FOM: categorical digression; the other one percent

[Stephen G Simpson]

Reply to Andrej Bauer's message of 22 Feb 2000 3 PM.

 > Thank you for the pointers. I think I've read most of those
 > discussions. I'll try to not go there.

Good.  Unless you have something new to say.

 > Let's stick to the discussion about adjoint functors.

Perhaps you have forgotten that the entire discussion of category
theory and adjoint functors is itself a digression.  

We began a few days ago with an earnest question raised by a student,
a question that is of great interest and importance for f.o.m.:

  It is often said that 99 percent of math is formalizable in ZFC.
  What about the other one percent?

But then, because of category-theoretic zeal, we got diverted into a
pointless discussion of adjoint functors, whether adjoint functors are
more or less pervasive than groups, whether mathematicians should be
taught adjoint functors, etc etc, all irrelevant to f.o.m.

 > And let's try to follow Todd Wilson's suggestion of "good will".

Yes, let's follow Todd Wilson's suggestion of "good will".

 > These arguments about what a five year old can and cannot understand,
 > are you serious about them?

Yes, of course.

 > I think they're completely irrelevant.

They are irrelevant to adjoint functors.  But they are relevant to
f.o.m.

 > Maybe you can explain why a five year old's grasp of mathematical
 > concepts has anything to do with f.o.m.

OK, I will explain it.

First, what is f.o.m., anyway?

Here is my tentative working definition.

F.o.m. is the study of the *most basic* mathematical concepts
(concepts like set, function, number, algorithm, mathematical axiom,
mathematical proof, mathematical definition, ...)  and the logical
structure of math, with an eye to the unity of human knowledge.

Now, part of what makes a mathematical concept *most basic* is that it
can be understood premathematically or nonmathematically. This helps
to account for the connections of math to the rest of human knowledge.

Hence, the understanding of children who know no math is a sometimes
relevant issue for f.o.m.  Not necessarily decisive, but sometimes
relevant.

 > (on a separate instance, or at least a separate posting, please. I
 > don't want this to become a main issue).

It's interesting that you are so concerned to ``stick to the subject''
if the subject is adjoint functors, but apparently not if the subject
is ``the other one percent'' or other important f.o.m. issues.

 > > As a non-category-theorist and a human being, I of course find this
 > > way of viewing quantifiers somewhat unnatural.
 > 
 > Yes, well, I think everyone does when they first see it.

I have seen it many times, and each time I find it bizarre.

 > This game is not about what is more fundamental. 

Excuse me?

An important issue for f.o.m. is, what are the *most basic* or most
fundamental mathematical concepts.  The logical structure of
mathematics is deeply conditioned by this issue.

 > But it seems you don't quite believe yet that adjoint functors are
 > far, far more general and omnipresent than groups.

Right, I don't believe it.  And even if I did, that wouldn't prove
that adjoint functors are of interest for f.o.m.

 > Is your work on reverse mathematics foundational

Yes, reverse mathematics is part of f.o.m.  That's because reverse
mathematics is a study of the logical structure of mathematics,
specifically what axioms are needed to prove what theorems, leading to
a classification of theorems.

There is no remotely comparable reason for thinking that adjoint
functors are part of f.o.m.  Adjoint functors are simply a specialized
concept belonging to a particular branch of algebra called category
theory.  Category theory is an interesting branch of math, and it has
organizational value in areas like algebraic geometry, but it is not
f.o.m.

 > Do you still think that adjoint functors can be compared to groups
 > fairly?

Yes.  I made such a comparison.

 > Can you make statements like the above about groups?

Yes.  The integers form a group under addition.  The automorphisms of
any structure form a group.  Etc etc.

 > > The pervasive interest and fundamental nature of sets and classes in
 > > f.o.m. is extremely well established.  The interest of adjoint
 > > functors in f.o.m. is much, much less well established, or maybe not
 > > established at all.
 > 
 > I am establishing it as I speak :-)

Have you already given it your best shot?

 > I hope this posting makes it clear that adjoint functors are not like
 > groups.

No, it doesn't.

I propose we end the discussion of adjoint functors here.  It is off
the topic of f.o.m.

-- Steve