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Philos Stud DOI 10.1007/s11098-016-0670-y An anti-realist account of the application of mathematics Otávio Bueno1 Ó Springer Science+Business Media Dordrecht 2016 Abstract Mathematical concepts play at least three roles in the application of mathematics: an inferential role, a representational role, and an expressive role. In this paper, I argue that, despite what has often been alleged, platonists do not fully accommodate these features of the application of mathematics. At best, platonism provides partial ways of handling the issues. I then sketch an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features of the application process. In this way, a better account of mathematical applications is, in principle, available. Keywords Application of mathematics Representation Inference, Realism Anti-realism Nominalism Platonism 1 Introduction Platonism, or a robust ontologically committing form of realism about mathematics, involves three basic claims: (a) Mathematical objects (such as sets, functions, and numbers) exist. (b) These objects are abstract, in the sense that they are causally isolated and they are not located in space–time. And (c) the theories that describe such objects—mathematical theories—are taken to be true (they express true claims & Otávio Bueno [email protected] 1 Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA 123 O. Bueno about mathematics objects and their relations), and their terms refer.1 One of the main (alleged) reasons for platonism emerges from the indispensable need for quantifying over mathematical objects in the formulation of our best theories of the world. Given that reference to mathematical objects is indispensable to such theories, and since the latter cannot be rewritten without quantification over these objects, the commitment to the existence of mathematical entities cannot be avoided (Quine 1960; Putnam 1971; Colyvan 2001). Mathematical notions seem to play three basic roles in the application of mathematics. (1) They play an inferential role in that mathematical principles provide additional models (or, syntactically, additional premises) in terms of which inferences can be carried out. For instance, one of the most significant uses of group theory in quantum mechanics came at the applied level, helping to obtain certain solutions to Schrödinger’s equation, and thus allowing inferences from that equation to be obtained (Wigner 1931). (2) Mathematical notions also play a representational role, given that mathematical structures are used to represent features of the physical world, not only by providing objects and relations that may be associated with aspects of the latter, but also by providing various mappings that provide different types of relations between mathematical posits and the world. (Such relations include, e.g., various kinds of isomorphism and homomorphism.) The representational role is found, for example, in von Neumann’s use of Hilbert spaces to represent certain states of quantum systems (von Neumann 1932). (3) Finally, mathematical notions play an expressive role, given that they can be used to express certain relations among physical objects. For instance, Weyl’s characterization of the nature of quantum particles in terms of certain sets of invariants illustrates one expressive role of group-theoretic notions in the foundation of quantum mechanics (Weyl 1931). There are, of course, close connections between these three uses of mathematics. Often, the expressive role is achieved by invoking a particular representation of certain mathematical structures. But, still, these roles are distinct, since certain representations are established without the aim of expressing connections among physical entities, but are introduced, in some cases, to make certain inferences possible. Consider, for instance, the use of maximization techniques in economics. One need not take economic agents to be attempting literally to maximize their profits or their budgets all the time. The maximization offers a regulative norm, which need not express what is going on at the actual economic setting, but which allows economists to draw inferences based on what is admittedly a very idealized representation. In other words, the distinction between these three roles—in particular, between the expressive and the representational roles—is not sharp. But, despite the lack of sharpness, there is still a distinction to be drawn. It is a distinction of a pragmatic sort. The representational role invokes mathematics as a proxy for certain nonmathematical objects. The expressive role invokes mathematics as a means of 1 Of course, mathematical theories also need to meet additional constraints, such as being informative and mathematically tractable. A true but uninformative mathematical theory won’t be of much use. Similarly, a mathematical theory that is so computationally intractable that hardly any results can be derived from it will not offer significant gains—even if it were true. But these constraints may be taken to be mostly pragmatic. 123 An anti-realist account of the application of mathematics expressing certain claims about the world. This is a form of representation, of course, and in this sense the representational role is more basic.2 But it is still important to highlight the expressive role. Consider, for example, the notion of a field. This is a mathematical notion, which drops out very naturally from the mathematical formalism, but which (suitably interpreted) is used to denote a physical object. In this way, having the mathematics of fields allows us to assert things about the physical world that we could not express otherwise. Moreover, we may be interested in representing mathematically certain physical situations, but without taking the point of that representation to be the expression of certain claims about the physical system in question (even though we might be able to make such claims). For example, we can represent the degrees of belief of an epistemic agent in terms of the continuum, given the inferential power this representation provides, without thereby aiming to express literally claims about the agent in terms of the mathematical model. No human agent literally has such finegrained beliefs. In this paper, I defend two claims. First, despite what has often been alleged, platonists do not fully accommodate these three features of the application of mathematics. At best, platonism provides partial ways of handling these issues. Second, I develop an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features of the application process. In this way, a way of undermining the indispensability of mathematics is provided, one in which although mathematical theories are taken to be indispensable to a variety of applications in the sciences, their truth plays no role whatsoever. The resulting account is thus different from three of the main anti-realist views in the philosophy of mathematics. As opposed to Hartry Field’s mathematical fictionalism (1980, 1989), it is granted that mathematics is indeed indispensable, but in contrast with Geoffrey Hellman’s modal structuralism (1989, 1996) and Jody Azzouni’s deflationary nominalism (2004), mathematical theories are not taken to be true. For the modal structuralist, mathematical theories come out true in a modalstructural reformulation, which does not require the existence of mathematical objects, but only the possibility of certain structures. For the deflationary nominalist, mathematical theories are true, but given the adoption of an ontologically neutral reading of the quantifiers, the truth of mathematical statements does not require the existence of mathematical objects. (A critical evaluation of these three nominalist views can be found in Bueno 2013.) On the view favored here, truth is just irrelevant for the application of mathematics. As a result, the view does not depend on the introduction of modal operators to reformulate mathematical theories (as does the modal-structural 2 In some cases, the representational and the expressive roles can, of course, overlap. For example, when we measure the temperature of a gas under pressure, we seem to be representing the temperature by using certain numbers as a proxy for a physical property. But we also seem to be expressing a claim about the temperature of the gas. (Thanks to Russell Marcus for raising this point.) What is important here is the function played by each usage: we establish a certain mathematical representation in order to express a particular claim about the temperature of the gas. 123 O. Bueno approach), since applied mathematical statements are not taken to be true. Nor is the proposed anti-realist view committed to the nonstandard interpretation of mathematical statements recommended by the deflationary nominalist, according to which, for instance, it is true that there are twice differentiable functions that are solutions to a wave equation, even though functions do not exist. After all, truth is not the relevant norm of assessment of applied mathematics. And since mathematical theories are typically taken to be indispensable to applications in the sciences, the anti-realist view advanced here is not committed to the reformulation of scientific theories without invoking mathematics—a crucial feature of Field’s mathematical fictionalism. 2 Problems for platonism in applied mathematics 2.1 The inferential role of mathematics Let us start by considering the inferential role of mathematics. Prima facie, platonists should be able to accommodate this aspect of applied mathematics without difficulty. After all, for platonists mathematical theories are (taken to be) true, and so they can be easily added as premises in an argument involving other premises, such as those describing physical features of the phenomena. In this way, accommodating the inferential role of mathematical theories, at least in this form, should be unproblematic. But there are three difficulties here. First, to accommodate fully the inferential role of mathematics, we will also need to accommodate ‘‘mixed’’ statements, statements connecting mathematical and physical statements. These statements would also need to be (taken to be) true. This need not be a problem for platonists, as long as they are also scientific realists, or are in a position to assert that scientific theories, and related claims about the physical world, are true. This is, however, a significant restriction, given the difficulties associated with asserting the truth of such theories in the presence of idealizations, simplifications, and other known approximations inherent in scientific theorizing and modeling. If we are not in a position to assert the truth of the scientific theories themselves, we will not be able to assert the truth of the conjunction of such theories and the relevant mathematical theories. Of course, it might be objected that, in many cases—particularly in mathematical physics—the formulation and expression of scientific theories already requires the use of mathematics. It is not a simple matter of adding mathematical principles to an already formulated theory about the physical world. The formulation of the theories themselves demands mathematics. This response correctly highlights the fact that the application of mathematics is no simple business. And it certainly is not just a matter of adding mathematical statements to a non-mathematical body of statements, given that the language that is used to express some scientific theories already presupposes some mathematics. But this means that the original objection is still harder to meet. How can we reconcile the truth of mathematics with the partial truth of most scientific theories? 123 An anti-realist account of the application of mathematics This brings us to the second difficulty, which is a consequence of the first. For platonists to make full sense of the inferential role of mathematics, they need to be able to know (or, at least, assert) that both mathematical and mixed statements are true. What are the grounds for that knowledge (or that assertion)? Note that for platonists to accommodate fully the inferential role, it is not sufficient to have true mathematical statements at one’s disposal. One needs to know (or be in a position to assert) that such statements are true, so that we can then infer statements about the physical world from the relevant mathematical and mixed statements. But, in this case, some ground needs to be provided for the knowledge (or for the assertibility conditions) that we are expected to have of such mathematical and mixed statements. If these grounds are not provided, the platonist would offer, at best, a conditional account of the inferential role: (*) If mathematical (and mixed) statements are true, then the role such statements play in applied mathematics can be accommodated. But unless we are in a position to know (or assert) which mathematical (and mixed) statements are true, we will not be in a position to infer the consequent of the conditional (*) above. It is unclear, however, that platonists are in a position to assert that both mathematical and applied statements are true. After all, since idealizations and simplifications cannot be taken to be true, they cannot be made compatible with the assertion that mathematical theories are true. Thus, it is not clear that platonists are able to accommodate fully the inferential role of mathematics. The third difficult emerges at this point. Let us grant that we can devise a strategy to assert that idealizations and simplifications in scientific practice are true. This would still leave the platonist proposal in trouble. After all, it is not sufficient for platonists to assert that any conjunction of mathematical and physical statements is true. They need to know which mathematical statements are relevant to the particular application so that the statements can be used in true mixed statements. To accommodate the inferential role of mathematics, it is not enough simply to make any odd conjunction of mathematical and physical statements. We need to accommodate the relevant ones—namely those that emerge in the context of scientific practice. However, there seems to be a tension here. Platonists presuppose the existence of mathematical objects. But to accommodate the inferential role of mathematics the existence of these objects is not required at all. In fact, to draw conclusions from mathematical statements no commitment to mathematical objects has to be in place. After all, to establish the relevant inferences, all that is needed is to establish relations of logical consequence between the statements in question. No platonist ontology is needed for that. It may be objected that the tension I am pointing out cannot be right. After all, even the notion of consequence, as traditionally formulated in terms of the modeltheoretic account, presupposes platonism. We say that A is a consequence of B if every model of B (that is, every interpretation in which B is true) is also a model of A. Clearly these models are abstract objects. Thus, even the statement of the notion of inference, according to the model-theoretic account, seems to require platonism. Hence, rather than being in some conflict with the platonist picture, the inferential role of mathematics seems to support this view. 123 O. Bueno In response, note first that the model-theoretic account of consequence, despite its widespread use, is of course not the only account available. There is also the modal account: A is a consequence of B if it is necessary that: if A, then B; here the notion of necessity is defined in terms of a primitive notion of logical possibility. Clearly, no reference to mathematical objects is required by this account. (I will return to the modal conception below.) Second, independently of the concerns surrounding platonism, the model-theoretic account of consequence faces a serious limitation. It does not apply to classical set theory, in particular to a set theory such as Zermelo–Fraenkel with the axiom of choice (ZFC). After all, to use the model-theoretic account, first a domain of interpretation needs to be in place. The domain is typically a set, and in this case it needs to include all sets, since these are the objects that comprise the relevant domain. Thus, a set of all sets seems to be needed. However, no such set exists in a classical set theory (presumed consistent), and hence there is no suitable domain of interpretation for the modeltheoretical account to get off the ground (see Field 1989). Suppose this difficulty is circumvented by the introduction of larger totalities that are not sets as the domain of interpretation. In this case, the consistency of the resulting set theory still remains an issue. Moreover, in invoking the model-theoretic concept of logical consequence, one needs to be able to consider the interpretations in which every axiom of the theory is true in order to determine whether a given statement is true as well (or not). But how can the truth of all the theory’s axioms be properly assessed while the consistency of the theory is still open? Now, set theory plays a significant role in contemporary mathematics, and it is widely used in applications to the sciences. If the model-theoretic account of consequence does not apply to this theory, platonists cannot argue that this account supports their attempt to accommodate the inferential role of mathematics. In fact, the limitation of the modeltheoretic account suggests a significant limitation to platonist attempts to accommodate the inferential role. In the end, this role is in tension with the model-theoretic approach to logical consequence: given that the abstract object that is required by the model-theoretic account (namely, the set of all sets) does not exist if classical set theories are consistent, it is unclear how the platonist can make sense of the inferential role in general. In other words, the platonist’s attempt to accommodate model-theoretically the consequence relation requires the existence of an object that cannot exist as long as set theory’s consistency is assumed. And typically, set theory is taken to be consistent when it is used—for example, in applied mathematics—and in particular when inferences are drawn from the theory. Thus, as opposed to the model-theoretic account and its underlying platonism, to draw consequences from set theory, we cannot be committed to the existence of a set of all sets. I conclude that platonism cannot fully accommodate the inferential role of mathematics. Can the platonist give up the model-theoretic account of consequence, and adopt the modal account instead? This is certainly a possibility. But since the modal account is not platonistic, the platonist cannot take the resulting account of the inferential role of mathematics as supporting platonism. And a significant motivation underlying the use of the success of applied mathematics in support of platonism is undermined. After all, the crucial work is done by a non-platonist view: modalism (for details, see Bueno and Shalkowski 2013, 2015). 123 An anti-realist account of the application of mathematics 2.2 The representational role of mathematics It might also be thought that platonists should be able to accommodate the representational role of mathematics. After all, they are committed to an ontology of mathematical objects and relations that can be immediately used to accommodate the way in which mathematical theories represent. Platonists would point out that there are mathematical structures (thought of as a domain of objects and relations defined on that domain) and various sorts of relations between such structures (such as isomorphism, homomorphism, and so on). The representational role of mathematics is then characterized by specifying particular structures and particular relations between such structures. Some of these structures are used to represent relations among physical objects and states of physical systems. To do that, information about the relevant physical objects and states are encoded in a model of set theory, which allows for the study of the relations among the representations that are thus generated. Given that platonists are genuinely quantifying over all of these objects (structures and relations between structures), they should have no trouble making sense of the representational role of mathematics. But even here it is unclear whether platonists fully succeed. Clearly, they will have all sorts of mathematical objects, structures and relations to quantify over. The trouble is that to account for the representational role, one has to select the right sort of structures and relations. The problem then emerges: how can the platonist choose, among the various mathematical objects and relations, those that provide the appropriate structure? The issue is especially delicate for the platonist since mathematical objects are abstract. So, how can we have access to these objects, and how should the selection process between the various abstract structures be carried out? The obvious answer is to argue that the choice is made based on the structures that emerge from the physical phenomena. After all, these structures are the relevant ones that need to be accommodated. This response, however, is not open for platonists, for two reasons. First, physical structures are importantly different from those dealt with in mathematics. Physical structures are typically finite, and so, a plurality of different mathematical structures will fit the same structures that emerge from the physical world. As a result, in trying to choose among the various mathematically possible structures, platonists face a problem of underdetermination. In other words, physical structures do not uniquely determine the huge plurality of mathematical structures that are still possible for platonists. Several mathematical structures that agree with physically observable phenomena differ when we move beyond them. Consider, for example, Hilbert spaces and von Neumann’s type II1 factor algebras. Both can be used to represent states of quantum systems, and will essentially agree on the representation of quantum states and probability assignments when restricted to quantum systems with finitely many degrees of freedom. However, they diverge in their probability assignments when systems with infinitely many degrees of freedom are considered (Rédei 1997). The choice between such structures cannot be made on empirical grounds. Second, to make sense of physical structures, and to characterize them properly, it is crucial to identify their distinctive features. To do that, mathematical structures 123 O. Bueno whose properties are well known are invoked, and we try to embed the physical structures in question into the relevant mathematical structures. What typically happens at this point is that several abstract structures turn out to be adequate to the same physical structure; that is, several mathematical structures can be used to generate the same physical structure. As a result, underdetermination emerges again. But the problem that underdetermination yields here is different from the one considered above. What is at stake at this point is the understanding of physical structures. The reason why mathematical structures are brought into play is to help making sense of exactly how to represent the physical structures under consideration. Now, several different mathematical structures do the job, and some of them may generate a different understanding of the physical structures in question, since they embed the latter into different mathematical settings. Hence, the physical structures end up having different meanings in each embedding, since they are ultimately embedded in different mathematical structures that have different properties. For example, in the case of Hilbert spaces and von Neumann’s type II1 factor algebras, depending on the degrees of freedom of the system we consider, different properties (regarding, e.g., probability assignments) will emerge. As a result, the understanding of the physical structures change depending on the particular embedding that is adopted—for instance, which probability assignment can be defined for the physical structures themselves? In the end, given the underdetermination, platonists are left with the question: which of the several embeddings (if any) provide the correct understanding of the physical structures? Once again, it is unclear that there is a unique answer to this question. It may be argued that platonists need not be committed to a unique answer to the question of the understanding of physical structures via mathematical ones. Each answer provides a different approach to the issue of how the physical world could be, and so, in each case, a different understanding emerges. (A related scenario occurs with the various interpretations of quantum mechanics in an empiricist view; see van Fraassen 1991). I am very sympathetic to this response. In particular, I agree that having a plurality of embeddings from physical to mathematical structures does increase our understanding. However, this response fails to support platonism. After all, the approach is compatible with a thoroughly anti-realist understanding of the mathematical structures invoked by the embeddings in question. The different mathematical structures need not capture all aspects of the physical world—and clearly they do not. For different mathematical structures are compatible with the same physical configurations. So, the various mathematical structures and their physical interpretations provide distinct descriptions of the physical phenomena. Consider, for instance, mathematical descriptions that are not equivalent, but which still agree with the predictions about the physical phenomena. Different mathematical objects are referred to in each mathematical description: some structures posit vectors in a Hilbert space; others posit elements of a von Neumann algebra. Since these are very different objects, and they both have successful empirical consequences (given suitable interpretations of the mathematical formalism), which 123 An anti-realist account of the application of mathematics of the resulting picture (if any) is true? It is unclear how to decide this issue. In fact, as will become clear soon, the mathematical objects in question need not even exist for this process to go through. (I will return to this point below.) 2.3 The expressive role of mathematics Similarly, the expressive role of mathematics is not clearly accommodated by platonists. To account for this role, we need to provide the right framework to express—that is, formulate and characterize—relations between objects in the physical world. It may look as though platonists are well positioned to make sense of this issue. After all, on their view, mathematical objects and their relations exist and we are able to refer to them. Thus, platonists seem to have resources to express mathematical relations between physical objects. The difficulty here is that reference to mathematical objects is left quite open in mathematical practice, in the sense that nothing in that practice requires the existence of the corresponding objects (see Azzouni 1994, 2004). Clearly, mathematicians quantify over mathematical objects, but this quantification, as I will discuss below, need not carry ontological commitment. This means that mathematical objects are not needed for us to make sense of the expressive role of mathematical language. Ultimately, what is required is a language that expresses certain relations, but this language does not require the existence of the corresponding objects. As an illustration, consider the fact that physicists often distinguish between the mathematical and the physical content of their theories, indicating explicitly their commitment to the physical rather than the mathematical parts (see, for instance, Dirac (1958), and Bueno (2005) for discussion). In this sense, platonists are unnecessarily adding extra ontology, which fails to play a role in mathematical practice. The additional ontology that platonists add does seem to play a role in semantics, since by quantifying over mathematical entities, platonists can provide a unified semantics for both mathematical and scientific discourses—taking both types of discourses literally. But this is, of course, a separate issue from the expressive role of mathematics. And as will become clear below, the semantic issue can also be accommodated without commitment to the existence of mathematical objects. 3 An anti-realist alternative 3.1 Anti-realism in applied mathematics Isthereananti-realistalternativetoaccommodatetheprocessofapplicationofmathematics? Ithinkso;atleastinprinciple.ThealternativeIsketchhereisanti-realistinthesensethatitis notcommittedtotheexistenceofmathematicalobjects.Morebroadly,thealternativemight beconsideredasprovidingaframeworktointerpret(applied)mathematicswithoutrequiring that mathematical entities exist, but leaving it open that they may. Given the lack of commitment, the resulting view still counts as a form of anti-realism about mathematics. 123 O. Bueno After all, the proposal distinguishes between two types of commitment: a quantifier commitment and an ontological commitment (Azzouni 2004).3 We incur a quantifier commitment by (existentially) quantifying over a given domain of objects. The factthat we quantify over certain objects does not require that such objects exist: we often quantify over objects whose existence we have no reason to believe, such as fictional entities. But the quantification indicates that the objects in question are in the range of the quantifiers. On the view proposed here, to have ontological commitment to a given object requires, besides quantification, the satisfaction of an existence predicate is required. The objects that satisfy the existence predicate exist. Various sorts of existence predicates could be proposed, from causal efficacy through ontological independence to observability. But to avoid begging the question against platonists, I provide only sufficient, but not necessary, conditions for the existence predicate. Interestingly, these conditions can be found in scientific practice, and so are motivated independently of the issue regarding platonism in mathematics. They involve having a particular form of access to an object, an access that is counterfactually dependent on the objects in question, in the sense that: (1) had the scene before us were different, the objects in question would have been correspondingly different, and (2) had the scene before us been the same, the objects would have been correspondingly the same (see Bueno 2011).4 From these conditions three important epistemic conditions follow: the relevant access (1) is robust, (2) can be refined, and (3) allows us to track the objects in question in space and time. (Azzouni 2004 calls these sorts of conditions thick epistemic access.) If these conditions are met, and we have at least good reason to believe that they are indeed met, we will have sufficient reason to claim that the corresponding objects exist. Clearly, mathematical objects fail to satisfy the existence predicate. There is no need for us to be ontologically committed to them. However, given that such conditions are taken only as sufficient (but not necessary) for existence, the issue of the existence of mathematical objects is left open. For all we know, mathematical objects may exist. But it is still possible that they do not. Since it is unclear how we could establish the existence of mathematical objects, or their non-existence, the best attitude toward this issue is to suspend judgment. Thus, we have here an agnostic view regarding the ontological status of mathematical objects. This makes the interpretation proposed here anti-realist.5 3 Azzouni (2004) explicitly introduces these two types of commitment, but on his view mathematical objects do not exist. In Bueno (2013) some difficulties are raised to his approach. 4 These conditions can be seen as a generalization of corresponding conditions on observation advanced by Lewis (1980). Since certain instruments also satisfy them, we have here a broader conception of instrument-mediated observation (see Bueno 2011). 5 Azzouni (2004) distinguishes quantifier commitment and ontological commitment, but refuses to adopt an agnostic interpretation. Thus, he does not seem to be in a position to claim that when we quantify over certain objects, we quantify over something, even though it might be something that does not exist (for further discussion, see Bueno and Zalta 2005; Azzouni 2009, 2010; Bueno 2013). The approach suggested here does not face this difficulty, since it allows one to talk about properties of nonexistent objects. We may quantify over certain objects, but this is not enough to claim that these objects exist: an existence predicate also needs to be met (Zalta 2000; Bueno and Zalta 2005). 123 An anti-realist account of the application of mathematics For Azzouni, if the three conditions above are met, we have thick epistemic access to the corresponding objects (Azzouni 2004). On his view, however, these are not conditions for existence. Instead, ontological independence (from our linguistic practices and psychological processes) provides the criterion for existence. As a result, on his view, given that mathematical objects are just made up by mathematicians, and are, therefore, dependent on mathematicians’ linguistic and psychological processes, they do not exist. The trouble is that the platonist would draw exactly the opposite conclusion! Given that, for the platonist, mathematical objects would exist even if no person had ever existed, these objects are actually ontologically independent from our linguistic practices and psychological processes, and hence exist. This illustrates the difficulty of establishing, in a non-question-begging way, a criterion of existence (a point Azzouni is, of course, completely aware of). However, if counterfactual dependence and thick epistemic access are understood as only sufficient conditions for existence, this difficulty is avoided. In the end, it is left open whether mathematical objects exist or not. But given these objects are quantified over they are in the scope of the relevant quantification, although they need not exist. In this way, no questions are begged against platonists, and one can still make sense of quantification over mathematical objects that is so significant—indeed indispensable—to scientific and mathematical practice. Note that the truth of mathematical theories plays no role in the account suggested here. In fact, the truth of these theories is often incompatible with scientific practice. As noted above, idealizations and simplifications are key components in scientific activity, and this includes those idealizations in which mathematical theories are involved. Let us call them mathematical idealizations. The use of group theory to obtain solutions to Schrödinger’s equation provides a familiar example. Since idealizations, including mathematical ones, are not taken to be true, the mathematical theories that are part of mathematical idealizations are not taken to be true either—even though such idealizations are compatible with the physical outcome in question. In other words, mathematical theories need not be taken to be true to be good (Field 1989). But why is it that mathematical theories seem to be true no matter what? The reason is that, without a physical interpretation, a mathematical theory is compatible with any physical outcome—except perhaps for cardinality claims about the domain. Interpreting the mathematical formalism of the theory in an empirically adequate way does the crucial work. What typically happens is that the physical interpretation of the mathematical formalism, rather than the formalism itself, is incompatible with certain physical results (Bueno 2005). So, the mathematical formalism is always preserved. What changes is its physical interpretation. It is this feature of mathematics—the preservation of the mathematical formalism—that gives the impression that mathematical theories are true. But this is only a superficial impression. If we distinguish carefully—as physicists such as Dirac do—the physical interpretation of the mathematical formalism and the mathematical formalism itself, it becomes clear that interpreted mathematical theories are not true; some of them are not even empirically adequate. This is the case, for instance, of Dirac’s original interpretation of his celebrated equation. This 123 O. Bueno interpretation led him to formulate the ‘‘hole’’ theory, which ended up assigning the same mass to protons and electrons—an empirically inadequate claim (see Bueno 2005 and references therein). This provides a further reason why mathematical theories need not be true to be good. Azzouni insists that mathematical theories should be taken to be true (2004, 15–28). On his view, without asserting the truth of mathematical theories, we cannot derive consequences from theories that cannot be explicitly formulated, for example, because they are not finitely axiomatizable. In response, of course we do not just derive blindly consequences from theories that cannot be explicitly formulated. We work with fragments of these theories—indeed, with certain models—and these models can be fully expressed. The consequences are drawn in these simplified models, rather than in the context of the original theory. And there is no need to assert the truth of the original theory in this process. 3.2 Anti-realism and the three roles of mathematics How does the agnostic account suggested here accommodate the three roles of the application of mathematics? With regard to the inferential role, there is no need for one to claim that mathematical theories are true, or to be taken to be true, for these theories to play an inferential role. After all, as noted above, one can easily characterize inferences in modal terms: B logically follows from A if necessarily (A . B), where ‘necessarily’ is a modal operator of logical necessity, that can be easily defined from a primitive modal operator of logical possibility (Field 1989). In this way, mathematical theories can be used as premises in an argument regarding the application of mathematics, or can be invoked as yielding additional models, without mathematical theories being taken to be true. The notion of consequence, characterized (modally) in the object language rather than (model-theoretically) in the metalanguage, simply expresses certain logical connections between statements. The representation role can be similarly accommodated. First, it should be noted that the various relations between mathematical structures mentioned above (isomorphism, homomorphism etc.) can all be formulated in second-order logic, and so they need not presuppose mathematical objects. (In this sense, this is a matter of the application of logic, rather than mathematics.) Moreover, by quantifying over mathematical objects, and by clearly distinguishing quantifier commitment from ontological commitment, we can make perfect sense of the way in which one can use mathematical objects (which, on the interpretation advanced here, are in the range of the quantifiers in question), but refuse to be committed to their existence. Finally, the expressive role of mathematics can be accommodated as well. This role is basically a matter of using predicates of mathematical theories to express relations among physical objects. This is, ultimately, a matter of providing suitable interpretations of the mathematical formalism (see Bueno and French 2012). But to make sense of this role, there is no need for one to be committed to the existence of mathematical objects. At issue is how to interpret mathematical theories so that they can have empirical implications about the world, and the fact that by quantifying over mathematical objects, suitably interpreted, we can express 123 An anti-realist account of the application of mathematics more easily certain relations among physical entities. However, as noted, quantifier commitment does not entail ontological commitment. There is, of course, far more to be said about each of these points. But my purpose is simply to indicate that the anti-realist approach sketched here is, in principle, in a better position than platonist accounts to make sense of the application of mathematics. 4 Conclusion If the three roles that mathematical theories play in applications can be accommodated without the commitment to the existence of mathematical objects, what emerges, in outline, is an anti-realist account of the application of mathematics. 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