The Quine-Putnam Indispensability Argument

Many years ago I finished my PhD, entitled "The Mathematicization of Nature" (1998, LSE), in which I discussed the applicability of mathematics, the Quine-Putnam indispensability argument and considered a number of nominalist responses to it, in the end rejecting them all. The monograph Burgess & Rosen 1997, A Subject with No Object, had appeared a year earlier. At the time, I'd considered the issue definitively settled. And so I decided not to bother publishing anything in the area, as it would be pointless. (I did publish Ch. 5, which was about truth theories and deflationism.)

Jeez was I wrong! In the last fourteen years, the debate about the indispensability argument has continued, taking off in many different directions. And I'm pretty baffled at the whole thing. Even the formulation of the Indispensability Argument often given is incorrect, as far as I can see. So, here is mine, and I think it is reasonably faithful to the intentions of both Quine and Putnam.

1. Nominalism

Nominalism (in mathematics) is the claim that there are no numbers, sets, functions, and so on. (In addition, nominalism normally implies also that there are no syntactical types: i.e., finite sequences of symbols. Consequently there is a problem for nominalism at the level of syntax, a problem discussed long ago by Quine & Goodman 1947, "Steps Toward a Constructive Nominalism".) In particular, there are no mixed sets and no mixed functions. A mixed set is a set of non-mathematical entities, and a mixed function is a function whose domain or range includes some non-mathematical entities.

However, modern science is up-to-its-neck in mixed sets and functions. All the various quantities invoked in science are mixed functions. Laws of nature express properties of such mixed functions, and express relations between them. A differential equation in physics usually expresses some property of some mixed function(s). For example, it might say that a function defined on time instants has a certain property.

2. The Quine-Putnam Indispensability Argument 

Quine and Putnam both gave versions of an argument, which I formulate like this:

The Quine-Putnam Indispensability Argument
(1) Mathematicized theories are inconsistent with nominalism.
(2) Our best scientific theories are mathematicized.
(C) So, if one accepts our best scientific theories, one must reject nominalism. 

(The name "Quine-Putnam Indispensability Argument" derives, I believe, from Hartry Field.)

The argument for the first premise (1) is based on the following kind of example. Maxwell’s Laws include the mathematicized law:

At any spacetime point $p$, $(\underline{\nabla} \cdot \underline{B})(p) = 0$.

This is often abbreviated "$(\underline{\nabla} \cdot \underline{B}) = 0$", but it is clear that quantification over spacetime points is implicitly intended.

Since $\underline{B}$ is a vector field on spacetime, it is a mixed function, whose domain is spacetime, and whose range is some vector space (one that is isomorphic to $\mathbb{R}^3$). If nominalism is true, it follows that $\underline{B}$ does not exist, and therefore that Maxwell's Law, "$(\underline{\nabla} \cdot \underline{B}) = 0$", is false. (A slightly fancier version of this would refer instead to the electromagnetic field tensor $F_{ab}$, whose components unify the $\underline{B}$-field and the $\underline{E}$-field; but the considerations are more or less the same.) In general, if nominalism is true, then any such mathematicized theory is false. This establishes (1).

If this is right, then we have a major worry: this shows that a certain philosophical theory (nominalism) contradicts science. This is probably the central reason I am suspicious  of nominalism.

The argument for the second premise (2) requires one to compare our working mathematicized theories (Maxwell’s theory; Schroedinger equation; Einstein’s field equations; Yang-Mills gauge theories, etc.) with proposed nominalistic replacements. Having done this, one then concludes that either there are insuperable technical obstacles to the nominalization of such theories; or, though there may be, for certain mathematicized theories, nominalized replacements, even so, the mathematicized original is always a scientifically better theory, by scientific standards. (This is the sort of point emphasized by John Burgess, who semi-hemi-demi-jokingly suggested that nominalists might submit articles with their replacement theories to The Physical Review.)

So, our best scientific theories are mathematicized and are inconsistent with nominalism. Hence, if one accepts such theories, one must reject nominalism. This conclusion is epistemic only in a conditional sense. It simply says that one cannot have one’s cake and eat it. One cannot be a nominalist and a scientific realist.

3. Responses 

3.1 Rejecting (1): The rough idea is that mathematicized theories are consistent with nominalism. So, such theories may be true even though there are no mathematical entities. So, the magnetic field $\underline{B}$ doesn’t exist, but, even so, Maxwell’s Laws are true. This kind of view is advocated by Jody Azzouni (2004, Deflating Existential Consequence: A Case for Nominalism), but I'm not sure I quite understand it.

3.2 Rejecting (2): Our working scientific theories can be nominalized, and such theories are epistemically better. The betterness consists in the advantage that issues from the elimination of mathematicalia. This is essentially Hartry Field’s approach (Field 1980, Science Without Numbers).

3.3 Accepting, but living with, the conclusion: a nominalist might accept the Quine-Putnam argument, conceding the premises, but insist that one may “accept” mathematicized scientific theories in a weaker sense, which involves only accepting their nominalistic content. This is essentially Mary Leng’s and Joseph Melia's approach (Leng 2010, Mathematics and Reality; and Melia 2000, "Weaseling Aaway the Indispensability Argument" (Mind)).