Reconstructive Nominalism and Representation Theorems

6 04 2005

In “Science Without Numbers”, Hartry Field shows how to give a nominalistic theory for Newtonian gravity that agrees with established, platonistic theory in all its nominalistic predictions. One part of showing that it agrees is by showing that (assuming ZFC) any model of the nominalistic theory is isomorphic to a submodel of the platonistic theory.

In “A Subject With No Object”, Burgess and Rosen reconstruct Field’s argument, along with those of Chihara and Hellman, by trying to show that each one of them is able to construct a nominalistic theory over which the platonistic theory is conservative. However, rather than accepting Field’s (admittedly somewhat weak) arguments for the conservatism of mathematics in general, they try to prove a reverse representation theorem, establishing that the real numbers can be represented by some k-tuples of physical objects actually countenanced by Field’s theory. If all reference to real numbers can be replaced by reference to (say) triples of space-time points, then clearly we can translate the platonistic theory into a purely nominalistic form and preserve all the standard results.

However, this was definitely not Field’s strategy. Burgess and Rosen note that with this strategy, Field would be able to define multiplication directly on triples of points, and thus wouldn’t need his cardinal comparison relations, which are not purely first-order definable. Thus, they suggest that if he had followed their strategy, he could avoid many of the logical worries that plague him towards the end of “Science Without Numbers” and in many of his exchanges with Stewart Shapiro (and more recently Otávio Bueno).

However, I think I can explain why Field used a different strategy. Field didn’t want to find surrogates for real numbers, so that (say) the weight function would return a tuple of points rather than a real number – he wanted to define weight comparison relations, so that there is no entity at all that can be said to be the weight of an object; we can just talk about when one object is heavier than another, and when the differences between the weights of two pairs of objects are the same. The particular surrogates Burgess and Rosen use are in fact quite problematic, because there seems to be no reason why particular space-time points should be connected with an object in the way that its weight would have to be. While it gets over the anti-platonist argument that there’s no way at all the weights could be causally connected to the objects, there still seems to be no plausible way in which the weights are connected to the objects. And similar only slightly weakened versions of epistemic arguments would still apply as well. In addition, Burgess and Rosen point out that such a strategy requires the existence of infinitely many (in fact, uncountably many) physical objects in order to represent all the real numbers.

Thus, there’s little sense in which the Burgess and Rosen style of reconstruction would be a scientific improvement over platonistic theories, and thus the arguments in their last chapter would have a lot of force. However, I think Field’s theory really does limit itself to primitives (like weight comparison and betweenness) that seem perspicuous, whereas Burgess and Rosen’s nominalistic theory has to have a primitive that says when a triple of points represents the weight of a particular object. Field’s theory actually seems to me to be an improvement on standard Newtonian theory in just the same way that Hilbert’s is an improvement on Euclidean geometry. Few people will actually want to work with the newly reconstructed system, but it is characterized in a much more purely internal way, and thus can be more easily generalized and more compactly axiomatized. (That is, there is no need to add extra axioms to spell out all the details of the mathematical apparatus that goes along with the physical part.)