[FOM] John Baez on David Corfield's book

[David Corfield]

I imagine the rest of this newsgroup is tiring of this discussion, so I'll
summarise my case and then call it a day.

Philosophy should treat a knowledge-acquiring discipline as the philosopher
R.G. Collingwood suggests:

There are two questions to be asked whenever anyone inquires into the nature
of any science: 'what is it like?' and 'what is it about?'.of these two
questions
the one I have put first must necessarily be asked before the one I have put
second,
but when in due course we come to answer the second we can only answer it by
a fresh and closer consideration of the first. (Collingwood, The Principles
of History,
OUP: 39-40). [I recommend the whole section pp. 39-44]

Taking mathematics to qualify as what Collingwood calls a 'going concern',
we
need to ask ourselves what it is like to think as a mathematician.  There
are, of
course, many things to be said in answer to this question. My interests lie
in
the way relatively recent mathematicians think, but we need to look at
earlier times
too. [See, eg., Mancosu's on 17th century mathematics.] I hope my book
stimulates philosophers to consider more of these ways than has been the
case in the past. From the NSF grants brought to our attention, we can see
that
understanding/explanation and applicability are on the agenda. Perhaps we
should stop to wonder why it has taken so long for this to be so.

One of the ways in which mathematicians think is to ask themselves whether
what they are doing is of any value. With the freedom gained by entering
Cantor's
paradise comes the responsibility of selecting important topics to think
about.
Von Neumann famously worries that
mathematicians may follow the path of least resistance and develop
abstractions
as art for art's sake. He recommends a return to empirical sources, unless
the path is being pursued by 'men of exquisite taste'. Rota tells us that
Ulam
would tease von Neumann that his continous geometries and vNeumann
algebras were pointless constructions far removed from the requirements
of empirical investigation. They turn out to have been the product of an
exquisite taste, with notable successes in Vaughn Jones' work on knots
and Connes' approach to the Riemann Hypothesis.

I was asked why philosophers should learn what a vector bundle is. Well,
given that modern particle physics is expressed in the language of gauge
theory, and that gauge theory uses the mathematical language of connections
on vector bundles, anyone interested in the changing relationship between
mathematics and physics could tell us a great deal by studying the
development of these concepts. The 1950-75 period is especially interesting
because of a lack of dialogue between the parties, in the West at any rate,
culminating in a reconciliation in the late 70s. Since then interaction has
been intense.

In the mid-70s the physicist Yang was very surprised to find that geometers
such as Chern had developed such similar concepts. Chern responded that
the notions he had devised were 'natural', forced upon him. This perception
of a lack of freedom is something that interests me greatly. Restrictions
are clearly much tighter than being logically consistent.

Now to Collingwood's second question. He's right to say that the questions
are linked. From considering how mathematicians engage themselves
on research programmes we come to realise their aims are not merely to
prove results, but rather operate at different levels. At the highest level,
we find aims such as to produce richer conceptions
of notions such as space and symmetry. I've indicated what I take to be the
best account by a mathematician of current conceptions: Cartier's
(2001) 'A Mad Day's Work: From Grothendieck To Connes And Kontsevich
- The Evolution Of Concepts Of Space And Symmetry', Bulletin of the
American Mathematical Society 38(4): 389-408. Symmetry and more
broadly duality reach right through contemporary mathematics. A gentle
introduction is Vafa's 'Geometric Physics' read at the ICM in 1998.

The study of symmetry broke free of the constraint of groups with the
advent of quantum groups and Hopf algebras. Another line takes us to
groupoids and on more generally to categories. These two lines converge, not
surprisingly as representations of quantum groups form a braided monoidal
category. This ties back very neatly to Jones' work on knots and Witten's on
3-D
topological field theories.

Now you may be thinking this sounds far too difficult. We'll leave all this
to the mathematician. But this decision is not forced upon us. Some
philosophers
still believe they have something to say about the set concept, and
necessarily have learnt a lot of set theory to be able to do so. As
philosophers, are we
to stop
thinking about mathematical space at the point Hausdorff left it? This would
seem
completely arbitrary to me. Even if we don't contribute, we could do some of
our
philosopher colleagues a favour by giving them a sense of what's going on.
A colleague of mine told me of a metaphysician who was shocked to hear from
him that it was possible to conceive of a space with no distance function on
it.
What would he say if we told him that Grothendieck conceived of spaces with
points possessing internal structure?

We have a big task ahead of us, since we've let ourselves get so far behind.
Fortunately, many mathematicians are helpful souls and keen to convey their
images of mathematics.

David Corfield