Categorical Foundation of Mathematics?

[Mikhail Katz]

Timothy Chow wrote, in the context of a discussion of  the concept of the
philosophical coherence of set theory, that nobody in the wider scientific
community cares about the foundations of mathematics.  One can possibly
assume that there is a small community of natural scientists interested in
such issues.  On the other hand, it would be helpful to understand the idea
of such "philosophical coherence" to begin with.  Otherwise how can one be
expected to understand the claim that, say, ZFC is philosophically coherent
whereas, say, category theory is less so?

To make this workable, one could start by comparing what a mathematician
means by saying that a certain concept is philosophically coherent, and
what he means by saying that a certain concept truly exists.  Surely
"philosophically coherent" and "truly existing" seem different, but it
would be helpful to understand the precise difference between them.  To
explain this further, one could adopt an unlikely working hypothesis that
what the mathematician means (by both) is merely that he is currently
working on, and is intensely interested in, the concept in question.

For example, a number theorist focusing on what would seem (to other
mathematicians) to be an obscure abstract concept, will be intimately
convinced of its patent reality as much as that of the Milky Way.  To give
another example, a specialist immersed in set-theoretic multiverses will be
convinced that those are the truly real ones, rather than the naive
set-theoretic universes once thought to be the real thing.  Similarly, a
specialist in ZFC and/or its weaker subsystems will be convinced of the
utter reality (and philosophical coherence) of the set-theoretic concepts
he is currently working on.

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.

Mikhail Katz

On Fri, May 20, 2022 at 2:53 AM Timothy Y. Chow <tchow at math.princeton.edu>
wrote:

> 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
>
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