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Constructive empiricism is the view that "Science aims to give us theories which are empirically adequate; and acceptance of the theory involves as belief it is empirically adequat. Where a theory is empirically adequate "if and only what is says about the observable things and events in this world, is true - exactly if it 'saves the phenomena." Constructive empiricism is therefore committed to both an empirical distinction between observable and unobservable phenomena and an epistemological distinction between acceptance and belief. However, it is unclear what position the constructive empiricist should take towards issues in the philosophy of mathematics such as the existence of numbers. In attempting to answer this challenge to constructive empiricism this essay will do four things. Firstly, it will establish why constructive empiricism requires a philosophy of mathematics. Secondly, it will examine the compatibility of constructive empiricism and philosophies of mathematics. Thirdly, it will consider whether the weaker epistemic attitude of acceptance is a suitable solution to the constructive empiricist's attitude towards modal entities. Finally, van Fraasen's critique of abduction will be considered in the context of mathematics.
The constructive empiricist owes us a philosophy of mathematics. This is true for two reasons. The first is that the constructive empiricist must explain the use of mathematics in theories. Secondly, abstract mathematical objects are used in the formulation of constructive empiricism and so need to be accounted for
Under Van Fraassen's epistemic voluntarism he is required to argue that constructive empiricism is no less rational than realism. The constructive empiricist view of science must show that the realist aim for truth and belief in the truth of theories can be removed without any loss for those who practice science. Note that this does not mean that the constructive empiricist must reflect the actual beliefs of practicing scientists, but he must demonstrate that it is possible for the constructive empiricist to do science. A philosophy of mathematics is required for it to be clear if constructive empiricism is rationally permissible. This is true for three reasons. The first is to allow the constructive empiricist scientist knows what his attitude is. The second is to allow for salient features of the role of mathematics in theories to be explained. Finally, how mathematics relates to the empirical success of science should be explained. Therefore, a philosophy of mathematics is necessary for the constructive empiricist to produce an adequate description of science.
The formulation of constructive empiricism is heavily mathematical. For Van Fraassen, constructive empiricism is a correct attitude towards theories, but what is the nature of these theories that Van Fraassen is telling us about? For Van Fraassen, belief in a theory is a belief concerning the theory's set theoretic models:
To present a theory is to specify a family of structures, its models; and secondly, to specify certain parts of those models (the empirical substructures) as candidates for the direct representation of observable phenomenaâ€¦ the theory is empirically adequate if it has some model such that all appearance are isomorphic to empirical substructures of that model. 
This demonstrates that abstract mathematical objects are centrally involved in the articulation of Van Fraassen's constructive empiricism. For the constructive empiricist to accept a theory is to have some beliefs about abstract mathematical objects. This is true because part of accepting a theory is a belief that is empirically adequate and empirical adequacy is a property of a theory's mathematical models. We then need a philosophy of mathematics from the constructive empiricist to account for what sorts of beliefs we are entitled to hold about these abstract models.
It is not clear that a constructive empiricist can accept a theory that contains statements about mathematical objects. This is because, as Rosen  notes, if anything is unobservable then mathematical objects surely are. However, Van Fraassen's flavour of empiricism has the principled justification that the knowledge of unobservable things is impossible. Therefore, anyone who takes up accepting a theory containing mathematical objects has instantly taken up exactly the sort of metaphysical commitment that Van Fraassen warns against. The only consistent way for the constructive empiricist to avoid this is not to believe any theories containing abstract mathematical objects at all. Yet, as we saw above mathematical models are crucial to Van Fraassen's definition of empirical adequacy and without this the constructive empiricist cannot accept constructive empiricism. This is hugely problematic for Van Fraassen's account.
There are three possible ways out for Van Fraassen here. The first is to deny that constructive empiricism leads to nominalism. The second is to attempt a nominalistic reconciliation of this commitment with his empiricism. The third is to adopt a different epistemic attitude to mathematical objects than he does to other observables.
Let us consider the first approach, marrying constructive empiricism with mathematical Platonism. We can almost dismiss this immediately on the grounds of incoherence. This is because experience, the foundation for all of the constructive empiricist's knowledge, cannot tell us whether these abstract objects can exist or not. Van Fraassen  makes this point in his paper "Platonism's Pyrrhic Victory" which begins with the tale of Oz and Id:
Once upon a time there were two possible worlds, Oz and Id. These worlds were very much alike, and indeed very much like our world. Specifically, their inhabitants developed exactly the mathematics and mathematical logic we have today. The main differences were two: (a) in Oz, sets really existed, and in Id no abstract entities existed, but (b) in Id, mathematicians and philosophers were almost universally Platonist, while in Oz they refused, almost to a man, to believe that there existed any abstract entities.
They all lived happily ever after.
The inhabitants of Oz and Id seem able to live happily ever after exactly because their experience is not affected by the existence or not of abstract mathematical objects. Additionally, as Ladyman  observes it would be "bizarre to suggest that we do not know about electrons merely because they are unobservable, but that we do know about non-actual possibilia." This appears to generalise from modal entities, Ladyman's concern, to abstract mathematical objects such as sets. Moreover, Van Fraassen  admits that if constructive empiricism has a philosophy of mathematics "it would have to be a fictionalist account" Therefore, there does not seem to be any ground to suggest Van Fraassen's commitment to demarcating observable and unobservable entities does not also apply to mathematical objects.
The second approach is to extend Van Fraassen's constructive empiricism with a nominalist philosophy of mathematics that allows the constructive empiricist not to be committed to abstract mathematical objects. Bueno  takes up this challenge in his attempt to adapt Harty Field's  fictionalist account. Field's fictionalist account was motivated by Benacerraf's  epistemological problem that we do not seem able to give a satisfactory account of mathematical truth while also explaining why we can have access to this truth. For instance, as Benacerraf's problem applied to Platonism is that the causal theory of knowledge is inconsistent with mathematical entities being abstract. The constructive empiricist would do well to keep Benacerraf's problem in mind as he is required to provide both an adequate account of the role of mathematics in theories, but also an explanation of how we can have knowledge of mathematical objects.
There are two issues with this approach. The first is that fictionalism holds mathematical objects not to exist, but constructive empiricism is agnostic about this. Secondly, the notion of conservativeness is inconsistent with Van Fraassen's view on the syntactic characterization of theories.
Constructive empiricism involves agnosticism about the existence of mathematical objects, but fictionalism takes them to be false. This is a contradiction that must be resolved for fictionalism to be a coherent philosophy of mathematics for constructive empiricism. In particular, on Van Fraassen's view of truth and falseness when a theory is true it has "a model which is a faithful replica, in all detail, of our world"  , but even a false theory would be true of some model. Mathematical objects are still central to van Fraassen's understanding of how a false theory can represent the world. Therefore, even if we accept that mathematically stated theories are false this is also a claim about the relationship between the world and abstract mathematical models so does little to remove the need for the view of the constructive empiricist about this to be explained. Moreover, any view that takes mathematical objects to not exist will then have issues in describing their role in the definition of constructive empiricism. Bueno introduces the notions of partial structure and quasi-truth in order to make Field's strategy consistent with constructive empiricism. Without discussing the technical details, Dicken  argues that this response fails.
The second issue here is that by appealing to Field's notion of conservativeness Bueno is making an appeal to a syntactic characterization of theories. Van Fraassen believes this approach to be overly language dependent and that "the syntactically defined relationships are simply the wrong ones"  Instead Van Fraassen prefers the 'models' of semantic view discussed earlier. Therefore, this does not seem to be an approach that Van Fraassen could endorse.
Despite this, that mathematical fictionalism appears to be incompatible with the semantic view of theories might be more appealing to the constructive empiricist. This is because it is the reference of the semantic view to models that has given the constructive empiricist the problem of explaining their attitude to abstract mathematical objects such as isomorphisms. If we reformulate the definition of constructive empiricism we may remove the need to explain the presence of these terms. However, it is simply not clear that it is possible to do this while maintaining the spirit of constructive empiricism. Even if we take a syntactic view of theories (for instance, holding that a scientific theory T is the set consisting of the deductive closure of its axioms) then believing a theory to be empirically adequate would still require that we have some beliefs about mathematical objects. It is difficult to see how the 'models' of the semantic view can be considered as anything other than sets. Moreover, even if we could the constructive empiricist would still need to account for the role that mathematics plays in theories. Therefore, this inconsistency with the semantic view of theories is not a virtuous one because we cannot reformulate our definition of constructive empiricism as a way out of this problem.
There are then issues with reconciling constructive empiricism with either a Platonist or a nominalist philosophy of mathematics. There is a third alternative here. Recall that the constructive empiricist has a committed agnosticism toward statements about unobservable physical entities. This is a substantive epistemological commitment, in the words of van Fraassen  "acceptance involves a commitment to confront any future phenomena by means of the conceptual resources of this theoryâ€¦willingness to answer questions ex cathedra." That is, the constructive empiricist does not believe theories about the unobservable, but is willing to treat some of them as true for what is required in normal scientific practice. It seems as though this is a suitable way out for the constructive empiricist as by remaining agnostic about the existence of mathematical objects he does not fall to making the sort of metaphysical commitment van Fraassen is guarding against, but makes sufficient epistemological commitments to allow him to use statements that contain mathematical objects and so avoid the difficulties associated with the use of mathematical entities in the formulation of constructive empiricism.
Moreover, this approach looks to generalise to solve other problems with constructive empiricism. Consider the issue of observability. Establishing a distinction that the constructive empiricist can use is necessary for any epistemic significance to be attached to it. Van Fraassen believes that the limits of observability will be spelled out "in detail in the final physics and biology". However, Musgrave's  objection to this is that these theories will contain statements about unobservables and so the constructive empiricist cannot accept the theory. This self-defeating objection is strikingly similar to Rosen's 'transcendental' challenge to constructive empiricism. Committed agnosticism offers a solution to Musgrave's objection by allowing the constructive empiricist to use statements from the final physics and biology without having a metaphysical commitment to statements about unobservables. Therefore, this acceptance strategy also appears to be a promising solution to a number of concerns.
However, this strategy on a distinction between belief and committed agnosticism rely being maintained. Horwich  argues that this is not the case. In particular, he claims that
If we tried to formulate a psychological theory of the nature of belief, it would be plausible to treat beliefs as states with a particular kind of causal role. This would consist in such features as generating certain predictions, prompting certain utterances, being caused by certain observations, entering in characteristic ways into inferential relations, playing a certain part in deliberation, and so on. But that is to define belief in exactly the way instrumentalists characterize acceptance.
In response to Horwich's challenge Dicken  notes that while drawing this distinction might be difficult, it seems preferable to the challenges facing adapting a fictionalist philosophy of mathematics to fit with constructive empiricism. This is especially true given van Fraassen's epistemic voluntarism sets the constructive empiricist the burden of only having a consistent position on this distinction, not having to convince Horwich that there is a clear difference between the two. Indeed, some philosophers such as Rosen  maintain that there is such a distinction and that "acceptance is not belief." Therefore, while drawing a concrete distinction between committed agnosticism and belief may require more work this does look a more likely way out for the constructive empiricist than constructing a philosophy of mathematics. Additionally, the bigger payoff of being able to generalise this strategy to other modal entities provides another warrant for the constructive empiricist to pursue this.
There is a final challenge for the constructive empiricist. That is to explain the role of mathematics in empirically successful theories. Realists provide the following abduction, commonly known as the "no miracles" argument, as an explanation:
The truth of scientific theories constitutes the best explanation of their success.
Therefore, we should believe in the truth of these theories.
However, van Fraassen rejects this reasoning in his alternative explanation of the success of scientific theories. Van Fraassen  offers a Darwinian account of the success of science: "I claim that the success of current scientific theories is no miracleâ€¦ For any scientific theory is born into a life of fierce competition, a jungle red in tooth and claw. Only the successful theories survive - the ones which in fact latched on to actual regularities in nature." The explanation is that scientific theories tend to have observed consequences that are true because they were selected on that basis. This mechanism appears to explain the observable success of scientific theories without reference to their truth and so is a challenge to the "no miracles" abduction.
However, this alternative explanation of the success is problematic. Note that inference to best explanation is contrastive. We should pick the more likely of competing theories as the one that is most likely to be true. The above explanation does not confirm or deny that successful theories are true. Therefore the realist can accept both the "no miracles" argument and Van Fraassen's explanation. There is a possible response from Van Fraassen to this. Lipton  gives the example of explaining why a computer does not work.. We could explain this by saying that the computer was not plugged or that the fuse had blown. Both explanations do the job. However, once we observe that the computer is unplugged this explanation pre-empts the blown fuse explanation and makes it redundant.
Yet, there is reason to doubt that Van Fraassen's account pre-empts the "no miracles" argument. This is because Van Fraassen's critique of Inference to the Best Explanation is analogous to a Darwinian explanation, but as Musgrave  notes the Darwinian explanation addresses a different question. Van Fraassen's is an explanation for why all of our current theories are observationally successful, but it does not explain why a particular theory has that feature. Contrastingly, the "no miracles" argument does the job of explaining why any given theory has the feature of being observationally successful by appealing to an intrinsic property of that theory, its truth. Therefore, the Darwinian explanation is a less satisfactory explanation of the success of theories than the "no miracles" argument.
There is a possible response for the constructive empiricist here. That would be to argue that mathematics is only useful as a representative tool and so he should not be committed to the existence of abstract mathematical objects. However, this ignores cases where we appear to have mathematical explanation such as Colyvan's  example of the impossibility of squaring the circle being explained by Ï€ being transcendental and Baker's  use of prime numbers to explain the life cycles of cicadas. Nonetheless, the constructive empiricist might be able to use van Fraassen's  pragmatics of explanation to solve this problem. In this account we value answers to why questions on the grounds on how probably they are in light of our knowledge, whether they favour the topic against the other members of their contrast class and whether they can be made wholly or partially irrelevant by other answers. A constructive empiricist strategy to apparent cases of mathematical explanation would then be to explain them away by selecting other answers on the basis of van Fraassen's criteria. This approach is likely to be problematic as van Fraassen's account appears to allow for different explanations in different contexts and so the burden must be to rule out mathematical explanation from any possible context. Therefore, the constructive empiricist approach to explanation seems to permit cases of mathematical explanation without the constructive empiricist being able to account for the role of abstract mathematical objects in an explanation.
Constructive empiricism owes us an approach to abstract mathematical objects that maintains the distinction between observable and unobservable phenomena and also accounts for features of science such as its empirical success. Constructive empiricism seems incompatible with both mathematical Platonism and fictionalism, but adopting an attitude of committed agnosticism towards mathematical objects appears to be the best strategy for the constructive empiricist. However, drawing the necessary distinction between belief and committed agnosticism may prove problematic. Moreover, the constructive empiricist must also account for the role of mathematics in empirically successful theories and its potential explanatory role. Van Fraassen's epistemic voluntarism lessens the burden placed on the constructive empiricist by these challenges, but despite this constructive empiricism does not add up.