Nominalism in the Philosophy of Mathematics First published Mon Sep 16, 2013

Transcription

1 Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia Free Nominalism in the Philosophy of Mathematics First published Mon Sep 16, 2013 Nominalism about mathematics (or mathematical nominalism) is the view according to which either mathematical objects, relations, and structures do not exist at all, or they do not exist as abstract objects (they are neither located in space-time nor do they have causal powers). In the latter case, some suitable concrete replacement for mathematical objects is provided. Broadly speaking, there are two forms of mathematical nominalism: those views that require the reformulation of mathematical (or scientific) theories in order to avoid the commitment to mathematical objects (e.g., Field 1980; Hellman 1989), and those views that do not reformulate mathematical or scientific theories and offer instead an account of how no commitment to mathematical objects is involved when these theories are used (e.g., Azzouni 2004). Both forms of nominalism are examined, and they are assessed in light of how they address five central problems in the philosophy of mathematics (namely, problems dealing with the epistemology, the ontology, and the application of mathematics as well as the use of a uniform semantics and the proviso that mathematical and scientific theories be taken literally). 1. Two views about mathematics: nominalism and platonism 2. Five Problems 2.1 The epistemological problem of mathematics 2.2 The problem of the application of mathematics 2.3 The problem of uniform semantics 2.4 The problem of taking mathematical discourse literally 2.5 The ontological problem 3. Mathematical Fictionalism 3.1 Central features of mathematical fictionalism 3.2 Metalogic and the formulation of conservativeness 3.3 Assessment: benefits and problems of mathematical fictionalism 4. Modal Structuralism Page 1 of 41

2 4.1 Central features of modal structuralism 4.2 Assessment: benefits and problems of modal structuralism 5. Deflationary Nominalism 5.1 Central features of deflationary nominalism 5.2 Assessment: benefits of deflationary nominalism and a problem Bibliography Academic Tools Other Internet Resources Related Entries 1. Two views about mathematics: nominalism and platonism In ontological discussions about mathematics, two views are prominent. According to platonism, mathematical objects (as well as mathematical relations and structures) exist and are abstract; that is, they are not located in space and time and have no causal connection with us. Although this characterization of abstract objects is purely negative indicating what such objects are not in the context of mathematics it captures the crucial features the objects in questions are supposed to have. According to nominalism, mathematical objects (including, henceforth, mathematical relations and structures) do not exist, or at least they need not be taken to exist for us to make sense of mathematics. So, it is the nominalist's burden to show how to interpret mathematics without the commitment to the existence of mathematical objects. This is, in fact, a key feature of nominalism: those who defend the view need to show that it is possible to yield at least as much explanatory work as the platonist obtains, but invoking a meager ontology. To achieve that, nominalists in the philosophy of mathematics forge interconnections with metaphysics (whether mathematical objects do exist), epistemology (what kind of knowledge of these entities we have), and philosophy of science (how to make sense of the successful application of mathematics in science without being committed to the existence of mathematical entities). These interconnections are one of the sources of the variety of nominalist views. Despite the substantial differences between nominalism and platonism, they have at least one feature in common: both come in many forms. There are various versions of platonism in the philosophy of mathematics: standard (or object-based) platonism (Gödel 1944, 1947; Quine 1960), structuralism (Resnik 1997; Shapiro 1997), and full-blooded platonism (Balaguer 1998), among other views. Similarly, there are also several versions of nominalism: fictionalism (Field 1980, 1989), modal structuralism (Hellman 1989, 1996), constructibilism (Chihara 1990), the weaseling-away view (Melia 1995, 2000), figuralism (Yablo 2001), deflationary nominalism (Azzouni 2004), agnostic nominalism (Bueno 2008, 2009), and pretense views (Leng 2010), among others. Similarly to their platonist Page 2 of 41

3 counterparts, the various nominalist proposals have different motivations, and face their own difficulties. These will be explored in turn. (A critical survey of various nominalization strategies in mathematics can be found in Burgess and Rosen (1997). The authors address in detail both the technical and philosophical issues raised by nominalism in the philosophy of mathematics.) Discussions about nominalism in the philosophy of mathematics in the 20 th century started roughly with the work that W. V. Quine and Nelson Goodman developed toward constructive nominalism (Goodman and Quine 1947). But, as Quine later pointed out, in the end it was indispensable to quantify over classes (Quine 1960). As will become clear below, responses to this indispensability argument have generated a significant amount of work for nominalists. And it is the focus on the indispensability argument that largely distinguishes more recent nominalist views in the philosophy of mathematics, which I will focus on, from the nominalism developed in the early part of the 20 th century by the Polish school of logic (Simons 2010). Mathematical nominalism is a form of anti-realism about abstract objects. This is an independent issue from the traditional problem of nominalism about universals. A universal, according to a widespread use, is something that can be instantiated by different entities. Since abstract objects are neither spatial nor temporal, they cannot be instantiated. Thus, mathematical nominalism and nominalism about universals are independent from one another (see the entry on nominalism in metaphysics). It could be argued that certain sets encapsulate the instantiation model, since a set of concrete objects can be instantiated by such objects. But since the same set cannot be so instantiated, given that sets are individuated by their members and as long as their members are different the resulting sets are not the same, it is not clear that even these sets are instantiated. I will focus here on mathematical nominalism. 2. Five Problems In contemporary philosophy of mathematics, nominalism has been formulated in response to difficulties faced by platonism. But in developing their responses to platonism, nominalists also encounter difficulties of their own. Five problems need to be addressed in this context: 1. The epistemological problem of mathematics, 2. The problem of the application of mathematics, 3. The problem of uniform semantics, 4. The problem of taking mathematical discourse literally, and 5. The ontological problem. Page 3 of 41

4 Usually, problems (1) and (5) are considered as raising difficulties for platonism, whereas problems (2), (3), and (4) are often taken as yielding difficulties for nominalism. (I will discuss below to what extent such an assessment is accurate.) Each of these problems will be examined in turn. 2.1 The epistemological problem of mathematics Given that platonism postulates the existence of mathematical objects, the question arises as to how we obtain knowledge about them. The epistemological problem of mathematics is the problem of explaining the possibility of mathematical knowledge, given that mathematical objects themselves do not seem to play any role in generating our mathematical beliefs (Field 1989). This is taken to be a particular problem for platonism, since this view postulates the existence of mathematical objects, and one would expect such objects to play a role in the acquisition of mathematical knowledge. After all, on the platonist view, such knowledge is about the corresponding mathematical objects. However, despite various sophisticated attempts by platonists, there is still considerable controversy as to how exactly this process should be articulated. Should it be understood via mathematical intuition, by the introduction of suitable mathematical principles and definitions, or does it require some form of abstraction? In turn, the epistemological issue is far less problematic for nominalists, who are not committed to the existence of mathematical objects in the first place. They will have to explain other things, such as, how can the nominalist account for the difference between a mathematician, who knows a significant amount of mathematics, and a non-mathematician, who does not? This difference, according to some nominalists, is based on empirical and logical knowledge not on mathematical knowledge (Field 1989). 2.2 The problem of the application of mathematics Mathematics is often successfully used in scientific theories. How can such a success be explained? Platonists allegedly have an answer to this problem. Given that mathematical objects exist and are successfully referred to by our scientific theories, it is not surprising that such theories are successful. Reference to mathematical objects is just part of the reference to those entities that are indispensable to our best theories of the world. This frames the problem of the application of mathematics in terms of the indispensability argument. In fact, one of the main reasons for belief in the existence of mathematical objects some claim this is the only non-question begging reason (Field 1980) is given by the Page 4 of 41

5 indispensable use of mathematics in science. The crucial idea, originally put forward by W. V. Quine, and later articulated, in a different way, by Hilary Putnam, is that ontological commitment should be restricted to just those entities that are indispensable to our best theories of the world (Quine 1960; Putnam 1971; Colyvan 2001a). Mark Colyvan has formulated the argument in the following terms: (P1) We ought to be ontologically committed to all and only those entities that are indispensable to our best theories of the world. (P2) Mathematical entities are indispensable to our best theories of the world. Therefore, (C) we ought to be ontologically committed to mathematical entities. The first premise relies crucially on Quine's criterion of ontological commitment. After regimenting our best theories of the world in a first-order language, the ontological commitments of these theories can be read off as being the value of the existentially quantified variables. But how do we move from the ontological commitments of a theory to what we ought to be ontologically committed to? This is the point where the first premise of the indispensability argument emerges. If we are dealing with our best theories of the world, precisely those items that are indispensable to these theories amount to what we ought to be committed to. (Of course, a theory may quantify over more objects than those that are indispensable.) And by identifying the indispensable components invoked in the explanation of various phenomena, and noting that mathematical entities are among them, the platonist is then in a position to make sense of the success of applied mathematics. However, it turns out that whether the platonist can indeed explain the success of the application of mathematics is, in fact, controversial. Given that mathematical objects are abstract, it is unclear why the postulation of such entities is helpful to understand the success of applied mathematics. For the physical world being composed of objects located in space-time is not constituted by the entities postulated by the platonist. Hence, it is not clear why the correct description of relations among abstract (mathematical) entities is even relevant to understand the behavior of concrete objects in the physical world involved in the application of mathematics. Just mentioning that the physical world instantiates structures (or substructures) described in general terms by various mathematical theories is not enough (see, e.g., Shapiro 1997). For there are infinitely many mathematical structures, and there is no way of uniquely determining which of them is actually instantiated or even instantiated only in part in a finite region of the physical world. There is a genuine underdetermination here, given that the same physical structure in the world can be accommodated by very different mathematical structures. For instance, quantum mechanical phenomena can be characterized by group-theoretic structures (Weyl Page 5 of 41

6 1928) or by structures emerging from the theory of Hilbert spaces (von Neumann 1932). Mathematically, such structures are very different, but there is no way of deciding between them empirically. Despite the controversial nature of the platonist claim to be able to explain the success of applied mathematics, to accommodate that success is often taken as a significant benefit of platonism. Less controversially, the platonist is certainly able to describe the way in which mathematical theories are actually used in scientific practice without having to rewrite them. This is, as will become clear below, a significant benefit of the view. Nominalism, in turn, faces the difficulty of having to explain the successful use of mathematics in scientific theorizing. Since, according to the nominalist, mathematical objects do not exist or, at least, are not taken to exist it becomes unclear how referring to such entities can contribute to the empirical success of scientific theories. In particular, if it turns out that reference to mathematical entities is indeed indispensable to our best theories of the world, how can the nominalist deny the existence of such entities? As we will see below, several nominalist views in the philosophy of mathematics have emerged in response to the challenge raised by considerations based on the indispensability of mathematics. 2.3 The problem of uniform semantics One of the most significant features of platonism is the fact that it allows us to adopt the same semantics for both mathematical and scientific discourse. Given the existence of mathematical objects, mathematical statements are true in the same way as scientific statements are true. The only difference emerges from their respective truth makers: mathematical statements are true in virtue of abstract (mathematical) objects and relations among them, whereas scientific statements are ultimately true in virtue of concrete objects and the corresponding relations among such objects. This point is idealized in that it assumes that, somehow, we can manage to distill the empirical content of scientific statements independently of the contribution made by the mathematics that is often used to express such statements. Platonists who defend the indispensability argument insist that this is not possible to do (Quine 1960; Colyvan 2001a); even some nominalists concur (Azzouni 2011). Moreover, as is typical in the application of mathematics, there are also mixed statements, which involve both terms referring to concrete objects and terms referring to abstract ones. The platonist has no trouble providing a unified semantics for such statements either particularly if mathematical platonism is associated with realism about science. In this case, the platonist can provide a referential semantics throughout. Of course, the platonist about mathematics need not be a realist about science although it's common to combine Page 6 of 41

7 platonism and realism in this way. In principle, the platonist could adopt some form of antirealism about science, such as constructive empiricism (van Fraassen 1980; Bueno 2009). As long as the form of anti-realism regarding science allows for a referential semantics (and many do), the platonist would have no trouble providing a unified semantics for both mathematics and science (Benacerraf 1973). It is not clear that the nominalist can deliver these benefits. As will become clear shortly, most versions of nominalism require a substantial rewriting of mathematical language. As a result, a distinct semantics needs to be offered for that language in comparison with the semantics that is provided for scientific discourse. 2.4 The problem of taking mathematical discourse literally A related benefit of platonism is that it allows one to take mathematical discourse literally, given that mathematical terms refer. In particular, there is no change in the syntax of mathematical statements. So, when mathematicians claim that There are infinitely many prime numbers, the platonist can take that statement literally as describing the existence of an infinitude of primes. On the platonist view, there are obvious truth-makers for mathematical statements: mathematical objects and their corresponding properties and relations (Benacerraf 1973). We have here a major benefit of platonism. If one of the goals of the philosophy of mathematics is to provide understanding of mathematics and mathematical practice, it is a significant advantage that platonists are able to take the products of that practice such as mathematical theories literally and do not need to rewrite or reformulate them. After all, the platonist is then in a position to examine mathematical theories as they are actually formulated in mathematical practice, rather than discuss a parallel discourse offered by various reconstructions of mathematics given by those who avoid the commitment to mathematical objects (such as the nominalists). The inability to take mathematical discourse literally is indeed a problem for nominalists, who typically need to rewrite the relevant mathematical theories. As will become clear below, it is common that nominalization strategies for mathematics change either the syntax or the semantics of mathematical statements. For instance, in the case of modal structuralism, modal operators are introduced to preserve verbal agreement with the platonist (Hellman 1989). The proposal is that each mathematical statement S is translated into two modal statements: (i) if there were structures of the suitable kind, S would be true in these structures, and (ii) it's possible that there are such structures. As a result, both the syntax and the semantics of mathematics are changed. In the case of mathematical fictionalism, in order to preserve verbal agreement with the platonist despite the denial of the existence of mathematical objects, fiction operators (such as, According to Page 7 of 41

8 arithmetic ) are introduced (Field 1989). Once again, the resulting proposal changes the syntax (and, hence, the semantics) of mathematical discourse. This is a significant cost for these views. 2.5 The ontological problem The ontological problem consists in specifying the nature of the objects a philosophical conception of mathematics is ontologically committed to. Can the nature of these objects be properly determined? Are the objects in question such that we simply lack good grounds to believe in their existence? Traditional forms of platonism have been criticized for failing to offer an adequate solution to this problem. In response, some platonists have argued that the commitment to mathematical objects is neither problematic nor mysterious (see, e.g., Hale and Wright 2001). Similarly, even though some nominalists need not be committed to mathematical objects, they may be committed to other entities that may also raise ontological concerns (such as possibilia). The ontological problem is then the problem of assessing the status of the ultimate commitments of the view. Three nominalization strategies will be discussed below: mathematical fictionalism (Field 1980, 1989), modal structuralism (Hellman 1989, 1996), and deflationary nominalism (Azzouni 2004). The first two reject the second premise of the indispensability argument. They provide hard roads to nominalism (Colyvan 2010), in the sense that the nominalist needs to develop the complex, demanding work of showing how quantification over mathematical objects can be avoided in order to develop a suitable interpretation of mathematics. The third strategy rejects the first premise of the argument, thus bypassing the need to argue for the dispensability of mathematics (in fact, for the deflationary nominalist, mathematics is ultimately indispensable). By reassessing Quine's criterion of ontological commitment, and indicating that quantification over certain objects does not require their existence, this strategy yields an easy road to nominalism. Although this survey is clearly not exhaustive, since not every nominalist view available will be considered here, the three views discussed are representative: they occupy distinct points in the logical space, and they have been explicitly developed to address the various problems just listed. 3. Mathematical Fictionalism 3.1 Central features of mathematical fictionalism In a series of works, Hartry Field provided an ingenious strategy for the nominalization of science (Field 1980, 1989). As opposed to platonist views, in order to explain the usefulness of mathematics in science, Field does not postulate the truth of mathematical Page 8 of 41

9 theories. In his view, it is possible to explain successful applications of mathematics with no commitment to mathematical objects. Therefore, what he takes to be the main argument for platonism, which relies on the (apparent) indispensability of mathematics to science, is resisted. The nominalist nature of Field's account emerges from the fact that mathematical objects are not assumed to exist. Hence, mathematical theories are false. (Strictly speaking, Field notes, any existential mathematical statement is false, and any universal mathematical statement is vacuously true.) By devising a strategy that shows how to dispense with mathematical objects in the formulation of scientific theories, Field rejects the indispensability argument, and provides strong grounds for the articulation of a nominalist stance. Prima facie, it may sound counterintuitive to state that there are infinitely many prime numbers is false. But if numbers do not exist, that's the proper truth-value for that statement (assuming a standard semantics). In response to this concern, Field 1989 introduces a fictional operator, in terms of which verbal agreement can be reached with the platonist. In the case at hand, one would state: According to arithmetic, there are infinitely many prime numbers, which is clearly true. Given the use of a fictional operator, the resulting view is often called mathematical fictionalism. The nominalization strategy devised by the mathematical fictionalist depends on two interrelated moves. The first is to change the aim of mathematics, which is not taken to be truth, but something different. On this view, the proper norm of mathematics, which will guide the nominalization program, is conservativeness. A mathematical theory is conservative if it is consistent with every internally consistent theory about the physical world, where such theories do not involve any reference to, nor quantification over, mathematical objects, such as sets, functions, numbers etc. (Field 1989, p. 58). Conservativeness is stronger than consistency (since if a theory is conservative, it is consistent, but not vice versa). However, conservativeness is not weaker than truth (Field 1980, pp ; Field 1989, p. 59). So, Field is not countenancing a weaker aim of mathematics, but only a different one. It is precisely because mathematics is conservative that, despite being false, it can be useful. Of course, this usefulness is explained with no commitment to mathematical entities: mathematics is useful because it shortens our derivations. After all, if a mathematical theory M is conservative, then a nominalistic assertion A about the physical world (i.e. an assertion which does not refer to mathematical objects) follows from a body N of such assertions and M only if follows from N alone. That is, provided we have a sufficiently rich body of nominalistic assertions, the use of mathematics does not yield any new nominalistic consequences. Mathematics is only a useful instrument to help us in the derivations. Page 9 of 41

10 As a result, conservativeness can only be employed to do the required job if we have nominalistic premises to start with (Field 1989, p. 129). As Field points out, it is a confusion to argue against his view by claiming that if we add some bits of mathematics to a body of mathematical claims (not nominalistic ones), we may obtain new consequences that could not be achieved otherwise (Field 1989, p. 128). The restriction to nominalistic assertions is crucial. The second move of the mathematical fictionalist strategy is to provide such nominalistic premises. Field has done that in one important case: Newtonian gravitational theory. He elaborates on a work that has a respectable tradition: Hilbert's axiomatization of geometry (Hilbert 1971). What Hilbert provided was a synthetic formulation of geometry, which dispenses with metric concepts, and therefore does not include any quantification over real numbers. His axiomatization was based on concepts such as point, betweenness, and congruence. Intuitively speaking, we say that a point y is between the points x and z if y is a point in the line-segment whose endpoints are x and z. Also intuitively, we say that the line-segment xy is congruent to the line-segment zw if the distance from the point x to the point y is the same as that from the point z to w. After studying the formal properties of the resulting system, Hilbert proved a representation theorem. He showed, in a stronger mathematical theory, that given a model of the axiom system for space he had put forward, there is a function d from pairs of points onto non-negative real numbers such that the following homomorphism conditions are met: i. xy is congruent to zw iff d(x, y) = d(z,w), for all points x, y, z, and w; ii. y is between x and z iff d(x, y) + d(y, z) = d(x, z), for all points x, y, and z. As a result, if the function d is taken to represent distance, we obtain the expected results about congruence and betweenness. Thus, although we cannot talk about numbers in Hilbert's geometry (there are no such entities to quantify over), there is a metatheoretic result that associates assertions about distances with what can be said in the theory. Field calls such numerical claims abstract counterparts of purely geometric assertions, and they can be used to draw conclusions about purely geometrical claims in a smoother way. Indeed, because of the representation theorem, conclusions about space, statable without real numbers, can be drawn far more easily than we could achieve by a deflationary proof from Hilbert's axioms. This illustrates Field's point that the usefulness of mathematics derives from shortening derivations (Field 1980, pp ). Roughly speaking, what Field established was how to extend Hilbert's results about space to space-time. Similarly to Hilbert's approach, instead of formulating Newtonian laws in terms of numerical functors, Field showed that they can be recast in terms of comparative predicates. For example, instead of adopting a functor such as the gravitational potential of x, which is taken to have a numerical value, Field employed a comparative predicate such Page 10 of 41

11 as the difference in gravitational potential between x and y is less than that between z and w. Relying on a body of representation theorems (which plays the same role as Hilbert's representation theorem in geometry), Field established how several numerical functors can be obtained from comparative predicates. But in order to use those theorems, he first showed how to formulate Newtonian numerical laws (such as, Poisson's equation for the gravitational field) only in terms of comparative predicates. The result (Field 1989, pp ) is the following extended representation theorem. Let N be a theory formulated only in terms of comparative predicates (with no recourse to numerical functors). For any model S of N whose domain is constituted by space-time regions, there exists: i. A 1 1 spatio-temporal co-ordinate function f (unique up to a generalized Galilean transformation) mapping the space-time of S onto quadruples of real numbers; ii. A mass density function g (unique up to a positive multiplicative transformation) mapping the space-time of S onto an interval of non-negative real numbers; and iii. A gravitational potential function h (unique up to a positive linear transformation) mapping the space-time onto an interval of real numbers. Moreover, all these functions preserve structure, in the sense that the comparative relations defined in terms of them coincide with the comparative relations used in N. Furthermore, if f, g and h are taken as the denotation of the appropriate functors, the laws of Newtonian gravitational theory in their functorial form hold. Notice that, in quantifying over space-time regions, Field assumes a substantivalist view of space-time, according to which there are space-time regions that are not fully occupied (Field 1980, pp ; Field 1989, pp ). Given this result, the mathematical fictionalist is allowed to draw nominalistic conclusions from premises involving N plus a mathematical theory T. After all, due to the conservativeness of mathematics, such conclusions can be obtained independently of T. The role of the extended representation theorem is then to establish that, despite the lack of quantification over mathematical objects, precisely the same class of models is determined by formulating Newtonian gravitational theory in terms of functors (as the theory is usually expressed) or in terms of comparative predicates (as the mathematical fictionalist favors). Thus, the extended representation theorem ensures that the use of conservativeness of mathematics together with suitable nominalistic claims (formulated via comparative predicates) does not change the class of models of the original theory: the same comparative relations are preserved. Hence, what Field provided is a nominalization strategy, and since it reduces ontology, it seems a promising candidate for a nominalist stance vis-à-vis mathematics. How should the mathematical fictionalist approach physical theories, such as perhaps string theory, that do not seem to be about concrete observable objects? One possible response, assuming the lack of empirical import of such theories, is simply to reject that Page 11 of 41

12 they are physical theories, and as such they are not the sorts of theories for which the mathematical fictionalist needs to provide a nominalistic counterpart. In other words, until the moment in which such theories acquire the relevant empirical import, they need not worry the mathematical fictionalist. Theories of that sort would be classified as part of the mathematics rather than the physics. 3.2 Metalogic and the formulation of conservativeness But is mathematics conservative? In order to establish the conservativeness of mathematics, the mathematical fictionalist has used metalogical results, such as the completeness and the compactness of first-order logic (Field 1992, 1980, 1989). The issue then arises as to whether the mathematical fictionalist can use these results to develop the program. At two crucial junctures, Field has made use of metalogical results: (a) in his reformulation of the notion of conservativeness in nominalistically acceptable terms (Field 1989, pp ; Field 1991), and (b) in his nominalist proof of the conservativeness of set theory (Field 1992). These two outcomes are crucial for Field, since they establish the adequacy of conservativeness for the mathematical fictionalist. For (a) settles that the latter can formulate that notion without violating nominalism, and (b) concludes that conservativeness is a feature that mathematics actually has. But if these two outcomes are not legitimate, Field's approach cannot get off the ground. I will now consider whether these two uses of metalogical results are acceptable on nominalist grounds Conservativeness and the compactness theorem Let me start with (a). The mathematical fictionalist has relied on the compactness theorem to formulate the notion of conservativeness in an acceptable way, that is, without reference to mathematical entities. As noted above, conservativeness is defined in terms of consistency. But this notion is usually formulated either in semantic terms (as the existence of an appropriate model), or in proof-theoretic terms (in terms of suitable proofs). However, as Field acknowledges, these two formulations of consistency are platonist, since they depend on abstract objects (models and proofs), and therefore are not nominalistically acceptable. The mathematical fictionalist way out is to avoid moving to the metalanguage in order to express the conservativeness of mathematics. The idea is to state, in the object-language, the claim that a given mathematical theory is conservative by introducing a primitive notion of logical consistency: A. Thus, if B is any sentence, B* is the result of restricting B to non-mathematical entities, and M 1,, M n are the axioms of a mathematical theory M, the conservativeness of M can be expressed by the following schema (Field 1989, p. 120): Page 12 of 41

13 (C) If B, then (B* M 1 M n ). In other words, a mathematical theory M is conservative if it is consistent with every consistent theory about the physical world B*. This assumes, of course, that M is finitely axiomatized. But how can we apply (C) in the case of mathematical theories that are not finitely axiomatizable (such as Zermelo-Fraenkel set theory)? In this case, we cannot make the conjunction of all the axioms of the theory, since there are infinitely many of them. Field has addressed this issue, and he initially suggested that the mathematical fictionalist could use substitutional quantification to express these infinite conjunctions (Field 1984). In a postscript to the revised version of this essay (Field 1989, pp ), he notes that substitutional quantification can be avoided, provided that the mathematical and physical theories in question are expressed in a logic for which compactness holds. For in this case, the consistency of the whole theory is reduced to the consistency of each of its finite conjunctions. There are, however, three problems with this move. i. One concern about the use of substitutional quantification in this context involves the nature of substitutional instances. If the latter turn out to be abstract, which would be the case if such substitutional instances were not mere inscriptions, they are not available to the nominalist. If the substitutional instances are concrete, the nominalist needs to show that there are enough of them. ii. The very statement of the compactness theorem involves set-theoretic talk: let G be a set of formulas; if every finite subset of G is consistent, then G is consistent. How can nominalists rely on a theorem whose very statement involves abstract entities? In order to use this theorem, an appropriate reformulation is required. iii. Let us grant that it is possible to reformulate this statement without referring to sets. Can then the nominalist use the compactness theorem? As is well known, the proof of this theorem assumes set theory. The compactness theorem is usually presented as a corollary to the completeness theorem for first-order logic, whose proof assumes set theory (see, for example, Boolos and Jeffrey 1989, pp ). Alternatively, if the compactness result is to be proved directly, then one has to construct the appropriate model of G which again requires set theory. So, unless mathematical fictionalists are able to provide an appropriate nominalization strategy for set theory itself, they are not entitled to use this result. In other words, far more work is required before a Field-type nominalist is able to rely on metalogical results. But maybe this criticism misses the whole point of Field's program. As we saw, Field does not require that a mathematical theory M be true for it to be used. Only its conservativeness is demanded. So, if M is added to a body B* of nominalistic claims, no new nominalistic Page 13 of 41

14 conclusion is obtained which was not obtained by B* alone. In other words, what Field's strategy asks for is the formulation of appropriate nominalistic bodies of claims to which mathematics can be applied. The same point holds for metalogical results: provided that they are applied to nominalistic claims, Field is fine. The problem with this reply is that it involves the mathematical fictionalist program in a circle. The fictionalist cannot rely on the conservativeness of mathematics to justify the use of a mathematical result (the compactness theorem) that is required for the formulation of the notion of conservativeness itself. For in doing that, the fictionalist assumes that the notion of conservativeness is nominalistically acceptable, and this is exactly the point in question. Recall that the motivation for Field to use the compactness theorem was to reformulate conservativeness without having to assume abstract entities (namely, those required by the semantic and the proof-theoretic accounts of consistency). Thus, at this point, the mathematical fictionalist cannot yet use the notion of conservativeness; otherwise, the whole program would not get off the ground. I conclude that, similarly to any other part of mathematics, metalogical results also need to be obtained nominalistically. Trouble arises for nominalism otherwise Conservativeness and primitive modality But perhaps the mathematical fictionalist has a way out. As we saw, Field spells out the notion of conservativeness in terms of a primitive notion of logical consistency: A. And he also indicates that this notion is related to the model-theoretic concept of consistency in particular, to the formulation of this concept in von Neumann-Bernays-Gödel finitely axiomatizable set theory (NBG). This is done via two principles (Field 1989, p. 108): (MTP # ) If (NBG there is a model for A ), then A (ME # ) If (NBG there is no model for A ), then A. I am following Field's terminology: MTP # stands for model-theoretic possibility, and ME # for model existence. The symbol # indicates that, according to Field, these principles are nominalistically acceptable. After all, they are modal surrogates for the platonistic principles (Field 1989, pp ): (MTP) If there is a model for A, then A (ME) If there is no model for A, then A. It may be argued that, by using these principles, the mathematical fictionalist will be entitled to use the compactness theorem. First, one should try to state this theorem in a Page 14 of 41

15 nominalistically acceptable way. Without worrying too much about details, let us grant, for the sake of argument, that the following characterization will do: (Compact # ) If T, then f A 1,, A n [ (A 1 A n )], where T is a theory and each A i, 1 i n, is a formula (an axiom of T). The expression f A 1 A n is to be read as there are finitely many formulas A 1 A n. (This quantifier is not first-order. However, I am not going to press the point that the nominalist seems to need a non-first-order quantifier to express a property typical of first-order logic. This is only one of the worries we are leaving aside in this formulation.) This version is parasitic on the following platonistic formulation of the compactness theorem: (Compact) If there is no model for T, then f A 1,, A n such that there is no model for (A 1 A n ). In order for mathematical fictionalists to be entitled to use the compactness theorem, they will have to show that the nominalistic formulation (Compact # ) follows from the platonistic one (Compact). In this sense, if the latter is adequate, so is the former. More accurately, what has to be shown is that (Compact # ) follows from a modal surrogate of (Compact). After all, since what is at issue is the legitimacy of the compactness theorem on nominalist grounds, it would be question-begging to assume the full platonistic version from the outset. As we will see, there are two ways to try to establish this result. Unfortunately, none of them works: both are formally inadequate. The two options start in the same way. Suppose that (1) T. We have to establish that (2) f A 1 A n (A 1 A n ). It follows from (1) and (MTP # ) that and thus (3) (NBG there is a model for T ), (4) (NBG there is no model for T ). Let us assume the modal surrogate for the compactness theorem: (Compact M ) (NBG if there is no model for T, then f A 1 A n such that Page 15 of 41

16 there is no model for (A 1 A n )). Note that, since the modal surrogate is formulated in terms of models (rather than in terms of the primitive modal operator), it is still not what mathematical fictionalists need. What they need is (Compact # ), but one needs to show that they can get it. At this point, the options begin to diverge. The first option consists in drawing from (4) and (Compact M ) that (5) ( f A 1 A n such that there is no model for (A 1 A n )). There are, however, difficulties with this move. First, note that (5) is not equivalent to (2), which is the result to be achieved. Moreover, as opposed to (2), (5) is formulated in modeltheoretic terms, since it incorporates a claim about the nonexistence of a certain model. And what is required is a similar statement in terms of the primitive notion of consistency. In other words, we need the nominalistic counterpart of (5), rather than (5). But (5) has a nice feature. It is a modalized formulation of the consequent of (Compact). And since (5) only states the possibility that there is no model of a particular kind, it may be argued that it is nominalistically acceptable. (As will be examined below, modal structuralists advance a nominalization strategy exploring modality along these lines; see Hellman (1989).) Field, however, is skeptical about this move. On his view, modality is not a general surrogate for ontology (Field 1989, pp ). And one of his worries is that by allowing the introduction of modal operators, as a general nominalization strategy, we modalize away the physical content of the theory under consideration. However, since metalogical claims are not expected to have physical consequences, the worry need not arise here. At any rate, given that (5) does not establish what needs to be established, it does not solve the problem. The second option consists in moving to (5$) instead of (5): (5$) (NBG f A 1 A n such that there is no model for (A 1 A n )). Note that if (5$) were established, we would have settled the matter. After all, with a straightforward reworking of (ME # ) (namely, If (NBG f A 1 A n such that there is no model for (A 1 A n )), then (A 1 A n )), it follows from (5$) and (ME # ) that (2) f A 1 A n (A 1 A n ), which is the conclusion we need. The problem here is that (5$) does not follow from (4) and (Compact M ). Therefore, we cannot derive it. Page 16 of 41

17 Clearly, there may well be another option that establishes the intended conclusion. But, to say the least, the mathematical fictionalist has to present it before being entitled to use metalogic results. Until then, it is not clear that these results are nominalistically acceptable Metalogic and the proof of the conservativeness of set theory I should now consider issue (b): Field's nominalistic proof of the conservativeness of set theory. Let us grant that the concept of conservativeness has been formulated in some nominalistically acceptable way. If Field's proof were correct, he would have proved that mathematics itself is conservative as long as one assumes the usual reductions of mathematics to set theory. How does Field prove the conservativeness of set theory? It is by an ingenious argument, which adapts one of the Field's platonistic conservativeness proofs (Field 1980). For our present purposes, we need not examine the details of this argument, but simply note that at a crucial point the completeness of first-order logic is used to establish its conclusion (Field 1992, p. 118). The problem with this move is that, even if mathematical fictionalists formulate the statement of the completeness theorem without referring to mathematical entities, the proof of this theorem assumes set theory (see, for instance, Boolos and Jeffrey 1989, pp ). Therefore, fictionalists cannot use the theorem without undermining their nominalism. After all, the point of providing a nominalistic proof of the conservativeness of set theory is to show that, without recourse to platonist mathematics, the mathematical fictionalist is able to establish that mathematics is conservative. Field has offered a platonist argument for the conservativeness result (Field 1980) an argument that explicitly invoked properties of set theory. The idea was to provide a reductio of platonism: by using platonist mathematics, Field attempted to establish that mathematics was conservative and, thus, ultimately dispensable. In contrast with the earlier strategy, the goal was to provide a proof of the conservativeness of set theory that a nominalist could accept. But since the nominalistic proof relies on the completeness theorem, it is not at all clear that it is in fact nominalist. Mathematical fictionalists should first be able to prove the completeness result without assuming set theory. Alternatively, they should provide a nominalization strategy for set theory itself, which will then entitle them to use metalogical results. But it may be argued that the mathematical fictionalist only requires the conservativeness of the set theory in which the completeness theorem is proved. It should now be clear that this reply is entirely question begging, since the point at issue is exactly to prove the conservativeness of set theory. Thus, the fictionalist cannot assume that this result is already established at the metatheory. Page 17 of 41

18 In other words, without a broader nominalization strategy, which allows set theory itself to be nominalized, it seems difficult to see how mathematical fictionalists can use metalogical results as part of their program. The problem, however, is that it is not at all obvious that, at least in the form articulated by Field, the mathematical fictionalist program can be extended to set theory. For it only provides a nominalization strategy for scientific theories, that is, for the use of mathematics in science (e.g., in Newtonian gravitational theory). The approach doesn't address the nominalization of mathematics itself. In principle, one may object, this shouldn't be a problem. After all, the mathematical fictionalists motivation to develop their approach has focused on one issue: to overcome the indispensability argument thus addressing the use of mathematics in science. And the overall strategy, as noted, has been to provide nominalist counterparts to relevant scientific theories. The problem with this objection, however, is that given the nature of Field's strategy, the task of nominalizing science cannot be achieved without also nominalizing set theory. Thus, what is needed is a more open-ended, broader nominalism: one that goes hand in hand not only with science, but also with metalogic. As it stands, the mathematical fictionalist approach still leaves a considerable gap. 3.3 Assessment: benefits and problems of mathematical fictionalism The epistemological problem Given that mathematical objects do not exist, on the mathematical fictionalist perspective, the problem of how we can obtain knowledge of them simply vanishes. But another problem emerges instead: what is it that distinguishes a mathematician (who knows a lot about mathematics) and a non-mathematician (who does not have such knowledge)? The difference here (according to Field 1984) is not about having or lacking mathematical knowledge, but rather it is about logical knowledge: of knowing which mathematical theorems follow from certain mathematical principles, and which do not. The epistemological problem is then solved as long as the mathematical fictionalist provides an epistemology for logic. In fact, what needs to be offered is ultimately an epistemology for modality. After all, on Field's account, in order to avoid the platonist commitment to models or proofs, the concept of logical consequence is understood in terms of the primitive modal concept of logical possibility: A follows logically from B as long as the conjunction of B and the negation of A is impossible, that is, (B A). However, how are such judgments of impossibility established? Under what conditions do Page 18 of 41

19 we know that they hold? In simple cases, involving straightforward statements, to establish such judgments may be unproblematic. The problem emerges when more substantive statements are invoked. In these cases, we seem to need a significant amount of mathematical information in order to be able to determine whether the impossibilities in question really hold or not. Consider, for instance, the difficulty of establishing the independence of the axiom of choice and the continuum hypotheses from the axioms of Zermelo-Fraenkel set theory. Significantly complex mathematical models need to be constructed in this case, which rely on the development of special mathematical techniques to build them. What is required from the mathematical fictionalist at this stage is the nominalization of set theory itself something that, as we saw, Field still owes us The problem of the application of mathematics Similarly to the epistemological problem, the problem of the application of mathematics is partially solved by the mathematical fictionalist. Field provides an account of the application of mathematics that does not require the truth of mathematical theories. As we saw, this demands that mathematics be conservative in the relevant sense. However, it is unclear whether Field has established the conservativeness of mathematics, given his restrictive way of introducing non-set-theoretic vocabulary into the axioms of set theory as part of his attempted proof of the conservativeness of set theory (Azzouni 2009b, p. 169, note 47; additional difficulties for the mathematical fictionalist program can be found in Melia 1998, 2000). Field was working with restricted ZFU, Zermelo-Fraenkel set theory with the axiom of choice modified to allow for Urelemente, objects that are not sets, but not allowing for any non-set-theoretic vocabulary to appear in the comprehension axioms, that is, replacement or separation (Field 1980, p. 17). This is, however, a huge restriction, given that when mathematics is actually applied, non-set-theoretic vocabulary, when translated into set-theoretic language, will have to appear in the comprehension axioms. As formulated by Field, the proof failed to address the crucial case of actual applications of mathematics. Moreover, it is also unclear whether the nominalization program advanced by the mathematical fictionalist can be extended to other scientific theories, such as quantum mechanics (Malament 1992). Mark Balaguer responded to this challenge by trying to nominalize quantum mechanics along mathematical fictionalist lines (Balaguer 1998). However, as argued by Bueno (Bueno 2003), Balaguer's strategy is incompatible with a number of interpretations of quantum mechanics, in particular with Bas van Fraassen's version of the modal interpretation (van Fraassen 1991). And given that Balaguer's strategy invokes physically real propensities, it is unclear whether it is even compatible with nominalism. As a result, the nominalization of quantum mechanics still remains a major problem for the mathematical fictionalist. Page 19 of 41