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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
- Two Dilemmas of Consciousness
In the last of those I talked about the fact that, as Hofstadter shows with great intellectual rigor, "our naive hopes of lassoing reality with our words, thoughts, and formal expressions are in vain. Moreover," I added to that, "they cannot be otherwise."
That is the lesson of the first of the two parts of Hofstadter's book. I am now in the early going of reading the second part. A bit further on, I can tell already, Hofstadter will give a yet more rigorous proof of how truth-as-knowable is ineluctably larger than the set of truths that can be proven to be true.
How strange it is, though. This intrinsic "incompleteness" — this shortfall of that which is rigorously provable, vis-à-vis the totality of all propositions that we otherwise are able to know to be true — can itself be proven with full analytic rigor!
The provability of the incompleteness of provability depends on a trick discovered by the mathematician Kurt Gödel. By means of it, any and all formal systems of logic can be morphed into theorems about ordinary, everyday numbers. Then number theory itself, Hofstadter shows, can be extended to include various theorems that refer to ... number theory itself!
That is, these "meta-theorems," like regular theorems such as A + B = B + A, qualify as statements about the arithmetic properties of "natural" numbers (the integers zero and above). After all, just because they also represent statements that say something about number theory itself doesn't mean they aren't equally interpretable as mere statements about numbers. This is because all theorems of number theory are statements about numbers, whatever else they may be statements about.
Once one has all that firmly in mind, then one can ask whether a particular meta-theorem — one that Hofstadter names G, and whose interpretation is "G [itself] is not a theorem" — is in fact a theorem of number theory.
Pay close attention: "G is not a theorem" could conceivably, yes, be a derivable theorem!
Yet, if we quite reasonably assume that number theory is free of such internal contradictions, then, according to Gödel's Incompleteness Theorem, the question of whether G is or is not a theorem is simply undecidable, at least within the confines of number theory per se. Hence number theory is, contrary to our naive expectations, incomplete with respect to the whole body of knowable truths about itself.
Because Gödel's trick can be used to turn any formal system of logic whatsoever into a body of number-theoretical constructs, all formal systems are likewise incomplete in the very same way.
At the same time, we whose minds operate outside the confines of such formal, mechanical systems of provability know perfectly well that G expresses a true statement: namely, that G is in fact not a theorem of number theory.
Hofstadter summarizes the situation thus:
A string [of the specific formal system called Typographical Number Theory] has been found; [this string, G] expresses, unambiguously, a statement about certain arithmetical properties of natural numbers; moreover, by reasoning outside the system we can determine not only that the statement is a true one, but also that the string fails to be a theorem of TNT [because if it was a theorem, TNT would contain an internal contradiction]. And thus, if we ask TNT whether the statement is true, TNT says neither yes nor no. (p. 272)
The mind thus transcends the merely mechanical or "typographical" workings that are the essence of all formal systems of logical provability. This is so despite the reasonable conjecture that at bottom, the mind is rooted in the deterministic workings of the brain, and those workings are presumably entirely "typographical."
Hofstadter uses the word "typographical" to express the idea that you can represent the workings of the brain, at a low neuronal level of operation, by the workings of any reasonably powerful formal system of logic such as his Typographical Number Theory. Such a system, however powerful it may be, does what it does by virtue of nothing but the rote manipulation of strings and symbols. It is nothing more than an (abstract) machine.
To wit, the strings of TNT and other mechanistic formal systems are made up entirely of symbols such as letters and numbers. Unambiguous rules determine which typographical manipulations are permissible within the system. By applying aptly chosen rules one at a time, starter strings called "axioms" can be turned into all manner of full-fledged theorems.
In addition to their vacuous significances within the system itself, these mechanistic theorems can have, outside the system per se, one or more true interpretations that present themselves quite readily to the mind. Still, all that is really going on within the formal system itself is that new strings are being mechanically derived on the basis of rules and old strings. Meanings and truths are extraneous to the inner workings of such formal, typographical systems.
Yet the mind deals in truth and meanings more readily than it is capable of dealing, computer-like, with symbol strings. The brain, which is at base mechanical — or so it would seem — is able to transcend the intrinsic limits of the mere mechanistic processing of symbol strings, à la formal systems.
String processing à la formal systems is presumably wholly analogous to what the brain does, and all that it does, at the low level of its neurons. But at a higher level, the symbols take on meanings that are simply unavailable at lower levels of processing.
What could this notion of the transcendence of mind over the matter of the brain mean? And how could it be so, that a mind is seemingly more than just the sum of its working parts?
Such questions are the principal topic of the second part of Hofstadter's book, at least as far as I have read into it to date. I'll return to them in the next post in this "Strange Loops" series.
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