non philosophy

non philosophy

[categories]

Date: Fri, 7 Mar 1997 14:24:15 -0500 (EST)
From: Peter Freyd <pjf@saul.cis.upenn.edu>

William James answered his own question when he asked:

  Does category theory, being mathematics, have no associated
  philosophy?

Yes, category theory is mathematics. Therefore its associated
philosophy is whatever philosophy one chooses to associate with
mathematics.

As for the latter, Dusko provided a pretty good answer. I would modify
it only to reflect that mathematics (which, as always in the absence
of a qualifier, means pure mathematics) is a subject matter. Most
mathematicians are sufficiently confident about their subject matter
that they feel no need for a semantics, much less a stated philosophy.
(Yes, it has been notoriously difficult to define intensionally, but
that's not special to mathematics. What's the subject matter of
physics? If either mathematicians or physicists were -- using Dusko's
language -- to spend a lot of time defining their subject, there never
would have been much mathematics or physics.)

Steve thinks that the existence of a philosophy of category theory is
an "of course". In one of the public meanings of the word "philosophy"
he's certainly correct but not, I think, in the sense that would
include something like constructivism. (The public meaning in question
has even less than type-checking to do with either love or wisdom.
Well, maybe it has something to do with love.)

May I suggest that the applied mathematician may have a very different
understanding of category theory from the mathematician. Steve says
that category theory is "all things are connected". But that's an
article of faith for almost any mathematician. He goes on to say, "you
cannot fully describe anything purely in itself but only by the way it
connects with others." This assertion about what "you cannot" do
sounds like it could be a good way of describing *applied*
mathematicians.

To begin with, Eilenberg and Mac Lane defined categories in order to
define functors and they defined functors in order to define natural
transformations. Immediately it was noted that a new tool existed to
pin down -- in a formal way -- how it is that some of the all things
are connected. It should be noted that categories -- and more to the
point, functors -- have always been considered tools for studying the
subject matter of mathematics. Tools, not the subject matter itself.

I am on record that the language of categories began to become
respectable when Frank Adams was able to count the number of
independent vector fields on each sphere using a construction that
quantified over functors: it produced an n-ary transformation on the
K-functor for every n-ary endofunctor on the category of finite
dimensional vector spaces (which he assembled into what are now known
as the Adams operations).

One of the better successes since then has been the use of categories
in finding connections between various foundational systems. Because
some of these systems are constructivist it has apparently caused some
to think that categories are intrinsically constructivist.  Strange.

There's another important aspect of category theory. Most categories,
in the beginning at least, were categories that naturally arose from
existing branches of mathematics. Some of these categories, though,
had never been lived in before they were invented as categories.  Joel
Cohen named one of these the "Freyd Category" (named not after Peter
but Jennifer): its an abelian category whose full subcategory of
projectives is the stable-homotopy category; all the other objects
have no easy description; the category can be described as the target
of the universal homology theory. But a much better example is in
Serre's dissertation. This work, hailed by many as the single most
substantive dissertation ever written, contends with the two abelian
categories that result when one starts with the category of abelian
groups and identifies with the zero group all finite groups in one
case, or all finitely generated groups in the second case.  Most
remarkably, Serre did all this without using category theory. (The
fact that the first non-trivial construction of a category occured
without benefit of category theory must be reckoned an embarrassment
for category theorists.)

But in recent applications I think a very different type of question
is being asked: "Is it possible that there is a category in which
... can take place?"  These questions are at the heart of many
approaches to programming semantics. And they are at the heart of many
of the uses of categories in theoretical physics. But the first
serious example came, in fact, a long time ago. In the late 60's
Lawvere's approach to differential geometry asked just this type of
question. Elementary topoi made their first appearance as just a
preliminary part of the answer.

So what's this all have to do with philosophy? 

Not much, of course.