[FOM] Alain Connes' approach to Analysis

[Timothy Y. Chow]

On the topic of f.o.m. versus f.o.X., I have observed a number of cases 
that fit the following general pattern.

1. Some prominent figure vocally criticizes standard foundations as being 
inadequate, and proposes new foundations.

2. Experts in foundations push back, pointing out that the criticisms are 
overstated.  As a result, they are largely unreceptive of the possibility 
that the new foundations offer any advantages.

As far as the advancement of science is concerned, I think that both 
attitudes leave something to be desired.  It is often true that the old 
foundations are capable of sustaining the proposed new ideas, at least 
initially.  At the same time, it could be that the new ideas do provide 
some kind of enormous practical advantage, such as making it 
psychologically easier to prove new results, or providing a kind of 
conceptual clarify for practitioners of X that was previously lacking. 
The critic may, in his or her own mind, link the misguided criticisms with 
the new ideas, but one should not assume that this link is a necessary 
one.  The new proposal may have significant merit in its own right, 
particularly from the viewpoint of f.o.X., whether or not the criticisms 
of the old approach are on target.

Voevodsky's objections about the consistency of PA are a case in point. 
They were not new, or even very coherently articulated.  The same might be 
said, though perhaps to a lesser degree, of many of his other criticisms 
of conventional foundations.  On the other hand, I believe that there is 
plenty of evidence that his ideas have been highly valuable, certainly 
from the point of view of f.o.X. where X = homotopy theory, and possibly 
even from the point of view of f.o.m.

I don't know enough about Connes's work to say for sure, but superficially 
it looks like it might be another example where his criticisms of 
nonstandard analysis are overstated, yet his new ideas have significant 
f.o.X. merit.

Tim