FOM: Arithmetic vs Geometry : Categoricity
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Thu Oct 8 21:54:43 EDT 1998
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.
Isn't that still a substantial contrast between the two kinds of theorizing?
Neil Tennant
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