Many long-standing problems pertaining to contemporary philosophy of mathematics can be traced back to different approaches in determining the nature of mathematical entities which have been dominated by the debate between realists and nominalists. Through this discussion conceptualism is represented as a middle solution. However, it seems that until the 20th century there was no third position that would not necessitate any reliance on one of the two points of view. Fictionalism, on the other hand, observes mathematical entities in (...) a radically different way. This is reflected in the claim that the concepts being used in mathematics are nothing but a product of human fiction. This paper discusses the relationship between fictionalism and two traditional viewpoints within the discussion which attempts to successfully determine the ontological status of universals. One of the main points, demonstrated with concrete examples, is that fictionalism cannot be classified as a nominalist position. Since fictionalism is observed as an independent viewpoint, it is necessary to examine its range as well as the sustainability of the implications of opinions stated by their advocates. (shrink)

The theory of truthmaking has long aroused skepticism from philosophers who believe it to be tangled up in contentious ontological commitments and unnecessary theoretical baggage. In this book, Jamin Asay shows why that suspicion is unfounded. Challenging the current orthodoxy that truthmaking's fundamental purpose is to be a tool for explaining why truths are true, Asay revives the conception of truthmaking as fundamentally an exercise in ontology: a means for coordinating one's beliefs about what is true and one's ontological (...) commitments. He goes on to show how truthmaking connects to analyticity, truth, and realism, and how it contributes to debates over nominalism, presentism, mathematical objects, and fictional characters. His book is the most comprehensive exploration to date into what truthmaking is and how it contributes to metaphysical debates across philosophy, and will interest a wide range of readers in metaphysics and beyond. (shrink)

The fundamental idea of a Neoaristotelian inherence ontology of mathematical entities parallels that of an Aristotelian approach to the ontology of universals. It is proposed that mathematical objects are nominalizations especially of dimensional and related structural properties that inhere as formal species and hence as secondary substances of Aristotelian primary substances in the actual world of existent physical spatiotemporal entities. The approach makes it straightforward to understand the distinction between pure and applied mathematics, and the otherwise enigmatic success (...) of applied mathematics in the natural sciences. It also raises an interesting set of challenges for conventional mathematics, and in particular for the ontic status of infinity, infinite sets and series, infinitesimals, and transfinite cardinalities. The final arbiter of all such questions on an Aristotelian inherentist account of the nature of mathematical entities are the requirements of practicing scientists for infinitary versus strictly finite mathematics in describing, explaining, predicting and retrodicting physical spatiotemporal phenomena. Following Quine, we classify all mathematics that falls outside of this sphere of applied scientific need as belonging to pure, and, with no prejudice or downplaying of its importance, ‘recreational’, mathematics. We consider a number of important problems in the philosophy of mathematics, and indicate how a Neoaristotelian inherence metaphysics of mathematical entities provides a plausible answer to Benacerraf’s metaphilosophical dilemma, pitting the semantics of mathematical truth conditions against the epistemic possibilities for justifying an abstract realist ontology of mathematical entities and truth conditions. (shrink)

Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘π is irrational’, are to be interpreted at face value and, thus interpreted, are false. Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Fictionalism is a nominalist (or anti-realist) account of mathematics in that it denies the existence of a realm of (...) abstract mathematical entities. It should be contrasted with mathematical realism (or Platonism) where mathematical statements are taken to be true, and, moreover, are taken to be truths about mathematical entities. Fictionalism should also be contrasted with other nominalist philosophical accounts of mathematics that propose a reinterpretation of mathematical statements, according to which the statements in question are true but no longer about mathematical entities. Fictionalism is thus an error theory of mathematical discourse: at face value mathematical discourse commits us to mathematical entities and although we normally take many of the statements of this discourse to be true, in doing so we are in error (cf. error theories in ethics). (shrink)

Mathematical practice prompts theories about aprioricity, necessity, abstracta, and non-causal epistemic connections. But it is not clear what to count as the data: mathematical necessity or the appearance of mathematical necessity, abstractness or apparent abstractness, a prioricity or apparent aprioricity. Nor is it clear whether traditional metaphysical theories provide explanation or idle redescription. This paper suggests that abstract objects, rather than doing explanatory work, provide codifications of the data to be explained. It also suggests that traditional rivals—conceptualism, nominalism, realism—engage different (...) explanatory projects, and thus are not rivals at all. (shrink)

Are there any universal entities? Or is the world populated only by particular things? The problem of universals is one of the most fascinating and enduring topics in the history of metaphysics, with roots in ancient and medieval philosophy. This collection of new essays provides an innovative overview of the contemporary debate on universals. Rather than focusing exclusively on the traditional opposition between realism and nominalism, the contributors explore the complexity of the debate and illustrate a broad range of (...) positions within both the realist and the nominalist camps. Realism is viewed through the lens of the distinction between constituent and relational ontologies, while nominalism is reconstructed in light of the controversy over the notion of trope. The result is a fresh picture of contemporary metaphysics, in which traditional strategies of dealing with the problem of universals are both reaffirmed and called into question. (shrink)

The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine's "Indispensability Argument", which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge (...) the target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance pure mathematics. I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field's nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue one should guard against. I shall support my reasons by providing grounds for rejecting both Field's fictionalism and also his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability Argument. I shall then give a refutation of Mark Balaguer's argument for his thesis that fictionalism is "the best version of anti-realistic anti-platonism". (shrink)

What is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal logic (...) it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist. (shrink)

The author reviews and summarizes, in as jargon-free way as he is capable of, the form of anti-platonist anti-nominalism he has previously developed in works since the 1980s, and considers what additions and amendments are called for in the light of such recently much-discussed views on the existence and nature of mathematical objects as those known as hyperintensional metaphysics, natural language ontology, and mathematical structuralism.

This rich book differs from much contemporary philosophy of mathematics in the author’s witty, down to earth style, and his extensive experience as a working mathematician. It accords with the field in focusing on whether mathematical entities are real. Franklin holds that recent discussion of this has oscillated between various forms of Platonism, and various forms of nominalism. He denies nominalism by holding that universals exist and denies Platonism by holding that they are concrete, not abstract - looking to (...) Aristotle for inspiration. (shrink)

This compact volume, belonging to the Cambridge Elements series, is a useful introduction to some of the most fundamental questions of philosophy and foundations of mathematics. What really distinguishes realist and platonist views of mathematics from anti-platonist views, including fictionalist and nominalist and modal-structuralist views?1 They seem to confront similar problems of justification, presenting tradeoffs between which it is difficult to adjudicate. For example, how do we gain access to the abstract posits of platonist accounts of arithmetic, analysis, geometry, (...) etc., including numbers, functions, sets, points, lines, spaces, etc., whether it be such objects themselves or the possibilities of such, as postulated by modal and modal structural accounts?2 What real difference does it make, whether it be the existence of such things or the mathematical possibility of such things? Even fictionalist views seem to confront analogous problems. After all, we rely on mathematics for myriad scientific applications; so such mathematics had better at least be coherent, even if not true. But coherence requires at least formal consistency; so we seem implicitly to be committed to things like formal derivations, viz. the absence of any such having contradictory consequences, framed within the appropriate system of axioms and rules. But derivations are themselves akin to mathematical objects, as derivations are strings of strings of symbols. Moreover, derivations provided by axioms and rules form an infinite class, such that, at any future time, infinitely many will not have been written down. Moreover, such strings, as Gödel famously showed, can be coded as natural numbers. Thus, even fictionalist accounts (such as that of Field in his Science without Numbers [2016]) seem to confront problems very much like those plaguing platonist accounts. (shrink)

This book is an exploration and defense of the coherence of classical theism’s doctrine of divine aseity in the face of the challenge posed by Platonism with respect to abstract objects. A synoptic work in analytic philosophy of religion, the book engages discussions in philosophy of mathematics, philosophy of language, metaphysics, and metaontology. It addresses absolute creationism, non-Platonic realism, fictionalism, neutralism, and alternative logics and semantics, among other topics. The book offers a helpful taxonomy of the wide (...) range of options available to the classical theist for dealing with the challenge of Platonism. It probes in detail the diverse views on the reality of abstract objects and their compatibility with classical theism. It contains a most thorough discussion, rooted in careful exegesis, of the biblical and patristic basis of the doctrine of divine aseity. Finally, it challenges the influential Quinean metaontological theses concerning the way in which we make ontological commitments. (shrink)

In An Aristotelian Realist Philosophy of Mathematics Franklin develops a tantalizing alternative to Platonist and nominalist approaches by arguing that at least some mathematical universals exist in the physical realm and are knowable through ordinary methods of access to physical reality. By offering a third option that lies between these extreme all-or-nothing approaches and by rejecting the ‘dichotomy of objects into abstract and concrete’, Franklin provides potential solutions to many of these traditional problems and opens up a whole new (...) terrain for debate in the philosophy of mathematics. (shrink)

Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed more (...) briefly in section 6. (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)

Aldo Antonelli offers a novel view on abstraction principles in order to solve a traditional tension between different requirements: that the claims of science be taken at face value, even when involving putative reference to mathematical entities; and that referents of mathematical terms are identified and their possible relations to other objects specified. In his view, abstraction principles provide representatives for equivalence classes of second-order entities that are available provided the first- and second-order domains are in the equilibrium dictated by (...) the abstraction principles, and whose choice is otherwise unconstrained.entities are the referents of abstraction terms: such referents are to an extent indeterminate, but we can still quantify over them, predicate identity or non-identity, etc. Our knowledge of them is limited, but still substantial: we know whatever has to be true no matter how the representatives are chosen, i.e., what is true in all models of the corresponding abstraction principles. This view is backed up by an “austere” conception of universals, according to which these are first-order objects, i.e., ways of collecting first-order objects. Antonelli thus claims that second-order logic does not import any novel ontological commitment beyond ontology of first-order, naturalistically acceptable, objects. Moreover, in the case of arithmetic, even if Hume’s Principle is construed as described above, Frege’s Theorem goes through unaffected, since there is nothing in its proof that depends on an account of the “true nature” of numbers. A viable construal of logicism is then given by the combination of semantic nominalism and a naturalistic conception of abstraction. [Editors note]. (shrink)

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)

Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics are considered and rejected. (...) Constructive empiricism cannot be realist about abstract objects; it must reject even the realism advocated by otherwise ontologically restrained and epistemologically empiricist indispensability theorists. Indispensability arguments rely on the kind of inference to the best explanation the rejection of which is definitive of constructive empiricism. On the other hand, formalist and logicist anti-realist positions are also shown to be untenable. It is argued that a constructive empiricist philosophy of mathematics must be fictionalist. Borrowing and developing elements from both Philip Kitcher's constructive naturalism and Kendall Walton's theory of fiction, the account of mathematics advanced treats mathematics as a collection of stories told about an ideal agent and mathematical objects as fictions. The account explains what true portions of mathematics are about and why mathematics is useful, even while it is a story about an ideal agent operating in an ideal world; it connects theory and practice in mathematics with human experience of the phenomenal world. At the same time, the make-believe and game-playing aspects of the theory show how we can make sense of mathematics as fiction, as stories, without either undermining that explanation or being forced to accept abstract mathematical objects into our ontology. All of this occurs within the framework that constructive empiricism itself provides the epistemological limitations it mandates, the semantic view of theories, and an emphasis on the pragmatic dimension of our theories, our explanations, and of our relation to the language we use. (shrink)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical (...) world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

The aim of this work is to respond to the following question: how did Charles S. Peirce find unity for his pragmatist philosophy through the formulation of Scholastic Realism? The author proposes the said doctrine to be a reading guide, leading us through the different stages of Peirce's work as a philosopher. By understanding his realist doctrine, we can see why he believed it was a viable theory for understanding the problem of Universals. This book demonstrates why, in Peirce's (...) mind, such a problem has pervaded the history of philosophy. The author's line of argument reveals that Scholastic Realism is crucial to the understanding of his philosophy, which is a new approach in Peirce scholarship. It provides a useful framework for asking questions about reality in the same way that Peirce himself did. As a result, the author shows that Peirce's realism addresses different yet related philosophical problems, leading Peirce to brand the final version of his philosophy as «Scientific Metaphysics». The conclusion offers an interpretation of the Scholastic Realism principle as a solution to Peirce's concerns - a useful idea to achieve a better theory of reality in his struggle to realize metaphysics a posteriori. Peirce's doctrine is presented alongside some of its uses, especially in the fields of abstraction theory, and also in the fundamental principles of mathematics. This work should advance our comprehension of the problems related to Peirce's philosophy as well as shedding light on pragmatism and its origins as well as the battle between realism and nominalism. (shrink)

This essay explores the use of platonist and nominalist concepts, derived from the philosophy of mathematics and metaphysics, as a means of elucidating the debate on spacetime ontology and the spatial structures endorsed by scientific realists. Although the disputes associated with platonism and nominalism often mirror the complexities involved with substantivalism and relationism, it will be argued that a more refined three-part distinction among platonist/nominalist categories can nonetheless provide unique insights into the core assumptions that underlie spatial ontologies, (...) but it also assists in critiquing alternative uses of nominalism, platonism, and both ontic and epistemic structural realism. (shrink)

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining (...) abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. (shrink)

PLATONISM or platonic realism in logic and mathematics is probably the most widespread contemporary view in the philosophy of mathematics. It has become popularly identified with the acceptance of an ontology of sets and/or classes as fundamental among the building materials of the cosmos and of all that is therein. Usually, also, these entities are regarded as "abstract" rather than "concrete," but no one has given us a sufficiently detailed and acceptable theory as to how this dichotomy is (...) to be drawn. Sets and classes are taken somehow as the paradigm of abstract entities, as over and against something called 'individuals', which are then the paradigm of concrete entities. This is all very unsatisfactory and in need of much analysis and clarification. (shrink)

Metaphysical anti-realism is a large and heterogeneous group of views that do not share a common thesis but only share a certain family resemblance. Views as different as mathematical nominalism—the view that numbers do not exist—, ontological relativism—the view that what exists depends on a perspective—, and modal conventionalism—-the view that modal facts are conventional—all are versions of metaphysical anti-realism. As the latter two examples suggest, relativist ideas play a starring role in many versions of metaphysical anti-realism. But what does (...) it mean for the existence of something to “depend on” a perspective, or for a modal fact to “depend on” a convention? We can distinguish between various dependence relations, giving rise to an array of drastically different forms of metaphysical anti-realism. This article offers a guided tour. I develop a systematic distinction between various forms of metaphysical anti-realism with a focus on the role of relativist ideas in this landscape. (shrink)

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining (...) abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. (shrink)

Peirce's realism is a sophisticated realism inherited from the Avicennian Scotistic tradition, which may be briefly characterized by its opposition to metaphysical realism (Platonism) and various forms of nominalism. In this chapter, I consider how Peirce's realism fits his approach to mathematics, which is often presented as a somewhat incoherent mixture of Platonistic and conceptualistic elements. Without denying these, I claim that Peirce's subtle position not only helps to clear up some of these so-called inconsistencies but offers many insights for (...) contemporary ways of dealing with the mathematical aspects of the problem of universals. (shrink)

A Critical Introduction to Fictionalism provides a clear and comprehensive understanding of an important alternative to realism. Drawing on questions from ethics, the philosophy of religion, art, mathematics, logic and science, this is a complete exploration of how fictionalism contrasts with other non-realist doctrines and motivates influential fictionalist treatments across a range of philosophical issues. Defending and criticizing influential as well as emerging fictionalist approaches, this accessible overview discuses physical objects, universals, God, moral properties, numbers and other fictional entities. (...) Where possible it draws general lessons about the conditions under which a fictionalist treatment of a class of items is plausible. Distinguishing fictionalism from other views about the existence of items, it explains the central features of this key metaphysical topic. Featuring an historical survey, definitions of key terms, characterisations of important subdivisions, objections and problems for fictionalism, and contemporary fictionalist treatments of several issues, A Critical Introduction to Fictionalism is a valuable resource for students of metaphysics as well as students of philosophical methodology. It is the only book of its kind. (shrink)

This volume is devoted to problems within analytic metaphysics. It defends an ontology and theory of categories inspired by Aristotle, but revised in such a way as to be compatible with modern science. The ontology of both natural and social reality is addressed, starting out from the view that universals exist but only in the spatiotemporal world. In attempting to bring Aristotle's ontology up-to-date, the author relies very much on the thinking of Edmund Husserl, conceiving the cement (...) of the universe as Husserlian relations of existential dependence and regarding intentionality as a non-reducible category in the ontology of mind. The work is thoroughly realistic in spirit, but large parts of it should nonetheless be of interest to conceptualists and nominalists, too. (shrink)

Editorial NoteThe following Discussion Note is an edited transcription of the discussion on G. Aldo Antonelli’s paper “Semantic Nominalism: How I Learned to Stop Worrying and Love Universals”, held among participants at the IHPST-UC Davis Workshop Ontological Commitment in Mathematics which took place, in memoriam of Aldo Antonelli, at IHPST in Paris on December, 14–15, 2015. The note’s and volume’s editors would like to thank all participants in the discussion for their contributions, and Alberto Naibo, Michael Wright and the personnel (...) at IHPST for their technical support. (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares (...) these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

The author gives a critical and comprehensive study of the fundamental problem of universals in Indian Philosophy. The centre of the study is the controversy between the Nyaya-Vaisesika and the Mimamsa realists on the one hand and the Buddhist nominalists on the other. The author discusses not only the epistemological and metaphysical approach to the problem of universals but also the semantic approach made by the various systems of Indian Philosophy. In this context the view of the Grammarions (...) with special reference to Bhartrhari has been discussed in some detail. A brief but critical analysis of some of the main trends of thought on universals in Western Philosophy--beginning from Pluto to the contemporary philosophers--has also been given. Besides his scholarly and eminently readable treatment of fundamental problem of universals, the author has attempted to give his own solution of the problem. It is based on the recurrent identities and similarities which are the principles of grouping and which form the foundation of our thought and speech. (shrink)

On the relations of universals and particulars, by B. Russell.--Universals and resemblances, by H. H. Price.--On concept and object, by G. Frege.--Frege's hidden nominalism, by G. Bergmann.--Universals, by F. P. Ramsey.--Universals and metaphysical realism, by A. Donagan.--Universals and family resemblances, by R. Bambrough.--Particular and general, by P. F. Strawson.--The nature of universals and propositions, by G. F. Stout.--Are characteristics of particular things universal or particular? By G. E. Moore and G. F. Stout.--The relation of resemblance, by P. Butchvarov.--Qualities, by N. (...) Wolterstroff.--On what there is, by W. V. Quine.--Empiricism, semantics, and ontology, by R. Carnap.--The languages of realism and nominalism, by R. B. Brandt.--Grammar and existence: a preface to ontology, by W. Sellars.--A world of individuals, by N. Goodman.--Bibliographical notes (p. [307]-308). (shrink)

From the perspective of mathematical practice, I examine positions claiming that mathematical objects are introduced by human agents. I consider in particular mathematical fictionalism and a recent position on social ontology formulated by Cole (2013, 2015). These positions are able to solve some of the challenges that non-realist positions face. I argue, however, that mathematical entities have features other than fictional characters and social institutions. I emphasise that the way mathematical objects are introduced is different and point to the (...) multifaceted role that relations and interconnections play in this context. Finally, I argue that mathematical entities can be considered to be pragmatically real. (shrink)

This monograph details a new solution to an old problem of metaphysics. It presents an improved version of Ostrich Nominalism to solve the Problem of Universals. This innovative approach allows one to resolve the different formulations of the Problem, which represents an important meta-metaphysical achievement. In order to accomplish this ambitious task, the author appeals to the notion and logic of ontological grounding. Instead of defending Quine’s original principle of ontological commitment, he proposes the principle of grounded ontological commitment. This (...) represents an entirely new application of grounding. Some metaphysicians regard Ostrich Nominalism as a rejection of the problem rather than a proper solution to it. To counter this, the author presents solutions for each of the formulations. These include: the problem of predication, the problem of abstract reference, and the One Over Many as well as the Many Over One and the Similar but Different variants. This book will appeal to anyone interested in contemporary metaphysics. It will also serve as an ideal resource to scholars working on the history of philosophy. Many will recognize in the solution insights resembling those of traditional philosophers, especially of the Middle Ages. (shrink)

Nominalism faces the task of explaining away the ontological commitments of applied mathematical statements. This paper reviews an argument from the philosophy of logic that focuses on this task and which has been used as an objection to certain specific formulations of nominalism. The argument as it is developed in this paper aims to show that nominalism in general does not have the epistemological advantages its defendants claim it has. I distinguish between two strategies that are available to the (...) nominalist: The Evaluation Programme, which tries to preserve the common truth‐values of mathematical statements even if there are no mathematical objects, and Fictionalism, which denies that mathematical sentences have significant truth‐values. It is argued that the tenability of both strategies depends on the nominalist’s ability to account for the notion of consequence. This is a problem because the usual meta‐logical explications of consequence do themselves quantify over mathematical entities. While nominalists of both varieties may try to appeal to a primitive notion of consequence, or, alternatively, to primitive notions of logical or structural possibilities, such measures are objectionable. Even if we are equipped with a notion of either consequence or possibility that is primitive in the relevant sense, it will not be strong enough to account for the consequence relation required in classical mathematics. These examinations are also useful in assessing the possible counter‐intuitive appeal of the argument from the philosophy of logic. (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)

What are words? What makes two token words tokens of the same word-type? Are words abstract entities, or are they (merely) collections of tokens? The ontology of words tries to provide answers to these, and related questions. This article provides an overview of some of the most prominent views proposed in the literature, with a particular focus on the debate between type-realist, nominalist, and eliminativist ontologies of words.

I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism.

Review of James Franklin: An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure, Palgrave Macmillan, 2014, x + 308 pp.

As the subtitle and consecutive division of contents indicate, these two volumes are integral parts of a single work and one may wonder why they were not published as such since the indices and bibliography in the second volume refers to both works. The basic tripartite thesis of the combined volumes may be stated thus. Both universal properties and universal relationships exist independently of the classifying mind, but not in factual independence of particulars; what universals in fact exist, however, must (...) be determined not a priori by considering what predicates are applied to particulars by linguistic convention, but a posteriori according to the norms of an empiricistic epistemology. Hence, Armstrong calls his theory "a posteriori," or simply, "scientific realism," since it is the task of total science, conceived of as a total inquiry, to determine which monadic or polyadic predicates do correspond to universals and which do not. The first volume is essentially a critical appraisal of classical approaches to the nominalism-realism controversy, whereas the second is a more positive presentation of the author’s own theory and grows out of the prior critique. Thus the second volume can be read independently of the first, especially since a brief recapitulation of the argument of volume one is given at its beginning. Similarly, a brief summary of the argument of volume two is appended to volume one. Thus the two volumes, which apparently grew out of materials presented in independent, but related, seminars, could still be used as such. Only this and the possibility of a student buying the overly expensive work by parts seems to justify the peculiar way in which this set was published. (shrink)

The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such (...) objects in our discourse as though they are independent of time and space; finding a place for such objects in a broader framework of thought is a central task of ontology, or metaphysics. The rigor and precision of mathematical language depends on the fact that it is based on a limited vocabulary and very structured grammar, and semantic accounts of mathematical discourse often serve as a starting point for the philosophy of language. Although mathematical thought has exhibited a strong degree of stability through history, the practice has also evolved over time, and some developments have evoked controversy and debate; clarifying the basic goals of the practice and the methods that are appropriate to it is therefore an important foundational and methodological task, locating the philosophy of mathematics within the broader philosophy of science. (shrink)

Introduction: mathematization and the language of nature -- Realists and nominalists : language and mathematics before the scientific revolution -- Ontology recapitulates epistemology : Gassendi, epicurean atomism, and nominalism -- British empiricism, nominalism, and constructivism -- Three mathematicians : constructivist epistemology and the new mathematical methods -- Conclusion: mathematization and the nature of language.

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up (...) naturalisation of Badiou’s philosophical approach to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (shrink)

In a lot of domains in metaphysics the tacit assumption has been that whichever metaphysical principles turn out to be true, these will be necessarily true. Let us call necessitarianism about some domain the thesis that the right metaphysics of that domain is necessary. Necessitarianism has flourished. In the philosophy of maths we find it held that if mathematical objects exist, then they do of necessity. Mathematical Platonists affirm the necessary existence of mathematical objects (see for instance Hale and (...) Wright 1992 and 1994; Wright 1983 and 1988; Schiffer 1996; Resnik 1997; Shapiro 1997 and Zalta 1988) while mathematical nominalists, usually in the form of fictionalists, hold that necessarily such objects fail to exist (see for instance Balaguer 1996 and 1998; Rosen 2001 and Yablo 2005). In metaphysics more generally, until recently it was more or less assumed that whatever the right account of composition—the account of under what conditions some xs compose a y—that account will be necessarily true (for a discussion of theories of composition see Simons 1987 and van Inwagen 1987 and 1990; the modal status of the composition relation is explicitly addressed in Schaffer 2007; Parsons 2006 and Cameron 2007). Similarly, it has generally been assumed that whatever the right account is of the nature of properties, whether they be universals, tropes, or whether nominalism is true, that account will be necessarily true (though see Rosen 2006 for a recent suggestion to the contrary). In considering theories of persistence it has been widely held that whether objects endure or perdure through time is a matter of necessity (Sider 2001; though see Lewis 1999 p227 who defends contingent perdurantism). And with respect to theories of time it is frequently held that whichever of the A- or B-theory is true is necessarily true. A-theorists often argue that there is time in a world only if the A-theory is true at that world (see for instance McTaggart 1903; Markosian 2004; Bigelow 1996; Craig 2001) while B-theorists often argue that the A-theory is internally inconsistent (Smart 1987; Mellor 1998; Savitt 2000 and Le Poidevin 1991). Once again, we find a few recent contingentist dissenters. Bourne (2006) suggests that it is a contingent feature of time that it is tensed, and thus that the A-theory is contingently true. Worlds in which there exist only B-theoretic properties are worlds with time, it is just that time in those worlds time is radically different to the way it is actually. Other defenders of the B-theory, though not expressly contingentists, do offer arguments against versions of the A-theory that try to show that such A-theories theories are inconsistent with the actual laws of nature (see for instance Saunders 2002 and Callender 2000); these arguments, at least, leave room for the possibility that the A-theories in question are contingently false (at least on the assumption that the laws of nature are themselves contingent, an assumption that not everyone accepts). Despite some notable exceptions, necessitarianism has flourished in many, if not most, domains in metaphysics. One such exception is Lewis’ famous defence of Humean supervenience as a contingent claim about our world. Lewis does not argue that necessarily, the supervenience base for all matters of fact in a world is nothing but a vast mosaic of local matters of particular fact. Rather, he thinks that we have reason to think that our world is one in which Humean supervenience holds (see Lewis 1986 p9-10 and 1994). Another exception to the necessitarian orthodoxy is to be found in the lively debate about the modal status of the laws of nature. Here, if anything, contingentism has been the dominating force, with it generally being held that there are possible worlds in which different laws of nature hold (this view is defended by, among others, Lewis 1986 and 2010; Schaffer 2005 and Sidelle 2002). Necessitarian dissenters hold that the laws of nature are necessary, frequently because they think it is necessary that fundamental properties have the causal or nomic profiles they do (see for instance Shoemaker 1980 and 1988; Swoyer 1982; Bird 1995; Ellis and Lierse 1994). Nevertheless, when it comes to thinking about the nature of the laws themselves, the necessitarian presumption is back on firm footing. Though there is disagreement about whether the laws are generalisations that feature in the most virtuous true axiomatisation of all the particular matters of fact (often known as the Humean view of laws and defended by Ramsey 1978; Lewis 1986 and Beebee 2000) or whether laws are relations of necessity that hold between universals (a view defended by Armstrong 1983; Dretske 1977; Tooley 1977 and Carroll 1990) no one has seriously suggested that it might be a contingent matter which of these is the right account of laws. The necessitarian orthodoxy is not surprising since metaphysics is largely an a priori process. While a priori reasoning may be used to determine whether a proposition is necessary or contingent, it is not well placed (in the absence of a posteriori evidence) to determine whether a contingent proposition is actually true or false. Since metaphysicians aim to tell us which principles are true in which worlds, on the face of it the discovery that metaphysical principles are contingent seems to make part of the task of metaphysics epistemically intractable. In what follows I consider two reasons one might end up embracing contingentism and whether this would lead one into epistemic difficulty. The following section considers a route to contingent metaphysical truths that proceeds via a combination of conceptual necessities and empirical discoveries. Section 3 considers whether there might be synthetic contingent metaphysical truths, and the final section raises the question of whether if there were such truths we would be well placed to come to know them. (shrink)

Fictionalism is the view that a serious intellectual inquiry need not aim at truth. It came to prominence in philosophy in 1980, when Hartry Field argued that mathematics does not have to be true to be good, and Bas van Fraassen argued that the aim of science is not truth but empirical adequacy. Both suggested that the acceptance of a mathematical or scientific theory need not involve belief in its content. Thus the distinctive commitment of fictionalism is that acceptance (...) in a given domain of inquiry need not be truth-normed, and that the acceptance of a sentence from the associated region of discourse need not involve belief in its content. In metaphysics fictionalism is now widely regarded as an option worthy of serious consideration. This volume represents a major benchmark in the debate: it brings together an impressive international team of contributors, whose essays represent the state of the art in various areas of metaphysical controversy, relating to language, mathematics, modality, truth, belief, ontology, and morality. (shrink)

Finally the book concludes with a discussion of the most recent debates between realists and nominalists.

Bridges’s “moderate realism” is really a misnomer, since Aquinas’s view was that mathematical objects and universals are mere entia rationis, making Bridges’s view antirealist. The metaphysical idleness of properties on van Inwagen’s view ought to motivate reexamination of his presumed criterion of ontological commitment. Regarding paraphrastic strategies, one can meet van Inwagen’s challenge to provide a nominalistically acceptable paraphrase of Euclid’s proof of exactly five Platonic solids. Concerning fictionalism, van Inwagen should allow the anti-Platonist to treat abstracta as he treats (...) supposed composite, inanimate objects. Finally, van Inwagen too quickly dismisses the absolute creationist view that abstracta can be effects, if not causes. (shrink)

Is structure a fundamental and indispensable part of the world? Is the question of ontology a question about structure? Structure is a central notion in contemporary metaphysics [Sider 2011. Writing the Book of the World. Oxford: Clarendon Press]. Realism about structure claims that the question of ontology is about the fundamental and indispensable structure of the world. In this paper, I present a criticism of the metaphysics of realism about structure based on a version of Russell’s famous regress (...) argument against nominalism [Russell 1911. “On the Relation of Universals and Particular.” In Logic & Knowledge. Reprint, London: George Allen & Unwin]. First, I argue that the three general tests for the fundamentality of structure proposed by realism about structure rely on a particular empirical test for structure, namely, the so-called ‘similarity test for structure.’ Second, I argue that the similarity test is not well-founded because it leads to a vicious regress. Third, I argue that the regress affects the whole metaphysics of realism about structure, and that no structural notion can be said to be fundamental in connection with any of the other tests. Lastly, I argue that the question of ontology as a question about structure is not substantive. (shrink)