Gödel, Escher, Bach: An Eternal Golden Braid

Douglas R. Hofstadter

Preface

Page: 6 (0.73%) @ 16 Mar 2025 06:12:31 PM

In a word, GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter. What is a self, and how can a self come out of stuff that is as selfless as a stone or a puddle? What is an "I", and why are such things found (at least so far) only in association with, as poet Russell Edson once wonderfully phrased it, "teetering bulbs of dread and dream"– that is, only in association with certain kinds of gooey lumps encased in hard protective shells mounted atop mobile pedestals that roam the world on pairs of slightly fuzzy, jointed stilts?

Page: 10 (1.22%) @ 16 Mar 2025 08:10:08 PM

In short, an "I" comes about – in my view, at least via a kind of vortex whereby patterns in a brain mirror the brain's mirroring of the world, and eventually mirror themselves, whereupon the vortex of "I" becomes a real, causal entity.

Page: 11 (1.34%) @ 16 Mar 2025 08:13:04 PM

I can't help but recall, at this point, a horribly elitist but very droll remark by one of mny favorite writers, the American "critic of the seven arts", James Huneker, in his scintillating biography of Frédéric Chopin, on the subject of Chopin's étude Op. 25, No. 11 in A minor, which for me, and for Huneker, is one of the most stirring and most sublime pieces of music ever written: "Small-souled men, no matter how agile their fingers, should avoid it."

Page: 16 (1.95%) @ 16 Mar 2025 08:36:48 PM

Only at this stage did the book's unusual stylistic hallmarks really emerge the sometimes-silly playing with words, the concocting of novel verbal structures that imitate musical forms, the wallowing in analogies of every sort, the spinning of stories whose very structures exemplify the points they are talking about, the mixing of oddball personaliues in fantastic scenarios. 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 that it had an intrinsic rightness to it. One of the key qualities that made me so believe in what I was doing is that this was a book in which form was being given equal billing with content - and that was no accident, since GEB is in large part about how content is inseparable from form, how semantics is of a piece with syntax, how inextricable pattern and matter are from each other

Introduction: A Musical-logical Offering

Page: 53 (6.46%) @ 20 Mar 2025 07:21:19 PM

There is one canon in the Musical Offering which is particularly unusual. Labelled simply "Canon per Tonos", it has three voices. The uppermost voice sings a variant of the Royal Theme, while underneath it, two voices provide a canonic harmonization based on a second theme. The lower of this pair sings its theme in C minor (which is the key of the canon as a whole), and the upper of the pair sings the same theme displaced upwards in pitch by an interval of a fifth. What makes this canon different from any other, however, is that when it concludes-or, rather, seems to conclude-it is no longer in the key of C minor, but now is in D minor. Somehow, Bach has contrived to modulate (change keys) right under the listener's nose. And it is so constructed that this "ending" ties smoothly onto the beginning again; thus, one can repeat the process and return in the key of E, only to join again to the beginning. These successive modulations lead the car to increasingly remote provinces of tonality, so that after several of them, one would expect to be hopelessly far away from the starting key. And yet magically, after exactly six such modulations, the original key of C minor has been restored! All the voices are exactly one octave higher than they were at the beginning, and here the piece may be broken off in a musically agreeable way. Such, one imagines, was Bach's intention; but Bach undoubtedly also relished the implication that this process could go on ad infinitum, which is perhaps why he wrote in the margin, "As the modulation rises, so may the King's Glory." To emphasise its potentially infinite aspect, I like to call this the "Endlessly Rising Canon

Page: 53 (6.46%) @ 20 Mar 2025 07:24:36 PM

In this canon, Bach has given us our first example of the notion of Strange Loops. The "Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started. (Here, the system is that of musical keys.)

Page: 64 (7.80%) @ 20 Mar 2025 07:56:08 PM

Russell and Whitehead did subscribe to this view, and accordingly, Principia Mathematica was a mammoth exercise in exorcising Strange Loops from logic, set theory, and number theory. The idea of their system was basically this. A set of the lowest "type" could contain only "objects" as members, not sets. A set of the next type up could only contain objects or sets of the lowest type. In general, a set of a given type could only contain sets of a lower type or objects. Every set would belong to a specific type. Clearly, no set could contain itself because it would have to belong to a type higher than its own type. Only "run-of-the-mill" sets exist in such a system; furthermore, old R—the set of all run-of-the-mill sets—no longer is considered a set at all, because it does not belong to any finite type. To all appearances, then, this theory of tybes, which we might also call the "theory of the abolition of Strange Loops", successfully rids set theory of its paradoxes, but only at the cost of introducing an artificial-seeming hierarchy, and of disallowing the formation of certain kinds of sets such as the set of all run-of-the-mill sets. Intuitively, this is not the way we imagine sets.

Page: 65 (7.92%) @ 20 Mar 2025 08:00:29 PM

Now in set theory, which deals with abstractions that we don't use all the time, a stratification like the theory of types seems acceptable, even if a little strange—but when it comes to language, an all-pervading part of life, such stratification appears absurd.

Page: 66 (8.04%) @ 20 Mar 2025 08:11:09 PM

It is of course important to try to maintain consistency, but when this effort forces you into a stupendously ugly theory, you know something is wrong.

Page: 69 (8.40%) @ 21 Mar 2025 03:26:56 AM

Here, one runs up against a seeming paradox. Computers, by their very nature, are the most inflexible, desireless, rule-following of beasts. Fast though they may be, they are nonetheless the epitome of unconsciousness. How, then, can intelligent behavior be programmed? Isn't this the most blatant of contradictions in terms? One of the major theses of this book is that it is not a contradiction at all. One of the major purposes of this book is to urge each reader to confront the apparent contradiction head on, to savor it, to turn it over, to take it apart, to wallow in it, so that in the end the reader might emerge with new insights into the seemingly unbreachable gulf between the formal and the informal, the animate and the inanimate, the flexible and the inflexible

Two-part Intervention

Page: 88 (10.72%) @ 26 Mar 2025 08:29:49 AM

Here the narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the much-enduring Tortoise, and was writing in his notebook, which appeared to be nearly full. The Tortoise was saying, "Have you got that last step written down? Unless I've lost count, that makes a thousand and one. There are several millions more to come. And WOULD you mind, as a personal favour, considering what a lot of instruction this colloquy of ours will provide for the Logicians of the Nineteenth Century-WOULD you mind adopting a pun that my cousin the Mock-Turtle will then make, and allowing yourself to be renamed TAUGHT-US

Page: 90 (10.96%) @ 26 Mar 2025 08:36:28 AM

As is typical of rules of production, the statement establishes a causal connection between the theoremhood of two strings, but without asserting theoremhood for either one on its own.

Page: 91 (11.08%) @ 26 Mar 2025 08:43:47 AM

There is, incidentally, a fact about the pq-system which would enable us to say with confidence that it has a decision procedure, even before finding the addition criterion. That fact is that the pq-system is not complicated by the opposing currents of lengthening and shortening rules; it has only lengthening rules. Any formal system which tells you how to make longer theorems from shorter ones, but never the reverse, has got to have a decision procedure for its theorems. For suppose you are given a string. First check whether it's an axiom or not (I am assuming that there is a decision procedure for axiomhood- otherwise, things are hopeless). If it is an axiom, then iţ is by definition a theorem, and the test is over. So suppose instead that it's not an axiom. Then, to be a theorem, it must have come from a shorter string, via one of the rules. By going over the various rules one by one, you can pinpoint not only the rules that could conceivably produce that string, but also exactly which shorter strings could be its forebears on the "family tree". In this way, you "reduce" the problem to determining whether any of several new but shorter strings is a theorem. Each of them can in turn be subjected to the same test. The worst that can happen is a proliferation of more and more, but shorter and shorter, strings to test. As you continue inching your way backwards in this fashion, you must be getting closer to the source of all theorems— the axiom schemata. You just can't get shorter and shorter indefnitely; therefore, eventually either you will find that one of your short strings is an axiom, or you'll come to a point where you're stuck, in that none of your short strings is an axiom, and none of them can be further shortened by running some rule or other backwards. This points out that there really is not much deep interest in formal systems with lengthening rules only; it is the interplay of lengthening and shortening rules that gives formal systems a certain fascination.

Chapter 2: Meaning and Form of Mathematics

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in a language, when we have learned a meaning for a word, we then make new statements based on the meaning of the word. In a sense the meaning becomes active, since it brings into being a new rule for creating sentences. This means that our command of language is not like a finished product: the rules for making sentences increase when we learn new meanings. On the other hand, in a formal system, the theorems are predefined, by the rules of production. We can choose "meanings" based on an isomorphism (if we can find one) between theorems and true statements. But this does not give us the license to go out and add new theorems to the established theorems. That is what the Requirement of Formality in Chapter I was warning you of

Page: 95 (11.57%) @ 26 Mar 2025 08:59:18 AM

In a formal system, the meaning must remain passive; we can read each string according to the meanings of its constituent symbols, but we do not have the right to create new theorems purely on the basis of the meanings we've assigned the symbols. Interpreted formal systems straddle the line between systems without meaning, and systems with meaning. Their strings can be thought of as expressing" things, but this must come only as a consequence of the formal properties of the system.

Page: 99 (12.06%) @ 26 Mar 2025 10:53:23 AM

Multiplicative commutativity and associativity are just the assumptions that when you rotate the solid in various ways, the number of cubes will not change. Now these assumptions are not verifiable in all possible cases, because the number of such cases is infinite. We take them for granted; we believe them (if we ever think about them) as deeply as we could believe anything. The amount of money in our pocket will not change as we walk down the street, jostling it up and down; the number of books we have will not change if we pack them up in a box, load them into our car, drive one hundred miles, unload the box, unpack it, and place the books in a new shelf. All of this is part of what we mean by number.

Chapter 3: Figure and Ground

Page: 115 (14.01%) @ 26 Mar 2025 11:44:45 PM

There exist recursively enumerable sets which are not recursive.

Chapter 4: Consistency, Completeness and Geometry

Page: 135 (16.44%) @ 29 Mar 2025 01:43:16 PM

You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone. I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. For here it is true that si paullum a sunmo discessit, vergit ad imum. I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind…. I have travelled past all reefs of this infernal Dead Sea and have always come back with a broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness—aut Caesar aut nihil.

Page: 140 (17.05%) @ 30 Mar 2025 08:11:24 PM

The preceding example, in which some symbols could have interpretations while others didn't, is reminiscent of doing geometry in natural language, using some words as undefined terms. In such a case, words are divided into two classes: those whose meaning is fixed and immutable, and those whose meaning is to be adjusted until the system is consistent (these are the undefined terms). Doing geometry in this way requires that meanings have already been established for words in the first class, somewhere outside of geometry. Those words form a rigid skeleton, giving an underlying structure to the system; filling in that skeleton comes other material, which can vary (Euclidean or non-Euclidean geometry).

Formal systems are often built up in just this type of sequential, or hierarchical, manner. For example, Formal System I may be devised, with rules and axioms that give certain intended passive meanings to its symbols. Then Formal System I is incorporated fully into a larger system with more symbols—Formal System II. Since Formal System I's axioms and rules are part of Formal System II, the passive meanings of Formal System I's symbols remain valid; they form an immutable skeleton which then plays a large role in the determination of the passive meanings of the new symbols of Formal System II. The second system may in turn play the role of a skeleton with respect to a third system, and so on. It is also possible—and geometry is a good example of this—to have a system (e.g., absolute geometry) which partly pins down the passive meanings of its undefined terms, and which can be supplemented by extra rules or axioms, which then further restrict the passive meanings of the undefined terms. This is the case with Euclidean versus non-Euclidean geometry.

Page: 145 (17.66%) @ 30 Mar 2025 08:29:20 PM

What does it mean to say, as I did above, that "completeness is the maximal confirmation of passive meanings"? It means that if a system is consistent but incomplete, there is a mismatch between the symbols and their interpretations. The system does not have the power to justify being interpreted that way. Sometimes, if the interpretations are "trimmed" a little, the system can become complete

Chapter 5: Recursive Structures and Processes

Page: 171 (20.83%) @ 03 Apr 2025 11:42:25 PM

In the preceding example, I have introduced some basic terminology of recursion at least as seen through the eyes of computer scientists. The terms are push, pop, and stack (or push-down stack, to be precise), and they are all related. They were introduced in the late 1950s as part of IPL, one of the first languages for Artificial Intelligence.

Page: 183 (22.29%) @ 04 Apr 2025 12:51:40 AM

I should not keep you too much in the dark about the origin of these beautiful graphs. INT—standing for "interchange" comes from a problem involving "Eta-sequences", which are related to continued fractions. The basic idea behind INT is that plus and minus signs are interchanged in a certain kind of continued fraction. As a consequence, INT(INT(x)) =x. INT has the property that if x is rational, so is INT(x); if x is quadratic, so is INT(x). I do not know if this trend holds for higher algebraic degrees. Another lovely feature of INT is that at all rational values of x, it has a jump discontinuity, but at all irrational values of x, it is continuous.

Page: 183 (22.29%) @ 04 Apr 2025 12:53:12 AM

Gplot comes from a highly idealized version of the question, "What are the allowed energies of electrons in a crystal in a magnetic field?" This problem is interesting because it is a cross between two very simple and fundamental physical situations: an electron in a perfect crystal and an electron in a homogeneous magnetic field. These two simpler problems are both well understood, and their characteristic solutions seem almost incompatible with each other. Therefore, it is of quite some interest to see how nature manages to reconcile the two. As it happens, the crystal without-magnetic-held situation and the magnetic-field-without-crystal situation do have one feature in common: in each of them, the electron behaves periodically in time. It turns out that when the two situations are combined, the ratio of their two time periods is the key parameter. In fact, that ratio holds all the information about the distribution of allowed electron energies—but it only gives up its secret upon being expanded into a continued fraction.

Page: 185 (22.53%) @ 04 Apr 2025 12:54:38 AM

Gplot shows that distribution. The horizontal axis represents energy, and the vertical axis represents the above-mentioned ratio of time periods, which we can call "a". At the bottom, a is zero, and at the top, a is unity. When a is zero, there is no magnetic field. Each of the line segments making up Gplot is an "energy band" that is, it represents allowed values of energy. The empty swaths traversing Gplot on all different size scales are therefore regions of forbidden energy. One of the most startling properties of Gplot is that when a is rational (say plq in lowest terms), there are exactly q such bands (though when q is even, two of them "kiss" in the middle). And when a is irrational, the bands shrink to points, of which there are infinitely many, very sparsely distributed in a so-called "Cantor set", another recursively defined entity which springs up in topology. You might well wonder whether such an intricate structure would ever show up in an experiment. Frankly, I would be the most surprised person in the world if Gplot came out of any experiment. The physicality of Gplot lies in the fact that it points the way to the proper mathematical treatment of less idealized problems of this sort. In other words, Gplot is purely a contribution to theoretical physics, not a hint to experimentalists as to what to expect to see! An agnostic friend of mine once was so struck by Gplot's infinitely many infinities that he called it "a picture of God", which I don't think is blasphemous at all

Page: 195 (23.75%) @ 04 Apr 2025 01:20:06 AM

Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.

Chapter 6: The Location of Meaning

Page: 213 (25.94%) @ 05 Apr 2025 11:57:24 PM

This happens because our intelligence is not disembodied, but is instantiated in physical objects: our brains. Their structure is due to the long process of evolution, and their operations are governed by the laws of physics. Since they are physical entities, our brains run without being told how to run. So it is at the level where thoughts are produced by physical law that Carroll's rule-paradox breaks down; and likewise, it is at the level where a brain interprets incoming data as a message that the message-paradox breaks down.

Chapter 7: The Propositional Calculus

Page: 230 (28.01%) @ 06 Apr 2025 04:35:14 PM

You probably have noticed that each theorem, when interpreted, says something absolutely trivial and self-evident. (Sometimes they are so self-evident that they sound vacuous and-paradoxically enough-confusing or even wrong!) This may not be very impressive, but just remember that there are plenty of falsities out there which could have been produced-yet they weren't. This system—the Propositional Calculus—steps neatly from truth to truth, carefully avoiding all falsities, just as a person who is concerned with staying dry will step carefully from one stepping-stone in a creek to the next, following the layout of stepping-stones no matter how twisted and tricky it might be. What is impressive is that—in the Propositional Calculus—the whole thing is done purely typographically. There is nobody down "in there", thinking about the meaning of the strings. It is all done mechanically, thoughtlessly, rigidly, even stupidly.

Page: 239 (29.11%) @ 06 Apr 2025 08:43:44 PM

It is intended only for use in connection with mathematical concepts-which are themselves quite rigid. As a rather interesting example of this, let us make a derivation in which a very peculiar string is taken as a premise in a fantasy: <PA-P>, At least its semi-interpretation is peculiar. The Propositional Calculus, however, does not think about semi-interpretations; it just manipulates strings typographically—and typographically, there is really nothing peculiar about this string. Here is a fantasy with this string as its premise:

(1) [ (2) (3) (4) 196 (5) (6) (7 (8 (9) (10) (11) (12) (13) | <PA~P> P P <~Q-~P> <~PDQ Q (14) <<PA~P>DQ>

push premise separation separation push premise carry-over line 3 double-tilde pop fantasy contrapositive detachment (Lines 4,11) Pop fantasy Now this theorem has a very strange semi-interpretation: P and not P together imply Q Since Q is interpretable by any statenment, we can loosely take the theorem to say that "From a contradiction, anything follows"! Thus, in systems based on the Propositional Calculus, contradictions cannot be contained; they infect the whole system like an instantaneous global cancer.

Chapter 8: Typographical Number Theory

Page: 250 (30.45%) @ 07 Apr 2025 12:26:37 AM

2 plus 2 is not equal to 3: ~(SSO+SS0) = SSS0

Page: 253 (30.82%) @ 07 Apr 2025 12:38:26 AM

Now these three translations of "6 is even" are quite different strings, and it is by no means obvious that theoremhood of any one of them is tied to theoremhood of any of the others. (Similarly, the fact that --p-q----- was a theorem had very little to do with the fact that its "equivalent" string -p--q--- was a theorem. The equivalence lies in our minds, since, as humans, we almost automatically think about interpretations, not structural properties of formulas.)

Page: 258 (31.43%) @ 07 Apr 2025 01:01:35 AM

Someone might suggest the following way of constructing TNT: (1) Do not have any rules of inference; they are unnecessary, because (2) We take as axioms all true statements of number theory (as written in TNT notation). What a simple prescription! Unfortunately, it is as empty as one's instantaneous reaction says it is.

Page: 259 (31.55%) @ 07 Apr 2025 01:07:52 AM

This harks back to a celebrated statement by the German mathematician and logician Leopold Kronecker, archenemy of Georg Cantor: "God made the natural numbers; all the rest is the work of man."

Page: 266 (32.40%) @ 07 Apr 2025 01:25:51 AM

This kind of inconsistency, created by the opposition of (1) a pyramidal family of theorems which collectively assert that all natural numbers have some property, and (2) a single theorem which seems to assert that not all numbers have it, is given the name of inconsistency. An ω-inconsistent system is more like the at-the-outset-distasteful-but-in-the-end-acceptable non-Euclidean geometry.

Page: 270 (32.89%) @ 07 Apr 2025 01:48:15 AM

The mathematician's sense of tension is intimately related to his sense of beauty, and is what makes mathematics worthwhile doing.

Page: 271 (33.01%) @ 07 Apr 2025 01:53:20 AM

As it is now formulated, TNT has reached "critical mass" (perhaps a strange metaphor to apply to something called "TNT"). It is of the same strength as the system of Principia Mathematica; in TNT, one can now prove every theorem which you would find in a standard treatise on number theory. Of course, no one would claim that deriving theorems in TNT is the best way to do number theory. Anybody who felt that way would fall in the same class of people as those who think that the best way to know what 1000 × 1000 is, is to draw a 1000 by 1000 grid, and count all the squares in it… No, after total formalization, the only way to go is towards relaxation of the formal system. Otherwise, it is so enormously unwieldy as to be, for all practical purposes, useless. Thus, it is important to embed TNT within a wider context, a context which enables new rules of inference to be derived, so that derivations can be speeded up. This would require formalization of the language in which rules of inference are expressed—that is, the metalanguage. And one could go considerably further. However, none of these speeding-up tricks would make TNT any more powerful; they would simply make it more usable. The simple fact is that we have put into TNT every mode of thought that number theorists rely on. Embedding it in ever larger contexts will not enlarge the space of theorems; it will just make working in TNT-or in each "new, improved version"-look more like doing conventional number theory.

A Mu Offering

Page: 276 (33.62%) @ 07 Apr 2025 02:07:56 AM

That matter has troubled me quite a bit. But I think I have finally worked out an answer. It seems to me that you may begin approaching Zen through any path you know-even if it is completely antithetical to Zen. As you approach it, you gradually learn to stray from that path. The more you stray from the path, the closer you get to Zen.

Page: 276 (33.62%) @ 07 Apr 2025 02:10:39 AM

I doubt it. However, in my opinion, a delight in kõans comes a million times closer to real Zen than reading volume after volume about Zen, written in heavy philosophical jargon

Chapter 9: Mumon and Godel

Page: 294 (35.81%) @ 10 Apr 2025 12:35:29 AM

If any kõan serves to bewilder, this one does. And most likely, causing bewilderment is its precise purpose, for when one is in a bewildered state, one's mind does begin to operate nonlogically, to some extent. Only by stepping outside of logic, so the theory goes, can one make the leap to enlightenment. But what is so bad about logic? Why does it prevent the leap to enlightenment?

Page: 294 (35.81%) @ 10 Apr 2025 12:39:16 AM

At the core of dualism, according to Zen, are wordsjust plain words. The use of words is inherently dualistic, since each word represents, quite obviously, a conceptual category. Therefore, a major part of Zen is the fight against reliance on words. To combat the use of words, one of the best devices is the kõan, where words are so deeply abused that one's mind is practically left reeling, if one takes the kõans seriously. Therefore, it is perhaps wrong to say that the enemy of enlightenment is logic; rather, it is dualistic. verbal thinking. In fact, it is even more basic than that: it is perception. As soon as you perceive an object, you draw a line between it and the rest of the world; you divide the world, artificially, into parts, and you thereby miss the Way.

Page: 297 (36.18%) @ 10 Apr 2025 12:43:49 AM

If words are bad, and thinking is bad, what is good? Of course, to ask this is already horribly dualistic, but we are making no pretense of being faithful to Zen in discussing Zen-so we can try to answer the question seriously. I have a name for what Zen strives for: ism, Ism is an antiphilosophy, a way of being without thinking. The masters of ism are rocks, trees, clams, but it is the fate of higher animal species to have to strive for ism, without ever being able to attain it fully. Still, one is occasionally granted glimpses of ism

Page: 297 (36.18%) @ 10 Apr 2025 12:46:42 AM

A master was asked the question, "What is the Way?" by a curious monk. "It is right before your eyes," said the master. 254 "Why do I not see it for myself?" "Because you are thinking of yourself." "What about you: do you see it?" "So long as you see double, saying '1 don't', and 'you do, and so on, your eyes are clouded," said the master. "When there is neither 'I' nor 'You', can one see it?" "When there is neither '1' nor 'You', who is the one that wants to see it?"

Page: 299 (36.42%) @ 10 Apr 2025 02:41:05 AM

Chiyono studied Zen for many years under Bukkõ of Engaku. Still, she could not attain the fruits of meditation. At last, one moonlit night, she was carrying water in an old wooden pail girded with bamboo. The bamboo broke, and the bottom fell out of the pail. At that moment, she was set free. Chiyono said, "No more water in the pail, no more moon in the water."

Page: 301 (36.66%) @ 10 Apr 2025 02:45:49 AM

The Buddhist allegory of "Indra's Net" tells of an endless net of threads throughout the universe, the horizontal threads running through space, the vertical ones through time. At every crossing of threads is an individual, and every individual is a crystal bead. The great light of "Absolute Being" illuminates and penetrates every crystal bead; moreover, every crystal bead reflects not only the light from every other crystal in the net, but also every reflection of every reflection throughout the universe

Page: 307 (37.39%) @ 10 Apr 2025 03:07:07 AM

CENTRAL PROPOSITIoN: If there is a typographical rule which tells how certain digits are to be shifted, changed, dropped, or inserted in any number represented decimally, then this rule can be rep resented equally well by an arithmetical counterpart which in volves arithmetical operations with powers of 10 as well as addi tions, subtractions, and so forth. More briefly: Typographical rules for manipulating numerals are actually arithmetical rules for operating on numbers.

Chapter 10: Levels of Description, and Computer Systems

Page: 335 (40.80%) @ 13 Apr 2025 03:21:50 AM

There is one interesting difference between the way interpreters work and compilers work. A compiler takes input (a finished Algol program, for instance) and produces output (a long sequence of machine language instructions). At this point, the compiler has done its duty. The output is then given to the computer to run. By contrast, the interpreter is constantly running while the programmer types in one LISP statement after another, and each one gets executed then and there. But this doesn't mean that each statement gets first translated, then executed, for then an interpreter would be nothing but a line-by-line compiler. Instead, in an interpreter, the operations of reading a new line, "understanding" it, and executing it are intertwined: they occur simultaneously. Here is the idea, expanded a little more. Each time a new line of LISP is typed in, the interpreter tries to process it. This means that the interpreter jolts into action, and certain (machine language) instructions inside it get executed. Precisely uhich ones get executed depends on the LISP statement itself, of course. There are many JUMP instructions inside the interpreter, so that the new line of LISP may cause control to move around in a complex way -forwards, backwards, then forwards again, etc. Thus, each LISP statement gets converted into a "pathway" inside the interpreter, and the act of following that pathway achieves the desired effect. Sometimes it is helpful to think of the LISP statements as mere pieces of data which are fed sequentially to a constantly running machine language program (the LISP interpreter). When you think of things this way, you get a different image of the relation between a program written in a higher-level language and the machine which is executing it.

Ant Fugue

Page: 354 (43.12%) @ 14 Apr 2025 01:04:33 AM

"MU" is an ancient Zen answer which, when given to a question, UNASKS the question

Chapter 11: Brains and Thoughts

Page: 380 (46.29%) @ 15 Apr 2025 01:21:39 AM

Descriptions can be manufactured and manipulated in themselves. We can invent nonexistent people by making descriptions of them; we can merge two descriptions when we find they represent a single entity; we can split one description into two when we find it represents two things, not one--and so on. This "calculus of descriptions" is at the heart of thinking. It is said to be intentional and not extensional, which means that descriptions can "float" without being anchored down to specific, known objects. 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.

Page: 386 (47.02%) @ 15 Apr 2025 09:06:04 PM

The retinal image is coded in a straightforward way in the firing patterns of the neurons in the lateral geniculate, despite the fact that the neurons there are not arranged on a two-dimensional surface in the form of the retina, but in a three-dimensional block. So two dimensions get mapped onto three, yet the information is preserved: an isomorphism.

Page: 393 (47.87%) @ 15 Apr 2025 10:02:33 PM

It seems reasonable to think that the brush strokes of language are also brush strokes of thought, and therefore that symbols represent concepts of about this size. Thus, a symbol would be roughly something for which you know a word or stock phrase, or with which you associate a proper name

Page: 402 (48.96%) @ 18 Apr 2025 02:29:05 AM

Our facility for making instances out of classes and classes out of instances lies at the basis of our intelligence, and it is one of the great differences between human thought and the thought processes of other animals. Not that I have ever belonged to another species and experienced at first hand how it feels to think their way but from the outside it is apparent that no other species forms general concepts as we do, or imagines hypothetical worldsvariants on the world as it is, which aid in figuring out which future pathway to choose. For instance, consider the celebrated "language of the bees"information-laden dances which are performed by worker bees returning to the hive, to inform other bees of the location of nectar. While there may be in each bee a set of rudimentary symbols which are activated by such a dance, there is no reason to believe that a bee has an expandable vocabulary of symbols. Bees and other insects do not seem to have the power to generalize—that is, to develop new class symbols from instances which we would perceive as nearly identical

Chapter 12: Minds and Thoughts

Page: 420 (51.16%) @ 19 Apr 2025 11:59:40 PM

aberrant" kinds of thoughts listed above are composed, at rock bottom, completely out of beliefs or pieces of knowledge. That is, any weird and snaky indirect route breaks up into a number of non-weird, non-snaky direct stretches, and these short, straightforward symbol-connecting routes represent simple thoughts that one can rely on--beliefs and pieces of knowledge

Page: 426 (51.89%) @ 20 Apr 2025 05:09:42 PM

Thus, a chunked description of a brain state would give a catalogue of beliefs which would be evoked conditionally, dependent on circumstances. Since not all possible circumstances can be enumerated, one would have to settle for those which one thinks are "reasonable". Furthermore, one would have to settle for a chunked description of the circumstances themselves, since they obviously cannot and should not--be specified down to the atomic level! Therefore, one will not be able to make an exact, deterministic prediction saying which beliefs will be pulled out of the brain state by a given chunked circumstance. In summary, then, a chunked description of a brain state will consist of a probabilistic catalogue, in which are listed those beliefs which are most likely to be induced (and those symbols which are most likely to be activated) by various sets of "reasonably likely" cir cumstances, themselves described on a chunked level. Trying to chunk someone's beliefs without referring to context is precisely as silly as trying to describe the range of a single person's "potential progeny" without referring to the mate.

Page: 427 (52.01%) @ 20 Apr 2025 05:15:07 PM

There is no reason to expect that "I", or "the self", should not be represented by a symbol. In fact, the symbol for the self is probably the most complex of all the symbols in the brain. For this reason, I choose to put it on a new level of the hierarchy and call it a subsystem, rather than a symbol. To be precise, by "subsystem", I mean a constellation of symbols, each of which can be separately activated under the control of the subsystem itself. The image I wish to convey of a subsystem is that it functions almost as an independent "subbrain", equipped with its own repertoire of symbols which can trigger each other internally. Of course, there is also much communication between the subsystem and the "outside" world-that is, the rest of the brain. "Subsystem" is just another name for an overgrown symbol, one which has gotten so complicated that it has many subsymbols which interact among themselves. Thus, there is no strict level distinction between symbols and subsystems.

Chapter 13: BlooP and FlooP and GlooP

Page: 468 (57.00%) @ 24 Apr 2025 03:29:46 PM

For readers who wish to see an elegant and simple presentation of the Turing approach, I recommend the article by Hoare and Allison, mentioned in the Bibliography

Chapter 15: Jumping Out of the System

Page: 513 (62.48%) @ 02 May 2025 02:26:13 AM

Once this ability for self-reference is attained, the system has a hole which is tailor-made for itself; the hole takes the features of the system into account and uses them against the system.

Page: 515 (62.73%) @ 02 May 2025 02:48:49 AM

The first comment is about a set of his drawings, all of which are concerned with "the conflict between the flat and the spatial"; the second comment is about Dragon in particular. 1. Our three-dimensional space is the only true reality we know. The two-dimensional is every bit as fictitious as the four-dimensional, for nothing is flat, not even the most finely polished mirror. And yet we stick to the convention that a wall or a piece of paper is flat, and curiously enough, we still go on, as we have done since time immemorial, producing illusions of space on just such plane surfaces as these. Surely it is a bit absurd to draw a few lines and then claim: "This is a house", This odd situation is the theme of the next five pictures [including Dragon]. II. However much this dragon tries to be spatial, he remains completely lat. Two incisions are made in the paper on which he is printed. Then it is folded in such a way as to leave two square openings., But this dragon is an obstinate beast, and in spite of his two dimensions, he persists in assuming that he has three; so he sticks his head through one of the holes and his tail through the other.

Page: 517 (62.97%) @ 02 May 2025 02:54:24 AM

Against. For the very fact that we cannot write a program to do "Gödelizing" must make us somewhat suspicious that we ourselves could do it in every case. It is one thing to make the argument in the abstract that Gödelizing "can be done"; it is another thing to know how to do it in every particular case. In fact, as the formal systems (or programs) escalate in complexity, our own ability to "Gödelize" will eventually begin to waver

Page: 519 (63.22%) @ 02 May 2025 03:03:26 AM

An amusing way to see the incorrectness of Lucas' argument is to translate it into a battle between men and women… In his wanderings, Loocus the Thinker one day comes across an unknown object—a woman. Such a thing he has never seen before, and at first he is wondrously thrilled at her likeness to himself; but then, slightly scared of her as well, he cries to all the men about him, "Behold! I can look upon her face, which is something she cannot do-therefore women can never be like me!" And thus he proves man's superiority over women, much to his relief, and that of his male companions. Incidentally, the same argument proves that Loocus is superior to all other males, as well-but he doesn't point that out to them. The woman argues back: "Yes, you can see my face, which is something I can't do—but I can see your face, which is something you can't do! We're even." However, Loocus comes up with an unexpected counter: "I'm sorry, you're deluded if you think you can see my face. What you women do is not the same as what we men do- it is, as I have already pointed out, of an inferior caliber, and does not deserve to be called by the same name. You may call it 'woman-seeing'. Now the fact that you can 'womansee' my face is of no import, because the situation is not symmetric. You see?" "I womansee," womanreplies the woman, and the womanwalks away…
Well, this is the kind of "heads-in-the-sand" argument which you have to be willing to stomach if you are bent on seeing men and women running ahead of computers in these intellectual battles.

Page: 519 (63.22%) @ 02 May 2025 03:08:30 AM

However, there is a lesser ambition which it is possible to achieve: that is, one can certainly jump from a subsystem of one's brain into a wider subsystem.

Page: 520 (63.34%) @ 02 May 2025 03:14:19 AM

This drive to jump out of the system is a pervasive onc, and lies behind all progress in art, music, and other human endeavors. It also lies behind such trivial undertakings as the making of radio and television commercials. This insidious trend has been beautifully perceived and described by Erving Goffman in his book Frame Analysis: For example, an obviously professional actor completes a commercial pitch and, with the camera still on him, turns in obvious relief from his task, now to take real pleasure in consuming the product he had been advertising. This is, of course, but one exanıple of the way in which TV and radio commercials are coming to exploit framing devices to give an appearance of naturalness that (it is hoped) will override the reserve auditors have developed. Thus, use is currently being made of children's voices, presumably because these seem unschooled; street noises, and other effects to give the impression of interviews with unpaid respondents: false starts, flled pauses, byplays, and overlapping specch to sinulate actual conversation; and, follow ing Welles, the interception of a firn's jingle commercials to give news of its new product, alternating occasionally with interception by a public interest spot, this presumably keeping the faith of the auditor alive. The more that auditors withdraw to minor expressive details as a test of genuineness, the more that advertisers chase after them. What results is a sort of interaction pollution, a disorder that is also spread by the public relations consultants of political figures, and, more modestly, by micro-sociology.® Here we have yet another example of an escalating "TC-battle"—the antagonists this time being Truth and Commercials.

Chapter 17: Church, Turing, Tarski and Others

Page: 601 (73.20%) @ 05 May 2025 11:44:42 PM

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.

Page: 608 (74.06%) @ 06 May 2025 12:46:27 AM

Hardy describes what he perceived as Ramanujan's outstanding intellectual attributes:

With his memory, his patience, and his power of calculation, he combined a power of generalisation, 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.

The part of this passage which I have italicized seems to me to be an excellent characterization of some of the subtlest features of intelligence in general

Page: 615 (74.91%) @ 06 May 2025 03:08:28 PM

Historically, people have been naive about what qualities, if mechanized, would undeniably constitute intelligence. 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. If intelligence involves learning, creativity, and emotional responses, a sense of beauty, a sense of self, then there is a long road ahead, and it may be that these will only be realized when we have totally duplicated a living brain.

Page: 626 (76.25%) @ 06 May 2025 03:46:34 PM

It is for this reason that I chose to link beauty, in the Magnificrab, with truth, which we have seen is also one of the most intangible notions in all of metamathematics.

Chapter 18: Artificial Intelligence: Retrospects

Page: 655 (79.78%) @ 06 May 2025 05:24:15 PM

In any case, the trouble is not that problem reduction per se leads to failures; it is quite a sound technique. The problem is a deeper one: how do you choose a good internal representation for a problem? What kind of "space" do you see it in? What kinds of action reduce the "distance" between you and your goal in the space you have chosen? This can be expressed in mathematical language as the problem of hunting for an appropriate metric (distance function) between states. You want to find a metric in which the distance between you and your goal is very small

Page: 667 (81.24%) @ 06 May 2025 10:56:25 PM

This idea of a language in which false statements are ungrammatical is an old one, going back to Johann Amos Comenius, in 1633. It is very appealing because you have a crystal ball embodied in your grammar: just write down the statement you want to know about, and check to see if it is grammatical… Actually, Comenius went even further, for in his language, false statements were not only ungrammatical-they were inexpressible!

Page: 668 (81.36%) @ 06 May 2025 11:03:57 PM

Why is some music so much deeper and more beautiful than other music? It is because form, in music, is expressive—expressive to some strange subconscious regions of our minds. The sounds of music do not refer to serfs or city-states, but they do trigger clouds of emotion in our innermost selves; in that sense, musical meaning is dependent on intangible links from the symbols to things in the world—those "things", in this case, being secret software structures in our minds. No, great music will not come out of such an easy formalism as an ATN-grammar. Pseudomusic, like pseudo-fairy tales, may well come out—and that will be a valuable exploration for people to make—but the secrets of meaning in music lie far, far deeper than pure syntax

Chapter 19: Artificial Intelligence: Prospects

Page: 720 (87.70%) @ 08 May 2025 03:14:47 AM

Question: Will there be chess programs that can beat anyone? Speculation: No. There may be programs which can beat anyone at chess, but they will not be exclusively chess players. They will be programs of general intelligence, and they will be just as temperamental as people. "Do you want to play chess?" "No, I'm bored with chess. Let's talk about poetry." That may be the kind of dialogue you could have with a program that could beat everyone. That is because real intelligence inevitably depends on a total overview capacity—that is, a programmed ability to "jump out of the system", so to speak—at least roughly to the extent that we have that ability. Once that is present, you can't contain the program; it's gone beyond that certain critical point, and you just have to face the facts of what you've wrought

Chapter 20: Strange Loops, or Tangled Hierarchies

Page: 735 (89.52%) @ 10 May 2025 02:16:50 AM

Then a seeming anarchy takes over; but anarchy has its own kinds of rules, no less than does civilized society; it is just that they operate from the bottom up, not from the top down. A student of anarchy could try to discover rules according to which anarchic situations develop in time, and very likely, there are some such rules.

Page: 740 (90.13%) @ 10 May 2025 03:25:29 AM

Consistent or inconsistent, no one is exempt from the mystery of the self. Probably, we are all inconsistent. The world is just too complicated for a person to be able to afford the luxury of reconciling all of his beliefs with each other. Tension and confusion are important in a world where many decisions must be made quickly. Miguel de Unamuno once said, "If a person never contradicts himself, it must be that he says nothing." I would say that we all are in the same boat as the Zen master who, after contradicting himself several times in a row, said to the confused Doko, "I cannot understand myself."