FOM: Arithmetic vs Geometry : Categoricity
Joe Shipman
shipman at savera.com
Thu Oct 8 15:58:52 EDT 1998
Are we really sure categoricity is the issue?
How about "suffiency of axiom sets"?
The discovery of non-Euclidean geometries showed that a well-known set
of axioms for geometry was incomplete and noncategorical. Later
axiomatizations of various geometries extended the earlier set in
different but *complete* ways.
Godel showed that a well-known set of axioms for arithmetic was
incomplete and noncategorical, and that it could *not* be extended to a
complete set. (If you go to second order axioms you can pick out a
unique "categorical" structure, but that doesn't help you with what you
really cared about in the first place, namely deriving the true
first-order sentences).
In the case of geometry, the sufficiency of the new foundations for
deriving the true first-order sentences meant that one could forget
about models and a priori intuitions and adopt a completely synthetic
viewpoint (though the infeasibility of the algorithm for truth presents
some problems).
In the case of arithmetic, our conviction that sentences ought to have
TRUTH VALUES makes the view that there is a real, unique model
attractive, and we don't have the synthetic alternative by Godel's
theorem.
Before you respond by saying "we have an intended model and we just want
to know the true sentences for this model", recall that there was an
"intended model" for geometry as well (R^n). The reason the discovery
of non-Euclidean geometries was a surprise was twofold:
1) The parallel postulate was indispensable after all
2) The non-Euclidean geometries were new candidates for the geometry of
the physical world, because the parallel postulate was never as apparent
to physical intuition as the others (that's the reason people tried to
dispense with it in the first place!).
(It is possible to disagree with this and maintain that the "intended
model" was physical space and not R^n; while this may have been true
before Descartes, after him it was possible to adopt a pure-mathematical
viewpoint and ignore physics entirely.)
So is there really a difference betwen the cases of arithmetic and
geometry? I submit that before Godel's theorem clarified the situation
there was not. Just as a subset of the axioms Hilbert et al laid down
for real geometry applied more generally to other structures, a subset
of the Peano axioms for arithmetic (the ring axioms, for example)
applied more generally to other structures. Hilbert found arithmetic in
the same state geometry had been in before he and Pasch and others
cleaned it up; but Godel showed arithmetic could not be "cleaned up" in
the same way.
-- Joe Shipman
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