Categorical Foundation of Mathematics?

Harvey Friedman hmflogic at
Tue May 24 09:42:57 EDT 2022

On Mon, May 23, 2022 at 8:42 PM Timothy Y. Chow <tchow at>

> Mikhail Katz wrote:
> > To make the idea of philosophical coherence clear and precise (and
> > therefore workable), it may be helpful for starters to distinguish it
> > from "truly existing".  Otherwise one will have to fall back on the
> > working hypothesis above, no matter how implausible it may seem.
> Harvey Friedman is the only one who can say exactly what he means by the
> term "philosophically coherent," but since I have interrogated him on this
> point in the past, I can give my approximate understanding as a starting
> point that may be useful for others.
> Chow goes on to give a very well written and essentially accurate account
of what I must mean by philosophically coherent in the present context.

Philosophical coherence is something that is illustrated over and over
again in any good first year Introduction to Philosophy course in any good
Philosophy Department.

I assume that any FOM reader has had such a freshman course or at least
operates at at least that level when thinking philosophically.

I hesitate to get involved in any in depth discussion of philosophical
coherence because of my belief that anyone who isn't familiar with this
notion enough to see the special status of the usual formal systems of
f.o.m. is never going to get anything out of such a discussion. This would
be like screaming at a color blind person about all the rich colors they
are not paying attention to.

Chow obviously took some time to explain philosophical coherence in the
context at hand - and rather well.

I would be VERY INTERESTED to get an idea of whether any appreciable number
of FOM readers understood and related to Chow's post - that didn't already
feel comfortable with philosophical coherence in this context before
reading Chow.

One reason I have been sceptical of even trying to do what Chow has done is
the following:

No one has challenged me on here to show how to present the usual f.o.m.
with philosophical coherence. And no one has tried to present categorical
foundations in a philosophically coherent way on here.

This suggests hardened positions that are not possible to budge. I'd be
curious if Chow is able to move this needle.

Harvey Friedman
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