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| Douglas R. Hofstadter's Gödel, Escher, Bach |
- Gödel, Escher, Bach
- Insight Through Isomorphism and Paradox
- A Picture of God?
- God as Genotype
- More on God as Genotype
- Hints As To the Nature of Reality
In the last of those I talked about human (and possibly, someday, machine) intelligence as able to know the truth of this unprovable statement: "I am conscious."
Consciousness is the brain's faculty of monitoring whatever other processes of awareness are going on inside it and knowing what it is like to undergo those processes. The processes are, at some level, presumably mechanical and "typographical," just like the formal systems Hofstadter so lovingly describes in his book. His Typographical Number Theory, or TNT, is one such formal system; it is an attempt to turn mathematical number theory into something completely mechanical, hopefully without flaw or omission.
Zen consciousness, on the other hand, is knowing that what we tell ourselves about our mental states, using words to describe what they are like, is necessarily a lie.
Or, rather, what we tell ourselves about reality, using expressions made of words, can be at once neither true nor false — in many cases a verbal expression's truth or falsity is ultimately undecidable, and moreover, whether it is true or false ist, to a practitioner of Zen, beside the point. Satori, or Zen consciousness, lies beyond all words and all the concepts embodied by words.
Hofstadter gives his readers the flavor of Zen awareness in Chapter IX, "Mumon and Gödel." Mumon, he says, was the thirteenth-century Zen master who wrote the Mumonkan ("The Gateless Gate"), in which many of the best kōans or Zen enigmas are to be found.
Here is a kōan from Mumon's collection (p. 251):
Shuzan held out his short staff and said: "If you call this a short staff, you oppose its reality. If you do not call it a short staff, you ignore the fact. Now what do you wish to call this?"
There is no "right" answer — that's the point! From a Zen perspective, you cannot capture the "reality" of anything — even something so prosaic as a short staff — by naming it, thus putting it in contradistinction to all which is not a short staff. Reality is entire. It cannot be subdivided and named this, this, and this. To say at any time "That is not this" is to shortchange the totality of what is real.
But there are brute facts about reality as well, and refusing to name them is equally "wrong." Naming, which is an act of separating reality into manageable chunks, is at once futile and absolutely indispensable.
"Thus," writes Hofstadter (pp. 252-3),
... words lead to some truth — some falsehood, perhaps, as well — but certainly not to all truth. Relying on words to lead you to the truth is like relying on an incomplete formal system to lead you to the truth. A formal system will give you some truths, but as we shall soon see, a formal system — no matter how powerful — cannot lead to all truths. The dilemma of mathematicians is: what else is there to rely on, but formal systems? And the dilemma of Zen people is: what else is there to rely on, but words? Mumon states the dilemma very clearly: "It cannot be expressed with words and it cannot be expressed without words."
I think it fair to say that Hofstadter, at some early point in his life as a mathematician, had implicit confidence that formal systems (or "axiomatic" systems, as they are also known) were a way around reality's obstinate refusal to be encompassed by words. If you began with a mathematical theory such as that of numbers qua numbers, and if you took that number theory out of the domain of fuzzy, slippery verbal expressions such as "the number of primes is infinite" — what exactly is meant by "infinite"? — and put it in the form of non-verbal (indeed, non-arithmetical) symbol strings that can be generated and manipulated by mechanical rules, you could (Hofstadter expected) capture truth — potentially all truth — simply by generating all the "theorems" of the formal system, one by one.
True, there would be truths that were unreachable by any particular formal system of number theory. A possible example Hofstadter gives in his Chapter III on "Figure and Ground" concerns the prime numbers. A number not divisible by any number except for itself and 1 is prime. For example, 7 is a prime number, because it has no divisors. All other numbers are composite. 6 is composite because 2 and 3 divide it.
Hofstadter describes a simple formal system, the "tq-system," which has an easy time of generating all the composite numbers. The question then is, is it fair to say that any number which it does not show to be composite is accordingly prime?
No, he answers. The "holes" in the tq-system's list of composites are only negatively defined — which isn't good enough, in a formal system. Hofstadter gives the pictorial analogy of a "figure" and its "ground." If something in the picture is not part of the object being limned by the artist, it must be part of the background. Right?
Not necessarily! In visual depictions by M.C. Escher and other artists which Hofstadter alludes to, the distinction between figure and ground is purely arbitrary. What is figure from one perspective serves just as well as ground from another.
So, is there a way to modify the tq-system so that primes become, not ground, but figure?
Turns out there is — and Hofstadter gives it on pages 73-74. Yet, he says, from the fact that figure and ground, positive and negative, are perfectly complementary in this one example, we are not entitled to generalize to a belief that figure and ground always carry precisely the same information — as do the negative of a photograph and a positive image developed from it!
Specifically, when it comes to formal systems,
There exist formal systems whose negative space (set of non-theorems) is not the positive space (set of theorems) of any [other] formal system. (p. 72)
Surprise! As a budding mathematician, Hofstadter says he found this upsetting of his initial intuitions — this obstinate refusal of reality to be tamely, predictably either this or that — quite astonishing.
It is as if, in the "picture" which is reality, there is figure, there is ground, and there is a no-man's land which is neither one. Hofstadter's Fig. 18 on p. 71 gives some idea what this picture looks like. It diagrams the "relationship between various classes of TNT strings."
The diagram contains a white tree of axioms and theorems to be found in Typographical Number Theory, and also a black tree of all their negations — which, as "non-axioms"/"non-theorems," are equivalent to falsehoods in the verbal world.
Surrounding the white and black trees is a large area, black in the neighborhood of the white tree and white in the neighborhood of the black. The black area surrounding the white tree represents truths of number theory that are "unreachable" by the formal TNT system, no matter how powerful it may be spruced up to be. The white area around the black tree depicts TNT's set of "unreachable" falsehoods. Together, these two areas contain all the "sentences" of TNT whose truth or falsity simply cannot be decided within the formal system.
Undecidability is very Zen. It is even very verbal, in the sense that sentences in human languages are typically to be suspected of allowing too much interpretive wiggle room to be declared absolutely true or absolutely false. But undecidability is something a naive young mathematician might hope to eliminate in formal systems which are designed to be "isomorphic" with — to capture the same truths as — mathematical theories.
Hofstadter shows that it is possible to "transfer the study of any formal system — in fact the study of all formal systems — into number theory" (p. 264). You just substitute for the original typographical symbols specified in the formal system — whatever symbols they may be — numbers which can be manipulated arithmetically, not just typographically. When you do that, you necessarily have a system that is subject to all of the benefits and constraints of number theory.
One of these constraints, as the Austrian mathematician Kurt Gödel proved in 1931, is incompleteness — some of the "sentences" of the formal system that has, by means of this symbol-substitution process that is so aptly called "Gödel-numbering," been turned into an arithmetical "isomorph" are necessarily undecidable in their truth or falsity.
Once you twig to how Gödel-numbering unfailingly maps any formal system into the domain of number theory, and once you understand that number theory is ineluctably incomplete in terms of the decidability of some of its well-formed "sentences," you lose hope that all truth is somehow expressible. Words fall short. Numbers fall short. Well-formed strings of symbols in formal systems fall short.
And all fall short in the same way, in fact. As Mumon wrote of reality, "It cannot be expressed with words and it cannot be expressed without words."
The two dilemmas of consciousness Hofstadter speaks of — the one pertaining to Zen people and the other pertaining to mathematicians — are really one. This is why the possibility of Gödel-numbering any formal system, which Hofstadter calls "the simple observation ... at the heart of Gödel's method" of proving his Incompleteness Theorem, has "an absolutely shattering effect" (p. 264).
It is shattering because it means our naive hopes of lassoing reality with our words, thoughts, and formal expressions are in vain. Moreover, they cannot be otherwise.
If heaven is a perfect place where mind can lasso matter and make it behave, then we are barred from it, no matter how cleverly we use our brains. We have a better chance of finding heaven by "stepping outside of logic" (p. 251), or by seeking to "break the mind of logic" (p. 249). That is satori. That is Zen consciousness.
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