Re: Categories and functors, query

["R Brown" <ronnie.profbrown@btinternet.com>]

Dear All,

I agree with Andre that part of the matter is sociological. It is also quite
fundamental, and is about the proper aims of mathematics. The need is for
discussion, rather than total agreement.

Miles Reid's infamous comment was "The study of category theory  for its own
sake (surely one of the most sterile of intellectual  pursuits) also
dates from this time; Grothendieck can't necessarily be  blamed
for this, [!!!] since his own use of categories was very successful in
solving problems. "  (My riposte in a paper was to suggest a game: `I can
think of a more intellectually sterile pursuit than you can!') This suggests
the view
that solving problems, presumably already formulated ones,  is the key part
of mathematics. (Miles did tell me he expected his student to use topoi or
whatever!)  A 1974 report on graduate mathematicians in employment suggested
they were good at solving problems but not so good at formulating them.

Grothendieck in one letter to me wrote on his aim for
`understanding'. (see my article on `Promoting Mathematics' on my
Popularisation web page) I believe many students come into mathematics
because they like finding out why things are true, they want to understand.
Loday told me he thought one of the strengths of French mathematics was to
try to realise this aim. By contrast, I  once
asked Frank Adams why he wrote that  a certain nonabelian cohomology was
trivial and he said `you just do a calculation' - Frank was a determined
problem solver!

So people have asked: "Where are the big theorems, the big problems,  in
category theory?" Are they there? Does it matter if they are not there?

Atiyah in his article on `20th century mathematics' (Bull LMS, 2001) talks
about the unity of mathematics, but the word `category' does not occur in
his
article. (Neither does groupoid.) He states a dichotomy between geometry
(good)and algebra (bad) but fails to recognise the combination given by,
say, Grothendieck's work, and also by higher categorical structures.
Indeed, underlying structures and processes may be of various types, all
very useful to know. I am *very* impressed by Henry Whitehead's finding so
many of these.

A word often omitted in mathematics teaching is `analogy'. Yet this is what
abstraction is about, and why it is so powerful. Category theory allows for
powerful analogies.

I am always puzzled, even horrified,  by mathematicians who use the word
`nonsense' to describe the work of others (as is all too common), yet often
themselves cannot well define professionalism in the subject. Indeed they
often cannot believe the direction others may take is chosen for good
professional reasons! They sometimes say `not mainstream'. Yet history shows
`the mainstream' shifts its course radically over the years. The lack is of
a consistent and well maintained mathematical criticism, recognising
historical trends and not just the `great man (or woman)', or famous
problem,  approach.

I believe we need to have prepared an answer to: What has category theory
done for mathematics? And indeed for evaluation of any subject areas. But a
good case is that category theory leads, or can lead,  and has led, to
clarity, to understanding and development of the rich variety of structures
there are and to be found.  However this does not rate for million $ prizes
(as it should, of course!).

When I see all the current fuss (rightly) about the LHC in Geneva, I do
wonder: who is going to speak up for mathematics, to attract students into
the subject, by getting over a message as to its value and achievements? and
also getting this message over to students studying the subject! (see
`Promoting Mathematics' and  Tim and my article on `the methodology of
mathematics')

Ronnie

www.bangor.ac.uk/r.brown/publar.html

----- Original Message -----
From: "jim stasheff" <jds@math.upenn.edu>
To: <categories@mta.ca>
Sent: Tuesday, September 09, 2008 11:22 PM
Subject: categories: Re: Categories and functors, query


> Walter,
>
> I beg to differ only with
>
> In my experience, skepticism towards category theory is often rooted in
> the fear of the "illegitimately large" size, till today.
>
> In my experience, disdain for cat theory is due to papers with a very
> high density of unfamiliar names
> reminiscent of the minutia of PST and the (in) famous comment (by some
> one) about something like:
> hereditary hemi-demi-semigroups with chain condition
>
> jim
> Tholen wrote:
>> There is another aspect to the E-M achievement that I stressed in my
>> CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given the
>> extent to which 20th-century mathematics was entrenched in set theory,
>> it was a tremendous psychological step to put structure on "classes" and
>> to dare regarding these (perceived) monsters as objects that one could
>> study just as one would study individual groups or topological spaces.
>> In my experience, skepticism towards category theory is often rooted in
>> the fear of the "illegitimately large" size, till today. By comparison,
>> Brandt groupoids lived in the cozy and familiar small world, and their
>> definition was arrived at without having to leave the universe. With the
>> definition of category (and functor and natural transformation)
>> Eilenberg and Moore had to do a lot more than just repeating at the
>> monoid level what Brandt did at the group level! In my view their big
>> psychological step here is comparable to Cantor's daring to think that
>> there could be different levels of infinity.
>>
>> Cheers,
>> Walter.
>