Contemporary Nominalism

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Nominalism: Contemporary James Henry Collin University of Edinburgh ᾡᾘᾤᾜᾪὥᾚᾦᾣᾣᾠᾥίᾜᾛὥᾘᾚὥᾬᾢ

Contents Contents What is nominalism? . . . . . . . . . . . . . . . . . . . . . Contemporary nominalism . . . . . . . . . . . . . . . . . The motivations for nominalism . . . . . . . . . . . . . . Three kinds of nominalism . . . . . . . . . . . . . . . . . The application of mathematics . . . . . . . . . . . From applied mathematics to mathematical objects The indispensability argument . . . . . . . . . . . . . . . The role of confirmation holism . . . . . . . . . . Reconstructive nominalism . . . . . . . . . . . . . . . . . Chihara . . . . . . . . . . . . . . . . . . . . . . . . Field . . . . . . . . . . . . . . . . . . . . . . . . . Hermeneutic nominalism . . . . . . . . . . . . . . . . . . Azzouni . . . . . . . . . . . . . . . . . . . . . . . . Instrumentalism . . . . . . . . . . . . . . . . . . . . . . . Leng . . . . . . . . . . . . . . . . . . . . . . . . . 1

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All that there is or might be can be divided into two exclusive and exhaustive categories: the concrete and the abstract. Those who hold that there are only concrete objects are called ‘nominalists’, and those who hold that there are abstract objects are called ‘platonists’. (Small-‘p’ platonists in this sense are not necessarily followers of Plato; for this reason some prefer the apophatic ‘antinominalist’.) Examples of concrete objects are tables, chairs, stars, human beings, molecules, microbes, as well as more exotic theoretical entities such as electrons, bosons, dark matter and so on. Paradigmatically, concrete objects are spatiotemporal, contingent, have causal powers and can themselves be affected, participate in events, and can be interacted with even if indirectly. Examples of abstract objects are mathematical objects (such as numbers, sets, functions, groups) propositions, types of things such as the North American Grizzly Bear (as opposed to individual North American grizzly bears), the average family, the key of C#m, the game of chess, the English language and so on. Paradigmatically, abstract objects are non-spatial, necessary, acausal, and cannot be interacted with, even indirectly. What is nominalism? Although there is general agreement over which individual things count as concrete or abstract objects, there is less consensus over what the necessary and sufficient conditions for abstractness are. Some (e.g. Wetzel (2009)), take nonspatiality to be the criterion of abstractness. However, though typical abstract objects like numbers and propositions have no spatial location, non-spatiality is not sufficient for abstractness. Many hold that if God exists, God is non-spatial; but a being with the kinds of powers attributed to God is not rightly considered abstract. Nor is non-spatiality clearly necessary for abstractness. The North American Grizzly Bear may be said to have a particular geographical range, and some have suggested (Maddy (1990)) that sets of concrete objects are located where their members are located; the set of grizzly bears for example being located where grizzly bears are located. More commonly, lacking causal powers is taken to be necessary and sufficient for abstractness. None of the abstract objects listed above have causal powers or enter into causal relations. Rosen (Rosen (2012)) notes that if causal inefficacy is sufficient for abstractness then we have the

3 counterintuitive consequence that epiphenomenal qualia—qualitative conscious states that are caused by the brain but have no causes of their own—would, if they exist, count as abstract objects. Though counterintuitive, those who define abstractness in terms of causal inefficacy may be happy to accept this consequence. The existence of abstract objects is a topic that reaches into many areas of philosophy. Getting clear about what is at stake will involve touching upon fundamental issues in epistemology, the nature of truth and meaning, philosophy of science, philosophical naturalism and the existence of God. Contemporary nominalism Though nominalism is historically associated with the medieval philosopher and theologian William of Ockham, the contemporary debate arose in out of Gottlob Frege’s platonistic conclusion that numbers, sets and functions belong to a “third realm” (the first and second realms being physical, empirically knowable objects and the inner realm of mental objects respectively). In the 1930s nominalism was defended by some of the Warsaw logicians, and in 1947 Goodman and Quine’s ‘Steps Towards a Constructive Nominalism’ (Goodman and Quine (1947)) (where, disappointingly, Goodman and Quine rest the motivation for nominalism on an “intuition”) was published in the Journal of Symbolic Logic with huge influence over the subsequent debate. Quine would subsequently, and somewhat reluctantly, abandon this nominalism, as will be discussed later. Contemporary nominalism is distinct from classical nominalism; the position that there are no universals. Here, the label ‘classical’ does not imply that the existence of universals is not currently debated, any more than calling the writings of Plato and Aristotle classical implies that these do not receive attention from contemporary philosophers. Rather, it adverts to the origins of the debate in antiquity, as contrasted with the more recent emergence of the ‘contemporary’ debate. Both kinds of nominalism deny the existence of “abstract” objects, but the meaning of “abstract” is different in each case. In the classical debate “abstract” objects are universals. Universals are properties which can be instantiated by individual objects. The property of being negatively charged would be instantiated by individual electrons, for example. The two debates are orthogonal to each other. Some conceive of universals as being both spatially located and causally efficacious, so one could accept the existence of “abstract” (non-particular) objects whilst rejecting the existence of “abstract” (non-concrete) objects. Conversely, in the classical debate, some versions of the view called ‘trope nominalism’ posit the existence of particular objects, such as the shape of an individual lily, which are abstract in the contemporary sense. One could then reject the existence of “abstract” (non-particular)

4 objects whilst accepting the existence of “abstract” (non-concrete) objects. The motivations for nominalism Although nominalism is the metaphysical claim that there are no abstract objects, the chief argument for nominalism is the epistemological claim that we cannot have knowledge of abstract objects, even if they do exist; or, more weakly, that it is a mystery how we could have knowledge of abstract objects. The claim is the de jure one that we ought not to believe in abstract objects. The epistemological problem with abstract objects arises from the difficulty in squaring what abstract objects are like, if they exist, with what we know about ourselves as enquirers with particular capacities, abilities and faculties for gaining knowledge of what the world is like. Abstract objects are not only the sort of things that cannot be touched or seen, they cannot be interacted with or manipulated in any way. They have no effects or participate in events that could, even in principle, impinge on one’s experience. Facts about abstract objects, their existence or non-existence and their properties, cannot make a difference to any data one might have or come to acquire. The canonical articulation of the epistemological problem came from the Princeton philosopher Paul Benacerraf in the 1970s, when he claimed that our account of knowledge of mathematical objects ‘must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have’. (Benacerraf (1973, 409)). The over-all account of knowledge Benacerraf had in mind was a causal theory of knowledge (Goldman (1967)). Goldman’s account is as follows: S knows that p if and only if S’s belief that p is causally connected in an appropriate way with the fact that p. A causal criterion for knowledge immediately rules out knowledge of abstract objects, since abstract objects are acausal. As the causal theory of knowledge waned in popularity so did Benacerraf’s particular formulation of the epistemological problem. However, it is too quick to conclude from the failure of Goldman’s causal analysis of knowledge that there is no sound causal-epistemological argument against the possibility of knowledge of abstract objects. This depends on whether the analysis fails because appropriate causal connections between S’s belief that p and the fact that p are not necessary for knowledge, or because such causal connections are not sufficient for knowledge. If appropriate causal connections can be shown to be unnecessary for knowledge, then a causal objection to knowledge of abstract objects would indeed be amiss. On the other hand, if

5 appropriate causal connections are insufficient but necessary for knowledge, then the causal objection would go through. The most influential objections to causal theories are of this latter sort: appropriate causal connections, it is argued, are insufficient to rule out a person’s belief being luckily true, and hence not known. (Goldman (1976)) Nor are causal theories extinct. Colin Cheyne (Cheyne (1998); Cheyne (2001)) claims that the causal criterion does apply to existential knowledge, arguing that the claim is supported by examples from empirical science. More recently, Hartry Field (Field (1989)) reformulated the epistemic problem as a challenge to explain how our beliefs about abstract—in particular, mathematical— objects could be reliable. Realists about mathematical objects think that their beliefs about mathematical objects are largely true. If so, those beliefs are highly correlated with the mathematical facts. We require then an explanation of how it is that those beliefs have come to reflect the facts so accurately, but no such explanation is forthcoming. Two issues arise. The first is that this is a puzzle, rather than an argument against knowledge of abstract objects. It could only be made into the latter if one accepts the conditional If a person has knowledge of abstract objects, then an explanation of the reliability of her beliefs will be forthcoming. Secondly, reliability is usually thought to be a necessary but not sufficient condition for knowledge. This leaves open the possibility that one could explain the reliability of beliefs about abstract objects without showing that knowledge of abstract objects is possible. In this vein, Mark Balaguer (Balaguer (1998)) has argued that so long as every consistent mathematical object actually exists Field’s challenge can be met. So long as every consistent mathematical object in fact exists, and so long as our mathematical beliefs are consistent, they will reliably reflect the mathematical facts. Though this “full blooded platonism” would entail the de facto reliability of mathematical beliefs, it is less clear that this would suffice for knowledge of abstract objects. If creatures like us cannot gain knowledge of abstract objects given the truth of platonism, the claim that increasing the number of abstract objects would improve our epistemic standing towards those objects seems dubious. No formulation of the epistemological objection is uncontroversial, but a felt sense that something is epistemically worrying about abstract objects remains in the philosophical literature. Pinning down exactly what the epistemological problem with abstract objects is remains an open task for nominalists. A second kind of motivation for nominalism is the thought that we can make sense of everything we need to make sense of without needing to posit or presuppose the existence of abstract objects. If this is the case then there is simply no reason to believe in abstract objects. In turning to the three broad ways of being a nominalist, we will examine three broad ways of defending this sort of claim.

6 Three kinds of nominalism Broadly speaking, there are three ways of being a nominalist: a reconstructive way, a hermeneutic way and an instrumentalist way. These will be discussed shortly. The three modes of nominalism arise as responses to the way in which abstract language shows up in our practices. Though abstract language is deployed in very diverse contexts, the majority of literature on contemporary nominalism focusses on mathematical language as it is applied in science. As such this is also where our focus will be, though other forms of abstract language will not be ignored. At a high level of generality, mathematics is applied within the sciences in the following way (see the entry on The Applicability of Mathematics for details). Scientists devise equations which can be used to model or represent concrete systems. Measurement procedures are used to assign values to aspects of the target concrete system and these values are “plugged in” to the equations. Different values within the equations correspond to different magnitudes of properties of the concrete system; the mass of a proton, for instance, is represented by the number 938 when given in units MeV/c2. By manipulating the equations one can then make predictions about the concrete systems. These predictions are testable when the mathematical results are associated with measurement procedures, and are indirectly testable when the mathematical results are indirectly associated with measurement procedures. Taking a “toy” example, simple harmonic motion can be described using a trigonometric function: The application of mathematics

x(t) = sin ω t where x(t) is a function from the real numbers to the real numbers, sin is the trigonometric sine function from the real numbers to the real numbers, and ω is a constant. The function x(t) = sin ω t can be used to us what position a system exhibiting simple harmonic motion will be in at any given time. Functions of this sort are used to describe the motion of oscillating particles, swinging pendulums, springs and so on. The larger the oscillation being described, the larger the value of the constant ω. Taking an example from contemporary cosmology, we have Friedmann’s equation: ) ( dR 2 8πGρR2 − = −K dt 3 where G is the gravitational constant, ρ is the total mass density and K is a constant. Values derived from observations give us the result that K = 0. As K

7 is related to the curvature of spacetime the equation has allowed cosmologists to predict that spacetime, on a large scale, is flat. Representing properties of the universe as numbers allows algebraic reasoning to be used to describe, make predictions about and explain features of the concrete world. What does the application of mathematics in the sciences have to do with the existence of mathematical objects? The issue arises out of a broadly representationalist understanding of language, in which the purpose of language to ‘picture’ the world. Singular terms are used to denote objects in the world, and predicates are used to ascribe properties to those objects. This view of language is found in the kind of formal semantics that make an appearance in logic textbooks. There interpreted languages are understood as a function from the set of elements of the language itself—constants, predicates, sentences—to the domain of that language—the set of things that language is about. The function assigns objects to constants, sets of objects to predicates and truth values to sentences. Call this function |...|v In logic textbooks one finds things such as the following: From applied mathematics to mathematical objects

|Φt1 ...tn |v = T ⇐⇒ ⟨|t1 |v , ..., |tn |v ⟩ ∈ |Φ|v Intuitively, this means that a sentence is true if and only if all the things it talks about have the properties it says they do. A sentence is true if and only if it is “satisfied” by objects in the domain of discourse. The sentence ‘The Forth Road Bridge is red’ is true just in case the thing referred to by ‘The Forth Road Bridge’ (which is to say, the Forth Road Bridge itself ) is in the set of red things. This formal semantics is taken to apply uniformly to all assertoric discourse; as such the same considerations apply to sentences whose subject matter is abstract. The sentence ‘7 is prime’ is true just in case the thing in the domain referred to by ‘7’ (which is to say, the number 7 itself ) is in the set of prime things. For ‘7 is prime’ to be true, there must be a number 7. A like story can be told of quantification. Quantifiers are terms or phrases such as ‘there are’, ‘some’, ‘all’, ‘many’, ‘a few’ that serve to indicate quantities. In predicate logic the phrases ‘there is an x such that…’ and ’for all x′ are formalised respectively as ∃x and ∀x. In logic textbooks we find things such as the following: |∃xϕ|v = T ⇐⇒ |ϕ|w = T for at least one variable assignment |...|w that differs from |...|v in x at most.

8 Intuitively, this means that a sentence of the form ‘There is some x such that P’ is true if and only if there is something which has the property P. The sentence ‘Something is red’ is true just in case there is some thing which is in the set of red things. According to these formal semantics and the representationalist view of language, all the things referred to in a sentence or quantified over in a sentence must exist in order for that sentence to be true. This has the consequence that sentences which are about abstract objects and predicate properties of abstract objects require the existence of abstract objects to be true. So, if there are true sentences about abstract objects then abstract objects exist. Take the example of simple harmonic motion, governed by the equation x(t) = sin ω t. According to this picture of semantics, the equation says that there are real numbers x, t and ω such that the value of x at t is given by sin ω t. The truth of our best mathematicised physical theories requires the existence of mathematical objects. The issue is not restricted to applied mathematics, it proliferates throughout many different areas of discourse. For instance in pure mathematics we find quantification over and reference to (abstract) numbers: • There are infinitely many prime numbers. • π is irrational. • 87539319 is the smallest number expressible as as the sum of two cubes in three different ways. In biology and linguistics we find reference to (abstract) types (Wetzel (2009)): • • • •

The ivory-billed woodpecker was declared extinct. The banded bog skimmer was found for the first time in Maine. The gene encodes the instructions for the so-called D4 dopamine receptor. The word ‘rose’ appears four times in the sentence ‘Rose is a rose is a rose is a rose.’ • Though there are five vowel letters in English, the number of vowel sounds is in the hundreds. The species the ivory billed woodpecker cannot be identified with any individual concrete ivory billed woodpecker, since concrete individuals are not the sort of thing that can become extinct. Similarly the gene referred to above is the abstract gene type; it being the same gene that many individuals can share. The word rose is contrasted with individual concrete tokens of the word. The vowel letters and vowel sounds here cannot be identified with individual concrete tokens as there are many millions of those. Finally, talk of the content or meaning of sentences makes reference to abstract objects:

9 • The sentences ‘Snow is white’ and ‘Schnee ist weiß’ both express the proposition that snow is white. • Two sentences s1 and s2 have the same meaning if and only if the proposition expressed by s1 = the proposition expressed by s2. Propositions are not objects with causal powers or spatial locations, and cannot be identified with any concrete object. According to the broadly representationalist account of language described here, much of what we say in mathematical science, pure mathematics, biology, linguistics and semantics can only be true if abstract objects exist. The indispensability argument Nominalists are sceptical about our ability to read the fundamental natural of reality off the structure of sentences in this manner; but they are left with the task of making sense of what is going on when we speak and write this way. Given the semantic assumptions discussed above, the truth of these claims from physics, pure mathematics, biology, linguistics and propositional attitude talk requires the existence of abstract objects. On the face of it, we have at least some justification for believing these sentences to be true: many of them are parts of our best scientific theories about the world. Considerations like these have brought about what is by far the most influential challenge to nominalism in the contemporary debate: the indispensability argument. In fact, to talk of the indispensability argument is misleading since there are a number of distinct arguments that fall under that rubric (see the entry on Indispensabilty Arguments for details). Something like a core indispensability argument can however be isolated, and, because the various forms that contemporary nominalism takes have come about largely in response to the premises of this core argument, describing it will allow us to produce a useful taxonomy of nominalisms. At its heart, the indispensability argument is designed to show that nominalism is incompatible with the claims of contemporary science. Contemporary science asserts the existence of abstract objects, so if its claims are true nominalism is false; if we are justified in believing its claims we are not justified in believing nominalism. The “core” indispensability argument has three premises: (Realism) Our best current scientific theories are true (or at least approximately true). (Indispensability) Our best current scientific theories indispensably quantify over or refer to abstract objects.

10 (Quine’s Criterion) The existential quantifier (∃x) expresses existence. Something should be said about each of the premises. (Realism) is not as straightforward as it looks, since the denial of realism, instrumentalism, can be characterised in a number of different ways. The anti-nominalists John Burgess and Gideon Rosen have characterised a rejection of (Realism) as amounting to the claim that ‘standard science and mathematics are no reliable guides to what there is’ (Burgess and Rosen (1997, 60–61)). However, the most fully developed instrumentalist nominalism, that of Mary Leng, provides an account of how a denial of realism is compatible with substantive scientific knowledge of the concrete world. (Quine’s Criterion) is motivated by broadly the sort of semantic considerations discussed earlier. The name comes from the great twentieth century Harvard philosopher V.W.O. Quine, who claimed that the existential quantifier (∃x) expresses existence. Curiously enough, the paper most famous for establishing this view as the philosophical orthodoxy, On What There Is (Quine (1948)), offers no argument to that end. Elsewhere however, Quine defends the claim that ∃x expresses existence on the grounds that the meaning of ∃x is given by the English phrase ’there is an object x such that…” and that this expresses existence (Quine (1986, 89)). Finally, (Indispensability) is also not wholly straightforward. The method of Craigian elimination, for instance, can transform a theory Γ that quantifies over two kinds of things into a theory Γ◦ with infinitely many primitives and axioms that quantifies over only one of those kinds of things. No nominalist however has recommended Craigian elimination as a response to (Indispensability), perhaps because the process is not thought to explain the success of mathematical theories (see Burgess and Rosen (1997 I.B.4.b) for details). A fourth premise Confirmation holism is also often thought to be crucial to the indispensability argument, both by those who defend and those who resist the argument (see Colyvan (2001), Maddy (1997), Sober (1993), Leng (2010), Resnik (1997)). Confirmation holism is the claim that confirmation accrues to theories as a whole rather than accruing only to parts of those theories. (Realism), (Indispensability) and (Quine’s Criterion) mutually entail the falsity of nominalism, so additional premises are strictly redundant. However, confirmation holism is sometimes thought to be required to establish the kind of scientific realism that the indispensability argument requires. The thought goes as follows: if confirmation holism were false, nominalists would be free to claim that only those parts of scientific theories that talk about the concrete world are confirmed by empirical evidence. Mathematicized scientific theories though, cannot divided into nominalistic and platonistic parts, as this The role of confirmation holism

11 thought presupposes. One can think of the language of mathematical science as being two-sorted, ranging over two kinds of thing: (1) concrete entities, using primary variables: x1 , x2 , ..., xn . (2) abstract entities, using secondary variables: y1 , y2 , ..., yn . and therefore containing three kinds of predicate: (i) concrete predicates, expressing relations between concreta: C1 , C2 , ... (ii) abstract predicates, expressing relations between abstracta: A1 , A2 , ... (iii) mixed predicates, expressing relations between concrete and abstract objects: M1 , M2 , ... The problem for the confirmation holism approach arises because mathematical scientific theories do not have two parts—a mathematical and a nominalistic part—they have three parts—a mathematical, a nominalistic and a mixed part; and in highly mathematicized theories much of what is said about the concrete world falls within the mixed part of the theory. Measurement is one clear example of this. Measurements describe physical quantities by associating them with numerical magnitudes. Take the claim ‘The mass of d1 is 5 kilogrammes’, where d1 is a some concrete object. This is expressed more formally as ’Mass ′ kg (d1 ) = 5 , which describes a function from a concrete object, d1 , to an abstract object, the number 5. Although this is, in one sense, about the concrete world, it refers to abstract objects. Often concreta get described in scientific theories without those theories referring to concrete objects at all. This sounds superficially paradoxical, but can be seen not to be if we reflect on the role that mathematics plays in modelling the world. In designing a mathematical model of a concrete system, the concrete system is represented as a mathematical structure; concrete systems are reconstituted in the theory as abstract structures and implications about these abstract structures are deduced. Planets are described as point masses, gasses as collections of extensionless molecules moving at uniform velocities on a two-dimensional plane. In much mathematicised science, there are simply no nominalistic parts of theories that contain substantive information about the concrete world. Even observation sentences are routinely couched in mathematical terms, for instance if they are measurements of physical quantities. As such, even if the only parts of theories that were subject to confirmation were observation sentences, the three premises (Realism), (Indispensability) and (Quine’s Criterion) would still jointly entail the existence of abstract objects.

12 Some nominalists reject (Indispensability) and attempt to show that we can (in certain important contexts) get by without talking about abstract objects. This is reconstructive nominalism. Others reject (Quine’s criterion): one can make true claims “about” abstract objects without there being any abstract objects. ‘There are infinitely many primes’ really can be true without any primes existing. This is hermeneutic nominalism. Still others reject (Realism). They take abstract sentences, even those that appear in our best scientific theories, to be strictly false, do not attempt to show that we can get by without them, but offer an account of why we speak this way and why it is useful to do so. This is instrumentalist nominalism. These are the three broad ways of being a nominalist. Reconstructive nominalism The first of the premises to be concertedly challenged by nominalists was (Indispensability), and there have been many attempts to discharge or partially discharge this aim (a useful overview can be found in Burgess and Rosen (1997 III.B.I.a)). These have typically focussed on mathematical objects, rather than types, propositions or other kinds of abstracta. Hartry Field’s efforts, and responses to them, have been dominant in the philosophical literature on indispensability, to the extent that some discussion of indispensability carries on as though the failure of Field’s project would amount to the failure of reconstructive nominalism. Here we examine two important and representative strategies of dispensing with reference to and quantification over mathematical objects in some detail: Charles Chihara’s modal strategy, and Field’s geometrical strategy. Originally Chihara responded to (Indispensability) by developing a predicative system of mathematics which avoided quantification over mathematical objects by using constructibility quantifiers instead of the standard quantifiers (Chihara (1973)). Concerned that not all of the mathematics needed for contemporary science could be reconstructed in a predicative system, he has since retained the use of constructibility quantifiers but developed a different system without these restrictions (Chihara (1990); Chihara (2003)). It is Chihara’s developed view we discuss here. Today, the “official claims of mathematics”, as it were, come in existential form: they are claims about what mathematical objects exist and what relations they bear to each other. Things were not always so. In Euclid’s Elements we find the following axioms of geometry: Chihara

1. A straight line can be drawn joining any two points. 2. Any finite straight line can be extended continuously in a straight line.

13 3. For any line a circle can be drawn with the line as radius and an endpoint of the line as centre. These axioms concern not what exists or is “out there”, but what it is possible to construct. The claims of Euclidean geometry are modal rather than existential and, as a result, do not have any obvious ontological commitments to abstract (or, for that matter, concrete) objects. Geometry was principally carried out in this modal language for thousands of years, though by the twentieth century it had become common to make geometrical claims in existential language. Hilbert in his 1899 Grundlagen der Geometrie (Foundations of Geometry) gives the following as his first three axioms of geometry: 1. For every two points A, B there exists a line L that contains each of the points A, B. 2. For every two points A, B there exists no more than one line that contains each of the points A, B. 3. There exist at least two points on a line. There exist at least three points that do not lie on a line. Hibert’s axioms, in contrast to Euclid’s, appear existential, describing which points and lines exist. This modal-to-existential shift took place without fanfare amongst practicing mathematicians. For the nominalist however, all this may be philosophically significant. It shows that it is possible to practice mathematics— at least one part of mathematics—without making any claims about the existence of abstract mathematical objects. Chihara’s nominalism takes its cue from Euclid’s modal geometry; it aims to do all mathematics—or all the mathematics we need— in the modal, rather than the existential, mode. His goal is to: develop a mathematical system in which the existential theorems of traditional mathematics have been replaced by constructibility theorems: where, in traditional mathematics, it is asserted that such and such exists, in this system it will be asserted that such and such can be constructed. (Chihara (1990, 25)) Although Chihara works out this project in a good deal of technical detail, the fundamental idea behind it is straightforward enough. Where Field, as we will see, attempts to replace mathematicized physics with nominalistic physics, Chihara attempts to replace standard pure mathematics with a system of mathematics that makes no claims about the existence of mathematical objects. This nominalistic surrogate for standard mathematics then could be true without mathematical objects existing.

14 Constructibility theory How is this to be carried out? Chihara works out a modal version of simple type theory (henceforth STT) called constructibility theory (henceforth Ct). The language of STT contains the standard quantifiers ∃x (meaning ‘there is an object x such that…’) and ∀x, (meaning ‘every object x is such that…’) and the set-theoretic membership relation ∈ which is used to express which entities are in a set. Thom ∈ {Thom, Jonny, Phil, Colin, Ed} √ means that Thom is in the set √containing Thom, Jonny, Phil, Colin and Ed, 2 ∈ R means that the number 2 is in the set of real numbers, and so on. As the language of STT contains ∃x, ∀x and ∈, STT is used to make assertions about which sets exist. Sets can contain ordinary objects, both concrete and abstract, and they can also contain other sets. The claims that can be made in STT about which sets exist are not wholly unrestricted; if they were one could claim that there is a set which contains all and only those sets that do not contain themselves ∃x∀y(y ∈ x ↔ y ∈ / y). Consider the set just described: does it contain itself? If it does not contain itself then it follows that it does contain itself, as it is the set that contains all sets that do not contain themselves. On the other hand, if it does contain itself then it follows that it does not contain itself because it is the set that contains only those sets that do not contain themselves. This is Russell’s paradox. To avoid this incoherence, sets in STT are on levels; a set can only contain objects or sets on a lower level than itself. On level-0 there are ordinary objects; on level-1, sets containing ordinary objects; on level-2 sets containing sets that contain ordinary objects; and so on. In Chihara’s system, the existential quantifier ∃x and universal quantifier ∀x are supplemented with modal constructibility quantifiers Cx and Ax, which, instead of making assertions about what exists, make assertions about which sentences are constructible. Corresponding to the existential quantifier ∃x is Cx. Claims of the form (Cϕ)ψϕ mean: It is possible to construct an open sentence ϕ such that ϕ satisfies ψ. Corresponding to the universal quantifier ∀x is Ax. Claims of the form (Aϕ)ψϕ mean: Every open sentence ϕ that it is possible to construct is such that ϕ satisfies ψ. To understand the constructibility quantifiers Cx and Ax one must understand what it is for an open sentence to be satisfied. Take the open sentence ′ x is the writer of Gormenghast’. This sentence is satisfied by Mervyn Peake—i.e. the person who wrote Gormenghast. So, open sentences can be satisfied by ordinary

15 objects, but they can also be satisfied by other open sentences. Consider the sentence ’There is at least one object that satisfies F ′ . This is satisfied by the open sentence ‘x is the writer of Gormenghast’. Open sentences, like the sets of STT, are on levels. At level-0 there are ordinary objects; at level-1 are open sentences that are satisfied by ordinary objects; at level-2 are open sentences that are satisfied by open sentences that are satisfied by ordinary objects; and so on. For a sentence to be constructible is just for it to be possible to construct. The sort of possibility at play here is not practical possibility; no particular person need be capable of constructing the relevant sentences. What Chihara has in mind is metaphysical possibility, which is sometimes (somewhat misleadingly) called “broadly logical possibility”. This is absolute possibility, concerning how the world could have been. (Chihara sometimes also talks in terms of “conceptual” possibility, although conceptual and metaphysical possibility are not generally thought by philosophers to be equivalent.) Notice that the constructibility quantifier is not epistemic in any way. That an open sentence ϕ is constructible does not mean that we know how to construct it, or even that it is possible in principle to know how it can be constructed. Similarly, it need not be the case that it is in principle knowable which objects or open sentences would satisfy ϕ. With STT and Ct thus sketched, one can see, in a general way, how Chihara’s strategy works. For every claim in STT about the existence of particular sets there corresponds a claim in Ct about the satisfiability of open sentences. Where STT says, for example, ’There is a level-1 set x such that no level-0 object is in x′ Ct can say ’It is possible to construct a level-1 open sentence x such that no level-0 object would satisfy x′ . The set-theoretic relation ∈ is replaced by the satisfaction relation between objects and open sentences (or open sentences and other open sentences); assertions about sets are replaced by assertions about the constructibility of open sentences. In this way, Chihara creates a branch of mathematics that does not require reference to or quantification over abstract objects. Everything one can do with STT one can do with Ct. STT is a foundational branch of mathematics, which is to say that other branches of mathematics can be reconstructed in it. Plausibly then, STT is sufficient for any applications of mathematics that might arise in the sciences. Since Ct is a modalised version of STT, Ct, plausibly, is itself sufficient for any application of mathematics that might arise in the sciences. According to Chihara (Indispensability) is therefore false. Constructibility theory and standard type theory Chihara thinks of Ct as a modal version of STT, but Stewart Shapiro (Shapiro (1997)) has claimed that Ct is in fact equivalent to STT, and so could have no epistemological (or other)

16 advantage over STT. In defence of this, Shapiro provides a recipe for transforming sentences of Ct into sentences of STT: first, replace all the variables of Ct that range over Level n open sentences with variables of STT that range over Level n sets; second, replace the symbol for satisfaction with the ∈ symbol for set membership; third, replace the constructibility quantifiers Cx and Ax with the quantifiers of predicate logic ∃x and ∀x. Call a sentence of STT ϕ and its Ct counterpart tr(ϕ). Shapiro claims that ϕ is a theorem of (provable in) STT if and only if tr(ϕ) is a theorem of Ct, and that ϕ is true if and only if tr(ϕ) is true. That sentences of Ct can be transformed this way into sentences of STT and that these transformations preserve theoremhood and truth shows, Shapiro claims, that the two systems are definitionally equivalent—that Ct is a mere “notational variant” of STT. Chihara responds by noting that the ability to translate sentences of STT into Ct does not show that they are equivalent. Though the ability to “translate” between the two theories would show that the sentences of STT and Ct share certain mathematically significant relationships, it would not show that these sentences had the same meaning or were true under the same circumstances. Sentences of STT entail the existence of sets and are true only if sets exist, whereas sentences of Ct do not and are not. Additionally, the two theories are confirmed in different ways. The Ct sentence ‘It is possible to construct an open sentence of Level 1 that is not satisfied by any object’ is supported by laws of modal logic, considerations about what is possible, coherent, and so on. The STT counterpart sentence ‘There exists a set of Level 1 of which nothing is a member’ is not supported by those considerations. The role of possible world semantics Chihara (1990) uses possible world semantics to spell out, in a precise way, the logic of Cx and Ax. Possible world semantics is a rigorous, formal way of characterising modal statements; statements about what is possible, impossible, necessary and contingent. In possible world semantics possibilities are represented set-theoretically. Intuitively one can understand the set-theoretic apparatus as representing possible worlds; one world for every way the world could have been. Something is possible just in case it is true in at least one possible world, necessary just in case it is true in all possible worlds, and impossible just in case it is true in no possible worlds. From this basis modal logics can be constructed. Is it legitimate to engage in possible-world talk without believing in possible worlds, or is this an instance of intellectual doublethink? Shapiro (1997) claims that possible world semantics is not available to the nominalist since it not only quantifies over abstract objects but is used in explanations. If possible world se-

17 mantics is just a myth then its falsehood precludes it from explaining anything; just as a story about Zeus cannot explain facts about the weather. Chihara responds by drawing a distinction between scientific explanations of natural phenomena and explications of ideas and concepts. The role of possible world semantics in Ct is not akin to a scientific explanation of an event, but an explication of a concept. Possible world semantics is used to spell out how to make inferences using constructibility quantifiers. Put more picturesquely, it shows one how to reason with the constructibility quantifiers in broadly the same way that an allegorical tale, such as Animal Farm shows one how to reason about totalitarian government (though the latter does so in a less rigorous but more open-ended way than the former). Just as a novel is capable of doing this without the things depicted in it really existing, so too possible world semantics is capable of doing this without possible worlds really existing. None of the prominent objections to Chihara’s brand of reconstructive nominalism are decisive. Although the view has received comparatively little attention in the literature, it remains a live option for the nominalist who denies indispensability. As was mentioned, Field’s reconstructive project has been utterly dominant in the contemporary literature on reconstructive nominalism, and remains so despite Field himself having said in print very little about it in print since the early nineties. Mathematicized science uses mathematical models, equations and so on to represent concrete systems. The first goal of Field’s project is showing that our best scientific theories can be restated to avoid using mathematics. Here is a point of contrast with Chihara. Whereas Chihara claims, not that mathematics per se is dispensable to science but only the sort of mathematics that quantifies over abstract objects, Field’s project is to formulate scientific theories so that they do not make use of mathematics of any sort. Field also wants to establish that mathematical language is dispensable in principle; that there is no context in which science would require mathematics to do something which it could, even in principle, do without mathematics. To this end, the second goal of his project is showing that adding mathematics to claims about the concrete world does not allow us to infer anything about the concrete world that knowledge of intrinsic facts about the concreta would not allow us to infer on their own. Field

Field’s programme Field calls the process of removing reference to and quantification over abstracta “nominalization”. Field does not nominalzse all of contemporary science—the task would be colossal—but one important theory: New-

18 tonian Gravitational Theory (NGT). His hope is that in doing so, he will have made it plausible that a complete nominalization of science is at least in principle accomplishable. How does he carry out this task? Some mixed claims, expressing relationships between concrete and abstract objects are easy to reformulate in a purely nominalistic way. ‘There are exactly two remaining Beatles’ can be parsed ∃x∃y(Bx ∧ By ∧ x ̸= y) ∧ ∀x∀y∀z(Bx ∧ By ∧ Bz → x = y ∨ y = z ∨ x = z) where Bx means ‘x is a remaining Beatle’. Our best scientific theories however go far beyond claims about how many of a particular kind of object there are, so the means of nominalizing these theories will be more complex. As a result, the details of Field’s project are highly technical. An overview of its general contours however can be given. NGT describes the world by assigning properties such as mass, distance and so on, to points in space-time numerically. ‘The mass of b is 5kg’ for example is understood as meaning that there exists a mass-in-kilograms function f from a domain of concrete objects C to the real numbers R, such that f (b) = 5. Instead of describing the concrete world by assigning it numerical values, Field’s theory (henceforth FGT) describes the concrete domain directly using comparative language. In particular, distance claims are made using a betweenness relation ‘y Bet xz ′ , a simultaneity relation ‘x simul y ′ and a congruence relation ‘xy Cong zw′ . These are primitives of the theory, but intuitively ‘y Bet xz ′ , means that y is between x, z, ‘x simul y ′ that x and y are simultaneous, and ‘xy Cong zw′ that the distance from x to y is the same as the distance from z to w. In the same way, mass claims are expressed using mass-betweenness and mass-congruence relations. From these building blocks Field develops a scientific theory capable of describing space-time and many of its properties, without quantifying over mathematical objects. Representation theorems The next step is showing that FGT really is a nominalistic counterpart to NGT. To this end, Field proves a “representation theorem”. Intuitively, what Field’s representation theorem shows is that the domain of concrete things represented by FGT using comparative predicates (a spacetime with mass-density and gravitational properties) has the same structural features as the abstract mathematical model of space-time with mass-density and gravitational properties given by NGT. NGT is a mathematical mirror image of FGT. In more detail, Field proves that 1. There is a structure-preserving mapping ϕ from the sort of space described by FGT onto ordered quadruples of real numbers

19 2. There is a structure-preserving mapping ρ from the mass-density properties FGT ascribes to space-time onto an interval of non-negative real numbers, and 3. There is a structure-preserving mapping from ψ from the gravitational properties FGT ascribes to space-time onto an interval of real numbers where ϕ is unique up to a generalised Galilean transformation, ρ is unique up to a positive multiplicative transformation and ψ is unique up to a positive linear transformation. (What this means, in essence, is that choice of measurement scales are conventional. Different measurement scales could be used, so long as they preserve the right structural features of the measurement scales they replace. Saying something is 95.6 kilograms or 15.2 stone are two different ways of representing the same concrete fact about mass; no unique significance attaches to the numbers 95.6 or 15.2.) The representation theorem explains the utility of false mathematical theories: if the abstract mathematical model they describe has the same structure as the concrete world (described by true nominalistic theories), then reasoning about the abstract mathematical model will not lead us astray when making inferences about the concrete world. The following picture of nominalistic physics and its relation to scientific practice emerges: Nominalistic claims N1 … Nn have abstract counterparts N*1 … Nn* which use mathematical methods to describe the same physical world described by N1 … Nn. One can ascend from N1 … Nn to N*1 … Nn*, carry out derivations within the mathematical theory to arrive at some mathematised conclusion A*, and then descend to its nominalistic counterpart A. Mathematics facilitates inferences about the physical world, but these inferences could, according to Field, be made without mathematics, albeit more laboriously. Conservativeness Having developed a nominalistic physics and proved a representation theorem Field now needs to show that mathematics is truly dispensable to physics. This involves showing that mathematised theories cannot be used to imply facts about the concrete world that purely nominalistic theories could not. In the jargon, applied mathematics must be conservative over nominalistic theories: A mathematical theory S is conservative over a nominalistic theory N if and only if any nominalistic claim A is not a consequence of S+N unless it is a consequence of N alone. Here consequence is meant in the ‘semantic’ sense: a claim ϕ is a consequence of some theory Γ just in case it is not logically possible for Γ to be true and ϕ

20 to be false. One way Field motivates the conservativeness claim is “informal”: mathematics, on its own, ought not to impose constraints on the way the concrete world is. Were it to be discover that it did, that would be reason to think it in need of revision: [I]f it were to be discovered that standard mathematics implied that there are at least 106 non-mathematical objects in the universe, or that the Paris Commune was defeated … all but the most unregenerate rationalists would take this as showing that standard mathematics needed revision. Good mathematics is conservative; a discovery that accepted mathematics isn’t conservative would be a discovery that it isn’t good. (Field 1980, p13) In addition to these informal considerations, Field proves an important conservativeness result. ZFC (Zermelo-Fraenkel set theory with the axiom of choice) is a theory of pure mathematics, within which all other branches of mathematics can be modelled. This can be made into an applied theory by adding supplementary axioms. In particular: 1. an axiom of urelements, allowing there to be sets of concreta 2. a comprehension scheme, allowing definable kinds of concreta to form sets 3. a replacement scheme, saying that if a function from a set of concreta to other concreta can be defined, then those latter concreta also form a set. Field calls this theory ZFCV(N) and provides a proof that it is conservative over N. Conservativeness is important to Field’s kind of reconstructive nominalism, firstly because if the mathematics we apply is not conservative then there are things that can be done with mathematics that could not be done without mathematics, showing that it is not dispensable after all; and secondly, because it offers one explanation of why we manage to use (false) platonistic theories so successfully—they have the same consequences about the concrete world as the true, nominalistic theories underlying them. Can the project be rolled out? Field provides a nominalistic reconstruction of one important theory of physics, but some philosophers have questioned whether his project could be rolled out with nominalistic reconstructions of other theories such as General Relativity and Quantum Theory given. The facts represented mathematically in NGT are nominalistic facts—facts about the concrete world and its properties—so a nominalisation of NGT will represent those facts

21 directly. QM works differently. The mathematical formalism of QM is sometimes used to represent probabilities of measurement events, and a probability is not a concrete object. Even if QM could be reformulated to avoid reference to mathematical objects, it would remain a theory about probabilities; which is to say, a theory that talks about entities the nominalist does not take to exist. Balaguer (Balaguer (1996); Balaguer (1998)) has suggested that the Fieldian nominalist could take these probabilities to represent the propensities of the concrete systems they model. As Balaguer admits, the details of any such nominalisation have not been worked out, but even if they were this would not provide a means of nominalising phase space theories. Phase space theories use vectors to represent possible states of a concrete system. A Fieldian reconstruction of a phase space theory which avoided quantification over vectors would still quantify over possible states of concrete systems (Malament (1982)). Nor could these abstract objects be taken to represent propensities of concrete systems in the way that, plausibly, probabilities do. A number of commentators have taken it that these considerations license pessimism about the prospects of nominalising our best contemporary scientific theories, but the Fieldian nominalist could contest this conclusion. In the first place, one can question the inference from the fact that mathematical language has not, at this point in time, yet been dispensed from our best scientific theories, to the modal conclusion that mathematical language is indispensable from our best scientific theories. One would not similarly conclude that because Goldbach’s conjecture has not, at this point in time, yet been proven, it is unprovable. In the second place, progress has been made since Science Without Numbers was first published in 1980. Arntzenius and Dorr (Arntzenius and Dorr (2012)) take on the task of nominalising general relativity, which uses differential equations to describe the behaviour of fields and particles in curved space-times and vector bundles. They express confidence that, given an interpretation of what concrete facts the mathematical formalism of QM represents, nominalising strategies could be extended to apply in these cases. Conservativeness Field’s conservativeness claim has been criticised on a number of grounds. The standard logic textbook accounts of logical consequence quantify over sets. According to these, a theory is logically possible just in case it has a “model”. A theory has a model just in case there exists a set of objects which bear the relations to one another described by the theory. A claim ϕ is a consequence of a theory Γ if and only if there is no model of Γ & ¬ϕ. Logic consequence, it is argued, can only be understood if one posits the existence of sets. Field has a response: he takes logical possibility to be a primitive notion,

22 not ultimately to be explained in terms of the existence of certain sets. There are considerations in favour of this. Explaining modal facts in terms of set-theoretic ones may get things backwards. As Leng (2007) argues, one ought to explain the fact that there exists no set of all sets on the grounds that there could not exist a set of all sets, rather than explain why there could not exist a set of all sets on the grounds that there is no set of all sets. Another objection concerns the scope of Field’s conservativeness proof. Jospeh Melia (Melia (2006)) points out that although Field provides a proof of the conservativeness of ZFCV(N), he does not provide an argument that all useful applied mathematics can be carried out in ZFCV(N). Absent reasons to believe that this is the case, it is hard to assess the significance of Field’s proof, as it is hard to assess whether future applications of mathematics will be carried out in ZFCV(N). The ‘best’ theory? Mark Colyvan argues that talk of mathematical objects is not dispensable to our best scientific theories because nominalistic versions of those theories will be worse than their mathematical counterparts. Good theories must both internally consistent and consistent with observations, but there are additional theoretical virtues that must be taken into consideration. Colyvan (2001) lists the following: 1. Simplicity / Parsimony: Given two theories with the same empirical consequences, we prefer the theory that is simpler to state and which has simpler ontological commitments. 2. Unificatory / Explanatory Power: We prefer theories that predict the maximum number of observable consequence with the minimum number of theoretical devises. 3. Boldness / Fruitfulness: We prefer theories that make bold predictions of novel phenomena over those that only account for familiar phenomena. 4. Formal Elegance: We prefer theories that are, in a hard-to-define way, more beautiful than other theories. Colyvan contends that mathematical theories are often more “virtuous” than nominalistic ones. (See the section ‘Unification, Explanation and Confirmation’ in the entry on The Applicability of Mathematics and the section ’The Explanatory Indispensability Argument in the entry on The Indispensability Argument for examples of the unificatory and explanatory power of mathematics). Field however takes nominalistic theories to have greater explanatory power in some respects. They provide intrinsic explanations of physical phenomena, rather than appealing to extrinsic mathematical facts; and they eliminate the arbitrariness in

23 the form of arbitrary choice of units of measurement that accompany mathematical theories. It is open to the Fieldian nominalist to argue that her theoretical virtues are somehow better or more fundamental than those enjoyed by mathematical theories. Since there is no agreed-upon metric for measuring theoretical virtues, nor agreement over their epistemic significance—are they really indicators of truth or mere pragmatic expediencies?—it is unlikely that resolution in this area will be easily reached. Hermeneutic nominalism In recent years attempts at reconstructive nominalism have ebbed. Many nominalists now accept (Indispensability) or see it as somehow orthogonal to ontological questions. As the indispensability argument is valid, this requires rejecting either (Realism) or (Quine’s Criterion). This section explores the latter. A number of philosophers have questioned the semantic assumptions driving (Quine’s Criterion). Many natural language sentences employ apparent reference or quantification even when commitment to the existence of the things apparently referred to or quantified over would be, for various reasons, implausible: • • • • • • •

There is a better way than this. His lack of insight was astounding. There are many similarities between Sellars and Brandom. There is a chance we will make it in time. I have a beef with the current administration. The view of Edinburgh Castle from my office is wonderful. She did it for your sake.

If one takes both the surface structure of these sentences and the semantics discussed earlier seriously, then in taking these sentences to be true one would be committed to the existence of ways, lacks, similarities, chances, beefs, views and sakes. Positing the existence of objects such as these is bizarre. One response then is to deny that ‘there is’ always expresses existence. Jody Azzouni has been the most prominent defender of this approach to nominalism. Azzouni’s view is that both quantifiers and the term ‘exists’ are neutral between ontologically committing and non-ontologically committing uses. Context decides whether they are being used in ontic or non-ontic ways. (Azzouni (2004); Azzouni (2007); Azzouni (2010b); Azzouni (2010a)) ‘God exists’ uttered in a discussion between an atheist and theist would (usually) express ontological commitment, but ‘Important similarities between Sellars and Brandom really Azzouni

24 do exit’ would not (usually) be intended to express ontological commitment to similarities. Thick, thin and ultra-thin posits Azzouni then draws a distinction between mere quantifier commitments and existential commitments—not all quantifier commitments are existential commitments—and replaces Quine’s semantic criterion for what a discourse is committed to with a metaphysical criterion for what exists. Something exists, according to Azzouni, if and only if it is mind- and language-independent. This requires both rejecting the Quinian criterion and motivating the metaphysical one. The former task has already been described. With respect to the latter, Azzouni does not give a metaphysical argument for the criterion, but appeals instead to the de facto practices of people in general. One should adopt mind- and language-independence as a criteria for what exists because of the sociological fact that our community of speakers takes ontologically dependent items not to exist. Given this criterion for existence, the heart of Azzouni’s account consists in spelling out the implications of mind- and language-independence for our knowledge-gathering behaviour. Language- and mind-independent objects cannot be stipulated into existence. Inventing a fictional character on the other hand involves nothing more than thinking of her. Fictional entities are paradigmatically mind-dependent and hence non-existent. This is to be contrasted with the way we form beliefs about mind-independent objects. In particular, our epistemic access to mind-independent posits possess the following salient features: robustness, refinement, monitoring and grounding: • Robustness: Epistemic access to a posit is robust if results about that posit are independent of our expectations about it. For example, Newtonian mechanics (in conjunction with some auxiliary assumptions) predicted that the planet Uranus would have a particular perihelion which observation revealed it not to have. • Refinement: Epistemic access to a posit exhibits refinement when there are means by which to adjust or refine access to that posit. For example, more powerful telescopes allow improved access to distant parts of the observable universe. • Monitoring: Epistemic access to a posit involves monitoring when what the posit does through time can be tracked or when different aspects of the posit can be explored. For example, C.T.R. Wilson’s experiments appeared to reveal the trajectory of atoms by their observable effects on water vapour. • Grounding: Epistemic access to a posit exhibits grounding when properties of the posit itself explain why we can discover what properties the posit

25 has. That stars emit light explains why they are visible to the naked eye at night. If the way in which we establish truths about a posit does not fit with the way we establish truths about mind- and language-independent posits, then we are treating, in practice, those posits as mind- and language-dependent. Given Azzouni’s criterion for existence, this amounts to treating those posits as nonexistent. The leading idea here is that an examination of scientific practice shows that we treat concrete posits—observables, but also theoretical posits such as subatomic particles—as mind- and language-independent, but treat abstract objects as mind- and language-dependent. When these robustness, refinement, monitoring and grounding are active in deciding whether a posit exists and what features it has, one has what Azzouni calls thick epistemic access to it. Not all scientific posits which Azzouni takes to exist enjoy thick epistemic access however. To take an example; because the expansion of our universe is accelerating, there are parts of our universe which are sufficiently distant from us that information from them will never reach observers on Earth—these regions are outside our past light cone. Accepted cosmology posits the existence of galaxies, stars, nebulae and suchlike outside our past light cone, yet, concrete entities in these regions of the universe clearly fail to exhibit, at the very least, monitoring and grounding. Azzouni calls the sort of access we have to entities such as these thin epistemic access. In Azzouni’s developed view thin posits are ‘the items we commit ourselves to on the basis of our theories about what the things we thickly access are like’ (Azzouni (2012, 963)). Moreover, thin posits require “excuse clauses”: explanations, stemming from the scientific theories that describe them, of why we fail to have thick access to them. In the case of galaxies outside our light cone, we have thick access to things inside our past light cone; widely accepted cosmological theories about these posit features of the early universe which, in conjunction with natural laws, entail the existence of galaxies outside our light cone. Hence, theories about the things we thickly access commit us to the existence of galaxies which we cannot thickly access. They also provide the needed excuse clause: special relativity does not allow entities to travel through space faster than the speed of light. Objections Some have objected that denying Quine’s criterion is incoherent. How can sense be made of the view that it is both true that there are infinitely many primes and that no primes exist? (See Burgess (2004)). It is open to Azzouni to claim that both ‘There are infinitely many primes’ and ‘There are no primes’ can be true so long as the context has changed so that ‘there are’ is being used ontically in the latter case but not in the former. There are other ways to motivate

26 similar claims. Hirsch (2011) develops the related thesis of quantifier variance, according to which there are different existential quantifiers and different senses of ‘exist’ such that according to one sense ‘numbers exist’ is true and according to another sense it is false. Graham Priest (2005), himself a nominalist, develops a rigorous formal semantics according to which the existential quantifier—Priest calls it the ‘particular quantifier’ to avoid biasing the issue—does not express existence. Priest’s semantics are representationalist and akin to the sort of “standard” formal semantics eluded to earlier, except that the domain of discourse includes non-existent as well as existent objects. When one talks about abstracta, what one says is true or false depending on how things stand with non-existent objects. What precisely is at stake between someone who believes in non-existent objects (a ‘noneist’) and someone who does not is a delicate issue. On Azzouni’s view, posits can fail to exhibit robustness, refinement, monitoring and grounding but still be included in our ontology so long as there is an excuse clause explaining why they fail to exhibit these features. Mark Colyvan (2010) has objected that abstract objects do have an excuse clause; namely that they are acausal. Azzouni (2012) responds by noting that Colyvan’s excuse is a philosophical gloss, rather than stemming from actual scientific and mathematical practices. It is not wholly clear, however, what the force of this response is. An account of why philosophical excuse clauses are not epistemically significant in the right way would be apposite. Sorin Bangu (2012, 28–30) has objected to Azzouni’s claim that our community of speakers treats ontological independence as the criterion for existence. Bangu points out that this is an empirical, statistical claim, but that Azzouni presents no empirical, statistical evidence for it. Were a study to be carried out, it may turn out that opinion on the matter is not uniform (and this much seems true in the philosophical community at least). Instrumentalism For a long time instrumentalism, the rejection of (Realism), was not a popular nominalist response. This may have been motivated by the thought that a rejection of (Realism) constituted a rejection of the ability of science to informatively represent the world. Burgess (1983, 93) characterises instrumentalism as the view that ‘science is just a useful mythology, and no sort of approximation to or idealization of the truth’. Contemporary instrumentalist nominalism aims to avoid this result whilst maintaining that the theories of mathematical science are not strictly true.

27 The most detailed and sophisticated view of this sort is developed by Mary Leng (Leng (2002); Leng (2005); Leng (2007); Leng (2010)). Leng draws on earlier work by Mark Balaguer and Steven Yablo—both of whom ultimately defend the view that there is no fact of the matter over whether mathematical objects exist (see Balaguer (1998) and Yablo (2009))—to defend a distinctive instrumentalist nominalism. Instrumentalist nominalism stands apart from all other forms of contemporary nominalism by rejecting the need for a strictly true scientific account of reality. One does not need to find replacement theories that do not quantify over abstract objects or show that the theories that do quantify over abstract objects are in fact true. Leng aims to explain the usefulness of mathematics directly by studying mathematical practices and seeing if those practices can be understood without assuming the existence of mathematical objects. If one can make sense of mathematical-scientific practices—how they are used to make describe, make predictions about and explain features of the concrete domain—without positing mathematical objects, then positing the existence of mathematical objects is unnecessary. Leng

Mathematics and make-believe Leng’s account is fictionalist. Mathematical objects can rationally be treated as fictional; doing so does not jar with mathematical practices. Leng adopts aesthetician Kendal Walton’s (Walton (1990); Walton (1993)) account of fiction as make-believe to spell out the details. In articulating a fiction, one generates a prescription to imagine that things are thus and so. When Dorothy Sayers writes that Lord Peter Wimsey earned a first at Oxford, we are invited to imagine that there is a person, Wimsey, who has this particular property. This prescription to imagine, and the subsequent imagining, do not require the existence of Wimsey. A text is a kind of “prop” which, in conjunction with the usual conventions and practices involving fiction, generates the content of the fiction. Other principles are involved in what one is prescribed to imagine: logical consequence, facts about the (real) world, its laws of nature, and so on also come into play. This is what prevents us from imagining, or being prescribed to imagine, that Wimsey both gained a first and did not gain a first from Oxford, or that he can fly by flapping his arms. We will say that, if we are prescribed to imagine that some claim S is true, S is fictional. While sentences such as ‘Lord Peter Wimsey plays cricket’ are strictly false (since no such person exists), there is something correct about it. This is what Walton’s notion of fictionality is supposed to capture. The sentence’s correctness is due to it’s fictionality: the writings of Sayer, along with our conventions regarding what we do with fictions, how we

28 use fictions in practice, prescribe us to imagine that Lord Peter Wimsey plays cricket. Just as there can be mixed mathematical-concrete sentences, such as measurement claims, that describe relations between concrete and abstract entities, so too there can be mixed fictional-real sentences describing relations between fictions and the real world (Walton (1990, 410)): • • • •

Oscar Wilde killed off Dorian Gray by putting a knife through his heart. Most children like E.T. better than Mickey Mouse Sherlock Holmes is more famous than any other detective. Vanquished by reality, by Spain, Don Quixote died in his native village in the year 1614. He was survived but a short time by Maguel de Cervantes.

Again, Walton rejects the strict truth of these sentences. Their correctness also has to do with fictionality, although not in as straightforward a sense as pure sentences of fiction. These sentences are not fictional within their respective fictions. The novel The Picture of Dorian Gray does not depict Oscar Wilde committing an act of murder, Mickey Mouse is not in E.T., and vice versa. Instead, in making utterances like these we are engaged in “unofficial” games of fictionmaking. In these contexts we are invited to imagine that there are worlds created by their authors, allowing us to imagine relations between those worlds and between them and the real world. Again, the correctness or incorrectness of these claims depends on whether they are fictional; that is to say, whether they are in accordance with what we are prescribed to imagine. Here, however, the fictionality of the sentences does not only depend on what the author’s writings prescribe one to imagine; they also depend on how things stand with the real world. The fictionality of ‘Sherlock Holmes is more intelligent than any detective I’ve met’ depends both on real world “props” and what the prop of Doyle’s writings prescribe one to make-believe about Holmes. One value of this intermingling of fiction and reality—what Walton calls “prop-oriented make believe”—is that it allows one to represent the real world indirectly. Saying ‘Sherlock Holmes is more intelligent than any detective I’ve met’ allows one to indirectly express something about the intelligence of real world detectives that one might not otherwise be easily able to express. Talk of fictional objects can be used to place restrictions on real objects. That sentences describing things that do not exist are strictly false does not disqualify them from being used to express or grasp facts about the real world. Leng appropriates Walton’s account of prop-oriented make-believe to make sense of mathematicised science. Mathematical make-believe can be used to place restrictions on non-mathematical objects, and hence to describe, indirectly, the

29 concrete world. If one imagines that the set of real numbers R exists, one can imagine that there are functions that map concreta onto different real numbers depending on their qualitative properties, and which would allow one to represent those qualitative properties quantitatively. Imagining that there is a mass function, one could say of a concrete object d1 that Mass kg (d1 ) = 5 and in doing so place restrictions on d1 thus representing it indirectly. When this goes right, the measurement ascription will be fictional or ‘nominalistically adequate’: right with respect to the facts about the concrete world. (Rosen (2001) calls a theory Γ nominalistically adequate so long as the ‘concrete core’ or largest wholly concrete part of a world W at which Γ is true is an exact intrinsic duplicate of the concrete core of the actual world.) The falsity of ‘Sherlock Holmes is more intelligent than any detective I’ve met’ does not prevent it from being capable of accurately representing the real world. According to Leng, in grasping sentences like these we grasp their nominalistic content: what they “say” about the real world. In an analogous way, when we grasp mixed mathematical-physical claims, we grasp their nominalistic content. Scientific instrumentalism, on this account, does not debar science from being an accurate guide to what the world is like. Treating mathematics as a form of make-believe is consistent with treating scientific theories as having the power to accurately represent the world. Explaining the success of mathematics This however is not the end of the story. One reason many philosophers accept (Realism) is that they take it to be the only way to explain the predictive success of science. If our scientific theories are false, would it not be a hugely improbable coincidence that the very precise predictions they make are correct? J.J.C. Smart (1963: 36) gave the most wellknown formulation of this thought: Is it not off that the phenomena of the world should be such as to make a purely instrumental theory true? On the other hand, if we interpret a theory in a realist way, then we have no need of such a cosmic coincidence: it is not surprising that galvanometers and cloud chambers behave in the sort of way they do, for if there really are electrons, etc., this is just what we should expect. A lot of surprising facts no longer seem surprising. Smart had in mind instrumentalism regarding (concrete) theoretical entities such as subatomic particles, but many have endorsed the idea that the same problem carries over to instrumentalism about mathematical entities (Putnam (1971)). Leng claims that the success of mathematicised scientific theories is best

30 explained in terms of their nominalistic adequacy, not their truth. Mathematicised scientific theories describe non-causal relations between mathematical and concrete objects, but the behaviour of concrete systems—the behaviour that results in the observable events the theory predicts—cannot be in virtue of these relationships, since mathematical objects are abstract and cannot effect the behaviour of concrete systems in any way. The explanation for the predictive success of mathematicised theories must be that they respect the underlying concrete facts: the fundamental regularities that hold between concrete objects. Predictive success, in other words, is explained by nominalistic adequacy. Similar reasoning leads Leng to claim that what is tested empirically is the nominalistic adequacy of scientific theories, not their truth. Leng notes that this line of reasoning applies to the platonist as well as the nominalist. Even platonists should deny that the concrete objects described in our theories are the way they are because of abstract objects. Whilst it follows from the truth of a theory that it will be predictively successful, the explanation for why it is successful must be in terms of its nominalistic adequacy, it’s doing justice to the regularities that hold in concrete systems. Recall that Field explained the success of (false) mathematicised theories M by showing that they respect the non-mathematical relations that hold between concreta. He does this this by creating a nominalistic theory N which describes the concrete world directly and proving a representation theorem which shows that N and M both place the same sorts of restrictions on the concrete world. If Leng’s reasoning is correct, Field’s project is superfluous: one does not have to go to the trouble of spelling out a nominalistic counterpart theory, since it must be the nominalistic adequacy of mathematicized theories which explains their success regardless of whether a nominalistic counterpart theory is available. Mathematical explanations In addition to the predictive success of mathematical science, Leng looks to account for the explanatory success of mathematical science. For many mathematical explanations this is straightforward; mathematical models can be explanatory as a result of their representational role. The nominalistic adequacy of a theory explains why the concrete phenomenon in need of explanation occurs. Sometimes, though, the explanatory work done by mathematics is not exhausted by the nominalistic content of those theories, as there are cases in which if one were able to represent the nominalistic content directly, explanatory power would actually be lost. Here is an example: the Honeycomb Conjecture—the claim that a hexagonal grid divides a surface into regions of equal area in a way that minimises the total perimeter of cells—was proven in 1999 by Thomas Hales. (The example is discussed in Aidan Lyon and Mark

31 Colvyan Lyon and Colyvan (2007) and Colyvan (2012).) This proof, in conjunction with the premise that bees have evolved to minimise the amount of wax they must use whilst maximising the amount of honey they can store, can be used to explain why bees build hexagonal honeycombs. Notice that no nominalistic explanation could do the same explanatory work as this abstract one. An explanation which quantified over concrete particular honeycombs would lack the scope of the abstract explanation, which allows us to see that any hives that respond to these evolutionary pressures will have a hexagonal structure. Leng (2012) claims that structural explanations of this sort can also be accommodated by her form of nominalism. Structural explanations explain phenomena by showing that they will result if certain structural features are in place. The Honeycomb Conjecture is about abstract structures, but theories about abstract hexagonal structures can be (re)interpreted as being about concrete (approximately) hexagonal objects. Roughly speaking, what goes for ideal abstract hexagonal structures goes (approximately) for imperfect, concrete hexagonal structures. Axioms Γ characterising abstract hexagonal structures will be approximately true of imperfect concrete hexagonal structures. If some claim δ (such as that hexagonal grids minimise the total perimeter of cells) is entailed by Γ then δ will also hold in concrete systems approximately characterisable by Γ. More generally, a model of a theory Γ is a domain of objects that satisfies the axioms of Γ. If Γ entails some claim δ then any model of Γ must also be a model of δ. Γ characterises an abstract structure, which means that mathematical methods can be used to determine whether any model of Γ is a model of δ. However, the axioms of Γ can also be interpreted as being about a concrete system (the terms of Γ will be taken to denote concrete objects), in which case Γ will characterise the structural features of that system. A mathematical proof that any model of Γ is a model of δ will then explain why any concrete system characterisable by Γ will exhibit δ (when δ is interpreted as being about a concrete system). Using mathematical explanations does not commit one to the existence of mathematical objects, only to the claim that the concrete system being modelled has the structural features which mathematical methods allow one to describe. The applicability of the explanation is accounted for not in terms of the existence of mathematical objects, but from our ability to interpret the axioms of Γ as being approximately true of concrete systems. Again, Leng’s take on nominalism is not to rid ourselves of abstract talk, or to show how abstract talk could be true without abstract objects, but to explain why engaging in abstract talk can achieve certain ends despite it’s literal falsehood.

32 Objections Leng claims that, although our best empirical theories are not true, they express a true nominalistic content, and this content is given by what those theories entail about the concrete domain. There are difficulties in explicating what this amounts to. The nominalistic content of a mixed abstract-concrete sentence ϕ would be given by a sentence ϕN that described the way the physical world would have to be for ϕ to be true. In that case however, ϕN would be a nominalisation of ϕ in the sense found in Field’s reconstructive project. This means that if one does not know how to carry out a Field-style nominalisation on a given theory Γ, then one does not know how to articulate what the nominalistic content of Γ is. There remains something of a puzzle for the Lengian nominalist: in what sense can someone be said to grasp the nominalistic content of a theory if s/he cannot articulate what that content is? Other issues We have seen three broad ways of being a contemporary nominalist. All are hotly contested today, though none face obviously insuperable difficulties. Exploring the motivations for and against these positions has involved issues in semantics, epistemology and the philosophy of science. We now turn to two other broad topics in philosophy that have important relationships to contemporary nominalism: naturalism and theism. Though the term “naturalism” is used in many different, sometimes conflicting ways, there are broadly two kinds of naturalism: metaphysical naturalism and methodological naturalism. Metaphysical naturalism is the claim that only natural things exist: there are no supernatural beings that, in some way, transcend the natural order. As platonism clearly violates metaphysical naturalism—abstract objects are nonmaterial, eternal, immutable, acausal, not in space-time or subject to natural laws—our focus will be on methodological naturalism. Methodological naturalism involves a deferential attitude towards scientific practice. This cannot be quite as strong as the claim that one can only come to know (or be justified in believing) any given claim by scientific means, since the claim One can only come to know (or be justified in believing) any given claim by scientific means is not something that one can come to know (or be justified in believing) by scientific means. It is not, after all, part of any empirically testable scientific hypothesis: no researcher at CERN could devise an experimental setup with the large hadron collider to confirm this claim. That version of naturalism is not thereby shown to be strictly inconsistent, but it would entail it’s own unknowability (or inability to be justified under any circumstances). Instead, methodological naturalism can be parsed as something like the following: in scientific domains Nominalism and naturalism

33 of inquiry we should defer to the epistemic standards employed by working scientists, since there is no higher or better perspective from which to inquire into the nature of the world or from which to assess the claims of science. If working scientists take themselves to be justified in holding that the universe is approximately 13 billion years old, for instance, we should not take ourselves to be in possession of philosophical reasons to reject this claim. Quinian naturalism W.V.O. Quine is probably the most famous advocate of philosophical naturalism, particularly as it pertains to contemporary nominalism. According to Quine, natural science is: an inquiry into reality, fallible and corrigible but not answerable to any supra-scientific tribunal, and not in need of any justification beyond observation and the hypothetico-deductive method. (Quine (1975 see Maddy Oxford Handbook 439)) As such, ontology should be decided by looking to the content of our best contemporary scientific theories. If our overall body of scientific theory, suitably regimented into the language of first-order logic, states that something exists (quantifies over something), then we are sufficiently justified in taking that thing to exist. (See the entry the Indispensability Argument for details of how this goes.) Quine takes empirical science to be the arbiter of what exists. Although he understands science broadly, including disciplines like psychology, economics, sociology and history, pure mathematics does not fall under this rubric. For Quine, we ought to believe the claims of mathematics, but only because they find application in empirical science. Quine changes his mind over what to make of mathematics that is not applied: at one point he claims the higher reaches of set theory are meaningless (Quine (1984 Ox 445)), but later that we regard them as meaningful because they are ‘couched in the same grammar and vocabulary’ as mathematics which does get applied (Quine (1990 Ox 445)). Maddy’s naturalism Penelope Maddy (1997) claims that the Quinain ignores some nuances of scientific practice that have a bearing on what the naturalist should take to be the real scientific standards of evidence. Maddy points out that a historical study of scientific practice reveals that, though atomic theory was entrenched to the point that quantification over atoms was indispensable to our best science by 1860, scientists did not believe in the existence of atoms until atoms themselves were detected directly by Jean Baptiste Perrin, who was able to experimentally verify the predictions made by Einstein’s 1905 account of

34 Brownian motion in terms of atoms. Similarly, general relativity treats spacetime as a continuous manifold—viz. one in which for any two space-time points there exists another point between them; meaning space-time is “smooth” rather than quantised—but scientists who model space-time this way do not assume that space-time itself really has this property. The choice of model is based on pragmatic factors such as convenience, effectiveness and computational tractability. There are then things that our best scientific theories indispensably quantify over but which working scientists do not take themselves to be justified believing in. With regards to nominalism, Maddy notes that, since mathematical objects could not be directly detectable in the manner of theoretical posits like atoms, the evidence for the existence of mathematical objects cannot be of the same sort as the evidence for theoretical posits. In addition scientists include the mathematical “posits” they do because of pragmatic factors like convenience, effectiveness and computational tractability. Maddy takes this to undermine the claim that their mere presence in our best scientific theories provides justification for taking them to exist. (Quine, arch ontological pragmatist, would not see the pragmatic nature of these considerations as undermining ontological commitment.) Burgess and Rosen’s naturalism John Burgess and Gideon Rosen (Burgess and Rosen (1997); Rosen and Burgess (2005)) have given an influential naturalistic argument against nominalism, and in response to the epistemological problem with abstract objects. The Burgess-Rosen-style naturalist makes no attempt to reconcile the picture of ourselves—given to us by biology, neuroscience, empirical psychology and so on—as embodied creatures availed of particular information-gathering abilities, with the claim that we have knowledge of a domain of abstract objects. Instead, he appeals to mathematical practices and the manner in which mathematicians come to accept mathematical claims. If one asks a working scientist why she believes in protons, she will cite the usual scientific evidence for the existence of protons. For the naturalist, this standard of evidence should be sufficient for belief in protons. In the same way, argue Burgess and Rosen, one should look to the usual means by which mathematicians provide evidence for mathematical claims—for the naturalist the usual mathematical standards of evidence should be enough for us to accept the mathematical claims they aim to establish. Burgess and Rosen argue that nominalism requires rejecting the standards of justification at play in the mathematical community. They look to the circumstances under which mathematicians come to accept mathematical claims. Consider the claim: There are numbers greater than 1010 that are prime.

35 Mathematicians accept this claim on the basis of the following reductio proof: Assume that there are finitely many primes. These can be represented in a list: p1, p2, p3, p4, …, pn. Consider the number N = p1 × p2 × p3 × p4 × … × pn + 1. Either N is a prime number or it is not. If N is prime this contradicts the assumption that the list p1, p2, p3, p4, …, pn includes all the primes, so N cannot be prime. If N is not prime then it must have prime divisors (all natural numbers are either prime or are the products of primes). But this divisor cannot be on the list p1, p2, p3, p4, …, pn since dividing N by numbers on the list would leave a remainder of 1. This also contradicts the assumption that the list includes all the primes, so N cannot not be prime. The assumption that there are finitely many primes entails that N cannot be prime and cannot be not prime, which is absurd. So there are infinitely many primes. QED That there are infinitely many prime numbers entails that there are numbers, and is incompatible with nominalism. Moreover, it is a claim that mathematicians take to be established by the reasoning above. If nominalists reject that claim they must be using standards of justification different to the mathematician. Philosophers though have no higher or better vantage point from which to assess mathematical claims than mathematicians, so one should defer to the justificatory standards of mathematicians and accept that there is (decisive) justification for believing in abstract objects. Burgess and Rosen (Rosen and Burgess (2005)) parse the argument the following way: 1. The claims of standard mathematics appear to assert the existence of mathematical objects. 2. Experts—mathematicians and scientists—accept these claims, using them in practical and theoretical reasoning. 3. These claims are acceptable by mathematical standards. The claims that are not taken as axioms are supplied with proofs. 4. The claims of standard mathematics not only appear to assert the existence of mathematical objects, they do assert the existence of mathematical objects. 5. Accepting a claim—assenting to it verbally without reservations, using it in practical and theoretical reasoning etc.—just is believing the claim to be true. 6. The claims of standard mathematics are not only acceptable by mathematical standards but are acceptable by scientific standards: empirical scientists

36 defer to mathematicians on mathematical matters and there are no empirical scientific arguments against the claims of standard mathematics. 7. There are no philosophical considerations which can override mathematical and scientific standards of acceptability. 8. From 1, 2, 4, 5, Competent mathematicians and scientists believe in the existence of mathematical objects. 9. From 3, 6, 7, 8, We are justified in believing in mathematical objects. An interesting feature of the argument is that, if successful, it shows that nominalism is implausible whether or not mathematics is in principle dispensable to science; the de facto endorsement of mathematical claims by mathematicians is enough to undermine nominalism. Parts of the argument however can be contested by nominalists. Should one hold, for instance, that the seemingly formidable mathematicians who have denied the existence of mathematical objects— such as Alfred Tarski, Paul Cohen, Timothy Gowers, Soloman Feferman, Abraham Robinson, and so on—are in fact incompetent, as premise 8. would entail? How have contemporary nominalists responded to the considerations that motivate the argument? Leng’s pragmatic nominalism aims at showing why strictly false mathematical claims can reasonably be asserted and relied on in practical and theoretical reasoning. If her project is successful then 5 is undercut since accepting a claim, in the relevant sense of being nominalistically adequate, is very different from believing it to be true. Chihara (2006) has questioned 4. The claims of standard mathematics may not be genuine claims about the world; they may, for instance, express what would be true in a structure, or may be only partially meaningful and lacking in complete truth conditions. Azzouni could also be understood as rejecting 1 and 4. The contemporary debate on nominalism also has a bearing on theism. Classical conceptions of God take God to be a se: selfsufficient and not dependent on anything else for existence. The doctrine of divine aseity can be found in Jewish, Christian and Islamic thought. In particular, God is often taken to be uniquely a se. God’s unique aseity means that all things (other than God) depend on God for their existence. Abstract objects are usually thought to exist necessarily, and so eternally, if they exist at all. (Though it is notable that Field (1993) and Colyvan (2001) both reject this—Field takes abstract objects to be contingently non-existent and Colyvan takes them to be contingently existent). As necessary existents God would have no say, as it were, over whether abstract objects exist. God would not be the uniquely uncreated creator of all things (other than God), but one of an infinite number of uncreated Nominalism and theism

37 beings. As such, God’s unique aseity would be impugned. Platonism is incompatible with the existence of God, or, at least, God understood in the classical way (Craig (2014)). One option for the theist platonist is simply to reject the classical conception of God; God may not be uniquely a se. Some religious traditions may be committed to this doctrine however. In biblical passages the claims can be found, for instance, that ‘All things came into being through [God], and without him not one thing came into being’ (John 1:3) and that ‘in [God] all things in heaven and on earth were created, things visible and invisible, whether thrones or dominions or rulers or powers—all things have been created through him and for him. He himself is before all things, and in him all things hold together’ (Colossians 1:16-17). Similarly, the Nicene Creed—the credal document drawn up by the Council of Nicea in 325AD, which is taken to express the central claims shared by all denominations of Christianity—states: I believe in one God, the Father Almighty, Maker of heaven and earth and of all things visible and invisible. Though it is unlikely that the biblical authors of John and Colossians had abstract objects in mind, it is plausible that their claims about God’s creation were intended to be unrestricted; i.e. inclusive of every thing that might exist (other than God). Moreover, the Nicene Church Fathers, who would have been familiar with the metaphysics of Plato and Pythagoras, rejected that anything other than God could be uncreated (Craig (2014)). A less theological/exegetical and more purely metaphysical matter concerns nominalism and the conception of God as a perfect or maximally great being. A maximally great being is one whose greatness is unsurpassable. William Lane Craig has claimed that a maximally great being would be the ground of being for all things other than itself (Craig (2014)). Given platonism, God would be just one of an infinite multitude of uncreated objects that exist independently of God. This has been disputed on the grounds that it involves a mistake regarding the nature of necessity. That God cannot force a person to perform an action of her own free will is not a limit on God’s power or maximal greatness, since it is necessarily the case that one cannot act freely whilst being coerced—freedom just is such that it (necessarily) excludes coercion. Similarly, even a maximally great being could not make it the case that P, P → Q and ¬Q where P and Q are claims, since it is necessarily the case that P, P → Q |= Q. Scott Shalkowski argues that the relationship between God and abstracta should be understood in the same way. Since abstract objects are necessary existents it does not make sense to see their existence as an affront to God’s unsurpassable greatness. God’s inability to

38 create or destroy abstract objects is not an genuine limit on God’s power for the same reason that God’s inability to force a person to perform an action of her own free will is not a genuine limit on God’s power (Shalkowski (2014)).

references Arntzenius, Frank, and Cian Dorr. 2012. “Calculus as Geometry.” In Space, Time, and Stuff. Oxford University Press. Azzouni, Jody. 2004. “Theory, Observation and Scientific Realism.” Philosophy of Science 55: 371–92. ———. 2007. “Ontological Commitment in the Vernacular.” Nous 41 (2): 204–26. ———. 2010a. Talking About Nothing: Numbers, Hallucinations, and Fictions. Oxford University Press. ———. 2010b. “Ontology and the Word ’Exist’: Uneasy Relations.” Philosophia Mathematica 18 (3): 74–101. ———. 2012. “Taking the Easy Road Out of Dodge.” Mind 121 (484): 951– 65. Balaguer, Mark. 1996. “A fictionalist account of the indispensable applications of mathematics.” Philosophical Studies 83 (3): 291–314. doi:10.1007/BF00364610. ———. 1998. Platonism and Anti-Platonism in Mathematics. Oxford University Press. Bangu, Sorin. 2012. The Applicability of Mathematics in Science: Indispensability and Ontology. New Directions in the Philosophy of Science. Palgrave Macmillan. Benacerraf, Paul. 1973. “Mathematical truth.” The Journal of Philosophy 70 (19): 661–79. Burgess, John. 2004. “Review of Deflating Existential Consequence: A Case for Nominalism.” The Bulletin of Symbolic Logic 10: 573–77. Burgess, John P. 1983. “Why I am not a nominalist.” Notre Dame Journal of Formal Logic 24 (1): 93–105. doi:10.1305/ndjfl/1093870223. Burgess, John, and Gideon Rosen. 1997. A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford: Oxford University Press. Cheyne, Colin. 1998. “Existence claims and causality.” Australasian Journal of Philosophy 76 (1): 34–47. doi:10.1080/00048409812348171. ———. 2001. Knowledge, cause, and abstract objects: Causal objections to platonism. Kluwer Academic Publishers. Chihara, Charles. 1973. Ontology and the Vicious Circle Principle. Cornell University Press. ———. 1990. Constructibility and Mathematical Existence. Clarendon Press.

39 ———. 2003. A Structural Account of Mathematics. Oxford University Press. ———. 2006. “Burgess’s ’Scientific’ Arguments for the Existence of Mathematical Objects.” Philosophia Mathematica 14: 318–37. Colyvan, M. 2010. “There is No Easy Road to Nominalism.” Mind 119 (474): 285–306. doi:10.1093/mind/fzq014. Colyvan, Mark. 2001. The Indispensability of Mathematics. Oxford University Press. ———. 2012. An Introduction to the Philosophy of Mathematics. Cambridge Introductions to Philosophy. Cambridge University Press. Craig, William Lane. 2014. “Anti-Platonism.” In Beyond the Control of God?, edited by Paul M. Gould. Bloomsbury Academic. Field, Hartry. 1989. Realism, Mathematics and Modality. Wiley-Blackwell. ———. 1993. “The conceptual contingency of mathematical objects.” Mind 102 (406): 285–99. Goldman, Alvin. 1967. “A causal theory of knowing.” The Journal of Philosophy 64: 357–72. ———. 1976. “Discrimination and Perceptual Knowledge.” The Journal of Philosophy 73: 771–91. Goodman, Nelson, and W.V.O. Quine. 1947. “Steps Toward a Constructive Nominalism.” Journal of Symbolic Logic 12 (4): 105–22. Hirsch, Eli. 2011. Quantifier Variance and Realism: Essays in Metaontology. Oxford University Press. Leng, Mary. 2002. “What’s Wrong with Indispensability? (Or, the Case for Recreational Mathematics).” Synthese 131: 395–417. ———. 2005. “Revolutionary Fictionalism: A Call to Arms.” Philosophia Mathematica (III) 13 (3): 277–93. ———. 2007. “What’s There to Know? A Fictionalist Account of Mathematical Knowledge.” In Mathematical Knowledge, edited by Alexander Paseau Mary Leng and Michael Potter. Oxford University Press. ———. 2010. Mathematics and Reality. Oxford University Press. ———. 2012. “Taking It Easy: A Response to Colyvan.” Mind 121 (484): 983–95. Lyon, a., and M. Colyvan. 2007. “The Explanatory Power of Phase Spaces.” Philosophia Mathematica 16 (2): 227–43. doi:10.1093/philmat/nkm025. Maddy, Penelope. 1990. Realism in Mathematics. Clarendon Press. ———. 1997. Naturalism in Mathematics. Oxford: Oxford University Press. Malament, David. 1982. “Review of Field’s Science Without Numbers.” Journal of Philosophy 79: 523–34. Melia, Joseph. 2006. “The conservativeness of mathematics.” Analysis 66 (291): 202–8. doi:10.1111/j.1467-8284.2006.00615.x.

40 Priest, Graham. 2005. Towards Non-Being: The Logic and Metaphysics of Intentionality. Oxford: Oxford University Press. Putnam, Hilary. 1971. Philosophy of Logic. Harper Torchbooks. Quine, W.V.O. 1948. “On What There Is.” Review of Metaphysics 2 (5): 21– 36. ———. 1975. “Five Milestones of Empiricism.” In Theories and Things. Harvard University Press. ———. 1984. “Review of Charles Parsons’ Mathematics in Philosophy.” Journal of Philosophy. ———. 1986. Philosophy of Logic. 2nd ed. Harvard University Press. ———. 1990. Pusuit of Truth. Harvard University Press. Resnik, Michael. 1997. Mathematics as a Science of Patterns. Clarendon Press. Rosen, Gideon. 2001. “Nominalism, Naturalism, Epistemic Relativism.” Philosophical Perspectives 15: 69–91. ———. 2012. “Abstract Objects.” Stanford Encyclopedia of Philosophy. Rosen, Gideon, and John Burgess. 2005. “Nominalism Reconsidered.” In The Oxford Handbook of Philosophy of Mathematics and Logic, edited by Stewart Shapiro. Oxford Handbooks in Philosophy. Oxford University Press. Shalkowski, Scott. 2014. “God with or Without Abstract Objects.” In Beyond the Control of God?, edited by Paul M. Gould. Bloomsbury Academic. Shapiro, Stewart. 1997. Philosophy of Mathematics: Structure and Ontology. New York: Oxford University Press. Sober, Eliot. 1993. “Mathematics and Indispensability.” The Philosophical Review 102 (1): 35–57. Walton, Kendall. 1990. Mimesis as Make-Believe. Harvard University Press. ———. 1993. “Metaphor and Prop Oriented Make-Believe.” European Journal of Philosophy 1 (39-57). Wetzel, Linda. 2009. Types and tokens. MIT Press. Yablo, Stephen. 2009. “Must Existence-Questions Have Answers?” In Metametaphysics: New Essays on the Foundations of Ontology, edited by David Manley David Chalmers and Ryan Wasserman. Oxford University Press.