FOM: categorical "foundations"?

Harvey Friedman friedman at
Fri Jan 23 18:12:14 EST 1998

Response to Diskin, 9:15PM, 1/23/98.

>No doubts, all this complexity of object identification and dynamics can be
>also described in the pure SET-terms. However, this would be a very bulky
>specification blurring some essential aspects of the phenomenon.

Depends on who's doing the description. A top f.o.m. person should have no
trouble with this, and would be expected to surprise you with how elegant
and simple it could be made.

>In contrast,
>the TOP-based specification just throws light on these non-trivial questions
>(I have checked this in my communication with real  database designers. Note
>also that, in fact, database designers and software engineers  rediscovered
>implicitly many of topos-theoretic concepts (in their own, terribly awkward,

I believe this only if people come in with a bias to prove a point. After
all, if anybody bothers to learn a lot of this stuff, they already have a
bias in favor of it, and an investment in it, and are understandably more
amenable to casting things in these terms, rather than to see how to use
conventional methods. This is just human nature.

And I never denied that some category like constructions may be useful, but
do not have anything like the character of "foundations of mathematics."
After all, categories and toposes are perfectly easily understood as set
theoretic objects, and in fact that's how they are ultimately viewed by
everyone I have talked to in this math dept.

I can't help but repeat the two main points yet again about categorical

>From: Silver
>What I'm looking for is a development of category theory, or
>perhaps topos theory, that is based on some underlying *conception*.

And I answered:

>Hopeless. As foundations of mathematics, it is conceptually completely
>incoherent. Proponents are zealots who want to hide this from you in the
>worst way.

Another basic issue is raised by the statement of Feferman 7:15PM 1/16/98:

>... the notion of topos is a relatively sophisticated mathematical
>notion which assumes understanding of the notion of category and that in
>turn assumes understanding of notions of collection and function. ...
>Thus there is both a logical and psychological
>priority for the latter notions to the former.  'Logical' because what a
>topos is requires a definition in order to work with it and prove theorems
>about it, and this definition ultimately returns to the notions of
>collection (class, set, or whatever word you prefer) and function
>(or operation). 'Psychological' because you can't understand what a topos
>is unless you have some understanding of those notions. Just writing down
>the "axioms" for a topos does not provide that understanding.

These points, which are related, are among many objections that can be made
to categorical "foundations". They are completely fatal for it as
"foundations of mathematics."

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