FOM: essentially algebraic; worse than Simpson

[Harvey Friedman]

Reply to Awodey 11:43PM 3/14/98.

Awodey writes the following to Simpson:

>The problem is exactly that your opinion is not professional, but personal.
>Your postings have made it clear that you know almost nothing about
>category theory, so it is impossible for you to have a responsible,
>professional opinion.  Your strong views can only be based on some personal
>dislike for category theory, or category theorists, or whatever, but not on
>rational considerations.  Despite your "scientific curiosity" posturing,
>you are clearly not interested in a serious, scholarly debate, but only in
>defaming the use of category theory in logic, and in slandering the hard
>work of its practitioners.  It's obviously not in my best interest to
>participate in such a discussion with you.
>        Am I only willing to talk to other category theorists?  Far from it
>- I'm very interested in discussions of the kind I took this one to be.
>That's why I joined in, until I realized that your motives are not genuine.
>Then there is the further issue of your uncivility; I doubt I would want
>to participate in such a rude discussion even if it were not so biased.

I'm trying to figure out why this kind of complaint is more often directed
at Simpson than at me, since I claim to be "worse" than Simpson. For
example, I claim to also use subject headers like "categorical pseudo
foundations." I also believe that in a fully rigorous treatment, it is best
to define relational structures with a definite "signature" or "relational
type," and to define relational structures as isomorphic if and only if
they have the same signature (relational type) and there exists a one-one
onto function such that ...; and definitely not to regard the isomorphism
relation between structures of different signatures (relational types) as
meaningless. In fact, in general in fully rigorous treatments of any
mathematics, one should definitely treat any relation (written atomically)
between objects as meaningful - true or false - whereas plenty of terms -
like 1/0 - should be regarded as undefined. This is by far and away the
very best way to do things. Of course, in the statement of propositions,
one generally avoids making use of this convention, but sometimes it
greatly simplifies things to make use of this convention. E.g., 1/0 is
undefined. But "1/0 is an even integer" is merely false. Also "1/0 is an
odd integer" is also false. Try it; you'll like it.

I have said on the fom that categorical foundationalists have a profound
misunderstanding of f.o.m. I claim that this is worse than Simpson.

Awodey writes:

>Informally, a first-order theory is "essentially algebraic" if it is
>equational in partial operations, the domains of which are themselves
>equationally defined. ... Anyway, it's a condition on the logical
>complexity of the axioms of
>a first-order theory - somewhat more general than equational, but certainly
>narrower than universal Horn. ... The notion was introduced in Freyd, P.:
>"Aspects of Topoi", Bull.
>Austr. Math. Soc. 7, pp. 1--76, 467--80.  There it is noted that the
>elementary theory of topoi has this property, and the fact is put to good
>use.  Indeed, this is what is responsible for some of the nice properties
>of topoi that Aczel (I think) mentioned in a much earlier posting.
>        My point in bringing it up was that it seems reasonable to regard
>the logical complexity of the axioms of a theory as a measure of simplicity
>that is at least as significant as the number of bytes occurring in those
>axioms, and so this is a sense in which the topos axioms are simpler than
>those for conventional elementary set theory.

Well, let's look at the logical complexity (in this sense) of the axioms
for topoi given by McLarty'posting of 9:21AM 2/6/98 Challenge axioms, final
draft.

1. A
2. AE
3. A
4. AEA
5. A
6. AEA
7. A
8. axiom missing or nonexistent
9. AEA
10. AEA
11. AE
12. AEAEA
13. EAEA
14. AE
15. AEA

This is definitely not "essentially algebraic."