Re: A modern view of the halting problem


Alex McDonald wrote:



I'm not a big fan of misapplied theorems, but neither am I a big fan of
hand-waving statements like "unexpected truths". I can't think of a
single example of an "unexpected truth" in all of mathematics, unless
by unexpected the writer means counter-intuitive. It's this kind of
semantic sloppiness that irritates me.

Oh well...
What can I say.
There is lots of anectodal evidence that humans can easily solve
problems that computers have been unable to solve, or find incredible
difficult to solve, or intractible to solve. Until I see a proof to the
contrary, I'll go with the concensous that Humans are more capable than
Turing complete machines.


Again, semantic sloppiness. All systems, even meat-based ones, are
bound by their limitations,

Absolutely. It just appears that the meat-based ones have fewer
restrictions. I'm not claiming that a meat-machine can compute anything
(there are definitely some functions that cannot be computed, period,
and I don't see how a meat CPU is going to solve those problems --
though they can, at least, recognize them (which TC machines can't do).

unless we allow for the supernatural. It's
just that the box limitations are (currently) greater than ours in some
respects. (Yet far superior in others; speed of calculation being an
obvious case.) I don't subscribe to the supernatural.

I agree. And there's nothing you've said here that contradicts the
issue that Humans aren't necessarily bound by the undecideability
problem.

There is no proof that a human being can compute any undecideable
problem, but the fact that no one has provided a proof demonstrating
that humans cannot solve such problems (or, at least, recognize them)
offers us a little hope. Computer programs have no such hope.

Hope of what? The four-colour theorem was proved by computer, and is
currently unproven (and is possibly unproveable) by humans.

Don't confuse intractibility with impossible.
Cheers,
Randy Hyde

.

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