FOM: Re: algebraic simplicity? the obliteration of logic

[Steve Awodey]

Dear Steve,
>
>Why are you unwilling to have a discussion with me?  Are you only
>willing to talk to other category theorists?  However much we
>disagree, we are both professional scientists.  My professional
>opinion is that the claims of "categorical foundations" are
>exaggerated and wrong.  I have given solid reasons for my opinion.
>When scientists disagree, it's appropriate to engage in dialogue to
>find out who is mistaken, or at least to illuminate the issues in
>question.  I am raising some legitimate and important scientific
>issues.  Why don't you want to discuss them?  Is it because you have
>nothing to say?

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.

>One specific issue that I wanted to discuss with you is your claim
>that the topos axioms are in some sense simpler than the ZFC axioms.
...
>In your posting of 26 Jan 1998 01:09:16 you said:
>
> > the topos axioms are essentially algebraic, a condition that can
> > arguably be interpreted as a kind of logical simplicity (and is so
> > regarded by category theorists).  By contrast, ZFC does not have
> > this property (not even close).
>
>Why do you think that being "essentially algebraic" is to be
>interpreted as "logical simplicity"?

Alright, I do owe you (and any readers of this thread there may still be,
although I doubt that) an explanation of this remark:

Informally, a first-order theory is "essentially algebraic" if it is
equational in partial operations, the domains of which are themselves
equationally defined.  For example, the theory of categories has this
property, where the axioms are equational in the operations of domain,
codomain, identity, and composition, but composition of arrows is only
partially defined on the set of pairs of arrows  (f, g)  with
codomain(g)=domain(f).  I think you get the idea - a precise definition can
of course be given, but it's a bit fussy.
        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.  Models of such theories enjoy many of the
properties of models of equational theories - closure under products,
homomorphic images, and so on.  So they behave like equationally defined
"algebras", whence the name.  (So e.g. the product of a family of rings is
again a ring, while this is not so for fields, which are not equational.)

        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.

>This seems strange on the face
>of it.  Traditionally, algebra and logic are regarded as two different
>subjects, each presumably with its own criterion of simplicity.
>(Recall the earlier discussion of the fallacy of metabasis.)  Do you
>disagree with this traditional view?  You seem to think that logic as
>a subject is obsolete and is to be replaced by algebra.  Why?  I don't
>think that obliteration of logic would be conducive to the unity of
>science.  Do you think it would be?

Do you see what I mean by unjustified opinions?  You inferred that I "seem
to think that logic as a subject is obsolete and is to be replaced by
algebra" from the mere occurrence of the word "algebra" in the (perfectly
logical) notion "essentially algebraic".  You then accussed me of favoring
the "obliteration of logic".  All this because I made a perfectly
reasonable proposal to consider a measure of simplicity other than your
"number of bytes" measure.  Why should I stand for this?

Steve Awodey