FOM: re: arithmetic and geometgry

William Tait wtait at
Tue Oct 6 12:40:06 EDT 1998

Steve S. wrote (10/5 at 9:10)

>How does this dovetail with Weierstrass's reduction of analysis and
>geometry to arithmetic?  Did Frege accept that idea?

 Frege did not like the construction of the real numbers by
Weierstrass---or Cantor, Dedekind, Heine, or Meray---for reasons that do
not seem to me valid. But he himself had a project, never completed, to
construct the real numbers.
>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:
> > 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.

Kant, Gauss, and---mysteriously, even as late as 1924---Frege, when they
spoke of geometry, meant the science of physical space.  Gauss, recognizing
the possibility of alternative geometries as coherent possible descriptions
of space, rejected geometry (in this physical sense) as a priori. Kant,
prior to this, and Frege, after, maintained that physical geometry is a
priori. The interesting question, though I think only an historical one, is
why Frege held this view.

>Even today, some thinkers want to view geometry as a branch of physics
>rather than mathematics.

 Why aren't they simply using the term `geometry' in  a different way, to
describe what they believe to be the structure of spacetime? I don't see
this as a conflict in beliefs.

Bill Tait

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