David Corfield david.corfield at philosophy.oxford.ac.uk
Fri Oct 17 12:32:08 EDT 2003

is that it's not always the case that mathematics gets conceptually harder
up at the frontiers of research. As you can see from the last chapter of my
book, I'm indebted to the expository work of John Baez. One of his views,
which he shared with the late Gian-Carlo Rota, is expressed in the
following:

"...it makes sense to spend at least a little time going back and thinking
about simple things. This can be a bit embarrassing, because we feel
we are supposed to understand these things completely already. But in
fact we do not. And if we never face up to this fact, we will miss out on
opportunities to make mathematics as beautiful and easy to understand
as it deserves to be." (Baez and Dolan, 'From Finite Sets to Feynman
Diagrams', Mathematics Unlimited - 2001 and Beyond, also available
online)

One small example  from the field of combinatorics, where the concept
known as a 'species' is beginning to take hold: one can understand
much about the exponential function by knowing it corresponds to
the species 'count set structures' on a finite collection. E.g., the fact
the
exponential function is unchanged under differentiation corresponds
to the fact that the number of set structures on a finite collection (i.e,
one)
is not changed by tossing another object into the collection before
counting set structures.

The complicated looking commutator relations for annihilation
and creation operators turn out to reduce to the fact that
there is one more way to toss an extra object into a collection
then select one, than there is to select an object and then toss
in an extra object.

Rota, whose advocacy of species was drawn to the attention of this
list, describes the "bottom line" of his mathematical interests as the
placing of balls in boxes. Continuing the species line, I had an
exchange with Baez on sci.math.resesearch where it became clear
that the appearance of the Bernoulli numbers in physics has much
to do with the idea of putting things into boxes.

Still, the gap between philosophers and research mathematicians is extremely
wide. As Friedman says,

>In the present culture, one can expect mathematicians to be about as good
>at, or interested in, philosophy, as philosophers are good at, or
interested
>in, mathematics.

Anyone making a serious effort to construct a bridge between the
disciplines should be actively encouraged.

Be especially generous to those putting down piles in the middle of the gap.
(Martin Krieger's 'Doing Mathematics' might be taken as an example.)

It is not necessarily the case, (I would say it is not the case) that the
most
promising piece of territory to build the philosophical end of the bridge
is analytic metaphysics or philosophy of language.

Let's learn as much as we can from parallel bridges, e.g. philosophy of
science. An opportunity was lost in the reverse direction when
philosophers ignored Polya's work. Whatever you think of the probabilistic
approach to epistemology known as Bayesianism, much effort
was expended by philosophers of science reinventing Polya's work.

We can be helped by the fact that as philosophers of physics continue to
press forward to the physics of the past 30 years, they are learning
plenty of frontier mathematics. We should combine to examine the
role of mathematics in physics.

Many mathematicians are very generous with their time.

The greater the number of people engaged on the project, the less daunting
it will seem.

David Corfield
http://users.ox.ac.uk/~sfop0076/