Re: Can Computers Have Incomputable Concepts?

On Jun 24, 8:23 pm, t...@xxxxxxxxxxxxx wrote:

One can certainly write down axioms for arithmetical truth. See, for
example, Torkel Franzen's book "Inexhaustibility" for an accessible
exposition. When human beings reason mathematically about arithmetical
truth, any valid arguments they produce can be mimicked using formal
deductions from these axioms.

Not exactly. You will never have a sound formal system able to derive
everything that can be said of arithmetical truth, since that would
contradict Gödel's theorem. Certainly you can formally define
arithmetical truth and add an axiom to PA aserting that whatever PA
derives is an arithmetical truth,ou a system as Franzén considers. But
this won't give you a system able to tell you what formulas are
arithmetical truths and what are not, you know.

A Gödel or a consistency sentence for that system can be constructed
and it will be an arithmetical truth not derivable in the system. It's
part of the phenomenon of inexhaustibility Franzén studies.

The possibility of a system able to prove whatever humans can
eventually come to know about arithmetical truth is neither proved nor
disproved, for all I know.



I think your puzzlement stems from the assumption that computable concepts
are somehow less problematic that uncomputable ones. But as far as the
kinds of questions you're currently asking are concerned, one can raise the
same concerns about computable concepts. Let's take the concept of a
"prime number." A computer program can be written that distinguishes prime
numbers from composite numbers. Does the computer now "have the concept of
a prime number"? If external behavior isn't enough, then it is just as
puzzling how (or whether) a computer or a person can "have a concept"
regardless of whether the concept is computable or not.


Yes, you're ultimately right on this. But I think that incomputable
concepts are more adequate to reveal this because it is them that
require an intensional non behavioral way of 'having' them.


Ah, yes, if you're using "strong AI" strictly in the way Searle meant it,
then my hypothetical computer program doesn't have strong AI. However, in
that case, arguments about computability don't have much to do with the
possibility of strong AI. And indeed, as far as I'm aware, Searle doesn't
make any appeal to uncomputable concepts to argue against strong AI.

Actually he doesn't. But he might have done. Famously he argued that
intentionality (to be distinguished from 'intensionality'), the
possession of a semantic dimension, distinguishes minds from
computers. Now, the link between intentionality or semantics and
intensional representation of concepts seems to suggest itself.

Regards

.