…entities? 2. How to be a nominalist 2.1. “Speak with the vulgar …” 2.2. “…think with the learned” 3. Arguments for nominalism 3.1. Intelligibility, physicalism, and economy 3.2. Causal..

The problem that motivates me arises from a constellation of factors pulling in different, sometimes opposing directions. Simplifying, they are: (1) The complexity of the world; (2) Humans’ ambitious project of theoretical knowledge of the world; (3) The severe limitations of humans’ cognitive capacities; (4) The considerable intricacy of humans’ cognitive capacities . Given these circumstances, the question arises whether a serious notion of truth is applicable to human theories of the world. In particular, I am interested in the questions: (...) (a) Is a substantive standard of truth for human theories of the world possible? (b) What kind of standard would that be? (shrink)

In this paper we explain our pretense account of truth-talk and apply it in a diagnosis and treatment of the Liar Paradox. We begin by assuming that some form of deflationism is the correct approach to the topic of truth. We then briefly motivate the idea that all T-deflationists should endorse a fictionalist view of truth-talk, and, after distinguishing pretense-involving fictionalism (PIF) from error- theoretic fictionalism (ETF), explain the merits of the former over the latter. After presenting the basic framework (...) of our PIF account of truth-talk, we demonstrate a few advantages it offers over T-deflationist accounts that do not explicitly acknowledge pretense at work in the discourse. In turning to the Liar Paradox, we explain how the quasi-anaphoric functioning that our account attributes to truth-talk provides a diagnosis of the Liar Paradox (and other instances of semantic pathology) as having no content—in the sense of not specifying any of what we call M-conditions. At the same time, however, we vindicate the intuition that we can understand liar sentences, thereby avoiding one standard objection to “meaningless strategy” responses to the Liar Paradox. With this diagnosis in place, we then, by way of treatment, introduce a new predicate, ‘semantically defective’, and show how the explanation we give for its application allows for a consistent, yet revenge-immune, (dis)solution of the Liar Paradox, and semantic pathology generally. (shrink)

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 volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics.

There’s the question of what there is, and then there’s the question of what ultimately exists. Many contend that, once we have this distinction clearly in mind, we can see that there is no sensible debate to be had about whether there are such things as properties or tables or numbers, and that the only ontological question worth debating is whether such things are ultimate (in one or another sense). I argue that this is a mistake. Taking debates about ordinary (...) objects as a case study, I show that the arguments that animate these debates bear directly on the question of which objects there are and cannot plausibly be recast as arguments about what’s ultimate. I then address the objection that, because they are easy answerable, questions about what there is cannot be a proper subject of ontological debate. (shrink)

Relying on inference to the best explanation requires one to hold the intuition that the world is ‘intelligible’, that is, such that states of affairs at least generally have explanations for their obtaining. I argue that metaphysical naturalists are rationally required to withhold this intuition, unless they cease to be naturalists. This is because all plausible naturalistic aetiologies of the intuition entail that the intuition and the state of affairs which it represents are not causally connected in an epistemically appropriate (...) way. Given that one ought to rely on IBE, naturalists are forced to pick the latter and change their world-view. Traditional theists, in contrast, do not face this predicament. This, I argue, is strong grounds for preferring traditional theism to naturalism. (shrink)

The timeless solution to the problem of divine foreknowledge and human freedom has many advantages. Still, the relationship between a timeless God and temporal beings is problematic in a number of ways. In this paper, we focus on the specific problems the timeless view has to deal with when certain assumptions on the metaphysics of time are taken on board. It is shown that on static conception of time God’s omniscience is easily accounted for, but human freedom is threatened, while (...) a dynamic conception has no problems with human freedom, but, on this view, some truths seem not to be knowable by a timeless God. We propose Fragmentalism as a metaphysics of time in which the divine timeless knowledge of temporal events and human freedom can be reconciled. (shrink)

The timeless solution to the problem of divine foreknowledge and human freedom has many advantages. Still, the relationship between a timeless God and temporal beings is problematic in a number of ways. In this paper, we focus on the specific problems the timeless view has to deal with when certain assumptions on the metaphysics of time are taken on board. It is shown that on static conception of time God’s omniscience is easily accounted for, but human freedom is threatened, while (...) a dynamic conception has no problems with human freedom, but, on this view, some truths seem not to be knowable by a timeless God. We propose Fragmentalism as a metaphysics of time in which the divine timeless knowledge of temporal events and human freedom can be reconciled. -/- . (shrink)

Little is known about the aetiology of philosophical intuitions, in spite of their central role in analytic philosophy. This paper provides a psychological account of the intuitions that underlie philosophical practice, with a focus on intuitions that underlie the method of cases. I argue that many philosophical intuitions originate from spontaneous, early-developing, cognitive processes that also play a role in other cognitive domains. Additionally, they have a skilled, practiced, component. Philosophers are expert elicitors of intuitions in the dialectical context of (...) professional philosophy. If this analysis is correct, this should lead to a reassessment of experimental philosophical studies of expertise. (shrink)

It is very contentious whether the features of the manifest image have a place in the world as it is described by natural science. For the advocates of strict naturalism, this is a serious problem, which has been labelled ‘placement problem’. In this light, some of them try to show that those features are reducible to scientifically acceptable ones. Others, instead, argue that the features of the manifest image are mere illusions and, consequently, have to be eliminated from our ontology. (...) In brief, the two options that are open to strict naturalists for solving the placement problem are ontological reductionism and eliminativism. Other advocates of naturalist philosophy, however, claim that both these strategies fail and, consequently, opt for ‘mysterianism’, the view according to which we cannot give up the recalcitrant features of the manifest image even if we are not able to understand the ways in which they could be reduced to the scientific features. Mysterianism has the merit of facing the difficulties that whoever wants to explain reductively, or explain away, the features of the manifest image encounters. It is also a defeatist philosophical view, though, since it considers the most important philosophical problems as unsolvable mysteries. For this reason, I argue that mysterianism can also be taken as a reductio of strict naturalism, given its presumption that all phenomena are either explainable by the natural sciences or to be rejected as illusory. In this article, it is argued that the failures of reductionism, eliminativism and mysterianism should teach us that both the scientific image and the manifest image of the world are essential and mutually irreducible but not incompatible with each other. To support this claim, in the second part of the article, the case of free will is discussed. (shrink)

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist (...) hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge? (shrink)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to (...) express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

Christopher Peacocke has presented an original version of the perennial philosophical thesis that we can gain substantive metaphysical and epistemological insight from an analysis of our concepts. Peacocke's innovation is to look at how concepts are individuated by their possession conditions, which he believes can be specified in terms of conditions in which certain propositions containing those concepts are accepted. The ability to provide such insight is one of Peacocke's major arguments for his theory of concepts. I will critically examine (...) this "fruitfulness" argument by looking at one philosophical problem Peacocke uses his theory to solve and treats in depth. Peacocke (1999, 2001) defines what he calls the "Integration Challenge." The challenge is to integrate our metaphysics with our epistemology by showing that they are mutually acceptable. Peacocke's key conclusion is that the Integration Challenge can be met for "epistemically individuated concepts." A good theory of content, he believes, will close the apparent gap between an account of truth for any given subject matter and an overall account of knowledge. I shall argue that there are no epistemically individuated concepts, and shall critically analyze Peacocke's arguments for their existence. I will suggest more generally that the possession conditions of concepts and their principles of individuation shed little light on the epistemology or metaphysics of things other than concepts. My broader goal is to shed light on what concepts are by showing that they are more fundamental than the sorts of cognitive and epistemic factors a leading theory uses to define them. (shrink)

According to Jody Azzouni’s “deflationary nominalism,” the singular terms of mathematical language applied or unapplied to science refer to nothing at all. What does exist, Azzouni claims, must satisfy the quaternary condition he calls “thick epistemic access” (TEA). In this paper I argue that TEA surreptitiously reifies some mathematical entities. The mathematical entity that I take TEA to reify is the Fourier harmonic, an infinite-duration monochromatic sinusoid applied throughout engineering and physics. I defend the reality of the harmonic, in Azzouni’s (...) account, not by satisfying all four TEA conditions with respect to it, but by showing that the harmonic renders satisfiable the TEA condition called “grounding,” specifically in Azzouni’s (Talking about nothing: numbers, hallucinations, and fictions. New York: Oxford University Press, 2010) example of a human visually perceiving a vase. The harmonic thereby plays what Azzouni calls an “epistemic role,” and merits inclusion in the deflationary nominalist ontology. Against the “coding” objection of Azzouni and Bueno (Br J Philos Sci 67:781–816, 2016), which would nominalize the harmonic to nonexistent status, I reply that in the context of grounding, the coding objection begs the question. (shrink)

In his stimulating new book The Construction of Logical Space , Agustín Rayo offers a new account of mathematics, which he calls ‘Trivialist Platonism’. In this article, we take issue with Rayo’s case for Trivialist Platonism and his claim that the view overcomes Benacerraf’s dilemma. Our conclusion is that Rayo has not shown that Trivialist Platonism has any advantage over nominalism.

Many contemporary philosophers rate error theories poorly. We identify the arguments these philosophers invoke, and expose their deficiencies. We thereby show that the prospects for error theory have been systematically underestimated. By undermining general arguments against all error theories, we leave it open whether any more particular arguments against particular error theories are more successful. The merits of error theories need to be settled on a case-by-case basis: there is no good general argument against error theories.

Do numbers exist? Do properties? Do possible worlds? Do fictional characters? Many metaphysicians spend time and effort trying to answer these and other questions about the existence of various entities. These inquiries have recently encountered opposition: a group of philosophers, drawing inspiration from Aristotle, have argued that many or all of the existence questions debated by metaphysicians can be answered trivially, and so are not worth debating. Our task is to defend existence questions from the neo-Aristotelians' attacks.

Many philosophers employ an intellectual division of labour. Philosophy tells us what the truth conditions of various philosophically interesting sentences are. For example, atomic sentences containing numerals are sentences containing singular terms putatively referring to numbers; sentences about what could be are sentences quantifying over possible worlds and so on. Some discipline outside of philosophy tells us that certain of these sentences are true. The purported result is that such philosophically controversial entities as numbers and possible worlds have been shown (...) to exist. I criticize this ‘twin-track strategy’ and try to show that the two components conflict rather than complement each other. (shrink)

Two epistemological critiques of non-naturalism are not always carefully distinguished. According to the Causal Objection, the fact that moral properties cannot cause our moral beliefs implies that it would be a coincidence if many of them were true. According to the Evolutionary Objection, the fact that evolutionary pressures have influenced our moral beliefs implies a similar coincidence. After distinguishing these epistemological critiques, I provide an extensive defense of the Causal Objection that also strengthens the Evolutionary Objection. In particular, I formulate (...) a “Master Causal Objection” featuring the controversial premise that non-naturalism can provide no adequate explanation for moral knowledge. I defend this premise by first narrowing down the range of candidate explanations to conceptual, constitutive, and evolutionary explanations, and then considering and eliminating each of these in turn. My discussion of evolutionary explanations suggests that non-naturalists must refute the Causal Objection in order to refute the Evolutionary Objection. (shrink)

The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.

In this paper I will analyze some philosophically relevant aspects involved in the dissolution of Benacerraf’s problem of fixing the identity of natural numbers by Shapiro’s structuralism. My fundamental aim is to present three criticisms to Shapiro’s position –to his conception of language, to his characterization of structures as ante rem, and to his dramatic conception of ontology. Some of these criticisms will also be directed to Benacerraf’s identification problem.

Arguments for some controversial positions in metaethics—typically moral scepticism or the moral error theory—are sometimes thought to overreach. They appear to entail sceptical or error‐theoretic views about non‐moral branches of thought in a sense that is costly or implausible. If this is true, those metaethical arguments should be rejected. This is the companions in guilt strategy in metaethics. In this article, the contemporary use of the companions in guilt strategy is explored and assessed. The methodology of the strategy is discussed, (...) and criteria for assessing specific instances of its use are identified. Prominent instances of its use in the contemporary literature are then examined. The focus is on those that take (a) epistemic judgment, (b) prudential judgment, and (c) mathematical judgment as “companions,” with a view to undermining the moral error theory and moral scepticism, respectively. (shrink)

It has been argued that the attempt to meet indispensability arguments for realism in mathematics, by appeal to counterfactual statements, presupposes a view of mathematical modality according to which even though mathematical entities do not exist, they might have existed. But I have sought to defend this controversial view of mathematical modality from various objections derived from the fact that the existence or nonexistence of mathematical objects makes no difference to the arrangement of concrete objects. This defense of the controversial (...) view of mathematical modality obviously falls far short of a full endorsement of the counterfactual approach, but I hope my remarks may serve to help keep such an approach a live option. (shrink)

Both literalism, the view that mathematical objects simply exist in the empirical world, and fictionalism, the view that mathematical objects do not exist but are rather harmless fictions, have been both ascribed to Aristotle. The ascription of literalism to Aristotle, however, commits Aristotle to the unattractive view that mathematics studies but a small fragment of the physical world; and there is evidence that Aristotle would deny the literalist position that mathematical objects are perceivable. The ascription of fictionalism also faces a (...) difficult challenge: there is evidence that Aristotle would deny the fictionalist position that mathematics is false. I argue that, in Aristotle's view, the fiction of mathematics is not to treat what does not exist as if existing but to treat mathematical objects with an ontological status they lack. This form of fictionalism is consistent with holding that mathematics is true. (shrink)

(2000). Four Epistemological Challenges to Ethical Naturalism: Naturalized Epistemology and the First-Person Perspective. Canadian Journal of Philosophy: Vol. 30, Supplementary Volume 26: Moral Epistemology Naturalized, pp. 30-74.

There is an incompatibility between the deflationist approach to truth, which makes truth transparent on the basis of an antecedent grasp of meaning, and the traditional endeavour, exemplified by Davidson, to explicate meaning through of truth. I suggest that both parties are in the explanatory red: deflationist lack a non-truth-involving theory of meaning and Davidsonians lack a non-deflationary account of truth. My focus is on the attempts of the latter party to resolve their problem. I look in detail at Davidson's (...) more recent work and suggest that it seeks to articulate a primitive notion of truth that may balance between a notion that collapses into deflationism and one that is wholly subsumed under a general theory of interpretation. I conclude that this tightrope walk is ultimately unsuccessful. Equally, however, some reasons are provided for thinking that deflationism might be equally unsuccessful with its problem. 'Truth or meaning?' remains an open question. (shrink)

Robert Hanna (Rationality and logic. MIT Press, Cambridge, 2006) articulates and defends the thesis of logical cognitivism, the claim that human logical competence is grounded in a cognitive faculty (in Chomsky’s sense) that is not naturalistically explicable. This position is intended to steer us between the Scylla of logical Platonism and the Charybdis of logical naturalism (/psychologism). The paper argues that Hanna’s interpretation of Chomsky is mistaken. Read aright, Chomsky’s position offers a defensible version of naturalism, one Hanna may accept (...) as far as his version of naturalism goes, although not one that supports the claim that cognitive science offers a place for logic that is somehow outside the natural, contingent order. (shrink)

Modal epistemologists parse modal conditions on knowledge in terms of metaphysical possibilities or ways the world might have been. This is problematic. Understanding modal conditions on knowledge this way has made modal epistemology, as currently worked out, unable to account for epistemic luck in the case of necessary truths, and unable to characterise widely discussed issues such as the problem of religious diversity and the perceived epistemological problem with knowledge of abstract objects. Moreover, there is reason to think that this (...) is a congenital defect of orthodox modal epistemology. This way of characterising modal epistemology is however optional. It is shown that one can non-circularly characterise modal conditions on knowledge in terms of epistemic possibilities, or ways the world might be for the target agent. Characterising the anti-luck condition in terms of epistemic possibilities removes the impediment to understanding epistemic luck in the case of necessary truths and opens the door to using these conditions to shed new light on some longstanding epistemological problems. (shrink)

The semantic inferentialist account of the social institution of semantic meaning can be naturally extended to account for social ontology. I argue here that semantic inferentialism provides a framework within which mathematical ontology can be understood as social ontology, and mathematical facts as socially instituted facts. I argue further that the semantic inferentialist framework provides resources to underpin at least some aspects of the objectivity of mathematics, even when the truth of mathematical claims is understood as socially instituted.

In this paper I examine Quine''s indispensability argument, with particular emphasis on what is meant by ''indispensable''. I show that confirmation theory plays a crucial role in answering this question and that once indispensability is understood in this light, Quine''s argument is seen to be a serious stumbling block for any scientific realist wishing to maintain an anti-realist position with regard to mathematical entities.

If correct, Christopher Peacocke’s [20] “manifestationism without verificationism,” would explode the dichotomy between realism and inferentialism in the contemporary philosophy of language. I first explicate Peacocke’s theory, defending it from a criticism of Neil Tennant’s. This involves devising a recursive definition for grasp of logical contents along the lines Peacocke suggests. Unfortunately though, the generalized account reveals the Achilles’ heel of the whole theory. By inventing a new logical operator with the introduction rule for the existential quantifier and the elimination (...) rule for the universal quantifier, I am able to show that Peacocke’s theory only avoids verificationism to the extent that it does not satisfy manifestationism. (shrink)

In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. There simply is no intelligible problem (...) that satisfies all of the constraints which have been placed on the Benacerraf Problem. The point generalizes to all arguments with the structure of the Benacerraf Problem aimed at realism about a domain meeting certain conditions. Such arguments include so-called "Evolutionary Debunking Arguments" aimed at moral realism. I conclude with some suggestions about the relationship between the Benacerraf Problem and the Gettier Problem. (shrink)

Ethics and mathematics have long invited comparisons. On the one hand, both ethical and mathematical propositions can appear to be knowable a priori, if knowable at all. On the other hand, mathematical propositions seem to admit of proof, and to enter into empirical scientific theories, in a way that ethical propositions do not. In this article, I discuss apparent similarities and differences between ethical (i.e., moral) and mathematical knowledge, realistically construed -- i.e., construed as independent of human mind and languages. (...) I argue that some are are merely apparent, while others are of little consequence. There is a difference between the cases. But it is not an epistemological difference per se. The difference, surprisingly, is that ethical knowledge, if it is practical, cannot fail to be objective in a way that mathematical knowledge can. One upshot of the discussion is radicalization of Moore’s Open Question Argument. Another is that the concepts of realism and objectivity, which are widely identified, are actually in tension. (shrink)

Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...) to determine what Benacerraf’s challenge could be such that this view is warranted. I argue that it could not be any of the challenges with which it has been traditionally identified by its advocates, like of Benacerraf and Field. Not only are none of the challenges easier for the pluralist to meet. None satisfies a key constraint that has been placed on Benacerraf’s challenge. However, I argue that Benacerraf’s challenge could be the challenge to show that our set-theoretic beliefs are safe – i.e., to show that we could not have easily had false ones. Whether the pluralist is, in fact, better positioned to show that our set-theoretic beliefs are safe turns on a broadly empirical conjecture which is outstanding. If this conjecture proves to be false, then it is unclear what the epistemological argument for set-theoretic pluralism is supposed to be. (shrink)

There is a long tradition comparing moral knowledge to mathematical knowledge. In this paper, I discuss apparent similarities and differences between knowledge in the two areas, realistically conceived. I argue that many of these are only apparent, while others are less philosophically significant than might be thought. The picture that emerges is surprising. There are definitely differences between epistemological arguments in the two areas. However, these differences, if anything, increase the plausibility of moral realism as compared to mathematical realism. It (...) is hard to see how one might argue, on epistemological grounds, for moral antirealism while maintaining commitment to mathematical realism. But it may be possible to do the opposite. (shrink)

What are intuitions? According to doxastic views, they are doxastic attitudes or dispositions, such as judgments or inclinations to make judgments. According to perceptualist views, they are—like perceptual experiences—pre-doxastic experiences that—unlike perceptual experiences—represent abstract matters as being a certain way. In this paper I argue against doxasticism and in favor of perceptualism. I describe two features that militate against doxasticist views of perception itself: perception is belief-independent and perception is presentational. Then I argue that intuitions also have both features. The (...) upshot is that intuitions are importantly similar to perceptual experiences, and so should not be identified with doxastic attitudes or dispositions. I consider a popular argument from the introspective absence of sui generis intuition experiences in favor of doxasticism. I develop a conception of intuition experiences that helps to defuse this argument. (shrink)

In this paper I assume that we have some intuitive knowledge—i.e. beliefs that amount to knowledge because they are based on intuitions. The question I take up is this: given that some intuition makes a belief based on it amount to knowledge, in virtue of what does it do so? We can ask a similar question about perception. That is: given that some perception makes a belief based on it amount to knowledge, in virtue of what does it do so? (...) A natural idea about perception is that a perception makes a belief amount to knowledge in part by making you sensorily aware of the concrete objects it is about. The analogous idea about intuition is that an intuition makes a belief amount to knowledge in part by making you intellectually aware of the abstract objects it is about. I expand both ideas into fuller accounts of perceptual and intuitive knowledge, explain the main challenge to this sort of account of intuitive knowledge (i.e. the challenge of making sense of intellectual awareness), and develop a response to it. (shrink)

Awareness is a two-place determinable relation some determinates of which are seeing, hearing, etc. Abstract objects are items such as universals and functions, which contrast with concrete objects such as solids and liquids. It is uncontroversial that we are sometimes aware of concrete objects. In this paper I explore the more controversial topic of awareness of abstract objects. I distinguish two questions. First, the Existence Question: are there any experiences that make their subjects aware of abstract objects? Second, the Grounding (...) Question: if an experience makes its subject aware of an abstract object, in virtue of what does it do so? I defend the view that intuitions, specifically mathematical intuitions, sometimes make their subjects aware of abstract objects. In defending this view, I develop an account of the ground of intuitive awareness. (shrink)

According to standard mathematical platonism, mathematical entities (numbers, sets, etc.) are abstract entities. As such, they lack causal powers and spatio-temporal location. Platonists owe us an account of how we acquire knowledge of this inaccessible mathematical realm. Some recent versions of mathematical platonism postulate a plenitude of mathematical entities, and Mark Balaguer has argued that, given the existence of such a plenitude, the attainment of mathematical knowledge is rendered non-problematic. I assess his epistemology for such a profligate platonism and find (...) it unsatisfactory because it lacks an adequate semantics, in particular, an adequate account of reference. (shrink)

The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their (...) dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. (shrink)

Since Peirce defined the first operators for three-valued logic, it is usually assumed that he rejected the principle of bivalence. However, I argue that, because bivalence is a principle, the strategy used by Peirce to defend logical principles can be used to defend bivalence. Construing logic as the study of substitutions of equivalent representations, Peirce showed that some patterns of substitution get realized in the very act of questioning them. While I recognize that we can devise non-classical notations, I argue (...) that, when we make claims about those notations, we inevitably get saddled with bivalent commitments. I present several simple inferences to show this. The argument that results from those examples is ‘pragmatic’, because the inevitability of the principle is revealed in use (not mention); and it is ‘semiotic’, because this revelation happens in the use of signs. (shrink)

During the past decade a new twist in the debate regarding the a priori has unfolded. A number of prominent epistemologists have challenged the coherence or importance of the a priori—a posteriori distinction or, alternatively, of the concept of a priori knowledge. My focus in this paper is on these new challenges to the a priori. My goals are to provide a framework for organizing the challenges, articulate and assess a range of the challenges, and present two challenges of my (...) own. (shrink)

I develop a non-representationalist account of mathematical thought, on which the point of mathematical theorizing is to provide us with the conceptual capacity to structure and articulate information about the physical world in an epistemically useful way. On my view, accepting a mathematical theory is not a matter of having a belief about some subject matter; it is rather a matter of structuring logical space, in a sense to be made precise. This provides an elegant account of the cognitive utility (...) of mathematics. Further, it makes explicit how the brand of non-representationalism I develop is compatible with there being substantive rationality constraints on our mathematical theorizing. (shrink)