Categorical Foundations of Mathematics?

[Harvey Friedman]

Rodriguez writes https://cs.nyu.edu/pipermail/fom/2022-May/023311.html

"So,
if we follow Gromov's point of view, any attack on category theory should
start by attacking the foundations of Grothendieck's work rather than other
authors who may have betrayed Grothendieck's original vision."

I'm not attacking category theory at all, nor am I attacking PDE, nor
am I attacking algebraic geometry, nor am I attacking analytic or
algebraic number theory.

I'm attacking the idea that category theory serves as any kind of
philosophically coherent foundation for mathematics -- as can readily
be tested by the general intellectual community.

This is not even a contest. To the general intellectual community,
category theory as a starting from nothing ground up foundation for
mathematics would be like a pile of meaningless gibberish recognized
as some sort of mathspeak. They might even admire those who can speak
mathspeak but very content to deal with the idea of putting two
objects together to form a conceptual unit, as they did when they were
in the crib. With modern f.o.m. tools, it is quite amazing just how
few additional steps beyond that are needed to philosophically
coherently present real foundations of mathematics.

To the extent that some sort of category theory can be shoehorned into
some sort of philosophically coherent foundations of mathematics, it
is simply a result of it borrowing ideas from set theory. THere isn't
any new idea beyond set theoretic conceptions that could stand on its
own, with proper philosophical coherence or even sufficient simplicity
for the general intellectual community.

HAVING SAID THAT I COULD AT LEAST IMAGINE THAT THERE COULD BE SUCH A
NEW IDEA THAT DOES THAT. I just haven't seen it, and certainly nowhere
near the level of how it works in modern formulations with set theory.

Furthermore, at least so far, category theory hasn't even been useful
for ANY of our dramatic foundational revelations, of which we have a
few. The really useful tool for uncovering the dramatic foundational
revelations has always been through the usual standard classical set
theoretic foundations.

There is a kind of weak foundational scheme that is quite interesting
in its own right and really stands independent of both set theoretic
foundations and categorical pseudo foundations. And that is formal
arithmetic, exemplified by PA and its variants. Even though it can be
viewed as a tiny piece of set theoretic foundations, and much more
awkwardly as a tiny piece of categorical pseudo foundations, it has an
independent intuitive path that does appeal to the general
intellectual community and does have philosophical coherence.

Of course, all of the great work on PA is from the usual f.o.m.
tradition as is all of the other top findings in f.o.m.

Now how does set theory compare to category in terms of its use and
its insights into algebraic geometry, algebraic number theory,
algebraic topology, differential topology, and differential geometry?

Set theory is a useless pile of utter crap compared to category theory
for those purposes.

Harvey Friedman
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