[FOM] Alain Connes' approach to Analysis

Timothy Y. Chow tchow at math.princeton.edu
Tue Aug 28 13:58:26 EDT 2018

MK wrote:
> The by now STANDARD reference on this is the article
> Sanders, S. "To be or not to be constructive, that is not the question."
> Indag. Math. (N.S.) 29 (2018), no. 1, 313-381.
> https://doi.org/10.1016/j.indag.2017.05.005

I'm not sure if Jose is asking about Connes's critique of nonstandard 
analysis or about Connes's noncommutative geometry program.  Sanders 
addresses the former but not the latter, as far as I can tell.

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.


More information about the FOM mailing list