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*To*: "'fom@math.psu.edu'" <fom@math.psu.edu>*Subject*: FOM: Geometry, arithmetic and nominalism*From*: "Ketland,JJ" <J.J.Ketland@lse.ac.uk>*Date*: Tue, 8 Jun 1999 10:58:04 +0100*Sender*: owner-fom@math.psu.edu

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

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