[FOM] Alain Connes' approach to Analysis

[Timothy Y. Chow]

Joe Shipman wrote:

> I don't understand why suspicion of nonmeasurable sets and ultrafilters 
> should make a number theorist distrust proofs which use them, it is well 
> known that they can be constructively eliminated from the proof of any 
> Arithmetical statement.

I can't speak for Connes or for any particular number theorist, but in 
general, there is a difference of philosophy between f.o.m. and f.o.X. 
that one should keep in mind.  In f.o.X., one may reject a concept or an 
approach as being "wrong" in the sense that it fails to track the true 
essence of the X under study, or presents some psychological impediment to 
understanding X more deeply, even if it does not lead to false theorems.

Actually, this sense of "wrong" is important in f.o.m. too.  It's just 
that in f.o.X., it's by far the most important sense of "wrong."  Except 
in the relatively rare cases where some important argument is being 
criticized for lack of *rigor*, it's almost always what people in f.o.X. 
are concerned about.

Tim