FOM: Arithmetic vs Geometry : Categoricity

Robert Black Robert.Black at
Fri Oct 9 12:25:50 EDT 1998

I claim that if one is structuralist about arithmetic there isn't much of
an asymmetry between arithmetic and geometry.  Neil Tennant objects that
there's still the difference between arithmetic as a unique (up to
isomorphism) structure, while sets of geometrical axioms typically have
many nonisomorphic models and are thus better compared to the axioms for,
say, groups or rings.

Of course if you take something like the general axioms of affine geometry,
the comparison will be with something like the general axioms for a group.
But if you take a *particular* affine geometry it's a structure unique up
to isomorphism.  If you're a structuralist about arithmetic then you'll say
that talking of *the* natural numbers is like talking of *the* octahedral
group, or of Euclidean 3-dimensional space.  That's all I meant.

Joe Shipman, if I understand him aright, makes basically the point that
first-order Euclidean geometry is (as Tarski showed) a decidable theory,
while first-order arithmetic isn't.  That's true, but I think it may be
less of a difference than it looks at first sight.  First-order Euclidean
geometry is decidable so long as its predicates are suitably restricted
(e.g to relations of colinearity, betweenness and congruence).  And this is
basically because the first-order theory of R as a field (i.e. with the
language restricted to addition and multiplication) is decidable.  If we
add to the first-order language of R as a field a predicate meaning 'is a
natural number' then of course the whole of first-order arithmetic's in
there, and it's no longer decidable.  And obviously we'll be able to
similarly make first-order geometry undecidable by adding suitable extra
predicates.  Is the fact that first-order geometry restricted to
colinearity/betweenness/congruence is decidable any more significant than
the fact that first-order arithmetic restricted to addition ('Presburger
arithmetic') is decidable?

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845

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