Re: Platonism
From: Eray Ozkural exa (examachine_at_gmail.com)
Date: 12/06/04
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Date: 5 Dec 2004 19:58:18 -0800
"robert j. kolker" <nowhere@nowhere.com> wrote in message news:<VfMsd.149519$5K2.120122@attbi_s03>...
> patty wrote:
> >
> > I don't know if its totally accurate to say that mathematics is not
> > empirical. Once you make a construction, you subject it to tests. If
> > the tests fail, you reject the construction. What is your criteria for
> > being empirical ?
>
> Measurement and observation vs formal deduction. Essentially all
> mathematical proofs are tautologies. The same is not true for
> empirically true statements.
>
> It is empirically true that the washington monument is 555 +- 1 foot
> high. No logical contradiction would be entialed if it were only 500
> feet high. An empirically true statement just so happens to be true. A
> logically true statement is necessarily true.
I think you are overlooking the fact that depicting the axioms are of
a quite empirical nature. You think of an axiom, and then you put it
to the test, to see if it's accepted by the mathematical community.
(Regardless of whether there is any "reality" to its truth!)
Plus, you should not overlook the "logical depth" of a proof, which
may sometimes be non-trivial! It takes ages to find all the
interesting theorems in a formal theory!
And a third point: you should consider what it means for a theorem to
be interesting. Again, that's an empirical process that does not
explained by the fact that (the truth of) a theorem is, formally, a
reduction to (truth of) axioms!
The first is in fact a point made by none other than Godel to support
his brand of Platonism, however, it is independent of Platonism
objectively speaking. By the incompleteness theorem, you can't reduce
mathematics to "a=a", that is why you have to admit axioms of greater
complexity. This is best exemplified in work of Chaitin: ZFC explains
only so few facts about the halting problem.
Regards,
-- Eray Ozkural
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