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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
In the last-but-one of those posts I discussed the notion, which Hofstadter seems to suggest, that maybe God can be thought of as the implicit meaning, the inner message, the "seed-genotype" of reality, where reality can be considered to be the "phenotype" that emerges as the "seed-genotype" — God — is worked out fully over the course of time.
The analogy here is to the relationship of DNA in the nucleus of a cell to the body of an organism. The DNA in each cell is the genotype, and the body with all its various hereditary characteristics is the phenotype.
In the last post I extended that thought by suggesting (with Hofstadter, I dare to imagine) that not all truths about reality can be "derived" in a mechanical way of a sort that is appropriate to working out a formal system of logical implication. Such a formal system is Hofstadter's own "Typographical Number Theory" or "TNT." Intelligence, however, can jump out of TNT or any other formal reasoning system and twig to these "unreachable" truths in a more intuitive way.
All formal reasoning systems, or at least those of palpable power, are thus "incomplete," in the sense that there are always going to be truths that they "ought to" be able to derive, but can't. This is what the Austrian mathematician Kurt Gödel proved in 1931. The only loophole to Gödel's Incompleteness Theorem, it seems, is that a formal system that is "inconsistent" — wherein theorems of the form x and ~x, where the former means "x is true" and the latter means "x is not true," co-occur — can in fact be complete. Not a very enticing loophole, that.
The reader is apt to lose sight of notions like those two — God as the seed of reality and intelligence as perceiving truths that are not formally provable — as Hofstadter goes deeper and deeper into the mechanics of formal systems of proof. Or, rather, formal systems of derivation, since proofs are not, strictly speaking, what formal systems do. Proofs are things that we do when we are engaged with informal systems, such as mathematical number theory itself. When number theory per se is rigorously formalized, it becomes a thing like Typographical Number Theory ... and proofs necessarily give way to derivations.
See? In coping with that last paragraph, you probably lost track of the fact that it began with notions about God, reality, and truth. By the time you finished parsing the paragraph's later clauses relating to formal and informal systems, proofs, and derivations, all the stuff about God, reality, etc., had very likely evaporated from your awareness.
That is exactly the position I find myself in at this juncture in my reading of the book: Hofstadter's Chapter VIII, on Typographical Number Theory or TNT, which is dedicated to showing how to construct a formal system in order to express and derive (many of) the truths of mathematical number theory from just five axioms.
For example,
∀a:(a + 0) = a
is an axiom that expresses the meaning "For all natural numbers, represented by a, a plus zero equals a." (A natural number is any whole number, including zero, that is not negative.)
From that and four other axiomatic starter strings can be derived, by the judicious application of a few powerful yet mechanical rules, a wealth of other strings, which together form the body of theorems of TNT. Hofstadter goes to some lengths to show that (in the absence of one particular rule he brings in expressly to remedy the situation) one of the strings that cannot be derived is, surprisingly,
∀a:(0 + a) = a
which means "For all natural numbers a, zero plus a equals a."
In other words, the formal string above which contains a + 0 is not identical to that which contains 0 + a. Hence it is conceivable that one of them is a theorem of TNT and the other one isn't. Never mind that we interpret the two expressions as meaning the same thing, mathematically speaking. Interpretations are something we do outside the confines of the formal system itself. A formal system qua formal system has no such interpretive content whatsoever! All it has, at bottom, is form, and rules for the generation of new forms based on old.
The specification of the extra rule saves TNT from something called ω-incompleteness, which means the formal system cannot derive a summarizing theorem to capture an infinitude of specific examples of what the theorem ought to "say," all of which are indeed theorems that the system is able to derive. (ω, by the way, is the Greek letter "omega.")
ω-incompleteness is not the same as incompleteness-without-the-ω, which happens when the formal system cannot derive as a theorem something we otherwise "know" is true. For example, if TNT were not as well-designed as it is, it might not be able to derive the TNT string for "There are infinitely many prime numbers." As it is, this truth of number theory, long known to mathematicians as Euclid's Theorem, since Euclid first proved it, can be derived within TNT — but Hofstadter says (p. 228) it would "probably have doubled the length of the book" to have shown the derivation.
So TNT is not incomplete with respect to Euclid's Theorem, and whether it is incomplete or not with respect to other "known" truths of number theory is apparently immaterial at this stage of the discussion. (It is in fact incomplete, Gödel showed — but for now Hofstadter wants his reader to focus on the notion that it might be complete, irrespective of whether or not it is in any way ω-incomplete. Got that?)
In some cases, Hofstadter shows, it is even possible to intentionally leverage ω-incompleteness into a (desirable, if offputting) ω-inconsistency: the "obvious" summarizing theorem is negated, quite on purpose, in the formal system order in order to allow for (in Hofstadter's image which I take from p. 223) "supernatural" numbers which do not necessarily obey the "obvious" formulas.
This type of inconsistency is not really as problematic as it might seem to the lay person. It is akin to that which separates non-Euclidean geometry from the straight-up Euclidean kind we learn in school. Non-Euclidean geometry is essential to some areas of science, and so we see we must steel ourselves to, and even welcome, various kinds of ω-inconsistency in the formal systems which underpin math and science.
We will presently be told, as the percipient reader already knows, that there are real inconsistencies that inexorably plague formal systems, however, and they can indeed be seen as problematic. In fact, Kurt Gödel showed that extirpating such inconsistencies completely comes at the cost of rendering the formal system (if it purports to be as powerful as Typographical Number Theory) incomplete-without-the-ω.
When wading through material such as this, the reader can easily lose sight of the forest for the trees. This is particularly true because the forest seems (at least to me, at this juncture) to pose a bit of a paradox in its broader outlines.
One of the paradoxes stems from the fact that Hofstadter seems to be at one and the same time extolling the beauty of formal systems and disparaging them as ineluctably inconsistent and/or incomplete.
TNT affords Hofstadter an elegant, 56-line derivation of a theorem that represents the number-theoretical truth that every number "commutes" with every other, meaning that it doesn't matter in which order two numbers bracket a plus sign; their sum is the same. Hofstadter breaks the lengthy derivation up into chunks whose last lines represent points of tension and resolution in the flow of the logic.
To wit, when a preliminary, subsidiary derivation augurs the final coda in the main derivation, a sense of inevitability is set up in one's mind. The situation is much like when music sets up an expectancy of eventual resolution at the keynote.
This alternation of tension and resolution is "typical of the structure," Hofstadter writes,
... not only of formal derivations, but also of informal proofs. The mathematician's sense of tension is intimately related to his sense of beauty, and it is what makes mathematics worthwhile doing [p. 227].
He then adds:
... that in TNT itself there seems to be no reflection of these tensions. In other words TNT doesn't formalize the notions of tension and resolution ... any more than a piece of music is a book about harmony and rhythm.
And then he leaves open the question, "Could one devise a much fancier typographical [i.e., formal] system which is aware of the tensions and goals inside derivations?"
If the answer to that question is no, one is forced to conclude that human intelligence is constituted so as to transcend all forms of purely mechanical reasoning, since our minds are in fact attuned to such tensions and resolutions. The next question then is, how so? How could a brain do things beyond mechanical and typographical rote — things that no machine, no matter how elaborate we make it, could ever hope to emulate?
On the other hand, if Hofstadter's question has an affirmative answer, and machines could be built that monitor the qualia of their own mechanical transactions, then those machines — we would be forced to say, would we not? — would be conscious. They would have subjective experience, just as we do.
Yet, though conscious like us, these super-machines would presumably be as unable as we are to prove — or in their case, derive — the truth of their own consciousness!
I think that is the general shape of the forest within which formal systems like TNT represent mere trees.
The nature of our reality is, then, one in which our consciousness transcends the mere mechanical operations of formal logic. It is deeply illogical, when you come right down to it. There is no path of reasoning by which one could derive the fact that we are conscious from anything else we know about ourselves and the world we live in. Or so it seems.
Now, it may or may not be the case that machine-based artificial intelligence can aspire to emulate our human version of consciousness. Yet, even if that AI goal is someday reachable, the brute unprovability of the fact of consciousness would remain. Machine-based consciousness would be just as deeply illogical as human consciousness is!
Accordingly, as Hofstadter lingers over the beauties of derivations in formal systems like TNT, I believe I can sense an inner tension to his discourse that can only be resolved by a conclusion like the one I just mentioned. Consciousness is deeply illogical, and therefore inexplicable.
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