Date: Sat, 23 May 1998 21:13:09 +0100
Subject: Re: Math/Metaphysics


In message <199805221014.FAA16226-AT-endeavor.flash.net>,
Anthony Crifasi <crifasi-AT-flash.net> writes
>
>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.

Of course, I agree with the claim that being axiomatic does not mean
that "scince is ... our most intimate and primordial way of encountering
beings."

However, concerning the other claims, I submit:
 
If the position were correct, namely, that if some given theory could not
be formulated axiomatically, then that theory could be neither a
mathematical theory nor a theory of the natural sciences, then the
position would make no sense of our making sense of the history of
these disciplines. For example, on one interpretation of this position, it
would follow that until some theory in question, T, could be formulated
in the canonical first-order language of the type that Quine envisages in
W&O, or in the structure in which Euclid formulated the Elements, or
that in which Newton the Principia of Natura, or that in which Russell
and Whitehead the PM, then T could not be correctly characterized as
either a mathematical or a natural scientific theory (I think that Kuhn's
views could be interpreted as reacting to just this sort of position,
couldn't it be?).

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.

My interpretation of the Godel result stems from an interpretation which
Rhush Rhees once presented in a seminar on Time. We were discussing
the concept of "now," and I suggested that we construct a formal
language with a Now-operator and investigate what kinds of semantic
interpretations are possible for it (Hans Kamp, Arthur Prior, James
Higginbotham do this kind of thing). In his typical later-Wittgensteinian
way, he bluntly asked me "Why do that?" His point was that the
formalization will not provide us with any understanding that we don't
already have, and that even if it could provide us with an understanding
of some arbitrarily designated alleged surrogate of the everyday-life
aspect which we were trying to understand, we still would be left with
our perplexity about this aspect of everyday life.

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.

Granted, axiomatization can provide for a sometimes needed clarity and
also for a very 'economical' statement of a theory, which statement
might allow us to investigate issues of completeness, provability,
independence, etc., it doesn't thereby confer upon the theory any
features, characteristics, properties which would render it mathematical.

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.

On a final note, I have grave doubts about the following claim:
>physics proper, which is 
>strictly scientific in the sense of deduction
Cheers,
jim


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