FOM: Set vs topos debate

[Corfield, David [CES]]

One of the benefits to arise from this debate is the provision of material
for anyone wanting to try out Kuhnian or Lakatosian concepts. I imagine
that Kuhn would come off better. It is hard to imagine anything like the
*honest score-keeping* between rival research programmes coming
about, when goal posts can't even be decided on. But is it full-fledged
incommensurability?

We now know that things are a little more involved than membership to
this or that paradigm - there are different levels of commitment to beliefs
operating at different levels. Philosophers of science today also tell us to 
be
more *reflexive* . So maybe we should see where this debate fits into the
larger scheme of things.

If we think back to the Felix Klein-Weierstrass debate, or at any rate
Weierstrass's comment 'But what have groups to do with geometry?',
this wasn't just a war of words. Klein's vision of a more intuitive and
inter-connected  math was realised in the kind of education programmes
he set up and it had an impact on the direction of mathematics. Does either
side of this debate have ideas about where it wants to see mathematics
going, or the way it is taught?

Even if you maintain that you are just interested in FOM, you are presumably
hoping to protect or expand your patch and no doubt would act against the
appointment of someone from the other side. So the debate could have
effects on people's lives.

But there are even larger ramifications of the debate. There have
been consequences (possibly unintended) of the 'set theory as
foundation of math' picture. It has worked its way into the minds of the 
larger part
of Anglo-American philosophers. Are FOM set theorists happy with that, or do
they take  their work to be misappropriated? If the latter, should they not 
say so?

As several people have noted, set theoretic semantics tends to accompany
an immediate realism. Has the height of absurdity not been reached when
 we are told that '2 + 2 = 4' is false because 2 does not refer, but is a 
useful
 fiction allowing us a shorthand so as to avoid having to talk about 
different
 space-time regions?

Fortunately Penelope Maddy has realised the error of her ways concerning
the relationship between set theory and realism, and now studies the
methodology of set theory. But, as she agreed, this means that the 
methodology
of the geometric branches is also worth exploring.

Note, I'm not blaming set theorists for this, you could develop equal silly
things out of category theory. I simply wonder whether broadcasting loud
and clear that set theory has little bearing on the rest of math and should
not be used by philosophers in certain ways might allow some departmental
space for philosophers interested in the development of, say, differential
geometry, and how this relates to Poincare and Einstein's comments about
thinking with the body image or musculature, and to the fact that some 
people
working on artificial vision systems seem to want to pack a lot of 
differential
geometry into their machines. In sum, to carry out the kind of programme 
Hersh
has suggested. While he points out that there are empirical aspects here, I 
am
sure he had no intention to exclude philosophers from the programme.
What we need is a naturalised phenomenology, as suggested by Joseph
Margolis.

Given the list of suggestions of what should be on the FOM list, I can see
that this programme might need a new home. Anyone for a NatPhenOM-
list?

David Corfield
School of Cultural Studies
Leeds Metropolitan University
U.K.

Interests: Historicist philosophy of mathematics, cognitive psychology




it like Rota to be people trying to pay with monopoly money. If they are
pleased with the analytic philosophy of mathematics