In this paper, I argue that for the purposes of ordinary reasoning, sentences about properties of concrete objects can be replaced with sentences concerning how things in our universe would be related to inscriptions were there a pluriverse. Speaking loosely, pluriverses are composites of universes that collectively realize every way a universe could possibly be. As such, pluriverses exhaust all possible meanings that inscriptions could take. Moreover, because universes necessarily do not influence one another, our universe would not be any (...) different intrinsically if there were a pluriverse. These two facts enable anti-realists about abstract objects to replace, e.g. talk of anatomical features with talk of the inscriptions concerning anatomical structure that would exist were there a pluriverse. The availability of such replacements enables anti-realists to carry out essential ordinary reasoning without referring to properties, thereby making room for a consistent anti-realist worldview. The inscriptions of the would-be pluriverse are so numerous and varied that sentences about them can play the roles in ordinary reasoning served by simple sentences about properties of concrete objects. (shrink)

A basic way of evaluating metaphysical theories is to ask whether they give satisfying answers to the questions they set out to resolve. I propose an account of “third-order” virtue that tells us what it takes for certain kinds of metaphysical theories to do so. We should think of these theories as recipes. I identify three good-making features of recipes and show that they translate to third-order theoretical virtues. I apply the view to two theories—mereological universalism and plenitudinous platonism—and draw (...) out their third-order virtues and vices. One lesson is that there is an important difference between essentially and non-essentially third-order vicious theories. I also argue that if a theory is essentially third-order vicious, it cannot be assessed for more standard “second-order” theoretical virtues and vices, like parsimony. This motivates the idea that third-order virtues are distinct from second-order ones. Finally, I suggest that the relationship between truth, progress, and third-order virtue is more complex than it seems. (shrink)

Recently, philosophers have tried to develop a version of truthmaker theory which ties the truthmaking relation closely to the notion of fundamentality. In fact, some of these truthmaker-fundamentalists, as I call them, assume that the notion of fundamentality is intelligible in part by citing, as central examples of fundamentals, truthmakers, which they understand necessarily as constituents of fundamental reality. The aim of this paper is first to bring some order and clarity to this discussion, sketching how far TF is compatible (...) with orthodox truthmaking, and then critically to evaluate the limits of TF. It will be argued that truthmaker theory cannot directly help with articulating the nature of fundamental reality and that T-REL does not necessarily relate truths with anything more fundamental, unless what is fundamental is what the truthbearers in question are about. I shall argue that TF faces a rather thorny dilemma and some general problems. I shall present two exhaustive types of fundamentalism on which a version of TF can be based: deflationary and inflationary. It will be argued that each version of TF runs into significant troubles accounting for all truth, specifically ordinary truths and metaphysical truths about the relations between ordinary facts and fundamental facts. I shall not attempt to solve these problems, but rather, at the end, diagnose the issues with TF as lying in the difficulties with reconciling the manifest image with the scientific and metaphysical images of reality. (shrink)

Abstract objects are standardly taken to be causally inert, however principled arguments for this claim are rarely given. As a result, a number of recent authors have claimed that abstract objects are causally efficacious. These authors take abstracta to be temporally located in order to enter into causal relations but lack a spatial location. In this paper, I argue that such a position is untenable by showing first that causation requires its relata to have a temporal location, but second, that (...) if an entity is temporally located then it is spatiotemporally located since this follows from the theory of Relativity. Since abstract objects lack a spatiotemporal location, then if something is causally efficacious, it is not abstract. (shrink)

In this paper I sketch some lines of response to Mark Colyvan’s indispensability arguments for the existence of inconsistent objects, being mainly concerned with the indispens ability of inconsistent mathematical entities. My response will draw heavily on Jody Azzouni’s deflationary nominalism.

In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...) to other philosophically problematic entities. The moral is that puzzle avoidance fails to motivate deflationary nominalism. (shrink)

Modal primitivists hold that some modal truths are primitively true. They thus seem to face a special epistemological problem: how can primitive modal truths be known? The epistemological objection has not been adequately developed in the literature. I undertake to develop the objection, and then to argue that the best formulation of the epistemological objection targets all realists about modality, rather than the primitivist alone. Furthermore, the moves available to reductionists in response to the objection are also available to primitivists. (...) I conclude by suggesting that extant theories of the epistemology of modality are not sensitive to the question of primitivism versus reductionism. (shrink)

Meinongians in general, and Routley in particular, subscribe to the principle of the independence of Sosein from Sein. In this paper, I put forward an interpretation of the independence principle that philosophers working outside the Meinongian tradition can accept. Drawing on recent work by Stephen Yablo and others on the notion of subject matter, I offer a new account of the notion of Sosein as a subject matter and argue that in some cases Sosein might be independent from Sein. The (...) question whether numbers exist, for instance, is not part of the question of how numbers are, which is the topic mathematicians are interested in. (shrink)

I will contrast two conceptions of the nature of mathematical objects: the conception of mathematical objects as preconceived objects, and heavy duty platonism. I will argue that friends of the indispensability argument are committed to some metaphysical theses and that one promising way to motivate such theses is to adopt heavy duty platonism. On the other hand, combining the indispensability argument with the conception of mathematical objects as preconceived objects yields an unstable position. The conclusion is that the metaphysical commitments (...) of the indispensability argument should be carefully scrutinized. (shrink)

Norms are standards against which actions, dispositions of mind and character, states of affairs and so forth can be measured. They also govern our behaviour, make claims on us, bind us and provide reasons for action and thought that motivate us. J. L. Mackie argued that the intrinsic prescriptivity, or to-be-pursuedness, of moral norms would make them utterly unlike anything else that we know of. Therefore, we should favour an error theory of morality. Mackie thought that the to-be-pursuedness would have (...) to be built into mind-independent moral reality. One alternative, however, is that the to-be-pursuedness is built into our faculty of moral sensibility. There is a large body of empirical evidence demonstrating that the emotions play a central role in making moral judgments. I shall argue that this helps to explain how normative judgments are reliably and non-accidentally related to motivation. I shall also argue that emotional experience has the right structure and properties to provide us with a defeasible warrant for normative knowledge. The role of the emotions in our moral psychology does not obviously support anti-realism. Rather, emotional experience can be intentional, evaluative, evaluable, and quasi-perceptual. This makes emotional experience a plausible candidate for constituting a non-queer faculty of moral sensibility. (shrink)

Wright claims that his and Hale’s abstractionist neologicist project is primarily epistemological in aim. Its epistemological aims include establishing the possibility of a priori mathematical knowledge, and establishing the possibility of reference to abstract mathematical objects. But, as Wright acknowledges, there is a question of how neologicist epistemology applies to actual, ordinary mathematical beliefs. I take up this question, focusing on arithmetic. Following a suggestion of Hale and Wright, I consider the possibility that the neologicist account provides an idealised reconstruction (...) of how people acquire cognition of numbers or arithmetical knowledge. I conclude that it is unlikely that this account can be used to reconstruct even idealised answers to pressing questions about our actual epistemology. (shrink)

Jody Azzouni argues that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use ‘exist’ that we can employ to argue for positions in ontology. I raise and defend a novel objection to Azzouni: his view has the counterintuitive consequence that the facts regarding what exists can and will change when users of the word ‘exist’ change what criteria they associate with its usage. Considering three responses, I argue Azzouni (...) has best reason to take one that ultimately renders unsuccessful his arguments against mathematical abstracta. (shrink)

The question of how we can know anything about ideal entities to which we do not have access through our senses has been a major concern in the philosophical tradition since Plato's Phaedo. This article focuses on the paradigmatic case of mathematical knowledge. Following a suggestion by Gödel, we employ concepts and ideas from Husserlian phenomenology to argue that mathematical objects – and ideal entities in general – are recognized in a process very closely related to ordinary perception. Our analysis (...) combines Husserl's insights into the nature of perception and the phenomena of evidence and justification with case studies of the role of intuition in axiomatic set theory. (shrink)

To the memory of Prof. Grigori Mints, Stanford UniversityBorn: June 7, 1939, St. Petersburg, RussiaDied: May 29, 2014, Palo Alto, California.

If you put chimpanzees from different communities together you can expect mayhem - they are not keen on treating each other nicely. There is closely related species of apes, however, whose members have countless encounters with unrelated specimen on a daily basis and yet almost all get through the day in one piece - that species is us, homo sapiens. But what makes us get along, most of the time? Morality as such is, perhaps surprisingly, not a mainstream research topic (...) in social psychology. Most textbooks in the field do not even mention the term morality at all (and focus on 'prosociality' instead). This relative lack of 'coverage' might have to do with the difficulty of operationalising 'morality' in a fitting way (more on this below). However, morality certainly is a social phenomenon and therefore the collection of Fargas, Jussim, and Van Lange is certainly a step in the right direction; it offers an extensive overview of the field and is suggestive of the vast potential of studying morality through the lens of social psychology. (shrink)

Pure mathematics can play an indispensable role explaining empirical phenomena if recent accounts of insect evolution are correct. In particular, the prime life cycles of cicadas and the geometric structure of honeycombs are taken to undergird an inference to the best explanation about mathematical entities. Neither example supports this inference or the mathematical realism it is intended to establish. Both incorrectly assume that facts about mathematical optimality drove selection for the respective traits and explain why they exist. We show how (...) this problem can be avoided, identify limitations of explanatory indispensability arguments, and attempt to clarify the nature of mathematical explanation. (shrink)

Conferring legal personhood on purely synthetic entities is a very real legal possibility, one under consideration presently by the European Union. We show here that such legislative action would be morally unnecessary and legally troublesome. While AI legal personhood may have some emotional or economic appeal, so do many superficially desirable hazards against which the law protects us. We review the utility and history of legal fictions of personhood, discussing salient precedents where such fictions resulted in abuse or incoherence. We (...) conclude that difficulties in holding “electronic persons” accountable when they violate the rights of others outweigh the highly precarious moral interests that AI legal personhood might protect. (shrink)

The debate concerning the ontological status of musical works is perhaps the most animated debate in contemporary analytic philosophy of music. In my view, progress requires a piecemeal approach. So in this article I hone in on one particular musical work concept – that of the classical Western art musical work; that is, the work concept that regulates classical art-musical practice. I defend a fictionalist analysis – a strategy recently suggested by Andrew Kania as potentially fruitful – and I develop (...) a version of such an analysis in line with a broad commitment to philosophical naturalism. (shrink)

It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about (...) metaphysical necessity, then paradigmatic metaphysical necessities would be necessary in one sense of “necessary”, not necessary in another, and that would be it. The question of whether they were necessary simpliciter would be like the question of whether the Parallel Postulate is true simpliciter – understood as a pure mathematical conjecture, rather than as a hypothesis about physical spacetime. In a sense, the latter question has no objective answer. In this article, I argue that paradigmatic questions of modal metaphysics are like the Parallel Postulate question. I then discuss the deflationary ramifications of this argument. I conclude with an alternative conception of the space of possibility. According to this conception, there is no objective boundary between possibility and impossibility. Along the way, I sketch an analogy between modal metaphysics and set theory. (shrink)

This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in category theory is identifiable with the Grothendieck Universe Axiom and the elementary embeddings in Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, (...) and thus the generic invariance of mathematical truth, modal coalgebras are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility. (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 comprise a materially adequate fragment of the language of thought, and endeavor to show how such algebras provide the resources necessary to resolve Russell's paradox of propositions. I demonstrate, then, how modal expressivism can be regimented by modal coalgebraic automata, to which the above epistemic modal algebras are dually isomorphic. I examine, in particular, 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)

Restrictionism is a response to the Liar and other paradoxes concerning truth. Restrictionists—as I will call proponents of the strategy—respond to these paradoxes by giving up instances of the schema -/- <p> is true iff p. -/- My aim is to show that the current unpopularity of restrictionism is undeserved. I will argue that, whilst cautious versions of the strategy may face serious problems, a radical and previously overlooked version of restrictionism provides a strong and defensible response to the paradoxes.

A number of philosophers have argued in recent years that certain kinds of metaphysical debates—e.g., debates over the existence of past and future objects, mereological sums, and coincident objects—are merely verbal. It is argued in this paper that metaphysical debates are not merely verbal. The paper proceeds by uncovering and describing a pattern that can be found in a very wide range of philosophical problems and then explaining how, in connection with any problem of this general kind, there is always (...) a non-verbal debate to be had. Indeed, the paper provides a recipe for locating the non-verbal debates that surround these philosophical problems. This undermines metametaphysical verbalist views of our metaphysical questions—i.e., views that say that there is no non-verbal debate to be had about some metaphysical question. Finally, the paper also provides a quick argument against actual-literature verbalist views of our metaphysical questions; in other words, the paper argues that in connection with all of our metaphysical questions, it is easy to find examples of non-verbal debates in the actual philosophical literature. (shrink)

This paper offers a neo-pragmatist account of the representational character of the emotions, for those emotions that have such a character. Put most generally, neo-pragmatism is the view that language should not be conceived primarily in terms of a robust relation of reference to or representation of antecedently given objects and properties. Rather, we should view it as a social practice that lets us do various quite different sorts of things. One of those things might be called ‘assessing representational accuracy’, (...) but this need not be thought of in terms of a metaphysically heavyweight relation. In applying neo-pragmatist techniques to the domain of the emotions, the result will be an alternative to currently popular accounts that individuate emotions partly in terms of what they represent. This alternative continues to allow us to use representational language in connection with some emotions. And it also helps to explain the awkwardness of representational talk in connection with other emotions: an awkwardness that was always a liability of monolithic representationalist views. (shrink)

I defend the view that our ontology divides into categories, each with its own canonical way of identifying and distinguishing the objects it encompasses. For instance, I argue that natural numbers are identified and distinguished by their positions in the number sequence, and physical bodies, by facts having to do with spatiotemporal continuity. I also argue that objects belonging to different categories are ipso facto distinct. My arguments are based on an analysis of reference, which ascribes to reference a richer (...) structure than it is normally taken to have. (shrink)

The modal antirealist, as presented here, aims to secure at least some of the benefits associated with talking in genuine modal realist terms while avoiding commitment to a plurality of Lewisian (or ersatz) worlds. The antirealist stance of agnosticism about other worlds combines acceptance of Lewis's account of what world‐talk means with refusal to assert, or believe in, the existence of other worlds. Agnosticism about other worlds does not entail a comprehensive agnosticism about modality, but where such agnosticism about modality (...) is enforced, the aim of the agnostic programme is to show that it is not detrimental to our modal practices. The agnostic programme consists in an attempt to demonstrate the rational dispensability of that disputed class of modal beliefs which the agnostic eschews, but which are held by the realist and the folk. Here I attempt to motivate, describe, and illustrate such an agnostic antirealist programme in modal philosophy. (shrink)

RÉSUMÉMalgré son usage très répandu, l'inférence à la meilleure explication a souvent été considérée avec suspicion par des théoriciens d'allégeances diverses. On lui a reproché à maintes reprises defaire reposer son recours à la simplicité et ses autres vertus explicatives sur des présuppositions métaphysiques douteuses. J'aborde ces questions, dans le présent article, dans le contexte d'une discussion large de l'usage de l'IME pour fonder notre croyance au monde extérieur. Distinguant entre la légitimité et l'efficacité de l'IME, je soutiendrai qu'elle constitue (...) bien une forme légitime d'inférence, mais que son efficacité dépend de la fonction de la théorie de la connaissance qui la supporte. (shrink)

Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...) facie favor a realist account of numbers. (shrink)

Mathematical concepts play at least three roles in the application of mathematics: an inferential role, a representational role, and an expressive role. In this paper, I argue that, despite what has often been alleged, platonists do not fully accommodate these features of the application of mathematics. At best, platonism provides partial ways of handling the issues. I then sketch an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features of the application (...) process. In this way, a better account of mathematical applications is, in principle, available. (shrink)

According to modalism, modality is primitive. In this paper, we examine the implications of this view for modal epistemology, and articulate a modalist account of modal knowledge. First, we discuss a theoretical utility argument used by David Lewis in support of his claim that there is a plurality of concrete worlds. We reject this argument, and show how to dispense with possible worlds altogether. We proceed to account for modal knowledge in modalist terms.

A complete philosophy of mathematics must address Paul Benacerraf’s dilemma. The requirements of a general semantics for the truth of mathematical theorems that coheres also with the meaning and truth conditions for non-mathematical sentences, according to Benacerraf, should ideally be coupled with an adequate epistemology for the discovery of mathematical knowledge. Standard approaches to the philosophy of mathematics are criticized against their own merits and against the background of Benacerraf’s dilemma, particularly with respect to the problem of understanding the distinction (...) between pure and applied mathematics and the effectiveness of applied mathematics in the natural sciences and engineering. The evaluation of these alternatives provides the basis for articulating a philosophically advantageous Aristotelian inherence concept of mathematical entities. An inherence account solves Benacerraf’s dilemma by interpreting mathematical entities as nominalizations of structural spatiotemporal properties inhering in existent spatiotemporal entities. (shrink)

Dans cet article je propose d'exprimer et de défendre une conception des pratiques et du domaine de discours mathématiques qui soit sensible, d'une part, au pluralisme des relations entre pratiques inférentielles et intérêts, et d'autre part, à la structure objective et déterminante des concepts mathématiques. J'ébauche tout d'abord une caractérisation générale des pratiques, pour ensuite préciser certains phénomènes propres aux pratiques mathématiques. Suit un recensement des idées qui se dégagent des arguments pluralistes, et de celles qui sont à retenir. Mais (...) je défends par la suite la nécessité d'une forme de réalisme mathématique, qui toutefois ne peut être le réalisme d'objets que prônent les partisans de l'argument d'indispensabilité. Je défends plutôt un réalisme des concepts, soutenu par ce que je baptise l'« argument de la contrainte ». (shrink)

Impure systems contain Objects and Subjects: Subjects are human beings. We can distinguish a person as an observer (subjectively outside the system) and that by definition is the Subject himself, and part of the system. In this case he acquires the category of object. Objects (relative beings) are significances, which are the consequence of perceptual beliefs on the part of the Subject about material or energetic objects (absolute beings) with certain characteristics.The IS (Impure System) approach is as follows: Objects are (...) perceptual significances (relative beings) of material or energetic objects (absolute beings). The set of these objects will form an impure set of the first order. The existing relations between these relative objects will be of two classes: transactions of matter and/or energy and inferential relations. Transactions can have alethic modality: necessity, possibility, impossibility and contingency. Ontic existence of possibility entails that inferential relations have Deontic modality: obligation, permission, prohibition, faculty and analogy. We distinguished between theorems (natural laws) and norms (ethical, legislative and customary rules of conduct). (shrink)

Paul Benacerraf’s argument that mathematical realism is apparently incompatible with mathematical knowledge has been widely thought to also show that a priori knowledge in general is problematic. Although many philosophers have rejected Benacerraf’s argument because it assumes a causal theory of knowledge, some maintain that Benacerraf nevertheless put his finger on a genuine problem, even though he didn’t state the problem in its most challenging form. After diagnosing what went wrong with Benacerraf’s argument, I argue that a new, more challenging, (...) version of Benacerraf’s problem can be constructed. The new version—what I call the Defeater Version—of Benacerraf’s problem makes use of a no-defeater condition on knowledge and justification. I conclude by arguing that the best way to avoid the problem is to construct a theory of how a priori judgments reliably track the facts. I also suggest four different kinds of theories worth pursuing. (shrink)

This book offers an historically-informed critical assessment of Dummett's account of abstract objects, examining in detail some of the Fregean presuppositions whilst also engaging with recent work on the problem of abstract entities.

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)

In everyday speech we seem to refer to such things as abstract objects, moral properties, or propositional attitudes that have been the target of metaphysical and/or epistemological objections. Many philosophers, while endorsing scepticism about some of these entities, have not wished to charge ordinary speakers with fundamental error, or recommend that the discourse be revised or eliminated. To this end a number of non-revisionary antirealist strategies have been employed, including expressivism, reductionism and hermeneutic fictionalism. But each of these theories faces (...) forceful objections. In particular, we argue, proponents of these strategies face a dilemma: either concedes that their theory is revisionary, or adopt an implausible account of speaker-meaning whereby the content of certain types of utterance is opaque to their speakers. In this paper we introduce a new type of antirealist strategy, which is thoroughly non-revisionary, and leaves speaker-meaning transparent to speakers. We draw on work on pragmatics in the philosophy of language to develop a theory we call ‘pragmatic antirealism’. The pragmatic antirealist holds that while the sentences of the discourses in question have metaphysically contentious truth conditions, ordinary utterances of them are pragmatically modified in context in such a way that speakers do not incur commitment to those truth conditions. After setting out the theory, we show how it might be developed for both mathematical and ethical discourse, before responding to some likely objections. (shrink)

The access problem for mathematics arises from the supposition that the referents of mathematical terms inhabit a realm separate from us. Quine’s approach in the philosophy of mathematics dissolves the access problem, though his solution sometimes goes unrecognized, even by those who rely on his framework. This paper highlights both Quine’s position and its neglect. I argue that Michael Resnik’s structuralist, for example, has no access problem for the so-called mathematical objects he posits, despite recent criticism, since he relies on (...) an indispensability argument. Still, Resnik’s structuralist does not provide an account of our access to traditional mathematical objects, and this may be seen as a problem. (shrink)

I argue that the most popular versions of naturalism imply nominalism in philosophy of mathematics. In particular, there is a conflict in Quine's philosophy between naturalism and realism in mathematics. The argument starts from a consequence of naturalism on the nature of human cognitive subjects, physicalism about cognitive subjects, and concludes that this implies a version of nominalism, which I will carefully characterize. The indispensability of classical mathematics for the sciences and semantic/confirmation holism does not affect the argument. The disquotational (...) theory of reference and truth is discussed but rejected. This argument differs from the Benacerrafian arguments against realism, because it does not rely on any specific assumption about the nature of knowledge or reference. It differs from the popular objections to the indispensability argument for realism as well, because it can admit both indispensability and holism. This argument motivates a new, radically naturalistic and nominalistic approach to philosophy of mathematics. (shrink)

This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerrafs work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a (...) Fregean account of the objectivity and our knowledge of abstract objects. It is then argued that the resulting view faces no insurmountable metaphysical or epistemic obstacles. (shrink)

Epistemology has recently witnessed a number of efforts to rehabilitate rationalism, to defend the existence and importance of a priori knowledge or warrant construed as the product of rational insight or apprehension (Bealer 1987; Bigelow 1992; BonJour 1992, 1998; Burge 1998; Butchvarov 1970; Katz 1998; Plantinga 1993). This effort has sometimes been coupled with an attack on naturalistic epistemology, especially in BonJour 1994 and Katz 1998. Such coupling is not surprising, because naturalistic epistemology is often associated with thoroughgoing empiricism and (...) the rejection of the a priori. In this paper, however, I shall present a conception of naturalistic epistemology that is entirely compatible with a priori justification or knowledge. The resulting conception, I claim, gives us a better appreciation of the respective merits of the rational and the empirical, as well as a better understanding of how moderate epistemological naturalism comports, at least in principle, with moderate rationalism. This paper defends moderate rationalism; but it does not defend everything rationalists have often wanted, only what it is reasonable to grant them. (shrink)

The present article aims at showing that it is possible to construct a realist philosophy of mathematics which commits one neither to dream the dreams of Platonism nor to reduce the word ''realism'' to mere noise. It is argued that mathematics is a science of patterns, where patterns are not objects (or properties of objects), but aspects, or aspects of aspects, etc. of objects. (The notion of aspect originates from ideas sketched by Wittgenstein in the Philosophical Investigations.).

One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers.

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)

NeoFregeanism is an intriguing but elusive philosophy of mathematical existence. At crucial points, it goes cryptic and metaphorical. I want to put forward an interpretation of neoFregeanism—perhaps not one that actual neoFregeans will embrace—that makes sense of much of what they say. NeoFregeans should embrace quantifier variance.

Fine's replies to critics, in a symposium on his book The Limits of Abstraction.

Hartry Field (1980) has developed an interesting nominalization strategy for Newtonian gravitation theory—a strategy that reformulates the theory without quantification over abstract entities. According to David Malament (1982), Field's strategy cannot be extended to quantum mechanics (QM), and so it only has a limited scope. In a recent work, Mark Balaguer has responded to Malament's challenge by indicating how QM can be nominalized, and by “doing much of the work needed to provide the nominalization” (Balaguer 1998, 114). In this paper, (...) I critically assess Balaguer's proposal, and argue that it ultimately fails. Balaguer's strategy is incompatible with a number of interpretations of QM, in particular with Bas van Fraassen's version of the modal interpretation. And given that Balaguer's strategy invokes physically real propensities, it is unclear whether it is even compatible with nominalism. I conclude that the nominalization of QM remains a major problem for the nominalist. (shrink)

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)