Chapter 5 The Epistemology of Modality and the Epistemology of Mathematics

Chapter 5 The Epistemology of Modality and the Epistemology of Mathematics Otávio Bueno 5.1 Introduction In this paper I explore some connections between the epistemology of modality and the epistemology of mathematics, and argue that they have far more in common than it may initially seem to be the case even though modality need not (in fact, should not) be characterized in terms of possible worlds (as the modal realist insists) and mathematics need not (in fact, should not) be understood in terms of abstract entities (as the platonist recommends). Let s see why. 5.2 Modality and Mathematics 5.2.1 Modality Modality deals with what is possible and what is necessary. But it need not be characterized, as is commonly done in contemporary discussions by modal realists, in terms of possible worlds (Lewis 1986). Worlds provide just one way to regiment modal discourse, but to invoke these objects also generates significant problems. In particular, it s not clear how illuminating worlds ultimately are for the understanding of what is possible and what is necessary (for some critical discussion, see Bueno and Shalkowski (2015), pp. 672 677). If one is interested in figuring out whether something is possible in the actual world, it is typically not of much use to establish that it is true in some other possible world causally isolated from ours assuming there is any to begin with. For the problem then becomes one of figuring out whether O. Bueno ( ) Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA e-mail: ; otaviobueno@me.com SpringerInternationalPublishingSwitzerland2017 B. Fischer, F. Leon (eds.), Modal Epistemology After Rationalism, Synthese Library 378, DOI 10.1007/978-3-319-44309-6_5 67

68 O. Bueno the statement in question is true in that other possible world. To determine what is true in the actual world is, in many cases, hard enough. Figuring out what is true in a causally inaccessible world is something of an entirely different order of magnitude, and not clearly relevant in any case to what goes on in the actual world, despite what the modal realist may claim. In fact, modal realists insist that modal discourse is ultimately replaceable with discourse about possible worlds. But the point remains that, prima facie, the fact that something holds in a possible, causally inaccessible, world is just irrelevant to what goes on, or even could go on, in the actual world. The fact that such worlds are constituted by the same kind of stuff and have a similar structure (they are all maximally connected spatiotemporal regions) is not enough to establish relevance. Being constituted by the same kind of stuff is insufficient to establish that what happens in another world speaks to what happens, or even could happen, in the actual world. Just consider this case: Can you run 500 miles nonstop? According to modal realism, the answer is affirmative: there is a counterpart of you who does just that in some other possible world. (Nothing in modal realism precludes the existence of both such a world and such a counterpart of you in it.) I take it, however, that this is not sufficiently reassuring for you to go lace your shoes right now for an extremely long run! The modal realist will, no doubt, point out that it is very easy to determine what is true in a possible world. Crucial to modal realism is the claim that for every way the world could be there is a world that is that way (Lewis 1986, pp.86 92). Thus, possibilities, it seems, are very easy to come by: assuming that each of the following items are ways the world could be, there are worlds that include talking donkeys, flying pigs, or invisible giraffes. The difficulty, however, is to determine which ways in fact the world could be.cantheworldbeinsuchawaythatdifferent physical objects occupy the same region of space? Can the world be in such a way that physical objects are non-self-identical? Can the world be in such a way that objects have inconsistent properties (being, simultaneously, F and non-f)? Clearly, some prior characterization of possibility needs to be in place if situations such as these are to be ruled out. But this means that not every possibility can legitimately be understood in terms of worlds: a prior characterization is required so that what corresponds to ways the world could be are indeed genuine possibilities. We are told that modal realism has a number of benefits (Lewis 1986, pp. 5 69). By adding worlds to our ontology, truth conditions for modal claims can be systematically assigned: what is possible is what is true in some world; what is necessary is what is true in every world. Given worlds, we also obtain unification in our metaphysics, with proper characterization of modality, properties, and propositions. With the introduction of worlds, the crucial issues then become epistemological: (i) How can we know that the postulated worlds do exist, instead of being just a useful fiction? (ii) How can we know that particular claims about the worlds in question (for instance, regarding ways the world could be) are indeed true? To answer these questions is the task of a proper epistemology of modality within a modal realist framework.

5 TheEpistemology of Modalityand theepistemology of Mathematics 69 But these don t seem to be the right modal questions. The questions should be about what is possible or about what is necessary. It is the imposition of an ontology of worlds that distorts the modal questions into something entirely different from what they were supposed to be. But if worlds are not posited, modality can be understood for what it is: an engagement with the possible and the necessary and a modalist proposal, which takes modality as primitive, can get off the ground (see Bueno and Shalkowski 2015, pp.677 689). Modal realists will, no doubt, complain. If modal realism is true, that is, if there is a plurality of worlds, and if the reduction of modal discourse to possible worlds they defend is successfully implemented, then modal discourse can be fully captured by (that is, expressed and articulated in terms of) discourse about possible worlds. What we talk about when we talk about modality is ultimately (to talk about) worlds. When we make modal claims, we make them ultimately in terms of worlds. It is perfectly fine, modal realists tell us, when we aim to assert a modal utterance, to invoke and refer to worlds instead. We are not changing the subject by employing one when we intend to speak of the other: modal discourse is, after all, reducible to worlds. Despite the modal realists admission that worlds are not indispensable (Lewis 1986,pp.3 5),itisacrucialfeatureoftheirprogramtoarguethatmodaldiscourseis indeed reducible to worlds. Lewis is thoroughly Quinean in this regard (for Quine s well-known skepticism about modality, see, for instance, Quine 1960, 1981). Given the reducibility of the modal to the worldly, and since worlds, on Lewis view, are ultimately spatiotemporal objects, modal discourse becomes discourse about perfectly acceptable, well-behaved, well-understood objects. So the reducibility is indeed central to the success of modal realism. But on what grounds do modal realists argue in favor of their view? In Lewis case, the fundamental argument for modal realism is an argument based on the theoretical utility of positing worlds (see Lewis 1986,pp.3 5;Divers 2002,p.151). Since by doing that one produces a theory that has a number of theoretical virtues (it is simple, unified, and it has expressive and explanatory power), and given that these virtues are (taken to be) truth conducive, by invoking these virtues, the resulting theory is (more likely to be) true. It is unclear, however, that theoretical virtues are epistemic; that is, connected to the truth. Nor is it clear, more generally, that they are truth conducive (see Bueno and Shalkowski 2015, pp.672 675).These virtues are best understood as being pragmatic; that is, they express the preferences of the users rather than an increase in the likelihood of the truth of the theories under consideration. After all, a theory, despite being simple, unified, rich in expressive and explanatory power, can still be false. A clear example is provided by Newtonian gravitational theory, which, although it exemplifies all of these virtues, turns out to be false: it s unable to provide the proper description of Mercury s perihelion and it posits gravity as a force among objects (rather than, for instance, as part of the structure of spacetime). Despite being false, Newtonian theory is still useful, though, in light of all of the theoretical virtues that it satisfies, and thus is valuable to the scientific community. It is also successful

70 O. Bueno in an impressive array of predictions it makes, not because it is true (which it isn t), but because it is adequate to the empirical domains it applies to. It might be argued, however, that Newtonian theory fails to exhibit all of the theoretical virtues. For instance, the argument goes, its explanatory power is not significantly high. But it is hard to make this case without thereby also undermining the explanatory power of all scientific theories. If a theory that had such an impressive explanatory power as Newtonian theory had which, for the first time in Western science, precisely accounted for the movement of bodies near the surface of the earth, the tides, and planetary motions (bringing together astronomy and the physics of a moving earth) does not have enough explanatory power, it is unclear which theory does. Asimilartroublearisesforrelativitytheoryandquantummechanics.Despite the fact that both exemplify all of the theoretical virtues above, given that they are incompatible with one another they cannot both be true (assuming, of course, that there are no true inconsistent descriptions of the world). The fact that these examples derive from empirical theories shouldn t suggest that the point is restricted to theories of this kind. It also goes through for theories of a non-empirical sort, which do not depend on empirical considerations to be justified. For example, Frege s original logicist reconstruction of arithmetic had anumberoftheoreticalvirtues.itwassimple:itcharacterizednumbersinterms of logical concepts and definitions (disregarding, of course, Frege s now unusual notation). It was unified: it provided the resources to accommodate additional mathematical theories, such as analysis. It was also rich in expressive power: Frege s reconstruction allowed for the expression of crucial arithmetic and analytic concepts, and the derivation, in a rigorous setting, of central theorems of the relevant fields. Finally, it was rich in explanatory power: the reconstruction avoided a number of conceptual confusions that Frege painstakingly pointed out in previous attempts at providing a foundation for arithmetic, such as the confusion of what numbers are (on his view, certain logical objects) with counting and measuring (activities for which numbers can be used, but which ultimately presuppose numbers rather than the other way around; see, for instance, Frege 1884; Boolos 1998). Despite all of these theoretical virtues, however, Frege s reconstruction turns out to be inconsistent, and thus cannot be true (assuming that there are no true inconsistent statements). So whether we are dealing with empirical or non-empirical theories, the satisfaction of theoretical virtues is not enough to guarantee the theories truth conduciveness. It may be argued that, although not always truth conducive, the satisfaction of theoretical virtues is truth conducive in most cases, or in typical cases. The problem with this suggestion is that we are then left entirely in the dark as to the conditions under which theoretical virtues do lead to the truth and those under which they don t. For all we can tell, the cases involving Newtonian physics, relativity theory and quantum mechanics, and Frege s logicism are just as good and typical as any regarding the satisfaction of theoretical virtues. If in such significant cases the satisfaction of theoretical virtues is not enough to secure the truth of the theories in question, one wonders whether a serious case can be made for the theoretical

5 TheEpistemology of Modalityand theepistemology of Mathematics 71 virtues at all. In the end, the satisfaction of these virtues just doesn t seem to be enough to establish the truth of the relevant theories. The same concerns also apply to Lewis modal realism. Even if we grant that the modal realist account of modality is simple, unified, and has expressive and explanatory power, this doesn t establish that it is thereby true (or more likely to be so). These theoretical virtues are not epistemic, but rather pragmatic. Theories that exemplify the virtues are good theories for us, and we value them for pragmatic reasons, as we should. But we shouldn t infer that the theories are, thus, more likely to be true. (Note that despite the rejection of the claim that theoretical virtues are truth conducive, one need not thereby endorse a radical form of (dogmatic) skepticism, according to which knowledge is impossible. Leaving aside the issue of the coherence of such a view (it is clearly self-undermining), there are ways of gathering and assessing evidence about various parts of the world that do not rely at all on theoretical virtues, but depend rather on forging ways of interacting with and detecting certain aspects of reality (see Bueno 2011a). Having said that, if serious, local skeptical considerations are carefully raised, it s unclear that they can be resisted.) Besides being committed to the existence of a plurality of worlds, the modal realist also needs to implement a reductionist strategy, according to which modal discourse can be cashed out entirely in terms of possible worlds. If successful, the strategy would ensure that no primitive modal notion is ultimately needed. But it is unclear whether the strategy does succeed (see Shalkowski 1994). For the reduction to go through properly it is crucial that two requirements be met: (a) A form of soundness should be in place, so that no impossibility is included among the worlds. Thus, for instance, in no world an object both has and fails to have a given property (at the same time, in the same place). (b) A form of completeness also needs to be satisfied, according to which no possibility is excluded from the worlds. Thus, for instance, a world that is not characterized in terms of spatiotemporal relations is impossible. This means that a logical space of a very particular sort is required for the modal realist reduction to be implemented. Inconsistent objects need to be excluded; otherwise, some impossible objects would be deemed possible. Perhaps no one, with the exception of dialetheists (Priest 2006), will complain. But the point stands that a particular form of possibility is assumed in the very constitution of the modal realist logical space in order that inconsistent objects are not taken to be possible. Moreover, abstract, non-spatiotemporal worlds, which are not constituted by spatiotemporal relations, are considered impossible. After all, possibility is restricted to spatiotemporal worlds. But why should abstract worlds be excluded from the outset? There is a long-standing tradition of considering worlds as abstract objects, as everything that is the case (Wittgenstein 1922; see also Zalta 1983). Clearly, this conception of worlds is not incoherent, although it would have to be, given the modal realist view of what a world is. As a result, the very notion of logical space that the modal realist assumes, the space of what is possible, already presupposes some modal notions: inconsistent

72 O. Bueno objects are impossible and so are non-spatiotemporal worlds. But this means that the modal realist is not in a position to do without some primitive modal notions, and the reductionist strategy is bound to fail. (This is, of course, part of a modalist critique of modal realism; see further Shalkowski 1994). If the modal realist s reductionist project doesn t succeed, a significant constraint and motivation for the proposal is lost. The modal realist would then be unable to claim that the overall balance of evidence favors modal realism, given that one of the key reasons to embrace the ontology of worlds, which other things being equal one would have no reason to be committed to, is precisely the elimination of primitive modality. But without reductionism, no such claim can be maintained. 5.2.2 Mathematics As typically understood, mathematics deals with concepts, objects, relations, and structures about a variety of mathematical domains, from algebra and geometry to analysis and set theory, including the various subfields generated by connecting these more basic ones. As a field of investigation, mathematics is the systematic study of these objects via the inferential relations that can be established between certain principles (in particular, comprehension principles that introduce the relevant concepts and relate them to one another) and particular mathematical results (theorems about the objects under study). Mathematical practice is typically silent about the fundamental ontology of mathematics. The practice itself does not settle, nor does it require settling, issues such as the ontological status of mathematical objects and structures: whether they are abstract or concrete, universal or particular, existent or not. Mathematical practice can be interpreted as being about existing, abstract, universal objects and structures, as platonists insist. But it can just as well be interpreted as being about nonexistent, abstract, universal objects and structures, as nominalists point out (on this interpretation, mathematical objects would be abstract universals had they existed). Alternatively, mathematical practice can also be interpreted as being about existent concrete particulars, as some constructivists defend (on this reading, mathematical objects are mind-dependent entities). The fact that each of these distinct philosophical interpretations of mathematics is compatible with mathematical practice indicates that the practice alone does not settle the ontological issue. In the end, adding a largely irrelevant ontology to the practice provides very little gain regarding its proper understanding. There are also associated costs. On a platonist understanding of mathematics, there are mathematical objects, relations and structures, which exist independently of one s beliefs and linguistic practices, and these objects are abstract, that is, they are causally isolated and are not located in space and time. The issue of how we have knowledge of these objects thus becomes pressing. Platonists have developed a variety of strategies to address the problem: some allow access to mathematical objects via intuition (Gödel 1964); others develop easy strategies to

5 TheEpistemology of Modalityand theepistemology of Mathematics 73 secure reference to the relevant objects via suitable definitions (Hale and Wright 2001). There is, however, no agreement that these strategies work: the reliability of object-oriented intuition is questionable (Bueno 2008, 2011b), and some of the proposed definitions in particular, certain abstraction principles advanced by the neo-fregeans rely on controversial set-theoretic assumptions (Linnebo and Uzquiano 2009). Nominalist views, in turn, face problems to account properly for the use of mathematics in science (for a survey, see Bueno 2013). And constructivist proposals, despite ingenious attempts (Bishop 1967), still face limitations as to how much mathematics can actually be constructed within the restrictions imposed by their program. Arguably Bishop s approach, although significantly more successful in securing more mathematics than other constructivist views, may be too lenient in the end, since it allows for any way of defining mathematical objects that classical mathematicians allow for, only restricting the underlying logic to an intuitionistic one. 5.3 Epistemological Connections In light of these considerations, it may seem that the epistemology of modality and the epistemology of mathematics have very little in common. Mathematical objects are abstract (at least when understood platonistically), and our knowledge of these objects does not depend on empirical traits of the world. How could it, given that the objects in question are not in space and time? However, to know what is possible or impossible, we need to rely quite heavily on empirical features of the world. We rely on particular details of the actual world in order to figure out which possibilities hold, which situations are possible or not. Thus, the two domains, the mathematical and the modal, and the means of knowing them, seem to be quite apart. Appearances, however, are often deceptive. In what follows, I explore some connections between the epistemology of mathematics and the epistemology of modality, indicating some significant similarities between our knowledge of modality (understood as knowledge of the possible and the necessary, independently of worlds) and our knowledge of mathematics (understood as knowledge of the mathematical, independently of a platonist metaphysics). Along the way, important differences between the resulting epistemologies also need to be recognized. I will focus on two complementary routes to approach these issues: (a) The modal route to mathematical epistemology:mathematicscanbeinterpreted in terms of modal logic (Putnam 1967/1979; Hellman 1989, 1996). On this view, mathematical knowledge is fundamentally modal knowledge, because the mathematical is understood modally. A direct link between the epistemology of the mathematical and the modal emerges. But this assumes a particular modalist conception of mathematics. (b) The mathematical route to modal epistemology: Modality, on a modalist view, can be interpreted as the domain of the possible and the necessary (see Bueno and Shalkowski 2009, 2013, 2015, and references therein). What is possible

74 O. Bueno is taken to be objective, that is, independent of one s linguistic practices and psychological processes. But what is possible need not be actual. Thus there need not be any causal process liking what is possible but not actual to what is actually the case. The route to knowledge of the modal is the route to knowledge of non-causal relations. Similarly, mathematics can be interpreted as the study of abstract objects, relations and structures, and that domain is objective, that is, independent of one s linguistic practices and psychological processes. In virtue of being abstract, this domain involves no causal relations. The route to mathematical knowledge, similarly to knowledge of the modal, is the route to knowledge of non-causal relations. But, as will become clear, some care is needed to articulate this strategy properly. 5.4 The Modal Route to the Epistemology of Mathematics Consider the statement: 1. There are infinitely many prime numbers. On a platonist reading, this statement seems to entail the existence of an infinity of abstract objects. But this reading of the statement is not forced upon us. For this statement can be expressed, in a modal second-order language, in terms of two statements (Putnam 1967/1979; Hellman 1989, 1996): 2. If there were structures satisfying the axioms of Peano Arithmetic, it would be true in those structures that there are infinitely many prime numbers. 3. It is possible that there structures satisfying the axioms of Peano Arithmetic. The result is a modal-structural interpretation of mathematics. Condition (3) is needed in order to avoid the trivialization of the translation scheme. For if it were impossible that there were structures satisfying the axioms of Peano Arithmetic, (2) would be vacuously true, and the translation would fail to recognize the difference between (1) and its negation. We have here the modal correlates to usual platonist mathematics, which preserve the objectivity of the original platonist statement without the commitment to the existence of mathematical objects. But this benefit would be lost if the modality invoked in (2) and (3) were understood in terms of possible worlds. After all, in this case we would simply be replacing one ontology with another: instead of mathematical objects, we would have possible worlds. The alternative is to take the modal operators involved as being primitive. But one needs a primitive notion of modality for independent reasons anyway. First, the primitive notion is required to make sense of the relation of logical consequence. In a valid argument, the conjunction of the premises and the negation of the conclusion is impossible. A model-theoretic understanding of consequence, which allegedly avoids the need for a primitive notion of modality, is limited in

5 TheEpistemology of Modalityand theepistemology of Mathematics 75 that it does not seem to apply to a classical set theory, such as Zermelo-Fraenkel set theory. After all, there is no set of all sets, and thus no domain of the relevant models for set theory itself (see Field 1989). Second, as noted above, primitive modality is needed to accommodate the structure of logical spaces; in particular, to ensure that every possibility corresponds to a location on that space, and that no impossibility does. This use, we saw, is not properly captured by a conception of possibility in terms of worlds. Given that for the modal realist worlds are spatiotemporal objects, the logical space is significantly restricted: a non-spatiotemporal world becomes impossible. This seems to be an artifact of the framework rather than a robust modal truth. Third, primitive modality is required to understand counterfactual reasoning (Williamson 2007). The notion of similarity among worlds that modal realism requires makes it very difficult to implement a proper assessment of counterfactuals, given the inherent indeterminacy of the similarity relation. How similar do worlds need to be to count as similar enough? Which world is more similar to ours: one in which armadillos have no blood in their veins but are otherwise just like armadillos in the actual world, or one in which armadillos, although full of blood, survive only by drinking beer? Judgments of similarity are often made on the basis of modal considerations. Consider whether armadillos can survive only on beer. Based on their actual biological constitution, and their digestive system, armadillos, let s take it, are unlikely to survive well on that diet. But suppose that their digestive system were different, so that they could properly absorb alcohol and whatever nutrients are available in beer. In this case, the answer would be, of course, quite different. At this point, it becomes clear that modal considerations are guiding one s reasoning about armadillos. In fact, consider whether a bloodless creature that looks like an armadillo and which has all the differences required by the lack of that scarlet fluid in its veins would still be an armadillo. It is unclear how this issue could be settled without there being features that are necessary for armadillos to be armadillos. Based on these features, similarity judgments can be made: too much difference in the entire animal or enough difference in its necessary traits, and the creature under consideration would fail to be an armadillo. Considerations of this sort are needed to determine how similar the creatures in question are to armadillos. Whether the similarity is based on internal features of these animals, external traits, or something else altogether, modal considerations are thereby needed to carry out judgments of similarity properly. Interestingly, primitive modality is in place also in the case of mathematics, and it is crucial to make sense of mathematical discourse. In fact, rather than artificially introducing modal operators, as recommended by the modal-structural interpretation, I want to argue that mathematics is in fact already modal. (So the modalism I favor is not committed to a modal-structural interpretation of mathematics, although this interpretation of mathematics does provide support for a significant role of modality in mathematics, and that s why I started with it.) First, note the modal import of reductio proofs: in establishing that certain combinations of assumptions leads to a contradiction, what is, in fact, established is their

76 O. Bueno impossibility. Nocorrespondingconstructionispossible.Similarly,directproofs also have modal import: they show the possibility of certain combinations of assumptions the possibility of constructing certain mathematical objects and their properties. When mathematicians consider that a certain mathematical object exists, they are typically asserting that it is possible to construct the object in question, given the specification of the relevant properties in the comprehension principles under consideration. This is implemented in terms of the possibility of deriving the corresponding result to the effect that there is such an object given the relevant assumptions invoked in its characterization, and whatever additional inferential resources that are available in the underlying mathematical background in which the proof is implemented. The idea here is not to apply modal operators to the mathematical principles in question, but to highlight the modal character of these principles themselves. They have a modal force given their content: mathematics is ultimately about what is possible (or not possible) to construct. On this view, in order for one to be in a position to assert the existence of certain mathematical objects, the possibility of constructing these objects is crucial, and that possibility of construction is articulated in terms of the possibility of deriving the corresponding existential statement in a suitable mathematical system. It is important to note that such a system need not be, and typically is not, formalized. Mathematics is implemented, and carried out, in an informal, but rigorous, context: that of natural language expanded with suitable symbols. With the exception of several branches of computer science (including automated theorem proving), in which formalization is required for the configuration and articulation of these fields, the norm in mathematics is to invoke informal, but rigorous, reasoning rather than focus on formal reconstructions. Just consider the way in which proofs are presented in mathematical journals. Typically they are not formalized; in fact, they are not even presented in full, with only the most significant or less familiar steps being offered. Despite that, mathematical proofs are rigorously structured, without ambiguity or vagueness, and they highlight the relevant connections among the concepts in question. Note also that, throughout these considerations, constructions should be understood very broadly: they need not be constrained by constructivist requirements, although it is of course perfectly acceptable if they are so constrained. (Constructivism ends up producing more mathematics, and the results are eventually incorporated into the overall body of mathematical statements and practice.) What is the significance of the modal content of mathematical statements? It is the case that every metric space is a normal space. It is then impossible for a metric space not to be normal, but it is possible for a normal space not to be metric. The import of such modal content prevents one for searching for non-normal metric spaces, while encouraging one to search for normal but non-metric spaces. The very practice of mathematics is informed, shaped, and constrained by the modal content of mathematical results. Even if one adopts the view according to which mathematical objects can be known via some form of mathematical intuition, the resulting view still relies on

5 TheEpistemology of Modalityand theepistemology of Mathematics 77 modal considerations. To know something via mathematical intuition is to be able to have certain phenomenological experiences, in particular, a suitable veridical seeming (see, e.g., Chudnoff 2013). On Kurt Gödel s view, the axioms [of set theory] force themselves upon us as being true (Gödel 1964, p. 485).This forcing generates a certain seeming in us, and the capacity of undergoing these experiences is required in order for this account to go through. So the possibility of having certain veridical seemings is needed, and such seemings should be accessible to us. Suppose, for the sake of argument, that veridical cognitive seemings cannot be had, since seemings about abstract objects, such as sets, functions, and other entities studied in mathematics, cannot be implemented. After all, we can only literally perceive, and have seemings about, objects to which we have some spatiotemporal access, and, thus, any form of cognitive phenomenology about the abstract is ultimately incoherent. To resist this line of argument requires establishing the very possibility of veridical seemings about abstract entities, and modality is crucial for that. As a result, the very adequacy of cognitive phenomenology depends on modal considerations regarding the possibility of veridical cognitive seemings. However the details are implemented, on this conception, mathematics is fundamentally modal: either because the content of mathematical statements is ultimately modal or because modal considerations are required to constrain the appropriate mode of access to the relevant mathematical facts (in terms of veridical cognitive seemings). As a result, the epistemology of mathematics is a chapter in the epistemology of modality, since what is at issue is the specification of what, in a particular mathematical context, is possible or impossible. This specification is determined by the relevant comprehension principles (in terms of which the objects in question are characterized), and the underlying logic (in terms of which the allowed inferences are specified). It is by figuring out what follows from the relevant mathematical principles (given the underlying logic) that mathematical knowledge is typically obtained. Given the modal content of mathematical proofs and mathematical statements, this way of knowing mathematics crucially involves awayofhavingsomeknowledgeofmodality,sinceitinvolvesknowingwhatis possible or what is necessary in the relevant mathematical context (determined, as noted, by the relevant principles and the logic in use). Interestingly, these considerations, which emphasize ways of knowing mathematics, take us directly to the next route to the epistemology of the modal. 5.5 The Mathematical Route to the Epistemology of Modality From a modalist perspective, the epistemology of modality, as the study of the conditions under which one has knowledge of what is possible and what is necessary (again, not involving worlds), has much in common with the epistemology of mathematics. Both involve reasoning from assumptions (regarding modal properties of the relevant objects or regarding mathematical properties of the relevant structures, respectively). Both are highly sensitive to the particular principles that are invoked

78 O. Bueno in each case (whether the principles involve modal traits or mathematical objects and structures). Both are inferential: in the sense that a key point in studying what one knows about a given domain, whether mathematical or modal, is to be in a position to draw inferences about the relevant objects. On the modalist view, to know that something, say P, is possible involves to be entitled to introduce a possibility operator: it is possible that P (see Bueno and Shalkowski 2015; what follows expands on the account presented in this paper by including explicit evidential conditions under which modal operators can be introduced). In some instances, it is very straightforward to do so: it is simply a matter of noting that P is actual, and therefore possible. In more interesting cases, however, P is not actual. So, how do we know, in this instance, that P is possible? What is needed is to derive it is possible that P from particular assumptions. Depending on these assumptions, it may be relatively unproblematic to obtain the target conclusion, by invoking the modal properties of the objects under consideration. One knows that a table is breakable, despite not having ever been broken, due to the material constitution of the table and the stress it can be put to. These are properties of the table in the actual world, which are the relevant properties to determine the table s breakability. In many instances, however, the properties invoked to determine whether something is possible or not will depend on more controversial assumptions. Would Descartes be the same person if he had different parents? One could confidently assert that he wouldn t, since a different sperm and egg would be involved. This assumes, however, essentialism about Descartes origins. And how do we know that this doctrine is, in fact, true? Typically, this involves a variety of theoretical virtues favoring essentialism. But, as noted above, theoretical virtues are hardly epistemic, and are better understood as being just pragmatic. As a result, they fail to provide reasons to believe that essentialism is true. Moreover, the denial of essentialism about origins also provides a perfectly coherent answer to the question regarding Descartes identity; one in which Descartes could still be the same person despite having different parents than those he actually had. It is unclear, however, how to settle the issue between these two possibilities, since they ultimately rely on dramatically different metaphysical assumptions. Being unable to know that these assumptions are true, and since they are required for an argument that establishes the possibility (or impossibility) in question, essentialists and anti-essentialists alike are then unable to have the relevant modal knowledge. (I ll return to this example below.) In general, the more contrived the required assumptions are, and the less reason one has to believe that they are true, the more reason one has to question whether the modal knowledge in question does obtain, or at least that we are in a position to assert the relevant modal claim. In many instances of alleged philosophical modal knowledge, it is unclear that one is entitled to it. Suspending the judgment, in these cases, seems to be the proper answer. When are we entitled to introduce a modal operator? When there are good reasons for doing so, which can be formulated in terms of two conditions: (a) Evidential support:thereisevidenceinsupportoftherelevantpossibilityclaim,that

5 TheEpistemology of Modalityand theepistemology of Mathematics 79 is, one is in a position to rule out likely alternatives that undermine the possibility under consideration. In particular, the claim is not ruled out by well-established and well-entrenched beliefs we already have (unless there are good, independent reasons to revise such beliefs). (b) Independent plausibility: there are independent considerations in favor of the possibility claim in question; that is, (i) there are arguments in its support, and (ii) the arguments are not undermined by equally plausible opposing arguments. Ordinary modal claims easily satisfy both conditions (leaving radical skepticism aside for the moment). There is evidence that the table in front of me is breakable, given the materials it is made of. There are also independent considerations that support the possibility that the table can be broken, in light of the various stresses it can be put to. Taken together, we do have good reasons for asserting the corresponding modal claim and introduce the modal operator accordingly. Extraordinary modal claims, particular of a philosophical kind, however, are far less secure. For the relevant modal claims invariably depend on controversial (metaphysical, epistemological) assumptions. Even if there were evidential support for them (a big if!), the independent plausibility requirement is usually not met. After all, the assumptions that philosophical modal claims have can be typically undermined by contrasting them with rival philosophical assumptions, which deny the initial assumptions, and produce, thus, a situation in which it is unclear that one actually has the required independent plausibility to have good reason to introduce the modal operator in question. For instance, essentialists about origins may think that their essentialist assumptions, which are required to deny that Descartes could have had different parents, are well supported and have independent plausibility. But anti-essentialists will immediately point out that there are good reasons to deny the essentialist assumptions. Needless to say, essentialists will say exactly the same about the assumptions made by anti-essentialists. The result is that neither essentialists nor anti-essentialists will be in a position to maintain that their views satisfy the independent plausibility requirement. No knowledge of extraordinary possibility claims will then emerge in this case. This is typical of philosophical claims about extraordinary possibilities. Similar considerations apply to knowledge of necessity, which also relies on the introduction of a suitable operator: in this case, a necessity one. Clearly, if it is atheoremthatp (in a given system), then P is necessary. But such necessity is restricted to the assumptions involved in the theorem. Consider the claim that every set can be well ordered. How do we know whether it is true? The answer, of course, depends on whether the axiom of choice holds or not. If the axiom does hold, then the claim is true, and necessarily so, given the axiom. Otherwise, the claim is false, and hence not known. In this case, both conditions above, namely, evidential support and independent plausibility, are clearly satisfied. The axiom of choice unquestionably entails that every set is well ordered, and in this way, there is evidential support for the relevant possibility (to the effect that that every set is well ordered). With regard to the independent plausibility of such possibility, it ultimately depends on the independent plausibility of the axiom of choice itself, which, after all, entails the set

80 O. Bueno ordering in question. The axiom has a number of extremely significant consequences for a variety of fields in classical mathematics (for details, see Moore 1982), and this fact lends independent plausibility to the adoption of the axiom. But certain considerations have also been raised against the axiom of choice. Awell-knowncaseisthatoftheTarski-Banachtheorem,aconsequenceofthis axiom, which may be thought of as casting doubt on the axiom itself given its counterintuitive nature. According to the Tarski-Banach theorem, a threedimensional sphere can always be decomposed into finitely many pieces (disjoint subsets), which, in turn, can be reassembled together to form two copies of the original sphere. The theorem, prima facie, may seem to be surprising. However, it eventually provides support for the axiom, as an additional consequence, among so many significant ones, that the axiom yields. After all, given the axiom s content, the Tarski-Banach consequence is not counterintuitive, particularly if interpreted, as it should, as a claim about abstract objects rather than physical ones. In particular, the disjoint subsets (the pieces ) involved in the reassembling process are not solids, but nonmeasurable sets, collections of scattered points that fail to have a volume (in the ordinary sense of the term). It is precisely at this point that the axiom of choice is needed, in order to guarantee that, based on un uncountable number of choices, these nonmeasurable sets can be reassembled into two copies of the original sphere. Rather than a counterintuitive result, we have a feature of the axiom of choice at work. In the end, there is independent plausibility for the possibility that every set can be well ordered. These are, of course, cases of ordinary knowledge of necessity. Extraordinary cases involve philosophical assumptions about what necessarily is the case. And here, once again, there is reason for skepticism. The assumptions that are needed in order to derive the relevant necessity claims, even if they had evidential support, will end up failing the independent plausibility condition. Given conflicting philosophical assumptions regarding philosophical claims about necessity (e.g., some assert, while others deny, that a set is necessarily constituted by their members), there typically are considerations that undermine the various arguments in support of the claims under consideration. For instance, is Socrates necessarily human? Essentialist assumptions could be invoked to support the conclusion that he is. But these assumptions are questioned in light of anti-essentialist considerations. As a result, the independent plausibility requirement is violated, and neither essentialists nor anti-essentialists are in a position to have the relevant modal knowledge. But perhaps instead of having categorical modal knowledge about philosophical issues (that is, knowledge of what is possible or necessary, independently of particular assumptions), we can at least have conditional modal knowledge about such issues (that is, knowledge of what is possible or necessary given certain philosophical assumptions). After all, although we may not be able to discharge the relevant philosophical assumptions, we can at least establish that if such assumptions were the case, such and such conclusions would also be. (I owe this point to Bob Fischer.) In each case, establishing such conditionals provide us with some understanding of what the various possibilities and necessities ultimately are, which assumptions they rely on, which considerations may be invoked to challenge

5 TheEpistemology of Modalityand theepistemology of Mathematics 81 them, and how plausible or questionable these assumptions turn out to be. This offers a rich structure of conditional modal knowledge via logical relations among the various assumptions, their potential challenges, and established conclusions. The resulting structure has much in common with Robert Nozick s approach to philosophical explanations. On his view: There are various philosophical views, mutually incompatible, which cannot be dismissed or simply rejected. Philosophy s output is the basketful of these admissible views, all together. One delimiting strategy would be to modify and shave these views, capturing what is true in each, to make them compatible parts of one new view. This book puts forward its explanations in a very tentative spirit; not only do I not ask you to believe that they are correct, I do not think it important for me to believe them correct, either. Still I do believe, and hope you will find it so, that these proposed explanations are illuminating and worth considering, that they are worth surpassing; also that the process of seeking and elaborating explanations, being open to new possibilities, the new wonderings and wanderings, the free explanation, is itself a delight. (Nozick 1981, p.21) The exploration of these possibilities is, of course, the crucial feature of how we end up obtaining conditional modal knowledge about philosophical issues, and the understanding that emerges as a result. (Thanks, again, to Bob Fischer.) But is such a conditional modal knowledge in philosophy enough for one s philosophical purposes? It ultimately depends on what these purposes are. If one were engaged in the philosophical enterprise of establishing metaphysical truths about reality, clearly this would not be enough. After all, one may not be able to settle the philosophical assumptions that are required for the task at hand. If, however, the goal is to understand, to figure out the modal constraints on various philosophical claims about the world, then the exploration of the relevant possibilities (and impossibilities) is clearly relevant, albeit no categorical modal knowledge of the extraordinary possibilities in question is forthcoming. On this modalist account, we may have a lot of ordinary modal knowledge (that is, knowledge of ordinary possibilities and necessities), but less of extraordinary categorical modal knowledge (that is, knowledge of philosophical possibilities and necessities independently of any assumptions), although we may have some conditional extraordinary modal knowledge, by taking notice of the relevant philosophical assumptions, determining their consequences, and resisting the temptation to discharge the assumptions. Interestingly, this conditional modal knowledge can be obtained without much commitment, since only the logical relations between assumptions and results are highlighted rather than any claim about the correctness of the assumptions in question. This is, I take it, as it should be. 5.6 Conclusion: Rationalism and Empiricism The proposal advanced here bears important connections with (a weak form of) rationalism, in the sense that modal knowledge involves reasoning from various assumptions made in a given context. But it goes beyond (weak) rationalism in that

82 O. Bueno empirical assumptions, which are crucial for the specification and determination of the objects and their modal properties in particular domains, are also invoked and relied on. In the end, a combination of both (weak) rationalist and empiricist considerations is needed in order to articular a proper, stable view. This should not be surprising, since both views work better together, with suitable adjustments as needed, to advance and develop further an epistemology of modality. Even a traditional empiricist, such as Hume, clearly acknowledged the role of rationalist features in his approach. For instance, Hume s emphasis on the significance of relations of ideas and his corresponding insistence, regarding matters of fact, that everything that does not involve an inconsistency is possible, can be thought of as rationalist traits (see Hume 1739/2000, 1748/1999). Descartes, in turn, voiced empiricist sentiments when he expressed, in the Principles of Philosophy (Part IV, Section 204), that with regard to what one cannot perceive it is enough to account for how it can be (Descartes 1644/1985). On reflection, some aspects of empiricism and rationalism should be carefully and properly integrated in the development of a proper epistemology of modality. In fact, with regard to the two routes discussed above, the modal route to the epistemology of mathematics and the mathematical route to the epistemology of modality, both incorporate rationalist and empiricist features. They each highlight the role that reasoning from principles play in the epistemology of mathematics and modality, thus emphasizing a rationalist trait. But they also invoke (when relevant) the role of modal properties of the objects under consideration, and particularly in the context of the epistemology of modality, when dealing with concrete objects, this amounts to an emphasis on an empiricist feature. Given the combination of empiricism and rationalism, the proposal sketched above establishes some connections between the epistemology of modality and the epistemology of mathematics. Notwithstanding the clear differences between the two domains (the mathematical and the modal), there is far more in common between their epistemology than it may initially meet the eyes. In the end, mathematical epistemology and modal epistemology, despite undeniable differences in what they are about, should go hand in hand. Acknowledgements Many thanks to Jacob Busch, Albert Casullo, Bob Fischer, Hannes Leitgeb, Daniel Nolan, Sonia Roca Royes, Scott Shalkowski, Asbjørn Steglich-Petersen, Anand Vaidya, and Tim Williamson for helpful discussions of the issues examined in this paper. Thanks, in particular, to Bob Fischer, Melisa Vivanco, and an anonymous referee for insightful comments on earlier versions of the work. Their comments led to substantial improvements. References Benacerraf, P., & Putnam, H. (Eds.). (1983). Philosophy of mathematics: Selected readings (2nd ed.). Cambridge: Cambridge University Press. Bernecker, S., & Pritchard, D. (Eds.). (2011). Routledge companion to epistemology. London: Routledge.