In response to comments on my book, Being Realistic about Reasons, by Justin Clarke-Doane, David Enoch and Tristram McPherson, and Gideon Rosen, I try to clarify my domain-based view of ontology, my understanding of the epistemology of normative judgments, and my interpretation of the phenomenon of supervenience.

W. V. Quine is arguably the intellectual father of contemporary naturalism, the idea that there is no distinctively philosophical perspective on reality. Yet, even though Quine has always been a science-minded philosopher, he did not adopt a fully naturalistic perspective until the early 1950s. In this paper, I reconstruct the genesis of Quine’s ideas on the relation between science and philosophy. Scrutinizing his unpublished papers and notebooks, I examine Quine’s development in the first decades of his career. After identifying three (...) commitments supporting his naturalism—viz. empiricism, holism, and realism—I piece together the evolution of Quine’s position by examining the origins of these commitments one by one, showing how his early views gradually evolved into the mature naturalistic position that would have such an enormous impact on post-war analytic philosophy. (shrink)

This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the set of observables of a physical system, be it classical or quantum, is described by a Jordan-Lie algebra. From the geometric point of view, the space of states of any system is described by a uniform Poisson space with transition probability. Both these structures (...) are here perceived as formal translations of the fundamental twofold role of properties in Mechanics: they are at the same time quantities and transformations. The question becomes then to understand the precise articulation between these two roles. The analysis will show that Quantum Mechanics can be thought as distinguishing itself from Classical Mechanics by a compatibility condition between properties-as-quantities and properties-as-transformations. -/- Moreover, this dissertation shows the existence of a tension between a certain "abstract way" of conceiving mathematical structures, used in the practice of mathematical physics, and the necessary capacity to specify particular states or observables. It then becomes important to understand how, within the formalism, one can construct a labelling scheme. The “Chase for Individuation” is the analysis of different mathematical techniques which attempt to overcome this tension. In particular, we discuss how group theory furnishes a partial solution. (shrink)

Quine is routinely perceived as saving metaphysics from Carnapian positivism. Where Carnap rejects metaphysical existence claims as meaningless, Quine is taken to restore their intelligibility by dismantling the former’s internal–external distinction. The problem with this picture, however, is that it does not sit well with the fact that Quine, on many occasions, has argued that metaphysical existence claims ought to be dismissed. Setting aside the hypothesis that Quine’s metaphysical position is incoherent, one has to conclude that his views on metaphysics (...) are subtler than is often presupposed; both the received view that Quine saved metaphysics and the opposite view that Carnap and Quine are on the same anti-metaphysical team seem too one-sided if we take seriously Quine’s own pronouncements on the issue. In this paper, I offer a detailed reconstruction of Quine’s perspective on metaphysical existence claims. Scrutinizing his published work as well as unpublished papers, letters, and notebooks, I show how Quine is able to both blur the boundary between scientific sense and metaphysical nonsense and to argue that we cannot ask what reality is really like in a distinctively philosophical way. I argue that although Quine’s position is much closer to Carnap’s than the received view suggests, it still differs in two crucial respects. (shrink)

This essay defends the neo-Quinean approach to ontology against the criticisms of two neo-Carnapians, Huw Price and Amie Thomasson.

It is known that the Restricted Predicate Calculus can be embedded in an elementary theory, the signature of which consists of exactly two equivalences. Some special models for the mentioned theory were constructed to prove this fact. Besides formal adequacy of these models, a question may be posed concerning their conceptual simplicity, "transparency" of interpretations they assigned to the two stated equivalences. In works known to us these interpretations are rather complex, and can be called "technical", serving only the purpose (...) of embedding. We propose a conversion method, which transforms an arbitrary model of RPC into some model of the elementary theory TR, which includes three equivalences. RPC is embeddable in TR, and it appears possible to assign some "natural" interpretations to three equivalences using the "Track of Relation " concept. (shrink)

Hybrid languages are introduced in order to evaluate the strength of “minimal” mereologies with relatively strong frame definability properties. Appealing to a robust form of nominalism, I claim that one investigated language \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {m}}$\end{document} is maximally acceptable for nominalistic mereology. In an extension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {gem}}$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {m}}$\end{document}, a modal analog (...) for the classical systems of Leonard and Goodman and Leśniewski is introduced and shown to be complete with respect to 0-deleted Boolean algebras. We characterize the formulas of first-order logic invariant for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {gem}}$\end{document}-bisimulations. (shrink)

This article examines how Quine and Sellars develop informatively contrasting responses to a fundamental tension in Carnap’s semantics ca. 1950. Quine’s philosophy could well be styled ‘Essays in Radical Empiricism’; his assay of radical empiricism is invaluable for what it reveals about the inherent limits of empiricism. Careful examination shows that Quine’s criticism of Carnap’s semantics in ‘Two Dogmas of Empiricism’ fails, that at its core Quine’s semantics is for two key reasons incoherent and that his hallmark Thesis of Extensionalism (...) is untenable. The tension in Carnap’s semantics together with Quine’s exposure of the severe limits of radical empiricism illuminate central features of Sellars’s philosophy: the fully general form of the myth of givenness, together with Sellars’s alternative Kantian characterisation of understanding; the full significance of Carnap’s distinction between conceptual analysis and conceptual explication, and its important methodological implications; the specifically pragmatic character of Sellars’s realism; and Sellars’s methodological reasons for holding that philosophy must be systematic and that systematic philosophy must be deeply historically and textually informed. This paper thus re-examines this recent episode of philosophical history for its philosophical benefits and systematic insights. (shrink)

When some objects are the parts of another object, they compose that object and that object is composite. This article is intended as an introduction to the central questions about composition and a highly selective overview of various answers to those questions. In §1, we review some formal features of parthood that are important for understanding the nature of composition. In §2, we consider some answers to the question: which pluralities of objects together compose something? As we will see, the (...) dominant answers are all of them and none of them. In §§3-4, we examine one of the main arguments that has driven philosophers to these extreme answers: the argument from vagueness. In §5, we turn to the question of whether composition is unique: is it sometimes the case that some things compose more than one thing? Finally, in §6, we turn from the question of which composites exist to the question of which composites exist fundamentally. (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)

For the sake of explaining the merits of a Leśniewski type nominalism, it should be made clear what is meant by „nominalism” and what the characteristics of this special type of nominalism are. To the first question we can find quite a lot of mutually inconsistent answers. Therefore I will just explain the distinction between two different nominalistic traditions which I hold to be fundamental. I think we should not just focus on the question which so-called abstract entities are rejected (...) but as well look for basic entities nominalists rely on. (shrink)

This volume covers a wide range of topics that fall under the 'philosophy of quantifiers', a philosophy that spans across multiple areas such as logic, metaphysics, epistemology, and even the history of philosophy. It discusses the import of quantifier variance in the model theory of mathematics. It advances an argument for the uniqueness of quantifier meaning in terms of Evert Beth’s notion of implicit definition, and clarifies the oldest explicit formulation of quantifier variance: the one proposed by Rudolf Carnap. -/- (...) The volume further examines what it means that a quantifier can have multiple meanings, and addresses how existential vagueness can induce vagueness in our modal notions. Finally, the book explores the role played by quantifiers with respect to various kinds of semantic paradoxes, the logicality issue, ontological commitment, and the behavior of quantifiers in intensional contexts. ​. (shrink)

Philosophers of information, according to Luciano Floridi (The philosophy of information. Oxford University Press, Oxford, 2010, p 32), study how information should be “adequately created, processed, managed, and used.” A small number of epistemologists have employed the concept of information as a cornerstone of their theoretical framework. How this concept can be used to make sense of seemingly intractable epistemological problems, however, has not been widely explored. This paper examines Fred Dretske’s information-based epistemology, in particular his response to radical epistemological (...) skepticism. We discuss the relationship between information, evidence and knowledge in relation to the problem of skepticism and the options available to an information-based epistemology for dealing with it. (shrink)

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on (...) Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

An encyclopedia entry which covers various revisionary conceptions of which macroscopic objects there are, and the puzzles and arguments that motivate these conceptions: sorites arguments, the argument from vagueness, the puzzles of material constitution, arguments against indeterminate identity, arguments from arbitrariness, debunking arguments, the overdetermination argument, and the problem of the many.

The dispute between nominalists and Platonic realists has been with us for a long time — long enough to have assumed many forms. I don't want to rehearse the history of these various debates, or even to look at the matter from a historical point of view. But I would like to begin by distinguishing two quite different skirmishes in the general battle, one of which is new, and one of which is very old. We begin with the new one, (...) which is the clearest. (shrink)

I consider the well-known criticism of Quine's characterization of first-order logical truth that it expands the class of logical truths beyond what is sanctioned by the model-theoretic account. Briefly, I argue that at best the criticism is shallow and can be answered with slight alterations in Quine's account. At worse the criticism is defective because, in part, it is based on a misrepresentation of Quine. This serves not only to clarify Quine's position, but also to crystallize what is and what (...) is not at issue in choosing the model-theoretic account of first-order logical truth over one in terms of substitutions. I conclude by highlighting the need for justifying the belief that the definition of first-order logical truth in terms of models is superior to its definition in terms of substitutions. (shrink)

Tarski, Carnap and Quine spent the academic year 1940?1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among them during the year. These conversations centred around semantical issues related to the analytic/synthetic distinction and on the project of a finitist/nominalist construction of mathematics and science. Carnap's Nachlaß in Pittsburgh contains a set of detailed notes, amounting to more than 80 typescripted pages, taken by Carnap while these discussions were taking (...) place. In my article, I present a survey of these notes with special emphasis on Tarski's rejection of the analytic/synthetic distinction, the passage from typed languages to first-order languages, Tarski's finitism/nominalism, and the construction of a finitist language for mathematics and science. (shrink)

In this book, Zalta attempts to lay the axiomatic foundations of metaphysics by developing and applying a (formal) theory of abstract objects. The cornerstones include a principle which presents precise conditions under which there are abstract objects and a principle which says when apparently distinct such objects are in fact identical. The principles are constructed out of a basic set of primitive notions, which are identified at the end of the Introduction, just before the theorizing begins. The main reason for (...) producing a theory which defines a logical space of abstract objects is that it may have a great deal of explanatory power. It is hoped that the data explained by means of the theory will be of interest to pure and applied metaphysicians, logicians and linguists, and pure and applied epistemologists. (shrink)

Arthur Pap’s work played an important role in the development of the analytic tradition. This role goes beyond the merely historical fact that Pap’s views of dispositional and modal concepts were influential. As a sympathetic critic of logical empiricism, Pap, like Quine, saw a deep tension in logical empiricism at its very best in the work of Carnap. But Pap’s critique of Carnap is quite different from Quine’s, and represents the discovery of limits beyond which empiricism cannot go, where there (...) lies nothing other than intuitive knowledge of logic itself. Pap’s arguments for this intuitive knowledge anticipate Etchemendy’s recent critique of the model-theoretic account of logical consequence. Pap’s work also anticipates prominent developments in the contemporary neo-Fregean philosophy of mathematics championed by Wright and Hale. Finally, Pap’s major philosophical preoccupation, the concepts of necessity and possibility, provides distinctive solutions and perspectives on issues of contemporary concern in the metaphysics of modality. In particular, Pap’s account of modality allows us to see the significance of Kripke’s well-known arguments on necessity and apriority in a new light. (shrink)

The discovery of the importance of mereology follows and does not precede the formalisation of the theory. In particular, it was only after the construction of an axiomatic theory of the part-whole relation by the Polish logician Stanisław Leśniewski that any attempt was made to reinterpret some periods in the history of philosophy in the light of the theory of parts and wholes. Secondly, the push for formalisation - and the individuation of mereology as a specific theoretical field - arise (...) from several particular theories, first from biology and then from psychology. Mereological considerations were explicitly developed within the school of Brentano, by Brentano himself- starting from the problem of the unity of consciousness - and by his students, particularly Stumpl, Twardowski and the Gestaltists. The most complete result of this tradition is undoubtedly the third of Husserl's Logical Investigations, called "the single most important contribution to realist (Aristotelian) ontology in the modern period". (shrink)

A brief review of the historicalrelation between logic and ontologyand of the opposition between the viewsof logic as language and logic as calculusis given. We argue that predication is morefundamental than membership and that differenttheories of predication are based on differenttheories of universals, the three most importantbeing nominalism, conceptualism, and realism.These theories can be formulated as formalontologies, each with its own logic, andcompared with one another in terms of theirrespective explanatory powers. After a briefsurvey of such a comparison, we argue (...) that anextended form of conceptual realism provides themost coherent formal ontology and, as such, canbe used to defend the view of logic as language. (shrink)

This paper presents a novel account of applied mathematics. It shows how we can distinguish the physical content from the mathematical form of a scientific theory even in cases where the mathematics applied is indispensable and cannot be eliminated by paraphrase.

Some realists claim that theoretical entities like numbers and electrons are indispensable for describing the empirical world. Motivated by the meta-ontology of Quine, I take this claim to imply that, for some first-order theory T and formula δ(x) such that T ⊢ ∃xδ ∧ ∃x¬δ, where δ(x) is intended to apply to all and only empirical entities, there is no first-order theory T′ such that (a) T and T′ describe the δ:s in the same way, (b) T′ ⊢ ∀xδ, and (...) (c) T′ is at least as attractive as T in terms of other theoretical virtues. In an attempt to refute the realist claim, I try to solve the general problem of nominalizing T (with respect to δ), namely to find a theory T′ satisfying conditions (a)–(c) under various precisifications thereof. In particular, I note that condition (a) can be understood either in terms of syntactic or semantic equivalence, where the latter is strictly stronger than the former. The results are somewhat mixed. On the positive side, even under the stronger precisification of (a), I establish that (1) if the vocabulary of T is finite, a nominalizing theory can always be found that is recursive if T is, and (2) if T postulates infinitely many δ:s, a nominalizing theory can always be found that is no more computationally complex than T. On the negative side, even under the weaker precisification of (a), I establish that (3) certain finite theories cannot be nominalized by a finite theory. (shrink)

This article provides a computational example of a mathematical explanation within science, concerning computational equivalence of programs. In addition, it outlines the logical structure of the reasoning involved in explanations in applied mathematics. It concludes with a challenge that the nominalist provide a nominalistic explanation for the computational equivalence of certain programs.

This unique anthology of new, contributed essays offers a range of perspectives on various aspects of ontic vagueness. It seeks to answer core questions pertaining to onticism, the view that vagueness exists in the world itself. The questions to be addressed include whether vague objects must have vague identity, and whether ontic vagueness has a distinctive logic, one that is not shared by semantic or epistemic vagueness. The essays in this volume explain the motivations behind onticism, such as the plausibility (...) of mereological vagueness and indeterminacy in quantum mechanics and they offer various arguments both for and against ontic vagueness; onticism is also compared with other, competing theories of vagueness such as semanticism, the view that vagueness exists only in our linguistic representation of the world. Gareth Evans’s influential paper of 1978, “Can There Be Vague Objects?” gave a simple but cogent argument against the coherence of ontic vagueness. Onticism was subsequently dismissed by many. However, in recent years, researchers have become aware of the logical gaps in Evans’s argument and this has triggered a new wave of interest in onticism. Onticism is now widely regarded as at least a coherent view. Reflecting this growing consensus, the present anthology for the first time puts together essays that are focused on onticism and its various facets and it fills in the lacuna in the literature on vagueness, a much-discussed subject in contemporary philosophy. (shrink)

Quine's views on ontology and naturalism are well-known but rarely considered in tandem. According to my interpretation the connection between them is vital. I read Quine as a global epistemic structuralist. Quine thought we only ever know objects qua solutions to puzzles about significant intersections in observations. Objects are always accessed descriptively, via their roles in our best theory. Quine's Kant lectures contain an early version of epistemic structuralism with uncharacteristic remarks about the mental. Here Quine embraces mitigated anomalous monism, (...) allowing introspection and the availability in principle of full physical descriptions of the perceptual states which get science off the ground. Later versions abandon these ideas. My epistemic-structural interpretation explains why. I argue first-personal introspective access to mental states is incompatible with global epistemic structuralism. (shrink)

Discussion of atomistic and monistic theses about abstract reality.

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)

Huw Price has argued that naturalistic philosophy inevitably leads to a deflationary approach to ontological questions. In this paper, I rebut these arguments. A more substantive, less language-focused approach to metaphysics remains open to naturalists. However, rebutting one of Price’s main arguments requires rejecting Quine’s criterion of ontological commitment. So, even though Price’s argument is unsound, it reveals that naturalists cannot rest content with broadly Quinean, “mainstream metaphysics,” which, I suggest, naturalists also have independent reasons to reject.

At the heart of my pragmatic theory of truth and ontology is a view of the relation between language and reality which I term internal justification: a way of explaining how sentences may have truth-values which we cannot discover without invoking the need for the mystery of a correspondence relation. The epistemology upon which the theory depend~ is fallibilist and holistic ; places heavy reliance on modal idioms ; and leads to the conclusion that current versions of realism and anti-realism (...) are deficient. Just as my theory avoids the need for an epistemic 'given', it avoids the need for a metaphysical 'given' or 'joints'. I offer a view of the nature of philosophy and what it can properly achieve with respect to ontological questions ; since those views lead me to believe that philosophical discussion about what exists should be restricted to 'entities' discussed in non-philosophical contexts, my views on how we should understand claims made about the existence of middle-sized physical objects, theoretical entities in science, and abstract entities in mathematics, give the thesis a schematic completeness. My theory leads me to a conception of inquiry which defends the cognitive status of moral statements whilst being critical of Kantian and utilitarian approaches to morality. Chapter 1 explores the views of my closest philosophical allies: William James and Nelson Goodman. (shrink)

This book offers a critical account of the fundamental elements of Leibniz's philosophy, as they manifest themselves in his metaphysics and philosophy of language. Emphasis is placed upon his hitherto neglected doctrine of nominalism, which states that only concrete individuals exist and that there are no such things as abstract entities – no numbers, geometrical figures or other mathematical objects, nor any abstractions such as space, time, heat, light, justice, goodness, or beauty. Using this doctrine as a basis, the book (...) considers the core of Leibniz's metaphysics and philosophy of language, including the well‐known principles of Contradiction, Sufficient Reason, Continuity and Individuation, and the concepts of Possible World, Individual Substance, Truth, and Necessity. (shrink)

"Philosophical discussion of the notion of existence, or being, has centered on two main problems which have not always been very clearly distinguished. First, there is the problem of what we are to say about the existence of fictitious objects, such as centaurs, dragons, and Pegasus; second, there is the problem of what we are t o say about the existence of abstract objects, such as qualities, relations, and numbers. Both problems have tempted philosophers to say that there are inferior (...) sorts of existence a s well as the ordinary straightforward sort, and they therefore often suggest that we use the word "being" to cover both kinds but restrict "existence" to "being" of the common, non-fictitious, non-abstract sort. (Sometimes the term "reality" is proposed for "existence" or for "being.") T he problems of fiction and abstraction are different, how ever, for there are both real and fictitious abstractions. For example, the integer between two and four is real, but the integer between two and three is fictitious. On the other hand, there are both concrete and abstract fictions; for example, the winged horse of Bellerophon and the integer between two and three. Accordingly, philosophers have often dealt with the two problems in quite different ways and perhaps ought to do so. While these are the two main problems, there are others, f or example, that of what we are to say of the being of objects which have not yet begun, or have now ceased, to exist. The history of this subject, moreover, has been tangled with theological issues, to which it will be necessary to refer at certain points.". (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)

This article attempts to motivate a new approach to anti-realism (or nominalism) in the philosophy of mathematics. I will explore the strongest challenges to anti-realism, based on sympathetic interpretations of our intuitions that appear to support realism. I will argue that the current anti-realistic philosophies have not yet met these challenges, and that is why they cannot convince realists. Then, I will introduce a research project for a new, truly naturalistic, and completely scientific approach to philosophy of mathematics. It belongs (...) to anti-realism, but can meet those challenges and can perhaps convince some realists, at least those who are also naturalists. (shrink)

1. 1. PROGRAM It will be our aim to reconstruct, with precision, certain views which have been traditionally associated with nominalism and to investigate problems arising from these views in the construction of interpreted formal systems. Several such systems are developed in accordance with the demand that the sentences of a system which is acceptable to a nominalist must not imply the existence of any entities other than individuals. Emphasis will be placed on the constructionist method of philosophical analysis. To (...) follow this method is to introduce the central notions of the subject-matter to be investigated into a system governed by exact rules. For example, the constructionist method of investigating the properties of geometric figures may consist in formulating a system of postulates and definitions which, together with the apparatus of formal logic, generates all necessary truths concerning geometric figures. Similarly, a constructionist analysis of the notion of an individual may take the form of an axiomatic theory whose provable assertions are just those which seem essential to the role played by the concept of an individual in system atic contexts. Such axiomatic theories gain in interest if they are supple mented by precise semantical rules specifying the denotation of all terms and the truth conditions of all sentences of the theory. (shrink)

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...) with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)

W. V. Quine assumed that in _Empiricism, Semantics, and Ontology Rudolf Carnap was attempting to dodge commitment to abstract entities--without either renouncing quantification over them or demonstrating their dispensability--by wielding the analytic/synthetic distinction against ontological issues. Quine's interpretation of Carnap's intent--and his criticism of it--is widely endorsed. But Carnap objected, I argue, not to abstract entities, but to his critics' suggestion that empiricism implies nominalism. Quine's and Carnap's views are therefore more akin than Quine ever suspected. Unfortunately, Quine's misinterpretation of (...) Carnap's position has led to a misunderstanding of the implications of Quine's own naturalistic turn for contemporary ontology. (shrink)