Re: Mathematics, Pseudomathematics, Artificial Intelligence

From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 02/14/05 

Date: Mon, 14 Feb 2005 22:02:42 GMT

"robert j. kolker" <nowhere@nowhere.net> wrote in message
news:37ch62F5b3q46U6@individual.net...
>
>
> david petry wrote:
>
>> FWIW, I'm quite certain the Kronecker was Jewish, so there can't be
>> any strong relationship between being against Cantor's Theory and
>> being against Jews.
>
> Nowadays most of the goyim in mathematics accept cantorian set theory.
>
> Bob Kolker
>

After reading Sol Feferman I'm not sure it makes any practical difference.
He mentions his system W and Harvey Friedman's (Steven G. Simpson)
Reverse Mathematics as adequate to the task of scientific mathematics.

"In The Light Of Logic" by Sol Feferman, page 287
3. What's Wrong with Set-Theoretical Foundations?

"Philosophically, set theory -- even in its "moderate" form given by
Zermelo's axioms -- require for its justification a strong form of
platonic realism. This is not without its defenders, most notably
Godel (1944 and 1947/1964) (cd. also Maddy 1990). For its
critics, however, the following are highly problematic features of
this philosophy:

(i) abstract entities are assumed to exist independently of any
    means of human definition or construction;

(ii) classical reasoning (leading to nonconstructive existence results)
     is admitted, since the statements of set theory are supposed to
     be about such an independently existing reality and thus have a
     determinate truth value (true or false);

(iii) completed infinite totalities and, in particular, the totality of all
      subsets of any infinite set are assumed to exist;

(iv) in consequence of (iii) and the Axiom of Separation,
      impredicative definitions of sets are routinely admitted;

(v) the Axiom of Choice is assumed in order to carry through the
     Cantorian theory of transfinite cardinals."

SH: I think the Law of the Excluded Middle is assumed which would
seem to have a consequence for probabilistic approaches to AI. I
don't think Feferman considers Category Theory an adequate substitute
for Cantorian set theory but he mentions it as a possibility.

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