Benaceraff 1965: "What Numbers Could Not Be" (In addition to these, there are also the classic articles setting out the Indispensablity argument against nominalism:Phil Review: here also)

Benacerraf 1973: "Mathematical Truth" (J. Phil: here also)

Putnam 1971: Philosophy of Logic (also in his Mathematics, Matter and Method, 1979)(Quine's articles are for the most part earlier: 40s and 50s.) And finally, the classic, and best, by orders of magnitude, response to this argument:

Putnam 1975: "What is Mathematical Truth?" (in his Mathematics, Matter and Method, 1979)

Field 1980:Field's book is out of print, to which my highly reasoned response is: $!!!****!!$%^&!!!. This is evidence of the intellectual philistinism of the modern world. Every physicist in the world should have a copy of this book.Science Without Numbers(selections photocopied here)

My own favourite article in philosophy of mathematics---since let's say 1980---is John Burgess's:

Burgess 1983: "Why I am not a Nominalist" (NDJFL)In this article, Burgess considers a number of ways that nominalists might respond to what follows from nominalism. Nominalism is the claim:

$\mathsf{Nom}$: There are no numbers, sets, functions, Lie groups, tensor fields, etc.This has very dramatic consequences. First, it implies that virtually all of mathematics---e.g., arithmetic, analysis, algebra, topology, geometry and set theory---is

*false*. Second, it implies that our best scientific theories are

*false*. For, as Quine and Putnam frequently pointed out, if $\Theta$ is one of our best scientific theories, then $\{\Theta, \mathsf{Nom}\}$ is inconsistent.

In his 1983 article, Burgess sets out three ways of responding to this (for definiteness, $\Theta$ might be Maxwell's theory of electromagnetism, which postulates various

*vector fields*, $A_{\mu}, J_{\mu}$, etc.)

- (Hermeneutic nominalism) $\Theta$ is
*literally*false, but can be*reinterpreted*as $\Theta^{h}$, which is true and is consistent with $\mathsf{Nom}$. - (Revolutionary nominalism) $\Theta$ is false, but can
*replaced*by a true, equally explanatory, theory $\Theta^{r}$ which is consistent with $\mathsf{Nom}$. - (Instrumentalist nominalism) $\Theta$ is false, but is
*instrumentally useful*because it is nominalistically adequate.

*Ontology and the Vicious Circle Principle*), 1991 (

*Constructibility and Mathematical Existence*) and Hellman 1989 (

*Mathematics Without Numbers*). I should mention Azzouni's view, which seems like a form of hermeneuticism, concerning the meaning of the English phrase "there is". A review of Azzouni's

*Deflating Existential Commitment*by Joseph Melia is here. And a review by John Burgess is here.

Until very recently, no one had really tried to defend the instrumentalist option. However, this view has become popular since around 2000. I think there are grounds for saying that the trend began with an important article in

*Mind*:

Melia 2000: "Weaseling Away the Indispensability Argument" (Since then, instrumentalist approaches have come in several shapes and sizes, including that of Yablo and of Leng. (The views are sometimes called "fictionalism", but it seems best to stick with Burgess's classification:Mind).

*instrumentalist nominalism*.) The crucial feature of instrumentalism is that it drops the requirement of providing either a reconstrual of mathematics (

*hermeneuticism*), or a purely nominalistic replacement for scientific theories (

*revolutionary nominalism*). For this reason, it has been called "Easy Road Nominalism". See:

Colyvan, 2010, "There is No Easy Road to Nominalism" (But it's clear that instrumentalism cannot come soMind)

*cheap*. There are quite complicated technical issues that arise, even if one does not aim to provide a full-blown reconstrual or replacement for our scientific (or mathematical) theories. That this is so can be seen by considering the analogous case for purely

*mathematical*instrumentalism. See, e.g.,

Caldon & Ignjatović 2005, "On Mathematical Instrumentalism",Part of the problem with evaluating instrumentalism concerning mathematicizedJ.Symb.Logic.

*science*is the lack of a

*precise formalization*of our best scientific theories. For no one can seriously pretend that we have anything like a good formalization of, say, Special Relativity or Quantum Field Theory.

Still, on the side of

*logic*, we do have many good ideas about formalization in general, and one tends to suppose that the problem in the case of our physical theories is largely one of

*effort*: an extremely sustained effort of a large research group, involving logicians and physicists, would be required, and this just doesn't happen.

A nice formalization technique for a theory which involves quantification over both concreta and abstracta is a 2-sorted one. The language $L$ used is an (interpreted) 2-sorted language. It has two styles of variables, so let's call them:

$c_0, c_1, c_2, \dots$ : ranging overThe language $L$ will also contain, in addition to $=$, primitive predicates, which then fall into three sorts:concreta.

$a_0, a_1, a_2, \dots$ : ranging overabstracta.

This formalization approach was devised, as far as I know, by John Burgess.Concretumpredicates: call them $C_i$.

Abstractumpredicates: call them $A_i$.

Mixedpredicates: call them $M_i$.

Being a 2-sorted language, $L$ is clearly reducible to a 1-sorted language by introducing two primitive unary predicates $\mathsf{Conc}(x)$ and $\mathsf{Abs}(x)$, meaning "$x$ is concrete" and "$x$ is abstract", and using the fairly obvious translation. E.g., let the variables $c_i$ and $a_i$ be mapped by $^{\dagger}$ as follows

$c_i \mapsto (c_i)^{\dagger} = x_{2i}$,and for atomic formulas,

$a_i \mapsto (a_i)^{\dagger} = x_{2i+1}$

$(P(t_1, \dots, t_n))^{\dagger} = P((t_1)^{\dagger}, \dots, (t_n)^{\dagger}).$and for quantified formulas,

$(\forall c_i \phi )^{\dagger} = \forall x_{2i}(\mathsf{Conc}(x_{2i}) \to \phi^{\dagger}).$However, the 2-sorted language $L$ has a certain conceptual appeal, and it is convenient for the purposes of proving certain conservation and reduction/elimination theorems.

$(\forall a_i \phi )^{\dagger} = \forall x_{2i+1}(\mathsf{Abs}(x_{2i+1}) \to \phi^{\dagger}).$

There are two important things worth pointing out, because these points seem to me to have been

*ignored*in some recent literature:

1. The applicability of mathematics depends on mixed predicates.Examples of mixed predicates are:

2. The Indispensability argument against nominalism depends on mixed predicates.

Membership: $c \in a$I think one can argue that, in fact,

Quantity value: $a = m_{kg}(c)$

*all*applications of mathematics can be reduced to these two cases.

In addition to mixed predicates, the abstract entities that the theory postulates can be:

*pure*abstracta*mixed*abstracta.

*mixed*abstracta are mixed sets and relations. For example, a

*set*of eggs is a mixed abstractum. The membership and quantity value relations are themselves mixed abstracta. Examples of

*pure*abstracta are

*cardinal numbers*and

*pure sets*.

And the theory may, in addition, have predicates relating mixed abstracta to pure abstracta. For example, the cardinality relation:

Cardinality: $a_2 = card(a_1)$This maps a set $a_1$, which may or may not be mixed, to a cardinal number, $a_2$, which is a pure abstractum. For example,

$44 = card(\{c \mid c \mbox{ is a US President in the years 1789 - 2014}\})$To understand how the formalization works, it is best to look at the models/interpreations for the language $L$. They have the 2-sorted form,

$((D_1, D_2); \{R^{C}_i\}, \{R^{M}_i\}, \{R^{A}_i\})$Here $D_1$ is the domain of (what the interpretation thinks are) concreta, and $D_2$ is the domain of (what the interpretation think are) abstracta. In addition to these various interpretations of $L$, I stated at the outset that $L$ is an interpreted language, and so there is an intended interpretation,

$\mathcal{I} = ((D_C, D_A); \{C_i\}, \{M_i\}, \{A_i\})$where $D_C$ is the domain of

*concreta*, $D_A$ is the domain of

*abstracta*, and the $C_i$ are the primitive concretum relations, the $M_i$ are the primitive mixed relations, and the $A_i$ are the primitive abstractum relations.

Suppose now that $\Theta$ is a theory in $L$. One can define truth in $L$ as follows:

One can define what it is for a model to be "nominalistically correct". LetDefinition(truth of a theory)

$\Theta$ is true in $L$ if and only if $\mathcal{I} \models \Theta$.

$\mathcal{A} = ((D_1, D_2); \{R^{C}_i\}, \{R^{M}_i\}, \{R^{A}_i\})$be some $L$-interpretation. Then:

And one can define "nominalistic adequacy" of a theory $\Theta$:Definition(nominalistic correctness of a model)

$\mathcal{A}$ isnominalistically correctif and only if $(D_1, \{R^{C}_i\}) \cong (D_C, \{C_i\})$.

With all this, we can now explain the philosophical view that treats abstract entities as useful fictions in scientific theories $\Theta$, but does not propose any reconstruction.Definition(nominalistic adequacy of a theory)

$\Theta$ isnominalistically adequate(in $L$) if and only if there is some $\mathcal{A} \models \Theta$ such that $\mathcal{A}$ is nominalistically correct.

So, for example, a scientific theory might imply a regularity amongst concreta, e.g.,Scientific Instrumentalism about Abstracta

The aim of scientific theories is to providenominalistically adequate theories.

$\Theta \vdash \forall c(C_1(c) \to C_2(c))$(that is, all concreta with property $C_1$ have property $C_2$, where "property-talk" is loose talk.)

One might then

*empirically*check this generalisation, $\forall c(C_1(c) \to C_2(c))$ to see if it holds in many cases. If this holds, then one will regard the theory $\Theta$ has having received some positive degree of justification or confirmation (or, if you are Popperian, as having survived a genuine test of its refutability).

The point about scientific instrumentalism is that this justification is taken to confirm the claim:

$\Theta$ isinstead of the claim:nominalistically adequate

$\Theta$ is (approximately)true.

*Instrumentalism*about abstracta is a view that clearly has similarity with van Fraassen's "constructive empiricism".

However, as things stand, there is not much of a technical literature on this approach. The work is technically difficult, and with a few honorable exceptions (Melia, Yablo, Pettigrew) contemporary nominalists have tried to avoid technical questions.

I occasionally keep an eye out for articles on this topic. The ones I know dealing with technical issues that arise connected to instrumentalism nominalism are these (one by yours truly, one by my fellow M-Phi contributor Richard Pettigrew):

Melia 2000: "Weaseling Away the Indispensability Argument" (Mind)

Ketland 2011: "Nominalistic Adequacy" (Proc. Arist. Soc.)

Pettigrew 2012: "Indispensability Arguments and Instrumental Nominalism" (RSL)

Yablo 2012: "Explanation, Extrapolation and Existence" (Mind)

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