From: "Anthony Crifasi" <crifasi-AT-flash.net>
Date: Tue, 26 May 1998 08:48:44 +0000
Subject: Re: Math/Metaphysics


Jim wrote:

> However, the birth of most of what we consider to be mathematical and
> natural scientific theories predated any such axiomatic formulations. For
> example, Euclid's Elements accumulated the 'knowledge' of Plane
> Geometry that existed up to his time (even though the Fifth Postulate
> was a still then widely argued "for" and "against"); and it wasn't until
> Dedekind or Frege that real or positive numbers received any kind of
> "rigorous" axiomatic definition, or until Cantor introduced sets --
> although his formulation suffered from Rusell's paradox -- that intervals
> (sets) of convergence received their axiomatic treatment, or until
> Bolzano,Weirastrauss, and Cauchy, that the epsilon-delta definition of
> "limit" or the reals received the kind of formulation required for
> axiomatization. However, despite that being the case, the "data," if you
> will, from which and for which these various formulations were being
> crafted were nonetheless, mathematical. 

The 'knowledge' of Plane geometry which predated Euclid was still in 
axiomatic form, even if it was in the form of individual propositions 
and proofs, as opposed to the whole system presented by Euclid. In 
other words, before Euclid, the need for the axiomatic form of 
geometrical arguments and proofs was still demanded. The 
Pythagoreans didn't just say, "the squares on the two legs of a right 
triangle equal the square on the hypotenuse" without simultaneously 
presenting an attempt at an axiomatic proof. This does not mean that 
they traced the proofs and axioms all the way back to the definitions 
of line, point, and circle, as Euclid did. But they still presented 
deductive arguments based on prior and posterior premises. The 
history of science only shows that axioms which were previously 
regarded as "fundamental" and "absolute" were later either rejected 
altogether, or limited to within a certain sphere, as Euclidean 
axioms were limited to Euclidean plane geometry after the advent of 
non-Euclidean geometry, just as Newtonian axioms were limited to 
inertial reference frames after the advent of relativity. So judging 
from this history, science does not seem to escape axiomatization; it 
only escapes *absolute* axiomatization.

> Similarly, over and above providing a clear, precise, and well-organized
> presentation of not only the rules of inference used in either a
> mathematical or natural scientific theory, but also the 'first' or more
> 'fundamental' assumptions of that theory, an axiomatization will not
> confer upon the theory any character which it doesn't already possess.

I think you may be confusing the "presentation" of the rules of 
inference with the rules themselves, which are the axioms. The 
"axioms" of science are not merely the "presentation" of the 
rules; they ARE the rules. So yes, the "presentation" of the axioms 
does not confer upon the theory any character which is doesn't 
already possess, but that is merely to say that the "presentation" of 
the axioms does not confer upon the theory the axioms themselves. The 
theory itself is by nature axiomatic; and those axioms may also be 
"presented." Without those axioms, there would be no "rules of 
inference" at all, and therefore no rules by which one scientific 
result could be judged better or worse than any other result, even 
WITHIN any particular system. Only by transcending science altogether 
can we transcend axiomatization, which is precisely what Heidegger 
does.

> As a positive alternative to this position, it seems more plausible that
> what renders a theory T mathematical or natural scientific is T's
> "history," i.e., with respect to what data, in relation to which work of
> which individual persons, in terms of what kind of vocabulary T was
> developed. This position acords well with Hanson's views on the 'theory
> ladenness' of scientific theories, and also the historical basis on which
> theories have been considered scientific.

But again, all that implies is that there are no "absolutely true" 
axioms, so that "axioms" in general must be studied in terms of the 
history of their development, not in terms of "absolute truth." For 
example, when Euclidean geometry was considered "absolutely true," 
the "history" of the development of Euclidean axioms was considered 
irrelevant (eg., Aristotle's treatment of Euclidean axioms); all that 
mattered to geometry qua geometry was the simple "truth" of the 
axioms. But with the advent of non-Euclidean geometry, Euclidean 
axioms could no longer be considered "absolute" for geometry in 
general. The investigation of those axioms could therefore not rest 
with simply apprehending their "absolute truth;" rather, their 
development and history became essential to the axioms themselves.

So in general, I don't see how Godel's argument results in the 
de-axiomatization of science as such.

Anthony Crifasi


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