The Quine-Putnam Indispensability Argument Again

An earlier post on the Quine-Putnam indispensability argument was linked on reddit and received some discussion there. It gets mentioned here occasionally because it was my thesis topic, and I tend to think the argument given by W.V. Quine and Hilary Putnam is not quite understood properly. It is sometimes expressed as an argument "for the existence of mathematical entities" -- as if the default position was nominalism (the non-existence of mathematical entities). It's certainly true that this was Quine's default up to around 1947, and perhaps that's why it generally gets formulated in that way. Quine was concerned with whether a "nominalistic" position - that is, a theory of reality which postulates only concrete entities - would be sufficient for the needs of science, and eventually concluded that it would not be. Scientific theories not only do, but also need to, refer to numbers, functions, sets, quantities, vector spaces, fibre bundles, Lie groups and so on. So, really the argument, as worked out by Quine and Putnam, is simply that science and nominalism don't combine well.

[I can't remember if I have mentioned this joke before: abstract entities include propositions. And nominalism is itself a proposition. Consequently, if nominalism is true, nominalism doesn't even exist.]

But there are other widely discussed formulations, the most widely known one being by Mark Colyvan, which begins from some sort of nominalistic view as the default, and then argues for an epistemological conclusion that we "ought to have ontological commitment" to mathematical objects because they are indispensable to science.

The lines of argument by Quine appeared in scattered writings over a period of forty years; and Hilary Putnam wrote two works focused on the topic: Philosophy of Logic (1971) and "What is Mathematical Truth?" (1975). In its strongest form, the argument pits modern science against nominalism.

For example, the magnetic field, $\mathbf{B}$, a physical quantity associated with all kinds of phenomena (e.g., light, which generates another joke: if you think the electromagnetic field is an invisible theoretical entity, then, since it actually is light, then your view implies that light is invisible: I think I heard this joke towards scientific anti-realism from Jeremy Butterfield). In technical terms, $\mathbf{B}$ is an axial vector field. In even more technical terms, the components $B_x,B_y,B_z$ are three of the components of the electromagnetic tensor field $F_{ab}$. In even more technical terms, the electromagnetic field $F_{ab}$ is a "curvature" of a "connection" on a fibre bundle; ... Putnam's own example was the gravitational field (or, more exactly, the potential $\Phi$), as appears in the field theoretic formulation of Newton's theory of gravity via Poisson's equation,

$\nabla^2 \Phi = 4 \pi G \rho$

The field $\mathbf{B}$ is a function, its domain the set of spacetime points (themselves usually thought of as concrete entities) and its range some vector space (the details aren't important: if you want, let it be $\mathbb{R}^3$). Nominalism is the claim that there are no numbers, sets, functions, propositions, sentences, Hilbert spaces, manifolds, etc., etc. Then a strong version of the argument of Quine and Putnam runs like this. The premises are:

(i) $\mathbf{B}$ is a function.
(ii) The existence of functions is inconsistent with nominalism.

So,

(C) If nominalism is true, there is no such thing as the magnetic field.

This argument doesn't get into details of theory formulations, or existential quantifiers, or claims about "ontological commitment". It simply says that, generally, physics and mathematics are mixed up together in a very deep way; and dispensing with physical quantities, like fields and wave functions and so on (which are functions with abstract value ranges), without a huge reformulation effort is no easy task.

There are many responses to such arguments, and the debate can get quite technical quickly. My co-blogger Richard Pettigrew has written an interesting detailed response, "Indispensability arguments and instrumental nominalism" (RSL, 2012) mentioned here before, along with some other similar approaches.