Date: Thu, 21 May 1998 21:14:27 +0100
From: jim <jmd-AT-dasein.demon.co.uk>
Subject: Re: Math/Metaphysics


In message <199805191833.NAA29935-AT-endeavor.flash.net>,
Anthony Crifasi <crifasi-AT-flash.net> writes
>
>I am not intimate with Godel's results, though I know them in general, so 
>please forgive any obvious inaccuracies. First, what exactly does Godel 
>prove? 
>Is is that every axiomatic system has axioms which cannot be proven within 
>that system? If so, then all this implies is that there are no "absolutely true" 
>axioms, not that any given mathematical system can lack axiomatic structure. 
>Even if they are not the "traditional" axioms, they are still axioms. All 
>Heidegger would be saying, then, is that mathematics qua mathematics is 
>axiomatic by nature, not that there is any "one true" mathematics in the 
>traditional sense.

Actually, the axioms are merely starting points for the theory; that is
their only claim to fame. They have no unimpeachable status at all. No
mathematician (probably) would nowadays ever claim that the axioms
from which he chooses to begin his theorizing are "absolutely true."

To be a little more precise, G's results tell us that any math theory
formulated in a language sufficient to express the numbers and also such
everyday arithmetic functions as plus, times, exponentiation, division,
etc., and also math induction -- basically, in a language sufficient to
express ordinary arithmetic, Peano Arithmetic -- that such a theory will
contain 'true' sentences which are not deducible from the theory's
axioms using its rules of inference.

This result gives rise to the question: why axiomatize mathematics
anyway? Surely, its a nice "bookeeping" method for helping us to
remember which claims we are accepting as "starting points" (which
claims are our axioms), but we accord no special epistemic status to
these starting points; we don't claim that they are in some sense self-
evident, or analytically true, or absolute, or unrevisable, etc. Indeed, we
can "do" our math without organizing it into any axiomatic structure at
all. Having such a structure is not an 'essential' trait of mathematical
theories or disciplines.

PS.
I'm currently doing some related research; but am now starting to think
that I might have been unfair to H's position, because I was too hasty.
However, I am still pursuing this issue.
Cheers
jim


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