Gödel, Escher, Bach (Part III)

[The following, except where otherwise noted, is by Douglas Hofstadter, from his book Gödel, Escher, Bach.]

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Achilles:  One day, while Joshu and a monk were standing together in the monastery, a dog wandered by. The monk asked Joshu, ‘Does a dog have Buddha-nature?’

Tortoise:  Whatever that is. So tell me ― what did Joshu reply?

Achilles:  ‘Mu’.

Tortoise:  ‘Mu’? What’s this ‘Mu’? What about the dog? What about its Buddha-nature? What’s the answer?

Achilles:  Oh, but ‘Mu’ is Joshu’s answer. By saying ‘Mu’, Joshu let the other monk know that only by not asking such questions can one know the answer to them.

Tortoise:  Joshu ‘unasked’ the question?

Achilles:  Exactly!”

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“In the 1940’s, the Dutch psychologist Adriaan de Groot made studies of how chess novices and chess masters perceive a chess situation. Put in their starkest terms, his results imply that chess masters perceive the distribution of pieces in chunks. There’s a higher-level description of the board than the straightforward ‘white pawn on K5, black rook on Q6’ type of description, and the master somehow produces such a mental image of the board. This was demonstrated by the high speed with which a master could reproduce an actual position taken from a game, compared with the novice’s plodding reconstruction of the position, after both of them had had five-second glances at the board. Highly revealing was the fact that masters’ mistakes involved incorrectly placing whole groups of pieces, which left the game strategically almost the same, but to a novice’s eyes, not at all the same. The clincher was to do the same experiment but with pieces randomly assigned to the squares on the board, instead of copied from actual games. The masters were found to be simply no better than the novices in reconstructing such random boards.

The conclusion is that in normal chess play, certain types of situation recur, and it’s 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. Nearly everyone is surprised to find out that in actual play, a master usually examines only a handful of possible moves! The trick is that his mode of perceiving the board is like a filter: he literally doesn’t 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.

The distinction can apply just as well to other intellectual activities ― for instance, doing mathematics. A gifted mathematician doesn’t usually think up and try out all sorts of false pathways to the desired theorem, as less gifted people might do; rather, he immediately just ‘smells’ the promising paths and takes them.”

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“Experts have a different sort of knowledge than novices do ― knowledge focused on higher-order patterns. As a consequence, experts can, in effect, think in larger units, tackling problems in big steps rather than small ones. This is evident, for example, in studies of chess players. Novice chess players think about a game in terms of the position of individual pieces; experts, in contrast, think about the board in terms of pieces organized into broad, strategic groupings (e.g. a kingside attack with pawns). This is made possible by the fact that the masters have a ‘chess vocabulary’ in which these complex concepts are stored as single memory chunks, each with an associated set of subroutines for how one should respond to the pattern. Some investigators estimate that the masters may have as many as 50,000 of these chunks in their memories, each representing a strategic pattern.” [Gleitman, Reisberg, & Gross]

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“Fortunately, one doesn’t have to know all about quarks to understand many things about the particles which they may compose. Thus, a nuclear physicist can proceed with theories of nuclei that are based on protons and neutrons, and ignore quark theories and their rivals. The nuclear physicist has a chunked picture of protons and neutrons ― a description derived from lower-level theories but which doesn’t require understanding the lower-level theories. Likewise, a chemist has a chunked picture of the electrons and their orbitals, and builds theories of small molecules, theories which can be taken over in a chunked way by the molecular biologist, who has an intuition for how small molecules hang together, but whose technical expertise is in the field of extremely large molecules and how they interact. Then the cell biologist has a chunked picture of the units which the molecular biologist pores over, and tries to use them to account for the ways that cells interact.

The point is clear. Each level is, in some sense, ‘sealed off’ from the levels above and below it. When thinking about the higher levels, the details of the lower ones can be abstracted away without issue.”

[1]

“Although there’s always some ‘leakage’ between the hierarchical levels of science, so that a chemist can’t afford to ignore lower-level physics totally, or a biologist to ignore chemistry totally, there’s almost no leakage from one level to a distant level. That’s why people can have intuitive understandings of other people without necessarily understanding the quark model. All that a person needs is a chunked model of how the highest level acts; and as all know, in everyday life, such models are very realistic and successful.”

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“There’s one significant negative feature of a chunked model: it usually doesn’t have exact predictive power. That is, we save ourselves from the impossible task of seeing people as collections of quarks (or whatever’s at the lowest level) by using chunked models: but of course such models only give us probabilistic estimates of how other people will feel, will react to what we say or do, and so on. In short, in using chunked high-level models, we sacrifice determinism for simplicity. Despite not being sure how people will react to a joke, we tell it with the expectation at they’ll do something such as laugh, or not laugh ― rather than, say, climb the nearest flagpole. A chunked model defines a ‘space’ within which behavior is expected to fall, and specifies probabilities of its falling in different parts of that space.”

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“This idea that the bottom level, though one hundred percent responsible for what’s happening is nonetheless irrelevant to what happens, sounds almost paradoxical, and yet it’s a commonly known truism. I’ll illustrate it with an example.

Consider the day when I first heard the fourth étude of Chopin’s Opus 25 on my parents’ record player, and instantly fell in love with it. Now suppose that my mother had placed the needle in the groove a millisecond later. One thing for sure is that all the molecules in the room would have moved completely differently. If you’d been one of those molecules, you would’ve had a wildly different life story. Thanks to that millisecond delay, you would’ve careened and bashed into completely different molecules in utterly different places, spun off in totally different directions, and on and on. But would any of that have made a difference to the experience of the kid listening to the music? No, not the tiniest difference. All that would have mattered was that Opus 25, number 4 got transmitted faithfully through the air, and that would most surely have happened ― no matter whether a second earlier or a second later.

Although the air molecules were crucial mediating agents for a series of high-level events involving a certain kid and a certain piece of music, their precise behavior wasn’t crucial. Indeed, saying it ‘wasn’t crucial’ is a ridiculous understatement. Those air molecules could’ve done exactly the same job in an astronomical number of different but humanly indistinguishable fashions.”

[Pannenkoek] “I’d never thought about that before. The closest concept to this in my mind is the ship of Theseus. It also reminds me of how it’s said that, after some number of years, every atom in your body has been replaced.”

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“A volcano is undeniably there, but who needs to talk about mountains and subterranean pressures and lava and such things? We can dispense with such emergent concepts altogether by shifting to the deeper level of atoms or elementary particles. The bottom line, at least for our physicist, is that emergent phenomena are just convenient shorthands that summarize a large number of deeper, lower-level phenomena; they’re never essential to any explanation. Reductionism ho!

The only problem is the enormous escalation in complexity when we drop all macroscopic terms and ways of looking at things. If we refuse to use any language that involves emergent phenomena, then we’re condemned to seeing only untold myriads of particles ― and we can’t draw neat boundary lines separating an entity such as a cell or a frog from the rest of the world in which it resides. The laws of particle physics don’t respect such notions as ‘frog’, ‘cell’, ‘gene’, ‘genetic code’, or even ‘amino acid’ ― they involve only particles, and larger macroscopic boundaries drawn for the convenience of thinking beings are no more relevant to them than voting precinct boundaries are to butterflies.

One gets into very hot water if one goes the fully reductionistic route. Not only do all the objects in ‘the system’ become microscopic and numerous, but also the system itself grows beyond bounds in space and time and becomes, in the end, the entire universe taken over all of time. There’s no comprehensibility left, since everything is shattered into a trillion trillion trillion invisible pieces that are scattered hither and yon. Reductionism is merciless.”

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“It’s much more natural for us to say that a war was triggered for religious or economic reasons than to try to imagine a war as a vast pattern of interacting elementary particles and to think of what triggered it in similar terms ― even though physicists may insist that that is the only ‘true’ level of explanation for it, in the sense that no information would be thrown away if we were to speak at that level.”

“We non-godlike beings are condemned not to speak at that level of no information loss. We necessarily simplify, and indeed, vastly so. But that sacrifice is also our glory. Drastic simplification is what allows us to reduce situations to their bare bones, to discover abstract essences, to put our fingers on what matters, to understand phenomena at amazingly high levels, to survive reliably in this world, and to formulate literature, music, and science.”

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“Let’s imagine a self-modifying game. Think of chess. But let’s invent a variation in which, on your turn, you can either make a move or change the rules. But how? At liberty? Can you turn it into checkers? Clearly such anarchy would be pointless. There must be some constraints. For instance, one version might allow you to redefine the knight’s move. Instead of being 1-and-then-2, it could be m-and-then-n, where m and n are arbitrary natural numbers; and on your turn you could change either m or n by ±1. So it could go from 1-2 to 1-3 to 0-3 to 0-4 to 0-5 to 1-5 to 2-5… Then there could be rules about redefining the bishop’s moves, and the other pieces’ moves as well. There could be rules about adding new squares, or deleting old squares…

Now we have two layers of rules: those which tell how to move pieces, and those which tell how to change the rules. So we have rules and metarules. The next step is obvious: introduce metametarules by which we can change the metarules. It’s not so obvious how to do this. The reason it’s easy to formulate rules for moving pieces is that pieces move in a formalized space: the checkerboard. If you can devise a simple formal notation for expressing rules and metarules, then to manipulate them will be like formally manipulating strings (sequences of characters), or even like manipulating chess pieces. To carry things to their logical extreme, you could even express rules and metarules as positions on auxiliary chess boards. Then an arbitrary chess position could be read as a game, or as a set of rules, or as a set of metarules, etc., depending on which interpretation you place on it. Of course, both players would have to agree on conventions for interpreting the notation.

Now we can have any number of adjacent chess boards: one for the game, one for rules, one for metarules, one for metametarules, and so on, as far as you care to carry it. On your turn, you may make a move on any one of the chess boards except the top-level one, using the rules which apply (they come from the next chess board up in the hierarchy). Undoubtedly both players would get quite disoriented by the fact that almost anything ― though not everything ― can change. By definition, the top-level chess board can’t be changed, because you don’t have rules telling how to change it. It’s inviolate. And let’s assume for this example that there’s more that’s inviolate: the conventions by which the different boards are interpreted, the agreement to take turns, and the agreement that each person may change one chess board each turn.

Now it’s possible to go considerably further in removing the pillars by which orientation is achieved. We begin by collapsing the whole array of boards into a single board. What’s meant by this? There’ll be two ways of interpreting the board: (1) as pieces to be moved; (2) as rules for moving the pieces. On your turn, you move pieces ― and inevitably you change rules! Thus, the rules dictate how the rules change. The distinction between game, rules, metarules, metametarules, has been lost. What was once a nice clean hierarchical setup has become a strange loop, or tangled hierarchy. The moves change the rules, the rules determine the moves, round and round the mulberry bush… Now, there are still different levels, though there’s no longer an inviolable top level. And more to the point, the distinction between ‘lower’ and ‘higher’ has been wiped out.

This will serve as a key metaphor for the brain/mind.”

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“In any system, however, there’s always some ‘protected’ level which is unassailable by the rules on other levels, no matter how tangled their interaction may be among themselves. An amusing riddle illustrates this idea:

There are three authors: A, B, and C. Now it happens that A exists only in a novel by B. Likewise, B exists only in a novel by C. And strangely, C, too, exists only in a novel by A, of course. Now, is such an ‘authorship triangle’ really possible?

Of course it’s possible. But there’s a trick… All three authors A, B, C, are themselves characters in another novel by H! You can think of the ABC triangle as a strange loop, or tangled hierarchy; but author H is outside of the space in which that tangle takes place ― author H is in an inviolate space. Although A, B, and C all have access ― direct or indirect ― to each other, and can do dastardly things to each other in their various novels, none of them can touch H’s life! If I were to draw author H, I’d represent him somewhere off the page. Of course, that would present a problem, since drawing a thing necessarily puts it onto the page… Anyway, H is really outside of the world of A, B, and C, and should be represented as being so.

The same applies to Escher’s Drawing Hands.

Here, a left hand draws a right hand, while at the same time, the right hand draws the left hand. Once again, levels which ordinarily are seen as hierarchical ― that which draws, and that which is drawn ― turn back on each other, creating a tangled hierarchy. But behind it all lurks the undrawn but drawing hand of M. C. Escher, creator of both the left and and the right hand. In my schematic version of the picture, Escher is outside of the two-hand space. The strange loop is at the top, and there’s the inviolate level below it, enabling it to come into being.”

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“Now let’s turn our attention to the brain and its neurons. Despite the complexity of its input, a single neuron can respond in only one of two very primitive ways ― by firing or not firing. This is a very small amount of information. Certainly, for large amounts of information to be carried or processed, many neurons must be involved. And therefore, one might guess that there exist larger structures/activities, composed from many neurons, which handle concepts, on a higher level. Let’s from now on refer to these hypothetical neural structures/activities ― whether they come in the form of pancakes, garden rakes, rattlesnakes, snowflakes, or even ripples on lakes ― as symbols.”

“But what’s the advantage of this high-level picture? Why is it better to say, ‘Symbols A and B triggered symbol C,’ than to say, ‘Neurons 183 through 612 excited neuron 75 and caused it to fire’? It’s better because symbols symbolize things, and neurons don’t. Symbols are the hardware realizations of concepts.”

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“It isn’t accurate to think of a symbol as simply ‘on’ or ‘off’. While this is true of neurons, it doesn’t carry upwards, to collections of them. In this respect, symbols are quite a bit more complicated than neurons ― as you might expect, since they’re made up of many neurons. The messages that are exchanged between symbols are more complex than the mere fact, ‘I’m now activated.’ That’s more like the neuron-level messages. Each symbol can be activated in many different ways, and the type of activation will be influential in determining which other symbols it tries to activate. How these intertwining triggering relationships can be represented in a pictorial manner ― indeed, whether they can be at all ― isn’t clear.”

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“There’s no reason to expect that ‘I’, or ‘the self’, shouldn’t be represented by a symbol. In fact, this is one of (if not the) most important symbol of all.”

“We aren’t born with an ‘I’ ― it emerges gradually as experience shapes our dense web of active symbols into a tapestry rich and complex enough to begin twisting back upon itself.” 

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“There exists within the cranium a whole world of diverse causal forces; what’s more, there are forces within forces within forces, as in no other cubic half-foot of universe that we know.” [2]

“The only way we can try 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’re struggling and having to come up with new concepts in our attempt to explain it.”

You could say that the human mind or consciousness is the highest of higher-order patterns.

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“What sorts of ‘rules’ could possibly capture all of what we think of as intelligent behavior? Certainly there must be rules on all sorts of different levels. There must be many ‘just plain’ rules. There must be ‘metarules’ to modify the ‘just plain’ rules; then ‘metametarules’ to modify the metarules, and so on. The flexibility of intelligence comes from the enormous number of different rules, and levels of rules. The reason that so many rules on so many different levels must exist is that, in life, a creature is faced with millions of situations of completely different types. In some situations, there are stereotyped responses which require ‘just plain’ rules. Some situations are mixtures of stereotyped situations ― thus they require rules for deciding which of the ‘just plain’ rules to apply. Some situations can’t be classified ― thus there must exist rules for inventing new rules… and on and on.”

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“Certainly it’s possible for a computer program to modify itself ― but such modifiability has to be inherent in the program to start with, so that can’t be counted as an example of ‘jumping out of the system’. No matter how a program twists and turns to get out of itself, it’s still following the rules inherent in itself. It’s no more possible for it to escape than it is for a human being to decide voluntarily not to obey the laws of physics. Physics is an overriding system, from which there can be no escape.”

[Pannenkoek] “This reminds me of the concept of the technological singularity.”

To paraphrase Hofstadter, the relationship the program has to itself is can never be exactly the same as the relationship the programmer has to the program, because the programmer is outside of the system.

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“There’s a fascinating connection between the problem of jumping out of the system and the quest for complete objectivity. When I read Jauch’s dialogues in his book, Are Quanta Real?, I found myself wondering why there were three characters participating: Simplico, Salviati, and Sagredo. Why wouldn’t two have sufficed: Simplicio, the educated simpleton, and Salviati, the knowledgeable thinker? What function does Sagredo have? Well, he’s supposed to be a sort of neutral third party, dispassionately weighing the two sides and coming out with a ‘fair’ and ‘impartial’ judgment.

It sounds very balanced, and yet there’s a problem: Sagredo is always agreeing with Salviati, not with Simplicio. How come ‘objectivity personified’ is playing favorites? One answer, of course, is that Salviati is enunciating correct views, so Sagredo has no choice but to take his side. But what then of fairness or ‘equal time’? By adding Sagredo, the author stacked the deck more against Simplicio, rather than less. Perhaps there should be added a yet higher level Sagredo ― someone who’ll be objective about this whole situation… You can see where it’s going. We’re getting into a never-ending series of ‘escalations in objectivity’, which have the curious property of never getting any more objective than at the first level: where Salviati is simply right, and Simplicio wrong.”

[Pannenkoek] “After all, it’s still just within a story that’s meant to convey the author’s point of view.”

“So the puzzle remains: why add Sagredo at all? The answer is that it gives the illusion of stepping out of the system, in some intuitively appealing sense.”

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“There are cases where only a rare individual will have the vision to perceive a system which governs many peoples’ lives, one which had never before even been recognized as a system. Then such people often devote their lives to convincing other people that the system really is there and that it ought to be exited from.”

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Tortoise:  Whenever you taste the ‘popping tonic’ (a drink which brings you up a meta-level), you feel a deep sense of satisfaction, as if you’d been waiting to taste it all your life.”

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“In Zen, too, we can see the preoccupation with the concept of transcending the system. Perhaps, self-transcendence is even the central theme of Zen. A student of Zen is always trying to understand more deeply what he is, by stepping more and more out of what he sees himself to be, by breaking every rule and convention which he perceives himself to be chained by ― needless to say, including those of Zen itself. Somewhere along this elusive path may come enlightenment. In any case, as I see it, the hope is that by gradually deepening one’s self-awareness, by gradually widening the scope of ‘the system’, one will in the end come to a feeling of being at one with the entire universe.”

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“To me, Zen is intellectual quicksand ― darkness, meaninglessness, chaos. It’s tantalizing and infuriating. And yet it’s humorous, refreshing, enticing. Zen has its own special kind of meaning, brightness, and clarity. One of the basic tenets of Zen Buddhism is that there’s no way to characterize what Zen is. No matter what verbal space you try to enclose Zen in, it resists, and spills over.

In general, the Zen attitude is that words and truth are incompatible, or at least that no words can capture truth. (Though we’re making no pretense of being faithful to Zen in discussing Zen).”

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“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’s 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.”

[Pannenkoek] “A way of being without thinking? So there’s more than one?” I think specifically he’s contrasting it with dreamless sleep and death.

Ism reminds me of the empty set.

I can also remember another author directly rejecting Hofstadter’s sentiment above: “Enlightenment means rising above thought, not falling back to a level below it ― the level of an animal or a plant. If that were the price, it would not be worth attaining. Rather, enlightenment is consciousness without thought; thought is only one small aspect of consciousness.” [3]

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“The Zen master Mumon Yamada penetrated into the Mystery of the Undecidable as clearly as any mathematician, in a concise poem:

‘Does a dog have Buddha-nature?
You must answer this question.
And if you answer yes or no,
you lose your own Buddha-nature.
So, what’s your answer?’

This type of paradox is quite characteristic of Zen. It’s an attempt to ‘break the mind of logic’.”

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“A formal system will give you some truths, but no formal system ― no matter how powerful ― can lead to all truths. So the dilemma of mathematicians is, what else is there to rely on but formal systems? And the dilemma of Zen is, what else is there to rely on but words?

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“Zen minds revel in this irreconcilability. Over and over again, they face the conflict between the Eastern belief, ‘The world and I are one, so the notion of my ceasing to exist is a contradiction in terms’ (my verbalization is undoubtedly too Westernized ― apologies to Zenists), and the Western belief, ‘I’m just part of the world, and I will die, but the world will go on without me.’”

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Citations

  1. Randall Munroe
  2. Roger Sperry, collected by Hofstadter
  3. Eckhart Tolle
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