Fictionalism has recently returned as a standard response to ontologically problematic domains. This article assesses moral fictionalism. It argues (i) that a correct understanding of the dialectical situation in contemporary metaethics shows that fictionalism is only an interesting new alternative if it can provide a new account of normative content: what is it that I am thinking or saying when I think or say that I ought to do something; and (ii) that fictionalism, qua fictionalism, does not provide us with (...) any new resources for providing such an account. (shrink)

Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine's argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to (...) appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam's writings reveals some neglected arguments for platonism which may be superior to the indispensability argument. (shrink)

Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.

During the past two decades, an interest among philosophers in fictitious and illusory beliefs has sprung up in fields ranging anywhere from mathematics and modality to morality.1 In this paper, we focus primarily on the view that Saul Smilansky has dubbed “free will illusionism”—i.e., the purportedly descriptive claim that most people have illusory beliefs concerning the existence of libertarian free will, coupled with the normative claim that because dispelling these illusory beliefs would produce negative personal and societal consequences, those of (...) us who happen to know the dangerous and gloomy truth about the non-existence of libertarian free will should simply keep quiet in the name of the common good. (shrink)

Many philosophical issues concern questions of objectivity and subjectivity. Of these questions, there are two kinds. The first considers whether something is objective or subjective; the second what it _means_ for something to be objective or subjective.

Using two decades worth of experimental philosophy (aka x-phi), Edouard Machery argues in Philosophy within its Proper Bounds (OUP, 2017) that philosophers’ use of the “method of cases” is unreliable because it has a strong tendency to elicit different intuitive responses from non-philosophers. And because, as Machery argues, appealing to such cases is usually the only way for philosophers to acquire the kind of knowledge they seek, an extensive philosophical skepticism follows. I argue that Machery’s “Unreliability” argument fails because, once (...) its premises are percisified, they are either self-defeating or without justification. This is a significant result because Machery’s arguments are the most widely cited and discussed x-phi arguments for philosophical skepticism and many hold that Machery provides the most empirically informed, convincing, and thus best case for this kind of skepticism. So, if my arguments are sound, then the best x-phi argument for philosophical skepticism fails. I further argue that this result provides strong reason to believe the more general conclusion that “negative” x-phi is likely doomed: x-phi likely can never support a substantive philosophical skepticism. Ultimately, I argue for the broad conclusion that all empirically minded arguments for philosophical skepticism are likely to fail for the same reasons that Machery’s does, i.e. they are (likely) self-defeating. (shrink)

Call the epistemological grounds on which we rationally should determine our ontological (or alethiological) commitments regarding an entity its arbiter of existence (or arbiter of truth). It is commonly thought that arbiters of existence and truth can be provided by our practices. This paper argues that such views have several implications: (1) the relation of arbiters to our metaphysical commitments consists in indispensability, (2) realist views about a kind of entity should take the kinds of practices providing that entity’s arbiters (...) to align with respect to their metaphysical dependencies, (3) if realists take a kind of practice to provide grounds on which to affirm the existence of a kind of entity, they should turn to those same grounds when seeking to provide an epistemology of the relevant domain. (shrink)

This paper examines two alleged limitations in the use of formal languages: on the one hand, the trade-offs between expressive and inferential power, and on the other, the phenomenon of system imprisonment. After reconceptualizing the issue, we consider the role played by modality in the understanding of certain aspects of mathematical structures and argue for its centrality.

Learning information about the etiology of one's beliefs can reduce the justification a thinker has for those beliefs. Learning information about the etiology of one's desires, emotions, or concepts can similarly have a debunking effect. In this chapter, I develop a unified account of etiological debunking that applies across these different kinds of cases. According to this account, etiological debunking arguments work by providing reason to think that there is no satisfying explanation of how it is that some part of (...) our mental life is fitting. The role that etiological information plays is to help to rule out (or render less plausible) potential explanations, given our background views about the world. This account suggests several potential morals for epistemology, the philosophy of mind, and metaethics. (shrink)

One common attitude toward abstract objects is a kind of platonism: a view on which those objects are mind-independent and causally inert. But there's an epistemological problem here: given any naturalistically respectable understanding of how our minds work, we can't be in any sort of contact with mind-independent, causally inert objects. So platonists, in order to avoid skepticism, tend to endorse epistemological theories on which knowledge is easy, in the sense that it requires no such contact—appeals to Boghossian’s notion of (...) epistemic analyticity are particularly common here, as are appeals to some broadly pragmatic account of the good standing of basic beliefs. I argue, though, that these appeals are hopeless: an argument adapted from the Benacerraf–Field challenge shows that, even if some such theory can deliver the verdict that our beliefs about abstract objects have some prima facie good standing, this good standing will inevitably be defeated. (shrink)

At the core of the recent debate over moral debunking arguments is a disagreement between explanationist and modalist approaches. Explanationists think that the lack of an explanatory connection between our moral beliefs and the moral truths, given a non-naturalist realist conception of morality, is a reason to reject non-naturalism. Modalists disagree. They say that, given non-naturalism, our beliefs have the appropriate modal features with respect to truth -- in particular they are safe and sensitive -- so there is no problem. (...) -/- There is something of a stand-off here. I argue, though, that by looking at the role explanatory and modal factors have to play in theory choice more generally, and, in particular, by considering the practice of theory choice in science, we can see that the explanationist is right. The lack of an explanatory connection between our moral beliefs and the moral truths is a reason to reject non-naturalist realism about morality. (shrink)

The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated (...) to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles stereotypical philosophy. We conclude by distinguishing two ideas that have long been associated -- realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be objective. One upshot of the discussion is that even questions of fundamental physics may fail to be objective in roughly the sense that the question, ‘Is the Parallel Postulate is true?’, understood as one of pure mathematics, fails to be. Another is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones. -/- . (shrink)

In this chapter, I make the case for platonic realism, the thesis that there are properties that lack spatial locations. After criticizing the one-over-many argument for realism and Lewis's argument for realism, I endorse a modal argument that derives the existence of platonic properties from considerations involving necessary truth. I then defend this argument from various objections. Finally, I argue that epistemic considerations and considerations of parsimony favor a weak form of platonic realism on which there are platonic properties, but (...) each property could have had an instance, and would have been located in its instances if it had any. (shrink)

Mathematical pluralism can take one of three forms: (1) every consistent mathematical theory consists of truths about its own domain of individuals and relations; (2) every mathematical theory, consistent or inconsistent, consists of truths about its own (possibly uninteresting) domain of individuals and relations; and (3) the principal philosophies of mathematics are each based upon an insight or truth about the nature of mathematics that can be validated. (1) includes the multiverse approach to set theory. (2) helps us to understand (...) the significance of the distinguished non‐logical individual and relation terms of even inconsistent theories. (3) is a metaphilosophical form of mathematical pluralism and hasn't been discussed in the literature. In what follows, I show how the analysis of theoretical mathematics in object theory exhibits all three forms of mathematical pluralism. (shrink)

Universals are putative objects like wisdom, morality, redness, etc. Although we believe in properties (which, we argue, are not a kind of object), we do not believe in universals. However, a number of ordinary, natural language constructions seem to commit us to their existence. In this paper, we provide a fictionalist theory of universals, which allows us to speak as if universals existed, whilst denying that any really do.

This work explores issues with the eliminativist formulation of ontic structural realism. An ontology that totally eliminates objects is found lacking by arguing, first, that the theoretical frameworks used to support the best arguments against an object-oriented ontology (quantum mechanics, relativity theory, quantum field theory) can be seen in every case as physical models of empty worlds, and therefore do not represent all the information that comes from science, and in particular from fundamental physics, which also includes information about local (...) interactions between objects. Secondly, by giving a critical assessment of the role of symmetries in these fundamental physical theories; and, lastly, by warning about unfounded metaphysical assumptions. An argument is made for a moderate form of structural realism instead, one in which objects play the fundamental role of representing symmetries and bearing their conserved charges, and of participating in the network of interactions observed in the world. (shrink)

This work deals with obstacles hindering a metaphysics of laws of nature in terms of dispositions, i.e., of fundamental properties that are causal powers. A recent analysis of the principle of least action has put into question the viability of dispositionalism in the case of classical mechanics, generally seen as the physical theory most easily amenable to a dispositional ontology. Here, a proper consideration of the framework role played by the least action principle within the classical image of the world (...) allows us to build a consistent metaphysics of dispositions as charges of interactions. In doing so we develop a general approach that opens the way towards an ontology of dispositions for fundamental physics also beyond classical mechanics. (shrink)

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...) one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

In this paper I explore irrealist alternatives to orthodox realism about grounding, and claim that at least some of these alternatives represent fertile areas for future discussion.

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet this challenge, (...) and we argue that all these strategies are untenable. (shrink)

In this paper, I develop a "safety result" for applied mathematics. I show that whenever a theory in natural science entails some non-mathematical conclusion via an application of mathematics, there is a counterpart theory that carries no commitment to mathematical objects, entails the same conclusion, and the claims of which are true if the claims of the original theory are "correct": roughly, true given the assumption that mathematical objects exist. The framework used for proving the safety result has some advantages (...) over existing nominalistic accounts of applied mathematics. It also provides a nominalistic account of pure mathematics. (shrink)

In this paper, I develop a theory on which each of a thing’s abundant properties is immanent in that thing. On the version of the theory I will propose, universals are abundant, each instantiated universal is immanent, and each uninstantiated universal is such that it could have been instantiated, in which case it would have been immanent. After setting out the theory, I will defend it from David Lewis’s argument that such a combination of immanence and abundance is absurd. I (...) will then advocate the theory on the grounds that it accomplishes all of Lewis’s “new work” while providing a gain in parsimony and a new account of fine-grained content. I will close with a discussion of how the theory also affords a new reply to two objections to uninstantiated universals: Armstrong’s charge that they are inconsistent with naturalism, and a Benacerraf-Field-style objection about epistemic access. (shrink)

Nominalists are confronted with a grave difficulty: if abstract objects do not exist, what explains the success of theories that invoke them? In this paper, I make headway on this problem. I develop a formal language in which certain platonistic claims about properties and certain nominalistic claims can be expressed, develop a formal language in which only certain nominalistic claims can be expressed, describe a function mapping sentences of the first language to sentences of the second language, and prove some (...) facts about that function and facts about some sound logics for those languages. In doing so, I prove that, given some plausible metaphysical assumptions, a large class of sentences about properties of concrete objects are “safe” on nominalistic grounds. Whenever some true sentences about concrete objects and some sentences belonging to this class that are true according to platonists collectively entail a conclusion about concrete objects, some nominalistically acceptable sentences are true and entail the same conclusion. Because the proof can itself be formulated without abstract objects, it provides a nominalistic explanation of the success of theories that invoke properties of concrete objects. (shrink)

In “The Evolutionary Debunking of Quasi-Realism,” Neil Sinclair and James Chamberlain present a novel answer that quasi-realists can pro-vide to a version of the reliability challenge in ethics—which asks for an explanation of why our moral beliefs are generally true—and in so doing, they examine whether evolutionary arguments can debunk quasi-realism. Although reliability challenges differ from EDAs in several respects, there may well be a connection between them. For the explanatory premise of an EDA may state that a particular theory (...) of beliefs of a certain kind does not, or cannot, provide a plausible account of why those beliefs might be generally true, and its epistemic premise may state that, if that is the case, then the beliefs in question have a negative epistemic status or the theory is false inasmuch as, if it were true, it would lead to those beliefs having such a negative epistemic status. The quasi-realist can answer the reli-ability challenge by claiming that, when we form our moral beliefs through a process of well-informed impartial reflection, we form them in response to the non-moral features of things on which depend the moral features they have. Hence, when we form beliefs by means of such a process, we are most likely forming true moral beliefs. (shrink)

Talk of metaphysical modality as “absolute” is ambiguous, as it appears to convey multiple ideas. Metaphysical possibility is supposedly completely unrestricted or unqualified; metaphysical necessity is unconditional and exceptionless. Moreover, metaphysical modality is thought to be absolute in the sense that it’s real or genuine and the most objective modality: metaphysical possibility and necessity capture ways things could and must have really been. As we disentangle these ideas, certain talk of metaphysical modality qua “absolute” turns out to be misguided. Metaphysical (...) possibility isn't completely unrestricted or most inclusive compared to the other modalities; metaphysical necessity, like all kinds of necessities, is relative to or conditional upon a specific framework of reference. Still, metaphysical modality captures how things could and must have really been most generally because it deals with reality and the nature of things or their essence. That’s the chief interest of metaphysics. Arguments against the alleged absoluteness of metaphysical modality may not thereby undermine its philosophical significance. (shrink)

I argue that the two-dimensional hyperintensions of epistemic topic-sensitive two-dimensional truthmaker semantics provide a compelling solution to the access problem. I countenance an abstraction principle for epistemic hyperintensions based on Voevodsky's Univalence Axiom and function type equivalence in Homotopy Type Theory. I apply, further, modal rationalism in modal epistemology to solve the access problem. Epistemic possibility and hyperintensionality, i.e. conceivability, can be a guide to metaphysical possibility and hyperintensionality, when (i) epistemic worlds or epistemic hyperintensional states are interpreted as being (...) centered metaphysical worlds or hyperintensional states, i.e. indexed to an agent, when (ii) the epistemic (hyper-)intensions and metaphysical (hyper-)intensions for a sentence coincide, i.e. the hyperintension has the same value irrespective of whether the worlds in the argument of the functions are considered as epistemic or metaphysical, and when (iii) sentences are said to consist in super-rigid expressions, i.e. rigid expressions in all epistemic worlds or states and in all metaphysical worlds or states. I argue that (i) and (ii) obtain in the case of the access problem. (shrink)

This book discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the problem of explaining the (defeasible) justification of our mathematical beliefs (‘the justificatory challenge’), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the problem of explaining their reliability (‘the reliability challenge’), arises to the extent that we could have easily had different beliefs. The book shows that mathematical facts are not, in general, empirically accessible, contra Quine, (...) and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is intelligible. The book concludes with a complementary ‘pluralism’ about modality, logic and normative theory, highlighting its revisionary implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap’s pragmatism, transposed to the key of realism. (shrink)

In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (...) (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5). (shrink)

In this paper, I argue that all expressions for abstract objects are meaningless. My argument closely follows David Lewis’ argument against the intelligibility of certain theories of possible worlds, but modifies it in order to yield a general conclusion about language pertaining to abstract objects. If my Lewisian argument is sound, not only can we not know that abstract objects exist, we cannot even refer to or think about them. However, while the Lewisian argument strongly motivates nominalism, it also undermines (...) certain nominalist theories. (shrink)

This article presents a challenge that those philosophers who deny the causal interpretation of explanations provided by population genetics might have to address. Indeed, some philosophers, known as statisticalists, claim that the concept of natural selection is statistical in character and cannot be construed in causal terms. On the contrary, other philosophers, known as causalists, argue against the statistical view and support the causal interpretation of natural selection. The problem I am concerned with here arises for the statisticalists because the (...) debate on the nature of natural selection intersects the debate on whether mathematical explanations of empirical facts are genuine scientific explanations. I argue that if the explanations provided by population genetics are regarded by the statisticalists as non-causal explanations of that kind, then statisticalism risks being incompatible with a naturalist stance. The statisticalist faces a dilemma: either she maintains statisticalism but has to renounce naturalism; or she maintains naturalism but has to content herself with an account of the explanations provided by population genetics that she deems unsatisfactory. This challenge is relevant to the statisticalists because many of them see themselves as naturalists. (shrink)

You see a cherry and you experience it as red. A textbook explanation for why you have this sort of experience is going to cite such things as the cherry’s chemical surface properties and the distinctive mixture wavelengths of light it is disposed to reflect. What does not show up in this explanation is the redness of the cherry. Many allege that the availability of color-free explanations of color experience somehow calls into question our beliefs about the colors of objects (...) around us. We explore how such explanations are supposed to undermine color beliefs, and in particular whether evolutionary considerations have any special role to play. (shrink)

Sensitivity has sometimes been thought to be a highly epistemologically significant property, serving as a proxy for a kind of responsiveness to the facts that ensure that the truth of our beliefs isn’t just a lucky coincidence. But it's an imperfect proxy: there are various well-known cases in which sensitivity-based anti-luck conditions return the wrong verdicts. And as a result of these failures, contemporary theorists often dismiss such conditions out of hand. I show here, though, that a sensitivity-based understanding of (...) epistemic luck can be developed that respects what was attractive about sensitivity-based approaches in the first place but that's immune to these failures. (shrink)

Are there genuine mathematical explanations of physical phenomena, and if so, how can mathematical theories, which are typically thought to concern abstract mathematical objects, explain contingent empirical matters? The answer, I argue, is in seeing an important range of mathematical explanations as structural explanations, where structural explanations explain a phenomenon by showing it to have been an inevitable consequence of the structural features instantiated in the physical system under consideration. Such explanations are best cast as deductive arguments which, by virtue (...) of their form, establish that, given the mathematical structure instantiated in the physical system under consideration, the explanandum had to occur. Against the claims of platonists such as Alan Baker and Mark Colyvan, I argue that formulating mathematical explanations as structural explanations in this way shows that we can accept that mathematics can play an indispensable explanatory role in empirical science without committing to the existence of any abstract mathematical objects. (shrink)

In recent years, several philosophers have argued that the a priori/a posteriori distinction is a legitimate distinction but does not carve at the epistemological joints and is theoretically unimportant. In this paper, I do two main things. First, I respond to the most prominent recent challenge to the significance of the a priori/a posteriori distinction – the central argument in Williamson (2013). Second, I discuss the question of what the theoretical significance of the a priori/a posteriori distinction is. -/- I (...) first present the a priori/a posteriori distinction as it is typically developed. I then turn to Williamson’s challenge to the significance of the distinction. Williamson points out that we often use the same cognitive mechanisms in coming to have a priori and a posteriori knowledge. So how could it be, asks Williamson, that there is a “deep epistemological difference” between the two? In response to this challenge, I argue that there is an important disanalogy between Williamson’s central example of a case of a priori knowledge and his central example of a case of a posteriori knowledge. Although the beliefs in the two cases are formed in similar ways, the ways in which their justification can be defeated are different. This suggests that there is an important epistemological difference between the two cases, one that cannot be captured in terms of the cognitive mechanisms used to form the beliefs. -/- Although Williamson’s argument is unsuccessful, there remains the question of just what the theoretical significance of the a priori/a posteriori distinction is. I argue that the point of the distinction is not to enable us to represent some joint in nature, but rather to help us to identify epistemological problem cases. We understand – more-or-less – the epistemology of simple perceptual knowledge. The epistemology of non-perceptual knowledge is far less clear. The purpose of labeling a case of knowledge as a priori is to claim that its epistemology should not be assimilated to the epistemology of perception. Instead, it is something of a puzzle case. -/- This proposal has an important implication. There are several ways in which a case of knowledge can be different from a simple case of perceptual knowledge. Two differences are perhaps the most important: (i) the justification of the belief does not involve phenomenality, and (ii) the belief does not stand in a causal relation to what the belief is about. When beliefs about some subject matter fit either (i) or (ii), an epistemological puzzle arises. So there is more than one kind of epistemological puzzle to solve. This suggests that there is an important theoretical role for (at least) two distinctions in the ballpark of the traditional a priori/a posteriori distinction. (shrink)

Indispensability arguments are used as a way of working out what there is: our best science tells us what things there are. Some philosophers think that indispensability arguments can be used to show that we should be committed to the existence of mathematical objects (numbers, functions, sets). Do indispensability arguments also deliver conclusions about the modal properties of these mathematical entities? Colyvan (in Leng, Paseau, Potter (eds) Mathematical knowledge, OUP, Oxford, 109-122, 2007) and Hartry Field (Realism, mathematics and modality, Blackwell, (...) Oxford, 1989) each suggest that a consequence of the empirical methodology of indispensability arguments is that the resulting mathematical objects can only be said to exist (or not exist) contingently. Kristie Miller has argued that this line of thought doesn’t work (Miller in Erkenntnis, 77 (3), 335-359, 2012). Miller argues that indispensability arguments are in direct tension with contingentism about mathematical objects, and that they cannot tell us about the modal status of mathematical objects. I argue that Miller’s argument is crucially imprecise, and that the best way of making it clearer no longer shows that the indispensability strategy collapses or is unstable if it delivers contingentist conclusions about what there is. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order PA and Zermelo’s quasi-categoricity (...) theorem for second-order ZFC—these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

It’s sometimes suggested that we can (in a sense) settle the truth-value of some statements in the language of number theory by stipulation, adopting either φ or ¬φ as an additional axiom. For example, in Clarke-Doane (2020b) and a series of recent APA presentations, Clarke-Doane suggests that any Σ01 sound expansion of our current arithmetical practice would express a truth. In this paper, I’ll argue that (given a certain popular assumption about the model-theoretic representability of languages like ours) we can’t (...) know ourselves to have any such freedom. (shrink)

Plausibly, mathematical claims are true, but the fundamental furniture of the world does not include mathematical objects. This can be made sense of by providing mathematical claims with paraphrases, which make clear how the truth of such claims does not require the fundamental existence of mathematical objects. This paper explores the consequences of this type of position for explanatory structure. There is an apparently straightforward relationship between this sort of structure, and the logical sort: i.e. logically complex claims are explained (...) by logically simpler ones. For example, disjunctions are explained by their (true) disjuncts, while generalizations are explained by their (true) instances. This would seem as plausible in the case of mathematics as elsewhere. Also, it would seem to be something that the anti-realist approaches at issue would want to preserve. It will be argued, however, that these approaches cannot do this: they lead not merely to violations of the familiar principles relating logical and explanatory structure, but even to reversals of these. That is, there are cases where generalizations explain their instances, or disjunctions their disjuncts. (shrink)

A Benacerraf–Field challenge is an argument intended to show that common realist theories of a given domain are untenable: such theories make it impossible to explain how we’ve arrived at the truth in that domain, and insofar as a theory makes our reliability in a domain inexplicable, we must either reject that theory or give up the relevant beliefs. But there’s no consensus about what would count here as a satisfactory explanation of our reliability. It’s sometimes suggested that giving such (...) an explanation would involve showing that our beliefs meet some modal condition, but realists have claimed that this sort of modal interpretation of the challenge deprives it of any force: since the facts in question are metaphysically necessary and so obtain in all possible worlds, it’s trivially easy, even given realism, to show that our beliefs have the relevant modal features. Here I show that this claim is mistaken—what motivates a modal interpretation of the challenge in the first place also motivates an understanding of the relevant features in terms of epistemic possibilities rather than metaphysical possibilities, and there are indeed epistemically possible worlds where the facts in question don’t obtain. (shrink)

According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then (...) propose a fallibilist account of mathematical justification. I show that the main function of mathematical justification is to guarantee that the mathematical community can correct the errors that inevitably arise from our fallible practices. (shrink)

According to Hartry Field, the mathematical Platonist is hostage of a dilemma. Faced with the request of explaining the mathematicians’ reliability, one option could be to maintain that the mathematicians are reliably responsive to a realm populated with mathematical entities; alternatively, one might try to contend that the mathematical realm conceptually depends on, and for this reason is reliably reflected by, the mathematicians’ (best) opinions; however, both alternatives are actually unavailable to the Platonist: the first one because it is in (...) tension with the idea that mathematical entities are causally ineffective, the second one because it is in tension with the suggestion that mathematical entities are mind-independent. John Divers and Alexander Miller have tried to reject the conclusion of this argument—according to which Platonism is inconsistent with a satisfactory epistemology for arithmetic—by redescribing the second horn of the dilemma in light of Crispin Wright’s notion of judgment-dependent truth; in particular they have contended that once arithmetical truth is conceived in this way the Platonist can have a substantial epistemology which does not conflict with the idea that the mathematical entities exist mind-independently. In this paper I analyze Wright’s notion of judgment-dependent truth, and reject Divers and Miller’s argument for the conclusion that arithmetical truth can be so characterized. In the final part, I address the worry that my argument generalizes very quickly to the conclusion that no area of discourse could be characterized as judgment-dependent. As against this conclusion, I indicate under what conditions—notably not satisfied in Divers and Miller’s case, but possibly satisfied in others—a discourse’s judgment-dependency can be successfully vindicated. (shrink)

This paper aims to provide a mathematically tractable background against which to model both modal cognitivism and modal expressivism. I argue that epistemic modal algebras, endowed with a hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics, comprise a materially adequate fragment of the language of thought. I demonstrate, then, how modal expressivism can be regimented by modal coalgebraic automata, to which the above epistemic modal algebras are categorically dual. I examine five methods for modeling the dynamics of conceptual engineering for intensions and (...) hyperintensions. I develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. I examine then the virtues unique to the modal expressivist approach here proffered in the setting of the foundations of mathematics, by contrast to competing approaches based upon both the inferentialist approach to concept-individuation and the codification of speech acts via intensional semantics. (shrink)

I argue that modal epistemology should pay more attention to questions about the structure and function of modal thought. We can treat these questions from synchronic and diachronic angles. From a synchronic perspective, I consider whether a general argument for the epistemic support of modal though can be made on the basis of modal thoughs’s indispensability for what Enoch and Schechter (2008) call rationally required epistemic projects. After formulating the argument, I defend it from various objections. I also examine the (...) possibility of considering the indispensability of modal thought in terms of its components. Finally, I argue that we also need to approach these issues from a diachronic perspective, and I sketch how to approach this task. (shrink)

Putnam’s model-theoretic arguments for the indeterminacy of reference have been taken to pose a special problem for mathematical languages. In this paper, I argue that if one accepts that there are theory-external constraints on the reference of at least some expressions of ordinary language, then Putnam’s model-theoretic arguments for mathematical languages don’t go through. In particular, I argue for a kind of descriptivism about mathematical expressions according to which their reference is “anchored” in the reference of expressions of ordinary language. (...) These anchors add enough to the content of mathematical expressions to forestall the radical kind of indeterminacy that model-theoretic arguments are purported to show, while still leaving room for a plausible, moderate kind of indeterminacy. (shrink)

The idea that there are some facts that call for explanation serves as an unexamined premise in influential arguments for the inexistence of moral or mathematical facts and for the existence of a god and of other universes. This book is the first to offer a comprehensive and critical treatment of this idea. It argues that calling for explanation is a sometimes-misleading figure of speech rather than a fundamental property of facts.

While there has been much discussion about what makes some mathematical proofs more explanatory than others, and what are mathematical coincidences, in this article I explore the distinct phenomenon of mathematical facts that call for explanation. The existence of mathematical facts that call for explanation stands in tension with virtually all existing accounts of “calling for explanation”, which imply that necessary facts cannot call for explanation. In this paper I explore what theoretical revisions are needed in order to accommodate this (...) phenomenon. One of the important upshots is that, contrary to the current consensus, low prior probability is not a necessary condition for calling for explanation. In the final section I explain how the results of this inquiry help us make progress in assessing Hartry Field’s style of reliability argument against mathematical Platonism and against robust realism in other domains of necessary facts, such as ethics. (shrink)

Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap. Abstractionists about numbers—those who think natural numbers are objects characterized by abstraction principles—face an acute version of this problem. Many abstraction principles appear to characterize the natural numbers. If each abstraction principle determines its own sort, then there is no single subject-matter of arithmetic—there are too many numbers. That is, unless objects can belong to more than one sort. (...) But if there are multi-sorted objects, there should be cross-sortal identity principles for identifying objects across sorts. The going cross-sortal identity principle, ECIA2 of Cook and Ebert (2005), solves the problem of too many numbers. But, I argue, it does so at a high cost. I therefore propose a novel cross-sortal identity principle, based on embeddings of the induced models of abstracts developed by Walsh (2012). The new criterion matches ECIA2’s success, but offers interestingly different answers to the more controversial identifications made by ECIA2. (shrink)

Philosophers of physics have long debated whether the Past State of low entropy of our universe calls for explanation. What is meant by “calls for explanation”? In this article we analyze this notion, distinguishing between several possible meanings that may be attached to it. Taking the debate around the Past State as a case study, we show how our analysis of what “calling for explanation” might mean can contribute to clarifying the debate and perhaps to settling it, thus demonstrating the (...) fruitfulness of this analysis. Applying our analysis, we show that two main opponents in this debate, Huw Price and Craig Callender, are, for the most part, talking past each other rather than disagreeing, as they employ different notions of “calling for explanation”. We then proceed to show how answering the different questions that arise out of the different meanings of “calling for explanation” can result in clarifying the problems at hand and thus, hopefully, to solving them. (shrink)

We are fallible creatures, prone to making all sorts of mistakes. So, we should be open to evidence of error. But what constitutes such evidence? And what is it to rationally accommodate it? I approach these questions by considering an evolutionary debunking argument according to which (a) we have good, scientific, reason to think our moral beliefs are mistaken, and (b) rationally accommodating this requires revising our confidence in, or altogether abandoning the suspect beliefs. I present a dilemma for such (...) debunkers, which shows that either we have no reason to worry about our moral beliefs, or we do but we can self-correct. Either way, moral skepticism doesn’t follow. That the evolutionary debunking argument fails is important; also important, however, is what its failure reveals about rational belief revision. Specifically, it suggests that getting evidence of error is a non-trivial endeavor and that we cannot learn that we are likely to be mistaken about some matter from a neutral stance on that matter. (shrink)