# Douglas Hofstadter - Gödel, Escher, Bach (Highlights)  ## Metadata **Review**:: [readwise.io](https://readwise.io/bookreview/35460217) **Source**:: #from/readwise #from/api_book **Zettel**:: #zettel/fleeting **Status**:: #x **Authors**:: [[Douglas Hofstadter]] **Full Title**:: Gödel, Escher, Bach **Category**:: #books #readwise/books **Category Icon**:: 📚 **Highlighted**:: [[2023-12-16]] **Created**:: [[2023-12-16]] ## Highlights ### Preface to GEB's Twentieth-anniversary Edition - 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 1) ^642362962 - GEB was inspired by my long-held conviction that the "strange loop" notion holds the key to unraveling the mystery that we conscious beings call "being" or "consciousness". (Page 1) ^642362963 - The Gödelian strange loop that arises in formal systems in mathematics (i.e., collections of rules for churning out an endless series of mathematical truths solely by mechanical symbol-shunting without any regard to meanings or ideas hidden in the shapes being manipulated) is a loop that allows such a system to "perceive itself", to talk about itself, to become "self-aware", and in a sense it would not be going too far to say that by virtue of having such a loop, a formal system acquires a self. (Page 1) ^642362964 - When and only when such a loop arises in a brain or in any other substrate, is a *person* — a unique new "I" — brought into being. (Page 1) ^642362965 - How constraints can enhance creativity. (Page 1) ^642362966 ### Introduction: A Musico-Logical Offering - One could probably liken the task of improvising a six-part fugue to the playing of sixty simultaneous blindfold games of chess, and winning them all. To improvise an eight-part fugue is really beyond human capability. (Page 7) ^642362968 #analogy - 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. (Page 10) ^642362969 #definition - And in particular, the Strange Loop is one of the most recurrent themes in Escher's work. (Page 11) ^642362970 - The idea of a code, in other words, is at the heart of his construction. In the Godel Code, usually called " Godel-numbering" , numbers are made to stand for symbols and sequences of symbols. That way, each statement of number theory, being a sequence of specialized symbols, acquires a Godel number, something like a telephone number or a license plate, by which it can be referred to. And this coding trick enables statements of number theory to be understood on two different levels: as statements of number theory, and also as statements about statements of number theory. (Page 18) ^642362971 - Without doubt, Strange Loops involving rules that change themselves, directly or indirectly, are at the core of intelligence. (Page 27) ^642362972 ### Chapter I: The MU-puzzle - 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. (Page 37) ^642362974 ### Chapter II: Meaning and Form in Mathematics - 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. (Page 48) ^642362976 - 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. (Page 50) ^642362977 ### Chapter III: Figure and Ground - The reason hesitating is that the holes are only negatively defined-they are the things that are left out of a list which is positively defined. (Page 67) ^642362979 ### Contracrostipunctus - But if there is a defect anywhere, is not in THEM, but in your expectations of what they should b able to do! And the Crab was just full of such unrealistic expectations. (Page 77) ^642362981 ### Chapter IV: Consistency, Completeness, and Geometry - If, however, you can divest yourself of those preconceived images, and merely let a "straight line" be something which satisfies the new propositions, then you have achieved a radically new viewpoint. (Page 92) ^642362983 - By changing the interpretations, we regained consistency! It now becomes clear that consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system. (Page 94) ^642362984 ### Chapter V: Recursive Structures and Processes - Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law. (Page 152) ^642362986 - 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? [...] This kind of " tangled recursion" probably lies at the heart of intelligence. (Page 152) ^642362987 #keypoint ### Chapter VI: The Location of Meaning - One view says that so much of the information is outside the DNA that it is not reasonable to look upon the DNA as anything more than a very intricate set of triggers, like a sequence of buttons to be pushed on a jukebox; another view says that the information is all there, but in a very implicit form. (Page 161) ^642362989 - The issue is whether any message has, per se, enough compelling inner logic that its context will be restored automatically whenever intelligence of a high enough level comes in contact with it. (Page 164) ^642362990 - Here, the trap is the idea that before you can understand any message, you have to have a message which tells you how to understand that message; in other words, there is an infinite hierarchy of levels of messages, which prevents any message from ever getting understood. (Page 170) ^642362991 - 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. (Page 170) ^642362992 - It seems that the clarity of the outer message resides in the sheer length of the message. This is not unexpected; it parallels precisely what happens in deciphering ancient texts. Clearly, one's likelihood of success depends crucially on the amount of text available. (Page 174) ^642362993 Also the reason why LLM works? - But there strange reversal here: intelligence loves patterns and balks at randomness For most people, the randomness in Cage's music requires much explanation. (Page 175) ^642362994 ### Chapter VII: The Propositional Calculus - Once again, we are up against the issue which Lewis Carroll so sharply set forth in his Dialogue: you can't go on defending your patterns of reasoning forever. There comes a point where faith takes over. (Page 192) ^642362996 - Even if a system can "think about itself', it still is not outside itself. You, outside the system, perceive it differently from the way it perceives itself. So there still is a metatheory–a view from outside-even for a theory which can "think about itself' inside itself. (Page 194) ^642362997 ### Chapter IX: Mumon and Gödel - But in general, the Zen attitude is that words and truth are incompatible, or at least that no words can capture truth. (Page 246) ^642362999 - 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 you thereby miss the Way. (Page 251) ^642363000 ### Chapter X: Levels of Description, and Computer Systems - The conclusion is that in normal chess play, certain types of situation recur–certain patterns–and it is to those high-level patterns that the master is sensitive. He thinks on a different level from the novice; his set of concepts is different. (Page 286) ^642363002 - The trick is that his mode of perceiving the board is like a filter: he literally does not see bad moves when he looks at a chess situation-no more than chess amateurs see illegal moves when they look at a chess situation. (Page 286) ^642363003 - The main cause of this level-confusion was that communication with all levels of the computer system was taking place on a single screen, on a single terminal. (Page 301) ^642363004 - In short, in using chunked high-level models, we sacrifice determinism for simplicity. (Page 306) ^642363005 ### … Ant Fugue - Certainly no one would insist that individual brain cells have to be intelligent beings on their own, in order to explain the fact that a person can have an intelligent conversation. (Page 315) ^642363007 - A description on that level would fall somewhere in between the low-level and symbol-level descriptions. It would contain a great deal of information about what is actually going on in specific locations throughout the colony, although certainly less than an ant-by-ant description, since teams consist of clumps of ants. A team-by-team description is like a summary of an ant-by-ant description. However, you have to add extra things which were not present in the ant-by-ant description-such as the relationships between teams, and the supply of various castes here and there. This extra complication is the price you pay for the right to summarize. (Page 326) ^642363008 - There is no natural mapping from the individual letters into the real world. The natural mapping occurs on a higher level between words, and parts of the real world. If you wanted to describe the book, therefore, you would make no mention of the letter level. (Page 326) ^642363009 - The activities of all symbols are strictly determined by the state of the full system in which they reside. Therefore, the full system is responsible for how its symbols trigger each other, and so it is quite reasonable to speak of the full system as the " agent" . As the symbols operate, the state of the system gets slowly transformed, or updated. But there are many features which remain over time. It is this partially constant, partially varying system which is the agent. (Page 327) ^642363010 ### Chapter XI: Brains and Thoughts - This is but one example of what is actually a common phenomenon-a sensation of something "crystallizing" in your mind at the moment of recognition, which takes place not when the light rays hit your retina, but sometime later, after some part of your intelligence has had a chance to act on the retinal signals. (Page 347) ^642363012 - The crucial step that needs to be taken is from a low-Ievel-neuron-by-neuron-description of the state of a brain, to a high-Ievel-module-by-module-description of the same state of the same brain. Or, to revert to the suggestive terminology of the Ant Fugue, we want to shift the description of the brain state from the signal level to the symbol, level. (Page 349) ^642363013 - Incidentally, the requirement that symbols should be able to pass sophisticated messages to and fro is probably sufficient to exclude neurons themselves from playing the role of symbols. (Page 350) ^642363014 - The next question-and an extremely important one it is, too concerns the nature and " size" of the concepts which are represented in the tin by single symbols. (Page 350) ^642363015 ### Chapter XII: Minds and Thoughts - What does it matter if two brains are isomorphic, or quasi-isomorphic, or not isomorphic at all? The answer is that we have an intuitive sense that, although other people differ from us in important ways, they are still "the same" as we are in some deep and important ways. It would be instructive to be able to pinpoint what this invariant core of human intelligence is, and then to be able to describe the kinds of "embellishments" which can be added to it, making each one of us a unique embodiment of this abstract and mysterious quality called "intelligence". (Page 375) ^642363017 - Warren Weaver, one of the first advocates of translation by computer, in the late 1940's: "When I look at an article in Russian, I say, 'This is really written in English, but it has been coded in some strange symbols. I will now proceed to decode.'" (Page 380) ^642363018 - From this follows an important conclusion: there is no information in the brain state itself which tells which route will be chosen. The external circumstances will play a large determining role in choosing the route. (Page 383) ^642363019 - What the brain state can provide, if properly read, is a conditional description of the selection of routes. (Page 383) ^642363020 - 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" circumstances, themselves described on a chunked level. (Page 384) ^642363021 ### Aria with Diverse Variations - If there WERE some arithmetical fact which were caused by an infinite collection of unrelated coincidences, then you could never give a finite proof for that truth. And that is ridiculous. (Page 398) ^642363023 - But there are certain properties of integers for which terminating tests can be proven to exist, and yet about which it can also be PROVEN that there is no way to predict in advance how long they will take. (Page 400) ^642363024 ### Chapter XIII: BlooP and FlooP and GlooP - In fact, it is widely believed that there cannot be any more powerful language for describing calculations than languages that are equivalent to FlooP. (Page 428) ^642363026 - This conclusion is so peculiar that it should cause us to investigate very carefully the pillars on which it rests. And one of them, you will recall, was our shaky assumption that there is a decision procedure which can tell terminating from nonterminating FlooP programs. (Page 429) ^642363027 - TNT is so powerful that not only are all primitive recursive predicates represented, but moreover all general recursive predicates are represented. (Page 430) ^642363028 ### Chapter XIV: On Formally Undecidable Propositions of TNT and Related Systems - 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. (Page 450) ^642363030 - One can see this kind of thing over and over again in physics, where a physicist uses a "nonreal" situation to learn about deeply hidden features of reality. (Page 456) ^642363031 - You fit your mathematics to the world, and not the other way around. (Page 457) ^642363032 ### Chapter XV: Jumping out of the System - No single scheme, no matter how complex, can name all ordinals. And from this, it follows that no algorithmic method can tell how to apply the method of Gödel to all possible kinds of formal systems. Aad unless one is rather mystically inclined, therefore one must conclude at any human being simply will reach the limits of his own ability to Gödelize at some point. From there on out, formal systems of that complexity, though admittedly incomplete for the Gödel reason, will have as much power as that human being. (Page 476) ^642363034 - 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 at best change some parts of itself by obeying its own instructions. This is reminiscent of the numerous paradoxical question, "Can God make a stone so heavy that he can't lift it? " (Page 478) ^642363035 ### Chapter XVI: Self-Ref and Self-Rep - Thus in this program, one string functions in two ways: first as program, and second as data. (Page 499) ^642363037 Lisp - Such recognition problems may remind you of the original, key problem about formal systems: How can you tell if a string has, or does not have, some property such as theoremhood? (Page 540) ^642363038 - One is that of cellular differentiation: how do different cells, sharing exactly the same DNA, perform different roles-such as a kidney cell, a bone marrow cell, and a brain cell? The other is that of morphogenesis ("birth of form"): how does intercellular communication on a local level give rise to large-scale, global structures and organizations (Page 544) ^642363039 ### Chapter XVII: Church, Turing, Tarski, and Others - 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. (Page 559) ^642363041 - Even Hardy-who would have been the first to deny that Ramanujan had any mystical powers-once wrote about one of Ramanujan's failures, "And yet I am not sure that, in some ways, his failure was not more wonderful than any of his triumphs." (Page 564) ^642363042 - Incidentally, one of the most obvious clues that no "hot line to God" is involved is the mere fact that when the numbers involved get bigger, the answers are slower in coming. (Page 567) ^642363043 - 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. (Page 573) ^642363044 - Perhaps they feel that the appreciation of beauty requires an element of irrationality, and therefore is incompatible with the very fiber of computers. (Page 575) ^642363045 - This illustrates how faultless functioning on one level may underlie symbol manipulation on a higher level-and the goals of the higher level may be completely unrelated to the propagation of Truth. (Page 575) ^642363046 - The overt behavior could appear either rational or irrational; but underneath it would be the performance of reliable, logical hardware. (Page 577) ^642363047 ### Chapter XVIII: Artificial Intelligence: Retrospects - The ineluctable core of intelligence is always in that next thing which hasn't yet been programmed. (Page 601) ^642363049 - What in one space looks like a retreat can in another space look like a revolutionary step forward. (Page 612) ^642363050 - But dogs in front of fences sometimes have a hard time doing that, especially when that bone is sitting there so close, staring them in the face, and looking so good. 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. (Page 612) ^642363051 - A pocket calculator does not store in its memory knowledge of how to add; that knowledge is encoded into its "guts" . (Page 616) ^642363052 - The answer is: it turns out that it is extremely useful to have the same information in several different forms for different purposes. (Page 616) ^642363053 - We see on Earth that there are two molecules, one of which is good for replication [DNA] and one of which is good for action [proteins]. (Page 617) ^642363054 - This is, it seems to me, a general principle: you get bored with something not when you have exhausted its repertoire of behavior, but when you have mapped out the limits of the space that contains its behavior. (Page 621) ^642363055 - If one path-in the tree fails to achieve the desired goal, then the PLANNER program will "backtrack" and try another route. "Backtracking" is the magic word as far as PLANNER is concerned. (Page 629) ^642363056 Step back - It is probably safe to say that writing a program which can fully handle the top five words of English-"the" , "of', "and" , "a" , and "to" -would be equivalent to solving the entire problem of AI, and hence tantamount to knowing what intelligence and consciousness are. (Page 629) ^642363057 - By contrast, in SHRDLU, all words were represented as programs. (Page 630) ^642363058 Functional programming - At first glance, it seems that there is too much structure in this program, as we don't like to think of the meaning of a simple phrase as explicitly containing loops, conditional tests, and other programming details. (Page 632) ^642363059 ### Chapter XIX: Artificial Intelligence: Prospects - The manufacture of "subjunctive worlds" happens so casually, so naturally, that we hardly notice what we are doing. We select from our fantasy a world which is close, in some internal mental sense, to the real world. We compare what is real with what we perceive as almost real. In so doing, what we gain is some intangible kind of perspective on reality. (Page 643) ^642363061 - When it is hard to build a template because the preprocessor finds too much diversity, that should serve as a clue that concepts on a higher level of abstraction are involved than the preprocessor knows about. (Page 657) ^642363062 - It is a common notion that randomness is an indispensable ingredient of creative acts. (Page 673) ^642363063 - What we see as randomness is often simply an effect of looking at something symmetric through a "skew" filter. An elegant example was provided by Salviati's two ways of looking at the number of π/4. Although the decimal expansion of π/4 is not literally random, it is as random as one would need for most purposes: it is "pseudorandom". Mathematics is full of pseudorandomness–plenty enough to supply all would-be creators for all time. (Page 673) ^642363064 - In fact, I have a favorite "meta-analogy", in which I liken analogies to chords. The idea is simple: superficially similar ideas are often not deeply related; and deeply related ideas are often superficially disparate. The analogy to chords is natural: physically close notes are harmonically distant (e.g., E-F-G); and harmonically close notes are physically distant (e.g., G-E-B). (Page 673) ^642363065 ### Chapter XX: Strange Loops, Or Tangled Hierarchies - In fact, machines get around the Tortoise's silly objections as easily as people do, and moreover for exactly the same reason: both machines and people are made of hardware which runs all by itself, according to the laws of physics. There is no need to rely on "rules that permit you to apply the rules" , because the lowest-level rules-those without any "meta'''s in front-are embedded in the hardware, and they run without permission. (Page 685) ^642363067 - Likewise, machines may someday have wills despite the fact that no magic program spontaneously appears in memory from out of nowhere (a "self-programmed program"). (Page 686) ^642363068 - A Tangled Hierarchy occurs when what you presume are clean hierarchical levels take you by surprise and fold back in a hierarchy-violating way. The surprise element is important; it is the reason I call Strange Loops "strange" . A simple tangle, like feedback, doesn't involve violations of presumed level distinctions. (Page 691) ^642363069 - Despite this argument, people have an intuitive sense of evidence. This is because-to repeat an old refrain-people have built-in hardware in their brains that includes some rudimentary ways of interpreting evidence. (Page 694) ^642363070 - Recall that the level-crossing which takes place at this exact translation point is what creates Gödel's incompleteness and the self-proving character of Henkin's sentence. I postulate that a similar level-crossing is what creates our nearly unanalyzable feelings of self. (Page 709) ^642363071 - Though no one of us will ever be able to step back far enough to see the "big picture", we shouldn't forget that it exists. (Page 710) ^642363072 ### Six-Part Ricercar - In any event, I find indirect self-reference a more general concept, and far more stimulating, than direct self-reference. Moreover, no reference is truly direct-every reference depends on SOME kind of coding scheme. It's just a question of how implicit it is. (Page 739) ^642363074 - Perhaps being a good improviser is incompatible with knowing how one does it. (Page 739) ^642363075