FOM: arithmetic vs geometry
Stephen G Simpson
simpson at math.psu.edu
Mon Oct 5 21:10:36 EDT 1998
Edwin Mares 30 Sep 1998 15:20:02 writes:
> Frege explicitly wanted to reduce arithmetic to logic to
> prove that arithmetical statements are analytically true. But he
> didn't think that this was true of geometry, say, on which he seems
> to have taken a rather Kantian line.
Absolutely. This is a good correction. Rather than say that Frege
tried to reduce mathematics to logic, we should say that Frege tried
to reduce arithmetic to logic. This seems truer to what Frege says.
However .....
How does this dovetail with Weierstrass's reduction of analysis and
geometry to arithmetic? Did Frege accept that idea?
Detlefsen's long posting of 2 Oct 1998 13:22:16 is fascinating. Just
to make a start on the many issues raised there:
Detlefsen asks:
> QUESTION IT WOULD BE PROFITABLE TO DISCUSS 1: Is or should the
> asymmetry between arithmetic and geometry that Gauss and nearly all
> other 19th century foundational thinkers believed in still be
> treated as a fundamental 'datum' of the foundations of mathematics
> today?
I guess the orthodox line today is that arithmetic and geometry are
all of a piece: geometry is a kind of real analysis (locally Euclidean
spaces, i.e. spaces that locally look like R^n, where R is the real
line) which is based on arithmetic (Dedekind or Cauchy construction of
the real line). But obviously this papers over a lot of difficulties.
Even today, some thinkers want to view geometry as a branch of physics
rather than mathematics.
The influence of Kant in all this is striking. Will we never rid
ourselves of these crippling dichotomies?
-- Steve
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