[FOM] Alternative foundations/philosophical

Timothy Y. Chow tchow at alum.mit.edu
Tue Mar 4 18:28:51 EST 2014

Friedman wrote:

> 2. In conversations about these matters, I have always asked for the 
> SIMPLEST IMPORTANT situation where the usual set theoretic foundations 
> is impossible to work with in a practical sense. I say that the burden 
> would then be on me to show how the usual foundations can be readily 
> adapted to do at least as good a job as alternatives. In each case, I 
> never got a sufficiently clear account of such a SIMPLE IMPORTANT 
> situation so that I could dig my teeth into how it could be properly 
> handled in the usual foundations.

The word "impossible" is a strong one.  What is the SIMPLEST IMPORTANT 
instance of a problem that is impossible to handle practically using 
probability distributions instead of random variables, or impossible to 
handle practically using ideal class groups instead of adeles or ideles, 
or impossible to handle practically using classical analysis instead of 
nonstandard analysis?  I think the answer in each case is, there isn't 
one.  Mathematical practice is a vague and amorphous thing that isn't 
subject to the well-ordering principle.  There's no sharp boundary between 
what is convenient and what is cumbersome.  Nevertheless there can be 
heaps of things that are handled better using new foundations even if we 
can't specify the minimum size of a heap.


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