FOM: Geometry, arithmetic and nominalism

[Ketland,JJ]

In an earlier posting, Steve Simpson mentioned an idea that Tarski's
elementary geometry augmented with a predicate Z for the integers forms a
system GZ bi-interpretable with second-order arithmetic Z_2. (I suppose that
lines and regions become "sets" of "integers").

I don't know about detailed mathematical work on this, but something vaguely
similar was developed by the philosopher Hartry Field in his 1980 book
"Science Without Numbers". Here the idea is to take a certain (monadic,
second-order) geometry as basic and use it to develop all the standard
mathematical apparatus. Field explains how certain ideas can be developed
within this system - e.g., modelling the integers as an infinite discretely
spaced sequence of points on a line.
Field explains how representation and uniqueness theorems for the formal
system can be proved (analogous to Tarski's), identifying models of the
geometry with standard geometric structures built up using the reals.
Field's idea is especially interesting since he argues that 
	(a) the system does not refer to specifically abstract mathematical
entities (i.e., the 	points and lines and so on are concrete elements of
spacetime)
	(b) we don't need to introduce any further abstract mathematical
apparatus (sets 	and functions) beyond what we've already got.
If this argument is right, it shows how the mathematical apparatus used in
scientific theories of physics (e.g., field theories) might be *dispensable*
in favour of a concrete geometrical ontology: Field argues that this
vindicates a version of (non-finitistic) nominalism. (I.e., abstract
mathematics is just a useful instrument, useful in deducing things about the
physical world, but always dispensable in principle).

A similar idea is developed by John Burgess in a paper "Synthetic Mechanics"
(1984: Journal of Philosophical Logic 13), and more fully in the 1997 book
cited below.

In fact, Field bases his nominalistic claim on a certain conservativeness
claim: given a nominalistic theory N of the physical world (no explicit
reference to numbers or sets etc.), the result of adding mathematical axioms
(e.g., some set theory) should always be a conservative extension of N.
This latter claim seems, under some conditions, to contradict Goedel's
second incompleteness theorem: since if N is recursively axiomatized,
consistent and contains an interpretation of Peano arithmetic say, then it
will be incomplete (and Con(N) will not be a theorem) and if standard set
theory *can* prove that N is consistent, then adding this set theory to N
will prove Con(N), so the extension is not conservative. (As has become
clear, this result depends upon a subtlety: one must expand the axiom
schemes in N to include the set-theoretical membership predicate). Field's
approach has been criticized along those lines, notably by Stewart Shapiro
(1983, Journal of Philosophy, "Conservativeness and Incompleteness") and
John Burgess (see Burgess and Gideon Rosen 1997, "A Subject With No Object:
Strategies for Nominalistic Interpretation of Mathematics", Oxford:
Clarendon Press, pp. 190-196).

Jeff Ketland

Dept of Philosophy, Logic and Scientific method
London School of Economics