From: "Anthony Crifasi" <crifasi-AT-flash.net>
Date: Fri, 22 May 1998 05:16:18 -0500
Subject: Re: Math/Metaphysics


Jim wrote:

> 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.

The last two sentences here do not seem to me to necessarily entail one 
another. It is true that we can "do" math, but it is difficult for me to see how that 
is mathematics qua mathematics anymore. For example, we can also "do" 
physics or any other Objective science, but that is not physics proper, which is 
strictly scientific in the sense of deduction, even if the scientific starting points 
of that deduction are not considered "unimpeachable" ones. So it seems to me 
that even if we can "do" mathematics or physics or any other science, it is still 
"an essential trait" of those sciences qua sciences to be axiomatic. To 
characterize them otherwise would be to destroy their status as science. This 
of course does not mean, as Heidegger points out, that science is the our most 
intimate and primordial way of encountering beings. It merely means that 
science is itself by nature axiomatic.

Anthony Crifasi

> 
> 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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