[FOM] Alain Connes' approach to Analysis

[Harvey Friedman]

On Tue, Aug 28, 2018 at 3:51 PM Timothy Y. Chow
<tchow at math.princeton.edu> wrote:

> It's an interesting question what the relationship is, or should be,
> between the "foundations of mathematics" and the "foundations of X" where
> X is some branch of mathematics (analysis, geometry, combinatorics, ...).
>
> The conventional view seems to be that f.o.m. does not have too much to
> with f.o.X.  When Gian-Carlo Rota initiated the famous series of papers
> "on the foundations of combinatorial theory," there was not much in those
> papers of interest to most practitioners of f.o.m.  Grothendieck's
> revolutionary approach to the foundations of algebraic geometry may have
> attracted some attention because of logical questions surrounding
> universes and topoi, but from the point of view of an algebraic geometer,
> such logical questions had almost nothing to do with the main point of
> what Grothendieck was doing.
>
> Sometimes f.o.m. and f.o.X. intersect if there is a question of *rigor*.
> The question of whether umbral calculus, or "Italian algebraic geometry,"
> was rigorous is of interest to both f.o.m. and practitioners of X.  But
> again, rigor is not usually the main motivation behind f.o.X.
>
> I know very little about noncommutative geometry, but my superficial
> impression is that this is another case of "f.o.X." where the main point
> has very little to do with f.o.m., or at least with f.o.m. as f.o.m. is
> commonly perceived.

Two quite different issues arise concerning the relationship between
f.o.m. and f.o.X. One is readily explained and reasonably well
understood, and the other is much more subtle. I have these two
aspects in mind:

1. f.o.m. normally concerns matters that are so basic as to resonate
with virtually all mathematical minds at some significant level. It
addresses general intellectual issues that are at the heart of
mathematical culture. At least this happens , episodically, at the
highest level. Of course, many general intellectual issues resonating
with many mathematical minds, even a majority of such minds, are not
yet adequately addressed by f.o.m. But such issues are at least high
on the f.o.m. agenda.

2. f.o.X normally concerns matters that resonate more or less only
with those mathematical minds concerned with and/or knowledgeable
about X. This is of course an incomparably smaller community than what
is addressed in 1. What is not so clear is how to state in clear
terms, for any given X, just what falls under f.o.X and what falls
under X. One idea is to simply imitate 1 above and say that f.o.X
addresses mattes that clearly resonate with virtually all of those
concerned with and/or knowledgeable about X.

I believe that it is interesting and important to go beyond thinking
about f.o.m. and f.o.X in terms of radical differences between sizes
and scopes of audiences. But I think it at least sets the stage for
further analysis.

There is an interesting asymmetric aspect to this. There is the
general question of what generally is f.o.m. and also what generally
is f.o.X, independently of X. Both of these questions are naturally
part of f.o.m., and not part of any f.o.X.

Harvey friedman