From: gudeman@cs.arizona.edu (David Gudeman)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 9 Apr 92 21:08:20 GMT
Organization: U of Arizona CS Dept, Tucson

In article  <1992Apr9.090844.12520@midway.uchicago.edu> Anil Trivedi writes:
]
]In mathematics, traditional arithmetic, algebra, geometry etc. are much
]more stable than any "foundations" anyone has provided for them. Make no
]mistake, it is those "foundations" that are going to come and go and 
]change with time: 
]
]THESE ARE CLEVERLY WORDED SUPER-PATTERNS, NOT FOUNDATIONS.

The term "foundational" in this sense does not refer to relatively
permanency but to relative primacy.  In other words, mathematics in
general is built upon a foundational theory, the foundational theory
is not built upon the rest of mathematics.  The word is taken from an
analogy with physical structures and like all analogies is imperfect.
Just because you can point out a particular imperfection in the
analogy is no reason to get snotty about the term.

Furthermore, while it may be that all foundations proposed to date
will pass away, it is presumptious to suggest that this is a necessary
character of such theories.  The obvious implication is that such
studies are superfluous and have no fundamental importance.  But there
are some serious questions about the nature of mathematics, the
reasoning used in mathematics, and the relationship of mathematics to
the real world that have never been satisfactorialy answered.  You may
not be personally interested in these questions, satisfied to just
"use what works", but that is what makes you a physicist rather than a
philosopher.  Some people are not satisfied to just use a tool without
understanding how it works and why it works the way it does.

In thermodynamics it is common to used infinitesimal methods on
properties that do not have infinitesimal meanings.  For example,

        pV        pV
  dV = ---- dT + ---- dP
        pT        pP

(V = volume, T = temperature, P = pressure, p = partial derivitive).
Now none of these quantities really has infinitesimal meanings.  When
you get to small enough volumes, for example, there are just a few
molecules, and it is not realy clear how much space they are taking
up.  Zemansky and Dittman [_Heat and Thermodynamics_, 6th edition,
page 34] say that these derivitives are assumed to be very small with
respect to the quantity being changed, but large with respect to
molecules.  This, of course, does not help at all, since any finite
change has a finite measurement which will, in general, be different
from what the equation predicts.  Physicists may be happy with such
psuedo-explanations, but others want to know _really_ why infinistimal
methods work on finite measurements.  (Actually, this question is
fairly easy to answer, and Zamansky and Dittman are not even close).

A more general question related to the one above (in the sense that I
think it has the same answer) is the following: Theoretical physics is
generally done using the rules of mathematics alone, without regard to
physical reality.  When you empirically determine some equation that
describes physical events, then do mathematical manipulations to
predict things that your empricial study did not reveal, why does it
work?  Just because it is _mathematically_ legimate to infer, for
example, from "2x = 4" to "x = 2", why does that actually work when x
refers to a physical property?  Why should physical properties follow
the laws of mathematics?

A more difficult and fundamental questions is: Is mathematical
reasoning purely logical, or do we somehow just "know" things about
mathematical objects so that we can make leaps of intuition?  A simple
example is induction over the natural numbers (the set of non-negative
integers).  Given that you can prove something is true of zero
(express this as p(0)) and that you can prove that p(n) implies
p(n+1), this seems to prove immediately that p is true of all natural
numbers.  Why?  Are there more fundamental rules of logic that we can
use in order to prove that the above works, or is this just something
that is immediately obvious and needs no further proof?

I'll note that there has been at least one case in mathematics where
everyone said that something is "obvious and needs no further proof",
and later it was discovered that, in fact, this thing was not true.
This is the case of naive set theory.  One must wonder whether such
problems lurk in the rest of mathematics, and only through
foundational studies will we ever (possibly) have any proof that
mathematics actually works.
-- 
					David Gudeman
gudeman@cs.arizona.edu
noao!arizona!gudeman
Xref: sparky sci.philosophy.tech:2081 comp.ai.philosophy:4232