track of quite easily by our subconscious mind. Probably the reason it is so easy is that each level is extremely different in flavor from each other level. If they were all similar, we would get confused in no time flat. An example of a more complex recursion is, of course, our Dialogue. There, Achilles and the Tortoise appeared on all the different levels. Sometimes they were reading a story in which they appeared as characters. That is when your mind may get a little hazy on what's going on, and you have to concentrate carefully to get things straight. "Let's see, the real Achilles and Tortoise are still up there in Goodfortune's helicopter, but the  
 
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secondary ones are in some Escher pictureand then they found this book and are reading in it, so it's the tertiary Achilles and Tortoise who wandering around inside the grooves of the Little Harmonic Labyrinth. wait a minuteI left out one level somewhere ..." You have to ha conscious mental stack like this in order to keep track of the recursion the Dialogue. (See Fig. 26.)  
 
 
FIGURE 26. Diagram of the structure of the Dialogue Little Harmonic Labyrinth Vertical descents are "pushes"; rises ore "pops". Notice the similarity of this diagram to indentation pattern of the Dialogue. From the diagram it is clear that the initial tension Goodfortune's threatnever was resolved; Achilles and the Tortoise were just left dangling the sky. Some readers might agonize over this unpopped push, while others might not ba eyelash. In the story, Bach's musical labyrinth likewise was cut off too soonbut Achilles d even notice anything funny. Only the Tortoise was aware of the more global dangling tension Stacks in Music While we're talking about the Little Harmonic Labyrinth, we should discuss something which is hinted at, if not stated explicitly in the Dialogue: that hear music recursivelyin particular, that we maintain a mental stack of keys, and that each new modulation pushes a new key onto the stack. implication is further that we want to hear that sequence of keys retrace reverse orderpopping the pushed keys off the stack, one by one, until the tonic is reached. This is an exaggeration. There is a grain of truth to it however. Any reasonably musical person automatically maintains a shallow with two keys. In that "short stack", the true tonic key is held and also most immediate "pseudotonic" (the key the composer is pretending t in). In other words, the most global key and the most local key. That the listener knows when the true tonic is regained, and feels a strong s of "relief". The listener can also distinguish (unlike Achilles) between a local easing of tensionfor example a resolution into the pseudotonic   
 
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and a global resolution. In fact, a pseudoresolution should heighten the global tension, not relieve it, because it is a piece of ironyjust like Achilles' rescue from his perilous perch on the swinging lamp, when all the while you know he and the Tortoise are really awaiting their dire fates at the knife of Monsieur Goodfortune. Since tension and resolution are the heart and soul of music, there are many, many examples. But let us just look at a couple in Bach. Bach wrote many pieces in an "AABB" formthat is, where there are two halves, and each one is repeated. Let's take the gigue from the French Suite no. 5, which is quite typical of the form. Its tonic key is G, and we hear a gay dancing melody which establishes the key of G strongly. Soon, however, a modulation in the Asection leads to the closely related key of D (the dominant). When the Asection ends, we are in the key of D. In fact, it sounds as if the piece has ended in the key of D! (Or at least it might sound that way to Achilles.) But then a strange thing happenswe abruptly jump back to the beginning, back to G, and rehear the same transition into D. But then a strange thing happenswe abruptly jump back to the beginning, back to G, and rehear the same transition into D. Then comes the Bsection. With the inversion of the theme for our melody, we begin in D as if that had always been the tonicbut we modulate back to G after all, which means that we pop back into the tonic, and the Bsection ends properly. Then that funny repetition takes place, jerking us without warning back into D, and letting us return to G once more. Then that funny repetition takes place, jerking us without warning back into D, and letting us return to G once more. The psychological effect of all this key shiftingsome jerky, some smoothis very difficult to describe. It is part of the magic of music that we can automatically make sense of these shifts. Or perhaps it is the magic of Bach that he can write pieces with this kind of structure which have such a natural grace to them that we are not aware of exactly what is happening. The original Little Harmonic Labyrinth is a piece by Bach in which he tries to lose you in a labyrinth of quick key changes. Pretty soon you are so disoriented that you don't have any sense of direction leftyou don't know where the true tonic is, unless you have perfect pitch, or like Theseus, have a friend like Ariadne who gives you a thread that allows you to retrace your steps. In this case, the thread would be a written score. This pieceanother example is the Endlessly Rising Canongoes to show that, as music listeners, we don't have very reliable deep stacks.  
 
Recursion in Language Our mental stacking power is perhaps slightly stronger in language. The grammatical structure of all languages involves setting up quite elaborate pushdown stacks, though, to be sure, the difficulty of understanding a sentence increases sharply with the number of pushes onto the stack. The proverbial German phenomenon of the "verbattheend", about which  
 
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Droll tales of absentminded professors who would begin a sentence, ramble on for an entire lecture, and then finish up by rattling off a string of verbs by which their audience, for whom the stack had long since lost its coherence, would be totally nonplussed, are told, is an excellent example of linguistic pushing and popping. The confusion among the audience outoforder popping from the stack onto which the professor's verbs been pushed, is amusing to imagine, could engender. But in normal ken German, such deep stacks almost never occurin fact, native speaker of German often unconsciously violate certain conventions which force verb to go to the end, in order to avoid the mental effort of keeping track of the stack. Every language has constructions which involve stacks, though usually of a less spectacular nature than German. But there are always of rephrasing sentences so that the depth of stacking is minimal. Recursive Transition Networks The syntactical structure of sentences affords a good place to present a of describing recursive structures and processes: the Recursive Transition Network (RTN). An RTN is a diagram showing various paths which can be followed to accomplish a particular task. Each path consists of a number of nodes, or little boxes with words in them, joined by arcs, or lines with arrows. The overall name for the RTN is written separately at the left, and the and last nodes have the words begin and end in them. All the other nodes contain either very short explicit directions to perform, or else name other RTN's. Each time you hit a node, you are to carry out the direct inside it, or to jump to the RTN named inside it, and carry it out. Let's take a sample RTN, called ORNATE NOUN, which tells how to construct a certain type of English noun phrase. (See Fig. 27a.) If traverse ORNATE NOUN purely horizontally, we begin', then we create ARTICLE, an ADJECTIVE, and a NOUN, then we end. For instance, "the shampoo" or "a thankless brunch". But the arcs show other possibilities such as skipping the article, or repeating the adjective. Thus we co construct "milk", or "big red blue green sneezes", etc. When you hit the node NOUN, you are asking the unknown black I called NOUN to fetch any noun for you from its storehouse of nouns. This is known as a procedure call, in computer science terminology. It means you temporarily give control to a procedure (here, NOUN) which (1) does thing (produces a noun) and then (2) hands control back to you. In above RTN, there are calls on three such procedures: ARTICLE, ADJECTIVE and NOUN. Now the RTN ORNATE NOUN could itself be called from so other RTNfor instance an RTN called SENTENCE. In this case, ORNATE NOUN would produce a phrase such as "the silly shampoo" and d return to the place inside SENTENCE from which it had been called. I quite reminiscent of the way in which you resume where you left off nested telephone calls or nested news reports. However, despite calling this a "recursive transition network", we have  
 
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FIGURE 27. Recursive Transition Networks for ORNATE NOUN and FANCY NOUN.  
 
not exhibited any true recursion so far. Things get recursiveand seemingly circularwhen you go to an RTN such as the one in Figure 27b, for FANCY NOUN. As you can see, every possible pathway in FANCY NOUN involves a call on ORNATE NOUN, so there is no way to avoid getting a noun of some sort or other. And it is possible to be no more ornate than that, coming out merely with "milk" or "big red blue green sneezes". But three of the pathways involve recursive calls on FANCY NOUN itself. It certainly looks as if something is being defined in terms of itself. Is that what is happening, or not? The answer is "yes, but benignly". Suppose that, in the procedure SENTENCE, there is a node which calls FANCY NOUN, and we hit that node. This means that we commit to memory (viz., the stack) the location of that node inside SENTENCE, so we'll know where to return tothen we transfer our attention to the procedure FANCY NOUN. Now we must choose a pathway to take, in order to generate a FANCY NOUN. Suppose we choose the lower of the upper pathwaysthe one whose calling sequence goes: ORNATE NOUN; RELATIVE PRONOUN; FANCY NOUN; VERB.  
 
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So we spit out an ORNATE NOUN: "the strange bagels"; a RELATIVE NOUN: "that"; and now we are suddenly asked for a FANCY NOUN. B are in the middle of FANCY NOUN! Yes, but remember our executive was in the middle of one phone call when he got another one. He n stored the old phone call's status on a stack, and began the new one nothing were unusual. So we shall do the same. We first write down in our stack the node we are at in the outer call on FANCY NOUN, so that we have a "return address"; then we jump t beginning of FANCY NOUN as if nothing were unusual. Now we h~ choose a pathway again. For variety's sake, let's choose the lower pat] ORNATE NOUN; PREPOSITION; FANCY NOUN. That means we produce an ORNATE NOUN (say "the purple cow"), then a PREPOSITION (say “without"), and once again, we hit the recursion. So we hang onto our hats descend one more level. To avoid complexity, let's assume that this the pathway we take is the direct one just ORNATE NOUN. For example: we might get "horns". We hit the node END in this call on FANCY NOUN which amounts to popping out, and so we go to our stack to find the return address. It tells us that we were in the middle of executing FANCY NOUN one level upand so we resume there. This yields "the purple cow without horns". On this level, too, we hit END, and so we pop up once more, this finding ourselves in need of a VERBso let's choose "gobbled". This ends highestlevel call on FANCY NOUN, with the result that the phrase "the strange bagels that the purple cow without horns gobbled" will get passed upwards to the patient SENTENCE, as we pop for the last time. As you see, we didn't get into any infinite regress. The reason is tl least one pathway inside the RTN FANCY NOUN does not involve recursive calls on FANCY NOUN itself. Of course, we could have perversely insisted on always choosing the bottom pathway inside FANCY NOUN then we would never have gotten finished, just as the acronym "GOD” never got fully expanded. But if the pathways are chosen at random, an infinite regress of that sort will not happen. "Bottoming Out" and Heterarchies This is the crucial fact which distinguishes recursive definitions from circular ones. There is always some part of the definition which avoids reference, so that the action of constructing an object which satisfies the definition will eventually "bottom out". Now there are more oblique ways of achieving recursivity in RTNs than by selfcalling. There is the analogue of Escher's Drawing (Fig. 135), where each of two procedures calls the other, but not itself. For example, we could have an RTN named CLAUSE, which calls FANCY NOUN whenever it needs an object for a transitive verb, and conversely, the u path of FANCY NOUN could call RELATIVE PRONOUN and then CLAUSE  
 
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whenever it wants a relative clause. This is an example of indirect recursion. It is reminiscent also of the twostep version of the Epimenides paradox. Needless to say, there can be a trio of procedures which call one another, cyclicallyand so on. There can be a whole family of RTN's which are all tangled up, calling each other and themselves like crazy. A program which has such a structure in which there is no single "highest level", or "monitor", is called a heterarchy (as distinguished from a hierarchy). The term is due, I believe, to Warren McCulloch, one of the first cyberneticists, and a reverent student of brains and minds. Expanding Nodes One graphic way of thinking about RTN's is this. Whenever you are moving along some pathway and you hit a node which calls on an RTN, you "expand" that node, which means to replace it by a very small copy of the RTN it calls (see Fig. 28). Then you proceed into the very small RTN,  
 
 
FIGURE 28. The FANCY NOUN RTN with one node recursively expanded When you pop out of it, you are automatically in the right place in the big one. While in the small one, you may wind up constructing even more miniature RTN's. But by expanding nodes only when you come across them, you avoid the need to make an infinite diagram, even when an RTN calls itself. Expanding a node is a little like replacing a letter in an acronym by the word it stands for. The "GOD" acronym is recursive but has the defector advantagethat you must repeatedly expand the `G'; thus it never bottoms out. When an RTN is implemented as a real computer program, however, it always has at least one pathway which avoids recursivity (direct or indirect) so that infinite regress is not created. Even the most heterarchical program structure bottoms outotherwise it couldn't run! It would just be constantly expanding node after node, but never performing any action.  
 
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Diagram G and Recursive Sequences Infinite geometrical structures can be defined in just this waythat is by expanding node after node. For example, let us define an infinite diagram called "Diagram G". To do so, we shall use an implicit representation. In two nodes, we shall write merely the letter `G', which, however, will stand for an entire copy of Diagram G. In Figure 29a, Diagram G is portrayed implicitly. Now if we wish to see Diagram G more explicitly, we expand each of the two G'sthat is, we replace them by the same diagram, only reduced in scale (see Fig. 29b). This "secondorder" version of Diagram gives us an inkling of what the final, impossibletorealize Diagram G really looks like. In Figure 30 is shown a larger portion of Diagram G, where all the nodes have been numbered from the bottom up, and from left to right. Two extra nodesnumbers  1 and 2 have been inserted at the bottom This infinite tree has some very curious mathematical properties Running up its righthand edge is the famous sequence of Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, discovered around the year 1202 by Leonardo of Pisa, son of Bonaccio, ergo "Filius Bonacci", or "Fibonacci" for short. These numbers are best  
 
FIGURE 29. (a) Diagram G, unexpanded. (c) Diagram H, unexpanded (b) Diagram G, expanded once. (d) Diagram H, expanded once  
 
 
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FIGURE 30. Diagram G, further expanded and with numbered nodes. defined recursively by the pair of formulas FIBO(n) = FIBO(n 1) + FIBO(n2) for n > 2 FIBO(l) = FIBO(2) = 1 Notice how new Fibonacci numbers are defined in terms of previous Fibonacci numbers. We could represent this pair of formulas in an RTN (see Fig. 31).  
 
 
FIGURE 31. An RTN for Fibonacci numbers.  
 
Thus you can calculate FIBO(15) by a sequence of recursive calls on the procedure defined by the RTN above. This recursive definition bottoms out when you hit FIBO(1) or FIBO(2) (which are given explicitly) after you have worked your way backwards through descending values of n. It is slightly awkward to work your way backwards, when you could just as well work your way forwards, starting with FIBO(l) and FIBO(2) and always adding the most recent two values, until you reach FIBO(15). That way you don't need to keep track of a stack. Now Diagram G has some even more surprising properties than this. Its entire structure can be coded up in a single recursive definition, as follows:  
 
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G(n) = n  G(G(n 1)) for n > 0 G(O) = 0 How does this function G(n) code for the treestructure? Quite simply you construct a tree by placing G(n) below n, for all values of n, you recreate Diagram G. In fact, that is how I discovered Diagram G in the place. I was investigating the function G, and in trying to calculate its values quickly, I conceived of displaying the values I already knew in a tree. T surprise, the tree turned out to have this extremely orderly recursive geometrical description. What is more wonderful is that if you make the analogous tree function H(n) defined with one more nesting than G— H(n) = n  H(H(H(n  1))) for n > 0 H(0) = 0 then the associated "Diagram H" is defined implicitly as shown in Figure 29c. The righthand trunk contains one more node; that is the difference. The first recursive expansion of Diagram H is shown in Figure 29d. And so it goes, for any degree of nesting. There is a beautiful regularity to the recursive geometrical structures, which corresponds precisely to the recursive algebraic definitions. A problem for curious readers is: suppose you flip Diagram G around as if in a mirror, and label the nodes of the new tree so they increase left to right. Can you find a recursive algebraic definition for this "fliptree. What about for the "flip" of the Htree? Etc.? Another pleasing problem involves a pair of recursively intertwined functions F(n) and M(n)  "married" functions, you might say  defined this way: F(n) = n  M(F(n 1)) For n > 0 M(n) = n  F(M(n 1)) F(0) = 1, and M(0) = 0 The RTN's for these two functions call each other and themselves as well. The problem is simply to discover the recursive structures of Diagram F; and Diagram M. They are quite elegant and simple. A Chaotic Sequence One last example of recursion in number theory leads to a small my Consider the following recursive definition of a function: Q(n) = Q(n  Q(n 1)) + Q(n  Q(n2)) for n > 2 Q(1) = Q(2) = 1.  
 
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It is reminiscent of the Fibonacci definition in that each new value is a sum of two previous valuesbut not of the immediately previous two values. Instead, the two immediately previous values tell how far to count back to obtain the numbers to be added to make the new value! The first 17 Qnumbers run as follows: 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, . . . . .. . 5 + 6 = 11 how far to move to the left New term To obtain the next one, move leftwards (from the three dots) respectively 10 and 9 terms; you will hit a 5 and a 6, shown by the arrows. Their sum1 lyields the new value: Q(18). This is the strange process by which the list of known Qnumbers is used to extend itself. The resulting sequence is, to put it mildly, erratic. The further out you go, the less sense it seems to make. This is one of those very peculiar cases where what seems to be a somewhat natural definition leads to extremely puzzling behavior: chaos produced in a very orderly manner. One is naturally led to wonder whether the apparent chaos conceals some subtle regularity. Of course, by definition, there is regularity, but what is of interest is whether there is another way of characterizing this sequenceand with luck, a nonrecursive way. Two Striking Recursive Graphs The marvels of recursion in mathematics are innumerable, and it is not my purpose to present them all. However, there are a couple of particularly striking examples from my own experience which I feel are worth presenting. They are both graphs. One came up in the course of some numbertheoretical investigations. The other came up in the course of my Ph.D. thesis work, in solid state physics. What is truly fascinating is that the graphs are closely related. The first one (Fig. 32) is a graph of a function which I call INT(x). It is plotted here for x between 0 and 1. For x between any other pair of integers n and n + 1, you just find INT(xn), then add n back. The structure of the plot is quite jumpy, as you can see. It consists of an infinite number of curved pieces, which get smaller and smaller towards the cornersand incidentally, less and less curved. Now if you look closely at each such piece, you will find that it is actually a copy of the full graph, merely curved! The implications are wild. One of them is that the graph of INT consists of nothing but copies of itself, nested down infinitely deeply. If you pick up any piece of the graph, no matter how small, you are holding a complete copy of the whole graphin fact, infinitely many copies of it! The fact that INT consists of nothing but copies of itself might make you think it is too ephemeral to exist. Its definition sounds too circular.  
 
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FIGURE 32. Graph of the function INT(x). There is a jump discontinuity at every rat value of x.  
 
How does it ever get off the ground? That is a very interesting matter. main thing to notice is that, to describe INT to someone who hasn't see it will not suffice merely to say, "It consists of copies of itself." The o half of the storythe nonrecursive halftells where those copies lie in the square, and how they have been deformed, relative to the full graph. Only the combination of these two aspects of INT will specify structure of INT. It is exactly as in the definition of Fibonacci number where you need two linesone to define the recursion, the other to de the bottom (i.e., the values at the beginning). To be very concrete, if make one of the bottom values 3 instead of 1, you will produce a completely different sequence, known as the Lucas sequence: 1, 3, 4 , 7, 11, 18, 29, 47, 76, 123, .. . the "bottom" 29 + 47 = 76 same recursive rule as for the Fibonacci numbers  
 
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What corresponds to the bottom in the definition of INT is a picture (Fig. 33a) composed of many boxes, showing where the copies go, and how they are distorted. I call it the "skeleton" of INT. To construct INT from its skeleton, you do the following. First, for each box of the skeleton, you do two operations: (1) put a small curved copy of the skeleton inside the box, using the curved line inside it as a guide; (2) erase the containing box and its curved line. Once this has been done for each box of the original skeleton, you are left with many "baby" skeletons in place of one big one. Next you repeat the process one level down, with all the baby skeletons. Then again, again, and again ... What you approach in the limit is an exact graph of INT, though you never get there. By nesting the skeleton inside itself over and over again, you gradually construct the graph of INT "from out of nothing". But in fact the "nothing" was not nothingit was a picture. To see this even more dramatically, imagine keeping the recursive part of the definition of INT, but changing the initial picture, the skeleton. A variant skeleton is shown in Figure 33b, again with boxes which get smaller and smaller as they trail off to the four corners. If you nest this second skeleton inside itself over and over again, you will create the key graph from my Ph.D. thesis, which I call Gplot (Fig. 34). (In fact, some complicated distortion of each copy is needed as wellbut nesting is the basic idea.). Gplot is thus a member of the INTfamily. It is a distant relative, because its skeleton is quite different fromand considerably more complex thanthat of INT. However, the recursive part of the definition is identical, and therein lies the family tie. I should not keep you too much in the dark about the origin of these beautiful graphs. INTstanding for "interchange"comes from a problem involving "Etasequences", which are related to continued fractions. The basic idea behind INT is that plus and minus signs are interchanged in a certain kind of continued fraction. As a consequence, INT(INT(x)) = x. INT has the property that if x is rational, so is INT(x); if x is quadratic, so is INT(x). I do not know if this trend holds for higher algebraic degrees. Another lovely feature of INT is that at all rational values of x, it has a jump discontinuity, but at all irrational values of x, it is continuous. Gplot comes from a highly idealized version of the question, "What are the allowed energies of electrons in a crystal in a magnetic field?" This problem is interesting because it is a cross between two very simple and fundamental physical situations: an electron in a perfect crystal, and an electron in a homogeneous magnetic field. These two simpler problems are both well understood, and their characteristic solutions seem almost incompatible with each other. Therefore, it is of quite some interest to see how nature manages to reconcile the two. As it happens, the crystal withoutmagneticfield situation and the magneticfieldwithoutcrystal situation do have one feature in common: in each of them, the electron behaves periodically in time. It turns out that when the two situations are combined, the ratio of their two time periods is the key parameter. In fact, that ratio holds all the information about the distribution of allowed electron energiesbut it only gives up its secret upon being expanded into a continued fraction.  
 
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Gplot shows that distribution. The horizontal axis represents energy, and the vertical axis represents the abovementioned ratio of time periods, which we can call "±". At the bottom, a is zero, and at the top a is unity. When a is zero, there is no magnetic field. Each of the line segments making up Gplot is an "energy band"that is, it represents allowed values of energy. The empty swaths traversing Gplot on all different size scales are therefore regions of forbidden energy. One of the most startling properties of Gplot is that when a is rational (say p/q in lowest terms), there are exactly q such bands (though when q is even, two of them "kiss" in the middle). And when a is irrational, the bands shrink to points, of which there are infinitely many, very sparsely distributed in a socalled "Cantor set"  another recursively defined entity which springs up in topology. You might well wonder whether such an intricate structure would ever show up in an experiment. Frankly, I would be the most surprised person in the world if Gplot came out of any experiment. The physicality of Gplot lies in the fact that it points the way to the proper mathematical treatment of less idealized problems of this sort. In other words, Gplot is purely a contribution to theoretical physics, not a hint to experimentalists as to what to expect to see! An agnostic friend of mine once was so struck by Gplot's infinitely many infinities that he called it "a picture of God", which I don't think is blasphemous at all. Recursion at the Lowest Level of Matter We have seen recursion in the grammars of languages, we have seen recursive geometrical trees which grow upwards forever, and we have seen one way in which recursion enters the theory of solid state physics. Now we are going to see yet another way in which the whole world is built out of recursion. This has to do with the structure of elementary particles: electrons, protons, neutrons, and the tiny quanta of electromagnetic radiation called "photons". We are going to see that particles arein a certain sense which can only be defined rigorously in relativistic quantum mechanics nested inside each other in a way which can be described recursively, perhaps even by some sort of "grammar". We begin with the observation that if particles didn't interact with each other, things would be incredibly simple. Physicists would like such a world because then they could calculate the behavior of all particles easily (if physicists in such a world existed, which is a doubtful proposition). Particles without interactions are called bare particles, and they are purely hypothetical creations; they don't exist. Now when you "turn on" the interactions, then particles get tangled up together in the way that functions F and M are tangled together, or married people are tangled together. These real particles are said to be renormalizedan ugly but intriguing term. What happens is that no particle can even be defined without referring to all other particles, whose definitions in turn depend on the first particles, etc. Round and round, in a neverending loop.  
 
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Figure 34. Gplot; a recursive graph, showing energy bands for electrons in an idealized crystal in a magnetic field, ± representing magnetic field strength, runs vertically from 0 to 1. Energy runs horizontally. The horizontal line segments are bands of allowed electron energies.  
 
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Let us be a little more concrete, now. Let's limit ourselves to only two kinds of particles: electrons and photons. We'll also have to throw in the electron's antiparticle, the positron. (Photons are their own antiparticles.) Imagine first a dull world where a bare electron wishes to propagate from point A to point B, as Zeno did in my ThreePart Invention. A physicist would draw a picture like this:  
 
 
There is a mathematical expression which corresponds to this line and its endpoints, and it is easy to write down. With it, a physicist can understand the behavior of the bare electron in this trajectory. Now let us "turn on" the electromagnetic interaction, whereby electrons and photons interact. Although there are no photons in the scene, there will nevertheless be profound consequences even for this simple trajectory. In particular, our electron now becomes capable of emitting and then reabsorbing virtual photonsphotons which flicker in and out of existence before they can be seen. Let us show one such process:  
 
 
Now as our electron propagates, it may emit and reabsorb one photon after another, or it may even nest them, as shown below:  
 
 
The mathematical expressions corresponding to these diagramscalled "Feynman diagrams"are easy to write down, but they are harder to calculate than that for the bare electron. But what really complicates matters is that a photon (real or virtual) can decay for a brief moment into an electronpositron pair. Then these two annihilate each other, and, as if by magic, the original photon reappears. This sort of process is shown below:  
 
 
The electron has a rightpointing arrow, while the positron's arrow points leftwards.  
 
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As you might have anticipated, these virtual processes can be inside each other to arbitrary depth. This can give rise to some complicatedlooking drawings, such as the one in Figure 35. In that man diagram, a single electron enters on the left at A, does some an acrobatics, and then a single electron emerges on the right at B. outsider who can't see the inner mess, it looks as if one electron peacefully sailed from A to B. In the diagram, you can see how el lines can get arbitrarily embellished, and so can the photon lines diagram would be ferociously hard to calculate.  
 
 
FIGURE 35. A Feynman diagram showing the propagation of a renormalized electron from A to B. In this diagram, time increases to the right. Therefore, in the segments where the electron’s arrow points leftwards, it is moving "backwards in time". A more intuitive way to say this is that an antielectron (positron) is moving forwards in time. Photons are their own antiparticles; hence their lines have no need of arrows. There is a sort of "grammar" to these diagrams, that only certain pictures to be realized in nature. For instance, the one be impossible:  
 
 
You might say it is not a "wellformed" Feynman diagram. The gram a result of basic laws of physics, such as conservation of energy, conservation of electric charge, and so on. And, like the grammars of l  languages, this grammar has a recursive structure, in that it allow' nestings of structures inside each other. It would be possible to drat set of recursive transition networks defining the "grammar" of the electromagnetic interaction. When bare electrons and bare photons are allowed to interact ii arbitrarily tangled ways, the result is renormalized electrons and ph Thus, to understand how a real, physical electron propagates from A to B,  
 
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the physicist has to be able to take a sort of average of all the infinitely many different possible drawings which involve virtual particles. This is Zeno with a vengeance! Thus the point is that a physical particlea renormalized particle involves (1) a bare particle and (2) a huge tangle of virtual particles, inextricably wound together in a recursive mess. Every real particle's existence therefore involves the existence of infinitely many other particles, contained in a virtual "cloud" which surrounds it as it propagates. And each of the virtual particles in the cloud, of course, also drags along its own virtual cloud, and so on ad infinitum. Particle physicists have found that this complexity is too much to handle, and in order to understand the behavior of electrons and photons, they use approximations which neglect all but fairly simple Feynman diagrams. Fortunately, the more complex a diagram, the less important its contribution. There is no known way of summing up all of the infinitely many possible diagrams, to get an expression for the behavior of a fully renormalized, physical electron. But by considering roughly the simplest hundred diagrams for certain processes, physicists have been able to predict one value (the socalled gfactor of the muon) to nine decimal places  correctly! Renormalization takes place not only among electrons and photons. Whenever any types of particle interact together, physicists use the ideas of renormalization to understand the phenomena. Thus protons and neutrons, neutrinos, pimesons, quarksall the beasts in the subnuclear zoo they all have bare and renormalized versions in physical theories. And from billions of these bubbles within bubbles are all the beasts and baubles of the world composed. Copies and Sameness Let us now consider Gplot once again. You will remember that in the Introduction, we spoke of different varieties of canons. Each type of canon exploited some manner of taking an original theme and copying it by an isomorphism, or informationpreserving transformation. Sometimes the copies were upside down, sometimes backwards, sometimes shrunken or expanded ... In Gplot we have all those types of transformation, and more. The mappings between the full Gplot and the "copies" of itself inside itself involve size changes, skewings, reflections, and more. And yet there remains a sort of skeletal identity, which the eye can pick up with a bit of effort, particularly after it has practiced with INT. Escher took the idea of an object's parts being copies of the object itself and made it into a print: his woodcut Fishes and Scales (Fig. 36). Of course these fishes and scales are the same only when seen on a sufficiently abstract plane. Now everyone knows that a fish's scales aren't really small copies of the fish; and a fish's cells aren't small copies of the fish; however, a fish's DNA, sitting inside each and every one of the fish's cells, is a very convo  
 
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FIGURE 36. Fish and Scales, by M. C. Escher (woodcut, 1959). luted "copy" of the entire fishand so there is more than a grain of truth to the Escher picture. What is there that is the "same" about all butterflies? The mapping from one butterfly to another does not map cell onto cell; rather, it m; functional part onto functional part, and this may be partially on a macroscopic scale, partially on a microscopic scale. The exact proportions of pa are not preserved; just the functional relationships between parts. This is the type of isomorphism which links all butterflies in Escher's wood engraving Butterflies (Fig. 37) to each other. The same goes for the more abstract butterflies of Gplot, which are all linked to each other by mathematical mappings that carry functional part onto functional part, but totally ignore exact line proportions, angles, and so on. Taking this exploration of sameness to a yet higher plane of abstraction, we might well ask, "What is there that is the `same' about all Esc l drawings?" It would be quite ludicrous to attempt to map them piece by piece onto each other. The amazing thing is that even a tiny section of an  
 
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FIGURE 37. Butterflies, by M. C. Escher (woodengraving, 1950). Escher drawing or a Bach piece gives it away. Just as a fish's DNA is contained inside every tiny bit of the fish, so a creator's "signature" is contained inside every tiny section of his creations. We don't know what to call it but "style"  a vague and elusive word. We keep on running up against "samenessindifferentness", and the question When are two things the same?  
 
It will recur over and over again in this book. We shall come at it from all sorts of skew angles, and in the end, we shall see how deeply this simple question is connected with the nature of intelligence. That this issue arose in the Chapter on recursion is no accident, for recursion is a domain where "samenessindifferentness" plays a central role. Recursion is based on the "same" thing happening on several differ  
 
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ent levels at once. But the events on different levels aren't exactly samerather, we find some invariant feature in them, despite many s in which they differ. For example, in the Little Harmonic Labyrinth, all stories on different levels are quite unrelatedtheir "sameness" reside only two facts: (1) they are stories, and (2) they involve the Tortoise and Achilles. Other than that, they are radically different from each other. Programming and Recursion: Modularity, Loops, Procedures One of the essential skills in computer programming is to perceive wl two processes are the same in this extended sense, for that leads modularizationthe breakingup of a task into natural subtasks. For stance, one might want a sequence of many similar operations to be cart out one after another. Instead of writing them all out, one can write a h which tells the computer to perform a fixed set of operations and then loop back and perform them again, over and over, until some condition is satisfied. Now the body of the loopthe fixed set of instructions to repeatedneed not actually be completely fixed. It may vary in so predictable way. An example is the most simpleminded test for the primality o natural number N, in which you begin by trying to divide N by 2, then 3, 4, 5, etc. until N  1. If N has survived all these tests without be divisible, it's prime. Notice that each step in the loop is similar to, but i the same as, each other step. Notice also that the number of steps varies with Nhence a loop of fixed length could never work as a general test primality. There are two criteria for "aborting" the loop: (1) if so number divides N exactly, quit with answer "NO"; (2) if N  1 is react as a test divisor and N survives, quit with answer "YES". The general idea of loops, then, is this: perform some series of related steps over and over, and abort the process when specific conditions are n Now sometimes, the maximum number of steps in a loop will be known advance; other times, you just begin, and wait until it is aborted. The second type of loop  which I call a free loop  is dangerous, because criterion for abortion may never occur, leaving the computer in a socal "infinite loop". This distinction between bounded loops and free loops is one the most important concepts in all of computer science, and we shall dev an entire Chapter to it: "BlooP and FlooP and G1ooP". Now loops may be nested inside each other. For instance, suppose t we wish to test all the numbers between 1 and 5000 for primality. We c write a second loop which uses the abovedescribed test over and over starting with N = I and finishing with N = 5000. So our program i have a "looptheloop" structure. Such program structures are typical – in fact they are deemed to be good programming style. This kind of nest loop also occurs in assembly instructions for commonplace items, and such activities as knitting or crochetingin which very small loops are  
 
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repeated several times in larger loops, which in turn are carried out repeatedly ... While the result of a lowlevel loop might be no more than couple of stitches, the result of a highlevel loop might be a substantial portion of a piece of clothing. In music, too, nested loops often occuras, for instance, when a scale (a small loop) is played several times in a row, perhaps displaced in pitch each new time. For example, the last movements of both the Prokofiev fifth piano concerto and the Rachmaninoff second symphony contain extended passages in which fast, medium, and slow scaleloops are played simultaneously by different groups of instruments, to great effect. The Prokofiev scales go up; the Rachmaninoffscales, down. Take your pick. A more general notion than loop is that of subroutine, or procedure, which we have already discussed somewhat. The basic idea here is that a group of operations are lumped together and considered a single unit with a namesuch as the procedure ORNATE NOUN. As we saw in RTN's, procedures can call each other by name, and thereby express very concisely sequences of operations which are to be carried out. This is the essence of modularity in programming. Modularity exists, of course, in hifi systems, furniture, living cells, human societywherever there is hierarchical organization. More often than not, one wants a procedure which will act variably, according to context. Such a procedure can either be given a way of peering out at what is stored in memory and selecting its actions accordingly, or it can be explicitly fed a list of parameters which guide its choice of what actions to take. Sometimes both of these methods are used. In RTN terminology, choosing the sequence of actions to carry out amounts to choosing which pathway to follow. An RTN which has been souped up with parameters and conditions that control the choice of pathways inside it is called an Augmented Transition Network (ATN). A place where you might prefer ATN's to RTN's is in producing sensibleas distinguished from nonsensicalEnglish sentences out of raw words, according to a grammar represented in a set of ATN's. The parameters and conditions would allow you to insert various semantic constraints, so that random juxtapositions like "a thankless brunch" would be prohibited. More on this in Chapter XVIII, however.  
 
Recursion in Chess Programs A classic example of a recursive procedure with parameters is one for choosing the "best" move in chess. The best move would seem to be the one which leaves your opponent in the toughest situation. Therefore, a test for goodness of a move is simply this: pretend you've made the move, and now evaluate the board from the point of view of your opponent. But how does your opponent evaluate the position? Well, he looks for his best move. That is, he mentally runs through all possible moves and evaluates them from what he thinks is your point of view, hoping they will look bad to you. But  
 
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notice that we have now defined "best move" recursively, simply maxim that what is best for one side is worst for the other. The procedure which looks for the best move operates by trying a move and then calling on itself in the role of opponent! As such, it tries another n calls on itself in the role of its opponent's opponentthat is, its This recursion can go several levels deepbut it's got to bottom out somewhere! How do you evaluate a board position without looking There are a number of useful criteria for this purpose, such as si number of pieces on each side, the number and type of pieces undo the control of the center, and so on. By using this kind of evaluation at the bottom, the recursive movegenerator can pop back upwards an( evaluation at the top level of each different move. One of the parameters in the selfcalling, then, must tell how many moves to look ahead. TI most call on the procedure will use some externally set value parameter. Thereafter, each time the procedure recursively calls must decrease this lookahead parameter by 1. That way, w parameter reaches zero, the procedure will follow the alternate pathway  the nonrecursive evaluation. In this kind of gameplaying program, each move investigate the generation of a socalled "lookahead tree", with the move trunk, responses as main branches, counterresponses as subsidiary branches, and so on. In Figure 38 I have shown a simple lookahead tree depicting the start of a tictartoe game. There is an art to figuring to avoid exploring every branch of a lookahead tree out to its tip. trees, peoplenot computersseem to excel at this art; it is known that toplevel players look ahead relatively little, compared to most chess programs – yet the people are far better! In the early days of compute people used to estimate that it would be ten years until a computer (or FIGURE 38. The branching tree of moves and countermoves at the start of c tictactoe.  
 
 
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program) was world champion. But after ten years had passed, it seemed that the day a computer would become world champion was still more than ten years away ... This is just one more piece of evidence for the rather recursive Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law. Recursion and Unpredictability Now what is the connection between the recursive processes of this Chapter, and the recursive sets of the preceding Chapter? The answer involves the notion of a recursively enumerable set. For a set to be r.e. means that it can be generated from a set of starting points (axioms), by the repeated application of rules of inference. Thus, the set grows and grows, each new element being compounded somehow out of previous elements, in a sort of "mathematical snowball". But this is the essence of recursionsomething being defined in terms of simpler versions of itself, instead of explicitly. The Fibonacci numbers and the Lucas numbers are perfect examples of r.e. setssnowballing from two elements by a recursive rule into infinite sets. It is just a matter of convention to call an r.e. set whose complement is also r.e. "recursive". Recursive enumeration is a process in which new things emerge from old things by fixed rules. There seem to be many surprises in such processesfor example the unpredictability of the Qsequence. It might seem that recursively defined sequences of that type possess some sort of inherently increasing complexity of behavior, so that the further out you go, the less predictable they get. This kind of thought carried a little further suggests that suitably complicated recursive systems might be strong enough to break out of any predetermined patterns. And isn't this one of the defining properties of intelligence? Instead of just considering programs composed of procedures which can recursively call themselves, why not get really sophisticated, and invent programs which can modify themselvesprograms which can act on programs, extending them, improving them, generalizing them, fixing them, and so on? This kind of "tangled recursion" probably lies at the heart of intelligence.  
 
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Canon by Intervallic Augmentation Achilles and the Tortoise have just finished a delicious Chinese banquet for two, at the best Chinese restaurant in town.  
 
Achilles: You wield a mean chopstick, Mr. T. Tortoise: I ought to. Ever since my youth, I have had a fondness for Oriental cuisine. And youdid you enjoy your meal, Achilles? Achilles: Immensely. I'd not eaten Chinese food before. This meal was a splendid introduction. And now, are you in a hurry to go, or shall we just sit here and talk a little while? Tortoise: I'd love to talk while we drink our tea. Waiter! (A waiter comes up.) Could we have our bill, please, and some more tea? (The waiter rushes off.) Achilles: You may know more about Chinese cuisine than I do, Mr.T, I'll bet I know more about Japanese poetry than you do. Have you ever read any haiku? Tortoise: I'm afraid not. What is a haiku? Achilles: A haiku is a Japanese seventeensyllable poemor minipoem rather, which is evocative in the same way, perhaps, as a fragrant petal is, or a lily pond in a light drizzle. It generally consists of groups of: of five, then seven, then five syllables. Tortoise: Such compressed poems with seventeen syllables can't much meaning ... Achilles: Meaning lies as much in the mind of the reader as i haiku. Tortoise: Hmm ... That's an evocative statement. (The waiter arrives with their bill, another pot of tea, and two fortune cookies.) Thank you, waiter. Care for more tea, Achilles? Achilles: Please. Those little cookies look delicious. (Picks one up, bites I into it and begins to chew.) Hey! What's this funny thing inside? A piece of paper? Tortoise: That's your fortune, Achilles. Many Chinese restaurants give out fortune cookies with their bills, as a way of softening the blow. I frequent Chinese restaurants, you come to think of fortune cookies  
 
 
less as cookies than as message bearers Unfortunately you seem to have swallowed some of your fortune. What does the rest say? Achilles: It's a little strange, for all the letters are run together, with no spaces in between. Perhaps it needs decoding in some way? Oh, now I see. If you put the spaces back in where they belong, it says, "ONE WAR TWO EAR EWE". I can't quite make head or tail of that. Maybe it was a haikulike poem, of which I ate the majority of syllables. Tortoise: In that case, your fortune is now a mere 5/17haiku. And a curious image it evokes. If 5/17haiku is a new art form, then I'd say woe, 0, woe are we ... May I look at it? Achilles (handing the Tortoise the small slip of paper): Certainly. Tortoise: Why, when I "decode" it, Achilles, it comes out completely different! It's not a 5/17haiku at all. It is a sixsyllable message which says, "0 NEW ART WOE ARE WE". That sounds like an insightful commentary on the new art form of 5/17haiku. Achilles: You're right. Isn't it astonishing that the poem contains its own commentary! Tortoise: All I did was to shift the reading frame by one unitthat is, shift all the spaces one unit to the right. Achilles: Let's see what your fortune says, Mr. Tortoise. Tortoise (deftly splitting open his cookie, reads): "Fortune lies as much in the hand of the eater as in the cookie." Achilles: Your fortune is also a haiku, Mr. Tortoiseat least it's got seventeen syllables in the 575 form. Tortoise: Glory be! I would never have noticed that, Achilles. It's the kind of thing only you would have noticed. What struck me more is what it sayswhich, of course, is open to interpretation. Achilles: I guess it just shows that each of us has his own characteristic way of interpreting messages which we run across ... (Idly, Achilles gazes at the tea leaves on the bottom of his empty teacup.) Tortoise: More tea, Achilles? Achilles: Yes, thank you. By the way, how is your friend the Crab? I have been thinking about him a lot since you told me of your peculiar phonographbattle. Tortoise: I have told him about you, too, and he is quite eager to meet you. He is getting along just fine. In fact, he recently made a new acquisition in the record player line: a rare type of jukebox. Achilles: Oh, would you tell me about it? I find jukeboxes, with their flashing colored lights and silly songs, so quaint and reminiscent of bygone eras. Tortoise: This jukebox is too large to fit in his house, so he had a shed specially built in back for it. Achilles: I can't imagine why it would be so large, unless it has an unusually large selection of records. Is that it? Tortoise: As a matter of fact, it has exactly one record.  
 
 
Achilles: What? A jukebox with only one record? That's a contradiction in terms. Why is the jukebox so big, then? Is its single record gigantic  twenty feet in diameter? Tortoise: No, it's just a regular jukeboxstyle record. Achilles: Now, Mr. Tortoise, you must be joshing me. After all, what I of a jukebox is it that has only a single song? Tortoise: Who said anything about a single song, Achilles? Achilles: Every jukebox I've ever run into obeyed the fundamental jukeboxaxiom: "One record, one song". Tortoise: This jukebox is different, Achilles. The one record sits vertically, suspended, and behind it there is a small but elaborate network of overhead rails, from which hang various record players. When push a pair of buttons, such as B1, that selects one of the record players. This triggers an automatic mechanism that starts the record player squeakily rolling along the rusty tracks. It gets shunted alongside the recordthen it clicks into playing position. Achilles: And then the record begins spinning and music comes out  right? Tortoise: Not quite. The record stands stillit's the record player which rotates. Achilles: I might have known. But how, if you have but one record to play can you get more than one song out of this crazy contraption? Tortoise: I myself asked the Crab that question. He merely suggested I try it out. So I fished a quarter from my pocket (you get three plays for a quarter), stuffed it in the slot, and hit buttons B1, then C3 then B10all just at random. Achilles: So phonograph B1 came sliding down the rail, I suppose, plugged itself into the vertical record, and began spinning? Tortoise: Exactly. The music that came out was quite agreeable, based the famous old tune BACH, which I believe you remember.  
 
 
Achilles: Could I ever forget it? Tortoise: This was record player B1. Then it finished, and was s rolled back into its hanging position, so that C3 could be slid into position. Achilles: Now don't tell me that C3 played another song? Tortoise: It did just that. Achilles: Ah, I understand. It played the flip side of the first song, or another band on the same side. Tortoise: No, the record has grooves only on one side, and has only a single band.  
 
 
Achilles: I don't understand that at all. You CAN'T pull different songs out of the same record! Tortoise: That's what I thought until I saw Mr. Crab's jukebox. Achilles: How did the second song go? Tortoise: That's the interesting thing ... It was a song based on the melody CAGE. Achilles: That's a totally different melody! Tortoise: True. Achilles: And isn't John Cage a composer of modern music? I seem to remember reading about him in one of my books on haiku. Tortoise: Exactly. He has composed many celebrated pieces, such as 4'33", a threemovement piece consisting of silences of different lengths. It's wonderfully expressiveif you like that sort of thing. Achilles: I can see where if I were in a loud and brash cafe I might gladly pay to hear Cage's 4'33" on a jukebox. It might afford some relief! Tortoise: Rightwho wants to hear the racket of clinking dishes and jangling silverware? By the way, another place where 4'33" would come in handy is the Hall of Big Cats, at feeding time. Achilles: Are you suggesting that Cage belongs in the zoo? Well, I guess that makes some sense. But about the Crab's jukebox ... I am baffled. How could both "BACH" and "CAGE" be coded inside a single record at once? Tortoise: You may notice that there is some relation between the two, Achilles, if you inspect them carefully. Let me point the way. What do you get if you list the successive intervals in the melody BACH? Achilles: Let me see. First it goes down one semitone, from B to A (where B is taken the German way); then it rises three semitones to C; and finally it falls one semitone, to H. That yields the pattern: 1, +3, 1. Tortoise: Precisely. What about CAGE, now? Achilles: Well, in this case, it begins by falling three semitones, then ten semitones (nearly an octave), and finally falls three more semitones. That means the pattern is: 3, +10, 3. It's very much like the other one, isn't it? Tortoise: Indeed it is. They have exactly the same "skeleton", in a certain sense. You can make CAGE out of BACH by multiplying all the intervals by 31/3, and taking the nearest whole number. Achilles: Well, blow me down and pick me up! So does that mean that only  
 
 
some sort of skeletal code is present in the grooves, and that the various record players add their own interpretations to that code? Tortoise: I don't know, for sure. The cagey Crab wouldn't fill me in on the details. But I did get to hear a third song, when record player B1 swiveled into place. Achilles: How did it go? Tortoise: The melody consisted of enormously wide intervals, and we BCAH.  
 
 
The interval pattern in semitones was: 10, +33, 10. It can be gotten from the CAGE pattern by yet another multiplication by 3%3, and rounding to whole numbers. Achilles: Is there a name for this kind of interval multiplication? Tortoise: One could call it "intervallic augmentation". It is similar to tl canonic device of temporal augmentation, where all the time values notes in a melody get multiplied by some constant. There, the effect just to slow the melody down. Here, the effect is to expand the melodic range in a curious way. Achilles: Amazing. So all three melodies you tried were intervallic augmentations of one single underlying groovepattern in the record: Tortoise: That's what I concluded. Achilles: I find it curious that when you augment BACH you get CAGE and when you augment CAGE over again, you get BACH back, except jumbled up inside, as if BACH had an upset stomach after passing through the intermediate stage of CAGE. Tortoise: That sounds like an insightful commentary on the new art form of Cage.  
 
 
CHAPTER VI  
 
The Location of Meaning When Is One Thing Not Always the Same? LAST CHAPTER, WE came upon the question, "When are two things the same?" In this Chapter, we will deal with the flip side of that question: "When is one thing not always the same?" The issue we are broaching is whether meaning can be said to be inherent in a message, or whether meaning is always manufactured by the interaction of a mind or a mechanism with a messageas in the preceding Dialogue. In the latter case, meaning could not said to be located in any single place, nor could it be said that a message has any universal, or objective, meaning, since each observer could bring its own meaning to each message. But in the former case, meaning would have both location and universality. In this Chapter, I want to present the case for the universality of at least some messages, without, to be sure, claiming it for all messages. The idea of an "objective meaning" of a message will turn out to be related, in an interesting way, to the simplicity with which intelligence can be described. InformationBearers and Information Revealers I'll begin with my favorite example: the relationship between records, music, and record players. We feel quite comfortable with the idea that a record contains the same information as a piece of music, because of the existence of record players, which can "read" records and convert the groovepatterns into sounds. In other words, there is an isomorphism between groovepatterns and sounds, and the record player is a mechanism which physically realizes that isomorphism. It is natural, then, to think of the record as an informationbearer, and the recordplayer as an informationrevealer. A second example of these notions is given by the pqsystem. There, the "informationbearers" are the theorems, and the "informationrevealer" is the interpretation, which is so transparent that we don't need any electrical machine to help us extract the information from pqtheorems. One gets the impression from these two examples that isomorphisms and decoding mechanisms (i.e., informationrevealers) simply reveal information which is intrinsically inside the structures, waiting to be "pulled out". This leads to the idea that for each structure, there are certain pieces of information which can be pulled out of it, while there are other pieces of information which cannot be pulled out of it. But what does this phrase  
 
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"pull out" really mean? How hard are you allowed to pull? There are c where by investing sufficient effort, you can pull very recondite piece of information out of certain structures. In fact, the pullingout may inv such complicated operations that it makes you feel you are putting in n information than you are pulling out. Genotype and Phenotype Take the case of the genetic information commonly said to reside in double helix of deoxyribonucleic acid (DNA). A molecule of DNA – a genotypeis converted into a physical organisma phenotypeby a complex process, involving the manufacture of proteins, the replication the DNA, the replication of cells, the gradual differentiation of cell types and so on. Incidentally, this unrolling of phenotype from genotype epigenesisis the most tangled of tangled recursions, and in Chapter we shall devote our full attention to it. Epigenesis is guided by a se enormously complex cycles of chemical reactions and feedback loops the time the full organism has been constructed, there is not even remotest similarity between its physical characteristics and its genotype. And yet, it is standard practice to attribute the physical structure of organism to the structure of its DNA, and to that alone. The first evidence for this point of view came from experiments conducted by Oswald A, in 1946, and overwhelming corroborative evidence has since been amassed Avery's experiments showed that, of all the biological molecules, only E transmits hereditary properties. One can modify other molecules it organism, such as proteins, but such modifications will not be transmitted to later generations. However, when DNA is modified, all successive generations inherit the modified DNA. Such experiments show that the only of changing the instructions for building a new organism is to change DNAand this, in turn, implies that those instructions must be cc somehow in the structure of the DNA. Exotic and Prosaic Isomorphisms Therefore one seems forced into accepting the idea that the DNA's structure contains the information of the phenotype's structure, which is to the two are isomorphic. However, the isomorphism is an exotic one, by w] I mean that it is highly nontrivial to divide the phenotype and genotype into "parts" which can be mapped onto each other. Prosaic isomorphic by contrast, would be ones in which the parts of one structure are easily mappable onto the parts of the other. An example is the isomorphism between a record and a piece of music, where one knows that to any so in the piece there exists an exact "image" in the patterns etched into grooves, and one could pinpoint it arbitrarily accurately, if the need arose Another prosaic isomorphism is that between Gplot and any of its internal butterflies.  
 
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The isomorphism between DNA structure and phenotype structure is anything but prosaic, and the mechanism which carries it out physically is awesomely complicated. For instance, if you wanted to find some piece of your DNA which accounts for the shape of your nose or the shape of your fingerprint, you would have a very hard time. It would be a little like trying to pin down the note in a piece of music which is the carrier of the emotional meaning of the piece. Of course there is no such note, because the emotional meaning is carried on a very high level, by large "chunks" of the piece, not by single notes. Incidentally, such "chunks" are not necessarily sets of contiguous notes; there may be disconnected sections which, taken together, carry some emotional meaning. Similarly, "genetic meaning"that is, information about phenotype structureis spread all through the small parts of a molecule of DNA, although nobody understands the language yet. (Warning: Understanding this "language" would not at all be the same as cracking the Genetic Code, something which took place in the early 1960's. The Genetic Code tells how to translate short portions of DNA into various amino acids. Thus, cracking the Genetic Code is comparable to figuring out the phonetic values of the letters of a foreign alphabet, without figuring out the grammar of the language or the meanings of any of its words. The cracking of the Genetic Code was a vital step on the way to extracting the meaning of DNA strands, but it was only the first on a long path which is yet to be trodden.)  
 
Jukeboxes and Triggers The genetic meaning contained in DNA is one of the best possible examples of implicit meaning. In order to convert genotype into phenotype, a set of mechanisms far more complex than the genotype must operate on the genotype. The various parts of the genotype serve as triggers for those mechanisms. A jukeboxthe ordinary type, not the Crab type!provides a useful analogy here: a pair of buttons specifies a very complex action to be taken by the mechanism, so that the pair of buttons could well be described as "triggering" the song which is played. In the process which converts genotype into phenotype, cellular jukeboxesif you will pardon the notion!accept "buttonpushings" from short excerpts from a long strand of DNA, and the "songs" which they play are often prime ingredients in the creation of further "jukeboxes". It is as if the output of real jukeboxes, instead of being love ballads, were songs whose lyrics told how to build more complex jukeboxes ... Portions of the DNA trigger the manufacture of proteins; those proteins trigger hundreds of new reactions; they in turn trigger the replicatingoperation which, in several steps, copies the DNAand on and on ... This gives a sense of how recursive the whole process is. The final result of these manytriggered triggerings is the phenotypethe individual. And one says that the phenotype is the revelationthe "pullingout"of the information that was present in the DNA to start with, latently. (The term "revelation" in this context is due to  
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Jacques Monod, one of the deepest and most original of twentiethcentury molecular biologists.) Now no one would say that a song coming out of the loudspeaker of jukebox constitutes a "revelation" of information inherent in the pair buttons which were pressed, for the pair of buttons seem to be mere triggers, whose purpose is to activate informationbearing portions of the jukebox mechanism. On the other hand, it seems perfectly reasonable to call t extraction of music from a record a "revelation" of information inherent the record, for several reasons: (1) the music does not seem to be concealed in the mechanism of the record player; (2) it is possible to match pieces of the input (the record) with pieces of the output (the music) to an arbitrary degree of accuracy; (3) it is possible to play other records on the same record player and get other sounds out; (4) the record and the record player are easily separated from one another. It is another question altogether whether the fragments of a smashed record contain intrinsic meaning. The edges of the separate pieces together and in that way allow the information to be reconstitutedt something much more complex is going on here. Then there is the question of the intrinsic meaning of a scrambled telephone call ... There is a vast spectrum of degrees of inherency of meaning. It is interesting to try place epigenesis in this spectrum. As development of an organism takes place, can it be said that the information is being "pulled out" of its DNA? Is that where all of the information about the organism's structure reside; DNA and the Necessity of Chemical Context In one sense, the answer seems to be yes, thanks to experiments li Avery's. But in another sense, the answer seems to be no, because so much of the pullingout process depends on extraordinarily complicated cellular chemical processes, which are not coded for in the DNA itself. The DNA relies on the fact that they will happen, but does not seem to contain a code which brings them about. Thus we have two conflicting views on the nature of the information in a genotype. One view says that so much of t information is outside the DNA that it is not reasonable to look upon the DNA as anything more than a very intricate set of triggers, like a sequence of buttons to be pushed on a jukebox; another view says that the information is all there, but in a very implicit form. Now it might seem that these are just two ways of saying the same thing, but that is not necessarily so. One view says that the DNA is quite meaningless out of context; the other says that even if it were taken out context, a molecule of DNA from a living being has such a compelling inner  
 
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logic to its structure that its message could be deduced anyway. To put it as succinctly as possible, one view says that in order for DNA to have meaning, chemical context is necessary; the other view says that only intelligence is necessary to reveal the "intrinsic meaning" of a strand of DNA. An Unlikely UFO We can get some perspective on this issue by considering a strange hypothetical event. A record of David Oistrakh and Lev Oborin playing Bach's sonata in F Minor for violin and clavier is sent up in a satellite. From the satellite it is then launched on a course which will carry it outside of the solar system, perhaps out of the entire galaxy just a thin plastic platter with a hole in the middle, swirling its way through intergalactic space. It has certainly lost its context. How much meaning does it carry? If an alien civilization were to encounter it, they would almost certainly be struck by its shape, and would probably be very interested in it. Thus immediately its shape, acting as a trigger, has given them some information: that it is an artifact, perhaps an informationbearing artifact. This ideacommunicated, or triggered, by the record itselfnow creates a new context in which the record will henceforth be perceived. The next steps in the decoding might take considerably longerbut that is very hard for us to assess. We can imagine that if such a record had arrived on earth in Bach's time, no one would have known what to make of it, and very likely it would not have gotten deciphered. But that does not diminish our conviction that the information was in principle there; we just know that human knowledge in those times was not very sophisticated with respect to the possibilities of storage, transformation, and revelation of information. Levels of Understanding of a Message Nowadays, the idea of decoding is extremely widespread; it is a significant part of the activity of astronomers, linguists, archaeologists, military specialists, and so on. It is often suggested that we may be floating in a sea of radio messages from other civilizations, messages which we do not yet know how to decipher. And much serious thought has been given to the techniques of deciphering such a message. One of the main problems perhaps the deepest problemis the question, "How will we recognize the fact that there is a message at all? How to identify a frame?" The sending of a record seems to be a simple solutionits gross physical structure is very attentiondrawing, and it is at least plausible to us that it would trigger, in any sufficiently great intelligence, the idea of looking for information hidden in it. However, for technological reasons, sending of solid objects to other star systems seems to be out of the question. Still, that does not prevent our thinking about the idea. Now suppose that an alien civilization hit upon the idea that the appropriate mechanism for translation of the record is a machine which  
 
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converts the groovepatterns into sounds. This would still be a far cry from a true deciphering. What, indeed, would constitute a successful deciphering of such a record? Evidently, the civilization would have to be able to ma sense out of the sounds. Mere production of sounds is in itself hart worthwhile, unless they have the desired triggering effect in the brains that is the word) of the alien creatures. And what is that desired effect? would be to activate structures in their brains which create emotional effects in them which are analogous to the emotional effects which experience in hearing the piece. In fact, the production of sounds cot even be bypassed, provided that they used the record in some other way get at the appropriate structures in their brains. (If we humans had a w of triggering the appropriate structures in our brains in sequential order, as music does, we might be quite content to bypass the soundsbut it see] extraordinarily unlikely that there is any way to do that, other than via o ears. Deaf composersBeethoven, Dvofák, Faureor musicians who can "hear" music by looking at a score, do not give the lie to this assertion, for such abilities are founded upon preceding decades of direct auditory experiences.) Here is where things become very unclear. Will beings of an alien civilization have emotions? Will their emotionssupposing they have somebe mappable, in any sense, onto ours? If they do have emotions somewhat like ours, do the emotions cluster together in somewhat the same way as ours do? Will they understand such amalgams as tragic beauty courageous suffering? If it turns out that beings throughout the universe do share cognitive structures with us to the extent that even emotions overlap, then in some sense, the record can never be out of its natural context; that context is part of the scheme of things, in nature. And if such is the case, then it is likely that a meandering record, if not destroyed en route, would eventually get picked up by a being or group of beings, at get deciphered in a way which we would consider successful.  
 
"Imaginary Spacescape" In asking about the meaning of a molecule of DNA above, I used t phrase "compelling inner logic"; and I think this is a key notion. To illustrate this, let us slightly modify our hypothetical recordintospa event by substituting John Cage's "Imaginary Landscape no. 4" for the Bach. This piece is a classic of aleatoric, or chance, musicmusic who structure is chosen by various random processes, rather than by an attempt to convey a personal emotion. In this case, twentyfour performers attar themselves to the twentyfour knobs on twelve radios. For the duration the piece they twiddle their knobs in aleatoric ways so that each radio randomly gets louder and softer, switching stations all the while. The tot sound produced is the piece of music. Cage's attitude is expressed in 14 own words: "to let sounds be themselves, rather than vehicles for man made theories or expressions of human sentiments."  
 
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Now imagine that this is the piece on the record sent out into space. It would be extraordinarily unlikelyif not downright impossiblefor an alien civilization to understand the nature of the artifact. They would probably be very puzzled by the contradiction between the frame message ("I am a message; decode me"), and the chaos of the inner structure. There are few "chunks" to seize onto in this Cage piece, few patterns which could guide a decipherer. On the other hand, there seems to be, in a Bach piece, much to seize ontopatterns, patterns of patterns, and so on. We have no way of knowing whether such patterns are universally appealing. We do not know enough about the nature of intelligence, emotions, or music to say whether the inner logic of a piece by Bach is so universally compelling that its meaning could span galaxies. However, whether Bach in particular has enough inner logic is not the issue here; the issue is whether any message has, per se, enough compelling inner logic that its context will be restored automatically whenever intelligence of a high enough level comes in contact with it. If some message did have that contextrestoring property, then it would seem reasonable to consider the meaning of the message as an inherent property of the message. The Heroic Decipherers Another illuminating example of these ideas is the decipherment of ancient texts written in unknown languages and unknown alphabets. The intuition feels that there is information inherent in such texts, whether or not we succeed in revealing it. It is as strong a feeling as the belief that there is meaning inherent in a newspaper written in Chinese, even if we are completely ignorant of Chinese. Once the script or language of a text has been broken, then no one questions where the meaning resides: clearly it resides in the text, not in the method of decipherment just as music resides in a record, not inside a record player! One of the ways that we identify decoding mechanisms is by the fact that they do not add any meaning to the signs or objects which they take as input; they merely reveal the intrinsic meaning of those signs or objects. A jukebox is not a decoding mechanism, for it does not reveal any meaning belonging to its input symbols; on the contrary, it supplies meaning concealed inside itself. Now the decipherment of an ancient text may have involved decades of labor by several rival teams of scholars, drawing on knowledge stored in libraries all over the world ... Doesn't this process add information, too? Just how intrinsic is the meaning of a text, when such mammoth efforts are required in order to find the decoding rules? Has one put meaning into the text, or was that meaning already there? My intuition says that the meaning was always there, and that despite the arduousness of the pullingout process, no meaning was pulled out that wasn't in the text to start with. This intuition comes mainly from one fact: I feel that the result was inevitable; that, had the text not been deciphered by this group at this time, it would have been deciphered by that group at that timeand it would have come  
 
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FIGURE 39. The Rosetta Stone [courtesy of the British Museum.  
 
out the same way. That is why the meaning is part of the text itself; it acts upon intelligence in a predictable way. Generally, we can say: meaning is part of an object to the extent that it acts upon intelligence in a predictable way. In Figure 39 is shown the Rosetta stone, one of the most precious of all historic discoveries. It was the key to the decipherment of Egyptian hieroglyphics, for it contains parallel text in three ancient scripts: hieroglyphic demotic characters, and Greek. The inscription on this basalt stele was firs deciphered in 1821 by Jean Francois Champollion, the "father of Egyptology"; it is a decree of priests assembled at Memphis in favor of Ptolemy Epiphanes.  
 
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Three Layers of Any Message In these examples of decipherment of outofcontext messages, we can separate out fairly clearly three levels of information: (1) the frame message; (2) the outer message; (3) the inner message. The one we are most familiar with is (3), the inner message; it is the message which is supposed to be transmitted: the emotional experiences in music, the phenotype in genetics, the royalty and rites of ancient civilizations in tablets, etc. To understand the inner message is to have extracted the meaning intended by the sender.. The frame message is the message "I am a message; decode me if you can!"; and it is implicitly conveyed by the gross structural aspects of any informationbearer. To understand the frame message is to recognize the need for a decodingmechanism. If the frame message is recognized as such, then attention is switched to level (2), the outer message. This is information, implicitly carried by symbolpatterns and structures in the message, which tells how to decode the inner message. To understand the outer message is to build, or know how to build, the correct decoding mechanism for the inner message. This outer level is perforce an implicit message, in the sense that the sender cannot ensure that it will be understood. It would be a vain effort to send instructions which tell how to decode the outer message, for they would have to be part of the inner message, which can only be understood once the decoding mechanism has been found. For this reason, the outer message is necessarily a set of triggers, rather than a message which can be revealed by a known decoder. The formulation of these three "layers" is only a rather crude beginning at analyzing how meaning is contained in messages. There may be layers and layers of outer and inner messages, rather than just one of each. Think, for instance, of how intricately tangled are the inner and outer messages of the Rosetta stone. To decode a message fully, one would have to reconstruct the entire semantic structure which underlay its creation and thus to understand the sender in every deep way. Hence one could throw away the inner message, because if one truly understood all the finesses of the outer message, the inner message would be reconstructible. The book After Babel, by George Steiner, is a long discussion of the interaction between inner and outer messages (though he never uses that terminology). The tone of his book is given by this quote: We normally use a shorthand beneath which there lies a wealth of subconscious, deliberately concealed or declared associations so extensive and intri  
 
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cate that they probably equal the sum and uniqueness of our status as an individual person.' Thoughts along the same lines are expressed by Leonard B. Meyer, in h book Music, the Arts, and Ideas: The way of listening to a composition by Elliott Carter is radically different from the way of listening appropriate to a work by John Cage. Similarly, a novel by Beckett must in a significant sense be read differently from one by Bellow. A painting by Willem de Kooning and one by Andy Warhol require different perceptionalcognitive attitudes.' Perhaps works of art are trying to convey their style more than an thing else. In that case, if you could ever plumb a style to its very bottom you could dispense with the creations in that style. "Style", "outer message "decoding technique"all ways of expressing the same basic idea. Schrodinger's Aperiodic Crystals What makes us see a frame message in certain objects, but none in other; Why should an alien civilization suspect, if they intercept an errant record that a message lurks within? What would make a record any different from a meteorite? Clearly its geometric shape is the first clue that "something funny is going on". The next clue is that, on a more microscopic scale, consists of a very long aperiodic sequence of patterns, arranged in a spiral If we were to unwrap the spiral, we would have one huge linear sequence (around 2000 feet long) of minuscule symbols. This is not so different from a DNA molecule, whose symbols, drawn from a meager "alphabet" of four different chemical bases, are arrayed in a onedimensional sequence, an then coiled up into a helix. Before Avery had established the connection between genes and DNA, the physicist Erwin Schrödinger predicted, o purely theoretical grounds, that genetic information would have to be stored in "aperiodic crystals", in his influential book What Is Life? In fact books themselves are aperiodic crystals contained inside neat geometric forms. These examples suggest that, where an aperiodic crystal is found "packaged" inside a very regular geometric structure, there may lurk a inner message. (I don't claim this is a complete characterization of frame messages; however, it is a fact that many common messages have frame messages of this description. See Figure 40 for some good examples.) Languages for the Three Levels The three levels are very clear in the case of a message found in a bottle washed up on a beach. The first level, the frame message, is found when one picks up the bottle and sees that it is sealed, and contains a dry piece c paper. Even without seeing writing, one recognizes this type of artifact an informationbearer, and at this point it would take an extraordinary almost inhumanlack of curiosity, to drop the bottle and not look further.  
 
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The Location of Meaning  176  
 
 
Next, one opens the bottle and examines the marks on the paper. Perhaps, they are in Japanese; this can be discovered without any of the inner message being understoodit merely comes from a recognition of 1 characters. The outer message can be stated as an English sentence: "I in Japanese." Once this has been discovered, then one can proceed the inner message, which may be a call for help, a haiku poem, a lover’s lament ... It would be of no use to include in the inner message a translation the sentence "This message is in Japanese", since it would take someone who knew Japanese to read it. And before reading it, he would have recognize the fact that, as it is in Japanese, he can read it. You might try wriggle out of this by including translations of the statement "This mess2 is in Japanese" into many different languages. That would help it practical sense, but in a theoretical sense the same difficulty is there. . Englishspeaking person still has to recognize the "Englishness" of the message; otherwise it does no good. Thus one cannot avoid the problem that one has to find out how to decipher the inner message from the outside the inner message itself may provide clues and confirmations, but those ; at best triggers acting upon the bottle finder (or upon the people whom enlists to help). Similar kinds of problem confront the shortwave radio listener. First he has to decide whether the sounds he hears actually constitute a message or are just static. The sounds in themselves do not give the answer, not e% in the unlikely case that the inner message is in the listener's own native language, and is saying, "These sounds actually constitute a message a are not just static!" If the listener recognizes a frame message in the soup then he tries to identify the language the broadcast is inand clearly, he is still on the outside; he accepts triggers from the radio, but they cam explicitly tell him the answer. It is in the nature of outer messages that they are not conveyed in any FIGURE 40. A collage of scripts. Uppermost on the left is an inscription in the un ciphered boustrophedonic writing system from Easter Island, in which every second lin upside down. The characters are chiseled on a wooden tablet, 4 inches by 35 inches. Mov clockwise, we encounter vertically written Mongolian: above, presentday Mongolian, below, a document dating from 1314. Then we come to a poem in Bengali by Rabindran Tagore in the bottom righthand corner. Next to it is a newspaper headline in Malayalam (II Kerala, southern India), above which is the elegant curvilinear language Tamil (F Kerala). The smallest entry is part of a folk tale in Buginese (Celebes Island, Indonesia). In center of the collage is a paragraph in the Thai language, and above it a manuscript in Rn dating from the fourteenth century, containing a sample of the provincial law of Scania (so Sweden). Finally, wedged in on the left is a section of the laws of Hammurabi, written Assyrian cuneiform. As an outsider, I feel a deep sense of mystery as I wonder how meanin cloaked in the strange curves and angles of each of these beautiful aperiodic crystals. Info there is content. [From Ham Jensen, Sign, Symbol, and Script (New York: G. Putnam's S. 1969), pp. 89 (cuneiform), 356 (Easter Island), 386, 417 (Mongolian), 552 (Runic); from Keno Katzner, The Languages of the World (New York: Funk & Wagnalls, 1975), pp. 190 (Bengali), (Buginese); from I. A. Richards and Christine Gibson, English Through Pictures (New Y Washington Square Press, 1960), pp. 73 (Tamil), 82 (Thai).  
 
The Location of Meaning  177  
 
 
explicit language. To find an explicit language in which to convey outer messages would not be a breakthroughit would be a contradiction in terms! It is always the listener's burden to understand the outer message. Success lets him break through into the inside, at which point the ratio of triggers to explicit meanings shifts drastically towards the latter. By comparison with the previous stages, understanding the inner message seems effortless. It is as if it just gets pumped in. The "Jukebox" Theory of Meaning. These examples may appear to be evidence for the viewpoint that no message has intrinsic meaning, for in order to understand any inner message, no matter how simple it is, one must first understand its frame message and its outer message, both of which are carried only by triggers (such as being written in the Japanese alphabet, or having spiraling grooves, etc.). It begins to seem, then, that one cannot get away from a "jukebox" theory of meaningthe doctrine that no message contains inherent meaning, because, before any message can be understood, it has to be used as the input to some "jukebox", which means that information contained in the "jukebox" must be added to the message before it acquires meaning. This argument is very similar to the trap which the Tortoise caught Achilles in, in Lewis Carroll's Dialogue. There, the trap was the idea that before you can use any rule, you have to have a rule which tells you how to use that rule; in other words, there is an infinite hierarchy of levels of rules, which prevents any rule from ever getting used. Here, the trap is the idea that before you can understand any message, you have to have a message which tells you how to understand that message; in other words, there is an infinite hierarchy of levels of messages, which prevents any message from ever getting understood. However, we all know that these paradoxes are invalid, for rules do get used, and messages do get understood. How come? Against the Jukebox Theory This happens because our intelligence is not disembodied, but is instantiated in physical objects: our brains. Their structure is due to the long process of evolution, and their operations are governed by the laws of physics. Since they are physical entities, our brains run without being told how to run. So it is at the level where thoughts are produced by physical law that Carroll's ruleparadox breaks down; and likewise, it is at the level where a brain interprets incoming data as a message that the messageparadox breaks down. It seems that brains come equipped with "hardware" for recognizing that certain things are messages, and for decoding those messages. This minimal inborn ability to extract inner meaning is what allows the highly recursive, snowballing process of language acquisition to take place. The inborn hardware is like a jukebox: it supplies the additional information which turns mere triggers into complete messages.  
 
The Location of Meaning  178  
 
 
Meaning Is Intrinsic If Intelligence Is Natural Now if different people's "jukeboxes" had different "songs" in then responded to given triggers in completely idiosyncratic ways, the would have no inclination to attribute intrinsic meaning to those tri; However, human brains are so constructed that one brain responds in much the same way to a given trigger as does another brain, all other t being equal. This is why a baby can learn any language; it responds to triggers in the same way as any other baby. This uniformity of "human jukeboxes" establishes a uniform "language" in which frame message outer messages can be communicated. If, furthermore, we believe human intelligence is just one example of a general phenomena naturethe emergence of intelligent beings in widely varying contexts then presumably the "language" in which frame messages and outer sages are communicated among humans is a "dialect" of a universal gauge by which intelligences can communicate with each other. Thus, would be certain kinds of triggers which would have "universal triggering power", in that all intelligent beings would tend to respond to them i same way as we do. This would allow us to shift our description of where meaning located. We could ascribe the meanings (frame, outer, and inner) message to the message itself, because of the fact that deciphering mechanisms are themselves universalthat is, they are fundamental f of nature which arise in the same way in diverse contexts. To make it concrete, suppose that "A5" triggered the same song in all jukeboxes suppose moreover that jukeboxes were not manmade artifacts, but w occurring natural objects, like galaxies or carbon atoms. Under such circumstances, we would probably feel justified in calling the universal triggering power of "A5" its "inherent meaning"; also, "A5" would merit: the name of "message", rather than "trigger", and the song would indeed "revelation" of the inherent, though implicit, meaning of "A5". Earth Chauvinism This ascribing of meaning to a message comes from the invariance c processing of the message by intelligences distributed anywhere ii universe. In that sense, it bears some resemblance to the ascribing of to an object. To the ancients, it must have seemed that an object's weight was an intrinsic property of the object. But as gravity became understood, it was realized that weight varies with the gravitational field the object is immersed in. Nevertheless, there is a related quantity, the mass, which not vary according to the gravitational field; and from this invariance the conclusion that an object's mass was an intrinsic property of the object itself. If it turns out that mass is also variable, according to context, then will backtrack and revise our opinion that it is an intrinsic property of an object. In the same way, we might imagine that there could exist other  
 
The Location of Meaning  179  
 
 
kinds of "jukeboxes"intelligenceswhich communicate among each other via messages which we would never recognize as messages, and who also would never recognize our messages as messages. If that were the case, then the claim that meaning is an intrinsic property of a set of symbols would have to be reconsidered. On the other hand, how could we ever realize that such beings existed? It is interesting to compare this argument for the inherency of meaning with a parallel argument for the inherency of weight. Suppose one defined an object's weight as "the magnitude of the downward force which the object exerts when on the surface of the planet Earth". Under this definition, the downward force which an object exerts when on the surface of Mars would have to be given another name than "weight". This definition makes weight an inherent property, but at the cost of geocentricity" Earth chauvinism". It would be like "Greenwich chauvinism"refusing to accept local time anywhere on the globe but in the GMT time zone. It is an unnatural way to think of time. Perhaps we are unknowingly burdened with a similar chauvinism with respect to intelligence, and consequently with respect to meaning. In our chauvinism, we would call any being with a brain sufficiently much like our own "intelligent", and refuse to recognize other types of objects as intelligent. To take an extreme example, consider a meteorite which, instead of deciphering the outerspace Bach record, punctures it with colossal indifference, and continues in its merry orbit. It has interacted with the record in a way which we feel disregards the record's meaning. Therefore, we might well feel tempted to call the meteorite "stupid". But perhaps we would thereby do the meteorite a disservice. Perhaps it has a "higher intelligence" which we in our Earth chauvinism cannot perceive, and its interaction with the record was a manifestation of that higher intelligence. Perhaps, then, the record has a "higher meaning"totally different from that which we attribute to it; perhaps its meaning depends on the type of intelligence perceiving it. Perhaps. It would be nice if we could define intelligence in some other way than "that which gets the same meaning out of a sequence of symbols as we do". For if we can only define it this one way, then our argument that meaning is an intrinsic property is circular, hence contentfree. We should try to formulate in some independent way a set of characteristics which deserve the name "intelligence". Such characteristics would constitute the uniform core of intelligence, shared by humans. At this point in history we do not yet have a welldefined list of those characteristics. However, it appears likely that within the next few decades there will be much progress made in elucidating what human intelligence is. In particular, perhaps cognitive psychologists, workers in Artificial Intelligence, and neuroscientists will be able to synthesize their understandings, and come up with a definition of intelligence. It may still be humanchauvinistic; there is no way around that. But to counterbalance that, there may be some elegant and beautifuland perhaps even simpleabstract ways of characterizing the essence of intelligence. This would serve to lessen the feeling of having  
 
The Location of Meaning  180  
 
 
formulated an anthropocentric concept. And of course, if contact were established with an alien civilization from another star system, we feel supported in our belief that our own type of intelligence is not just a fluke, but an example of a basic form which reappears in nature in contexts, like stars and uranium nuclei. This in turn would support the idea of meaning being an inherent property. To conclude this topic, let us consider some new and old ex; and discuss the degree of inherent meaning which they have, by ourselves, to the extent that we can, in the shoes of an alien civilization which intercepts a weird object ...  
 
Two Plaques in Space Consider a rectangular plaque made of an indestructible metallic alloy which are engraved two dots, one immediately above the another preceding colon shows a picture. Though the overall form of the might suggest that it is an artifact, and therefore that it might conceal some message, two dots are simply not sufficient to convey anything. (Can before reading on, hypothesize what they are supposed to mean suppose that we made a second plaque, containing more dots, as follows.  
 
• • • • • • ••••••• • •••••••••••••a • •• •••••••••••• ••••••••••• • •••••••••••••••••••••••••••••••••••••a  
 
Now one of the most obvious things to doso it might seer terrestrial intelligence at leastwould be to count the dots in the successive rows. The sequence obtained is: 1, 1, 2, 3, 5, 8, 13, 21, 34.  
 
Here there is evidence of a rule governing the progression from one the next. In fact, the recursive part of the definition of the Fib numbers can be inferred, with some confidence, from this list. Supp think of the initial pair of values (1,1) as a "genotype" from which the "phenotype"the full Fibonacci sequenceis pulled out by a recursive rule. By sending the genotype alonenamely the first version plaquewe fail to send the information which allows reconstitution phenotype. Thus, the genotype does not contain the full specification of  
 
The Location of Meaning  181  
 
 
the phenotype. On the other hand, if we consider the second version of the plaque to be the genotype, then there is much better cause to suppose that the phenotype could actually be reconstituted. This new version of the genotypea "long genotype"contains so much information that the mechanism by which phenotype is pulled out of genotype can be inferred by intelligence from the genotype alone. Once this mechanism is firmly established as the way to pull phenotype from genotype, then we can go back to using "short genotypes"like the first plaque. For instance, the "short genotype" (1,3) would yield the phenotype 1, 3, 4, 7, 11, 18, 29, 47, .. . the Lucas sequence. And for every set of two initial valuesthat is, for every short genotypethere will be a corresponding phenotype. But the short genotypes, unlike the long ones, are only triggersbuttons to be pushed on the jukeboxes into which the recursive rule has been built. The long genotypes are informative enough that they trigger, in an intelligent being, the recognition of what kind of "jukebox" to build. In that sense, the long genotypes contain the information of the phenotype, whereas the short genotypes do not. In other words, the long genotype transmits not only an inner message, but also an outer message, which enables the inner message to be read. It seems that the clarity of the outer message resides in the sheer length of the message. This is not unexpected; it parallels precisely what happens in deciphering ancient texts. Clearly, one's likelihood of success depends crucially on the amount of text available. Bach vs. Cage Again But just having a long text may not be enough. Let us take up once more the difference between sending a record of Bach's music into space, and a record of John Cage's music. Incidentally, the latter, being a Composition of Aleatorically Generated Elements, might be handily called a "CAGE", whereas the former, being a Beautiful Aperiodic Crystal of Harmony, might aptly be dubbed a "BACH". Now let's consider what the meaning of a Cage piece is to ourselves. A Cage piece has to be taken in a large cultural settingas a revolt against certain kinds of traditions. Thus, if we want to transmit that meaning, we must not only send the notes of the piece, but we must have earlier communicated an extensive history of Western culture. It is fair to say, then, that an isolated record of John Cage's music does not have an intrinsic meaning. However, for a listener who is sufficiently well versed in Western and Eastern cultures, particularly in the trends in Western music over the last few decades, it does carry meaningbut such a listener is like a jukebox, and the piece is like a pair of buttons. The meaning is mostly contained inside the listener to begin with; the music serves only to trigger it. And this "jukebox", unlike pure intelligence, is not at all universal; it is highly earthbound, depending on idiosyncratic se  
 
The Location of Meaning  182  
 
 
quences of events all over our globe for long period of time. Hoping that John Cage's music will be understood by another civilization is like hoping that your favorite tune, on a jukebox on the moon, will have the same buttons as in a saloon in Saskatoon. On the other hand, to appreciate Bach requires far less cultural k edge. This may seem like high irony, for Bach is so much more con and organized, and Cage is so devoid of intellectuality. But there strange reversal here: intelligence loves patterns and balks at randomness For most people, the randomness in Cage's music requires much explanation; and even after explanations, they may feel they are missing the messagewhereas with much of Bach, words are superfluous. In sense, Bach's music is more selfcontained than Cage's music. Still, it is clear how much of the human condition is presumed by Bach. For instance, music has three major dimensions of structure (me harmony, rhythm), each of which can be further divided into small intermediate, and overall aspects. Now in each of these dimensions, there is a certain amount of complexity which our minds can handle before boggling; clearly a composer takes this into account, mostly unconsciously when writing a piece. These "levels of tolerable complexity" along different dimensions are probably very dependent on the peculiar conditions of our evolution as a species, and another intelligent species might have developed music with totally different levels of tolerable complexity along these many dimensions. Thus a Bach piece might conceivably have to be accompanied, by a lot of information about the human species, which simply could not inferred from the music's structure alone. If we equate the Bach music a genotype, and the emotions which it is supposed to evoke with the phenotype, then what we are interested in is whether the genotype con all the information necessary for the revelation of the phenotype. How Universal Is DNA's Message? The general question which we are facing, and which is very similar t questions inspired by the two plaques, is this: "How much of the co necessary for its own understanding is a message capable of restoring? can now revert to the original biological meanings of "genotype" "phenotype"DNA and a living organismand ask similar quest Does DNA have universal triggering power? Or does it need a "biojukebox" to reveal its meaning? Can DNA evoke a phenotype without being embedded in the proper chemical context? To this question to answer is nobut a qualified no. Certainly a molecule of DNA in a vacuum will not create anything at all. However, if a molecule of DNA were set to seek its fortune in the universe, as we imagined the BACH and the CAGE were, it might be intercepted by an intelligent civilization. They might first of all recognize its frame message. Given that, they might to try to deduce from its chemical structure what kind of chemical environment it seemed to want, and then supply such an environment. Succes  
 
The Location of Meaning  183  
 
 
sively more refined attempts along these lines might eventually lead to a full restoration of the chemical context necessary for the revelation of DNA's phenotypical meaning. This may sound a little implausible, but if one allows many millions of years for the experiment, perhaps the DNA's meaning would finally emerge. On the other hand, if the sequence of bases which compose a strand of DNA were sent as abstract symbols (as in Fig. 41), not as a long helical molecule, the odds are virtually nil that this, as an outer message, would trigger the proper decoding mechanism which would enable the phenotype to be drawn out of the genotype. This would be a case of wrapping an inner message in such an abstract outer message that the contextrestoring power of the outer message would be lost, and so in a very pragmatic sense, the set of symbols would have no intrinsic meaning. Lest you think this all sounds hopelessly abstract and philosophical, consider that the exact moment when phenotype can be said to be "available", or "implied", by genotype, is a highly charged issue in our day: it is the issue of abortion. FIGURE 41. This Giant Aperiodic Crystal is the base sequence for the chromosome of bacteriophage OX174. It is the first complete genome ever mapped out for any organism. About 2,000 of these boustrophedonic pages would be needed to show the base sequence of a single E. Coli cell, and about one million pages to show the base sequence of the DNA of a single human cell. The book now in your hands contains roughly the same amount of information as a molecular blueprint for one measly E. Coli cell.  
 
CCGT CACGATTC ACACC CTCC CAATTCTAT GTTTTCATGCCTCC AAAT CTTCGAGCCTTTTTTATGGTT CCTT CTTATT ACC CTT GTG AA T GTG AGG CTG , rAGGAATACCTTCGGTTCCTAAGCCCTAACTCTTTCTCArCTTTACGCTCTTCGGAGTIATCGTCCAAATICICCGACaATnCCAGTTTCACTTTTATTA^ » C AT CGATAACCG CAT C AAG CT CTTGG AAGAGATTCTGTC TTTT CGT AT G CAGCCCCT T GAGTTCG AT AATGGTG ATATGTA TGTTGACG GC C A TAAGG CT . ^ AC AATMTTATAGTT C AACCCCCTCCTG T AACAT CGTAACA CG<S TT AACTAGGTAATTCAAGACTCATTGTC TATCTTTG ACTAGTC CT T GCAGT CTTGG ^ > CT ATAGACCACCC CC C CG AAG CCCAC C AAAAATGGTTTTTAC ACAACC AG AAGACGC TTACGCAGTTT TG CC GCAACCT CC CT CCTC AA C CCC CTCTTAA i f TTTCCGAC ATC CGCTATAG AATCACCT C C G GACCTCGTTAG AAC TIGTG AGTAGC AATTATCC AAAG AAAAACC CCATTAA T ATCACTAC CCGT TATAGG * ^ C CTATTC AG C CT ITG ATG AATGCAA TCCGACAGGCTCATC CTC ATGGTTGGTITAT CGTTTTTCACACTCT CA CCTTCCCTC AC GACCG AT T AC AGG CGT •) f GTC AC CCGCAC ACTTACT AATCCC AAC G CTCC GAG CCCTCCTT CTTGGTATGCTCC TT ATAGTGCTTTTA ICACTGCCTTT CC TAAGG CT AATAGT A TT T* * GT A rC ACTATTTTTGTCTCCCTG ACTATCC TACAC CTAATGG C CCTCTT CATTTCCATGCGGTCCACTTT ATCCCC AC ACTT CCTAC ACC TAG CCT TG AC s < CGCACA rCG CTTC AC C CT ACC CGTATGACAITCCTATTCCGG TG CAT AAAACGTT CG AT AAATTG AC C GC CGCT AACGCAT AC CCTC CT C C TTT TAAT CC' * AGGACCCTTTTT CAC GTTCTGGTTGG TTGTGGCCTGTTCATC CT AAAGG TG AC C CG CT TAAACCTAC C AGTIATATCGCT GTTGGTTTCTATGTGG CTAA1 rCAACCCTTCATCC CTCTC GAACC AAAAATCACTC AA C AACGT AAG AAATCG AGG ATCTGGA AAT CCT C GTTCCAGCTATAG ACTG AAAAACAAT TGCA T A ' ^ AAC CTGT TC AG AAT CAC AATC AG C CC CAACTT CG GCA TG AAAATGCT C ACAAT GAC AAAT CT CTC C ACGGACT CCTTAAT CC AACTT AC CAACCTGGGTT > f ACCCCGCGCTTTT GCACCCGATCTC ATTGAAAAGGG TCCGACTTACAGTAGACAGAA AAACG CAAG AC GAAGTTATAGACC AACTTC C C CCACCC CAC CA * * ACCTGTGACC ACAAAT GTGCTCAAATTTATGCG CG CTICG AT AAAAATGAITGG CGTATCCAACCTGCAGACTTTTAICGCTTCCATGACGCAGAAGTTAl f AAGG C GCTC GT C AGGTG AACCT AAATT AAGCATTTGT TCCTCATCATTAAGGACG AAAT AGTT CTATT AAAAACCTC ACT AGTCTTT A TACGCTT T CAC A' AATG AC AAAATTCGAC CTAICCTTC CGCAG CT CG ACAAGCT C TTAC TTTC CC AC CTTTCC CCATCAACTAACC A TT CTCTC AAA AA CT CAC C C CTTCCA T ) r AATT TTAC ACT TCT TCTCTTAG AGATGGTACTTGT T T T AC ACTGACT ATAG AT TTGGT C AGG AACTGCTTGCACGGTT CCT ATA ATT CC GTG AAG AGG AC * ' AAG AGCCTCGATTACTATCIGAGTCCGATGCTCTTCAACC ACT AATAGCT AACAAAT CAT CAGTCAAGTTACTG AACA A T CCGTA C C T TT CCACAC C G CT i r CC AGT C AT C GTTACGTT TGAAAC AATCACCACT C TTTT AG CTTTAGTACAAGCC AATTTAC GTTTTGCCGTCTTCGG ACT TACT CC AA TT AT CT CC GG TT » ^GCTCTCCTCCTCCTCGCTCCGTTGAGCCTTCCGTrTATGGTACGCTGGACTTTGTAGCATACCCTCGCTTTCCTGCTCCTGTTGAGTTTATTCCTCCCGTj < G GCTC CCAGCT GCGGT AATTA TTAC AAAAGG CATTTAACTCGCGGAAGGTACTACTCTGTCCCG C AAACTTAC AACT GC CCT AC TTG TATT ATT CCT TAG * CTT AAAGCCGCTG AATTGTTCG CGTTTAC CTTG CGTG TA CGCGC ACG AAA CACT GACGT T C TT ACTGACC CAG AAG AA A ACGTC CGT CA AA AATTAC GTG ; rTCGCGGAAATGCCAACGGAAATCAIGCACCCTTCCCGACGCCTGCTGCTCCCCCICCCGGTCTTGCAAAAAATGGAAATCTGTAATCTACTCAGGAAGCC' * CCT CTTTCCTATCTAG CTGCTC AACAATTTT AATTGCAG GGCCTTCGGC CCCTTACTTC ACC AT AAATTATCTCT AAT ATTC A AACTGGC GCC GAG CCTA) < TAG ACCT TC CTC ACCCCTCG CT ATTGCCCTC ATCAACT TTACC ATTATTC TCCIGG TT AG ACTGGTCGTTCCTT CCC TT CTAC CCTTTC C AGT AC GCCCT * "GGACCCCGTTGGCGCTCTCCCTCTTTCTCCATTCCCTCGTGGCCTTCCTATTGACTCTACTGTACACATTTTTACTTTTTArGTCCCTCATCGTCAGGTTj fGTICTTTTC GCCCT ACCAGTT AT ATTGGT C AI CAC AATTG TCAG CCCTCTC C TC ACCCT AAITGTCGTAGG AAGT AC TT G AA TT AG GTG AC AACTCG TAT* C CACG ATTAACCCTGATACC AAT AAAAT C C C TAAGCATTTGTTTCACCGTT ATTTC AAIATCTAT AAC AACTArrTTAAAqCCCCCTGG ATG CC TG ACCG ) f C ACTC CTC CTTCC CC TCGTC ACC TTTAC AAAAACTCTAC CG TCCTTG CCT TTCGTATTCCTCG TAGTAG AACTAATT CG ACT AATCCC AAT CC CAG CCAT' AGTCAGCTTT CTGCCC AAAT CACCACTTCTACCACATCTATTGACATTATGGCTCTGCAAGCTGCTTATGCT AATTTG GAT ACTG AGG AAG AAC GT G ATT) (. GTCT CT AAT CTCCCGTACTCTTCArTTCCTGC C AACAGT CG CAGTAT ACTCCAAAAT CC AGC TTTACTTCTTTATT CTA CT A C CATTGC G AC GT AC T T C A* GCCAICTCGCTATGATCTTCATGCAACTGAC C AAACGTCGTTAGCCC AGT TTTCTGG TC GTG TTCAA C AC AC CTATAA ACATTCTGTG CCC CGT TT CTTT1 ( TTCAGTTTCGTGGAAATCCCAAnCCArGACTTAGACAAATCACCGTCATCCGCCITTTGCITGTTCCCCTTCTCAITTGTATCACGGTACGAGTCCTTG^ A r A C C CATATTG CTGGCC A C C CTGTTTT GTATGG CAACTTC C CC C CC C GTGAAATTT CTATCAAGGATGTTTT CCCTTCTCG TGATT CG T C TAAG AACTT} t rCTrCCCCCAACGACTTACTTACCCTTCCGAACTTCTTCCACTATTCGTCCTCTTTGTATGCTTCCCCCTATTGCTATGGTCACTGCGACTCCTTAGAAT'' ^{1}GGTC ArnCC AAC AACC C GTACTI ATT CC C AAC CATG ATT AT C ACC ACTG TTTC AGT C GTTCAGTTCTTCCAGTGC ATAGT C TTACCT C ATGTGACCTTT ; r C d C C GT TTTT AATTTT AAAAAT CG CG AA G C C AATATTCGAGTGTC AGTT AGAA AAT ACTCCITC AGTACT AACTT AC CCCT C ACC AC CC CT CT AACG CT A* TC ACC CC TTGACCAAG C C MG CGCGG TAG G T T TT CTCCTTACG ACTTT AAT CATCTT TCAG AC TTTTATT T CTCCCCACAATTCAAA CT TTTTTTCTG AI ) C ATTCC CAGT TTG ATAGT TTTA TAITCCAACT CCTACATCGAA ATCC AC AGACATTTTCTCCACCCCTT CTT CG ACCTCATTC TCTT CAC T CTT GCTCG AA' ^ATGCTCGTAATGCTCCTTTTCTTCATTCCATTCACATCGATACATCTCTCAACGCCGCTAATCAGGTTGTTTCACTTGGTGCTCATATTGCTTTTCATGCj rTAC C C CTGCTACGTTTCCTATT TGTAGTATCCGTCACCCCTCCCATCAC CCTTGGCTTCTTCTCACTTT CGCTTCGTTTGTCCCTTTTT TAAATCCC AG C * CArCCTGCTTATTATACCGTC AAGGACT GTG TGACTATTC ACGT C CT T CCC CGT AC GCCCCCC AAT AA CG TCT A CGTTGGTT TCATGC TT TCG TCT AACT *) f re GIG CTTTCT ATTTACTGC ACTG AATT CACC GACCTCTGTTTATT AG AGAAA TTATTGG AGT AACTCGCTTTGGTTAGCC GCCGTAA A TCA TCCCC ATT' A TIG CT C CCCCTATTC CTT CTGCICTTC CT GG TGGCG CC A TG TC TAAAI TCTTTCC ACGCGGTC AA AAAG CC GCCTCCCC TCG CAIT CAAG GTG ATGT C C 1 (rTTCATCCCCGCCGGACTAGTCCCAATCCTTCTAATCTCCCAACTTACCGTCTAAATTATfKTCGfACTGGGTACGGATGTCATAACAAT^GGCATCCTT' ^{1}CGTTTCTGCTGCTATCCCTAAAGCTGGTAAACCACTTCTTGAAGGTACGTTCCAGGGTGGCACTTCTCCCGTTTCTGATAACTTCCTTGATTTGCTTGGAi 'CT A CT C CTG GTCCTC C G AGGG TTCGTAATTCG AGTCC TTTACCTCGT CGTTCT A TTAGTC CTCA r AGG AAAG G AAAT AGTC CCCCTCT C AACCGTCCTT C * CTT C CTC TC CTGG TATCC TT GAG CCCCC ATT TCACAAT CAA AAAC ACCT TACT A AAATGC AACTCC AC AATCA CAAAG AC AT TG CCC ACATCC AAA ATG A") C C AAGACAACT ATTCCTT CC TAG AGT AAAACACGTATAICC AC C AG AAAC CAT AAC AC CCCACTTCAC CG C C T GACTTACCG TCCTT AG AC AA AAACT CAC' C ACT C TACTG CTCGC CTTC CCT CT ATT AT CC AAAACACCAATCTTTCC AACCAAC ACCACGTTTCCG AG A TTATGCGCCAAATC CTT AC T CAACCTCAA A} rATGGCGACTAACACGCAAACCACTACTTCATTCACTTCCACTCGTGATTCCAACCCTCAGTAAACAAACTAAACCAGTAACCATTTTArCACTCCTCCGC' * TGC CTCT TCTCATArTCG CG CT ACTC C AAACCATA rTTCT AATCTCGTC ACTG ATC CTCCTTCTGGTCTGCTTGAIATTTTTCATCCTATTC AT AAAC CT 1 AATA AAG GAT CTGTrTAATCTCC CTTATGCTACTCCAAATCCCAC AAACGTCTTT AACAACCTTCATAGCCCTTC *  
 
The Location of Meaning  184  
 
 
Chromatic Fantasy, And Feud. Having had a splendid dip in the pond, the Tortoise is just crawling out and shaking himself dry, when who but Achilles walks by. Tortoise: Ho there, Achilles. I was just thinking of you as I splash around in the pond. Achilles: Isn't that curious? I was just thinking of you, too, while I meandered through the meadows. They're so green at this time of year. Tortoise: You think so? It reminds me of a thought I was hoping to share with you. Would you like to hear it? Achilles: Oh, I would be delighted. That is, I would be delighted as long you're not going to try to snare me in one of your wicked traps of log Mr. T. Tortoise: Wicked traps? Oh, you do me wrong. Would I do anything wicked? I'm a peaceful soul, bothering nobody and leading a gent; herbivorous life. And my thoughts merely drift among the oddities and quirks of how things are (as I see them). I, humble observer phenomena, plod along and puff my silly words into the air rather unspectacularly, I am afraid. But to reassure you about my intention I was only planning to speak of my Tortoiseshell today, and as you know, those things have nothingnothing whatsoeverto do with logic! Achilles: Your words Do reassure me, Mr. T. And, in fact, my curiosity quite piqued. I would certainly like to listen to what you have to say even if it is unspectacular. Tortoise: Let's see ... how shall I begin? Hmm ... What strikes you me about my shell, Achilles? Achilles: It looks wonderfully clean! Tortoise: Thank you. I just went swimming and washed off several layers of dirt which had accumulated last century. Now you can see ho green my shell is. Achilles: Such a good healthy green shell, it's nice to see it shining in sun. Tortoise: Green? It's not green. Achilles: Well, didn't you just tell me Tortoise: I did. Achilles: Then, we agree: it is green. Tortoise: No, it isn't green. Achilles: Oh, I understand your game. You're hinting to me that what you say isn't necessarily true; that Tortoises play with language; that your statements and reality don't necessarily match; that   
 
 
Tortoise: I certainly am not. Tortoises treat words as sacred. Tortoises revere accuracy. Achilles: Well, then, why did you say that your shell is green, and that it is not green also? Tortoise: I never said such a thing; but I wish I had. Achilles: You would have liked to say that? Tortoise: Not a bit. I regret saying it, and disagree wholeheartedly with it. Achilles: That certainly contradicts what you said before! Tortoise: Contradicts? Contradicts? I never contradict myself. It's not part of Tortoisenature. Achilles: Well, I've caught you this time, you slippery fellow, you. Caught you in a fullfledged contradiction. Tortoise: Yes, I guess you did. Achilles: There you go again! Now you're contradicting yourself more and more! You are so steeped in contradiction it's impossible to argue with you! Tortoise: Not really. I argue with myself without any trouble at all. Perhaps the problem is with you. I would venture a guess that maybe you're the one who's contradictory, but you're so trapped in your own tangled web that you can't see how inconsistent you're being. Achilles: What an insulting suggestion! I'm going to show you that you're the contradictory one, and there are no two ways about it. Tortoise: Well, if it's so, your task ought to be cut out for you. What could be easier than to point out a contradiction? Go aheadtry it out. Achilles: Hmm ... Now I hardly know where to begin. Oh ... I know. You first said that (1) your shell is green, and then you went on to say that (2) your shell is not green. What more can I say? Tortoise: Just kindly point out the contradiction. Quit beating around the bush. Achilles: Butbutbut ... Oh, now I begin to see. (Sometimes I am so slowwitted!) It must be that you and I differ as to what constitutes a contradiction. That's the trouble. Well, let me make myself very clear: a contradiction occurs when somebody says one thing and denies it at the same time. Tortoise: A neat trick. I'd like to see it done. Probably ventriloquists would excel at contradictions, speaking out of both sides of their mouth, as it were. But I'm not a ventriloquist. Achilles: Well, what I actually meant is just that somebody can say one thing and deny it all within one single sentence! It doesn't literally have to be in the same instant. Tortoise: Well, you didn't give ONE sentence. You gave TWO. Achilles: Yestwo sentences that contradict each other! Tortoise: I am sad to see the tangled structure of your thoughts becoming so exposed, Achilles. First you told me that a contradiction is some thing which occurs in a single sentence. Then you told me that you  
 
 
Found a contradiction in a pair of sentences I uttered. Frankly, it’s just as I said. Your own system of thought is so delusional that you manage to avoid seeing how inconsistent it is. From the outside, however plain as day. Achilles: Sometimes I get so confused by your diversionary tactics tl can't quite tell if we're arguing about something utterly petty, or something deep and profound! Tortoise: I assure you, Tortoises don't spend their time on the petty. Hence it's the latter. Achilles: I am very reassured. Thank you. Now I have had a moment to reflect, and I see the necessary logical step to convince you that you contradicted yourself. Tortoise: Good, good. I hope it's an easy step, an indisputable one. Achilles: It certainly is. Even you will agree with it. The idea is that you believed sentence 1 ("My shell is green"), AND you believed sentence 2 ("My shell is not green"), you would believe one compound( sentence in which both were combined, wouldn't you? Tortoise: Of course. It would only be reasonable ... providing just that the manner of combination is universally acceptable. But I'm sure we'll agree on that. Achilles: Yes, and then I'll have you! The combination I propose is Tortoise: But we must be careful in combining sentences. For instance you'd grant that "Politicians lie" is true, wouldn't you? Achilles: Who could deny it? Tortoise: Good. Likewise, "Castiron sinks" is a valid utterance, isn't it? Achilles: Indubitably. Tortoise: Then, putting them together, we get "Politicians lie in cast iron sinks". Now that's not the case, is it? Achilles: Now wait a minute ... "Politicians lie in castiron sinks?" N no, but Tortoise: So, you see, combining two true sentences in one is not a policy, is it? Achilles: But youyou combined the twoin such a silly way! Tortoise: Silly? What have you got to object to in the way I combined them Would you have me do otherwise? Achilles: You should have used the word "and", not "in". Tortoise: I should have? You mean, if YOU'D had YOUR way, I should h; Achilles: Noit's the LOGICAL thing to do. It's got nothing to do with personally. Tortoise: This is where you always lose me, when you resort to your L and its highsounding Principles. None of that for me today, plea Achilles: Oh, Mr. Tortoise, don't put me through all this agony. You k very well that that's what "and" means! It's harmless to combine true sentences with "and"! Tortoise: "Harmless", my eye! What gall! This is certainly a pernicious plot  
 
 
to entrap a poor, innocent, bumbling Tortoise in a fatal contradiction. If it were so harmless, why would you be trying so bloody hard to get me to do it? Eh? Achilles: You've left me speechless. You make me feel like a villain, where I really had only the most innocent of motivations. Tortoise: That's what everyone believes of himself... Achilles: Shame on metrying to outwit you, to use words to snare you in a selfcontradiction. I feel so rotten. Tortoise: And well you should. I know what you were trying to set up. Your plan was to make me accept sentence 3, to wit: "My shell is green and my shell is not green". And such a blatant falsehood is repellent to the Tongue of a Tortoise. Achilles: Oh, I'm so sorry I started all this. Tortoise: You needn't be sorry. My feelings aren't hurt. After all, I'm used to the unreasonable ways of the folk about me. I enjoy your company, Achilles, even if your thinking lacks clarity. Achilles: Yes ... Well, I fear I am set in my ways, and will probably continue to err and err again, in my quest for Truth. Tortoise: Today's exchange may have served a little to right your course. Good day, Achilles. Achilles: Good day, Mr. T.  
 
 
CHAPTER VII  
 
The Propositional Calculus Words and Symbols THE PRECEDING DIALOGUE is reminiscent of the TwoPart Invention by Lewis Carroll. In both, the Tortoise refuses to use normal, ordinary in the normal, ordinary wayor at least he refuses to do so when it is his advantage to do so. A way to think about the Carroll paradox was given last Chapter. In this Chapter we are going to make symbols dc Achilles couldn't make the Tortoise do with his words. That is, we are to make a formal system one of whose symbols will do just what A wished the word `and' would do, when spoken by the Tortoise, and ail of whose symbols will behave the way the words 'if... then . . .' ought to behave. There are only two other words which we will attempt to deal with `or' and `not'. Reasoning which depends only on correct usage of these words is termed propositional reasoning. Alphabet and First Rule of the Propositional Calculus I will present this new formal system, called the Propositional Calculus, like a puzzle, not explaining everything at once, but letting you things out to some extent. We begin with the list of symbols: < > P Q R ' A V Z> ~ The first rule of this system that I will reveal is the following: RULE OF JOINING: If x and y are theorems of the system, then so is the string < xay >. This rule takes two theorems and combines them into one. It s remind you of the Dialogue. WellFormed Strings There will be several other rules of inference, and they will all be pres shortlybut first, it is important to define a subset of all strings, namely the  
 
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well formed strings. They will be defined in a recursive way. We begin with the ATOMS: P, Q, and R are called atoms.. New atoms are formed by appending primes onto the right of old atomsthus, R', Q", P"', etc. This gives an endless supply of atoms. All atoms are wellformed. Then we have four recursive FORMATION RULES: If x and y are wellformed, then the following four strings are also wellformed: (1) ~x (2) < xAy> (3) < xvy> (4) < xz)y> For example, all of the following are wellformed: P atom ~P by (1) —P by (1) Q' atom ~Q1 by (1) <Pa~Q' > by (2) ~<Pa~Q' > by (1) —<P3~Q' > by (4) <~<Pa~Q' >v—<P=>~Q' » by (3) The last one may look quite formidable, but it is built up straightforwardly from two componentsnamely the two lines just above it. Each of them is in turn built up from previous lines ... and so on. Every wellformed string can in this way be traced back to its elementary constituentsthat is, atoms. You simply run the formation rules backwards until you can no more. This process is guaranteed to terminate, since each formation rule (when run forwards) is a lengthening rule, so that running it backwards always drives you towards atoms. This method of decomposing strings thus serves as a check on the wellformedness of any string. It is a topdown decision procedure for wellformedness. You can test your understanding of this decision procedure by checking which of the following strings are wellformed: (1) <P> (2) (2) <~P> (3) <PaQaR> (4) <PaQ> (5) «PaQ>aQ~aP» (6) <Pa~P> (7) «Pv<Q=>R»a<~Pv~R'» (8) <PaQ>a< QaP:  
 
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(Answer: Those whose numbers are Fibonacci numbers are not formed. The rest are wellformed.) More Rules of Inference Now we come to the rest of the rules by which theorems of this system constructed. A few rules of inference follow. In all of them, the symbols ´x´ and 'y' are always to be understood as restricted to well formed strings RULE OF SEPARATION: If < x'y> is a theorem, then both x and theorems. Incidentally, you should have a pretty good guess by now as to concept the symbol `A' stands for. (Hint: it is the troublesome word the preceding Dialogue.) From the following rule, you should be a figure out what concept the tilde ('~') represents: DOUBLETILDE RULE: The string '~~' can be deleted from any theorem. It can also be inserted into any theorem, provided that the rest string is itself wellformed.  
 
The Fantasy Rule Now a special feature of this system is that it has no axiomsonly rule you think back to the previous formal systems we've seen, you may w( how there can be any theorems, then. How does everything get started? The answer is that there is one rule which manufactures theorems from out of thin airit doesn't need an "old theorem" as input. (The rest of the do require input.) This special rule is called the fantasy rule. The reason I call it that is quite simple. To use the fantasy rule, the first thing you do is to write down an wellformed string x you like, and then "fantasize" by asking, "What if string x were an axiom, or a theorem?" And then, you let the system give an answer. That is, you go ahead and make a derivation with x ; opening line; let us suppose y is the last line. (Of course the derivation must strictly follow the rules of the system.) Everything from x to y (inclusive) is the fantasy; x is the premise of the fantasy, and y is its outcome. The next step is to jump out of the fantasy, having learned from it that out. If x were a theorem, y would be a theorem. Still, you might wonder, where is the real theorem? The real theorem is the string <xƒy> Notice the resemblance of this string to the sentence printed above To signal the entry into, and emergence from, a fantasy, one uses the  
 
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square brackets `[' and ']', respectively. Thus, whenever you see a left square bracket, you know you are "pushing" into a fantasy, and the next line will contain the fantasy's premise. Whenever you see a right square bracket, you know you are "popping" back out, and the preceding line was the outcome. It is helpful (though not necessary) to indent those lines of a derivation which take place in fantasies. Here is an illustration of the fantasy rule, in which the string P is taken as a premise. (It so happens that P is not a theorem, but that is of no import; we are merely inquiring, "What if it were?") We make the following fantasy: [ push into fantasy P premise ~~~P outcome (by double tilde rule) ] pop out of fantasy The fantasy shows that: If P were a theorem, so would ~~P be one. We now "squeeze" this sentence of English (the metalanguage) into the formal notation (the object language): <Pƒ~~P>. This, our first theorem of the Propositional Calculus, should reveal to you the intended interpretation of the symbol `ƒ'. Here is another derivation using the fantasy rule:  
 
 
 
It is important to understand that only the last line is a genuine theorem, hereeverything else is in the fantasy. Recursion and the Fantasy Rule As you might guess from the recursion terminology "push" and "pop", the fantasy rule can be used recursivelythus, there can be fantasies within fantasies, thricenested fantasies, and so on. This means that there are all sorts of "levels of reality", just as in nested stories or movies. When you pop out of a moviewithinamovie, you feel for a moment as if you had reached the real world, though you are still one level away from the top. Similarly, when you pop out of a fantasywithinafantasy, you are in a "realer" world than you had been, but you are still one level away from the top. Now a "No Smoking" sign inside a movie theater does not apply to the  
 
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characters in the moviethere is no carryover from the real world in fantasy world, in movies. But in the Propositional Calculus, then carryover from the real world into the fantasies; there is even carry from a fantasy to fantasies inside it. This is formalized by the following rule: CARRYOVER RULE: Inside a fantasy, any theorem from the "reality level higher can be brought in and used. It is as if a "No Smoking" sign in a theater applied not only to a moviegoers, but also to all the actors in the movie, and, by repetition of the same idea, to anyone inside multiply nested movies! (Warning: There carryover in the reverse direction: theorems inside fantasies cannot be exported to the exterior! If it weren't for this fact, you could write any as the first line of a fantasy, and then lift it out into the real world as a theorem.) To show how carryover works, and to show how the fantasy rule can be used recursively, we present the following derivation: [ push P premise of outer fantasy [ push again Q premise of inner fantasy P carryover of P into inner fantasy <P'Q> joining ] pop out of inner fantasy, regain outer fantasy <Qƒ<P'Q>> fantasy rule ] pop out of outer fantasy, reach real world! <Pƒ<Qƒ<P'Q>>> fantasy rule Note that I've indented the outer fantasy once, and the inner fantasy twice, to emphasize the nature of these nested "levels of reality". One to look at the fantasy rule is to say that an observation made about the system is inserted into the system. Namely, the theorem < xƒy> which gets produced can be thought of as a representation inside the system of the statement about the system "If x is a theorem, then y is too". To be specific, the intended interpretation for <PƒQ> is "if P, then Q equivalently, "P implies Q". The Converse of the Fantasy Rule Now Lewis Carroll's Dialogue was all about "ifthen" statements. In particular, Achilles had a lot of trouble in persuading the Tortoise to accept the second clause of an "ifthen" statement, even when the "ifthen" state itself was accepted, as well as its first clause. The next rule allows y infer the second "clause" of a'ƒ'string, provided that the `ƒ'string it a theorem, and that its first "clause" is also a theorem.  
 
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RULE OF DETACHMENT: If x and < xz>y> are both theorems, then y is a theorem. Incidentally, this rule is often called "Modus Ponens", and the fantasy rule is often called the "Deduction Theorem". The Intended Interpretation of the Symbols We might as well let the cat out of the bag at this point, and reveal the "meanings" of the rest of the symbols of our new system. In case it is not yet apparent, the symbol `A' is meant to be acting isomorphically to the normal, everyday word `and'. The symbol '' represents the word 'not'it is a formal sort of negation. The angle brackets '<' and `>' are grouperstheir function being very similar to that of parentheses in ordinary algebra. The main difference is that in algebra, you have the freedom to insert parentheses or to leave them out, according to taste and style, whereas in a formal system, such anarchic freedom is not tolerated. The symbol V' represents the word `or' ('vel' is a Latin word for `or'). The `or' that is meant is the socalled inclusive `or', which means that the interpretation of <xvy> is "either x or yor both". The only symbols we have not interpreted are the atoms. An atom has no single interpretationit may be interpreted by any sentence of English (it must continue to be interpreted by the same sentence if it occurs multiply within a string or derivation). Thus, for example, the wellformed string <Pa~P> could be interpreted by the compound sentence This mind is Buddha, and this mind is not Buddha. Now let us look at each of the theorems so far derived, and interpret them. The first one was <P3—P>. If we keep the same interpretation for P, we have the following interpretation: If this mind is Buddha, then it is not the case that this mind is not Buddha. Note how I rendered the double negation. It is awkward to repeat a negation in any natural language, so one gets around it by using two different ways of expressing negation. The second theorem we derived was «PaQ>3<QaP». If we let Q be interpreted by the sentence "This flax weighs three pounds", then our theorem reads as follows: If this mind is Buddha and this flax weighs three pounds, then this flax weighs three pounds and this mind is Buddha. The third theorem was <Pz<Q3<PaQ>>>. This one goes into the following nested "ifthen" sentence:  
 
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If this mind is Buddha, then, if this flax weighs three pounds, then this mind is Buddha and this flax weighs three pounds. You probably have noticed that each theorem, when interpreted, something absolutely trivial and selfevident. (Sometimes they are so s evident that they sound vacuous andparadoxically enoughconfusing or even wrong!) This may not be very impressive, but just remember there are plenty of falsities out there which could have been produced they weren't. This systemthe Propositional Calculussteps neatly ft truth to truth, carefully avoiding all falsities, just as a person who is concerned with staying dry will step carefully from one steppingstone creek to the next, following the layout of steppingstones no matter I twisted and tricky it might be. What is impressive is thatin the Propositional Calculusthe whole thing is done purely typographically. There is nobody down "in there", thinking about the meaning of the strings. It i! done mechanically, thoughtlessly, rigidly, even stupidly. Rounding Out the List of Rules We have not yet stated all the rules of the Propositional Calculus. The complete set of rules is listed below, including the three new ones. JOINING RULE: If x and y are theorems, then < xay> is a theorem. SEPARATION RULE: If < xay> is a theorem, then both x and y are theorems. DOUBLETILDE RULE: The string '~~' can be deleted from any theorem can also be inserted into any theorem, provided that the result string is itself wellformed. FANTASY RULE: If y can be derived when x is assumed to be a theorem then < xz>y> is a theorem. CARRYOVER RULE: Inside a fantasy, any theorem from the "reality" c level higher can be brought in and used. RULE OF DETACHMENT: If x and < xz>y> are both theorems, then y is a theorem. CONTRAPOSITIVE RULE: <xz)y> and <~yz>~x> are interchangeable DE MORGAN'S RULE: <~xA~y> and ~< xvy> are interchangeable. SWITCHEROO RULE: <xvy> and <~xz>y> are interchangeable. (The Switcheroo rule is named after Q. q. Switcheroo, an Albanian railroad engineer who worked in logic on the siding.) By "interchangeable" in foregoing rules, the following is meant: If an expression of one form occurs as either a theorem or part of a theorem, the other form may be  
 
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substituted, and the resulting string will also be a theorem. It must be kept in mind that the symbols ‘x’ and ‘y’ always stand for wellformed strings of the system. Justifying the Rules Before we see these rules used inside derivations, let us look at some very short justifications for them. You can probably justify them to yourself better than my examples  which is why I only give a couple. The contrapositive rule expresses explicitly a way of turning around conditional statements which we carry out unconsciously. For instance, the “Zentence” If you are studying it, then you are far from the Way Means the same thing as If you are close to the Way, then you are not studying it. De Morgan’s rule can be illustrated by our familiar sentence “The flag is not moving and the wind is not moving”. If P symbolizes “the flag is not moving”, and Q symbolizes “the wind is moving”, then the compound sentence is symbolized by <~Pa~Q>, which, according to Morgan’s law, is interchangeable with ~<PvQ>. whose interpretation would be “It is not true that either the flag or the wind is moving”. And no one could deny that it is a Zensible conclusion to draw. For the Switrcheroo rule, consider the sentence “Either a cloud is hanging over the mountain, or the moonlight is penetrating the waves of the lake,” which might be spoken, I suppose, by a wistful Zen master remembering a familiar lake which he can visualize mentally but cannot see. Now hang on to your seat, for the Swircheroo rule tells us that this is interchangeable with the thought “If a cloud is not hanging over the mountain, then the moonlight is penetrating the waves of the lake.” This may not be enlightenment, but it is the best the Propositional Calculus has to offer. Playing around with the system Now, let us apply these rules to a previous theorem, ands see what we get: For instance, take the theorem <P3—P>: <P=>—P> old theorem <_P_{3}~P>: contrapositive <~P3~P> doubletilde <Pv~P> switcheroo This new theorem, when interpreted, says:  
 
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Either this mind is Buddha, or this mind is not Buddha Once again, the interpreted theorem, though perhaps less than mind boggling, is at least true. SemiInterpretations It is natural, when one reads theorems of the Propositional Calculus out loud, to interpret everything but the atoms. I call this semiinterpreting. For example, the semiinterpretation of <Pv~P> would be P or not P. Despite the fact that P is not a sentence, the above semisentence still sounds true, because you can very easily imagine sticking any sentence in for P  and the form of the semiinterpreted theorem assures you that however you make your choice, the resulting sentence will be true. And that is the key idea of the Propositional Calculus: it produces theorems which, when semiinterpreted, are seen to be “universally true semisaentences”, by which is meant that no matter how you complete the interpretation, the final result will be a true statement. Ganto’s Ax Now we can do a more advanced exercise, based on a Zen koan called “Ganto’s Ax”. Here is how it began. One day Tokusan told his student Ganto, “I have two monks who have been here for many years. Go and examine them.” Ganto picked up an ax and went to the hut where the two monks were meditating. He raised the ax, saying “If you say a word, I will cut off your heads; and if you do not say a word, I will also cut off your heads.”^{1} If you say a word I will cut off this koan, and if you do not say a word, I will also cut off this koan  because I want you to translate some of it into our notation. Let us symbolize “you say a word” by P and “I will cut off your heads” by Q. Then Ganto’s ax threat is symbolized by the string «P3Q>a<~ P=>Q» What if this ax threat were an axiom? Here is a fantasy to answer that question.  
 
 
 
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The power of the Propositional Calculus is shown in this example. Why, in but two dozen steps, we have deduced Q: that the heads will be cut off! (Ominously, the rule last invoked was "detachment" ...) It might seem superfluous to continue the koan now, since we know what must ensue ... However, I shall drop my resolve to cut the koan off; it is a true Zen koan, after all. The rest of the incident is here related: Both monks continued their meditation as if he had not spoken. Ganto dropped the ax and said, "You are true Zen students." He returned to Tokusan and related the incident. "I see your side well," Tokusan agreed, "but tell me, how is their side?" "Tõzan may admit them," replied Ganto, "but they should not be admitted under Tokusan."2 Do you see my side well? How is the Zen side? Is There a Decision Procedure for Theorems? The Propositional Calculus gives us a set of rules for producing statements which would be true in all conceivable worlds. That is why all of its theorems sound so simpleminded; it seems that they have absolutely no content! Looked at this way, the Propositional Calculus might seem to be a waste of time, since what it tells us is absolutely trivial. On the other hand, it does it by specifying the form of statements that are universally true, and this throws a new kind of light onto the core truths of the universe: they are not only fundamental, but also regular: they can be produced by one set of typographical rules. To put it another way, they are all "cut from the same cloth". You might consider whether the same could be said about Zen koans: could they all be produced by one set of typographical rules? It is quite relevant here to bring up the question of a decision procedure. That is, does there exist any mechanical method to tell nontheorems from theorems? If so, that would tell us that the set of theorems of the  
 
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Propositional Calculus is not only r.e., but also recursive. It turns out that there is an interesting decision procedurethe method of truth u would take us a bit afield to present it here; you can find it in almost any standard book on logic. And what about Zen koans? Could there conceivably be a mechanical decision procedure which distinguishes genuine Zen koans from other things? Do We Know the System Is Consistent? Up till now, we have only presumed that all theorems, when interpreted as indicated, are true statements. But do we know that that is the case' we prove it to be? This is just another way of asking whether the intended interpretations ('and' for `'', etc.) merit being called the "passive meanings” of the symbols. One can look at this issue from two very different points of view, which might be called the "prudent" and "imprudent" points I will now present those two sides as I see them, personifying their as "Prudence" and "Imprudence". Prudence: We will only KNOW that all theorems come out true un intended interpretation if we manage to PROVE it. That is the c: thoughtful way to proceed. Imprudence: On the contrary. It is OBVIOUS that all theorems will come out true. If you doubt me, look again at the rules of the system. You will find that each rule makes a symbol act exactly as the word it represents ought to be used. For instance, the joining rule makes the symbol ‘'’ act as `and' ought to act; the rule of detachment makes `ƒ' act as it ought to, if it is to stand for 'implies', or 'ifthen'; and so on. Unless you are like the Tortoise, you will recognize in each rule a codification of a pattern you use in your own thought patterns. So if you trust your own thought patterns, then you HAVE to believe that all theorems come out true! That's the way I see it. I don't need any further proof. If you think that some theorem comes out false, then presumably you think that some rule must be wrong. Show me which one. Prudence: I'm not sure that there is any faulty rule, so I can't point one out to you. Still, I can imagine the following kind of scenario. You, following the rules, come up with a theorem  say x. Meanwhile I, also following the rules, come up with another theoremit happens to be ~x. Can't you force yourself to conceive of that? Imprudence: All right; let's suppose it happened. Why would it bother you? Or let me put it another way. Suppose that in playing with the MIUsystem, I came up with a theorem x, and you came up with xU Can you force yourself to conceive of that? Prudence: Of coursein fact both MI and MIU are theorems. Imprudence: Doesn't that bother you? Prudence: Of course not. Your example is ridiculous, because MI and MIU are not CONTRADICTORY, whereas two strings x and ~x in the Propositional Calculus ARE contradictory.  
 
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Imprudence: Well, yes  provided you wish to interpret `~' as `not'. But what would lead you to think that '~' should be interpreted as `not'? Prudence: The rules themselves. When you look at them, you realize that the only conceivable interpretation for '~' is 'not'and likewise, the only conceivable interpretation for `'' is `and', etc. Imprudence: In other words, you are convinced that the rules capture the meanings of those words? Prudence: Precisely. Imprudence: And yet you are still willing to entertain the thought that both x and ~x could be theorems? Why not also entertain the notion that hedgehogs are frogs, or that 1 equals 2, or that the moon is made of green cheese? I for one am not prepared even to consider whether such basic ingredients of my thought processes are wrong because if I entertained that notion, then I would also have to consider whether my modes of analyzing the entire question are also wrong, and I would wind up in a total tangle. Prudence: Your arguments are forceful ... Yet I would still like to see a PROOF that all theorems come out true, or that x and ~x can never both be theorems. Imprudence: You want a proof. I guess that means that you want to be more convinced that the Propositional Calculus is consistent than you are convinced of your own sanity. Any proof I could think of would involve mental operations of a greater complexity than anything in the Propositional Calculus itself. So what would it prove? Your desire for a proof of consistency of the Propositional Calculus makes me think of someone who is learning English and insists on being given a dictionary which definers all the simple words in terms of complicated ones... The Carroll Dialogue Again This little debate shows the difficulty of trying to use logic and reasoning to defend themselves. At some point, you reach rock bottom, and there is no defense except loudly shouting, "I know I'm right!" Once again, we are up against the issue which Lewis Carroll so sharply set forth in his Dialogue: you can't go on defending your patterns of reasoning forever. There comes a point where faith takes over. A system of reasoning can be compared to an egg. An egg has a shell which protects its insides. If you want to ship an egg somewhere, though, you don't rely on the shell. You pack the egg in some sort of container, chosen according to how rough you expect the egg's voyage to be. To be extra careful, you may put the egg inside several nested boxes. However, no matter how many layers of boxes you pack your egg in, you can imagine some cataclysm which could break the egg. But that doesn't mean that you'll never risk transporting your egg. Similarly, one can never give an ultimate, absolute proof that a proof in some system is correct. Of course,  
 
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one can give a proof of a proof, or a proof of a proof of a proof  but the validity of the outermost system always remains an unproven assumption, accepted on faith. One can always imagine that some unsuspected subtlety will invalidate every single level of proof down to the bottom, and tI "proven" result will be seen not to be correct after all. But that doesn’t mean that mathematicians and logicians are constantly worrying that the whole edifice of mathematics might be wrong. On the other hand, unorthodox proofs are proposed, or extremely lengthy proofs, or proofs generated by computers, then people do stop to think a bit about what they really mean by that quasisacred word "proven". An excellent exercise for you at this point would be to go back Carroll Dialogue, and code the various stages of the debate into our notation  beginning with the original bone of contention: Achilles: If you have «AaB>3Z>, and you also have <AaB>, then surely you have Z. Tortoise: Oh! You mean: ««AaB>3Z>a<AaB»3Z>, : don't you? (Hint: Whatever Achilles considers a rule of inference, the Tortoise immediately flattens into a mere string of the system. If you use or letters A, B, and Z, you will get a recursive pattern of longer and strings.) Shortcuts and Derived Rules When carrying out derivations in the Propositional Calculus, one quickly invents various types of shortcut, which are not strictly part of the system For instance, if the string <Qv~Q> were needed at some point, and <Pv~P> had been derived earlier, many people would proceed as if <Qv~Q> had been derived, since they know that its derivation is an exact parallel to that of <Pv~P>. The derived theorem is treated as a "theorem schema" a mold for other theorems. This turns out to be a perfect valid procedure, in that it always leads you to new theorems, but it is not a rule of the Propositional Calculus as we presented it. It is, rather, a derived rule, It is part of the knowledge which we have about the system. That this rule keeps you within the space of theorems needs proof, of coursebut such a proof is not like a derivation inside the system. It is a proof in the ordinary, intuitive sense  a chain of reasoning carried out in the Imode. The theory about the Propositional Calculus is a "metatheory", and results in it can be called "metatheorems" Theorems about theorems. (Incidentally, note the peculiar capitalization in the phrase "Theorems about theorems". It is a consequence of our convention: metatheorems are Theorems (proven results) concerning theorems (derivable strings).) In the Propositional Calculus, one could discover many metatheorems, or derived rules of inference. For instance, there is a De Morgan's Rule:  
 
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<~x(~y> and ~<x'y> are interchangeable. If this were a rule of the system, it could speed up many derivations considerably. But if we prove that it is correct, isn't that good enough? Can't we use it just like a rule of inference, from then on? There is no reason to doubt the correctness of this particular derived rule. But once you start admitting derived rules as part of your procedure in the Propositional Calculus, you have lost the formality of the system, since derived rules are derived informallyoutside the system. Now formal systems were proposed as a way to exhibit every step of a proof explicitly, within one single, rigid framework, so that any mathematician could check another's work mechanically. But if you are willing to step outside of that framework at the drop of a hat, you might as well never have created it at all. Therefore, there is a drawback to using such shortcuts. Formalizing Higher Levels On the other hand, there is an alternative way out. Why not formalize the metatheory, too? That way, derived rules (metatheorems) would be theorems of a larger formal system, and it would be legitimate to look for shortcuts and derive them as theoremsthat is, theorems of the formalized metatheorywhich could then be used to speed up the derivations of theorems of the Propositional Calculus. This is an interesting idea, but as soon as it is suggested, one jumps ahead to think of metametatheories, and so on. It is clear that no matter how many levels you formalize, someone will eventually want to make shortcuts in the top level. It might even be suggested that a theory of reasoning could be identical to its own metatheory, if it were worked out carefully. Then, it might seem, all levels would collapse into one, and thinking about the system would be just one way of working in the system! But it is not that easy. Even if a system can "think about itself", it still is not outside itself. You, outside the system, perceive it differently from the way it perceives itself. So there still is a metatheorya view from outsideeven for a theory which can "think about itself" inside itself. We will find that there are theories which can "think about themselves". In fact, we will soon see a system in which this happens completely accidentally, without our even intending it! And we will see what kinds of effects this produces. But for our study of the Propositional Calculus, we will stick with the simplest ideasno mixing of levels. Fallacies can result if you fail to distinguish carefully between working in the system (the Mmode) and thinking about the system (the Imode). For example, it might seem perfectly reasonable to assume that, since <P(~P> (whose semiinterpretation is "either P or not P") is a theorem, either P or ~P must be a theorem. But this is dead wrong: neither one of the latter pair is a theorem. In general, it is a dangerous practice to assume that symbols can be slipped back and forth between different levelshere, the language of the formal system and its metalanguage (English).  
 
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Reflections on the Strengths and Weaknesses of the System You have now seen one example of a system with a purposeto re part of the architecture of logical thought. The concepts which this handles are very few in number, and they are very simple, precise co But the simplicity and precision of the Propositional Calculus are the kinds of features which make it appealing to mathematicians. There are two reasons for this. (1) It can be studied for its own properties, ex geometry studies simple, rigid shapes. Variants can be made on it, employing different symbols, rules of inference, axioms or axiom schemata on. (Incidentally, the version of the Propositional Calculus here pr is related to one invented by G. Gentzen in the early 1930's. The other versions in which only one rule of inference is useddetachment usuallyand in which there are several axioms, or axiom schemata study of ways to carry out propositional reasoning in elegant formal systems is an appealing branch of pure mathematics. (2) The Propositional Calculus can easily be extended to include other fundamental aspects of reasoning. Some of this will be shown in the next Chapter, where the Propositional Calculus is incorporated lock, stock and barrel into a much larger and deeper system in which sophisticated numbertheoretical reasoning can be done. Proofs vs. Derivations The Propositional Calculus is very much like reasoning in some w one should not equate its rules with the rules of human thought. A proof is something informal, or in other words a product of normal thought written in a human language, for human consumption. All sorts of complex features of thought may be used in proofs, and, though they may “feel right", one may wonder if they can be defended logically. That is really what formalization is for. A derivation is an artificial counterpart of and its purpose is to reach the same goal but via a logical structure whose methods are not only all explicit, but also very simple. If  and this is usually the case it happens that a formal derivation is extremely lengthy compared with the corresponding "natural" proof that is just too bad. It is the price one pays for making each step so simple. What often happens is that a derivation and a proof are "simple" in complementary senses of the word. The proof is simple in that each step sounds right", even though one may not know just why; the derivation is simple in that each of its myriad steps is considered so trivial that it is beyond reproach, and since the whole derivation consists just of such trivial steps it is supposedly errorfree. Each type of simplicity, however, brings along a characteristic type of complexity. In the case of proofs, it is the complexity of the underlying system on which they rest namely, human language  and in the case of derivations, it is their astronomical size, which makes them almost impossible to grasp. Thus, the Propositional Calculus should be thought of as part of a  
 
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general method for synthesizing artificial prooflike structures. It does not, however, have much flexibility or generality. It is intended only for use in connection with mathematical conceptswhich are themselves quite rigid. As a rather interesting example of this, let us make a derivation in which a very peculiar string is taken as a premise in a fantasy: <P'~P>. At least its semiinterpretation is peculiar. The Propositional Calculus, however, does not think about semiinterpretations; it just manipulates strings typographicallyand typographically, there is really nothing peculiar about this string. Here is a fantasy with this string as its premise:  
 
 
 
Now this theorem has a very strange semiinterpretation: P and not P together imply Q Since Q is interpretable by any statement, we can loosely take the theorem to say that "From a contradiction, anything follows"! Thus, in systems based on the Propositional Calculus, contradictions cannot be contained; they infect the whole system like an instantaneous global cancer. The Handling of Contradictions This does not sound much like human thought. If you found a contradiction in your own thoughts, it's very unlikely that your whole mentality would break down. Instead, you would probably begin to question the beliefs or modes of reasoning which you felt had led to the contradictory thoughts. In other words, to the extent you could, you would step out of the systems inside you which you felt were responsible for the contradiction, and try to repair them. One of the least likely things for you to do would be to throw up your arms and cry, "Well, I guess that shows that I believe everything now!" As a joke, yesbut not seriously. Indeed, contradiction is a major source of clarification and progress in all domains of lifeand mathematics is no exception. When in times past, a  
 
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contradiction in mathematics was found, mathematicians would immediately seek to pinpoint the system responsible for it, to jump out of it, to reason about it, and to amend it. Rather than weakening mathematics, the discovery and repair of a contradiction would strengthen it. This might take time and a number of false starts, but in the end it would yield fruit. For instance, in the Middle Ages, the value of the infinite series 1  1 + 1  1 + 1 . .. was hotly disputed. It was "proven" to equal 0, 1, ½, and perhaps other values. Out of such controversial findings came a fuller, deeper about infinite series. A more relevant example is the contradiction right now confronting usnamely the discrepancy between the way we really think, and t the Propositional Calculus imitates us. This has been a source of discomfort for many logicians, and much creative effort has gone into trying to patch up the Propositional Calculus so that it would not act so stupidly and inflexibly. One attempt, put forth in the book Entailment by A. R. Anderson and N. Belnap,^{3} involves "relevant implication", which tries to make the symbol for "ifthen" reflect genuine causality, or at least connect meanings. Consider the following theorems of the Propositional Calculus <P3<Q3P» <P=><Qv~P» «Pa~P>3Q> «P3Q>v<Q=>P»  
 
They, and many others like them, all show that there need be no relationship at all between the first and second clauses of an ifthen statement for it to be provable within the Propositional Calculus. In protest, "relevant implication" puts certain restrictions on the contexts in which the rules of inference can be applied. Intuitively, it says that "something can only be derived from something else if they have to do with each other”. For example, line 10 in the derivation given above would not be allowed in such a system, and that would block the derivation of the <<P'~P >ƒQ> More radical attempts abandon completely the quest for completeness or consistency, and try to mimic human reasoning with all its inconsistencies. Such research no longer has as its goal to provide a solid underpinning for mathematics, but purely to study human thought processes. Despite its quirks, the Propositional Calculus has some feat recommend itself. If one embeds it into a larger system (as we will do next Chapter), and if one is sure that the larger system contains no contradictions (and we will be), then the Propositional Calculus does all that one could hope: it provides valid propositional inferences  all that can be made. So if ever an incompleteness or an inconsistency is uncovered, can be sure that it will be the fault of the larger system, and not of its subsystem which is the Propositional Calculus.  
 
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FIGURE 42. “Crab Canon”, by M. C. Escher (~1965)  
 
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Crab Canon Achilles and the Tortoise happen upon each other in the park one day while strolling. Tortoise: Good day, Mr. A. Achilles: Why, same to you. Tortoise: So nice to run into you. Achilles: That echoes my thoughts. Tortoise: And it's a perfect day for a walk. I think I'll be walking home soon. Achilles: Oh, really? I guess there's nothing better for you than w Tortoise: Incidentally, you're looking in very fine fettle these days, I must say. Achilles: Thank you very much. Tortoise: Not at all. Here, care for one of my cigars? Achilles: Oh, you are such a philistine. In this area, the Dutch contributions are of markedly inferior taste, don't you think? Tortoise: I disagree, in this case. But speaking of taste, I finally saw that Crab Canon by your favorite artist, M. C. Escher, in a gallery the other day, and I fully appreciate the beauty and ingenuity with which he made one single theme mesh with itself going both backwards and forwards. But I am afraid I will always feel Bach is superior to Escher.  
 
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Achilles: I don't know. But one thing for certain is that I don't worry about arguments of taste. De gustibus non est disputandum. Tortoise: Tell me, what's it like to be your age? Is it true that one has no worries at all? Achilles: To be precise, one has no frets. Tortoise: Oh, well, it's all the same to me. Achilles: Fiddle. It makes a big difference, you know. Tortoise: Say, don't you play the guitar? Achilles: That's my good friend. He often plays, the fool. But I myself wouldn't touch a guitar with a tenfoot pole! (Suddenly, the Crab, appearing from out of nowhere, wanders up excitedly, pointing to a rather prominent black eye.) Crab: Hallo! Hulloo! What's up? What's new? You see this bump, this lump? Given to me by a grump. Ho! And on such a fine day. You see, I was just idly loafing about the park when up lumbers this giant fellow from Warsawa colossal bear of a manplaying a lute. He was three meters tall, if I'm a day. I mosey on up to the chap, reach skyward and manage to tap him on the knee, saying, "Pardon me, sir, but you are Poleluting our park with your mazurkas." But wow! he had no sense of humornot a bit, not a witand POW!he lets loose and belts me one, smack in the eye! Were it in my nature, I would crab up a storm, but in the timehonored tradition of my species, I backed off. After all, when we walk forwards, we move backwards. It's in our genes, you know, turning round and round. That reminds meI've always wondered, "Which came firstthe Crab, or the Gene?" That is to say, "Which came last the Gene, or the Crab?" I'm always turning things round and round, you know. It's in our genes, after all. When we walk backwards, we move forwards. Ah me, oh my! I must lope along on my merry wayso off I go on such a fine day. Sing "ho!" for the life of a Crab! TATA! iOle! (And he disappears as suddenly as he arrived.) Tortoise: That's my good friend. He often plays the fool. But I myself wouldn't touch a tenfoot Pole with a guitar! Achilles: Say, don't you play the guitar? Tortoise: Fiddle. It makes a big difference, Achilles: Oh, well, it's all the same to me. Tortoise: To be precise, one has no frets. Achilles: Tell me, what's it like to be your age? Is it true that one has no worries at all? Tortoise: I don't know. But one thing for certain is that I don't worry about arguments of taste. Disputandum non est de gustibus.  
 
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FIGURE 43. Here is a short section one of the Crab's Genes, turning round and round. When the two DNA strands are raveled and laid out side by side, they read this way: ….TTTTTTTTTCGAAAAAAAAA ….AAAAAAAAGCTTTTTTTTTT Notice that they are the same, only one forwards while the other goes backwards This is the defining property of the form called "crab canon" in music. It is reminiscent of, though a little different from palindrome, which is a sentence that reads the same backwards and forwards ,In molecular biology, such segments of DNA are called "palindromes "a slight misnomer, since "crab canon" would be more accurate. Not only is this DNA segment crabcanonicalbut moreover its base sequence codes for the Dialogue's structure Look carefully!  
 
Achilles: I disagree, in this case. But speaking of taste, I finally heard that Crab Canon by your favorite composer, J. S. Bach, in a concert other day, and I fully appreciate the beauty and ingenuity with which he made one single theme mesh with itself going both backwards and forwards. But I'm afraid I will always feel Escher is superior to Bach Tortoise: Oh, you are such a philistine. In this area, the Dutch contributions are of markedly inferior taste, don't you think? Achilles: Not at all. Here, care for one of my cigars? Tortoise: Thank you very much. Achilles: Incidentally, you're looking in very fine fettle these days, I must say.  
 
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Tortoise: Oh, really? I guess there's nothing better for you than walking. Achilles: And it's a perfect day for a walk. I think I'll be walking home soon. Tortoise: That echoes my thoughts. Achilles: So nice to run into you. Tortoise: Why, same to you. Achilles: Good day, Mr. T  
 
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CHAPTER VIII  
 
Typographical Number Theory  
 
The Crab Canon and Indirect SelfReference THREE EXAMPLES OF indirect selfreference are found in the Crab Canon. Achilles and the Tortoise both describe artistic creations they knowand, quite accidentally, those creations happen to have the same structure as the Dialogue they're in. (Imagine my surprise, when I, the author, noticed this!) Also, the Crab describes a biological structure and that, too, has the same property. Of course, one could read the Dialogue and understand it and somehow fail to notice that it, too, has the form of a crab canon. This would be understanding it on one level, but not on another. To see the selfreference, one has to look at the form, as well as the content, of the Dialogue. Gödel’s construction depends on describing the form, as well as the content, of strings of the formal system we shall define in this Chapter  Typographical Number Theory (TNT). The unexpected twist is that, because of the subtle mapping which Gödel discovered, the form of strings can be described in the formal system itself. Let us acquaint ourselves with this strange system with the capacity for wrapping around. What We Want to Be Able to Express in TNT We'll begin by citing some typical sentences belonging to number theory; then we will try to find a set of basic notions in terms of which all our sentences can be rephrased. Those notions will then be given individual symbols. Incidentally, it should be stated at the outset that the term "number theory" will refer only to properties of positive integers and zero (and sets of such integers). These numbers are called the natural numbers. Negative numbers play no role in this theory. Thus the word "number", when used, will mean exclusively a natural number. And it is important  vitalfor you to keep separate in your mind the formal system (TNT) and the rather illdefined but comfortable old branch of mathematics that is number theory itself; this I shall call "N". Some typical sentences of Nnumber theoryare: (1) 5 is prime. (2) 2 is not a square. (3) 1729 is a sum of two cubes. (4) No sum of two positive cubes is itself a cube. (5) There are infinitely many prime numbers. (6) 6 is even.  
 
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Now it may seem that we will need a symbol for each notion such as "prime” or "cube" or "positive"  but those notions are really not primitive. Primeness, for instance, has to do with the factors which a number has, which in turn has to do with multiplication. Cubeness as well is defined in terms multiplication. Let us rephrase the sentences, then, in terms of what seem to be more elementary notions. (1) There do not exist numbers a and b, both greater than 1. such that 5 equals a times b. (2) There does not exist a number b, such that b times b equals 2. (3) There exist numbers b and c such that b times b times b, plus c times c times c, equals 1729. (4') For all numbers b and c, greater than 0, there is no number a such that a times a times a equals b times b times b plus c times c times c. (5) For each number a, there exists a number b, greater than a, with the property that there do not exist numbers c and d, both greater than 1, such that b equals c times d. (6') There exists a number e such that 2 times e equals 6. This analysis has gotten us a long ways towards the basic elements of language of number theory. It is clear that a few phrases reappear over a over: for all numbers b there exists a number b, such that greater than equals times plus 0, 1, 2, . . Most of these will be granted individual symbols. An exception is "greater than", which can be further reduced. In fact, the sentence "a is greater than b" becomes there exists a number c, not equal to 0, such that a equals b plus c. Numerals We will not have a distinct symbol for each natural number. Instead, we have a very simple, uniform way of giving a compound symbol to e natural number  very much as we did in the pqsystem. Here is notation for natural numbers:  
 
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zero: 0 one: SO two: SSO three: SSSO etc. The symbol S has an interpretation"the successor of". Hence, the interpretation of SSO is literally "the successor of the successor of zero". Strings of this form are called numerals. Variables and Terms Clearly, we need a way of referring to unspecified, or variable, numbers. For that, we will use the letters a, b, c, d, e. But five will not be enough. We need an unlimited supply of them, just as we had of atoms in the Propositional Calculus. We will use a similar method for making more variables: tacking on any number of primes. (Note: Of course the symbol "'read "prime"is not to be confused with prime numbers!) For instance: e d' c" b´´´... a´´´´ are all variables. In a way it is a luxury to use the first five letters of the alphabet when we could get away with just a and the prime. Later on, I will actually drop b, c, d, and e, which will result in a sort of "austere" version of TNTaustere in the sense that it is a little harder to decipher complex formulas. But for now we'll be luxurious. Now what about addition and multiplication? Very simple: we will use the ordinary symbols `+' and `•'. However, we will also introduce a parenthesizing requirement (we are now slowly slipping into the rules which define wellformed strings of TNT). To write "b plus c" and "b times c", for instance, we use the strings (b+c) (b • c) There is no laxness about such parentheses; to violate the convention is to produce a nonwellformed formula. ("Formula"? I use the term instead of "string" because it is conventional to do so. A formula is no more and no less than a string of TNT.) Incidentally, addition and multiplication are always to be thought of as binary operationsthat is, they unite precisely two numbers, never three or more. Hence, if you wish to translate "1 plus 2 plus 3", you have to decide which of the following two expressions you want:  
 
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(SO+(SSO+SSSO)) ((SO+SSO)+SSSO) The next notion we'll symbolize is equals. That is very simple: we use ´=´.The advantage of taking over the standard symbol used N  nonformal number theory  iis obvious: easy legibility. The disadvantage is very much like the disadvantage of using the words "point" a "line" in a formal treatment of geometry: unless one is very conscious a careful, one may blur the distinction between the familiar meaning and strictly rulegoverned behavior of the formal symbol. In discuss geometry, I distinguished between the everyday word and the formal to by capitalizing the formal term: thus, in elliptical geometry, a POINT was 1 union of two ordinary points. Here, there is no such distinction; hen mental effort is needed not to confuse a symbol with all of the association is laden with. As I said earlier, with reference to the pqsystem: the string  is not the number 3, but it acts isomorphically to 3, at least in the context of additions. Similar remarks go for the string SSSO. Atoms and Propositional Symbols All the symbols of the Propositional Calculus except the letters used making atoms (P, Q, and R) will be used in TNT, and they retain their interpretations. The role of atoms will be played by strings which, when interpreted, are statements of equality, such as SO=SSO or (SO • SO) Now, we have the equipment to do a fair amount of translation of simple sentences into the notation of TNT: 2 plus 3 equals 4: (SSO+SSSO)=SSSSO 2 plus 2 is not equal to 3: ~(SSO+SSO)=SSSO If 1 equals 0, then 0 equals 1: <SO=OJO=SO> The first of these strings is an atom; the rest are compound formulas (Warning: The `and' in the phrase "I and 1 make 2" is just another word for `plus', and must be represented by `+' (and the requisite parentheses).) Free Variables and Quantifiers All the wellformed formulas above have the property that their interpretations are sentences which are either true or false. There are, however, wellformed formulas which donot have that property, such as this one (b+SO)=SSO Its interpretation is "b plus 1 equals 2". Since b is unspecified, there is way to assign a truth value to the statement. It is like an outofcontext statement with a pronoun, such as "she is clumsy". It is neither true nor false; it is waiting for you to put it into a context. Because it is neither true nor false, such a formula is called open, and the variable b is called a free variable.  
 
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One way of changing an open formula into a closed formula, or sentence, is by prefixing it with a quantifiereither the phrase "there exists a number b such that , or the phrase "for all numbers b". In the first instance, you get the sentence There exists a number b such that b plus 1 equals 2. Clearly this is true. In the second instance, you get the sentence For all numbers b, b plus 1 equals 2. Clearly this is false. We now introduce symbols for both of these quantifiers. These sentences are translated into TNTnotation as follows: b:(b+SO)=SSO ('' stands for `exists'.)
It is very important to note that these statements are no longer about unspecified numbers; the first one is an assertion of existence, and the second one is a universal assertion. They would mean the same thing, even if written with c instead of b: c:(c+SO)=SSO ` A variable which is under the dominion of a quantifier is called a quantified variable. The following two formulas illustrate the difference between free variables and quantified variables: (b.b)=SSO (open) b:(b•b)=SSO (closed; a sentence of TNT) The first one expresses a property which might be possessed by some natural number. Of course, no natural number has that property. And that is precisely what is expressed by the second one. It is very crucial to understand this difference between a string with a free variable, which expresses a property, and a string where the variable is quantified, which expresses a truth or falsity. The English translation of a formula with at least one free variablean open formulais called a predicate. It is a sentence without a subject (or a sentence whose subject is an outofcontext pronoun). For instance, "is a sentence without a subject" "would be an anomaly" "runs backwards and forwards simultaneously" "improvised a sixpart fugue on demand" are nonarithmetical predicates. They express properties which specific entities might or might not possess. One could as well stick on a "dummy  
 
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subject", such as "soandso". A string with free variables is like a predicate with "soandso" as its subject. For instance, (SO+SO)=b is like saying "1 plus 1 equals soandso". This is a predicate in the variable b. It expresses a property which the number b might have. If one wet substitute various numerals for b, one would get a succession of forms most of which would express falsehoods. Here is another example of difference between open formulas and sentences: ` The above formula is a sentence representing, of course, the commutativity of addition. On the other hand, ` is an open formula, since b is free. It expresses a property which unspecified number b might or might not have  namely of commuting with all numbers c. Translating Our Sample Sentences This completes the vocabulary with which we will express all num theoretical statements! It takes considerable practice to get the hang of expressing complicated statements of N in this notation, and converse] figuring out the meaning of wellformed formulas. For this reason return to the six sample sentences given at the beginning, and work their translations into TNT. By the way, don't think that the translations given below are uniquefar from it. There are many  infinitely many  ways to express each one. Let us begin with the last one: "6 is even". This we rephrased in to of more primitive notions as "There exists a number e such that 2 times e equals 6". This one is easy : e:(SSO. e)=SSSSSSO Note the necessity of the quantifier; it simply would not do to write (SSO . e)=SSSSSSO alone. This string's interpretation is of course neither true nor false; it expresses a property which the number e might have. It is curious that, since we know multiplication is commutative might easily have written e:(e  SSO)=SSSSSSO instead. Or, knowing that equality is a symmetrical relation, we might 1 chosen to write the sides of the equation in the opposite order:  
 
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e:SSSSSSO=(SSO • e) Now these three translations of "6 is even" are quite different strings, and it is by no means obvious that theoremhood of any one of them is tied to theoremhood of any of the others. (Similarly, the fact that pq was a theorem had very little to do with the fact that its "equivalent" string pq was a theorem. The equivalence lies in our minds, since, as humans, we almost automatically think about interpretations, not structural properties of formulas.) We can dispense with sentence 2: "2 is not a square", almost immediately: b:(b • b)=SSO However, once again, we find an ambiguity. What if we had chosen to write it this way? Vb: (b • b) =SSO The first way says, "It is not the case that there exists a number b with the property that b's square is 2", while the second way says, "For all numbers b, it is not the case that b's square is 2." Once again, to us, they are conceptually equivalentbut to TNT, they are distinct strings. Let us proceed to sentence 3: "1729 is a sum of two cubes." This one will involve two existential quantifiers, one after the other, as follows: b:c:SSSSSS…………SSSSSO=(((b • b) • b)+((c • c) • c)) 1729 of them There are alternatives galore. Reverse the order of the quantifiers; switch the sides of the equation; change the variables to d and e; reverse the addition; write the multiplications differently; etc., etc. However, I prefer the following two translations of the sentence: b:c:(((SSSSSSSSSSO.SSSSSSSSSSO).SSSSSSSSSSO)+ ((SSSSSSSSSO • SSSSSSSSSO) • SSSSSSSSSO))=(((b • b) • b)+((c • c) • c)) and b:c:(((SSSSSSSSSSSSO.SSSSSSSSSSSSO). SSSSSSSSSSSSO)+ ((SO •SO) • SO))=(((b •b) •b)+((c • c) •c)) Do you see why? Tricks of the Trade Now let us tackle the related sentence 4: "No sum of two positive cubes is itself a cube". Suppose that we wished merely to state that 7 is not a sum of two positive cubes. The easiest way to do this is by negating the formula  
 
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which asserts that 7 is a sum of two positive cubes. This will be just like the preceding sentence involving 1729, except that we have to add in the proviso of the cubes being positive. We can do this with a trick: prefix variables with the symbol S, as follows: b:c:SSSSSSSO=(((Sb • Sb) • Sb)+((Sc • Sc) Sc)) You see, we are cubing not b and c, but their successors, which must be positive, since the smallest value which either b or c can take on is zero. Hence the righthand side represents a sum of two positive cubes. In( tally, notice that the phrase "there exist numbers b and c such that…..”) when translated, does not involve the symbol `n' which stands for ‘and’. That symbol is used for connecting entire wellformed strings, not for joining two quantifiers. Now that we have translated "7 is a sum of two positive cubes", we wish to negate it. That simply involves prefixing the whole thing by a single (Note: you should not negate each quantifier, even though the desired phrase runs "There do not exist numbers b and c such that ...".) Thus we get: b:c:SSSSSSSO=(((Sb • Sb) • Sb)+((Sc Sc) Sc)) Now our original goal was to assert this property not of the number of all cubes. Therefore, let us replace the numeral SSSSSSSO by the ((aa)a), which is the translation of "a cubed": b:c:((a •a) •a)=(((Sb •Sb) • Sb)+((Sc Sc) Sc)) At this stage, we are in possession of an open formula, since a is still free. This formula expresses a property which a number a might or might not haveand it is our purpose to assert that all numbers do have that property. That is simple  just prefix the whole thing with a universal quantifier
An equally good translation would be this: a:b:c:((aa) a)=(((Sb•Sb)•Sb)+((Sc•Sc)•Sc)) In austere TNT, we could use a' instead of b, and a" instead of c, and the formula would become: a: a': a":((a • a) • a) =(((Sa' • Sa') • Sa') +((Sa" • Sa") • Sa")) What about sentence 1: "5 is prime"? We had reworded it in this way "There do not exist numbers a and b, both greater than 1, such equals a times b". We can slightly modify it, as follows: "There do not exist numbers a and b such that 5 equals a plus 2, times b plus 2". This is another tricksince a and b are restricted to natural number values, this is an adequate way to say the same thing. Now "b plus 2" could be translated into  
 
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(b+SSO), but there is a shorter way to write it  namely, SSb. Likewise, "c plus 2" can be written SSc. Now, our translation is extremely concise: b: c:SSSSSO=(SSb • SSc) Without the initial tilde, it would be an assertion that two natural numbers do exist, which, when augmented by 2, have a product equal to 5. With the tilde in front, that whole statement is denied, resulting in an assertion that 5 is prime. If we wanted to assert that d plus e plus 1, rather than 5, is prime, the most economical way would be to replace the numeral for 5 by the string (d+Se): b: c:(d+Se)=(SSb SSc) Once again, an open formula, one whose interpretation is neither a true nor a false sentence, but just an assertion about two unspecified numbers, d and e. Notice that the number represented by the string (d+Se) is necessarily greater than d, since one has added to d an unspecified but definitely positive amount. Therefore, if we existentially quantify over the variable e, we will have a formula which asserts that: There exists a number which is greater than d and which is prime. e: b:3c:(d+Se)=(SSb • SSc) Well, all we have left to do now is to assert that this property actually obtains, no matter what d is. The way to do that is to universally quantify over the variable d: Vd:3e:3b:3c:(d+Se)=(SSb •SSc) That's the translation of sentence 5!  
 
Translation Puzzles for You This completes the exercise of translating all six typical numbertheoretical sentences. However, it does not necessarily make you an expert in the notation of TNT. There are still some tricky issues to be mastered. The following six wellformed formulas will test your understanding of TNT notation. What do they mean? Which ones are true (under interpretation, of course), and which ones are false? (Hint: the way to tackle this exercise is to move leftwards. First, translate the atom; next, figure out what adding a single quantifier or a tilde does; then move leftwards, adding another quantifier or tilde; then move leftwards again, and do the same.) 
 
 
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(Second hint: Either four of them are true and two false, or four false and two true.) How to Distinguish True from False? At this juncture, it is worthwhile pausing for breath and contempt what it would mean to have a formal system that could sift out the true from the false ones. This system would treat all these stringswhich look like statementsas designs having form, but no content. An( system would be like a sieve through which could pass only designs v special stylethe "style of truth". If you yourself have gone through ti formulas above, and have separated the true from the false by this about meaning, you will appreciate the subtlety that any system would to have, that could do the same thingbut typographically! The bout separating the set of true statements from the set of false statements written in the TNTnotation) is anything but straight; it is a boundary with many treacherous curves (recall Fig. 18), a boundary of which mathematicians have delineated stretches, here and there, working over hundreds years. Just think what a coup it would be to have a typographical m( which was guaranteed to place any formula on the proper side o border! The Rules of WellFormedness It is useful to have a table of Rules of Formation for wellformed formulas This is provided below. There are some preliminary stages, defining numerals, variables, and terms. Those three classes of strings are ingredients of wellformed formulas, but are not in themselves wellformed. The smallest wellformed formulas are the atoms; then there are ways of compounding atoms. Many of these rules are recursive lengthening rules, in that they take as input an item of a given class and produce a longer item of the class. In this table, I use `x' and 'y' to stand for wellformed formulas, and `s', `t', and `u' to stand for other kinds of TNTstrings. Needless to say, none of these five symbols is itself a symbol of TNT. NUMERALS. 0 is a numeral. A numeral preceded by S is also a numeral. Examples: 0 SO S50 SSSO SSSSO SSSSSO  
 
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VARIABLES. a is a variable. If we're not being austere, so are b, c, d and e. A variable followed by a prime is also a variable. Examples: ah' c" d^{m} a"" TERMS. All numerals and variables are terms. A term preceded by S is also a term. If s and t are terms, then so are (s+ t) and (s • t). Examples: 0 b SSa' (SO • (SSO+c)) S(Sa • (Sb • Sc)) TERMS may be divided into two categories: (1) DEFINITE terms. These contain no variables. Examples: 0 (SO+SO) SS((SSO.SSO)+(SO.SO)) (2) INDEFINITE terms. These contain variables. Examples: b Sa (b+SO) (((SO+SO)+SO)+e) The above rules tell how to make parts of wellformed formulas; the remaining rules tell how to make complete wellformed formulas. ATOMS. If s and t are terms, then s = t is an atom. Examples: SO=0 (SS0+SS0)=5SSS0 5(b+c)=((c«d).e) If an atom contains a variable u, then u is free in it. Thus there are four free variables in the last example. NEGATIONS. A wellformed formula preceded by a tilde is wellformed. Examples: ~S0=0 ~3b:(b+b)=SO <O=03S0=O> ~b=SO The quantification status of a variable (which says whether the variable is free or quantified) does not change under negation. COMPOUNDS. If x and y are wellformed formulas, and provided that no variable which is free in one is quantified in the other, then the following are all wellformed formulas: < xa y>, < xv y>, < xz> y>. Examples: <O=Oa~0=0> <b=bv~3c:c=b> <SO=03¥c:~3b:(b+b)=0 The quantification status of a variable doesn't change here. QUANTI FI CATIONS. If u is a variable, and x is a wellformed formula in which u is free then the following strings are wellformed formulas: 3u: x and ¥u: x. Examples: ¥b:<b=bv~3c:c=b> vc:~3b:(b+b)=c ~3c:Sc=d OPEN FORMULAS contain at least one free variable. Examples: c=c b=b <¥b:b=bn—c=c> CLOSED FORMULAS (SENTENCES) contain no free variables. Examples: 50=0 ~¥d:d=0 3c:<Vb:b=bA~c=c>  
 
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This completes the table of Rules of Formation for the wellformed formulas of TNT. A Few More Translation Exercises And now, a few practice exercises for you, to test your understanding of the notation of TNT. Try to translate the first four of the following Nsentences into TNTsentences, and the last one into an open formed formula. All natural numbers are equal to 4. There is no natural number which equals its own square. Different natural numbers have different successors. If 1 equals 0, then every number is odd. b is a power of 2. The last one you may find a little tricky. But it is nothing, compared to this one: b is a power of 10. Strangely, this one takes great cleverness to render in our notation. I would caution you to try it only if you are willing to spend hours and hours on it  and if you know quite a bit of number theory! A Non typographical System This concludes the exposition of the notation of TNT; however, we still left with the problem of making TNT into the ambitious system which we have described. Success would justify the interpretations which we given to the various symbols. Until we have done that, however, particular interpretations are no more justified than the "horseapple happy" interpretations were for the pqsystem's symbols. Someone might suggest the following way of constructing TNT: (1) Do not have any rules of inference; they are unnecessary, because (2) We take as axioms all true statements of number theory (as written in TNTnotation). What a simple prescription! Unfortunately it is as empty as instantaneous reaction says it is. Part (2) is, of course, not a typographical description of strings. The whole purpose of TNT is to figure out if and how it is possible to characterize the true strings typographically. The Five Axioms and First Rules of TNT Thus we will follow a more difficult route than the suggestion above; we will have axioms and rules of inference. Firstly, as was promised, all of the rules of the Propositional Calculus are taken over into TNT. Therefore one theorem of TNT will be this one:  
 
<S0=0v~S0=0>  
 
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which can be derived in the same way as <P(P> was derived. Before we give more rules, let us give the five axioms of TNT: Axiom 1: Axiom 2: Axiom 3: Axiom 4: Axiom 5: (In the austere versions, use a' instead of b.) All of them are very simple to understand. Axiom 1 states a special fact about the number 0; Axioms 2 and 3 are concerned with the nature of addition; Axioms 4 and 5 are concerned with the nature of multiplication, and in particular with its relation to addition. The Five Peano Postulates By the way, the interpretation of Axiom 1"Zero is not the successor of any natural number"is one of five famous properties of natural numbers first explicitly recognized by the mathematician and logician Giuseppe Peano, in 1889. In setting out his postulates, Peano was following the path of Euclid in this way: he made no attempt to formalize the principles of reasoning, but tried to give a small set of properties of natural numbers from which everything else could be derived by reasoning. Peano's attempt might thus be considered "semiformal". Peano's work had a significant influence, and thus it would be good to show Peano's five postulates. Since the notion of "natural number" is the one which Peano was attempting to define, we will not use the familiar term "natural number", which is laden with connotation. We will replace it with the undefined term djinn, a word which comes fresh and free of connotations to our mind. Then Peano's five postulates place five restrictions on djinns. There are two other undefined terms: Genie, and meta. I will let you figure out for yourself what usual concept each of them is supposed to represent. The five Peano postulates: (1) Genie is a djinn. (2) Every djinn has a mesa (which is also a djinn). (3) Genie is not the mesa of any djinn. (4) Different djinns have different metas. (5) If Genie has X, and each djinn relays X to its mesa, then all djinns get X. In light of the lamps of the Little Harmonic Labyrinth, we should name the set of all djinns "GOD". This harks back to a celebrated statement by the German mathematician and logician Leopold Kronecker, archenemy of Georg Cantor: "God made the natural numbers; all the rest is the work of man."  
 
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You may recognize Peano's fifth postulate as the principle of mathematical inductionanother term for a hereditary argument. Peano he that his five restrictions on the concepts "Genie", "djinn", and "mesa" so strong that if two different people formed images in their minds o concepts, the two images would have completely isomorphic structures. example, everybody's image would include an infinite number of distinct djinns. And presumably everybody would agree that no djinn coins with its own meta, or its meta's meta, etc. Peano hoped to have pinned down the essence of natural numbers in his five postulates. Mathematicians generally grant that he succeeded that does not lessen the importance of the question, "How is a true statement about natural numbers to be distinguished from a false one?" At answer this question, mathematicians turned to totally formal systems, as TNT. However, you will see the influence of Peano in TNT, because all of his postulates are incorporated in TNT in one way or another. New Rules of TNT: Specification and Generalization Now we come to the new rules of TNT. Many of these rules will allow reach in and change the internal structure of the atoms of TNT. In sense they deal with more "microscopic" properties of strings than the of the Propositional Calculus, which treat atoms as indivisible units. example, it would be nice if we could extract the string SO=O from the first axiom. To do this we would need a rule which permits us to di universal quantifier, and at the same time to change the internal strut of the string which remains, if we wish. Here is such a rule: RULE OF SPECIFICATION: Suppose u is a variable which occurs inside string x. If the string (Restriction: The term which replaces u must not contain any vat that is quantified in x.) The rule of specification allows the desired string to be extracted Axiom 1. It is a onestep derivation:
~S0=0 specification Notice that the rule of specification will allow some formulas which co: free variables (i.e., open formulas) to become theorems. For example following strings could also be derived from Axiom 1, by specification: Sa=0 ~S(c+SSO)=0 There is another rule, the rule of generalization, which allows us to put  
 
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back the universal quantifier on theorems which contain variables that became free as a result of usage of specification. Acting on the lower string, for example, generalization would give:
Generalization undoes the action of specification, and vice versa. Usually, generalization is applied after several intermediate steps have transformed the open formula in various ways. Here is the exact statement of the rule: RULE OF GENERALIZATION: Suppose x is a theorem in which u, a variable, occurs free. Then ( Restriction: No generalization is allowed in a fantasy on any variable which appeared free in the fantasy's premise.) The need for restrictions on these two rules will shortly be demonstrated explicitly. Incidentally, this generalization is the same generalization as was mentioned in Chapter II, in Euclid's proof about the infinitude of primes. Already we can see how the symbolmanipulating rules are starting to approximate the kind of reasoning which a mathematician uses. The Existential Quantifier These past two rules told how to take off universal quantifiers and put them back on; the next two rules tell how to handle existential quantifiers. RULE OF INTERCHANGE: Suppose u is a variable. Then the strings Vu: and 3u: are interchangeable anywhere inside any theorem. For example, let us apply this rule to Axiom 1:
By the way, you might notice that both these strings are perfectly natural renditions, in TNT, of the sentence "Zero is not the successor of any natural number". Therefore it is good that they can be turned into each other with ease. The next rule is, if anything, even more intuitive. It corresponds to the very simple kind of inference we make when we go from "2 is prime" to "There exists a prime". The name of this rule is selfexplanatory: RULE OF EXISTENCE: Suppose a term (which may contain variables as long as they are free) appears once, or multiply, in a theorem. Then any (or several, or all) of the appearances of the term may be replaced by a variable which otherwise does not occur in the theorem, and the corresponding existential quantifier must be placed in front. Let us apply the rule to as usualAxiom 1:  
 
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You might now try to shunt symbols, according to rules so far giver produce the theorem ~ Rules of Equality and Successorship We have given rules for manipulating quantifiers, but so far none for symbols `=' and 'S'. We rectify that situation now. In what follows, r, s, t all stand for arbitrary terms. RULES OF EQUALITY: SYMMETRY: If r = s is a theorem, then so is s = r. TRANSITIVITY: If r = s and s = t are theorems, then so is r = t. RULFS OF SUCCESSORSHIP: ADD S: If r = t is a theorem, then Sr = St is a theorem. DROP S: If Sr = St is a theorem, then r = t is a theorem. Now we are equipped with rules that can give us a fantastic variet theorems. For example, the following derivations yield theorems which pretty fundamental:  
 
 
 
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Illegal Shortcuts  
 
Now here is an interesting question: "How can we make a derivation for the string 0=0?" It seems that the obvious route to go would be first to derive the string  
 
 
 
I gave this miniexercise to point out one simple fact: that one should not jump too fast in manipulating symbols (such as `=') which are familiar. One must follow the rules, and not one's knowledge of the passive meanings of the symbols. Of course, this latter type of knowledge is invaluable in guiding the route of a derivation. Why Specification and Generalization Are Restricted Now let us see why there are restrictions necessary on both specification and generalization. Here are two derivations. In each of them, one of the restrictions is violated. Look at the disastrous results they produce:  
 
 
 
So now you can see why those restrictions are needed. Here is a simple puzzle: translate (if you have not already done so) Peano's fourth postulate into TNTnotation, and then derive that string as a theorem.  
 
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Something Is Missing Now if you experiment around for a while with the rules and axioms of TNT so far presented, you will find that you can produce the following pyramidal family of theorems (a set of strings all cast from an identical mold, differing from one another only in that the numerals 0, SO, SSO, and s have been stuffed in): (0+0)=0 (O+SO)=S0 (O+SSO)=SSO (O+SSSO)=SSSO (O+SSSSO)=SSSSO etc. As a matter of fact, each of the theorems in this family can be derived the one directly above it, in only a couple of lines. Thus it is a so "cascade" of theorems, each one triggering the next. (These theorem very reminiscent of the pqtheorems, where the middle and right] groups of hyphens grew simultaneously.) Now there is one string which we can easily write down, and v summarizes the passive meaning of them all, taken together. That un sally quantified summarizing string is this:
Yet with the rules so far given, this string eludes production. Ti produce it yourself if you don't believe me. You may think that we should immediately remedy the situation the following (PROPOSED) RULE OF ALL: If all the strings in a pyramidal family are theorems, then so is the universally quantified string which summarizes them. The problem with this rule is that it cannot be used in the Mmode. people who are thinking about the system can ever know that an infinite set of strings are all theorems. Thus this is not a rule that can be stuck i any formal system. ÉIncomplete Systems and Undecidable Strings So we find ourselves in a strange situation, in which we can typographically produce theorems about the addition of any specific numbers, but even a simple string as the one above, which expresses a property of addition in general, is not a theorem. You might think that is not all that strange, we were in precisely that situation with the pqsystem. However, the pqsystem had no pretensions about what it ought to be able to do; and ii fact  
 
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there was no way to express general statements about addition in its symbolism, let alone prove them. The equipment simply was not there, and it did not even occur to us to think that the system was defective. Here, however, the expressive capability is far stronger, and we have correspondingly higher expectations of TNT than of the pqsystem. If the string above is not a theorem, then we will have good reason to consider TNT to be defective. As a matter of fact, there is a name for systems with this kind of defectthey are called Éincomplete. (The prefix 'É''omega' comes from the fact that the totality of natural numbers is sometimes denoted by `É'.) Here is the exact definition: A system is Éincomplete if all the strings in a pyramidal family are theorems, but the universally quantified summarizing string is not a theorem. Incidentally, the negation of the above summarizing string ~ is also a nontheorem of TNT. This means that the original string is undecidable within the system. If one or the other were a theorem, then we would say that it was decidable. Although it may sound like a mystical term, there is nothing mystical about undecidability within a given system. It is only a sign that the system could be extended. For example, within absolute geometry, Euclid's fifth postulate is undecidable. It has to be added as an extra postulate of geometry, to yield Euclidean geometry; or conversely, its negation can be added, to yield nonEuclidean geometry. If you think back to geometry, you will remember why this curious thing happens. It is because the four postulates of absolute geometry simply do not pin down the meanings of the terms "point" and "line", and there is room for different extensions of the notions. The points and lines of Euclidean geometry provide one kind of extension of the notions of "point" and "line"; the POINTS and LINES of nonEuclidean geometry, another. However, using the preflavored words "point" and "line" tended, for two millennia, to make people believe that those words were necessarily univalent, capable of only one meaning. NonEuclidean TNT We are now faced with a similar situation, involving TNT. We have adopted a notation which prejudices us in certain ways. For instance, usage of the symbol `+'tends to make us think that every theorem with a plus sign in it ought to say something known and familiar and "sensible" about the known and familiar operation we call "addition". Therefore it would run against the grain to propose adding the following "sixth axiom":
 
 
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It doesn't jibe with what we believe about addition. But it is one possible extension of TNT, as we have so far formulated TNT. The system which uses this as its sixth axiom is a consistent system, in the sense of not has, two theorems of the form x and  x. However, when you juxtapose this "sixth axiom" with the pyramidal family of theorems shown above, you will probably be bothered by a seeming inconsistency between the family and the new axiom. But this kind of inconsistency is riot so damaging as the other kind (where x and x are both theorems). In fact, it is not a true inconsistency, because there is a way of interpreting the symbols so that everything comes out all right. ÉInconsistency Is Not the Same as Inconsistency This kind of inconsistency, created by the opposition of (1) a pyramidal family of theorems which collectively assert that all natural numbers have some property, and (2) a single theorem which seems to assert that not all numbers have it, is given the name of winconsistency. An winconsistent system is more like the attheoutsetdistastefulbutintheendaccept nonEuclidean geometry. In order to form a mental model of what is going on, you have to imagine that there are some "extra", unsuspected numberslet us not call them "natural", but supernatural numberswhich have no numerals. Therefore, facts about them cannot be represented in the pyramidal family. (This is a little bit like Achilles' conception GODas a sort of "superdjinn", a being greater than any of the djinn This was scoffed at by the Genie, but it is a reasonable image, and may I you to imagine supernatural numbers.) What this tells us is that the axioms and rules of TNT, as so presented, do not fully pin down the interpretations for the symbol TNT. There is still room for variation in one's mental model of the notions they stand for. Each of the various possible extensions would pin d, some of the notions further; but in different ways. Which symbols we begin to take on "distasteful" passive meanings, if we added the "s axiom" given above? Would all of the symbols become tainted, or we some of them still mean what we want them to mean? I will let you tt about that. We will encounter a similar question in Chapter XIV, discuss the matter then. In any case, we will not follow this extension r but instead go on to try to repair the wincompleteness of TNT. The Last Rule The problem with the "Rule of All" was that it required knowing that all lines of an infinite pyramidal family are theorems  too much for a finite being. But suppose that each line of the pyramid can be derived from its predecessor in a patterned way. Then there would be a finite reason accounting for the fact that all the strings in the pyramid are theorems. The trick then, is to find the pattern that causes the cascade, and show that  
 
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pattern is a theorem in itself. That is like proving that each djinn passes a message to its meta, as in the children's game of "Telephone". The other thing left to show is that Genie starts the cascading messagethat is, to establish that the first line of the pyramid is a theorem. Then you know that GOD will get the message! In the particular pyramid we were looking at, there is a pattern, captured by lines 49 of the derivation below.  
 