FOM: Categorical pseudofoundations

[Vaughan Pratt]

From: Harvey Friedman <friedman at math.ohio-state.edu>
>Graph theorists and general topologists
>are completely content with the obvious and standard set theoretic
>foundations of graph theory and general topology. So is Grothendieck. Why
>aren't category/topos theorists? Very very peculiar.

Harvey raises a very nice question.  For graph theory the answer is
clear: graphs are positioned midway between sets and categories, being
simultaneously discrete and connected.  Graphs are equationally definable
in set theory, and categories are equationally definable in graph theory,
but categories are not equationally definable in set theory, the canonical
counterexample to the transitivity of monadicity.

So it's a coin flip, but you weight the coin according to the audience
you hope to reach.  The Dalai Lama does not address Berkeley students
in Tibetan.  Furthermore this situation is unlikely to change since most
students have neither the opportunity nor the motivation to learn Tibetan.

But it does not follow that Tibetan is an intrinsically bad language, and
I'm less sure as to whether topology is better viewed set-theoretically
or categorically (though Harvey is probably right that most topologists
prefer starting from sets, on account of both training and audience
for starters).

The former seems mysterious, as Robert Black nicely soliloquized.
The latter however makes good sense, topological spaces work essentially
the same way vector spaces do, but with a shift in emphasis from linear
combinations to limits.  But then that darned accessibility problem
rears its ugly head again...  :(

Vaughan Pratt