[FOM] John Baez on David Corfield's book

[David Corfield]

Stephen G Simpson wrote

> David Corfield, Sat, 27 Sep 2003 16:44:22 +0100 writes:
>
>  > I don't recall using the word 'foundationalism'. The 'foundationalist
>  > filter' I object to is [...]
>
> Aren't you evading the point here?  Even if you didn't use the word
> "foundationalism", you certainly used the derivative word
> "foundationalist".

It's not evasion. It's clarification. I'm not objecting here to
foundationalism,
but the view that philosophers should treat mathematics solely via the
foundational
disciplines.

> The question remains, do you agree or not with my point that the vast
> majority of mathematicians choose to organize their mathematical work
> in a foundationalist manner, with rigorous definitions, theorems,
> proofs, etc?  And, do you agree or not that the purpose of
> f.o.m. research is to analyze this logical, hierarchical structure?

Even if I answer yes to both, I still maintain that there's more to
mathematics of philosophical interest than that f.o.m picks up.
The excess is most clearly revealed in expository writing.

>  > the conception common to the majority of philosophers of
>  > mathematics that the ONLY aspects of mathematics of philosophical
>  > interest can be detected by proof theory, model theory, set theory,
>  > recursion theory.
>
> This seems remarkable.  Is this really a common conception among
> philosophers of mathematics?  Have any of them actually said this, or
> is this merely an inference on your part?  Aren't you putting words
> into their mouths?

The problem is an institutional one. I confess that I have received a
considerable amount of encouragement over the years, but it's
usually coloured by the sentiment "You brave but crazy fool, you're
committing academic suicide".

Graduate training requires huge investment in learning logic, set theory,
etc.
with no encouragement to find out what a vector bundle is. Getting
published requires you to develop the dominant programmes, hence so too
does getting a job. As for funding, I don't know how the money for
philosophy of math
is distributed in your country, but over here, a huge chunk goes to the
Neo-Fregean
programme.

The two small glimmers of hope that it's possible to do something different
that
gets noticed are those doing history of philosophy of mathematics,
who tell us we've got a lop-sided view of Frege, Hilbert, etc., and those
trying to
formulate theories of understanding and explanation in mathematics.

>  > Notice that this does NOT imply that these theories are of no
>  > philosophical interest.
>
> OK, good.  Thank you for that ambigous concession, in the form of a
> double negative.  Maybe you are not unrelentingly hostile to
> f.o.m. after all.

I never said I was.

> However, you are still attacking f.o.m. in ways that seem highly
> dubious.  For example:
>
>  > I meant f.o.m. research has no bearing on choice between rigorously
>  > defined concepts.
>
> Again, I have to disagree.  It seems to me that f.o.m. research has
> often had a decisive influence on choices among rigorously defined
> concepts.
>
> An example is the pervasive concept of topological space,
[snip] If set-theoretical f.o.m. were
> not in vogue, then mathematicians would surely have chosen some other
> concept.
>
> Don't you agree?

When you read a sentence expressed in the present tense it is necessary to
decide which form is being used. There are sentences such as
 'cats are mammals' which one takes to be timeless and sentences such as
'young people are disrespectful' which come with a sense that young
people once weren't ("In my day,...). I have written in another post that
foundational and methodological issues were interwoven in the 1880-1930
period and that they have since come apart, and how Frege would have
been disappointed by this. My claim was obviously time restricted then.
So invoking point set topology and Dedekind cuts won't do.

[By the way, I briefly treat Baldwin's point (one also made by Kreisel) that
algebraic topology has proved far more important for math than point-set
topology in chapter 8 of my book.]

What I doubt is the profound impact of the f.o.m activity of the past
twenty or so years on the conceptual decision making of: Connes, Deligne,
Drinfeld, Kontsevich, Manin, Arnold, Yau, Donaldson, Givental, Gromov,
Gowers, Borcherds, Thurston, Chern,...

> Maybe I missed it in the flurry of postings today, but I think we are
> still waiting for someone to present an example of a piece of
> non-foundational mathematics that is of philosophical interest.  I'm
> not asserting that there is no such thing, but surely it is desirable
> to have at least one good example on the table.

> In particular, I would like to know what philosophical questions are
> addressed.  Are they "real" or "core" philosophical questions, or are
> they merely peripheral ones?  And, how compelling are the answers to
> these questions?

When philosophers treat a knowledge-acquiring discipline, a large part
of what they do is to study the means by which this knowledges grows.
In this case, there's no need to point to a particular concept or theorem,
except by way of a case study to illustrate some of the forms this growth
takes. A large part of philosophy of science concerns this task, e.g., at
the
moment the role of modelling in science is very prominent. Scientists have
of course been using models for centuries, but it's only recently that
they've
been given an adequate treatment. In, say, 'Models as Mediators' you
will see the contributors treating case studies, not presuming that their
particular model is intrinsically interesting for the part of the world that
is modelled. There's an enormous amount of work for philosophers of math
to do in this vein.

Another side of the philosophy of X is to look at the basic concepts of X.
So,
concerning physics, we have discussions of time, cause, symmetry, etc.
The kinds of things we might look in math are space (take a lead from the
Cartier article I mentioned before), symmetry, and duality (from projective
geometry and Fourier analysis right up to mirror symmetry).

The only vaguely plausible way at present of treating these kind of concepts
in
a formal, linguistic framework is category theory. I would be happy to call
this
sort of study 'foundational', but I somehow don't think all the members of
this
list would be too happy.

David Corfield