A wonderful, beautiful work. Ask me about it, and I'll start nattering at you about sphex wasps, fugues, isomorphisms and "jumping out of the system." And my voice will trail off and you'll see me get a faraway look in my eyes.
It's actually quite difficult to describe what this book is about--at least, impossible to describe in a few short sentences.[1] But there are so many ways to read Godel, Escher, Bach, and such a wide range of ideas and insights one can get out of it, that it becomes a different book for every reader.
And let me confess, if you haven't read GEB yet, I am jealous of you.
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First of all this book can be understood on many levels. You can read it as an attempt to understand AI (a timely topic these days), and what it is precisely that makes up what we call "intelligence." You can read it as a deep discussion of the varied and beautiful consiliences that exists across art, music and mathematics. You can read it as a layperson's explanation of logic and formal systems.
GEB also serves as a master class on how to think by analogy, how to see patterns, and how to solve problems in wildly creative ways by loosening or even exiting the parameters of the problem ("jumping out of the system" as the author puts it). A book is of course about what it is literally about, but it also is about what it gets you to think about.
This is one of very few books that merit rereading, and rereading again. Books like this appear--if we're lucky--once in a generation.
If it's your first time reading GEB, I have a few pieces of advice. Each chapter is preceded by an "interlude," a short, allegorical story that touches on the material in the chapter to follow. As you read through each interlude/chapter pair, I suggest going back to skim the interlude again to groove the ideas and catch a few extra nuances. Over the course of the book this will help you connect up the various central ideas--and you'll also find more of the author's various subtleties and Easter eggs he places here and there throughout the text. I missed much of this on my first reading.
Elements of the book are technical and can be a bit difficult (see Chapters 13 and 14, for example, which deal with formal systems), at least they were difficult for me. But it is not necessary to understand these sections on an advanced level to get value from the book. Be patient and keep at it!
And if you're the kind of reader who only wants the "executive summary," then read Chapters 17-20, which take you to the book's punchline via discussions of the Church-Turing Thesis, various isomorphisms between our brains and machine brains, and a discussion of tangled hierarchies, strange loops, and the future prospects of AI. But if you're an "executive summary"-type reader, this book will likely be lost on you. You either get GEB or you don't.
And the reading list! It's a gift when an insightful author tells his readers about the various books that sparked his interest and triggered his insights, and author Douglas Hofstadter sprinkles wonderful reading ideas throughout GEB: biographies, books of poetry, even surprisingly readable books about mathematics. The reader can't help but write them all down excitedly, and then wonder how he's going to get to all of this reading when life is so short. I hope the readers of this blog (and there seem to be many more of you in recent months, welcome!) enjoy the reading list that I've put together below.
One last thing: a thought on rereading books. The best books change you. But you, as you change, change the reading experience too. Sometimes you grow into a book: when I read Turgenev's Fathers and Sons as a teenager it made no sense, when I read it in my fifties, it made all the sense in the world. With some books, a second reading allows you to see much more. When I read GEB for the first time, it had me sitting with my mouth hanging open. Now, rereading it, I realize how much more there is here to mine. I'll be reading this book a third time, and a fourth.
Footnote:
[1] In the 20th anniversary edition of this book, the author writes a 20-page preface explaining how no one seems to know what his book is about, including quoting a hilariously off-base one-sentence blurb the New York Times wrote to try to describe it: "A scientist argues that reality is a series of interconnected braids."
To Read [the descriptions in quotes here are Hofstadter's own comments about the book]:
Poetry of Russell Edson
Ernest Nagel and James R. Newman: Gödel's Proof
James Huneker: Chopin: The Man and His Music (public domain link) ["scintillating"]
John Chadwick: The Decipherment of Linear B
Howard DeLong: A Profile of Mathematical Logic ["superb"]
***Scott Kim: "The Impossible Skew Quadrilateral: A Four-Dimensional Optical Illusion" [paper]
Scott Kim: Inversions
Eric Temple Bell: Men of Mathematics ["He makes every life story read like a short novel."]
Douglas Hofstadter and Daniel Dennett: The Mind's I
Douglas Hofstadter's monthly columns for Scientific American: 1981-1983
Douglas Hofstadter: Metamagical Themas: Questing for the Essence of Mind and Pattern
***Douglas Hofstadter: Le Ton Beau de Marot: In Praise of the Music of Language
H.T. David and A. Mendel: The Bach Reader
***H.T. David: J.S. Bach's Musical Offering: History, Interpretation, and Analysis
***Johann Nikolaus Forkel: Johann Sebastian Bach: His Life, Art, and Work
Benoit Mandelbrot: Fractals
Warren McCulloch: Embodiments of Mind
***George Steiner: After Babel
***Leonard B. Meyer: Music, the Arts, and Ideas
Erwin Schrodinger: What Is Life? ["Influential"]
Paul Reps: Zen Flesh, Zen Bones
Gyomay M. Kubose: Zen Koans
Mumon: Mumonkan (The Gateless Gate)
Miranda Seymour: In Byron's Wake: The Turbulent Lives of Byron's Wife and Daughter: Annabella Milbanke and Ada Lovelace
E.O. Wilson: The Insect Societies
***Steven Rose: The Conscious Brain ["an excellent book...only his ideas on AI are way off"]
***Dean Wooldridge: Mechanical Man: The Physical Basis of Intelligent Life
***Marvin L. Minsky: "Matter, Mind, and Models" [paper]
***J. M . Jauch: Are Quanta Real?
M.C. Escher: The Graphic Work of M.C. Escher
***Erving Goffman: Frame Analysis
***Stanislaw Ulam: Adventures of a Mathematician ["Not only fun, but serious."]
Willard Van Orman Quine: The Ways of Paradox, and Other Essays
Jacques Monod: Chance and Necessity: An Essay on the Natural Philosophy of Modern Biology ["the book excited me deeply"]
***M. Bongard: Pattern Recognition ["magnificent" "invariably stimulating"] (See also this full online index of Bongard problems)
Suzi Gablik: Magritte
Music:
Chopin: Waltz opus 42
Chopin: Etude opus 25 #2
Bach: Tempo de Menuetto from Partita #5
Bach: The Finale of Sonata #1 for Unaccompanied Violin in G Minor
Bach: Aria with Diverse Variations (The Goldberg Variations)
Bach: Birthday Cantata
Bach: Musical Offering
Bach: Crab Canon
Bach: Art of the Fugue
Bach: Fantasia and Fugue in G minor, for Organ
Bach: Six-Part Ricercar
Bach: Six-Part Ricercar
[Readers: a severe warning: the notes that follow here are gigantically long. Please stop reading and value your lives! In fact they are so long that I've put the "To Read" section on top (as well as some music selections) to save you (and me) from having to scroll all the way to the bottom of the post.]
Notes:
Preface to the 20th Anniversary Edition
P-1ff The author talks about how no one understands this book, how major publishers like the New York Times got it all completely wrong; he then talks about what the book is really about: self-referentiality and the emergence of the soul under loops and strange circumstances; then a discussion of Kurt Gödel and Gödel's Theorem; then an engaging discussion of how Hofstadter came up with the idea of the book, and his intellectual journey writing it as he came up with the idea of including allegorical portions to set off the main themes of the book, which would explain the ideas in another way; also on how Hofstadter found a way to find "play" in writing this book, how it grew and grew as he wrote it until it became the book it is today.
P-12: "As I was writing, I certainly knew that my book would be quite different from other books on related topics, and that I was violating quite a number of conventions. Nonetheless I blithely continued, because I felt confident that what I was doing simply had to be done, and it had an intrinsic rightness to it."
P-14ff The author begs forgiveness for his book being sexist for having only male characters, and for using forbidden words like "mankind." On his fix in the French translation with "Madame Tortue."
P-18ff On Hoftstadter's intellectual path after GEB came out: he wrote a book in 1981 called The Mind's I with philosopher Daniel Dennett, he wrote a monthly column in Scientific American, he published more books: Metamagical Themas, Ambigrammi; on his delving into the world of language translation in his book Le Ton beau de Marot, which covers many domains, including creativity, meaning and what it means to think in another language [note: I'll be reviewing LTBDM shortly, it turned out to be extremely thought-provoking but also overwhelming]; how he translated Pushkin's poem Eugene Onegin; then a discussion of whether to tamper with Gödel, Escher, Bach--in particular the predictions in it that turned out to be wrong, like about chess-playing computers.
Introduction: A Musico-Logical Offering
3ff On Frederick the Great and his meeting with the elder Bach; on the Musical Offering Bach first improvised and then wrote out for Frederick the Great in the form of a puzzle (Quaerendo invenietis: by seeking you will discover) with multiple possible solutions under Bach's rules of canon and fugue; a discussion of isomorphisms in this musical genre, and then a discussion of a type of "strange loop" existing in one of the canons in the Musical Offering where the piece continues modulating higher and higher until after six go-rounds it arrives back at the same key all over again.
10ff Extending the strange loop idea to the "looping" images of Escher; on representations of infinity, along with elements of paradox and tangled hierarchies.
15ff Now on to Kurt Gödel and the underlying intuition that there seems to be something mathematical involved in all of these things; on the Epimenides Paradox or the "liar paradox": the statement "All Cretans are liars" (when said by a man from Crete), or the statement "I am lying" are examples of one step strange loops. Gödel's idea "was to use mathematical reasoning in exploring mathematical reasoning itself." It made mathematics introspective, leading to Gödel's Incompleteness Theorem, which the author paraphrases here in normal English: "all consistent axiomatic formulations of number theory include undecidable propositions," a self-referential mathematical statement, which, according to the author, "took genius merely to connect the idea of self-referential statements with number theory." On statements of number theory versus statements about statements of number theory--self-referentiality; on the idea of proof of a statement of number theory as defined within a fixed system of propositions: in Gödel's particular case he was referring to the system of reasoning in Principia Mathematica by Bertrand Russell and Alfred North Whitehead. Thus we can render Gödel's sentence again as "This statement of number theory does not have any proof in the system of Principia Mathematica" which the author calls "Godel sentence G" for shorthand. Gödel then proves that not only is Principia Mathematica incomplete, but that all axiomatic systems like Principia Mathematica are incomplete. This had an electrifying effect on logicians, mathematicians and philosophers, because "it showed that no fixed system, no matter how complicated, could represent the complexity of the whole numbers: 0, 1, 2, 3, ..." See as another example Euclidean geometry in light of the later discovery of other types of geometries: suddenly the mathematical community was shocked by the idea that you could have different kinds of points and lines in one single reality.
20ff On a theory of different types of infinities, known as the theory of sets: unearthing a variety of set-theoretical paradoxes: a set of things is not the thing itself. On "run-of-the-mill sets" vs "self-swallowing" sets: "the set of all sets" or "the set of all things except Joan of Arc" are self-swallowing sets, but nothing stops us from inventing "the set of all run-of-the-mill sets" which is a paradox: is it self-swallowing set or a run-of-the-mill set? It is neither--so what is it then? Can we make a rigorous theory of sets which is intuitive but skirts these paradoxes? This concept can be translated into language using Grelling's paradox, if we divide adjectives in English into two categories: 1) autological/self-descriptive adjectives (pentasyllabic, recherché [exotic, carefully sought out]) or 2) heterological/not self-descriptive (edible, incomplete, bisyllabic). If we consider "non-self-descriptive" as an adjective, to which class does it belong? This is a paradox! Thus in Principia Mathematica, Russell and Whitehead basically banned self-reference from number theory and set theory. The author then talks about how it's silly to ban self-reference in any form, because then you couldn't talk about yourself or the author couldn't make a reference to his own book for example. In an extreme example you can't even discuss the theory itself.
24ff On to a discussion of the early computers; on Charles Babbage and his (imagined) analytical engine with a store (memory) and a mill (calculating/decision-making); interestingly it was inspired by the Jacquard loom that could weave "amazingly complex patterns." Also on metamathmatics, appearing in the theory of computation discovered by Alan Turing.
26 "One of the major theses of this book is that it is not a contradiction at all [that intelligent behavior can be programmed]. 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 unreachable gulf between the formal and the informal, the animate and the inanimate, the flexible and the inflexible. This is what artificial intelligence research is all about." [And it is what thinking itself is about: finding and enjoying contradictions, savoring them, rolling them around in your mind, and emerging with new insights and mental models. I think this is why this book is so hypnotizing, even though the mathematical ideas here are way over my head.]
27 "The flexibility of intelligence comes from the enormous number of different rules, and levels of rules."
Three-Part Invention
29ff [I didn't properly understand this allegory when I read this book for the first time. But it is a self-referential story where a person referred to by the characters in the story (the author) enters the story. The author "knows" the story as well as the characters themselves because he created them; but even though they are his creations, the characters can both talk about him and he can talk directly to his characters. Both are examples of "jumping out of the system"--a critical idea in GEB--and self-referentiality. On the surface level, the conversation is about proving Xeno's Paradox of Motion, as the Tortoise and Achilles begin their race to demonstrate the proof. Note that you can find examples of self-referential metafictional scenarios like this in books like Don Quixote, or in practically anything written by Jorge Luis Borges.]
Chapter 1: The MU-Puzzle
33ff On formal systems. [This chapter was likewise over my head when I first read this book, but now I see it as a useful pedagogical exercise, the author gives the reader an exercise using a small formal system with four relatively simple rules, and then he assigns the reader to try to come up with a specific result using the permitted rules. The thing is, you can just "tell" that it's impossible to arrive at the final requested result, but if you stay "inside" the rules of the system and just follow them, you can experiment forever and never actually show that you can't arrive at the solution. In other words, knowledge of the impossibility of the task requires you to "leave the system" and look at it from the outside. This is the central idea of the chapter.
36 On playing with the rules inside a system but then making an observation about the system itself, which is something a computer cannot do. Basically this is a form of insight that a computer program lacks: "It is inherent in human consciousness to notice some facts about the things one is doing." The author arrives at a major point here. "It is an inherent property of intelligence that it can jump out of the task which it is performing, and survey what it has done; it is always looking for, and often finding, patterns."
37 "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 had never before been ever been recognized as a system; and such people often devote their lives to convincing other people that the system really is there, and that it ought to be exited from!" [What a beautiful quote! We can analogize it to fiat monetary systems, capitalism, communism's class struggle, neo-feudalism, W-2 hell versus financial independence/ERE, etc. If you cannot see the underlying system you can never escape from it.]
37ff Can a computer system jump out of a system? The author gives a surprising example of a chess program "that had the redeeming quality" of quitting a chess game when it was in a hopeless position. "It is very important when studying formal systems to distinguish working within the system from making statements or observations about the system." And here the author talks about his simple "MU-puzzle" where you work on it for a while, but then exit the system realizing you cannot produce MU--this is literally thinking about the system.
38ff On mechanical mode, M-mode, versus intelligent mode, I-mode, versus Un-mode, U-mode. [This last one is eventually going to turn into a joke about Zen much later in the book]; On building rules about the system from outside the system such that you can tell that the system will never yield the answer you're looking for.
Two-Part Invention
43ff Again, another allegory: here Achilles and the Tortoise have finished their race and recognized that there was no Zeno's Paradox because the distances they were covering were constantly diminishing; the paradox only applies if the distances constantly increase. Then they go over a syllogism about the transitive property of equality, but then include a fourth element to the syllogism that makes it infinitely recursive and never-ending; thus this is an allegorical example of the prior chapter, where you have a formal system with infinite recursion: you can only escape if by exiting the system and looking at it from outside. You'll also that the author, the narrator of this allegory, goes to the bank and returns to the Tortoise and Achilles some months later and sees from the outside that they're stuck in recursion, this is another allegorical example of observing a system from the outside. Creative!
Chapter 2: Meaning and Form in Mathematics
46ff "Do words and thoughts follow formal rules, or do they not?" "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." Discussion of the pq-system; on top-down vs bottom-up decision procedures to see if a statement is true in that formal system; on seeing an isomorphism with another system which produces "meaning," e.g.,: --p---q----- means "2 plus 3 equals 5"... or does it?
49ff An isomorphism as an information preserving transformation where two complex structures can be mapped to each other. "The perception of an isomorphism between two known structures is a significant advance in knowledge--and I claim that it is such perceptions of isomorphism which create meanings in the minds of people." And then on a two-tiered correspondence ("which is typical of all isomorphisms ") in the pq-system. Thus exploring and trying to understand a new formal system is like cracking a code; see The Decipherment of Linear B by John Chadwick.
51ff On meaningful versus meaningless interpretations; on distinguishing between interpretations and meanings; also "symbols of a formal system, though initially without meaning, cannot avoid taking on meaning of sorts, at least if an isomorphism is found." In language, when we have learned a meaning, we can make new statements based on that meaning; thus this is an "active" form of meaning because it produces new rules; but in true formal systems we can't just create new rules, thus in those systems the meaning must remain passive.
52ff On double meanings: there isn't necessarily "the" meaning only. Also on formal systems being based on a portion of reality and mimic that reality, but the reality and the formal system are independent, each one stands by itself. The PQ system is a simple formal system that mimics a system of addition, mimicking it with a meaningless symbol set with typographical rules. The question here is "Can all of reality be turned into a formal system?" Or consider the even more strongly worded statement "Reality is itself nothing but one very complicated formal system. Its symbols do not move around on paper, but rather in a three-dimensional vacuum (space); they are the elementary particles of which everything is composed... 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... The sole axiom is (or perhaps, was) the original configuration of all the particles at the 'beginning of time'."
54 Now the author offers a thought experiment: we have a formal system that appears to make every "theorem" come out true and every non-therem come out false, but how can we know this for sure if we can only check a finite number of cases? How will we know that all theorems express truth in this formal system unless we know everything there is to know about both the system and the corresponding domain of interpretation? For example, how do we assume the symbolic language for addition and multiplication hold true for all numbers? A discussion here about commutative and associative qualities of addition and multiplication, and how we take these qualities for granted, we hardly think about them at all, and yet "numbers as abstractions are really quite different from the everyday numbers which we use."
58ff The author then moves into non-counting number-based domains, taking the assertion "there are infinitely many prime numbers" which can't be confirmed or refuted using a counting process. Also on Euclid's Theorem (whatever number you pick there is a prime number larger than it), which is considered true due to "reasoning" but also due to a generalization [the author tells us we will meet this term again later in a more formal context]. The author then argues that such a proof has a patterned structure that binds the statements of the proof together, and such a proof can be represented and understood by finding a new vocabulary consisting of symbols that are suitable only for expressing statements about numbers. "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 abilities--or, more generally, whether it is theoretically possible to attain the level of our thinking abilities, by using some formal system."
Sonata for Unaccompanied Achilles
61ff [Another allegorical section here that turns out to be more subtle than I thought when I read this book for the first time.] There author offers a figure-ground puzzle here that exists on the verbal level (a word that begins and ends with HE); but on the story's literal level, Achilles is talking on the phone: we only hear half the conversation, and thus have to deduce what the Tortoise is saying to him as they make references to figure-ground challenges in various domains--both with "the mystery word that begins and ends in HE" as well as in the Escher drawing Mosaic II. Yet Achilles, despite having a print of that exact drawing hanging on his wall, can't seem draw the connection between the two (visual and typographical) domains.
Chapter 3: Figure and Ground
64ff Here the author produces another formal system, like the earlier "PQ" system, that typographically represents multiplication, and then shows another system that would capture number primeness; and then he shows how thinking about the various rules for these systems causes you to move up and down from "mechanical mode" (where you just follow rules) to "intelligent mode" (where you think about the rules: why they're in place, how they work, whether they make sense, and so on).
67 The author then illustrates how this is also a sort of figure/ground problem [!] where you look at "holes" in the series of theorems in a system and think about those as being prime numbers, they are things left out of a list which is positively defined. And then transferring this to a discussion of figure-ground-type images varying in ambiguity (which part is the "figure" and which part is the "ground"?); then The author gives us a hint of what's coming in a future chapter where he's going to create a typographical number theory that itself has sort of figure-ground problem.
70ff On figure ground issues in music: melody/countermelody, downbeat/upbeat, etc.
73 A discussion of figure/ground in the context of numbers (figure) and primes (ground); we can easily "see" and characterize rules for numbers, not so easily for primes, they are like holes, they don't have "form" in the same way. Then on what is now called a Hofstadter sequence: 1, 3, 7, 12, 18, 26, 35, 45, 56, 69... which is recursive in the sense that the next number is the next number not in the set added to the last number (thus the next number in the series is 83, or 14 + 69). [I didn't really grasp how beautifully recursive this is in my prior reading.]
74 Finally a solution to formally define primes by using a formal system without recursion or going backwards. "...it is this potential complexity of formal systems to involve arbitrary amounts of backwards-forwards 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."
Contracrostipunctus
75ff Another conversation between Achilles and the Tortoise on "music to break phonographs by." Discussion of the "perfect" phonograph that could reproduce any and all sounds, and what would happen if you played a record of sounds that would shatter and break a phonograph on that phonograph (Hofstadter describes it as the record entitled "I Cannot Be Played on Record Player 1"). The thought experiment here [which the book will return to repeatedly later on] is to consider a record player with such perfect fidelity that it could even play its own self-breaking sound--but if it can play its own self-breaking sound it is not a perfect record player! You could only play that record successfully on a "lower fidelity" or imperfect record player. Thus there is no such thing as a perfect record player. [Again, if you've read this book already, you'll see hints of Gödel's Theorem here with the self-referentiality; if it's your first time reading this book, this story just seems cute and thought-provoking, but it's not quite clear in what way it's supposed to be thought-provoking.]
78ff Then an iteration: one of the characters obtains Record Player Omega, a self-referential record player that can read and understand the phonograph before it plays it [e.g., it can "jump out of the system"], and thus it will recognize that the record will break the record player; it then responds by rebuilding itself and changing its own structure to be safe from the record's dangerous sound, and only then would it attempt to play the record.
79ff Achilles gives the Tortoise a goblet with the letters B A C H etched in it; then a discussion of Bach writing The Art of the Fugue, where Bach wrote the letters of his own name into the music, and then, according to a notation from his son C.P.E., Bach senior died right then. Thus Bach wrote his own "self-destructing record player." And then a conversation about acrostics, where the reader goes back and realizes that this entire story here is itself an acrostic. And then the Tortoise plays his violin, making a sound that shatters the B A C H goblet. [Basically here are still more analogies for Gödel's Theorem.] "For each record player there are records which it cannot play because they will cause its indirect self-destruction." The Tortoise "Gödelizes" the record players, while also the musical notes B-A-C-H likewise Gödelizes the crystal B A C H goblet.
Chapter 4: Consistency, Completeness, and Geometry
82ff On thinking more broadly about isomorphisms, in language in particular; also thinking about the medium by which we perceive the isomorphism, and thinking about meaning as mediated by isomorphisms; on the prior Contracrostipunctus story and its layers of explicit and implicit meaning [in other words, what the story is "about" on various levels, even to the ultra-literal level of marks on the paper]; and then on levels of "meaning" of the grooves on a record player, where the implicit meaning of the record--once "transcribed" via the player--then destroys it.
84 See photo for two recursive isomorphisms, giving a "visual rendition of the principle underlying Gödel's Theorem":
84ff Then onto implicit meanings of the dialog: the dialog is itself isomorphic (record player = goblet; see also the Tortoise's boomerang collection, see even the Tortoise's "devilish method of exploiting implicit meaning to cause backfires" itself backfires). And then of course the story itself is an isomorphism of Gödel's Theorem. See photo:
86 [Long quote here] "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. Gödel 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 [e.g., Principia Mathematica]. They found out otherwise in 1931 [when Gödel presented his theorem]. 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 reasoning methods which seemingly cannot be 'encapsulated'--they resist 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 non-living 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."
87ff On formal language, and making sure a word means a specific thing, so your isomorphisms stay stable, thus supporting the structure of your form system; see Euclid's Elements; on Euclid's four postulates ("absolute geometry") and his fifth postulate; on Girolamo Saccheri (1667-1733) and his near miss of non-Euclidean geometry, followed by another near miss of J.H. Lambert, and then a simultaneous discovery by the Hungarian Janos Bolyai and the Russian Nikolay Lobachevskiy in 1823. The central idea here is to rethink what a "straight line" actually is--it is something different in flat space versus spherical space.
94 "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."
94ff A quick discussion of what "consistency" and "inconsistency" mean, and what "internal inconsistency" means as we change the meaning of a symbol in a formal language. Also on imagining a hypothetical world where you might imagine whether or not your formal system is compatible and internally consistent.
97 On embedding one formal system in another, or nesting multiple systems in each other.
97 On layers of stability in visual perception; on Escher's impossible drawings, where we move from stable forms (staircases, figures) to unstable levels (where staircases intersect strangely and impossibly) which analogizes to Euclidean and non-Euclidean geometry.
99ff On assuming things across "every conceivable world": math, logic, number theory. Is mathematics the same in every conceivable world? Is there a conceivable world where there are not infinitely many prime numbers?
100ff On formal system "completeness" where consistency means "everything produced by the system is true" and completeness means "every true statement is produced by the system" (in that system's domain, not in all of reality itself).
101 "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 well-formed strings which express true statements of number theory, but which are not theorems. (There are truths belonging to number theory which are not provable within the system.)
Little Harmonic Labyrinth
103ff Achilles and the Tortoise are on a Ferris wheel, from which they are suddently reeled up into a helicopter by a cable [thus they experience a literal "jumping into and out of a system"] where another character "Goodfortune" threatens to eat them. Achilles and the Tortoise they read a book about themselves doing the things they are about to do, while they also experiment with jumping into impossible Escher prints. This allegory is filled with self-referential jokes and comments. Further, there is a looping, recursive story here about a genie that needs to ask meta-genies about whether a "metawish" can be granted, but the recursive loop isn't endless, there's a recursive acronym (GOD, which stands for "GOD over Djinn" and thus recursively includes the word "GOD" in its own acronym) that can be expanded (GOD Over Djinn, Over Djinn...), so it enables you to make a request of the Meta-Genie along with all Djinns above her. "GOD is the tower of djinns above any given djinn."
115ff Achilles says, "I wish my wish would not be granted" and a paradox occurs and the system crashes. They find themselves in Escher's drawing Reptiles, and the Tortoise begins reading one of the books in the drawing which is a recursive tale about them in a labyrinth, which is actually a groove of a record of a Bach piece, Little Harmonic Labyrinth. The piece has a "false" musical resolution, and Achilles is "fooled" by it. Note that this part of the story is a self-reference/isomorphism of the false resolution itself [!] because the earlier part of this story where Goodfortune was threatening to eat them was never actually resolved.
Chapter 5: Recursive Structures and Processes
127ff "What is recursion?" A specific example given here, in an AI context, of an executive mechanically going from call to call on his phone, putting people on hold, pushing to the next call, then popping back afterwards; here we also have some computer science terms used by analogy to explain nested jobs or nested tasks: to "push" is to go to the next task (usually a task on a lower level), while suspending operations on the current task; to "pop" is the reverse: to close the lower level operation and return to the higher level where you left off. You store memory of where you were on a "stack." Another example of "nesting" is in a news station: going from the studio to a recording of a reporter in the field, playing a recording of an interview, etc. We don't even notice the nesting here; our subconscious easily keeps track of what's going on.
129 On "stacks" in music: tonic (both global and local) and pseudotonic/pseudoresolution, resolution; see Bach's Gigue from the French Suite No. 5, or the Little Harmonic Labyrinth, or the Endlessly Rising Canon.
130ff On recursion in language: see for example the stereotypical long German sentence with a verb at the end; the author even includes a fairly witty syntactic example of the phenomenon of pushing and popping with this sentence: "The confusion among the audience that out-of-order popping from the stack onto which the professor's verbs had been pushed, is amusing to imagine, could engender." [It takes some decipherment to work out this sentence for sure]; on the Recursive Transition Network (RTN): a diagram which shows the various paths, connected by nodes and arrows, that can be followed to accomplish a task.
133ff On the difference between recursion and infinite regress; or the difference between a recursive definition and a circular definition: the recursive definition eventually bottoms out.
134ff Back to discussing RTNs: when you hit a node you can expand that node, this replaces it with a copy of the RTN it calls, thus the node is recursively expanded. But when you pop out of it you automatically are in the same correct place in the pathway you were on before, so you don't need to make an infinite diagram. "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 defect--or advantage--that you must repeatedly expand the 'G'; thus it never bottoms out." Other examples given here: Fibonacci numbers, numbers that are defined in terms of previous Fibonacci numbers (thus it makes recursive calls on itself). Also a discussion here of recursive algebraic definitions which render themselves in beautiful recursive geometrical structures: see the author's discussion of diagram G and diagram H on pages 135ff. See also an example here of a Q-sequence, a recursive number string where you count backward to obtain two numbers added together that make the new value, these types of strings can lead to "extremely puzzling behavior" and a type of chaos.
138ff Finally a discussion of two recursive functions which the author calls INT(x) and Gplot: when graphed, the INT(x) function seems to create something out of nothing with infinite nested copies of the function represented across the chart at an infinite range of scale; the Gpolot is a graphical representation of the relationship between an electron's energy levels in a magnetic field as well as in a crystalline environment. The author even goes so far to argue that elementary particles also are nested inside each other in a form of recursion, tangled up in their interactions with each other, showing a graphical rendition of this using "Feynman diagrams." Note also that Feynman diagrams have their own form of "grammar" in the sense of "well-formed" or "not well formed" diagrams: this is a lot like the example given several pages before about well-formed strings in Chapter 4.
146ff Now a discussion of copies and sameness: returning to the example first of the musical canon, taking a theme and copying it by isomorphism; also in some of Escher's art, see for example Fishes and Scales and Butterflies; also on the idea that we can easily recognize Escher's or Bach's "signatures" (their respective style) in their works. What is it that is the "same" about all of Escher's or Bach's works? Which takes us to the concept of "sameness-in-differentness," where the author asks the question "When are two things the same? [This question] 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."
148ff On recursion with sameness: you can have similar recursive things, but because they're nested they're not exactly the same; then a discussion of looping computer programs, which can be either bounded or free: the "bounded loop" has a set maximum number of steps; the "free loop" has a risk of becoming an infinite loop and never aborting; also loops can be nested inside each other. [The author will return to these concepts in Chapter 8.]
152 On recursively enumerable sets, like Fibonacci numbers or Lucas numbers, where they snowball, via a recursive rule from two elements into an infinite set. "Recursive enumeration is a process by which new things emerge from old things by fixed rules. There seem to be many surprises in such processes--for example the unpredictability of the Q-sequence [recall Q-sequences were chaotic sequences discussed on p137-8]. 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?" Next the other suggest the idea of a program that doesn't just recursively call itself but also can recursively modify itself--extend itself, improve itself, generalize about itself or fix itself. "This kind of 'tangled recursion' probably lies at the heart of intelligence."
Canon by Intervallic Augmentation
153ff "Achilles and the Tortoise have just finished a delicious Chinese banquet for two, at the best Chinese restaurant in town." They begin talking about haikus--in haikus.
Meaning lies as much
in the mind of the reader
as in the haiku.
The waiter arrives with fortune cookies, and one of the fortunes is encoded in the poem that contains its own self-referential commentary. Achilles and the Tortoise then talk about the Crab, who has obtained a rare type of jukebox: a jukebox that plays only one record, but uses different phonograph machines to play that record, where the record is held perfectly still and the phonograph machine spins around it; and yet each record player inside the jukebox will play a differently encoded version of the B A C H song. By using different encodings using the multiplication interval 3 and 1/3 ("one could call it intervallic augmentation") the jukebox can "pull" different songs based on the same underlying pattern. [The underlying issue here, which the author discusses in the next chapter, is literally "where is the actual song?" or more broadly, "where is the "meaning" in a given message?]
Chapter 6: The Location of Meaning
158ff On the idea of "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 message--as in the preceding dialogue. In the latter case, meaning could not be 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... 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." Discussion here of a record as an "information bearer," while the record player is an "information revealer." See also the PQ system where the information revealer is the interpretation and the information bearers are the theorems; also on the idea that information is intrinsically inside a structure waiting to be pulled out, so how hard are you allowed to pull? What can be pulled and what is intrinsic? Further examples: a discussion of genotype and phenotype in the DNA of an organism, see also the highly tangled recursion of epigenesis [we will see more about this in Chapter 16]. Basically the DNA structure and the phenotype of the organism are isomorphic, but in a very exotic way; in contrast see more prosaic isomorphisms, where the one structure is easily mappable onto parts of the other: like the isomorphism between a record and a piece of music, the record is a type of "image" of the music and we can pinpoint its "location" accurately and easily.
160 On the idea of "genetic meaning" which is encoded into a molecule of DNA; this is an example of implicit meaning, where "a set of mechanisms far more complex than the genotype must operate on the genotype." The author returns to the jukebox analogy where a combination of instructions would cause the DNA to "play songs" that would lead to the creation of further jukeboxes: thus the DNA triggers manufacturing of proteins, which trigger hundreds of new reactions, copying the DNA, creating new reactions, etc., with the final result being the individual.
161 Offhand mention here of Jacques Monod, "one of the deepest and most original of twentieth century molecular biologists." [Recently I posted a review of Monod's book Chance and Necessity.]
161ff Other questions: what about the intrinsic meaning of a smashed record, or of a scrambled telephone call; where would we place epigenesis on 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 resides?" The answer here is "it depends": there's so much information outside the DNA that we can't really look at DNA as anything more than a very intricate set of triggers, but we can also perhaps argue that the information is all "there" but in an extremely implicit form. Also, it's intriguing to think about the two following views, which seem similar but aren't: "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." See yet more examples: a record being sent out into space where perhaps it could be deciphered; or if such a record were sent back in time to Bach's era no one would know what to do with it.
162 On levels of understanding a message, how do you recognize there's a message at all? What if the aliens that receive the record don't have emotions mappable to ours? [This is something Stanislaw Lem considers deeply in his novels Solaris and Fiasco.]
163ff Another thought experiment using John Cage's "aleatoric" or "chance music," and a record sent into space containing this type of music where it would be nearly impossible to decipher the frame message ("I am a message; decode me") as well as the chaos of the inner structure (aliens would lack the necessary context to understand confused sounds from 20th century radio for example). Consider the contrast to a Bach piece where there are more intrinsically recognizable patterns, as well as patterns of those patterns. The question here is whether any message has enough inner logic that its context can be restored automatically when intelligence comes into contact with it.
164ff Next an example of deciphering unknown languages and alphabets; the Rosetta Stone; also, on seeing Chinese or Egyptian heiroglypics we somehow intuitively know there is information inherent in such texts even if we can't "read" them. One nuance about decoding: decoding does not add any meaning to the message that they take as input, they merely reveal the meaning.
166ff On the three layers of any message: 1) the frame message, 2) the outer message, 3) the inner message. "To understand the inner message is to have extracted the meaning intended by the sender." The frame message says "I am a message; decode me" thus "To understand the frame message is to recognize the need for a decoding-mechanism." Finally on the outer message: "To understand the outer message is to build, or know how to build, the correct decoding mechanism for the inner message." Note that this also is an implicit message, thus "the outer message is necessarily a set of triggers, rather than a message" because you can't read the message until you've decoded it.
166ff Note more potentially good reading ideas here as the author cites George Steiner's book After Babel as well as Leonard B. Myer's book Music, the Arts, and Ideas, and Erwin Schrodinger's "influential" book What Is Life? discussing various perceptual and cognitive tools required to understand different types of messages in music and language. The author suggests that "works of art are trying to convey their style more than anything else" thus style is an "outer message" or a "decoding technique."
167ff See for example a message in a bottle found on the beach: inside the bottle (which is sealed) we can see a dry piece of paper, thus we know for sure that it's an information-bearing message, and we can look at the marks on the paper and recognize that it's written say in Japanese. Once we discover this we can then proceed to decode the inner message. There's no point in including in the message a translation of the sentence "This message is in Japanese" because you'd have to know Japanese to read it: "one has to decipher the inner message from the outside." Note also that outer messages are not conveyed in any explicit language, this is simply the nature of an outer message.
170ff On the "jukebox theory of meaning": no message contains inherent meaning because before any message can be understood it has to be used as the input to some "jukebox," and this means the information contained in the jukebox must be added to the message before it can have meaning. The author then goes back to the prior dialogue where there is a trap: before you can use any rule, you have to have a rule which tells you how to use that rule, thus there's an infinite hierarchy of levels of rules, and this prevents any rule from ever getting used. The trap here is in order to understand a message you have to have a message which tells you how to understand the message. So how is it then that we can actually use rules and understand messages? It's "because our intelligence is not disembodied, but is instantiated in physical objects: our brains." Our brains have hardware for recognizing that certain things are messages and for decoding those messages; also the decoding mechanisms are universal and the deciphering mechanisms are universal in the intelligences that we have in our minds. Also an interesting thought here about comparing the inherency of meaning with the inherency of weight, but the weight of something will change depending on the gravitational field (the mass won't), and so perhaps there's an earth chauvinism if we think about the weight of something, just like we might have a similar chauvinism with respect to intelligence and thus also with respect to meaning. So we would call any brains sufficiently like our own "intelligent" while refusing to recognize other types of objects as "intelligent." The problem here, however, is that this definition is circular.
174ff Comparing the music of Bach with the music of John Cage: on understanding that John Cage's music does not have an intrinsic meaning, and it requires significant cultural context because it's a "revolt" against certain kinds of musical traditions; thus to understand John Cage we need to know not just music and how to hear it but also we have to have an understanding of Western culture, various conventions of classical music, etc. Thus a John Cage listener would be like a jukebox, and the piece itself is like a pair of buttons ("play record C-3"), but the meaning is mostly in the "jukebox" (the listener); it is the music that triggers the meaning. By contrast Bach can be understood with far less cultural knowledge. "Intelligence loves patterns and balks at randomness."
175 "How much of the context necessary for its own understanding is a message capable of restoring?" The author imagines DNA molecule and its proper chemical context as read by an intelligent civilization, and then compares this to sending DNA in the form of an abstract sequence of letters (e.g., "CTCCTCGGCGGGCACGTAG"), and thus the chance of understanding or decoding this message would be very low.
176 Interesting comment here: "Lest you think this all sounds hopelessly abstract and philosophical, consider that the exact moment when a 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." [Heavy! He's right.]
Chromatic Fantasy, And Feud
177ff Achilles and the Tortoise have a "tangled hierarchy" of a conversation where they're contradicting and recontradicting themselves and each other; also this is a discussion of operations in a linguistic context: say for example aligning two sentences with "and" versus "in." "It's harmless to combine two true sentences with 'and'!" So now the dialogue is itself a nested recursion: it's both a review of what's been discussed so far in the book and it's also a preview of the "propositional calculus" the author is about to teach the reader in the next chapter.
Chapter 7: The Propositional Calculus
181ff The author tells the reader he's going to develop a formal language to help do what Achilles couldn't get the Tortoise to do in the last dialogue: the Tortoise was able to get away with using language in a "normal" way when it was in his advantage to do so, but then use it in a non-normal way whenever that was in his advantage. On the idea of creating a formal language which limits this: "propositional reasoning." The author then goes through his system of symbols strings and rules. The idiosyncrasy here is that there are no axioms for this system: instead there's a special rule that manufactures theorems out of thin air--the "Fantasy Rule": you ask with any well-formed string "What if this string X were an axiom, or a theorem? And then you let the system itself give its answer. Thus you can push into a fantasy and pop out of a fantasy using the symbolic language of the system.
184ff The author gives a metaphor here of a "No Smoking" sign in a movie theater that clearly should not apply to the characters in the movie: in such situation there's no carryover from the real world into the fantasy world. But in his propositional calculus here there is a carryover: thus you can take any theorem from one reality into another, or from one level of the symbolic language to another. "One way to look at the fantasy rule is to say that an observation made about the system is inserted into the system." What's interesting here is the system is working mechanically, thoughtlessly; there is no one in there thinking about the meaning of the strings, the system just prints out strings.
189ff Interesting use of the Zen koan Ganto's Ax, which explores a paradox of speaking or not speaking. Note that the monks continue meditating, and Ganto tells them you are true Zen students, but when Ganto relates the incident to another teacher, Tokusan, who asks about the situation from the perspective of the students. Ganto then tells him that perhaps another teacher can teach these students, but they should study under Tokusan! [The primary idea here is to illustrate how the author's formal system might handle--or might fail to handle--the self-contradicting ideas in a Zen koan. The second part of the koan (I think) is a joking comment about Tokusan, who apparently used harsh treatment on his students, but it doesn't have anything to do with Hoftstadter's formal system example here.]
191ff Discussion here of a "prudent" and "imprudent" point of view on whether a system is consistent or not; the imprudent system assumes that any well-formed string from this formal system is true. The prudent system doubts that they're true because some statements come out of the system that are contradictory. But then the imprudent interpretation says "how do you know they are contradictory? If a system comes up with both strings x and ~x, you only think they are contradictory because you're assuming the only conceivable meaning of '~"'is 'not.' But if the system produces these two strings, you have to exit the system and use a more complicated set of operations from outside the formal system in order to prove they are inconsistent.
193ff On derived rules, which are "about" the system: they need proof but the proof is intuitive, it comes from a chain of reasoning carried out in I-mode, rather than M-mode: in other words it is metatheory. "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 informally--outside the system." But then the author asks, "Why not formalize the metatheory too? In other words can we have these derived rules be theorems of a larger formal system; this is interesting, but suddenly it starts you down a road of metametatheories and yet still there will be a desire to make shortcuts at the topmost level. "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." Also note the comments here on fallacies that can emerge if you fail to distinguish between working inside the system (M-mode) and working outside the system (I-mode). The idea here is it is dangerous to assume symbols can be shipped back and forth from the formal system and the metalanguage (in this case English) used to describe the system.
196ff More on reasoning within a system, finding a contradiction, and then jumping out of that system to reason about the contradiction: which would then help you strengthen the domain. This is much like what happened with mathematics over the centuries. See for example the debate in the Middle Ages about the value of the infinite series 1 - 1 + 1 - 1 + 1... [This is called Grandi's series] which helped produce a deeper and fuller theory about infinite series.
Crab Canon
199ff Achilles and the Tortoise meet up again and begin a conversation. They are interrupted suddenly by the Crab, who speaks briefly and then disappears. And then the conversation between Achilles and the Tortoise happens exactly backwards in a mirror image of what was said before the Crab arrived. Interestingly it actually does sound like a real dialogue in both directions.
Chapter 8: Typographical Number Theory
[I suspect that most English major-type readers like myself will more or less blank out during this chapter, it's mathy and somewhat intimidating. Keep going!]
204 Brief comments here on the prior dialogue, and how it had various self-references inside of it including the fact that the Crab Canon was itself in the form of a crab canon. "To see the self-reference, 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 the strange system with the capacity for wrapping around."
204ff Various mechanics of the Typographical Number Theory: specific symbols, the types of ways to express things that we want; on predicates, both verbal and arithmetical; then trying out some of the strings like "6 is even" and "2 is not a square" in this new notation; then using different forms of notation in this TNT formal system to say the same thing--which seems like the same in our minds but is actually examples of different strings that are distinct.
216ff Discussion of the five postulates of Giuseppe Peano, who made an attempt to formalize a small set of properties of natural numbers from which everything else could be derived by reasoning.
221ff Here is where the author starts to scratch at some of the problems Gödel talks about: "(Proposed) rule of all: 'If all the strings in a pyramidal family are theorems, then so is the universal quantified string which summarizes them.' The problem with this rule is that it cannot be used in the M-mode. Only 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 inside any formal system." [So to review a bit: the author has already shown in the much more limited "PQ system" that you couldn't produce certain theorems like the one above; here, even in this far richer, far more complex formal system TNT, where we have much higher expectations, we still have the same problem.] "As a matter of fact, there is a name for systems with this kind of defect--they are called Omega-incomplete." "The string is undecidable within the system." Just like within absolute geometry, Euclid's Fifth Postulate is undecidable. Also comments here on how a set of theorems may assert a certain property for all natural numbers, but then has the inconsistency of a theorem that asserts that not all numbers have it, including non-natural numbers, or, as the author calls them "supernatural" numbers.
225ff The author does a very long derivation in the TNT system, likening it to a passage of music where it becomes clearer and clearer where the thing is going, confirming the readers (or the listener's) intuition, and then accelerating with tension to an "almost there" feeling where the resolution is now clear... and then you play it out until the ending with the final tension resolved at the solution. "This is typical of the structure not only a formal derivations, but also of informal proofs. The mathematician's sense of tension is intimately related to his sense of beauty, and that is what makes mathematics worthwhile doing." The author then notes that in the language itself there's no reflection of these tensions: the tension exists in the doer of the derivation. "Could one devise a much fancier type of graphical system which is aware of the tensions and goals inside derivations?"
229ff Dealing with problems of circularity when we try to prove that a typographical number system like TNT is consistent. "If we all use the same equipment in a proof about our system as we have inserted into it, what will we have accomplished?" Finally a statement of Gödel's work which says: "Any system that is strong enough to prove TNT's consistency is at least as strong as TNT itself. And so circularity is inevitable."
A Mu Offering
231 "The Tortoise and Achilles have just been to hear a lecture on the origins of the genetic code, and are now drinking some tea at Achilles' home." Note this quote from the dialog: Achilles: "Molecular biology is filled with peculiar convoluted loops which I can't quite understand, such as the way that folded proteins, which are coded for in DNA, can loop back and manipulate the DNA which they come from, possibly even destroying it. Such strange loops always confuse the daylights out of me. They're eerie, in a way."
232ff The two then have a misunderstanding about the meaning of the word "enlightenment": one of them uses a loose meaning and the other uses the more formal meaning. Also an example of recursion here where their conversation mixes up a couple of letters [kind of like DNA would!] where the author has one of the characters say "The sixth patriarch was Eno" while the other one hears this as "Zeno."
233 Achilles then tells the koan of Joshu, where Joshu uses the word "MU" to "unask" the question, indicating that the question has no meaning--that "only by not asking such questions can one know the answer to them." The Tortoise replies, "MU sounds like a handy thing to have around. I'd like to unask a question or two, sometimes."
233ff Then two more self-contradicting two-line koans:
A monk asked Baso: "What is Buddha?"
Baso said: "This mind is Buddha."
Followed by:
A monk asked Baso: "What is Buddha?"
Baso said: "This mind is not Buddha."
Thus we have a Zen version of the various contradictory phrases from much earlier in the book ["The following statement is true. The previous statement is a lie" or "My shell is green and not green"].
Then a discussion about the decision procedure for the "genuineness" of a koan [read: "a decision procedure for the theoremhood of a string"]; the author likens it to looking at an orchard from one angle but then from another, where you can see a form of order that wasn't visible from the other angle. Then a discussion "about" the decision procedure that addresses how the system uses forms of transcription and translation, with obvious parallels here to DNA replication/transcription.
237 A well-known koan, Gutei's Finger, which addresses a variety of concepts: how a representation of something is not the same as that something; the finger is not the explanation, Zen itself is not a concept, or a word, or a thing; this is also a metadiscussion about the bond between a teacher and student and the idea of transmitting truth or teachings; that the student and teacher have a permanent bond, etc. The Tortoise and Achilles have a discussion about whether or not to take these stories seriously: "I would guess that if you tool all such stories entirely seriously, you would miss the point as often as you would get it."
238 And then a discussion of "going against the arrows": taking a translation language and using it in a direction you're not supposed to; this produces surprises, unusual things, also sometimes nonsense, but sometimes you get what seems to be a koan. [Again this is a metaphor for using a formal system.]
240 Another koan: "Joshu Investigates an Old Woman": on the idea that enlightenment is "straight ahead" and involves not letting yourself get diverted, and not being self-deceptive.
242ff Achilles and the Tortoise then begin working on different "strings" [literally strings in this case, but the obvious metaphor is "strings" produced in a formal system], finding one with a knot at each end. But when the line is pulled, both knots disappear, like a double negation; note other things they do here, like passing strings back and forth that they arrange with their feet, etc; Then Achilles "translates" the string into English, yielding a self-referential koan, although the Tortoise and Achilles don't quite recognize it as such "There is something might funny going on here, and I'm not sure I like it." Finally, the story ends with a koan from the Zen master Kyogen: "Zen is like a man hanging in a tree by his teeth over a precipice. His hands grasp no branch, his feet rest on no limb, and under the tree another person asks him: "Why did Bodhidharma come to China from India?" If the man in the tree does not answer, he fails; and if he does answer, he falls and loses his life. Now what shall he do?"
Chapter 9: Mumon and Gödel
246ff The author comments on Zen being "intellectual quicksand," tantalizing and infuriating and yet humorous and enticing; he hopes in this chapter to get some of these same reactions across to his reader, leading us to Gödelian matters. On Mumon and his work Mumonkan or The Gateless Gate, Mumon lived around the same time as Fibonacci, interestingly. On the opaqueness of both his koans and his commentary; perhaps his commentaries are deliberate meaningless, to show how useless it is to spend one's time chattering about Zen; but also on the multi-level understanding of Mumon's language and metalanguage. On how koans place the mind in a bewildered state where it begins to operate non-logically. But then the author asks, "But what is so bad about logic? Why does it prevent the leap to enlightenment?"
251ff On Zen and its struggle against dualism; on escaping the world of categories and conceptual division of the world; also on the use of words, words are inherently dualistic because each word represents a conceptual category; in other words the enemy of enlightenment is not logic but rather verbalist and dualistic thinking. [This is quite a beautiful chapter! There are all these Zen koans interspersed with some of Escher's most beautiful lithographs and woodcuts, like Three Worlds (1955), Dewdrop, (1948), Another World (1947), Day and Night (1938), Rind (1955), Puddle (1952), Rippled Surface (1950). It's quite an experience reading this part of the book.]
M.C. Escher, Three Worlds, 1955
252 "Relying on words to lead you to the truth is like relying on an incomplete formal system to leave you to the truth. A formal system will give you some truths, but as we shall soon see, a formal system--no matter how powerful--cannot lead to all truths. The dilemma of mathematicians is: what else is there to rely on, but formal systems? And the dilemma of Zen people is: what else is there to rely on, but words? Mumon states the dilemma very clearly: 'It cannot be expressed with words and it cannot be expressed without words.'... Zen is a philosophy which seems to have embraced the notion that the road to ultimate truth, like the only surefire cure for hiccups, may bristle with paradoxes."
254 An intriguing idea from the author here where he says "I have a name for what Zen strives for: ism. Ism is an antiphilosophy, a way of being without thinking." [It is likely that the author, a highly perceptive man, is also aware of the interesting coincidence that many ideologies end with the suffix -ism, and ideological "thinking" also tends to encourage not thinking...]
255 "Zen is a system and cannot be its own metasystem; there is always something outside of Zen, which cannot be fully understood or described within Zen."
255ff Now the author explicitly discusses some of the parallels between M.C. Escher and Zen, where Escher plays with reality and paradox and "delights in setting up contradictory pictures" that play with reality and unreality in the same way Zen does.
258 A comparison here of Escher's lithograph Three Spheres II (1946) with the Buddhist allegory of Indra's Net, an endless net of threads throughout the universe, in Escher's lithograph all the Spheres reflect each other, the table on which they sit, and even the artist who is drawing everything in the image.
259ff Now to analyze the MU system to see if it has theorem nature: it does not.
261ff On Gödel numbering: Thanks to the discovery by Gödel of a certain special kind of isomorphism, "there is a way to embed all problems about any formal system, in number theory." The author illustrates it with the MIU system (from Chapter 1) by "Gödel numbering" it: you give each symbol a number that we map it to, in an information preserving transformation; note also the dual nature of the Gödel numbered examples, we can see them both typographically and arithmetically: see for example how the author uses "multiplied by 10" or "divided by 10" versus adding (or removing) a zero to the right of the symbol 1. The idea here is you are stepping out of a purely typographical system into an isomorphic system that is mathematical or part of number theory. "It is as if somebody had no musical scores all his life, but purely visually--and then, all of a sudden, someone introduced him to the mapping between sounds and musical scores. What a rich, new world! ... The discovery of Gödel-numbering has been likened to the discovery, by Descartes, of the isomorphism between curves in a plane and equations in two variables: incredibly simple, once you see it--and opening onto a vast new world."
264ff "Typographical rules for manipulating numerals are actually arithmetical rules for operating on numbers. This simple observation is at the heart of Gödel's method, and it will have an absolutely shattering effect. It tells us that once we have a Gödel-numbering for any formal system, we can straightaway form a set of arithmetical rules which complete the Gödel isomorphism. The upshot is that we can transfer the study of any formal system--in fact the study of all formal systems--into number theory."
265ff Comments here on producible numbers in any given system, they are defined by a recursive method; given numbers we know are producible, we have rules telling us how to make more producible numbers, and the class of numbers is thus constantly extending itself like a list of Fibonacci numbers; thus the set of producible numbers of any system is a recursively innumerable set. Then the author asks what about the set of non-producible numbers: is that set also always recursively innumerable? Thus the idea here is when you transpose a formal system into number theory you can ask both 1) Can we characterize producible numbers in a simple way, and 2) Can we characterize nonproducible numbers in a recursively innumerable way?
266ff Effectively we are coding a number that is a theorem in one system, and looking at it after translating it into another system. The author says here: we could object and just say that this "coded message," unlike an uncoded message, doesn't express anything on its own--it requires knowledge of the code. "But in reality there is no such thing as an uncoded message, there are only messages written in more familiar codes, and messages written in less familiar codes." The new Gödel isomorphism is a new information-revealer like the decipherment of an ancient script. [Think of Gödel-numbering as decoding something we didn't think was coded in the first place.]
267ff Next, the author Gödel-numbers the TNT system, like he did with the MIU system.
270ff Now we see that the TNT system contains strings which talk about other strings of TNT; the architecture of any formal system can be mirrored inside number theory just like "the vibrations induced in a record player when it plays a record... And it is in the nature of any formalization of number theory that its metalanguage is embedded within it." "A string of TNT has an interpretation in N [number theory]; and a statement of N may have a second meaning as a statement about TNT." [e.g., the system "wraps around itself" and can talk about itself.] "We are now at the stage where the Tortoise was when he realized that a record could be made which would make the phonograph playing it break--but now the question is: 'Given a record player, how do you actually figure out what to put on the record?' That is a tricky matter. We want to find a string of TNT--which we'll call 'G'--which is about itself, in the sense that one of its passive meanings is a sentence about G. In particularly the passive meaning will turn out to be 'G is not a theorem of TNT.'"
271ff "The ingenious method of creating G, and some important concepts relating to TNT, will be developed in chapters 13 and 14; for now it is just interesting to glance ahead, a bit superficially, at the consequences of finding a self-referential piece of TNT." The idea here is G expresses a truth: we are asking if G is or is not a theorem: if the systems claims it is a theorem it would have to be a falsity, therefore G is not a theorem, but still G expresses a truth. Therefore we have found situation in TNT of a string which expresses a true statement, and yet that string is not a theorem; but yet when we ask TNT whether the statement is true TNT says neither yes nor no. Further, we ask what about ~G [not G or negative G] which says the opposite of what G says, and in this case it says G is a theorem. Therefore we have the paradox:
G: "I am a theorem of TNT"
~G: "My negation is a theorem of TNT"
Thus the string speaks about its own opposite.
Part II
Prelude...
275 Achilles and the Tortoise are at the Crab's house and they meet the Anteater. Achilles and the Tortoise give the Crab a gift: two records never heard before except when Bach played them; the discussion then turns to Fermat's Theorem [note also the image here of Mobius Strip II by M.C. Escher], and then a discussion of trying to find either proof of Fermat's claim or a counterexample [note that supposedly Fermat's theorem was proven in 1995, a few years before the 20th anniversary edition came out].
M.C. Escher, Mobius Strip II (Red Ants), 1963
278ff A discussion of the field of "acoustico-retrieval," reconstructing a sound by vibrations; the Tortoise has been trying to find a way to reconstruct the sounds of Bach playing his harpsichord via calculations of the motions of all molecules in the atmosphere in the 200 years since. A discussion then follows on whether those "motions" are gone forever, vs the idea that something either exists or it doesn't exist as a solution--and you can therefore work from both directions and prove one of the other. It turns out that the Tortoise has managed to capture Bach's works on two recordings, so they decide to play them. The music is beautiful. [I think there's even a reference here to Glenn Gould's famous recording of The Well-Tempered Clavier, where Gould sings (actually mumbles) to himself while playing. Neat Easter Egg!]
280ff The Crab then pulls out a unique score of The Well-Tempered Clavier (in other words an encoded or Gödel-numbered version of the music); they then have a discussion about the thrill of first experiencing Bach and how that thrill is gone once you know the music well; likewise there is a meta-discussion here about reliving that thrill if it can be encoded and re-experienced in some way in the structure of one's brain. Then a reference to M.C. Escher's Cube with Magic Ribbons, where the bumps/dents on the ribbon's surface look either like dents or bumps, but not at the same time: in other words here there are two mutually exclusive modes that can be perceived. The characters discuss how this is analogous to listening to a Fugue and following one of the voices or listen to the total effect altogether without disentangling them: one mode shuts out the other. Another more general way to think about is by considering a thing as a whole vs a thing by its component voices; this is a type of dichotomy that "applies to many kinds of structures built up from lower levels." They contemplate this dichotomy of a whole vs a collection of parts, and then the group looks at an image, staring at it in different ways in puzzlement. Here the author writes "ATTACCA" ["begin immediately" in music terms] at the end of the interlude, and we'll see this thread picked up again after Chapter 10.
Chapter 10: Levels of Description, and Computer Systems
285ff On things like a fugue or Gödel's string G that can have the property of being understood on different levels; how this confuses us sometimes but other times we handle it with no problem. See for example how we know that humans are built of cells but also in terms of molecules, etc. On getting computers and other artificial intelligence systems to accept information from one level of description but yet produce output on another: see for example a chess program run by an artificial intelligence program: these were initially were built to "look out" more moves into the future; but then the truly great human chess player has certain "chunking" abilities: he can look at patterns on the board as a totality; thus as a result chess programs were designed with this capacity too, and thus began to be more effective at the game. On the concept of "implicit pruning" versus "explicit pruning": knowing certain moves are not part of what you would do versus systematically considering a move and then discarding that move. The latter is more brute force, which is how chess computers worked [at least in the day when this author wrote this book]. Implicit pruning means "sitting on a different level and thinking about the game from a different level."
287ff On computers and how they too run on various levels: the binary level, which represents functions and data, also machine language versus assembly language programs, and then compilers, even compilers that can be used to compile extensions of themselves, thus using a sort of bootstrapping technique.
294ff On computer bugs that may happen on a level different from where they're seen; see also biological bugs that can show up in the phenotype but can be used to trace genetic difference in the genotype. It's the same problem and it shows up at different levels.
296ff On a computer having various "levels" on which it functions; e.g., the operating system, machine language, even down to the semiconductor design; the analogy the author gives is like an airline where the customer doesn't really care how full the fuel tanks are or how many chicken dinners are to be served, but if his bags don't arrive then he has to descend into multiple layers to figure out what went wrong. The system protects the user from knowledge of its various levels... unless there's a problem!
297ff On rigidity of computers, how they tend to be used to do precise tasks, but then a discussion of programming languages that allow some forms of imprecision.
299ff On conceptualizing AI as a series of levels of hardware and software, where descriptions of a process will sound different at different levels; the author thinks that there will be dozens or possibly several dozens of layers between machine language level and true intelligence.
300-1 Cute story here where some non-computer friends of the author were working with a simple computer conversation program when that computer's operating system overwrote a message on the screen. These friends then asked the computer "Why are you overtyping what's on the screen?" as if the computer program they were using could "know" a totally different function operating on a different level. To them it was all one level: "the computer." As an analogy the author likens it to asking a person "Why are you making so few red blood cells today?" since people do not know the "operating system" level of their bodies.
301-2 Discussion of software and hardware: the author uses the framework software equals anything you can send over the telephone lines and hardware as anything else (a piano is hardware, printed music is software; a telephone is hardware but a telephone number is software, etc.), but the distinction breaks down with a human brain where there are certain limitations: like we can't make our neurons fire faster or slower, and we can't redesign the interior of a neuron, but we can control how we think; thus we basically live with our limitations and don't really even think about them that much. The author says, "To suggest ways of reconciling the software of mind with the hardware of brain is a main goal of this book."
302ff Other examples where "levels" are unclear or not fully decomposable: like the weather, or a team of athletes (distinct yet different due to the individuals' composability as a team) or even the nucleus of an atom.
305ff On how levels can be "sealed off" from each other: an example here is a how biologist doesn't care about nuclear theory [he "chunks" at different levels]. Also, we can know a person without having to know biology or particle physics; to some extent these levels can still have leakage from level to level but in general they can be sealed off from each other, and we can still function and perform highly effective and explanatory "chunking" operations on the level (or levels) that we are at.
307ff Then comments on two types of systems, one where lower levels cancel out like atomic particles or gases, which are non-deterministic as the lowest level but on a higher level (like a wall or a container of gas) will be "chunked" and deterministic. But then on the other type of system where lower-level effects are magnified into enormous high-level consequences (like a pinball machine). The computer is a combination of both types of systems.
308-9 On epiphenomena: like a system slowing down when too many users are using it, or where is the "9.3" stored that enables a sprinter to run 100m in 9.3 seconds? Another good example: gullibility (is there a place in the brain where gullibility can be found and removed?): then finally on mind vs brain: the mind is the brain's top level, can it be understood without understanding the lower levels on which it both depends and does not depend? To what extent is it "sealed off"? Or can it be "skimmed off" and transplanted onto other systems? Can the brain be unraveled into modular subsystems?
...Ant Fugue
311ff This dialogue is (quite creatively) structured like a fugue, but the various "levels" of the fugue have trouble communicating with each other, they don't understand each other's expressions. And then the dialogue participants have a meta-debate about the difference between reductionism and holism, what they mean, whether they exist, etc., and then the idea of transcending this question arises, which is incomprehensible to some of the dialog/fugue participants, because they are stuck on their "level." The Anteater describes the intelligence of an ant colony, but he describes it in a way that doesn't really make sense to the other participants, certain terms are warped or misunderstood.
314 Tortoise: "Probably the rest of you were too engrossed in the discussion to notice the lovely stretto which just occurred in this Bach fugue."
315ff Note the discussions of the nature of an ant colony, how they communicate in "writing"--what the Anteater means here is communication via pheromones; on how "Aunt Hillary" is a collective of ants; how they're like neurons as part of a human brain; how their apparently random activity at the colony level amalgamate into overall trend-like behavior which emerge from apparent chaos.
317ff Going still deeper here, where there are intermediate levels of organized activity in an ant colony: think of different types of ants, castes, etc., also on how the Anteater is a close friend to the collective but a grave enemy of the individual ants. Also on how the ants perform in teams based on signals that reach a certain critical mass to direct activity, like gathering food. And then a debate on whether this is purposeful behavior or not--or both: what looks like completely non-purposeful behavior at the ant level appears to be purposeful at a colony level. And the anteater takes the discussion up to the evolutionary level, where looking at a past perspective drains the whole colony of meaning and purpose, it become just a response to the stimulus of evolution over billions of years.
322 Here the group has a self-referential (and fugue-like) conversation, as the characters refer to signals passing through each other as they pass messages over and through each other during their own conversation in a verbal version of a rapid-fire musical stretto.
324ff More discussion here of the various components of the larger system; active subsystems like a group of ants that may strengthen a signal by their group being above a certain critical size. Also analogizing this to the difference between words and letters: words are meaning-carrying entities composed of letters, but letters themselves carry no meaning. One layer is passive, the other layer is active. And then thinking about it at different levels: for example looking at a colony system using an ant-by-ant description would require astounding amounts of information and would explain very little, while looking at it at a higher level like groups of ants or various castes of ants explains much more and gives you a more intuitive picture of the colony--but yet this perspective leaves out the individual ants, which you would think would be the most important feature. The irony is the ants are not the most important feature! "I'm sure no ant would embrace your theory with eagerness." "Well, I never met an ant with a high-level point of view." The idea here is describing a structure but "omitting any mention of its fundamental building blocks." And then the Anteater gives another analogy of reading The Pickwick Papers letter by letter, which would be an exercise in meaninglessness because the natural mapping occurs on the word level between words and the real world; therefore if you want to explain or describe the book you would never make any mention of the letter level.
327ff On symbol manipulation and who does it, who is the agent? It's reasonable to speak of the full system as the agent because the symbols can only act within certain parameters dictated by the system, and as the symbols operate the state of the system becomes transformed or updated. "It is this partially constant, partially varying system which is the agent. One can give a name to the full system." [Which in this case the Anteater has called "Ant Hillary."]
328ff Further thoughts on the essence of consciousness as being aware of your own thoughts, reading your brain directly at the symbol level, while not having awareness of the lower levels such as the signal levels. "It is like being able to read a Dickens novel by direct visual perception, without ever having learned the letters of the alphabet." At this point the group realizes the message they've been looking at together has another "level": the letters are composed of dozens of little "MU" symbols, such that the highest level and the lowest level are the same.
329 Here again the Tortoise makes a self-reference to where the conversation "is" by referring to what structural element of the fugue is coming up. And then Achilles repeats himself in a textual replication of a fugue structure. Cute! Note also that Achilles actually performs the organ point "Gee!" without realizing it. "How can I not have noticed it, if it was so blatant?" "Perhaps you were so wrapped up in what you were saying that you were completely unaware of it. Ah, what a pity--a supreme irony in fact--that you missed it." [Again this maps self-referentially to the group's discussion of consciousness and meta-consciousness.]
332ff a discussion here of how the ant colony previously operated under a different identity, "J.S. Fermant," before the colony was destroyed in a freak thunderstorm; this reoriented the colony so that it lost its identity; but then later the ant colony regrouped, but under a identity, Aunt Hillary, which emerged spontaneously. Even though the ants are the same, the collective identity of the colony is different at the higher order. Aunt Hillary was just a new "sum" of the old parts. Finally, see the intriguing picture of an "ant" bridge on page 334.
Chapter 11: Brains and Thoughts
337 "In the coming two chapters, then, we will try to unite some insights gleaned from attempts at computer intelligence with some of the facts learned from ingenious experiments on living animal brains, as well as with results from research on human thought processes done by cognitive psychologists."
337ff "We saw, in the pq-system and then in other more complicated systems, how meaning, in a limited sense of the term, arose as a result of an isomorphism which maps typographical symbols on to numbers, operations, and relations; and strings of typographical symbols onto statements. Now in the brain we don't have typographical symbols, but we have something even better: active elements which can store information and transmit it and receive it from other active elements. Thus we have active symbols, rather than passive typographical symbols. In the brain, the rules are mixed right in with the symbols themselves, whereas on paper, the symbols are static entities, and the rules are in our heads."
338 On thinking being "intensional" (meaning: it is defined by its properties, not anchored to specific known objects, thus descriptions can "float" without being anchored down to specific objects) and not "extensional" (defined by the set of things it refers to; not defining it by its properties). "The intensionality of thought is connected to its flexibility; it gives us the ability to imagine hypothetical worlds, to amalgamate different descriptions or chop one description into separate pieces, and so on."
339ff Discussion of neurons (the brain's "ants") and the yes/no aspect of whether they fire or not based on a certain threshold of inputs; on larger structures in the brain that can handle concepts rather than the very limited information of a neuron firing or not firing.
341ff "If thinking does take place in the brain, then how are two brains different from each other? How is my brain different from yours?... How far does this identity of brains extend?" The author gives an example of the earthworm where a specific neuron corresponds exactly to the same neuron in another earthworm of the same species, so their brains are isomorphic to the point where one could say there was only "one earthworm." Note however this mappability does not exist with humans. Note also Karl Lashley's rat experiments which showed that the memory of how to traverse a maze seem to be equipotential, located across all regions of the cortex, thus there was no "place" where it resided. But then this contrasts dramatically with Wilder Penfield's examinations of humans that showed a very small section of the brain that could be triggered with electrodes that would unleash entire memories, or entire specific images or sensations, which implies local areas are responsible for specific memories after all.
346ff On the puzzling lack of direct correspondence between large-scale hardware and high level software in the brain; see the visual cortex, a large-scale piece of hardware, but yet there's nothing approaching the "recognition of objects" function localized in that cortex; on a sensation of something crystallizing in your mind when you recognize something at that moment, but it doesn't take place when light rays hit your retina, but sometime later, after your intelligence has had a chance to act on the retinal signals.
348ff On shifting our thinking and descriptions of the brain state from a signal level to a symbol level; on symbols being either dormant or awake/activated by some threshold of neurons firing; a high level description of when a symbol is awakened would be it sends out messages to trigger or awaken other symbols. Note also that referring to these symbols as "streams of nerve impulses by neurons" is a low level way of looking at things, we want to look at it from the higher level in the same way we think about a clock's behavior being "sealed off" from the laws of quantum mechanics; the idea here is that "symbols symbolize things but neurons don't." See also E.O. Wilson's point in his book The Insect Societies about messages propagating around an ant colony, where he describes it as a transfer of information among groups that a single individual could not pass to another.
350ff Discussion here of the "size" of a symbol: categories, individuals, classes of things or instances of things. Note that most symbols play multiple roles depending on context; a discussion of the prototype principle: a very specific event can serve as a general example of things, because they're vivid or they imprint themselves strongly on the memory. Also there is a generality in the specific, which is a type of paradox. Further on the "splitting off" of an instance from its default class: think about a specific football player that you know who you mentally "split off" from his football team, thus this player is somehow separate from the default class "football player" in the sense that you know things about him as distinguished from other football players who you don't know. The author uses a fictional player "Palindromi" as an example as he figures in specific plays, as you start to use his name, etc., thus he goes from being an instance of the default/parent class "football player" to being an instance of a separate class, while the parent class remains dormant. This is a process of growth and eventual detachment of an instance from a class, and there is not a clear boundary here of when the one instance becomes separate.
358ff On how low-level traffic of neuron firings gives rise to high-level traffic of symbol activations; further, if we can explain high-level traffic of symbol activation in its own terms with a theory that does not talk about low-level neural events, if these two problems can be solved, then possibly intelligence can be lifted out of the hardware in which it resides; and this is a big part of what would drive true artificial intelligence, it would mean that it was a software property. Note however if there's no way to do this without having the underlying hardware of neurons, thus this means intelligence is a brain-bound phenomenon and not "liftable." But then the author refers again to the example of an ant colony where no ant has the ability to carry information about nest structure, so then the question is how does the nest get created and where does that information reside? "Could there be an Artificial Ant Colony?" [Meaning could the ants' collective intelligence likewise be "lifted"?]
360ff Discussion of the sphex wasps' reproductive process of paralyzing a cricket and leaving it with its eggs for its young to feed on; this implies intelligence on one level until you examine the details, the behavior is fully hardwired. The other uses this as an example of where humans have the ability to see several instances as instances of an as yet unformed class and then to make the class symbol, where is the insect doesn't have any of these abilities to change levels or manipulate symbols. It's a type of hypothetical thinking in reclassifying "something" as "something else." [Note in particular here the experiment on page 360 where the researcher interrupts the sphex wasp's routine, and the wasp mechanically repeats its pattern, over and over and over again, without any ability to iterate. Thus "sphex wasp" is an incredibly useful mental shorthand for me to describe pre-programmed behavior without awareness or meta-intelligence applied to that behavior. We can often see this in human behavior, where a person is clearly following a script--a behavioral pattern or a mental pattern--without employ any capacity for iteration or any ability to jump out of that system and conceptualize their behavior from a different level. We shouldn't be surprised to see a sphex wasp operate at this "level" but we should expect ourselves not to operate on the sphex wasp level!]
362ff On how our "mind's eye" allows us to represent events, real or imagined, in ways that follow the laws of physics, using in the form of chunked intuitive laws that we have in our minds in order to survive. But at the same time we can also voluntarily violate physical law in our mental sequences of events if we want to. For example, imagine two cars approaching each other and then passing right through each other: this is something we can easily do in our minds. But how we do it is a whole other question.
363ff On "procedural knowledge" versus "declarative knowledge": declarative is explicitly stored knowledge whereas procedural knowledge is a program: it is knowledge not encoded as facts, but rather as programs. On purely procedural knowledge as an epiphenomenon. Then comments on shades of gray between declarative and procedural knowledge: what about a melody: how is it stored? Is it stored note by note? What about that melody in different keys? This indicates that tone relationships are stored, also aspects of a melody are activated by symbols triggering others. Also, consider the question "What is the population of Chicago?" It has sort of a paradoxical mix of procedural and declarative knowledge as you try to dredge the answer out of your "mental almanac."
English French German Suite
366ff [This section of the book was another example of something that had no impact on me at all when I first read it--I missed the point. Now that I'm rereading this book and also now that I've also read Hofstadter's book on language translation Le Ton Beau de Marot, I get it: it's an attempted to show how to preserve "information" as you translate playful nonsense language from one language to another. What liberties can you take and not take? And what exactly are you trying to preserve? The literal language? The rhyme or meter? The style and "feel" of the text? The translation task here has a lot more going on than you might initially think.]
Chapter 12: Minds and Thoughts
369ff On isomorphisms between brains on the symbolic level; examples might be similarities in the mode of thinking, on the repertoire of symbols uses, or what it is that triggers patterns of symbols in different people's minds. Comments of what could be a person's semantic network of nodes and connections between symbols: see the mind map on page 370; on "conceptual nearness"; on where we would see isomorphisms in a conceptual network; considering an analogy for this in wordplay in different languages--see for example a poem which activates other verbal symbols and is not mapped word by word literally in translation; finally on the translated poems in the prior interlude: there are still rough isomorphisms here, partly global and partly local.
375ff The thought experiment of the ASU, your own "in your head" version of the USA that maps isomorphically to other people's maps based on certain geological aspects or major cities that we all would think of or have in common, even if all the minor details of each person's map are totally different. We can still guide each other on our "relatively isomorphic" semantic maps. The idea here is that humans are different but they are also the same in certain deep and important ways, and thus it would be useful to pinpoint what those ways might be.
377ff Here the author tells the reader that his "ASU mapping" idea is actually an analogy for thought itself: see how a thought maps to a trip through a semantic network of symbols, just as we make our mental ASU maps between towns or cities.
379ff On types of problems and issues that show up with language translation [the author will go into this subject in exhaustive detail in his book Le Ton Beau de Marot... exhaustive]. see for example in Dostoevsky's novel Crime and Punishment, and the idea of translating a street name in the first sentence of the novel using various ways to convey aspects of the street itself, with different levels of fidelity: literal fidelity or "flavor-based" fidelity. The author gives all kinds of examples on how translators "capture" the sense of the street in English. And this is just the novel's first sentence! See also for example how to "translate" broken Russian spoken with a German accent, but into English. Then on "translating" this idea into translations between two computer languages. The author says we have to step away from a lower-level view and toward a higher, more chunked view to look at chunks of a program that may fit together and allows you to perceive the goals of the programmer, and then translate that into the other computer language. Then the author arrives at the underlying question: is there a well-defined, high-level description of a brain? "Surely, to answer this question would be of the highest importance if we seek to know whether we can ever understand ourselves."
384ff "These speculations about brain and mind are all well and good, but what about the feelings involved in consciousness? These symbols may trigger each other all they want, but unless someone perceives the whole thing, there's no consciousness. This makes sense to our intuition on some level, but it does not make much sense logically. For we would then be compelled to look for an explanation of the mechanism which does the perceiving of all the active symbols, if it is not covered by what we have described so far." [This is the "homonculus problem."] A "soul" as an explanation vs a "non-soulist" explanation: the non-soulist explanation would say that that consciousness is that property of a system that arises whenever there exists symbols in the system which obey triggering patterns like the ones described in the past several sections. But that doesn't explain what the sense of "I", the "self" is. On thinking of the self as a subsystem, a constellation of symbols like almost a sub-brain.
387ff Some additional shared vocabulary and shared concepts here between brains and computers: see for example shared code, or say a shared conception of a mountain shared by two close friends. Note that activation of one symbol can have different results on different subsystems, because in the computer they're processed by different interpreters.
388 On the self as a sort of "monitoring system" of brain activity; a subsystem of the brain itself; note that a computer with this kind of structure would make statements about itself which would resemble statements people make about themselves.
388ff A long quote here from the philosopher J.R. Lucas in a 1961 article "Minds, Machines, and Gödel": the idea here of us being tangled up in questions about "when you know something"; do you know that you know it, and then seeing this problem as sort of implicit, so we drop it; also on the idea that self-awareness of a conscious being isn't really divisible: we don't have selves, superselves, and super-super selves, thus a conscious being can deal with a Gödelian question in a way a machine cannot "because a conscious being can both consider itself and its performance and yet not be other than that which did the performance. A machine can be made in a manner of speaking to 'consider' its performance, but it cannot take this 'into account' without there by becoming a different machine, namely the old machine with a 'new part' added." Further on Turing's idea that--like with nuclear fission--there's a certain level below which nothing happens, but above which everything happens: brains and machines can be subcritical, and all machines are like this at present. But at a certain level of sophistication something might happen that would involve a type of consciousness, above some certain level of complexity. At this level of complexity it would not be the sum of its parts in the sense of how we think about a machine. "In fact we should say briefly that any system which was not floored by the Gödel question was eo ipso ["by that very quality"] not a Turing machine, i.e. not a machine within the meaning of the act." The author comments here on how he is boggled by this passage, as well as by Lucas's full article, and he will come back to many of the topics "in this odd passage."
Aria with Diverse Variations
391ff Achilles is unable to sleep and the Tortoise keeps him company. The Tortoise tells him a story about a count in Saxony who was suffering from sleeplessness who commissioned a musician, Goldberg, to compose a set of variations to be played by the count's court harpsichordist to help pass the time during his sleepless nights [now we have nested recursion in this story already]. And then a discovery of 14 new Goldberg Variations, making 44 rather than the original 30. They decide to name the number of Goldberg Variations "G" and decide that it is certain to be finite, although they concede that this is not the same as knowing how Big G is. Then they "push" into yet another story, here talking about even numbers being a sum of two odd primes, noting that coincidence that the year this was discovered, 1742, was the same year the Goldberg Variations were written. And this number itself is sum of two odd primes, 1729 and 13, as well as two other odd primes, 1747 and 5.
394 Then Achilles and the Tortoise begin discussing the "Goldbach Conjecture" ["every even number is a sum of two odd primes"] which has not been proven; then a discussion of Vinogradov numbers: "every sufficiently large odd number can be represented as a sum of no more than three odd primes."
395 [Here we have a self-referential discussion of recursion and bounded vs unbounded loops, a topic first touched on in Chapter 5 and discussed in depth in the next chapter, Chapter 13.] Achilles and the Tortoise then consider both 1) an even number as the sum of two odd primes or 2) an even number that is the difference of two odd crimes, and on choosing which one to work with they see that one of them is a search that is guaranteed to terminate: if you look at say numbers up to 1 trillion the "sum of two primes" characteristic is guaranteed to terminate. But the "difference of two primes" characteristic does not have any bound; one of the searches is detectable by brute force, the other one is not. And then a discussion of the question whether there might be some other way to prove whether or not the search might terminate: can you think about whether there might exist such a search method? We can search for such a search, but we can't guarantee that a "meta-search" like this would terminate either! The next step in the argument is to think about the idea "29 is a prime" and realize that it includes an infinity of subfacts, like "2 * 2 is not 29" and "4,000 * 3,000 is not 29," etc. "You don't see an explicit infinity because it is captured implicitly inside the images you manipulate."
400ff Then a discussion of a "predictably terminating test" and a discussion of the so-called "wondrous numbers" to see if you eventually arrive at 1 in a roundabout way. [This part of the interlude seems to reference the MIU problem at the beginning of the book.]
402 Note a funny Easter egg here of a book called Copper, Silver, Gold: An Indestructible Metallic Alloy, by an unnamed author who is actually the author himself: he cites himself in the Index and the Bibliography as "Egbert Gebstatter," with a pretty funny meta-reference describing this book as "remarkably similar to the present work... Of particular interest is a reference in its well-annotated bibliography to an isomorphic, but imaginary, book."
402 Another analogy here regarding a novel, where you can just tell by the "physicality" of a book whether you're approaching the ending and the climax; but an author may add an element of surprise by adding padding to the end of the story so as to not make it so obvious that the climax is coming. But if you make that padding too obviously extraneous, or too obviously part of some other story, you'll give it away that way too, thus the author may want to make that "padding" so it looks just like the story, so that the reader can't tell that it's a different story at all. Thus it will be a sort of post-ending ending, but you won't be able to tell when the real ending comes, it'll just blend right in. [Achilles and the Tortoise have this back and forth in what is again an isomorphism for the story they are in right now, because the story is in its own post-ending ending. Likewise this conversation maps as an analogy to earlier discussions they were having on primes and Vinogradov numbers.] Finally Achilles suggests solving this padding problem by putting in some clue that a "sufficiently assiduous reader" would find that would give away when the ending came, some sort of telltale feature--like introducing an extraneous character--that signals the end.
404 The story now introduces the "very gold Asian box" with mathematicians' names engraved on it in a diagonal where you subtract 1 from this diagonal: this refers to Georg Cantor, a mathematician who dealt with uncountable sets using a proof based on diagonals [this idea is way over my pay grade.]
Chapter 13: BlooP and FlooP and GlooP
406ff [This chapter is also largely over my head, but even if you're not comfortable with computer language you can still mostly follow along, and the author helps the reader significantly. Keep going!] The author introduces computer languages invented specifically for this chapter to explain aspects of the word "recursive": primary recursivity and general recursivity, "clarifying the machinery of self-reference in TNT. "...we are going to scrutinize 'self-awareness' in more formal settings." On a system needing to be sufficiently complex or sufficiently powerful to be tested for completeness.
408ff "The discovery of Gödel-numbering showed that any search for a string having a special typographical property has an arithmetical cousin: an isomorphic search for an integer with a corresponding special arithmetical property." Note also the long quote here from a dialogue from the book Are Quanta Real? by J.M. Jauch, discussing the idea that if we try to understand nature we should look at phenomena as if they were messages to be understood, knowing that each message appears to be random until we establish a code to read it. Also on codes as a claim to reality.
409ff Next: defining the precise meaning of the term "predictably long search" using one of the author's computer languages that he calls "BlooP" ["bounded loop"]; on the idea of a programmer including an upper bound or ceiling of a looping program, analogous to using 1 trillion in prior dialogue. Discussion of quit and abort commands, to either quit a block or abort a loop; and then on the idea of defining procedure but then calling that procedure inside later procedure definitions, thus defining "calling a procedure" as a primordial step as well; also the language allows automatic chunking and can output both integers and yes/no functions.
414ff The author gives an example of this program running looping "N" times in a procedure to test numbers for the Goldbach property, outputting a YES or NO.
417 On the idea of representability and expressibility: "expressing a predicate" is just translating something into a strict formalism from English, "representability" is a stronger notion that means all true instances of the predicate are theorems and all false instances are non-theorems.
418ff On thinking about whether every property of numbers can be detected by some suitable BlooP program; also asking can upper bounds always be given for the length of calculations, or are we unable to know the calculation length to be predictable in advance? The author is about to show us that the latter is the case, using the diagonal method of Georg Cantor, the founder of set theory. The author starts with imagining the pool of all possible BlooP programs: "Pool B," an infinite set. Next, consider a sub-pool of Pool B of programs that calculate functions of exactly one input parameter, and give that subpool a name, "Blue Programs," and assign it an index number, listed in order of length of the program. The idea is that this list is extremely well defined, at least in the abstract. And: "each Blue BlooP program has a unique and definite index number. This is the crucial idea." Then he applies the "twist" using Cantor's diagonal method, and he takes the catalog of blue programs and uses it to define a new function Bluediag[N], which feeds each looping procedure with its own index number and then adds 1 to the output; the author takes care to note here: "The peculiar thing about Bluediag[N] is that it is not represented in the catalog of Blue Programs. It cannot be." The reason here is because all Blue Programs have an index number; the actual output you get is that Bluediag[X] is equal to a number and also equal to the successor of that number [these statements are like the self-contradicting example from Chapter 9: G is a theorem of TNT/NotG is a theorem of TNT]. So first of all we've shown that not every number theoretical function must be calculable within a predictable number of steps.
421ff Here a brief discussion on Cantor's original diagonal argument: Cantor wanted to show that some item is not in a certain list, and also show that a "complete directory" of real numbers is a contradiction in terms. This refers to finite directories but also infinite directories. "The essence of Cantor's result is that there are (at least) two distinct types of infinity: one kind of infinity describes how many entries there can be in an infinite directory or table, and another describes how many real numbers there are (i.e., how many points there are on a line, or line segment)--and this latter is 'bigger', in the sense that the real numbers cannot be squeezed into a table whose length is described by the former kind of infinity." What this process does is it takes a list of numbers and changes each one of them to create a new number d, which cannot be on the list by definition. What does this argument prove? "It may become clearer if we apply it to the alleged 'List of All Great Mathematicians' in the dialogue--a more concrete example. The diagonal itself is 'Dboups'. If we perform the desired diagonal-subtraction, we will get 'Cantor'. Now two conclusions are possible. If you have an unshakable belief that the list is complete, then you must conclude that Cantor is not a Great Mathematician, for his name differs from all those on the list. On the other hand if you have an unshakable belief that Cantor is a Great Mathematician, then you must conclude that the List of All Great Mathematicians is incomplete, for Cantor's name is not on the list! (Woe to those who have unshakable beliefs on both sides!)" Thus the idea here is using an integer in two different ways or looking at an integer on two different levels, and thanks to this we can construct an item which is outside of some predetermined list (even if the list is infinite). Think of it like a vertical index or a horizontal index.
424 "The repeatability of Cantor's diagonal method is similar to the repeatability of the Tortoise's diabolic method for breaking the Crab's phonographs, one by one, as they get more and more 'hi-fi' and--at least so the Crab hoped--more 'Perfect'. This method involves constructing, for each phonograph, a particular song which that phonograph cannot reproduce. It is not a coincidence that Cantor's trick and the Tortoise's trick share this curious repeatability... Moreover, as the Tortoise subtly hinted to the innocent Achilles, the events in the Contracrostipunctus are a paraphrase of the construction which Gödel used in proving his Incompleteness Theorem; it follows that the Gödel construction is also very much like a diagonal construction. This will become quite apparent in the next two chapters."
424 Can we improve the BlooP program to make the diagonal function representable in it? Note that BlooP had bounded loops, but what if we had a second language with Free loops, FlooP? This program would answer either YES, NO or the program never halts and thus gives no answer (because it's an endlessly rising progression), and yet the answer always has the value NO, but it is always inaccessible [the author likens it to Joshu's nonanswer "MU"]. Thus we can separate FlooP procedures into two classes: terminators and non-terminators; and then can we build a test for termination or not? note that Alan Turing figured out a trick for feeding a program its own Gödel number to avoid infinite regress (and this will be addressed in the next chapter) but here we're going to take a different route to the same goal to prove that a termination tester is impossible; the author then directs us to a paper by Hoare and Allison in the Bibliography, which proves the theorem "any language containing conditionals and recursive function definitions which is powerful enough to program its own interpreter canopy used to program its own 'terminates' function."
426ff The whole idea is that if this were to work you could examine a series of numbers for some sort of property not by examining the numbers themselves, but by coding them into a program and then testing that program for termination. The author goes through this by considering the full set of all FlooP programs, and diagonal numbering them; the author goes one step further and [hypothetically] assumed we have a termination tester, and lists the set of all terminating FlooP programs, calling them Red programs, and diagonal numbers them. We arrive at the same problem as in the BlooP program when we diagonal numbered them: thus this catalog of Red programs also is not calculable even in the more powerful language FlooP.
428ff The author takes one last step here by positing a more powerful language, "GlooP"--but it turns out that this hypothesized GlooP language is a myth, as Alan Turing and Alonzo Church in the 1930s (separately) worked out in what we now call the Church-Turing Thesis. So now we have problem with unshackling a program to calculate a function for any value of another function, but then also the idea that we can tell a terminating from a non-terminating program, which would allow us to solve all problems of number theory in a uniform way. Brief discussion here of the Church-Turing Thesis [with more discussion on it in Chapter 17]; the author here discusses three ways to state the thesis:
1) What is human-computable is machine-computable.
2) What is machine-computable is FlooP-computable.
3) What is human-computable is FlooP-computable (i.e., general or partial recursive).
429 Note here that "BlooP-computable" means "primitive recursive" while FlooP-computable functions can be divided into two realms: 1) those which are computable by terminating FlooP programs, or general recursive, or 2) those which are computable only by nonterminating FlooP programs, or partial recursive.
Air on G's String
431 Achilles tells the Tortoise about an obscene phone call he got where someone was incoherent over the line, the person shouted something, then shouted it again and then hung up: "Yields falsehood when preceded by its quotation! Yields falsehood when preceded by its quotation!" Next the two of them are walking "inside" the lithograph Above and Below by MC Escher, and Achilles encourages the Tortoise to walk on the underside of the stairs in the image; the concepts of "up" and "down" start to take on a paradoxical meaning.
434 They continue discussing the phrase "...preceded by its quotation" and ruminate on the idea of imagining something preceded by its quotation. Then a discussion of the use-mention distinction [e.g., "philosophers make a lot of money" versus the "word 'philosopher' has five letters"] when the word is put in quotes we subtract out its meaning and connotation, nothing about the word matters except its typographical aspects. This is sort of meta-reference and a recursion combined.
435ff The Tortoise decides he will give this phenomenon a name, "Quine," named after Willard van Orman Quine, and they explore what happens when you "quine" a phrase: "'Is a sentence fragment' is a sentence fragment." You take a sentence fragment, quine it, and you've made a sentence--a true sentence in this case. Next they stumble into a self-referencing example: "'WHEN QUINED, YIELDS A TORTOISES LOVE SONG' WHEN QUINED, YIELDS A TORTOISE'S LOVE SONG.'" This sentence is talking to itself/referring to itself. Finally, Achilles and the Tortoise go back and quine the phrase "yields falsehood when preceded by its quotation" which is what the caller said over the telephone; you can't quite make out if this quined statement is a truth or a falsehood.
Chapter 14: On Formally Undecidable Propositions of TNT and Related Systems
438 [Note that this chapter is also mathematically challenging: it is one of the most difficult chapters of the book for non-math people.] This chapter's title is an intuitive adaptation of Gödel's famous paper from 1931; the TNT language is being used as a placeholder for mathematics and Principia Mathematica. On the first key idea: that there are strings of TNT which can be interpreted as speaking about other strings of TNT--that it is capable of introspection or self-scrutiny and that this property comes from Gödel-numbering. The second key idea is that "the property of self-scrutiny can be concentrated into a single string." On the idea that Gödel's proof consists of a fusion of these two ideas; and on the idea that Gödel-numbering "is related to the whole notion of what meaning and reference are, in symbol-manipulating systems. This is an idea which goes far beyond the confines of mathematical logic..."
438ff On "proof-pairs": a pair of natural numbers related in a particular way; what we do in any formal system has its parallel in arithmetical manipulations. Then the author asserts that "the property of being a proof-pair is a primitive recursive number-theoretical property and can therefore be tested for by a BlooP [bounded loop] program." Furthermore this property is consequently "represented in TNT by some formula having two free variables." In a formal system, it is always possible to tell in a predictably terminating way whether a given sequence of lines forms a proof, or not.
445ff On "arithmoquining": here the author considers the idea of substituting a formula's own Gödel number into itself just like quining; this allows you to make a TNT sentence which is about itself. "...the ultimate trick necessary for achieving self-reference in Quine's way is to quine a sentence which itself talks about the concept of quining. It's not enough just to quine--you must quine a quine-mentioning sentence!" And then the parallel trick to arithmoquine a formula which itself is talking about the notion of arithmoquining.
447ff The author gradually arrives at the formula G, whose Gödel number is "the arithmoquinification of u" and then further that "The formula whose Gödel number is the arithmoquinification of u is not a theorem of TNT"... but yet that formula is G itself, thus G is not a theorem of TNT. We have gradually pulled a high-level Interpretation--a sentence of meta-TNT--out of what was originally a low-level interpretation--a sentence of number theory."
448ff "The main consequence of this amazing construction has already been delineated in Chapter 9: it is the incompleteness of TNT." Basically G asserts a truth, but the truth that G asserts is its own non-theoremhood, thus from its theoremhood would follow its non-theoremhood, a contradiction. However, what if we assert that G is a nontheorem? This is acceptable because it's not a contradiction. However G's non-theoremhood is what G asserts, therefore G asserts a truth, and since G is not a theorem there exists at least one truth which is not a theorem of TNT. "Let us pause for breath for a moment, and review what has been done." The author then sets out an analogy going over the Epimenides paradox [recall this is when a person from Crete says "all Cretans are liars"] lined up with the paradox of G being both a theorem and a nontheorem of TNT [see photo]:
449ff "We have found a 'hole' in our system--an undecidable proposition." "Gödel found a simple way to express the statement "TNT is consistent" in a TNT formula; and then he showed that this formula (and all others which express the same idea) are only theorems of TNT under one condition: that TNT is inconsistent."
452ff A discussion here of infinitely large integers which the author calls "supernatural numbers" that can make a TNT proof pair with the Gödel number for G; further discussion of supernatural numbers under addition and multiplication as well as fractions, and how disorienting this whole thing is; likening it to Euclidean and non-Euclidean geometry in the sense that the two types of geometries don't talk about the same concepts even though they share words like "point" and "line," but these meanings are determined by the axiomatic system within which they are used. Likewise for number theory.
457 Just like with different types of geometries, physicists use different conceptions of space [e.g., 3-D space, Hilbert space, etc.]; none is "true" in any specific way versus all the others; likewise number theory likewise doesn't say that 2 + 2 is not equal to 4, it is just a mode for dealing with concepts of the infinite. "You fit your mathematics to the world, and not the other way around. For instance, we don't apply number theory to cloud systems, because the very concept of whole numbers hardly fits. There can be one cloud and another cloud, and they will come together and instead of there being two clouds, there will still only be one. This doesn't prove that 1 plus 1 equals 1; it just proves that our number-theoretical concept of 'one' is not applicable in its full power to cloud-counting."
459ff Finally, a brief discussion of Hilbert's Tenth Problem which includes Diophantine equations that may or may not have an integer solution: using Gödel numbering and the Gödel G number you can prove in reverse that the equation has no solution. "This is what the Tortoise did in the Prelude, using Fermat's equation as his Diophantine equation.
Birthday Cantatatata...
461ff The Tortoise repeatedly asks Achilles whether it's his birthday--just to be certain. The Tortoise doesn't want to make a wild guess, he wants to make only educated guesses. Achilles, in frustration, then says, "The answer to all the previous questions, and to all the succeeding ones which you will ask along the same line, is just this: YES."
462ff Basically, the Tortoise calls that meta-answer "Infinity" and then rolls up all those answers, and then asks Achilles the question "infinity plus one," to which Achilles says "all of your infinity-plus questions also have the answer yes. But each time the Tortoise describes the infinity of answers as an "Answer Schema" and rolls it up, then asking a new question which involves Answer Schema #2, #3, etc. Achilles then answers with "an Answer Schema to end all Answer Schemas--the meta-Answer Schema, in which he has jumped out of the whole system--and now we are done. But then the Tortoise responds that there could be answer schema w squared + 1 or w squared + 2 or w + w, or w to the third or fourth power, or w to the power of w, to the power of w again, ad infinitum; no matter how many infinite answers you roll up you can call it an Answer Schema and add more to it.
Chapter 15: Jumping out of the System
465ff Did Gödel just take advantage of a hidden defect in a particular formal system [TNT] such thata there could be a formal system superior to TNT which did not have any exposure to this Gödelian trick? Or if the basic problem is that TNT has a hole ("I cannot be proven in formal system TNT") why can't we just plug up the hole by tacking G on to TNT as a sixth axiom? Thus the new system, TNT + G, would be a superior formal system and therefore complete! But the problem is that the new system will succumb as well: "I cannot be proven in formal system TNT + G." The author likens this to singing a tune again, but in a higher key, basically: "I cannot be proven in formal system X."
468ff Going back to the Contracrostipunctus story where the Tortoise kept talking about the Crab's "record player Omega": now we know that the problem is endless just like with this formal system problem. "As you probably suspected, even this fantastic advance over TNT suffers the same fate. And what makes it quite weird is that it is still for, in essence, the same reason." The Gödel number always captures the typographical string; "...the system's own properties are reflected inside it and can be used against it... Any system, no matter how complex or tricky it is, can be Gödel-numbered...and this is the petard by which it is hoist." "TNT is therefore said to suffer from essential incompleteness because the incompleteness here is part and parcel of TNT; it is an essential part of the nature of TNT and cannot be eradicated in any way, whether simple-minded or ingenious. What's more, this problem will haunt any formal version of number theory, whether it is an extension of TNT, a modification of TNT, or an alternative to TNT." "...the system's own richness brings about its own downfall. The downfall occurs essentially because the system is powerful enough to have self-referential sentences."
471ff Now on to the argument by J.R. Lucas that the Gödel argument can be used to show that there's some ineffable quality to human intelligence, because Lucas believes that machines cannot do "Gödelization" the way people can; the author believes this argument is wrong but "fascinatingly so" and it was one of the forces driving him to think through matters in this book, and then he's going to rebut it in this chapter and in chapter 17. [There's an insight here: this is why you can get a ton of mileage from reading things you fundamentally disagree with--it causes you to think, if you're willing to do so!] Lucas argues that the human mind can always go one better than any formal system and the mechanical model is by definition finite and cannot Gödelize itself, it cannot stand outside its own system.
473ff The author starts with using M.C. Escher's engraving Dragon from 1952, where the dragon pokes his head through holes in a two-dimensional paper to bite his own tail. But he's still in two dimensions, no matter what happens he can't be in three dimensions; the author uses this as a metaphor to refer to how in two dimensions you're missing some essence of three-dimensionality; we're talking about trying to encapsulate a "Gödelizing operator" inside the program itself. But us outside the system can still zap it in a way it can't do itself--so then are we arguing for or against Lucas? The author says against: he says that the fact that we cannot write a program to do Gödelizing should make us suspicious that we [humans] can definitely Gödelize ourselves in every case; the idea here is that also a human eventually will reach the limits of its own ability to Gödelize at some point; this is one refutation of J.R. Lucas.
476ff Another reputation basically talks about how humans likewise are no less vulnerable to Gödel-like paradoxes: see for example the Epimenides paradox-type statement "Lucas cannot consistently assert this sentence": it's true and Lucas also cannot consistently assert it.
477ff The author now asserts that "it is entirely conceivable that a partial ability to 'step outside of itself' could be embodied in a computer program." On the distinction between perceiving oneself and transcending oneself, you can look at yourself in a mirror or on tape or movies but you can't break out of your own skin; TNT can talk about itself, but it cannot jump out of itself; a computer program can modify itself but it cannot violate its own instructions, it can only change parts of itself by obeying its own instructions; this is like the paradox "Can God make a stone so heavy that even he cannot lift it?" See also the long quote here from Erving Goffman and his book Frame Analysis, where in order to create an effective genuineness in, say, a TV commercial, there is a constantly escalating battle between the viewer and the commercial of effects to affect this genuineness, and Goffman frames it as a sort of interaction pollution. [This sent me down a rabbit hole of learning about Erving Goffman's works.] Also comments here on Zen and its preoccupation with transcending a system/jumping out of a system via koans, paradoxes, etc.
Edifying Thoughts of a Tobacco Smoker
480ff Achilles visits the Crab at his house and they look at paintings by Magritte, and the Crab literally pulls a pipe out from Magritte's painting The Shadows and begins smoking it. They also read a poem by Bach about his pipe as a metaphor for his own life, and then the Crab plays a record--on a monstrously complex contraption--of himself singing one of his own songs. Achilles is fascinated by the contraption that plays the record. the Crab tells him this is my Tortoise-chomping record player that chomps up records produced by Mr. Tortoise. They talk about their unending competition and the fact that the Crab designed a record player that disassembled and reassembled itself but the Tortoise designed a record to destroy the disassembly/reassembly subunit of that record player--which could not take itself apart and rebuild itself [in other words it could not Gödelize]. Then the Crab begins talking about ribosomes and other biological structures that possess the baffling ability to spontaneously self-assemble without being directed by some other unit or subunit, and that he considered developing a record player with that property: a grand self-assembling record player. But it still didn't work!
487ff Then the Crab tries a different approach, where he has defenses around his record player that only let certain records through, or it can only play a few particular records; the Tortoise then will need to devise a record which can slip past the Crab's record player censors; the Crab's record player will chomp all records which do not bear the proper label as read by an attached camera; Achilles and the Crab start playing with the camera and try to point the camera at the TV screen that it broadcasts to, and it produces a whole long "corridor" a set of nested copies of the screen itself, the image becomes self-engulfing; also on the delay of an image rendering itself through all the nested screens over time because of the slight delay in the circuitry, a sort of visual echo.
493ff Then a discussion of a "total self-engulfing," which would be a picture of the TV camera and the screen using a mirror showing both the TV camera and the screen. Note that a screen showing multiple screens is only part of the system, you need the camera too. But the problem here is you can't totally self-engulf because the image also doesn't include the back of the mirror, the inside of the television, or the back of the television, etc. In other words a total self-engulfing is impossible.
494 Finally a reference to Magritte's painting The Air and the Song, which has an image of a pipe and the text "Ceci n'est pas une pipe"; the Crab and Achilles have a debate as to whether the word "ceci" refers to the pipe, the whole painting, or what, and to what extent it is likewise self-engulfing.
Chapter 16: Self-Ref and Self-Rep
495ff On looking at mechanisms of self-reference and self-reproduction; implicitly and explicitly self-referential sentences; a discussion of the Epimenides paradox "this sentence is false" and the cognitive processes required to understand that specific type of self-reference; the author likens it to an iceberg with most of the activity going on below the surface; also an interesting paradox with "The sentence 'The sentence contains five words' contains five words." has to fail because, the sentence quoted within a sentence can't be shorter than the quoted sentence, and then a version using an infinitely long sentence: "The sentence 'The sentence... is infinitely long' is infinitely long." can work, but this cannot work for a finite sentence. "For the same reason, Gödel's string G could not contain the explicit numeral for its Gödel number: it would not fit."
497ff On quined sentences; and then a discussion of how a computer program can use a quining technique to reproduce itself.
500ff What is a copy? Think about a jukebox where you press a number that plays a song that tells you to put another nickel into the jukebox. Or a crab program which copies itself backwards; also a quined self-reference sentence that is a part in English preceded by a part in quotes in French--in other words it translates itself or is sort of like a canon that transposes itself into another key. And then getting back to the jukebox example: what about a program that can print out its own Gödel number? This would be much closer example of a true copy of itself than the earlier jukebox example.
503 On reproduction by class rather than instance: see for example a parent creating a child in a sort of coarse-grained isomorphism, they belong to the same species.
503ff What is an original? The author gives three examples: a program that prints itself out; a program that prints itself out along with its interpreter; a program that instructs a machine to put something together, where the original is the program, but the self-rep is the compound system. The rest of this chapter will discuss various intertwined self-replication examples of program, intepreter and processor.
504ff On typogenetics: the author's word for a descriptive system of the genetic process for didactic purposes; describing the strings or strands of DNA that are sequences of letters, and also enzymes or the mobile machines that perform operations on strands of letters; once he's discussed the various rules of this system he likens it to formal system like the MIU system; an interesting nuance here, though, that the enzymes are translated from each strand, thus the strands dictate the operations which will be performed on them. and then those operations will in turn produce new strands which will dictate further enzymes etc, "This is mixing levels with a vengeance!"
513ff More on mixing of levels: DNA strands are acted upon and therefore play the role of data; on the other hand they also dictate the actions which are to be performed on the data, and therefore they play the role of programs; thus neither strands nor enzymes can be thought of as being on a higher level than the other, this contrasts from the MIU system where the rules of inference belong to a higher level than strings; likewise this is true for TNT and all formal systems. But note however that in TNT levels do get mixed when statements about the system get mirrored inside the system, and "if we make a diagram showing the relationship between TNT and its metalanguage, we will produce something which resembles in a remarkable way the diagram which represents the central dogma of a molecular biology [by this the author means the double helix DNA model]."
517ff On messenger RNA: on the enzymes inside the nucleus which faithfully copy long stretches of the DNA's base sequence onto a new strand of messenger RNA, which then departs from the nucleus and wanders out into the cytoplasm, encounters a ribosome, and then the "translation" takes place: this is at the heart of all life, the author offers a metaphor here where the mRNA acts like a piece of magnetic recording tape, finding the ribosome which acts like a tape recorder, which reads it and converts it, producing amino acids (musical notes) and making up a protein (a piece of music); the author cites the production of codons as a type of "hemiola" in biochemistry.
519ff When the protein is emerging from a ribosome it gets longer and longer and then folds itself up into a three-dimensional shape, a tertiary structure; the amino acid sequence per se is the primary structure but the tertiary structure is implicit in the primary structure. Note however that this does not translate into also knowing the function of a protein.
525ff Returning to the image of a ribosome as a tape recorder, mRNA as tape, and protein as music; and then on to the idea of a polyribosome: imagine tape recorders in a row, evenly spaced, playing a single long tape passing serially through all the playing heads of the component recorders, a polyrecorder; the output then would be a many-voiced Canon; such things exist in cells where polyribosomes all play the same strand of DNA, producing identical mRNA strands staggered in time.
529ff On how DNA needs to have a sufficiently strong support system to self-replicate; this metaphor translates to formal systems that need to be "sufficiently powerful" to have self-reference capability; more on this in a little bit, but first the author here finishes his description of how stranded DNA can be a self-rep, using an unzipping molecule that then copies the two intertwining sides/strands of the DNA, with no self-knowledge, but just with rules of inference; this looks a lot like a quine sentence describing how to construct a copy of itself, with two copies of the same information, where one copy acts as instructions and the other as a template; in DNA the process is "vaguely parallel."
531ff On levels of meaning of DNA: there is a translation that happens, each DNA strand codes for an equivalent RNA strand (translation on top of transcription), and then the DNA is readable as a code for a set of proteins which manifests in the physical "pulling out" of proteins from genes, which is gene expression. But the author argues that in theory we could learn to read DNA like a piece of music and not actually go through the actual physical process of epigenesis. "The output of such a pseudo-epigenesis program would be a high level description of the phenotype." And then he speculates that there could be even then the ability to read the phenotype off of the genotype, without doing even a simulation of epigenesis, but finding some simpler form of decoding mechanism, calling it shortcut pseudo-epigenesis.
532ff Now the author maps Gödel's theorem onto DNA in Crick's molecular biological model, he calls it the "Central Dogmap" [see photo below]. "There is something almost mystical in seeing the deep sharing of such an abstract structure by these two esoteric, yet fundamental, advances in knowledge achieved in our century."
537ff Next a discussion of a virus, which injects its own viral DNA into a cell, hoping that it will get transcribed and translated by the cell, and thus direct the synthesis of its own proteins; basically it transports alien proteins "in code" into a cell, causing it to replicate the alien DNA, which then forms "bodies" of new viruses. The author then playfully parallels this to the Tortoise's game of finding a record that will destroy the Crab's record player. Then a discussion of recognition, disguises, and labeling: asking how molecules or higher level structures "recognize" each other? In the case of a cell or e. coli's tricks for outwitting a virus, it uses methylated DNA, which is basically DNA with a label on it, and the host cell will have an enzyme system which looks for unlabeled DNA and destroys it. Thus the virus that successfully gets itself reproduced can have the high level interpretation of "I can be reproduced by cells of type X."
542ff On Henkin sentences (sentences which assert their own producibility in a specific formal system; but without information on its derivation or proof), and the idea of a self-assembling virus versus the traditional non-self assembling virus; some viruses, if sitting in the right medium--see for example the tobacco mosaic virus which can self-assemble as long as it's sitting in "the rich chemical brew of a cell"; the author likens these two to explicit and implicit Henkin sentences, respectively. And then he takes this example of a self-assembling biological structure like the tobacco mosaic virus to raise the possibility of such a thing existing in the world of self-assembling machines: the idea here is "the information for the total conformation of the organism (or machine) is spread about in its parts; it is not concentrated in some single place."
543ff Next the author talks about cellular differentiation and what produces morphogenesis--the birth of form. Then he talks about feedback and feedforward mechanisms that exist both within cells and between cells, which tell when a cell should turn on (or turn off) production of various proteins. Concepts here of inhibition and repression, negative feedforward or negative feedback; also on the idea of inducers which produce positive feedback or feedforward [You can definitely see how Hoftstadter got all these ideas right out of Jacques Monod's Chance and Necessity].
545ff A comparison of feedback with strange loops, leading to discussion of how cell differentiation can happen in, say, the human body, where there are radically different types of cells in different organs; the other gives a simple metaphor of a chain letter that goes from person to person where each new person is asked to propagate the message faithfully, but also add some personal touch. "Eventually, there will be letters which are tremendously different from each other."
546ff On level-mixing inside the cell and in biology. "Although it is possible (to some extent) to draw boundaries and separate out the levels, it is just as important--and just as fascinating--to recognize the level-crossings and mixings." DNA can be viewed as a program written in a higher level language which is translated or interpreted into the "machine language" of the cell and its proteins; but at the same time DNA is a passive molecule which undergoes manipulation at the hands of various enzymes--thus in this sense, DNA is exactly like a long piece of data. Third, DNA contains templates on which RNA flashcards are "rubbed": thus DNA also contains the definition of its own higher level language. Likewise proteins also act as both programs, processors, data and interpreters. And so on with ribosomes and transfer RNA. The author's point here is that "nature feels quite comfortable in mixing levels which we tend to see is quite distinct." Finally a question about how the whole thing--the genetic code with all of its mechanisms--bootstrap itself up in the first place; this is a question that has no answer.
The Magnificrab, Indeed
549ff A discussion of the Crab receiving letters from an Indian fellow Najunamar [Ramanujan spelled backwards: this is the famous and not formally educated Indian mathematician who arrived at all sorts of important mathematical ideas, his Wikipedia page is very much worth a read]. Then the Crab pulls out his flute and begins playing music, but he doesn't have any songs memorized; he plays from a sheet of paper with what appears to be mathematical notation on it, and this unlocks certain musical tunes that he believes are "piano postulates." But Achilles believes that these are statements of number theory, and so Achilles writes out certain mathematical notations of his own in number theory, and the Crab "plays" them, one of which apparently sounds like the music of John Cage, another one which is not playable, and the last one that he's actually afraid to play, and makes an excuse to not play it, saying, "Whew! That was a close call!" [Probably here Achilles wrote out a type of Epimenides paradox in number theory, or worse, a "record player-breaking" string of notation that if played would blow up the Crab?]
Chapter 17: Church, Turing, Tarski, and Others
559 "We have come to the point where we can develop one of the main theses of this book: that every aspect of thinking can be viewed as a high-level description of a system which, on a low level, is governed by simple, even formal, rules. The 'system', of course, is a brain--unless one is speaking of thought processes flowing in another medium, such as a computer's circuits. The image is that of a formal system underlying an 'informal system'--a system which can, for instance, make puns, discover number patterns, forget names, make awful blunders in chess, and so forth. This is what one sees from the outside: its informal, overt, software level. By contrast, it has a formal, hidden, hardware level (or 'substrate') which is a formidably complex mechanism that makes transitions from state to state according to definite rules physically embodied in it, and according to the input of signals which impinge on it. A vision of the brain such as this has many philosophical and other consequences, needless to say. I shall try to spell some of them out in this chapter... The only way to understand such a complex system as a brain is by chunking it on higher and higher levels, and thereby losing some precision at each step. What emerges at the top level is the 'informal system' which obeys so many rules of such complexity that we do not yet have the vocabulary to think about it. And that is what Artificial Intelligence research is hoping to find."
560ff On how artificial intelligence/AI could also stand for "artificial intuition" or even "artificial imagery"; the idea is to get at what happens when one's mind invisibly chooses what makes the most sense in a very complex situation; see for example when deductive reasoning would be useless because it produces all kinds of irrelevant statements, while the mind can see immediately what is important here and what is not with a sort of sense of simplicity and beauty. "Where do these intuitions come from? How can they emerge from an underlying formal system?" References to the Crab in the prior story where he can distinguish between "beautiful" and "non-beautiful" music, but Achilles redefines this via number theory into categories "true" and "false"; but yet the Crab argues that he's incompetent in mathematics. Next the author goes into Church's Theorem ("There is no infallible method for telling theorems of TNT from nontheorems") and the Tarski-Church-Turing Theorem (there is no infallible method for telling true from false statements of number theory).
561ff Next, versions of the Church-Turing Thesis which the author calls "certainly one of the most important concepts in the philosophy of mathematics, brains, and thinking." He gives a tautological version: "mathematics problems can be solved only by doing mathematics." And then various more strongly worded versions, and then a version that is the "public processes version": solving a number problem can be both performed and communicated from one sentient being to another by means of language, thus there exists a recursive program that can give the same answers as the sentient beings' method does.
562ff On Srinivasa Ramanujan and his correspondence with the English mathematics professor G.H. Hardy, who upon receiving his work said some of it "defeated me completely"; on Ramanujan's lack of rigor (or his beautiful use of intuition, depending on how you want to frame it); on his almost autistic "friendship" with numbers; the author thinks about how these men think, and playfully restates the Church-Turing Thesis as "all mathematicians are isomorphic"; then an interesting quote from Hardy describing Ramanujan's intellectual attributes, where the author italicizes part of it describing it as "an excellent characterization of some of the subtlest features of intelligence in general": "With his memory, his patience, and his power of calculation, he combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own field, without a rival in his day."
567 Then a discussion of idiots savants: people who had the ability to calculate and multiply numbers far in excess of their abilities to explain the results.
567ff Finally the author states the standardized strengthened version of the Church-Turing Thesis, the "isomorphism version": that there is a method which a sentient being follows to sort numbers into two classes which yields an answer in a finite amount of time and always gives the same answer, thus some terminating FlooP-recursive function exists which gives exactly the same answers; moreover, the mental process and the program are isomorphic and there is a correspondence between the steps being carried out in both computer and brain. It asserts that when one computes, something one's mental activity can be mirrored isomorphically in some finite loop program; it's not that the brain is literally running such a program, it's just that these steps are taken in the same order as they could be in such a program, and the logical structure can be mirrored in a program like this. Note the isomorphic activity is taking place on the highest level, not at the level of neurons or bits, which are lower levels of brain and computer, respectively. Thus a high-level computer language maps to the human brain's symbol level, and both map onto a universe of natural numbers.
569ff On the difference between the well-defined narrow world of number theory and the poorly-defined area of real-world problems; number theory problems don't ask you, while changing a burned out light bulb, to move a garbage bag, and when you accidently spill a box of pills (which means you have to sweep the floor so the pet dog doesn't eat those pills) you are now facing a range of problems that are not related at all, yet they are one some level related. Number theory problems don't have extraneous things like pills or dogs. "It seems that a large amount of knowledge has to be taken into account in a highly integrated way for 'understanding' to take place." On mental structures for things like "dog" or "broom": what symbols exist and are they available for conscious inspection? And then how would we map all this to a machine? Note the Anteater's comment in the Ant Fugue of the indescribably boring nightmare of trying to understand a book on the "letter" level: the idea here is that computers, in order to map reality, would have to have certain layers that are not really understandable to us and are there only for their catalytic relation to layers above them rather than some direct connection to the outside world. While deductive reasoning can be programmed into essentially one level, imagery and analogical thought processes require several layers and are intrinsically non-skimmable.
571ff On higher-level meaning as an optional feature of a neural network; on how the "microscopic version" of the Church-Turing Thesis says that brain processes are no more mysterious than any other level of organization like stomach processes: digestion is an ordinary chemical process thus the same reasoning occurs in a brain process; thus the brain is understandable in principle. Which takes us to a "reductionist version" of the Church-Turing Thesis: all brain processes are derived from a computer substrate. "This statement is about the strongest theoretical underpinning one could give in support of the eventual possibility of realizing Artificial Intelligence."
572ff Note that you don't necessarily want to copy the brain: the actual hardware of AI should look quite different; the question is to what depth you have to go in copying a human brain, or what symbolic levels can be "skimmed off" of a brain's neural substrate and implemented in other media like the electronic substrate of a computer. "Sometimes it seems as though each new step towards AI, rather than producing something which everyone agrees is real intelligence, merely reveals what real intelligence is not." The author refers back to the previous story of the Magnificrab, asking could any brain process distinguish completely reliably between true and false statements of TNT without being in violation of the Church-Turing Thesis, and is perception of beauty a brain process? The Crab can't communicate his ability for example. Then there's a version that many people hold that the author calls the "soulist's version" of the Church-Turing Thesis: basically some kinds of things a brain can do can be vaguely approximated on a computer, but not most and certainly not the most interesting ones, and either way nothing explains the soul and computers don't have any bearing on this. The underlying idea here is that the appreciation of beauty is not possible for mere machines like it is for humans.
575ff Discussion of irrationality and rationality: the idea of irrationality being incompatible with computers sits on a confusion of levels, computers are not faultlessly functioning machines and are not bound to be logical; then again you can instruct a computer to print out a bunch of illogical statements but of course the computer in this case would not be making mistakes. A mistake would be involved if printed out something other than the statements it was instructed to print. But likewise a brain is a collection of faultlessly functioning elements: neurons, and neuron threshold is surpassed or not and then it fires. But at the same time neurons support high level behavior that is sometimes illogical on the most amazing level. In other words software can be running in a mind supported by faultlessly functioning hardware. Meaning can exist on two or more levels of a simple handling system "and along with meaning, rightness and wrongness can exist on all those levels."
579 Finally an "AI version" of the Church-Turing Thesis, that, according to the author, AI researchers rely on as "an article of faith": "As the intelligence of machines evolves, its underlying mechanisms will gradually converge to the mechanisms underlying human intelligence."
580 On Tarski's theorem: "Tarski asked whether there could be a way of expressing in TNT the concept of number-theoretical truth... More specifically, he wished to determine whether there is any TNT-formula with a single free variable a which can be translated thus: 'The formula whose Gödel number is a expresses a truth.'" The problem here is this reproduces the Epimenides paradox all over again saying "I am a falsity." Then, the author asks what is so bad about reproducing the Epimenides paradox, we already have it in English and it's not like the English language went up in smoke! The point is that in Tarski's formulation it is a statement about natural numbers that is both true and false at the same time; while in the English language the subject matter is abstract, this is a lot different when we're talking about concrete statements about numbers. And thus there's no way of expressing the notion of truth inside TNT. Thus truth is a far more elusive property than theoremhood.
581ff On thinking about the word "form" as it applies to constructions of arbitrarily complex shapes: what is the "form" of something that causes us to see beauty in a painting? It is lines and dots that hit our retina? Is that even form? Somehow we're responding to some inner meaning inside the picture, somehow we "pull out" from a two-dimensional picture some high dimensional notion of meaning; the same could be said about how we respond to music. Next the author considers two types of form a "syntactic form" or a "semantic form": the latter requiring more of an open-ended test, mainly because the object's meaning is not localized within the object itself. The author next takes this into the idea of music and how the meaning is obviously derived from the music, and thus it must be localized somewhere inside the music; but then when we think about the mechanism which does the "pulling out" of meaning--in this case the mental mechanism which interprets that music--somehow it sets up some sort of multidimensional cognitive structure with multiple connections, connections between pre-existing information, between links with other mental structures, etc., and through this process full meaning gradually unfolds. We could say for example that music or a piece of text "are partly triggers, and partly carriers of explicit meaning." Likewise with beauty: is it a property that is syntactic or semantic, or can it be something that changes over time, or is it the beholder who has changed over time?
SHRDLU, Toy of Man's Designing
586ff This is a whole dialogue between a person and an AI system with a twist: the AI system is dying to have someone try out his "newly developed human being" who can--in a limited way--analyze a conversation. Thus we have a dialog between a person and an AI, along with a running commentary about the nature of the various commands, questions, vocabulary words, and semantics of the conversation as analyzed by this "newly developed human being."
Chapter 18: Artificial Intelligence: Retrospects
594ff On Alan Turing's prophetic and provocative 1950 article "Computing Machinery and Intelligence"; remarks about Turing himself; his 1937 paper on the unsolvability of the halting problem; his building of computers in the 1940s; his pioneering influence in the field of computer science. Turing's 1950 article begins with the sentence: "I propose to consider the question 'Can machines think?'" On his "imitation game" which came to be called the Turing test; on how computers might behave on purpose like humans in order to imitate a human, but how would they go about doing this?
597ff Turing anticipates the nine objections to the notion that computers could think:
* The theological objection [God gave a soul to man, that is the way thinking happens]
* The "heads in the sand" objection [AI is too dreadful to think about, let's just pretend it can't happen]
* The mathematical objection [similar to J.R. Lucas' objection]
* The argument from consciousness
* Arguments from various disabilities ["you'll never make a computer do X"]
* Lady Lovelace's objection [computers can only do things we know how to order them to do, it never originates anything]
* The argument from continuity in the nervous system
* The argument from informality of behavior
* The argument from extra-sensory perception
600ff The author gives a brief history of AI, starting with Boole and De Morgan, who devised "laws of thought" (which are essentially the Propositional Calculus) in the 19th century. This gave us the first step towards AI software; Charles Babbage's "calculating engine": a precursor to computer hardware, and the first adding machines that performed "tasks previously performable only by human minds."
601ff Note also the paradox about "progress" and AI, that once something becomes programmed into a machine, we stop thinking of it as "essential to real thinking": in other words AI is whatever hasn't been done yet. Then the author gives a surprisingly long list of the various domains to which AI has been applied: mechanical translation, games, vision, symbolic manipulation of mathematical expressions, hearing, understanding natural language, producing natural language, analogical thinking, learning, etc.
603ff On some of the unexpected problems that came up with various tasks; mechanical language translation turned out to be much more difficult than anyone expected: it wasn't just dictionary lookup and word rearranging but it was having a mental model of the world being discussed and manipulating symbols in that world; see also in chess-playing programs you can use a brute force look-ahead method but this also requires heuristics and chunking of higher level strategy in the game. [It's fascinating to read these problems today when Google translate does a fairly good job translating language, it's not perfect, but a lot of the work is really crowdsourced iterations to translation results over time; also chess programs are currently operating at a far higher level than anything possible back when this book came out]. See also Arthur Samuel and his checkers playing program, which used both look-ahead and non-look-ahead ways of evaluating a board position; thinking about a dynamic evaluation of a tree of possible future moves. The author talks about the interesting recursion in this program to look seven moves ahead and flatten that look ahead into a simple static recipe of what to do.
606ff On the idea of "originality" in AI, when the AI "outdoes" its programmer: see the example of a proof of a geometry problem with an isosceles triangle: "where" was the proof? Was it in the computer, or in the human who programmed the computer? If the AI comes up with an original idea, who or where was the idea? In the programmer of the AI? Who gets the credit? On thinking of the computer as a "meta-author": the author of the author of the result; then taking this meta-author idea even further with composing computer music, especially if the composer the human is "surprised" at the result.
609ff On "problem reduction": breaking a longer-range problem into small subgoals or subproblems, possibly in a recursive fashion with sub-subproblems. But remember the Zeno paradox will you keep moving halfway towards the goal and then it becomes an infinite goal stack; see for example with a chess program where look-ahead techniques don't work because they're not based on planning and thus would explore a huge number of pointless alternatives; also an interesting example here of throwing your dog's bone over a chain link fence into another yard: the dog has the subgoal of going to the fence and then getting through the fence, and then running to the bone, or seeing the subproblems as going to the open gate, going through the gate, and then running to the bone. So the question is: how does the dog view the problem space? Because the correct solution increases the distance between the initial situation and the desired situation, you actually have to run away from the bone to get to it; thus here we have an instance where you can confuse "physical distance" with "problem distance": you have to shift your perception about what actually brings you "closer" to the bone. Of course humans perform variations of this problem all the time like when they leave their office to go home from work: they're going in a whole range of directions, but we don't think of anything of doing this at all. "And when the problem space is just a shade more abstract than physical space, people are often just as lacking in insight about what to do as the barking dogs. In some sense all problems are abstract versions of the dog-and-bone problem."
612ff Interesting insight here about choosing and changing your problem space: most problems exist in conceptual space, not literally in physical space, thus you can "run directly to the fence" and then rethink the problem space all over again; or you can think about the problem in the abstract, imagining that there is no "fence" separating you from your goal, such that you can proceed directly to the goal in this new space. Basically you're restructuring the problem, and this sort of solution sometimes appears as a flash of insight, rather than a process of slow, deliberate, obvious steps. And this takes us to yet another question: How do you choose a good internal representation for a problem? This of course is it itself a type of problem, and you might think of turning the technique of problem reduction back on it! The whole idea is that AI lacks programs which can step back and take a look at what is going on and then reorient. "It is the difference between the Sphex wasp, whose wired-in routine gives the deceptive appearance of great intelligence, and a human being observing a Sphex wasp." Basically "Mechanical Mode versus Intelligent Mode all over again.
614 [Fascinating thoughts here on "sameness" and meta-awareness here]: on how the sphex wasp can't tell that it's doing "the same thing" over and over again because to "notice" such a thing would be to jump out of the system. But then it's worth thinking about how does this happen to ourselves, highly repetitious situations that we handle in the same stupid way each time because we don't have enough of an overview to perceive the sameness.
619ff More thoughts on visualizing a problem space: on deductive versus analogical awareness; the ability to spot similarities and compare situations; see for example in a chess program, looking at the board situation from a strategic standpoint.
620ff Discussions of the "behavior space" of a program: the author built a program to create English language sentences out of the blue; but once he could see the entire behavior space of the program the program's output rapidly became uninteresting. Contrast this with the "behavior space" of a person, which is complex enough to often surprise people continually. Thus his program needed more "subtlety," but simple juxtaposition of words wasn't subtle; this takes us to the problem of representations of knowledge: in this case of language the author had to classify each word type in different semantic dimensions, in different classes and then superclasses (classes of classes); the author shares a series of sentence outputs from his program along with human-written sentences from an academic journal; it's interesting to read these sentences: some of them sound literally retarded, some sound somewhat poetic. The author talks about his emotions when he read these outputs coming out of the program: he was proud of the achievement; he felt like there was a machine working in a complicated way, moving symbols around according to rules, but then he also could clearly see that there was no real "understanding" behind those words. [Note this comment from the author that reminds me of midwit corporate types] "One definitely gets the feeling that the output is coming from a source with no understanding of what it is saying and no reason to say it." Note also how the author talks about how the computer uses the word "serf" or "person" but with no idea what those things were. "The words were empty formal symbols. [Note however that you can also see this same behavior among fully formed humans: see for example someone using a word or a phrase (or a term of jargon) correctly in a sentence, but without any understanding of the underlying meaning of that word or phrase. This is a mode whereby people can ape the behavior of an intelligent person by using intelligent-sounding words; this is an interesting analogy between computer behavior and human behavior!]
627 On Terry Winograd and his SHRDLU program, working with joint problems of language and understanding; the author asks what would it take for us to admit that a program had some understanding, such that we would not feel intuitively that there is nothing there? Winograd's problem dealt with the "blocks world," a simplified realm for testing a computer's vision and language-handling capability. The program had various problems with understanding questions in English about the situation, with giving answers, with understanding both steps in each operation as well as what with understanding what it had already done and why. Note the example here of the instructions the programmer had to write for the various possible meanings of the word "the". [!] Then a discussion on how syntax and semantics are so deeply intertwined in language, they merge into each other in natural language. Language and sentences do not divide up neatly into syntactic aspects and semantic aspects; finally a discussion of how many levels a system should have, and what kind of intelligence should be placed at each level. "Since we know so little about natural intelligence, it is hard for us to figure out which level of an artificially intelligent system should carry out what part of a task." [Note also the phrase "ETAOIN SHRDLU" which is a nonsense collection of letters from old-style typography: googling this phrase takes you into a whole rabbit hole on the history of hot metal typesetting.]
Contrafactus
633ff The crab invites a group of friends over to watch a football game on television; one of the guests is the Sloth who speaks in nonsense/paradoxical sentences that are grammatically correct, while the different guests answer him with likewise nonsensical sentences:
Sloth: Oh. how conventional. I never root for the home team. The closer a team lives to the antipodes, the more I root for it.
Achilles: Oh, so you live in the Antipodes? I've heard it's charming to live there, but I wouldn't want to visit them. They're so far away."
634ff We learn that the Crab owns a "Subjunc-TV", a TV that can go into the "subjunctive mode" and show replays where something that almost happened doesn't happen and vice versa using "subjunctive instant replay." Achilles plays with the dials and shows a game where the football is round instead of football-shaped. The Crab teases him for such tame and uninteresting subjunctives: he shows how the last play would have looked if the game had been baseball instead of football. He then runs the Subjunc-TV as if the game were played in four dimensions. The interlude closes with the announcer--subjunctively!--describing the story as if Mr. Crab had actually won a Subjunc-TV when it turns out that he actually didn't.
Chapter 19: Artificial Intelligence: Prospects
641ff On "almost" situations, counterfactuals: what makes some more realistic rather than others, or why do we think of certain "could have happened" type situations more than others; the author goes through examples comparing the statement "I don't know Russian" to "I would like to know Russian" or "I wish I knew Russian" or even "the rich counterfactual" "If I knew Russian, I would read Chekhov and [Mikhail] Lermontov in the original." "A live mind can see a window into a world of possibilities." The author then quotes the linguist George Steiner from his book After Babel, describing it as a counterfactual hymn to counterfactuality: "It is unlikely that man, as we know him, would have survived without the fictive, counter-factual, anti-determinist means of language, without the semantic capacity, generated and stored in the 'superfluous' zones of the cortex, to conceive of, to articulate possibilities beyond the treadmill of organic decay and death." Further comments here also on how we seem to slip into the world of "what ifs" without any conscious direction: also comments here on the prior dialogue which went further and further into less and less realistic subjunctive circumstances.
644ff Discussion of how we think about layers of stability inside a hierarchy of variability: we see certain things as either constants or variables depending on our framing of the situation. See for example how we wouldn't think about hypothetically changing the three-dimensionality of our world because it's so ingrained in our minds--it's a "constant constant"; but then there are background assumptions which are also "constants" like the parameters or rules of the game of football; then there are other somewhat more variable parameters like the weather, the opposing team, and probably several other layers of parameters; but then you get to more shaky aspects of your mental representation of the situation: things that could easily be hypothetical variables, like one of the players not stepping out of bounds; this is a more mentally loose event that we can let slip away from our mental reality when we imagine a counterfactual. Also on having nested frames; see also the notion of frames and frame theory from Marvin Minsky, for how we slip into default assumptions when we construct our mental representations of reality.
646ff On Bongard problems from Bongard's book Pattern Recognition: where we have different stages of classing boxed figures; discussion here of a "salient feature vocabulary" you have to develop for these groups of images, then using language about shapes, and then descriptions at a higher level, all of these tend to filter how we see the box; also we make a tentative description for the boxes and then we compare them with the other boxes and then restructure those descriptions by adding things discarding things or looking at the information from another angle; we then iterate this process until we figure out what what makes the two classes differ. Discussion here of "mental templates" and "sameness detectors" that sit in our mind; also ideas of slippage and tentativity, we can have tentative concepts or concepts that slip from one description to another. [See photos below for examples of typical Bongard problems.]
656ff On meta-descriptions and levels of abstraction that happen as a vital part of the recognition process. On a level of variety which tells you that the distinction exists on a higher level of abstraction than just appearance, a description of descriptions. On focusing vs filtering, both as ways to throw away information.
659ff on Bongard problems as a tiny form of science: finding patterns in the world, and as these patterns are sought we make mental templates, then remake them; we shift from one level of generality to another, etc. See for example Bongard problems #70-71, which are "the same problem" when looked at On a sufficiently abstract level:
Note this link for a full index of Bongard problems. It is literally hypnotic to look at these different problems!
661ff Comments on how these types of problems are related to other aspects of cognition: first on the intuition required to know when it makes sense to blur distinctions, or try redescriptions, or backtrack, or shift levels: these capabilities probably come only with experience in thought in general; also note that real world experience may allow some of these intuitions like for example with problem #70: if you know how trees are structured, it might give you an intuition about solving the problem. Thus it would be very difficult to define heuristics for these types of things.
662ff Discussion here on "message passing" used as a way to develop a form of intelligence between computers; if you think about it, a frame plus an actor equals a symbol (which is basically a message plus something acting on that message), which creates a more complex message or interpretation of that message; the author applies this metaphor to the cell and ribosomes being sought out in the cytoplasm; see another of the author's metaphors: the postal service and the various messages and methods one can use to send a message; even a telephone system. He describes how enzymes each have their own "address" in effect and is programmed at that address by a message that arrives and interacts chemically with that address. A further metaphor: how enzymes sit around waiting to be triggered by an incoming substrate, and when the substrate arrives the enzyme springs into action. This type of hair-trigger program has been used in AI by the name of "demon" where you might have many species of triggerable subroutines in a program just waiting to be triggered.
665 Discussion of the dialogue Crab Canon, how it illustrates how far a single idea can be pushed, as well as the symbolic recombination or fusion of ideas; he describes it as a form of meiosis/cell division: starting with a prophase (the simple idea that a piece of music could be imitated verbally) and he then begins mapping voices, thinking about how the musical reversal will take place, etc. Then a metaphase (where he took various stabs at filling in the dialogue), he stumbles onto the idea of having the Tortoise talk about Bach and Achilles talking about Escher in parallel language. At that point he has a flash of insight that the dialogue was becoming self-referential without even intending it, a form of indirect self-reference in that the characters didn't talk directly about their own dialogue, but rather about structures isomorphic to it; thus his dialogue had a conceptual skeleton similar to Gödel's string G; then the anaphase: he has a flash of insight viewing Escher's Crab Canon image, that it is a visual form of Bach's mucical Crab Canon; the author also learned the word "palindrome" around the same time and saw "crab canonical" structures in DNA. Finally a telophase: another insight where he realized ATC from the DNA molecule could be mapped to Achilles, the Tortoise and the Crab; thus he was able to map the dialogue's structure to the structure of DNA as well.
668ff Interesting discussion here on "conceptual skeletons" and "conceptual mapping": for example, The author has the idea of a structure having two parts doing the same thing, only moving in opposite directions, a concrete geometrical image that can be manipulated in the mind in various ways like a Bongard problem; and he sees that it maps as two chromosomes joined by a central mirror, an image directly drawn from meiosis. He then extends this metaphor to "recombinant ideas": place two ideas next to each other and have them exchange parts, kind of like how chromosomes exchange genes. "It is always of interest, and possibly of importance, when two or more ideas are discovered to share a conceptual skeleton." And then also thinking about a conceptual skeleton in a very loose way: for example at different levels of abstraction or on an isomorphic-type level. "The Vice President is the spare tire on the automobile of government." This phrase has a mapping at a fairly high level of abstraction; a car and an automobile are very different, however, we already have a conceptual skeleton for "Vice President" as well as for "car" and so "the forced mapping works comfortably."
670ff On how we use "handles" or "ports of access" to events, objects or ideas: the author talks about his car radio that he reaches down to turn on and finds that it's already on; thus he has a mental representation of a the radio as both "music producer" but also "boredom reliever"; and he's aware of the music is on but he's bored anyway, and so before he thinks too much, his reflex to turn it on is triggered; also he goes through various mental "ports of access" of how you might think about a radio: like "a thing that makes buzzing noises" or "has knobs" or "plays music," etc... all of these are attached to his mental symbol for a car radio. Then a discussion of "mental partitions" that prevent him from thinking too sloppily, or having symbols spill over into each other in a sloppy way: thus when he's repairing the radio he doesn't think about the time that he heard The Art of the Fugue on that radio; see also the form of mental partitions that happen in a rigid way where words for the same thing are "sealed off" in different languages: "If these partitions were not strong a bilingual person would constantly slip back and forth between languages." But then again, adults who learn languages do tend to have words slip from one group to another [that is for sure the case for me!]; finally, on simultaneous translators, who somehow can negate these partitions to allow rapid access from one language to another to translate. [Once again you can see inklings of why the author had an entire book he needed to write about the problems of translations, and why Le Ton Beau de Marot had to come out of him.]
673ff "It is obvious that we are talking about mechanization of creativity. But is this not a contradiction in terms? Almost, but not really. Creativity is the essence of that which is not mechanical. Yet every creative act is mechanical--it has its explanation no less than a case of the hiccups does. The mechanical substrate of creativity may be hidden from view, but it exists. Conversely, there is something unmechanical in flexible programs, even today. It may not constitute creativity, but when programs cease to be transparent to their creators, then the approach to creativity has begun." Thoughts here on the value of adding randomness both to our own brains but also to a computer program.
674ff "Bongard problems were chosen as a focus in this chapter because when you study them, you realize that the elusive sense for patterns which we humans inherit from our genes involves all the mechanisms of representation of knowledge, including nested contexts, conceptual skeletons and conceptual mapping, slippability, descriptions and meta-descriptions and their interactions, fission and fusion symbols; multiple representations (along different dimensions and different levels of abstraction), default expectations, and more."
674ff Comments here on the flexibility of language: how the computer program in the SHRDLU dialogue had a very rigid sense of language, whereas humans can handle hazy language and imprecise language much more easily. The author shares a story about a little girl who lost her balloon and cried and cried, and he posits that a computer can never understand all the various subtle references in the story, and certainly not understand what it means to be in the child's shoes "until it, too, has cried and cried."
676ff The author closes the chapter with 10 questions and speculations about AI, including "Will a computer program ever write beautiful music?" (Yes, but not soon); "Will emotions be explicitly programmed into a machine?" (No, ridiculous); also an interesting question here: "Will a thinking computer be able to add fast?" (The author describes how the idea of changing levels often addle-brains you, and paraphrases Descartes: "I think; therefore I have no access to the level where I sum."); Also note this question: "Will there be chess programs that can beat anyone?" (The author argues no: interesting here to see him, as a huge optimist about AI's capabilities, get this one very wrong.)
Sloth Canon
681ff Achilles and the Tortoise are visiting the Sloth; Achilles sits down to the piano and plays, but the piano is upside down and reversed: it's a "sloth piano" appropriate for playing inverted melodies.
Chapter 20: Strange Loops, or Tangled Hierarchies
684ff Can machines possess originality? Do they have "intentions" or are those simply the intentions of the human programmer specified in advance? Comments from Arthur Samuel [see Chapter 18, above], the inventor of the successful checkers program, saying how computers cannot do anything that it hasn't been instructed as to how to proceed. Samuel makes an interesting argument here: that there has to be a "complete hiatus" between the program (which is the programmer's wishes) and any development inside the machine of will of its own. "To believe otherwise is either to believe in magic or to believe that the existence of man's will is an illusion and that man's actions are as mechanical as the machine's. Perhaps Wiener's article [Norbert Weiner, who wrote an article suggesting computers can possess originality] and my rebuttal have both been mechanically determined, but this I refuse to believe." The author then frames Samuel's argument to mean that a mechanical form of will would require infinite regress; he argues that something is wrong with Samuel's argument, that it's based on the assumption that a machine can't do anything without having a rule telling it to do that thing. But the real truth is machines and people are made of hardware which runs all by itself according to the laws of physics. The lowest level rules are embedded in the hardware. Samuel's point is that a computer can't "want" to do anything because it was programmed by someone else, or said another way, a computer could only have a sense of desire if it could program itself, which is an absurdity. However Hofstadter says this therefore means that "Unless a person designed himself and chose his own wants (as well as choosing to choose his own wants, etc.) he cannot be said to have a will of his own... The Samuel argument doesn't say anything about the differences between people and machines, after all."
686ff On asking to what extent when we humans think we can change our mental rules, as well as our rules for changing our rules; but we can't change the rules at the bottom, because neurons run the way they run, you can have access to your thoughts but not your neurons; hardware rules cannot change, in fact their rigidity is what the software of our brains is built on: it's not a paradox but an actual fact about the mechanisms of intelligence. Thus this chapter talks about "self-modifiable software and inviolate hardware." [Is this true though? After all, we do grow new neuronal pathways in a recursive way depending on "modifications" we make to our software.]
687 On the author's hopes here to communicate some of his intangible intuitions about this problem of mind [using a combination of isomorphisms and self-reference by the way]: "I could not hope for more than that my own mind's blurry images of mines and images should catalyze the formation of sharper images of mines and images in other minds."
687ff On rules--say for example about chess--and then rules about how to change the rules (metarules), and then metametarules, which are less obvious to think about, because you start to lose distinctions between games rules. metarules, rules about rules about rules, etc., and you're now in a tangled hierarchy; however there are still certain things about the game that are implicitly inviolate (like that you take turns). So now the author talks about the Inviolate level (I-level) which governs what happens on the Tangled hierarchy level (T-level) but not the other way around; the idea here is that any system always has some protected level which is unassailable by the rules no matter how tangled the interactions may be among levels and rules [much like Gödelization applied to more and more powerful TNT systems]; the author gives an example of three authors: Z existing inside a novel by T, and T existing inside a novel by E, and E, amazingly, exists inside a novel by Z, but the whole thing is authored by H who's outside this loop: thus author H is in the inviolate space. None of the Z, T or E authors can touch H's life, they can't even imagine him. See also Escher's painting Drawing Hands, where the right hand and the left hand draw each other but the drawing itself is done by Escher who is outside of the two hand space. In other words, you have a strange loop at the top but an inviolate level below it.
691ff The author next takes these ideas and discusses brains and minds, talking about the illusion of a tangled hierarchy with (apparently) no inviolate level; but we only think that because that level is shielded from our view. The level of mind is a true tangled hierarchy, but the substrate of neurons and axons (which ironically itself is a tangle, literally) is the inviolate hardware layer; then discussing this in the domain of language and self-referentiality: when in language or in our actions we talk about ourselves or act on ourselves, where something in the system jumps out and acts on the system as if it were outside that system. When we think about our minds, metaphorically speaking, we forget about Escher in the Drawing Hands image, we feel like we don't have an inviolate level because "we are shielded from the lower levels, the neural tangle."
692-3 The author gives yet another analogy here, of strange loops in government: see for example during the Nixon Administration, when the Supreme Court (Note that the court system can change its own laws) and the executive branch were fighting over who had the right to decide what is "definitive" (and then you hit your head against the sort of ceiling where you're prevented from jumping out of the system to get a higher authority). At the end of the day the court system as well as all government institutions are sitting on a substrate of the millions of people in society, and the legal system "can be overridden just as easily as a river can overflow its banks." The society and the implicit permission it gives to honor these institutions is really the inviolate substrate here. Other examples would be the FBI investigating its own wrongdoings, or a parliament self-applying parliamentary rules of procedure, a sheriff going to jail while in office. Also examples in fringe science, where you challenge the objectivity of science itself, or examples of evidence and meta-evidence where you have an infinite regress paradox for layers of evidence, and yet people still have an intuitive sense of evidence.
695ff "One of the most severe of all problems of evidence interpretation is that of trying to interpret all the confusing signals from the outside as to who one is." On dealing with a constant flow of evidence from the outside affecting ones self-image and the individual's internal need for self-esteem; thus information has a very tangled hierarchy as we try "to reconcile what is and what we wish were." The total picture of "who I am" is integrated in the entire mental structure, producing "much of the dynamic tension which is so much a part of being human."
696ff Comment here about Gödel's Theorem mapping to psychology, which Hofstadter argues can be done as a metaphor but absolutely not as an exact mapping: "It would be a large mistake to think that what has been worked out with the utmost delicacy in mathematical logic should hold without modification in a completely different area." However in a metaphorical/analogical sense he talks about introspection: for example, figuring out if you are sane--which is a strange loop once you begin to question your own sanity. "I am just reminded of Gödel's Second Theorem, which implies that the only versions of Formal Number Theory which assert their own consistency are inconsistent..." Also applying Gödel's Theorem to understanding our own minds, and wondering what that means and on what level: To the level of knowing a complete wiring diagram of the brain? Or knowing oneself so perfectly that everything--including subconscious and intuitions--are out in the open? Obviously it's a trivial idea to know that we can't know our own actual brain state in its detail, but think about the age-old goal of "knowing yourself" in the sense of knowing your psychic structure. But is there a limit to which level of depth this Gödelian loop into our own psyche can go? The author refers to limitative theorems of meta-mathematics that say once you have the ability to represent your own structure to a certain critical point it is guaranteed you can never represent yourself totally; see Turings Halting Theorem, Church's Undecidability Theorem etc.
698ff Also on considering personal non-existence, which on some level makes no sense at all to us, although it makes sense if we think of ourselves as "just another human being"; further, on mixing subject and object which was a dualistic aspect of Western science which has been blurred by AI research; see also Gödelian aspects in quantum mechanics where the observer interferes with the observed. The author muses on how all of these results are limited; see for example Heisenberg's Uncertainty Principle.
699ff On the blurring of symbol versus object in music and art: see Wittgenstein's work, see also the use-mention concepts discussed earlier in the book, as well as quining: Quine wrote about the connection between signs and what they stand for. In music and in art the author describes "crises that reflect a profound concern with this problem" as domains that typically used a "vocabulary of symbols" but now we might be looking at pure globs of paint or pure sound; see for example John Cage's music and his "mentions" of sounds, but not using the "formulated code" that we would think of normally for music (see for example his polyradio piece Imaginary Landscape no. 4). "Art in this century has gone through many convulsions of this general type." On abandonment of representation in abstract art, then surrealism, see Magritte and his work on the symbol-object mystery or the use-mention distinction, for example in his "Ceci n'est pas une pipe" paintings "where the verbal message of the painting self-destructs in a most Gödelian way." See also where Mozart, in Don Giovanni, wrote into the score the sound of an orchestra tuning up. [And so we can see that it isn't as if only moderns thought up this kind of stuff! Note also how Cervantes, in part II of Don Quixote, has his characters discover that a book detailing their previous adventures (which is Part I of the story) has been read by people they meet.]
702ff Comments here on modern art and music intending to break down the notion that art is one step removed from reality, and that we need to interpret it through a code such that the viewer must have some substrate of knowledge about art to understand it. [This maps interestingly to the early 20th century German Expressionism painting movement(s), where artists wanted exactly this: a more direct connection emotionally between the viewer and the art.] "The idea was to eliminate the step of interpretation and let the naked object simply be, period. ('Period'--a curious Case of use-mention blur.)" The author makes an interesting argument here about how this actually backfires: the more viewers are told to look at objects without a code or without mystification the more the viewers try to look for a code and get more mystified. "After all, if a wooden crate on a museum floor is just a wooden crate on a museum floor, then why doesn't the janitor haul it out back and throw it in the garbage? Why is the name of an artist attached to it?" But then if the purpose is to instill a Zen-like sense of a world without categories and meanings, then we're encouraging viewers to look at a philosophy which rejects inner meanings; but yet viewers do ponder about inner meaning, thus there is still a code by which ideas are conveyed to the viewer--in fact this code is a more complex thing because it involves statements about the absence of codes, therefore it's part code, part meta-code.
704ff another interesting point here where the author talks about Leonard Meyer, in his book Music, the Arts, and Ideas, describing "transcendentalism" as a word to describe the breaking down of boundaries in modern art and modern music; the author prefers to use the word "ism" to describe this: a suffix without a prefix, suggesting an ideology without ideas. also "ism" embraces whatever it is [and it's a little bit like one of the self-expanding acronyms used earlier in the book if you think about it] also: "In 'ism' the word 'is' is half mentioned, half used: what could be more appropriate?"
706 "Can we ever hope to understand our minds/brains? Or does some marvelous diabolical Gödelian proposition preclude our ever unraveling our minds?" Gödel's theorem doesn't stop us from understanding our mind like we understand a car; but it's a wholly different thing to understand how to replicate our brains in a computer. "Gödel's theorem doesn't ban our reproducing our own level of intelligence via programs any more than it bans our reproducing out own level of intelligence via transmission of hereditary information in DNA, followed by education." Note here the author's mention that the DNA mechanism is itself a strange loop and a Gödelian mechanism! The author instead makes the case that understanding the process of understanding Gödel's proof, with its isomorphisms, its levels and self mirroring, gets us to think more deeply about mental structures on different levels. Gödel gets you to look at systems from higher level, including the ideas that there might be some higher level way of viewing the mind/brain, and that some facts could be explained on the high level quite easily, but not on lower levels at all. Much like something can't be explained in TNT.
711ff A discussion here considering free will: the author suggests instead of asking "Does system X have free will?" to ask "Does system X make choices?" And then evaluating that question for, say, a marble going down a hill, a calculator calculating the square root of 2, or a program playing chess, or a robot in a maze, or a human being confronting a dilemma. The marble clearly doesn't make "choices" and we can think about the calculator and the chess program being a "fancy marble" rolling down a "fancy hill." But the problem gets increasingly subjective as we proceed, because the chess program looks ahead at the various possible paths. The robot in the maze (in the author's example here) actually uses a random number to choose which way to go, and it has its own internal symbols, as well as a sort of a "self" symbol that affects the decisions it makes, thus we are unable to predict which way it goes; we can argue some level of meaning happens in this situation just like as we manipulate meaning with our own minds. The author thinks that self-reference is going to be at the core of AI. "And that is why Gödel is so deeply woven into the fabric of my book." [Note that p713 to the end of this chapter are a good summary of the entire underlying idea behind this very complex book, he explains what it is about Godel, Escher, Bach--as well as the three of them together--that makes them such important elements of his book.]
715ff On Escher's 1956 lithograph Print Gallery, with three kinds of "in-ness" [see photo below]: a boy in an art gallery in the town, where the town is artistically/depicted in the picture and the picture is mentally in the person; but we as observers of the artwork itself are not sucked into the recursion because we are outside the system. We can see things that the the young man can't see, like Escher's signature or the central blemish.
717 On how Print Gallery is "a pictorial parable for Gödel's Incompleteness Theorem. And that is why the strands of Gödel and Escher are so deeply interwoven in my book."
717ff On Bach and his Endlessly Rising Canon, on the idea of "Shepherd tones" with parallel scales but with a volume weighting, so that the scales seem to go up and up forever; also on how Bach actually included this general principle in some of his music, see for example halfway through the "Fantasia" from the Fantasia and Fugue in G Minor, for Organ; also comments here on the beauty and extreme depth of emotion in Bach's music, and the tangled hierarchy of it, just like those of Escher and Gödel, and how it reminds him of the human mind in its "beautiful many-voiced fugue." "And that is why in my book the three strands of Gödel, Escher, and Bach are woven into an Eternal Golden Braid."
Six-Part Ricercar
[I cannot even imagine all the work that must have been involved in constructing this intricate interlude. It's like a verbal Bongard problem, and it's hypnotic to read.]
720ff Achilles brings his cello to the Crab's residence, and in the Crab's music room there's a radio; Achilles turns it on to hear a discussion on free will and determinism, but then scornfully turns it off, saying "I can get along very well without such a program." The dialogue here is a fugue, as the Crab, the Tortoise and Achilles repeat each other in different contexts
721 The Crab has various computers sitting around that he calls "smart-stupids": they can be smart or stupid, depending on how skillfully they are instructed by their user.
722ff Self-referencing discussion quoting Martin Minsky about how intelligent machines will be just as confused and stubborn about free will as humans, which itself is a self-referential comment; they then talk about how they might just be characters in some dialogue; they discuss the previous Crab Canon story where they said the same things in reverse after the Crab showed up, thus the strange coincidence that they said the exact same things with symmetry indicates evidence of an "author" [You can see here the isomorphism to considering the existence of God because of the unbelievable miracles of life]. Achilles becomes very uncomfortable thinking that some author had worked out all this in detail, "programming" all the statements he made that day. And then the author enters[!], carrying a giant manuscript, himself reciting the same line "I can get along very well without such a program" [thus we all probably have less free will than we think...!] They all have a Borgesian self-referential conversation here, which is fun to read and to think about.
726ff One of the smart-stupids flashes a message that Babbage has arrived, they consider offering him a performance of all the celebrated canons from the Musical Offering, but Babbage says, "I can get along very well without such a program." [Again announcing himself with the same opening line in a fugue style]. The Crab offers Babbage the chance to work with one of his smart-stupids; Babbage does all kinds of interesting things with it; then the Crab asks Babbage if he is familiar with philosopher La Mettrie, who was a champion of materialism and wrote a book called L'Homme Machine, considering the idea of man being a mental machine but also considering the idea of imbuing a machine with a human-like faculty of intelligence; he suggests to Babbage the idea of programming one of the smart-stupids to play a reasonable game of chess.
729 The Crab: "My strongest point is simply that I seem to be able to formulate Themes whose potential for being developed is beyond my own capacity." [This is also witty, the Crab does this in multiple instances throughout the book, including in the Crab Canon where he literally does such a thing musically.] The chess program can run three simultaneous games and Babbage instructs two of them to play each other and the third to play the Crab, "a three-part chess fugue."
731ff More fugue-style writing: the phrase "the grounds are excellent" applied to different contexts: around the house, coffee, electric connections, figure-ground questions on an Escher painting, logic, etc.
733ff The Crab offers another idea: suggesting to Babbage the idea creating a smart-stupid that six times smarter than the Crab himself; Babbage creates a simulated human being that he calls Alan Turing, and its first sentence is "I can get along very well without such a program." And on the screen of the smart-stupid there is an image of the room they are all sitting in while peering out at them is a human face. And then Turing creates a smart-stupid that can simulate a human being with six times than his own, and he calls it "Charles Babbage" using the same fugue-style words staying this as Babbage did just before. The group has a [meta?]debate about who programmed who--Alan Turing or Charles Babbage--and which of them has free will. Babbage suggests performing a type of Turing test; Turing claims the idea was his, and is shocked that Babbage, "a program written by me... harbors the illusion of having inventing it all on its own!" They actually speak to each other as if it were a Turing test between computers; then a reversal happens and Turing enters the room while Babbage is on the computer; everyone is confused by this reversal, Babbage can understand why everyone's on the screen now from his perspective, the situation imitates an Escher print and a strange loop.
737ff The author actually breaks out his manuscript, describing how he makes various self-references using various devices, as well as how he imitates Gödel's self-referential construction; the author's own quote here is a fugue-style self-reference talking about "indirect self-reference is my favorite topic"; this is the author's verbal equivalent of modulating between keys; and it itself self-refers in the sense that it is isomorphic to its own mechanism; the conversation then circles back to Achilles wondering whether he has free will all over again! The group discusses how the dialogue copies the Bach piece it is based on, and the author tells them about the Crab's theme, which is the notes "CBABBAGE" spelled backwards. Turing then asks the author what his book is about, and Hofstadter says "Combining Escher, Gödel, and Bach, Beyond All Belief [which also is "CBABBAGE" spelled backwards, also, again fugue-like].
742 At this point the author, on the urging of the other characters, begins to talk about his book, and then the story goes to infinite regress with a self expanding acronym...
