FOM: Arithmetic vs Geometry : Categoricity
Charles Silver
csilver at sophia.smith.edu
Fri Oct 9 06:28:18 EDT 1998
On Thu, 8 Oct 1998, Neil Tennant wrote:
> Thanks to Joe Shipman for an interesting rejoinder. So now:
> 1) There are competing (axiomatized) geometrical theories. Some are complete,
> hence decidable. Some are (?) categorical in the power of the continuum.
> And it is an empirical question which of them describes physical space
> (or spacetime), upon a suitable operationalization of the notion of a
> geodesic.
> The *geometrical* problem has this flavor: Out there is, presumably, a
> unique geometrical structure that we call SPACE (or SPACETIME). Let's
> conduct empirical investigations to find which (from among many possible
> competing theories) truly describes it.
>
> 2) There is a unique, a priori structure known as the standard natural
> numbers. There are no seriously "competing" theories trying to describe it.
> There is only one theory, whose determination should be completely a priori;
> and that theory is Th(N). The trouble is, as G"odel showed, every sufficiently
> strong axiomatized subtheory of Th(N) is incomplete; whence Th(N) is
> unaxiomatizable.
> The *arithmetical* problem has this flavor: "Out there" (metaphorically, now!)
> is a unique structure that we call THE NATURAL NUMBERS. Let's conduct purely
> a priori investigations to axiomatize successively stronger and more inclusive
> subtheories of its theory, knowing that the job will never be complete.
I take it that Hersh and others would think you are radically
misdescribing mathematical activity, though I believe in their
descriptions of mathematical activity they'd have to include somewhere the
fact that mathematicians may *think* there's a unique structure out there.
I'm a little puzzled as to how a sophisticated version of this would go.
Would one who was describing mathematical activity, but totally rejected
Platonism (and perhaps supported something like "humanistic mathematics"),
have to say something to the effect that many of the poor slobs who *do*
mathematics think their work is guided by the existence of silly things
like unique structures? But we, acting as cultural anthropologists
observing their behavior, can see that they are really just engaged in a
lot of scribbling exercises, besides hanging out together and speaking in
a contrived language which seems to give them pleasure?
Charlie Silver
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