From: hal@finney.org
Date: Thu Jan 17 2002 - 16:07:26 MST
Andrew Clough writes:
> I have a question about Penrose's Arguments regarding the Turing Halting
> Theorem. I see that the program T sub n of n can never successfully prove
> that it will never halt, when, in fact, a human can see and prove that it
> will never halt. However, I fail to see how the humans task is the same as
> the computer's task. If the analogous task, a human asked to create a
> proof(n) that they could not create the proof(n), were attempted, the human
> would fail just as completely as the program. Of course, a human would
> give up after a while, as I'd expect an AI to terminate that subroutine.
This is similar to the notion captured in the claim: "Penrose cannot
consistently believe in the truth of this sentence." Anyone of us can
see it's true, but not poor Penrose!
But actually Penrose's claim is not that people can prove contradictions,
but rather that they can prove all true theorems. Computers which operate
via strict logic cannot, because they get stuck on their Godel sentence,
T_n(n) in Penrose's notation. Humans never get stuck, according to
Penrose, therefore they don't have Godel sentences, therefore they do
not operate via logic as computers do. Hence computers based on logic
can never reproduce human thought.
IMO the flaw is in the first step, the claim that humans can prove all
mathematical truths. This is self-evidently absurd since most truths
are far too complex for the entire human race ever to understand (just
as most numbers are far too big for any human to comprehend).
Hal
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