FOM: applied model theory; foundations of mathematics

Josef Mattes mattes at
Wed Oct 22 20:38:36 EDT 1997

Dear colleagues,

There have been a number of messages discussing the question of how
'foundational' Faltings' theorem on rational points is. I have to admit
that despite being a mathematician I never could get myself to get excited
about rational points on curves. 

More importantly, these rational points seem to have overshadowed some
earlier remarks which seem to me to deserve a reply: 

On Fri, 17 Oct 1997, Stephen G Simpson wrote:

> > This is confirmed by what happened in more recent times when
> > subareas of mathematics did undergo an overhaul of their
> > foundations, as shown by the examples of probability theory
> > (Kolmogorov), and algebraic geometry (Zariski, Weil, Grothendieck).
>These examples confirm nothing of the sort.  Overhauling a specialized
>area such as algebraic geometry is very different from overhauling the
>most basic concepts of mathematics and the relationship of mathematics
>as a whole with the rest of human knowledge.  The former is in the
>domain of pure math; the latter is the domain of f.o.m.

I think this seriously underestimates the consequences of the
work of Grothendieck et al. For example, these ideas are at the very
foundation of 'noncommutative geometry' and 'quantum groups' with
deep and important relations to measure theory, von Neumann algebras,
knots, string theory, etc. 
In another message we read that the foundational approach regards the
unity of human knowledge as paramount. Well, nocommutative geometry seems
to be doing quite well in this respect: 1.) They arose out of statistical
mechanics 2.) Connes came up with a new, elegant model unifying gravity
with the other fundamental forces of nature. Surely the problem of fitting
gravity into the rest of physics is of importance even on a general
intellectual scale?

Goedel's theorem has been quoted as an example of being of interest to
non-mathematicians. I just wonder, how much interest would there be if it
were not a statement about mathematics? Without interest in
mathematics/computation, who would be interested in Goedel?

There has been a question as to whether working mathematicians still
discuss what a proof is. Manin wrote somewhere that every mathematician
who thought about it knows what a proof is. Nevertheless, there is
discussion as to what the status of conjectures and other unproven
statements is, surely a foundational question I would think; see
Jaffe-Quinn in BullAMS 29#1 and Atiayh et al. in Bull.AMS 30#2.
Interestingly, I don't see sets mentioned there at all.

Josef Mattes

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