FOM: categorical mis-foundations; essentially algebraic theories

Stephen G Simpson simpson at
Sun Mar 15 01:32:41 EST 1998

Steve Awodey writes:
 > Your postings have made it clear that you know almost nothing about
 > category theory

Now now, don't get testy.  I wouldn't say I know almost nothing about
it.  I estimate I know the equivalent of 2 or 3 graduate courses in
it.  That's less than some people and more than others.

 > so it is impossible for you to have a responsible, professional
 > opinion.

Now now, calm down.  Actually, I do have a responsible, professional
opinion, and it's an opinion based on knowledge, not prejudice.  For
example, I pointed out in specific detail why McLarty's claim about
real analysis in a topos was phony, and I pointed out that there is no
comprehensive topos-theoretic f.o.m. other than slavish copying of
set-theoretic f.o.m.  Why do you try to dismiss these arguments by
attacking my alleged lack of knowledge?  Why don't you point out the
defects in my arguments, if you think you can?

 > Your strong views can only be based on some personal dislike for
 > category theory, or category theorists, or whatever,

I don't know why you say that.

Category theory: Actually I like it a lot, as an organizational tool,
in algebraic topology, algebraic geometry, and even in certain
technical areas of mathematical logic (e.g. Makkai/Reyes).  But I
haven't seen any instances where category theory really helps in the
investigation of what I regard as the big, important questions of
f.o.m.  So on the whole I regard category theory as interesting but
not a high priority for me.

Category theorists: Perhaps there are some category theorists for whom
I have what you might call a personal dislike, because I regard them
as dishonest charlatans.  But that isn't restricted to category
theory, and it's not really personal.  It's more a matter of
intellectual style.

Whatever: I don't know.

 > you are clearly not interested in a serious, scholarly debate

That's exactly what I *am* interested in.  That's why I started
the FOM list.

 > only in defaming the use of category theory in logic, and in
 > slandering the hard work of its practitioners.

Not at all.  If you can do something good with category theory,
I'd love to hear about it, and I promise I won't slander anybody's
hard work.

 > It's obviously not in my best interest to participate in such a
 > discussion with you.

That may be.  You decide.

 > Informally, a first-order theory is "essentially algebraic" if it is
 > equational in partial operations, the domains of which are themselves
 > equationally defined.

OK, thanks for the explanation.  I hadn't realized that "essentially
algebraic" is a specific technical notion.  I thought you were merely
saying that topos theory has a vaguely algebraic flavor.  This notion
that you are describing sounds potentially interesting.

 > the elementary theory of topoi has this property

Technical question: Do the first-order topos axioms *as finally stated
by McLarty on the FOM list* have this property?  If not, what
modifications are needed?  Same questions for McLarty's final set of
first-order axioms for elementary topos plus natural number object
plus well-pointedness plus Boolean plus choice.  (Apparently you need
all that to get a decent foundation for real analysis and other
standard mathematical topics.)

 > 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.

OK, good, thanks for the explanation.

If you want axioms of low logical complexity, I think we can easily
write them down for bounded ZC set theory without infinity and
foundation, provided appropriate primitives are chosen.  Note however
that it's usual to classify set-theoretic formulas by counting the
number of alternations of *unbounded* quantifiers, ignoring bounded
ones (Levy hierarchy).  So we need to exercise care in comparing this
to the fact that you mentioned, that first-order topos theory is
"essentially algebraic."

 > You inferred that I "seem to think that logic as a subject is
 > obsolete and is to be replaced by algebra" 

OK, I'm glad you *don't* think that logic is obsolete and is to be
replaced by algebra.  But then, why are you so interested in
catogorical logic?  Isn't that the point of categorical logic?  If
not, then what *is* the purpose of categorical logic?  Tell me, I'd
like to know.

 > All this because I made a perfectly reasonable proposal to consider
 > a measure of simplicity other than your "number of bytes" measure.

You didn't explain your proposal in enough detail.  It sounded for all
the world as if you were saying that topos theory is better than set
theory because it looks more like algebra.  I think this is a
wrong-headed argument.  And I still think that's the argument you are
making, to some extent.  But please feel free to explain yourself.

 > Why should I stand for this?

Because that's the price of being involved in a lively discussion of
issues and programs in f.o.m.  You've got to have a thick skin.

-- Steve Simpson

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