Subject: categories: pots and kettles

From: Paul Taylor <[email-masked]>
Date: Thu, 16 Mar 2006 15:56:55 +0000

I wonder what Eduardo Dubuc means by "the Benabou-Taylor confrontation".
Could he possibly be referring to the Christmas flame-war to which Eduardo
was himself a major contributor, but which I attempted to settle with him
privately and amicably?

I suggest that he and Marta Bunga should go for a long walk to calm down.

I'm not pointing any fingers at Eduardo or Marta, but those who were active
in category theory in the 1960s and 70s would be well advised to think twice
before accusing those who came later of bringing category theory into disrepute.

I wonder when the phrases "generalised abstract nonsense" and "empty set
theory" came into circulation?   I seem to remember hearing them when I was
an undergraduate.

I wonder which generation it was that got prestigeous jobs in mathematics
departments at a time when (at least in Britain) they were being given away
in corn flakes packets?   Which generation is it that struggles to get jobs
in computer science departments?  Which subject was it, according to the
participants of a parallel conference in Vancouver in 2004, whose followers
were "easily recognisable because they're all so old"?

Which generation was it that alienated other mathematicians by making
outrageous claims about the foundations of mathematics that it never
backed up with theorems?   Which generation actually got its hands dirty
and proved the theorems that relate category theory to other foundational
disciplines?

I have no idea what John Baez's standing is amongst physicists. Therefore
Marta is quite correct in warning me that it would be foolish for me to
base public claims of the value of category theory on its applications
to physics.   John is nevertheless very welcome in our community, in my
humble opinion, BECAUSE he brings in new ideas from physics, BECAUSE
he exports some of our ideas in return, and because he is good at
presenting ideas of either kind.

Also, I don't need to be specific about them to say that the applications
to physics, computation, topology, geometry, analysis, ... are the reasons
why I consider that category theory is a Good Thing.   In my experience,
category theory as a tool usually points me in the right direction, whereas
set theory usually points me in the wrong one.  (To give credit where
it is due, Jamie Gabbay's work on alpha-equivalence is a counterexample.)

Category theory has been successful on many many occasions - starting in
algebraic topology - in removing the clutter of ad hoc definitions that
have no conceptual anchor.   For example, I was once told that "a matrix
is an array of numbers".   I am currently looking into Interval Analysis,
in which all of the textbooks begin with a chapter of miscellaneous notation
based on defining an interval as a pair [a,b] or as the subset of classical
real numbers that lie between them - "analysis with double vision", as
I call it.   (To give credit again where it is due, the "Interval Newton"
algorithm is a genuine insight into solving equations - it does not just
do what you might imagine it does.)

So I urge Marta, Eduardo and everybody else to stop trying to "police"
category theory, and instead celebrate its achievements in many areas of
mathematics.

Paul Taylor
www.cs.man.ac.uk/~pt - new web pages and new stuff.