Gödel, Escher, Bach: A Mental Space Odyssey Justin M. Curry July 1, 2007
Contents 1 Welcome 4 1.1 Introduction......................................... 4 1.2 Class Philosophy...................................... 4 1.3 What is this document?.................................. 5 1.4 Acknowledgements..................................... 5 2 Tools for Thinking 6 2.1 Introduction......................................... 6 2.2 Formal Systems....................................... 6 2.3 Isomorphisms........................................ 7 2.3.1 Purpose....................................... 7 2.3.2 Physical Analogies................................. 7 2.4 Recursion.......................................... 8 2.4.1 Recursion in Math................................. 8 2.4.2 Recursion in Other Fields............................. 9 2.5 Paradox........................................... 9 2.5.1 Definition of Paradox............................... 9 2.5.2 Examples...................................... 11 2.5.3 Etymology..................................... 12 2.5.4 Common Themes.................................. 12 2.5.5 Types of Paradoxes................................ 12 2.6 Infinity............................................ 13 3 Introduction: A Musico-Logical Offering 14 3.1 Introduction to the Introduction............................. 14 3.2 Abstract........................................... 14 3.3 Bach............................................. 14 3.4 Canons and Fugues..................................... 15 3.5 An Endlessly Rising Canon................................ 15 3.6 Escher............................................ 17 3.7 Gödel............................................ 17 3.8 Mathematical Logic: A Synopsis............................. 17 3.9 Banishing Strange Loops.................................. 18 3.10 Consistency, Completeness, Hilbert s Program...................... 18 3.11 Babbage, Computers, Artificial Intelligence......................... 18 1
3.12... and Bach......................................... 18 3.13 Gödel, Escher, Bach................................... 19 3.14 Study Questions...................................... 19 4 Three Part Invention 21 4.1 Abstract........................................... 21 4.2 Study Questions...................................... 21 5 Chapter I: The MU Puzzle 23 5.1 Abstract........................................... 23 5.2 Formal Systems....................................... 23 5.3 Theorems, Axioms, Rules................................. 23 5.4 Inside and Outside the System.............................. 24 5.5 Jumping out of the System................................ 24 5.6 M-Mode, I-Mode, U-Mode................................. 24 5.7 Decision Procedures.................................... 25 5.8 Study Questions...................................... 25 6 Two-Part Invention 27 6.1 Abstract........................................... 27 6.2 Lewis Carroll s Paradox.................................. 27 7 Chapter II: Meaning and Form in Mathematics 28 7.1 Abstract........................................... 28 7.2 The pq-system....................................... 28 7.3 The Decision Procedure.................................. 28 7.4 Bottom-up vs. Top-down................................. 29 7.5 Isomorphisms Induce Meaning.............................. 29 7.6 Active vs. Passive Meanings................................ 29 7.7 Meaningless and Meaningful Interpretations....................... 29 7.8 Double-Entendre!...................................... 29 7.9 Formal Systems and Reality................................ 29 7.10 Mathematics and Symbol Manipulation......................... 30 7.11 The Basic Laws of Arithmetic............................... 30 7.12 Ideal Numbers....................................... 30 7.13 Euclid s Proof........................................ 30 7.14 Getting Around Infinity.................................. 30 7.15 Study Questions...................................... 30 7.15.1 The pq- System................................... 31 7.15.2 The Decision Procedure; Bottom-up versus Top-down............. 31 7.15.3 Isomorphisms Induce Meaning; Meaningless and Meaningful Interpretations. 31 7.15.4 Active versus Passive Meanings and Double-Entendre............. 32 7.16 Study Questions...................................... 32 7.16.1 Isomorphisms Induce Meaning; Meaningless and Meaningful Interpretations. 32 7.16.2 Active vs. Passive Meanings and Double-Entendre............... 32 2
8 Sonata for Unaccompanied Achilles 33 8.1 Abstract........................................... 33 8.2 Study Questions...................................... 33 9 Chapter III: Figure and Ground 34 9.1 Abstract........................................... 34 9.2 Primes vs. Composites................................... 34 9.3 The tq-system....................................... 34 9.4 Capturing Compositeness................................. 34 9.5 Illegally Characteriziing Primes.............................. 34 9.6 Figure and Ground..................................... 34 9.7 Figure and Ground in Music................................ 35 9.8 Recursively Enumberable Sets vs. Recursive Sets.................... 35 9.9 Primes as Figure Rather than Ground.......................... 35 9.10 Study Questions...................................... 35 9.10.1 Primes vs. Composites in the tq- systems.................... 35 9.10.2 Capturing Compositeness and Illegally Characterizing Primes......... 36 9.10.3 Figure and Ground................................. 36 10 Contracrostipunctus 37 10.1 Abstract........................................... 37 10.2 Secrets............................................ 37 10.3 Questions.......................................... 38 11 Chapter IV: Consistency, Completeness, and Geometry 39 11.1 Abstract........................................... 39 11.2 Questions.......................................... 39 3
Chapter 1 Welcome I don t think we re in Kansas anymore. 1.1 Introduction My name is Justin Curry and I will be the facilitator for your learning in this course. I say facilitator, because learning is a cooperative and democratic effort. You are here under (supposedly) your own free will, so you should make the most out of the time we have. I titled this course Gödel, Escher, Bach: A Mental Space Odyssey for a variety of reasons. The first part is not necessarily obvious. Although the three names allude to the very famous mathematician Kurt Gödel, the even more prominent artist Maurits Cornelis Escher, and finally the musical genius Johann Sebastian Bach, this course will NOT be about these three remarkable individuals. Rather, we will find that their spirits aboard our mental ship as we explore the ideas that surround the contents of this course. So what is this course about? It will be a course in climbing mental mountains and crossing intellectual oceans; with Douglas R. Hofstadter s Pulitizer Prize-winning book Gödel, Escher, Bach: An Eternal Golden Braid often called GEB, as our guide book. I will not go any further in describing what GEB is about and save that for a later time. Finally, you should really consider this class a Mental Space Odyssey an ode to Stanley Kubrick s 2001: A Space Odyssey. I want you all to become very used to the idea of packing your bags and going on an intellectual vacation. 1.2 Class Philosophy Your brain is about to go on a serious trip. I have already alluded to the fact that this class is going to be a democratic one, and as such I offer you the following deal: When ever I am going to fast, too slow, too math-y, too lofty, I want you to yell STOP! Everyone learns very differently and has a certain set of interests that gets them pumped up, and another set of turn-offs that puts them to sleep. Hofstadter talks about GEB as a book which illuminates a central concept from multiple angles, so I need to know what angles work for you guys, and I will try my best to accommodate them. In short, What gets you pumped up? 4
There is one more thing. Whenver you hear something which you want to challenge, I invite you to call me out on it. If it will benefit the discussion, I will pause, hear your case and we will lock horns until the issue is resolved. 1.3 What is this document? The course notes collected here represents a particular instance of a continuously evolving set of notes on GEB. It is intended to be a mildly cohesive, but more importantly comprehensive source for thinking about GEB, its nuances, its themes, and its inspired thinking (ha, recursive!). Although GEB is a self-contained education, the sole purpose of this book is to supplant GEB with outside resources! For the time being, it will only cover certain key chapters of GEB. This reflects that fact that these course notes were orginally designed for a 10 and an 8-week lecture course. Certain areas are necessarily sacrificed, but there is always room in the future! 1.4 Acknowledgements This is actually the product of many hands at work. Professor Jeff Elhaij at the Virginia Commonwealth University, taught a course in the Spring of 2001 based on GEB. Without his efforts, key parts of these course notes would be non-existent. Almost all of the study questions for the first three chapters and dialogues are his, and the logical content of this book is in part inspired by his course. I would also like to thank Wikipedia and all of the other GEB fans that have been infected by Hofstadter s thinking and have devoted a small (or large) part of their lives to GEB. This book has changed many people s lives, and I hope it will impact you! Finally I would like to offer special thanks to the following people for the following reasons in no particular order: Rob Speer (MIT) for teaching his own seminar on GEB. Agustin Rayo (MIT) for teaching me about Paradoxes, Infinities, and Language Daniel Rothman (MIT) for teaching me about Chaos and why beauty exists in nature Gerald Sacks (MIT) for teaching me Math Logic and Recursion Theory and regaling me in his personal stories of Gödel. Gerald Jay Sussman (MIT) for being enlightened. Matthew Gordon (MIT) for being the Lewis to my Clark in exploring the intellectual unknown. Curran Kelleher (UMass Lowell) for being the Intrepid Traveler! Sasha Rahlin (MIT) for being my first mate aboard my own personal odyssey. To M. and D. for obvious reasons. 5
Chapter 2 Tools for Thinking Always Be Prepared. Boy Scout Motto 2.1 Introduction The premise for this chapter is that before we take off on our journey, we need to read the travel guide, so that we know what kind of creatures to expect to encounter. Also in a weak allusion to the fugal structure of both GEB and this class, I want to lay the theme down for you now (or part of it), so that you might understand it when it is inverted, played backwards, forwards, upside-down, and across the universe. After intense speculation and discussion, I feel that the following concepts might help aid in your understanding of GEB and the content of this course. 2.2 Formal Systems From Wikipedia, the free encylopedia: In the formal sciences of logic and mathematics, together with the allied branches of computer science, information theory, and statistics, a formal system is a formal grammar used for modeling purposes. Formalization is the act of creating a formal system, in an attempt to capture the essential features of a real-world or conceptual system in formal language. In mathematics, formal proofs are the product of formal systems, consisting of axioms and rules of deduction. Theorems are then recognized as the possible last lines of formal proofs. The point of view that this picture encompasses mathematics has been called formalist. The term has been used pejoratively. On the other hand, David Hilbert founded metamathematics as a discipline designed for discussing formal systems; it is not assumed that the metalanguage in which proofs are studied is itself less informal than the usual habits of mathematicians suggest. To contrast with the metalanguage, the language described by a formal grammar is often called an object language (i.e., the object of discussion - this distinction may have been introduced by Carnap). It has become common to speak of a formalism, more-or-less synonymously with a formal system within standard mathematics invented for a particular purpose. This may not be much more than a notation, such as Dirac s bra-ket notation. Mathematical formal systems consist of the following: 6
1. A finite set of symbols which can be used for constructing formulae. 2. A grammar, i.e. a way of constructing well-formed formulae out of the symbols, such that it is possible to find a decision procedure for deciding whether a formula is a well-formed formula (wff) or not. 3. A set of axioms or axiom schemata: each axiom has to be a wff. 4. A set of inference rules. 5. A set of theorems. This set includes all the axioms, plus all wffs which can be derived from previously-derived theorems by means of rules of inference. Unlike the grammar for wffs, there is no guarantee that there will be a decision procedure for deciding whether a given wff is a theorem or not. 2.3 Isomorphisms From Wikipedia, the free encyclopedia: In mathematics, an isomorphism (Greek:isos equal, and morphe shape ) is a bijective map f such that both f and its inverse f 1 are homomorphisms, i.e. structure-preserving mappings. Informally, an isomorphism is a kind of mapping between objects, which shows a relationship between two properties or operations. If there exists an isomorphism between two structures, we call the two structures isomorphic. In a certain sense, Isomorphic sets are structurally identical, if you choose to ignore finer-grained differences that may arise from how they are defined. According to Douglas Hofstadter: 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 parts play similar roles in their respective structures. (Gödel, Escher, Bach, p. 49) 2.3.1 Purpose Isomorphisms are frequently used by mathematicians to save themselves work. If a good isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to solid ground, where the problem is easier to understand and work with. 2.3.2 Physical Analogies Here are some everyday examples of isomorphic structures: A solid cube made of wood and a solid cube made of lead are both solid cubes; although their matter differs, their geometric structures are isomorphic. A standard deck of 52 playing cards with green backs and a standard deck of 52 playing cards with brown backs; although the colours on the backs of each deck differ, the decks are 7
structurally isomorphic if we wish to play cards, it doesn t matter which deck we choose to use. The Clock Tower in London (that contains Big Ben) and a wristwatch; although the clocks vary greatly in size, their mechanisms of reckoning time are isomorphic. A six-sided die and a bag from which a number 1 through 6 is chosen; although the method of obtaining a number is different, their random number generating abilities are isomorphic. This is an example of functional isomorphism, without the presumption of geometric isomorphism. 2.4 Recursion Recursion is probably the most fundamental concept to GEB! In fact recursion has become so tightly associated with GEB that one Andrew Plotkin once defined recursion as: If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer to Douglas Hofstadter than you are; then ask him or her what recursion is. But what exactly is recursion? Wikipedia offers us the following definition: In mathematics and computer science, recursion specifies (or constructs) a class of objects or methods (or an object from a certain class) by defining a few very simple base cases or methods (often just one), and then defining rules to break down complex cases into simpler cases. This is an OK definition of recursion, but actually understanding what recursion is will take considerable experience and practice recognizing recursive structures everywhere in thought and nature. 2.4.1 Recursion in Math Recursion probably got its start in mathematics, especially in defining interesting sequences of numbers such as the Fibonacci sequence, which can be defined by {1, 1, 2, 3, 5, 8, 13, 21,...} f(n) = f(n 1) + f(n 2) n 2 (2.1) f(0) = f(1) = 1 (2.2) However, in my humble opinion some of the most interesting applications of recursion (besides Gödel s Incompleteness Theorem) are fractals. Fractals appear everywhere in nature and are selfsimilar. They exist in a fractional number of dimensions (thus Fractal) and look really cool. You might be confused by the notion of fractional (i.e. 1.7, 2.5, etc.) number of dimensions, and there many possible ways of rewiring your brain to think differently about dimensions, but here is one simple way. 8
Imagine a line. When you double its length, you now have two copies of the original. Imagine a square. When you double its sides, you now have four copies of the original. Imagine a cube. When you double its sides, you now have eight copies of the original. If you are perceptive enough, maybe you ll notice that the number of dimensions, d of each object, and the number of copies, N followed the following relationship: 2 d = N d 0 (2.3) However consider the sierpinski triangle (or gasket) a very famous fractal of dimension 1.585. It has the very strange property that if you double its sides (or scale them by 1 2 ) you have only 3 copies. Thus we need a dimension that satisfies 2 d = 3 d = log(3) 1.585... (2.4) log(2) Although this argument is very imprecise and informal, it will give you a flavor for fractals. If you want to study a more rigorous method for calculating dimension, look up the Box Counting Dimension or the Minkowski-Bouligand Dimension. 2.4.2 Recursion in Other Fields Recursion is everywhere! Whether it is developing an algorithm in computer science, or understanding how language works, or how evolution works, recursion appears to have a deep importance in the universe. 2.5 Paradox We will also find tightly knit with our exploration of recursion, are encounters with paradox. Paradoxes are a genuinely difficult thing to study, and we will explore several famous paradoxes, and try to understand its role in GEB. One tool which Hofstadter finds useful for considering paradoxes is the Zen Koan an apparently contradictory statement or story to help quiet the mind and I have attached an appendix of cool Zen Koans. 2.5.1 Definition of Paradox From Wikipedia A paradox is an apparently true statement or group of statements that seems to lead to a contradiction or to a situation that defies intuition. Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. The recognition of ambiguities, equivocations, and unstated assumptions underlying known paradoxes has led to significant advances in science, philosophy and mathematics. The word paradox is often used interchangeably and wrongly with contradiction; but where a contradiction by definition cannot be true, many paradoxes do allow for resolution, though many remain unresolved or only contentiously resolved, such as Curry s paradox. Still more casually, the 9
Figure 2.1: Part of the Mandelbrot Set A Picture of God? From Wikipedia 10
Figure 2.2: Boyle s perpetual motion scheme From Wikipedia term is sometimes used for situations that are merely surprising, albeit in a distinctly logical manner, such as the Birthday Paradox. This is also the usage in economics, where a paradox is an unintuitive outcome of economic theory. 2.5.2 Examples Sometimes supernatural or science fiction themes are held to be impossible on the grounds that result in paradoxes. The theme of time travel has generated a whole family of popular paradoxes, supposed to arise from a person s interference with the past. Suppose Jones, who was born in 1950, travels back in time to 1900 and kills his own grandfather. It follows that neither his father nor he himself will be born; but then he would not have existed to travel back in time and kill his own grandfather; but then his grandfather would not have died and Jones himself would have lived; etc. This is known as the Grandfather paradox. Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. This sentence is false is an example of the famous liar paradox: it is a sentence which cannot be consistently interpreted as true or false, because if it is false it must be true, and if it is true it must be false. Russell s paradox, which shows that the notion of the set of all those sets that do not contain themselves, was instrumental in the development of modern logic and set theory. 11
2.5.3 Etymology The etymology of paradox can be traced back the use of the word paradoxo, used in Plato s Parmenides by the Greek philosopher Zeno of Elea, who lived at 490-430 BC. The word was used to describe seminal philosophic ideas posited by Zeno, known as Zeno s paradoxes, which exerted a poignant effect on Greek thinkers that has survived to modern day. Zeno sought to illustrate that equal absurdities followed logically from the denial of Parmenides views. There were apparently 40 paradoxes of plurality and other paradoxes that Zeno used to attack the Greek understanding of the physical world. In fact, Zeno s paradoxes of multiplicity and motion revealed some problems in space and time that cannot be resolved without the mathematical methods discovered in the 19th century and perhaps beyond. Although it is unknown if Zeno coined the word, he can certainly be attributed as popularizing it. It is unknown if incarnations of paradox were used before Zeno of Elea. Later and more frequent usage of the word has been traced to the early Renaissance. Early forms of the word appeared in the late Latin paradoxum and the related Greek paradoxos meaning contrary to expectation, incredible. The word is composed of the preposition para which means against conjoined to the noun stem doxa, meaning belief. Compare orthodox (literally, straight teaching ) and heterodox (literally, different teaching ). The liar paradox and other paradoxes were studied in medieval times under the heading insolubilia. 2.5.4 Common Themes Common themes in paradoxes include direct and indirect self-reference, infinity, circular definitions, and confusion of levels of reasoning. Paradoxes which are not based on a hidden error generally happen at the fringes of context or language, and require extending the context or language to lose their paradox quality. In moral philosophy, paradox plays a central role in ethics debates. For instance, an ethical admonition to love thy neighbour is not just in contrast with, but in contradiction to an armed neighbour actively trying to kill you: if he or she succeeds, you will not be able to love him or her. But to preemptively attack them or restrain them is not usually understood as loving. This might be termed an ethical dilemma. Another example is the conflict between an injunction not to steal and one to care for a family that you cannot afford to feed without stolen money. 2.5.5 Types of Paradoxes W. V. Quine (1962) distinguished between three classes of paradoxes. A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless. Thus, the paradox of Frederic s birthday in The Pirates of Penzance establishes the surprising fact that a person may be more than Nine years old on his Ninth birthday. Likewise, Arrow s impossibility theorem involves behaviour of voting systems that is surprising but all too true. A falsidical paradox establishes a result that not only appears false but actually is false; there is a fallacy in the supposed demonstration. The various invalid proofs (e.g. that 1 = 2) are classic examples, generally relying on a hidden division by zero. Another example would be the Horse paradox. 12
A paradox which is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, the Grelling-Nelson paradox points out genuine problems in our understanding of the ideas of truth and description. 2.6 Infinity We will discuss infinity at great lengths in this course and will find that there exists a lovely trinity between recursion, paradox, and infinity. Meditate on the following quote from Douglas Adams Hitchhiker s Guide to the Galaxy as we read GEB. Bigger than the biggest thing ever and then some, much bigger than that, in fact really amazingly immense, a totally stunning size, real Wow, that s big! time. Infinity is just so big that by comparison, bigness itself looks really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of concept we are trying to get across here. 13
Chapter 3 Introduction: A Musico-Logical Offering Normal is a Distribution Unknown 3.1 Introduction to the Introduction As we have finally reached the beginning of the book proper, these notes should mirror the book as best as possible. Let us not forget that these notes are an ode to GEB, not to propositional calculus! As such there will be a chapter for each chapter and dialogue, and sections for each section of the chapters found in GEB.Here on out, this section will be occupied normally by Hofstadter s own abstract for each chapter found in the overview of GEB. 3.2 Abstract 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. Self-reference and its 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 book particularly the why and wherefore of the Dialogues. GEB pp. viii 3.3 Bach Of all the angles GEB uses to illuminate the same concept, music is perhaps my weakest side. As such, I will down play the role of music, but still try to give its proper place. I think the important thing to gather from this section is the type of intellectual playing that Hofstadter 14
likes to engage in. In many ways all of GEB is one big puzzle; much like a fugue. The kind of high-brow cleverness that Hoftstadter speaks of, will inspire the rest of the book (pp 7). 3.4 Canons and Fugues Let s go ahead and record some definitions and facts about canons and fugues (pp. 8-9). A canon is a piece of music where a single theme is repeated and played against itself. Hofstadter gives us several ways in which a canon can have complexity: 1. A copy of the theme is played a fixed time later. 2. The theme is staggered in time and pitch. 3. The theme is played at different speeds. 4. The theme is inverted. 5. The theme is played backwards Crab Canon! A fugue is a canon with more flexibility and opportunity for creative expression. The two quotes of the section seems to be: Such an information-preserving transformation is often called an isomorphism, and we will have much traffic with isomorphisms in this book. (p. 9) It is itself one large intellectual fugue, in which many ideas and forms have been woven together, and in which playful double meanings and subtle allusions are commonplace. (pp 9-10) The second quote is a classic example of Hofstadter talking about GEB within GEB! Even if you can t pick up all of GEB s clever layers, at least it has the decency to tell you that multiple layers do exist! 3.5 An Endlessly Rising Canon Here we meet out first definitions of Strange Loops and Tangled Hierarchies. In this canon [Musical Offering], 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 hierarchial system, we unexpectedly find ourselves right back where we started. (pp 10) Sometimes I use the term Tangled Hierarchy to describe a system in which a Strange Loop occurs. (pp 10) 15
Figure 3.1: All M.C. Escher works Cordon Art-Baarn-the Netherlands. Used by permission. All rights reserved. 16
3.6 Escher As far as the introduction goes, I believe this section is the most significant. By using images of Escher we begin to muse on what the actual structure of thought and existence is. 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?...in some of his drawings, one single theme can appear on different levels of reality...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 mind-boggling 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 half-real, half-mythical worlds, worlds filled with Strange Loops, which he seems to be inviting his viewers to enter. When reflecting on this quote, Escher reflects in Hofstadter s mind s eye, but Mandelbrot flashes in mine. Fractals and Fractal Geometry is such an apparent part of nature, Hofstadter s quote gains even more legitimacy than previously thought. Please see the appendix for a tour of the Mandelbrot set. Perhaps it will convince you of Hofstadter s thinking. 3.7 Gödel The essence of what Gödel did, was to use number theory to encode statements about provability of statements within a specific formal system. He then, in a very rigorous and brilliant way, encoded the statement: This statement is not provable. What if this statement is provable? If it is provable, then it is true since provable statements are necessarily true. If it is true, then it is not provable! Contradiction! What if this statement is not provable? That is exactly what the statement asserts, so it is true AND not-provable. 3.8 Mathematical Logic: A Synopsis Here we meet the founders of logic. The important example to take away is Russell s Paradox. The paradox is phrased in multiple ways. One version considers the set Ω of all sets not members of themselves. Now ask yourself, Is Ω a member of itself? If it is, then it isn t and if it isn t then it is. Some examples of self-containing sets might be the set of all sets, the set of all things not George Bush, etc. Perhaps that still seems convoluted. Consider a popular variant known as the Barber s Paradox. There is a town in Tumbolia which has a barber who shaves all and only people who don t shave themselves. Does our barber shave himself or not? 17
3.9 Banishing Strange Loops The idea of a hierarchy was thought to prevent paradoxes from creeping into mathematics. The idea is that we keep normal non-self referential language and restrict it to an object language call it L1. Then if we want to talk about statements in T1, we need to talk in an expanded metalanguge call it L2. But if we want to talk about L2 we need to operate in an expanded metalanguage L3, and so on... This way sentences like This sentence is not true. could never be expressed in L1, because the sentence it self isn t in the domain of L1. However a sentence like Snow is white. is perfectly acceptable for L1. We could then say something in L2 like Snow is white. is true. 3.10 Consistency, Completeness, Hilbert s Program Just remember the definitions. A system is consistent if it cannot prove both a statement P true and its negation not-p true. A system is complete if every true statement if provable. 3.11 Babbage, Computers, Artificial Intelligence... Convergence of three disciplines: 1. Theory of Axiomatic Reasoning 2. Study of Mechanical Computation 3. Psychology of Intelligence What is intelligence? How can it be programmed? Hofstadter comes up with a list of essential abilities for intelligence to have (p. 26): to respond to situations very flexibly; to take advantage of fortuitous circumstances; 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 which may link them; to synthesize new concepts by taking old concepts and putting them together in new ways; to come up with ideas which are novel. 3.12... and Bach Do you think it is possible for a computer to compose music? 18
3.13 Gödel, Escher, Bach Each dialogue is patterned on a different piece of Bach. The dialogues are meant to provide an intuitive understanding of the concepts before the technical understanding in the following chapter. Similarly in this course we will have dialogues and meta-dialogues preceding each reading assignment. 3.14 Study Questions 1. What property does Row, Row, Row Your Boat and Frere Jacques have in common? Would any tune work just as well? Suggest one as a test case. 2. Hofstadter claims that Good King Wenceslas works as an inversion. You can decide for yourself. (ask for music if you really want) 3. What s the connection between Bach s endlessly rising canon and Escher s Waterfall? 4. A quote from GEB (p15): 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? There s a lot to think about here. Try to elaborate on your conception of infinity, either musically, artistically, or through language. Consider using Escher s Metamorphosis, Mobius Strip I, or Mobius Strip II (p276) to assist you. For additional help, you might recall the Autumn Floods section of Chuang-Tzu: The Lord of the River said, Men who debate such matters these days all claim that the minutest thing has no form and the largest thing cannot be encompassed. Is this a true statement? Are there things about infinity that you find paradoxical? What are they? 5. Are the following statements true or false? What s the problem? All Cretans are liars. Uttered by Epimenides the Cretan I always lie. This statement is false. 6. Translate the following sentence into language the guy in the street could readily understand: All consistent axiomatic formulations of number theory include undecidable propositions. 7. Accept for the moment Hofstadter s statement that This statement of number theory does not have any proof in the system of Principia Mathematica can indeed be translated into a statement of number theory. We are interested in two questions: 1) Is it a true statement? and 2) Can it be proved in the system of Principia Mathematica? Describe the possibilities. 8. Consider R, the set of all run-of-the-mill sets (as defined in the text). Is R itself a run-ofthe-mill set or is it a self-swallowing set? 9. Which Box Contains the Gold? Two boxes labeled A and B. A sign on box A says The sign on box B is true and the gold is in box A. A sign on box B says The sign on box A is false and the gold is is in box A. Assuming there is gold in one of the boxes, which box contains the gold? 10. What is Hofstadter s main objection to the Hierarchical Theory of Types? 19
11. Identify the hierarchy of object language and metalanguage in the following statement: The definition of ambiguous is not ambiguous. Parse the statement in both sensible and paradoxical ways. Would punctuation help to clarify the intended meaning? 12. heterological: Is heterological (as defined in the text) heterological? 13. meta...: Compose a metasentence. 14. strange loop: Think of an example of a strange loop, besides what you ve read. How tight is it? 15. consistent system: What would be necessary to make a system inconsistent? 16. complete system: What would be necessary to make a system incomplete? 17. infinity: What is its essential property? 20
Chapter 4 Three Part Invention I hope I didn t brain my damage Homer Simpson 4.1 Abstract Bach wrote fifteen three-part inventions. In this three-part Dialogue, the Tortoise and Achilles the main fictional protagonists in the dialogues are 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. GEB pp. viii 4.2 Study Questions 1. To what Escher print does Achilles refer at the beginning of the dialogue (I mean, what does that print look like)? 2. What is a Möbius strip? To what print does Achilles refer? 3. What is the relationship between the hole in the flag and the Mbius strip? As soon as the Tortoise introduces the term Capitalized Essences, capitalized terms seem to abound. What are their significance? Write down those whose significance you dont see immediately on your Open Mysteries page. 4. Is Zeno the sixth patriarch or is he not? If he isn t, then why does Achilles think he is? 5. What story is recreated in this dialogue? 6. In what ways is this dialogue self-referential? 7. Do you understand the crux of the paradox (Achilles paradox) that Zeno relates? 8. Are you familiar with the Dichotomy Paradox to which the Tortoise refers? 9. Is there any significance in positioning the Tortoise upwind of Achilles? 10. Use ordinary algebra to find how long it will take for Achilles to catch the Tortoise. Assume that Achilles runs with a speed of 1 rod per second, so that his position is given by: 21
A(t) = position of Achilles over time = 0 rods from start + (1 rod/sec)(t sec) Assume also that the Tortoise runs with a speed of 1/2 rod per second and has a head start of 10 rods, so that his position is given by: T(t) = position of Tortoise over time = 10 rods from start + (1/2 rod/sec)(t sec) Play around with different values for the speeds, then see if you can derive a formula for the time at which Achilles catches the Tortoise in terms of their running speed A (for Achilles) and T (for the Tortoise). The Tortoise always gets a 10 rod head start. 11. What (if anything) is wrong with Zeno s argument? 12. What is a rod anyway? 22
Chapter 5 Chapter I: The MU Puzzle Has the dog Buddha-nature? MU! Zen Koan 5.1 Abstract A simple formal system (the MIU-system) is presented, and the rader 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. GEB pp. viii 5.2 Formal Systems See chapter on Tools for Thinking In regards to this particular formal system, we find that we have the following pieces of the puzzle.in order to distinguish which levels we are working on, we will employ the typewriter font. Remember the rules only go one way! LETTERS: M, I, U OUR SOUL POSSESSION: MI RULE I: If you possess a string who last letter s I, you can add on a U at the end. RULE II: Suppose you have Mx. then you may add Mxx to your collection. 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. RULE IV: If UU occurs inside one of your strings, you can drop it. 5.3 Theorems, Axioms, Rules Let s keep track of definitions for easy reference. A string is simply an ordered sequence of M s, I s, and U s. A theorem is a string produced (proved) by the rules of the formal system. An axiom is a starting point assumed to be true in a formal system. 23
The rules detailed above are rules of production or rules of inference. A derivation is a demonstration of how to produce one theorem from another theorem. 5.4 Inside and Outside the System What are some more difference between people and machines? Hofstadter talks a lot about observing patterns, but who is doing the observing and from where? 5.5 Jumping out of the System Do you really think that being able to jump out of a task and look for patterns is an inherent property of intelligence? What do you think of the following? Of course, there are cases where only a rare individual will have the vision to perceive a system which governs many people lives, a system which had 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! (pp. 37) What or who does this make you think of? Karl Marx and Communism Anarchism Socialism today and working peoples The Media The Government The Church The School Culture It all has to with cycles and loops! Is Hofstadter advising us to avoid repetition in thought and action? 5.6 M-Mode, I-Mode, U-Mode Mechanical Mode (M-Mode) Intelligent Mode (I-Mode) Un-Mode (U-Mode) Hofstadter calls the U-Mode a Zen way of approaching things. (pp. 39) What does this mean? 24
5.7 Decision Procedures OUR THEOREMHOOD TEST: Wait until the string in question is produced; when that happens, you know it is a theorem and if it never happens, you know that it is not a theorem. Is this a good test? NO! We want something that can terminate in finite time! 5.8 Study Questions 1. In principle, any three letters could have been chosen for the puzzle. Why were M, I, and U employed? 2. On the first page, four strings of letters are given to be strings of the MIU-system. What generality can you draw as to what constitutes such a string? 3. What does it mean to be in possession of a string? 4. What is the product of RULE I acting on the string MUUUIUI? What about the string MIIIUIU? 5. What is the product of RULE II acting on the string MUUUIUI? 6. The text states that from MU you can get MUU. Doesnt that imply that you can also get MU? If so, then the puzzle is solved! 7. What is the product of RULE III acting on the string MUUUIUI? On MIIIUIU? 8. What is the product of RULE IV acting on the string MUUUIUI? On MIIIUIU? 9. How does a theorem differ from an axiom (both in mathematics and the sense used in GEB)? 10. Is the notion of truth different for a theorem than an axiom? 11. How sure are you that if MI is the sole axiom, the MIU-system can produce no theorem that does not begin with M? 12. Is it possible for humans to act unobservantly? 13. Hofstadter suggests that the numbers 3 and 2 play important roles in the MIU-system. What roles? 14. Provide an example of how you used the M-mode in considering the puzzle. 15. Provide an example of how you used the I-mode in considering the puzzle. 16. Provide an example of how you used the U-mode in considering the puzzle. 17. Will the decision tree shown in Figure 11 produce every theorem of the MIU-system from the sole axiom MI? 25
18. Suppose we leave MIU and take on the observable universe. If there existed a decision tree that could generate every true statement (modeled after Figure 11), could I say that all truth is knowable? Would it help if I told you that the decision tree reached a decision after no more than 1000 steps? By the way, if each decision is yes-no, then the tree could conceivably deal with 2 1000 questions, a number close to 1 with 300 zeros after it, far more than the number of electrons in the universe. 19. Does there exist a litmus test for theoremhood in the MIU-system? 26
Chapter 6 Two-Part Invention I am the Walrus The Beatles 6.1 Abstract Bach also wrote fifteen two part inventions. This two-part 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. GEB pp. viii 6.2 Lewis Carroll s Paradox The basic axiom of reasoning is call Modus Ponens. Failing to agree to this axiom leads to infinite regress. What distinguishes these paradoxes? How do we resolve these paradoxes? Zeno should that Motion Is Inherently Impossible and thus Motion Unexists (MIII MU), but now Carroll seems to show that reasoning is impossible too! 27
Chapter 7 Chapter II: Meaning and Form in Mathematics But what does it all mean Basel? Austin Powers 7.1 Abstract A new formal system (the pq- system) is presented, even simpler than the MIU-system 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. GEB pp. viii 7.2 The pq-system DEFINITION: xp-qx- is an axiom, whenever x is composed of hyphens only. RULE: Suppose x,y, and z all stand for particular strings containing only hyphens. And suppose that xpyqz is known to be a theorem. Then xpy-qz- is a theorem. Hofstadter recommeds we try to develop a decision procedure a test that when applied to any theorem of the pq-, either verifies it is a theorem or shows that it is not. 7.3 The Decision Procedure 1. Test first that the theorem is a well-formed formula. 2. Count the number of hyphens in the first two hyphen-groups. 3. Count the number of hyphens in the third hyphen-group. 4. If the numbers in two and three are equal, you have a theorem! 28
7.4 Bottom-up vs. Top-down One starts from a trunk of axioms (the bottom) and derives up to all the theorem-leaves, the other takes a leaf of the theorem-tree and then tries to follow the branches all the way back to the trunk. 7.5 Isomorphisms Induce Meaning Recall the definition of an isomorphism from the Tools for Thinking chapter. Let us now write down the following definition: An interpretation is a symbol-word correspondence. How does interpretation work? Do you really think that the pq- system gains meaning because of the coincidence with addition rules? Where does meaning exist? 7.6 Active vs. Passive Meanings Hofstadter tells us the most important thing to get out of the chapter (p. 51): The pq-system 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. 7.7 Meaningless and Meaningful Interpretations Is meaning subjective? 7.8 Double-Entendre! The role of intuition in discovering truths must then be accompanied by the formality of string manipulation. Where does this break down? Does intution lead us astray or is it essential? 7.9 Formal Systems and Reality What do you think of the following excerpt? (GEB pp. 53-54) 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 system. Its symbold do not move around on paper, but rather in a three-dimensional 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 29
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) cast 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. 7.10 Mathematics and Symbol Manipulation How do we prove that isomorphisms with the real world are perfect? 7.11 The Basic Laws of Arithmetic In what physical contexts are basic truths like 1 + 1 = 2 in fact false? Consider water drops merging and splitting, clouds, and other continuous physical situations. How do we know that the math-reality isomorphism holds? 7.12 Ideal Numbers If numbers seem so fundamental, are they in some respects more real than things in reality, such as atoms and birds. Consider Descarte s skepticism of the senses, and consider the philosophical school of Platonism. 7.13 Euclid s Proof Once again, consider which do you more strongly believe: There are infinitely many primes The Sky is blue How should our belief system reflect this choice? 7.14 Getting Around Infinity Abstraction as a tool for thinking. 7.15 Study Questions Let s get a little more specific here. 30