From: mls@cbnewsm.cb.att.com (mike.siemon)
Subject: Re: Report on Philosophies of Physicists
Organization: AT&T
Date: Tue, 15 Sep 1992 03:58:32 GMT
Followup-To: sci.math
Summary: a topological perspective

I find myself a hundred or so articles behind, so this may be otiose;
however, I'd like to make one amplificiation on Torkel's remarks:

In article <TORKEL.92Sep13094850@bast.sics.se>, torkel@sics.se (Torkel Franzen) writes:

>   As for general topology, naturally it's founded on set theory. That,
> indeed, was the point of my invoking it. It is founded on set theory in
> the sense that it uses the basic concepts and results of set theory. It
> wouldn't exist without set theory. It is an area of mathematics that
> grew out of set theory. What I am claiming is that set theory became
> generally accepted by mathematicians, not as a result of their finding
> that it provided and attractive and workable foundation for mathematics -
> i.e. an axiomatic system within which the concepts of ordinary mathematics
> are definable and within which theorems of ordinary mathematics have
> corresponding formal derivations - but because of the mathematical uses
> of set theory, in particular in the creation of such subjects as
> point set topology and measure theory.

The point is well illustrated by the 20th century career of topology and
of analysis.  Poincare's reservations about point-set methods was merited;
very little of importance in topology has derived from them -- the main
lines being homological, homotopic and other developments that work with
cell complexes or other objects far closer to geometric intuition than
the matter studied in a standard point set topology text (e.g. Kelley)

Topology instruction takes a point-set course as prerequisite (or an early
emphasis) for two reasons, one negative and the other positive:

	- it is VERY important to "break" students from reliance on the
	  geometry and algebra of their earlier studies as *the* only
	  matter of a generalized "geometrical" reasoning.  There is such
	  a wealth of counter-example to expectations in the basic stuff
	  of a point-set topology course as to open previously closed minds.

	+ the notion of compactness, which Torkel has alluded to as one
	  of those things that is inherently set-theoretic (any cover of
	  a space having a finite subcover) turns out to be massively
	  significant, even if most (geometric/algebraic/differential)
	  topologists NEVER deal again with any of the other concepts of
	  the "basic" course.

I have a feeling that analysts find the point-set stuff much more central
than topologists do -- all my higher analysis courses refered constantly
to things like Baire category, and to measure-theoretic manipulations of
sets, and to approximation results like Stone-Weierstrass and beyond.  I
*never* saw so much reference to "standard" point-set theorems in topology
(the main access of such things came through differential geometry, by way
of Sard's Theorem -- a measure-theoretic result, basically.)

To reiterate Torkel's point -- ALL of this, as seen by a working mathe-
matician, has little or nothing to do with "foundations" even if some
of us maintain (also) an active interest in that.  And it seems to me,
though I haven't really looked at the history, that *foundational* use
of sets tended to be a *response* to Hilbert in the post 1900 time-frame,
rather than what had already by then become a MAJOR industry of "real"
math built on arguments that we first learned from Cantor in the context
of the rigorization/generalization of analysis.

-- 
Michael L. Siemon		   "Oh, stand, stand at the window,
					As the tears scald and start;
mls@usl.com			   You shall love your crooked neighbor
standard disclaimer			With your crooked heart."
Newsgroups: sci.physics,sci.math