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Saturday, June 12, 2010

This Gödel is Killing Me


I really wish I could take credit for that pun; but no...
Some years ago, J.R. Lucas formerly a mathematician and philosopher of Merton College, Oxford, wrote in "Minds, Machines and Gödel" (and later in The Freedom of the Will) that Gödel's Incompleteness Theorems imply that mechanistic theories of the mind are false. 
Gödel's theorem states that any consistent system strong enough to produce simple arithmetic contains unprovable, though perfectly meaningful formulae, some of which we, standing outside the system, can see to be true.

Example: Consider the statement G: {This formula is unprovable-within-the-system}. 
  • If G is false, then the formula is provable-in-the-system and G is true.  A contradiction.
  • If G is true, then G is unprovable-in-the-system.
  • If G is provable-in-the-system, then it is false; but if it is false then it must be provable; again a contradiction. 
  • If G is unprovable-in-the-system, then G is true. 
Thus we see that G must be true and must also be unprovable-in-the-system
(It could be argued that G might be true and unprovable in a system S, yet provable in some larger system S*.  But then there must be a Gödel sentence in S*, and so on.)
In "The Implications of Gödel's Theorem" Lucas notes that the theorems apply to "First-Order Arithmetic", i.e. to Elementary Number Theory formulated in "First-Order Logic", in which we have:
  1. the sentential connectives, negation, disjunction, conjunction, implication, (roughly equivalent to 'not', 'and/or', 'and', 'if') etc,
  2. the two quantifiers (Ax), "for all x," and (Vx), "for some x," ranging over individual variables
  3. a symbol (=) for identity.
So the theorems apply to any physical deterministic system.  A machine can be reduced to such a system by writing down all the rules that the machine employs.  Any such system contains a Gödel sentence.

A mechanistic system cannot recognize its Gödel sentence as true, because it can see truth only as a matter of provability. But human beings can see that Gödel-type sentences are true.

Therefore, human beings cannot be explained entirely in physical, deterministic terms. Nor can any physical system, like a computer, duplicate the human mind: since it would be unable to recognize its Gödel sentences as true. 
This is much condensed.  For details, refer to Lucas' "Gödelian Papers"


SFnal Implications

The argument means that "true" is a larger set than "provable."  If you don't believe it, try proving that the objective universe exists.  If you kicked a rock, you lose.  That is evidence only if the rock (and your boot, etc) exist, which is assuming that which is to be proved.  Some things must be taken on faith: the objective universe, the continuum hypothesis, etc.
Alas, a favorite SF trope: downloading one's mind into a computer is revealed as an impossibility.  Not merely as impossible as Faster Than Light (FTL) travel, but a higher order of impossibility.  It is possible that another physical theory may one day replace general relativity; but Gödel's Theorems are mathematical and thus proven with incontrovertible rigor.

An interesting implication is that even if the brain can be modeled by a mechanistic system, then to recognize its own Gödel sentences as true would require the mind to be something more than the brain. 
 
A second disappointment: The machine that thinks like a human is not possible.  Such a machine might be built to simulate any facet of human thinking, but cannot simulate every facet of human thinking.  It may simulate the brain, but not the mind.  It could not recognize its own Gödel sentences as true. 
But combine that with Searle's famous Chinese Room and we have a new take on the Turing Test.  
 
A third, but unrelated implication, first apparently noted by Jaki, is that the Theory of Everything in physics is unattainable.  That is, even if it is achieved, we cannot know afterward if we have achieved it.  Insofar as physics is spoken in the language of mathematics, it will be possible to say true but unprovable things in physics.  The good news is that there will always be new stuff to be learned. 



1 comment:

  1. I have written a fair amount about Gödel and Jaki so I encourage you to pick through my blog (search) for anything that catches your eye on this topic. G. Chaitin is a big fan of Gödel as well and I might even call him a neo-Gödelian in that he wants to run with the incompleteness so mathematicians can indulge in boundless theoretic freedom like never before. Interestingly, Wittgenstein had a low opinion of Gödel's "proofs," since, for him, they were just cliches in one language game. But as Fr. Jaki put it, with the wave of his hand, over coffee and brunch when I met him in Princeton, not long before he died, "Who cares what Wittgenstein thinks?"

    See you at Doc Feser's, heh!

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