Categorical Foundation of Mathematics?

[Timothy Y. Chow]

Harvey Friedman wrote:

> Generally speaking, philosophical coherence is required in order for a
> foundational scheme to have any traction in the wider intellectual and
> scientific community. Set theory has this, when presented using modern
> f.o.m. tools, but anything like that PNAS article certainly does not.

Let's be honest here.  In the "wider intellectual and scientific 
community" today, nobody cares about the foundations of mathematics.  In 
the arts and humanities (except for philosophy), nobody cares about 
mathematics, period.  In the sciences and engineering (except for 
theoretical computer science, which is more or less mathematics), people 
care about mathematics only insofar as it is a useful tool.  In 
philosophy, only specialists in the philosophy of mathematics care about 
the foundations of mathematics.

The last time there was a development in the foundations of mathematics 
that carried "general intellectual interest" was when Goedel proved his 
incompleteness theorems.  (Well, maybe a bit later, if the foundations of 
computer science count as the foundations of mathematics.)  The only 
audience for this topic is a small subset of mathematicians and 
philosophers.  And within that subset, an even tinier fraction care a lot 
about "philosophical coherence" in the sense you mean it.

Tim