# FOM: category theory and f.o.m.

holmes@catseye.idbsu.edu holmes at catseye.idbsu.edu
Tue Feb 22 13:26:54 EST 2000

```Thus Simpson:

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.

My response:

Category theory is in fact a study of one of these basic notions: it
is motivated by analysis of the concept of "function" in a very
general sense.  Further, it appears to be motivated by an analysis of
the concept of "structure" in the most general sense, which is
arguably exactly the subject of mathematics in general!

It should be noted that I am not a category theorist, not especially
fond of category theory, and not at all friendly to the idea of
replacing set theory with category theory as a foundation for
mathematics in the sense in which mathematics is commonly held to be
founded on set theory.  But category theory has _obvious_ foundational
relevance; Bauer's example is right on target for the sense in which I
see this to be true.  There is no more reason to expect a child to be
able to see this than to expect a child to be able to prove the
Cantor-Schroder-Bernstein theorem.  And children do know something
about "structure" and probably could be taught elementary category
theory by a creative teacher (in fact, I think this has been done).

Thus Simpson:

No, it doesn't.

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

My response:

This is not proven.  It certainly doesn't follow from Simpson's
definition of f.o.m.  Category theory is motivated by the analysis of
a notion of foundational (not just general mathematical) importance
(which is on Simpson's list of examples!).  On the face of it,
category theory _is_ f.o.m. by the definition given; a claim to the
contrary must be demonstrated.  (the fact that category theory is
formally a branch of algebra does not prove that it is not f.o.m.  -
synthetic combinatory logic is also a branch of algebra and is
obviously f.o.m.)

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.boisestate.edu
not glimpse the wonders therein. | http://math.boisestate.edu/~holmes

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