FOM: FOM

[Graham White]

I want to make a few remarks about the tone of discussion in FOM, which
I've been finding pretty damn awful. I've been considering formulating
an academic argument against you guys, but I think it's probably more
direct and accessible if I simply let off steam in this manner.

First, my background is in category theory; I have a doctorate in
algebraic geometry, and I'm now working in computer science. Along
the way I've done a lot of work in philosophy. Now philosophically
I'm not a foundationalist; I *don't* believe, that is, that the question
of foundations is *necessarily* the first, or the most important, in
a particular area of philosophy. That doesn't mean that I'm a post-modernist,
or an anti-realist, or a relativist, or any of those positions; it just
means that I disagree with the foundationalists on that particular point
of philosophical methodology. (And, in fact, I'm a realist, I'm also
not a relativist,
and I think that post-modernism is to philosophy as Madonna is to music; but
none of that stuff is relevant to what I'll be saying.)

Given that, I also think that some things in category theory are of
considerable philosophical interest (but not because they're foundational).
Namely, I think that category theory gives an interesting formal account of
mathematical practice; it tells us extremely interesting things about identity;
and it may also give us an interesting formal account of intensionality.
Now all of this is philosophically interesting, beyond the bounds of
philosophy of mathematics; philosophers in general would like to know
about practice, about indentity, and about intensionality. It seems
to be an interesting philosophical furrow to plough.

However, I've not got the least interest in converting you guys; it seems
to me that our projects ought, ideally, not to clash. I've subscribed to
this list because I like to keep up with what's going on in reverse
mathematics. If someone could come up with a good logical account of
foundations, that would, of course, be interesting, and the stuff that one
discovers in an attempt could also be interesting (and maybe not even
interesting for the reasons that you try to do it; an awful lot of things
in mathematics have turned out to be interesting for different reasons than
those for which they were first attempted).

Under the circumstances, then, I find the tone of this list rather
distressing. Here are some examples:

I) From the exchange between
Thomas Forster and Stephen Simpson:
>[Forster]
> > My feeling is that an important source of hostility to set theory
> > arises from mathematicians interpreting the foundational claims of
> > set theory as somehow deconstructing their activity, and nobody
> > likes being deconstructed!
>[Steven]
>Please define what you mean by "deconstruct".  This vague term
>(borrowed from modern literary theory of the politically correct
>variety) blurs a lot of important distinctions.
>
>Do you mean that many mathematicians don't like outsiders to analyze
>what they do in terms of general intellectual interest?

This seems to beg the question: it assumes that the only way of
"analysing [mathematics] in terms of general intellectual interest"
is a *foundational* analysis. Maybe one could argue that the connection between
mathematics and matters of general intellectual interest is fairly
innocuous, but that foundationalism is perceived as being too pre-emptive?

II) Stephen again:
>it's the job of f.o.m. to explain the meaning of
>mathematics.  So I prefer to confront these hostile mathematicians
>head on, rather than meekly surrender to them or try to get along with
>them, as Forster seems willing to do.
Seems unnecessarily monopolistic; *only* fom "explain[s] the meaning of
mathematics", and anyone who disagrees is "hostile".

III) From the same email:
>From [one] point of view, set theory is a respectable
>but narrow niche; it is probably not of much interest to most
>mathematicians, and it certainly does not have much in the way of
>general intellectual interest.
>
...
>On the other hand, there is the viewpoint that set theory is *not*
>just another branch of mathematics.  Rather, set theory has a special
>significance as a very successful foundation for *all* of mathematics;
>this is the view of G"odel, Cohen, Friedman, ....  From this point of
>view, set theory has tremendous general intellectual interest and is
>potentially of great significance to all mathematicians.  This is
>obviously a much loftier and more ambitious view of set theory.
Again, a rather arbitrary dichotomy: either set theory is a narrow
mathematical niche, or it is of "general intellectual interest" precisely
because it's foundational.


III) Stephen:
>In fact, most of the recursion
>theorists are boycotting FOM.  Only a few of them are subscribers, and
>they have posted hardly anything.  I wonder why.  Do they explicitly,
>consciously regard foundations of mathematics as irrelevant to what
>they are doing?

"Boycotting"? Is this something the recursion theorists have *organised*
among themselves? Or do they just individually find this an unpleasant list
to be on?

There seems to be a false dichotomy here: either you think of fom as
completely irrelevant to your activity, or you subscribe to the fom list.
But there are lots of positions in between; you might, for example, regard
fom as, in principle, relevant, but you might not want to refer to fom at
every turn.

IV) Stephen (replying to Thomas Forster):
>I like Adrian's article "The Ignorance of Bourbaki", which has been
>discussed here on FOM.  In particular, my posting of 17 Nov 1997
>16:20:11 contains a review from Mathematical Reviews.  But I don't
>think Bourbaki is the only source of hostility to f.o.m.  The full
>explanation of the hostility is unclear, but my best analysis is that
>it's part of a general trend toward anti-foundationalism and
>compartmentalization in academia.
OK. Two points here.
First, I just don't believe that Bourbaki is part of a general trend to
"anti-foundationalism
and compartmentalisation"  Andr'e Weil a compartmentaliser? Dieudonn'e a
compartmentaliser? Serre a compartmentaliser?
This is so daft I can't take it seriously, and maybe you didn't mean it
in that sense.

The second point is that there's again this dichotomy: either you're a
foundationalist, or you're a compartmentaliser.

V) Stephen:
>My problem with the category theorists, Bourbakians, and other
>structuralists is precisely that they fail to recognize this.  They
>delude themselves into thinking that structure alone is important.
>This has resulted in further fragmentation and isolation of
>mathematics from the rest of human knowledge.  Jeremy, you are trying
>to make friends with the structuralists, but is it worth it?
>
Do we? I don't. This seems to be a confusion of a mathematical area
(category theory) with an ontological position (structuralism). Well, you might
take the view (as I do) that category theory is philosophically interesting
without necessarily being a structuralist. (You might also be a
structuralist without also believing that "structure alone is
important", just as you might believe that everything is material without
also believing that quantum theory alone is important.) I also can't see
why this results in "fragmentation and isolation".

Well, enough of this. I'd really like to be a member of this list and enjoy
the interesting things you people have to say about reverse mathematics,
and also I would not like to feel ranted at every time I get a message
from this list; but the second seems not to be possible. Ideally I would
also like to have some reasoned, technically aware,
 *philosophical* discussion about the relation between mathematics and
human culture in general; but the discussion on this list seems never to
go into the philosophical arguments at all, merely to use philosophical
slogans as a stick with which to beat imagined opponents with. Shame.

Still, it's sociologically interesting; only list I've ever been on when
the list owner did more flaming than the rest of the list put together.

Yrs,
Graham White