Re: Can Computers Have Incomputable Concepts?

In article <1182671039.695295.178560@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
LauLuna <laureanoluna@xxxxxxxx> wrote:
But I see you propose that having a concept tshould be understood as
being able of some behavior. The fact that we have undecidable
concepts suggests there is something else to it; it suggests having a
concept implies having some representation of it. For undecidable
concepts this representation cannot be interpreted as an ability to
make out which objects the concept applies to.

So, what I'm asking is whether there is something corresponding to
intensional representation of concepts in computers.

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.

Really, this is no different from the formalization of *any* mathematical
concept, computable or not. We write down axioms for them and deduce
consequences. Everything is always intensional.

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.

And this is not even to mention the fact that the computer will fail
when the number is too large, and exceeds its computational resources.

As for Strong AI, your concept of it seems to differ from the one
proposed by Searle, who introduced the expression. For Searle Strong
AI implies computers being conscious in the sense of having strictly
mental states (semantical states), not just human-like behavior.
Remember the Chinese Room.

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.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
.

Relevant Pages

  • Re: Deep Thoughts # 7: A New Kind of Mathematics
    ... chvol@aol.com (Charlie-Boo) writes: ... >> representation issues which would presumably cause this to cycle)? ... Axioms in some thoroughly unspecified manner. ...
    (sci.math)
  • Re: Deep Thoughts # 7: A New Kind of Mathematics
    ... chvol@aol.com (Charlie-Boo) writes: ... >> representation issues which would presumably cause this to cycle)? ... Axioms in some thoroughly unspecified manner. ...
    (sci.logic)
  • Re: Why does everyone do it?
    ... what we can observe (it implies the existence of objects that cannot ... actual number (not just a representation of a number). ... For example, given the group axioms, it really doesn't matter whether ... west of the intellectual field and the hottest field on the ...
    (sci.math)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... You will find a description of the formal axioms of ZFC in any number ... the task of writing a proof checker for any common ... Which representation would you use to write your trivial proof ...
    (sci.logic)