[FOM] Bays' review of my book

[David Corfield]

I hope I may be permitted to air on this list a response to a recent review
of my book by Timothy Bays
(http://ndpr.icaap.org/content/archives/2004/1/bays-corfield.html ),
inasmuch as in so doing I shall be continuing a discussion we had here on
FOM late last year. I recognise that some FOMers are not interested in this
discussion. I respect this sentiment and would be only too pleased if
someone were to set up a new forum for such matters. In the meantime, I'm
maintaining a web-page to which I would be happy to add contributions from
other people (http://users.ox.ac.uk/~sfop0076/phorem.htm).

After a fair summary of the contents of the book, Bays arrives at two
conclusions, one negative and one half-positive:

That I do not give sufficient arguments to establish the inadequacy of
contemporary approaches to philosophy of mathematics.

That my philosophically suggestive work could well encourage philosophers to
learn more contemporary mathematics (a good thing), but that as it stands it
not fully-fledged philosophy.

Thinking back to the not-so-golden age of last Autumn on FOM, I think I
begin to recognise what is at stake here: differences in respective
philosophical bottom lines. My bottom line is that I want to gain a clearer
understanding of what mathematics is like as a knowledge-acquiring
discipline. Foundational studies have contributed greatly over the past
century, but increasingly over recent years it has become evident that
important dimensions of mathematics have been overlooked. These dimensions
include some of excellent philosophical pedigree such as the sense we can
make of the notion of developing concepts correctly, which it has taken us a
devil of a long time to realise motivated founding fathers such as Frege
(see several papers by Tappenden on this). If at times I appear negative
towards the status quo, it's brought about by the sense that we're depriving
a generation of young people of exciting areas of research. I get young
graduates writing to me for advice on what research topics they might
pursue, but have to warn them of the dangers of proceeding along a path
where recognition will be hard to come by. [Philosophy of physics may not be
different in this respect. See the end of John Earman's article
(http://philsci-archive.pitt.edu/archive/00000878/) encouraging young people
to take a serious look at how laws and symmetry work in physics despite
undoubted opposition.]

[By the way, the term "silly questions", cited as evidence of my antipathy,
is quoted out of context. I was referring to the work of Makkai to construct
a language in which it would be impossible to ask mathematically silly
questions. E.g., Which is the symmetry group of the sphere?, Which simple
Lie group corresponds to Dynkin diagram B_1?. "Are they the same?", "Yes",
"OK, they're isomorphic, but are they identical?". Similarly, when asked
"Are these two categories the same?" you want to know whether they're
equivalent rather than isomorphic or identical.]

As for my work not being philosophically achieved, I could argue that it is
hardly to be expected that one could produce novel approaches fully-armed,
as Athena emerged from the head of Zeus. But this is to miss the point,
because however well I'd done in my own eyes I don't think I would have
satisfied someone like Bays. This points to a difference of opinion on the
nature of philosophy. That his view is not universal I know from the
appreciative comments I have had from other philosophers, mostly outside the
philosophy of mathematics fold, for each of the chapters. Even chapter 9 on
naturalness, one of those singled out by Bays as not yet philosophy, has had
its admirers, including Jack Smart in a very kind letter. One might have
thought he was someone who could recognise the real McCoy.

We should accept that there are many different ways of doing philosophy. I
feel sure that my list of philosophical heroes does not overlap
substantially with Bays'. Something which characterises my list is an
absence of problem-setters or solvers. This no doubt marks a difference from
Bays:

"If Corfield's book inspires young philosophers to take more courses in core
mathematics, then it will have performed a genuine service. If they are able
to apply their knowledge of core mathematics to generate new philosophical
problems and/or solve old ones, then Corfield's book will turn out to be
something of a watershed."

When I discuss the mathematicians' notion of naturalness it isn't to turn it
into a problem. I don't want to give necessary and sufficient conditions for
its use and I'm not desperate to tie it to some formal system (although they
category theoretic notion of 'natural transformation' is an important
reference). I want to understand the presuppositions operating in a
community which could allow such talk to be sustained. Is it one in which
ideas can be said to force themselves upon practitioners, demanding that
they be used in specific ways? If I avoid posing philosophical problems, am
I thus engaging in merely philosophically suggestive description, as Kuhn's
work was once described? Jack Smart didn't think so. Perhaps members of that
generation have experience of a broader range of styles of philosophy.

Of the responses to Kuhn's 'Structure' which went beyond mere dismissal as
historical description, one attempted to tame it by viewing it as just
Carnapianism with some history added. Lakatos, however, was more concerned,
detecting that Kuhn was making a serious case for a novel (to Anglo-American
ears) sociological form of relativism: knowledge is a social kind. Lakatos
resists this relativism by fashioning Popper's World 3 in his own
historically-minded way. Neither Kuhn nor Lakatos were problem solvers.

Let me reveal one person on my list of heroes, who appears there for the
reason that he is the philosopher I most frequently 'bump into'
intellectually speaking, namely, Ian Hacking. Hacking is gloriously
pluralistic as his 'Historical Ontology' shows. (You'll find there the
interesting thesis that philosophy constituted as a series of problems is an
invention of relatively recent date, namely early 20th century.) Hacking's
pluralism is broad enough to include other exponents of "knowledge is a
social kind", Pickering and Latour, noted figures from Science Studies, for
their use of the history of recent science to make the philosophical case
for the contingency of scientific concepts, a parallel position to the one
which would find the naturalness of a mathematical concept to be always
socially constructed.

The question then arises as to whether my preference for more
historically-aware styles of philosophy, rather than the search for timeless
solutions to problems, and my wish to broach alternative dimensions of
mathematics are linked. Perhaps. But feel free any of you to treat topics
such as plausibility and naturalness in your own fashion. Just let's be very
careful about playing the "That's not philosophy" card. It has a nasty habit
of rebounding. Someone who used it on me once, was galled to have it branded
at a paper of his on Ramsey's logic by some very prominent UK philosopher. A
discipline which can embrace Plato, Hegel, Carnap, Rorty, Habermas, Lakatos
and David Lewis must know that it does not operate within very tight
boundaries.

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