[FOM] Alain Connes' approach to Analysis
Timothy Y. Chow
tchow at math.princeton.edu
Thu Aug 30 20:58:33 EDT 2018
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
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