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*)).