In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.

Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it is argued that the modality (...) of MS or a close cousin appears at crucial junctures in both STS and SGS, so that the above outcome is not obviously tendentious. (shrink)

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

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

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

Words form a fundamental basis for our understanding of linguistic practice. However, the precise ontology of words has eluded many philosophers and linguists. A persistent difficulty for most accounts of words is the type-token distinction [Bromberger, S. 1989. “Types and Tokens in Linguistics.” In Reflections on Chomsky, edited by A. George, 58–90. Basil Blackwell; Kaplan, D. 1990. “Words.” Aristotelian Society Supplementary Volume LXIV: 93–119]. In this paper, I present a novel account of words which differs from the atomistic and platonistic (...) conceptions of previous accounts which I argue fall prey to this problem. Specifically, I proffer a structuralist account of linguistic items, along the lines of structuralism in the philosophy of mathematics [Shapiro, S. 1997. Philosophy of Mathematics: Structure and Ontology. Oxford University Press], in which words are defined in part as positions in larger linguistic structures. I then follow Szabò [1999. “Expressions and Their Representations.” The Philosophical Quarterly 49 (195): 145–163] and Parsons [1990. “The Structuralist View of Mathematical Objects.” Synthese 84: 303–346] in further defining words as quasi-concrete objects according to a representation relation. This view aims for general correspondence with contemporary generative linguistic approaches to the study of language. (shrink)

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

How do axioms, or first principles, in ethics compare to those in mathematics? In this companion piece to G.C. Field's 1931 "On the Role of Definition in Ethics", I argue that there are similarities between the cases. However, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.

The term ‘continuous’ in real analysis wasn’t given an adequate formal definition until 1817. However, important theorems about continuity were proven long before that. How was this possible? In this paper, I introduce and refine a proposed answer to this question, derived from the work of Frank Jackson, David Lewis and other proponents of the ‘Canberra plan’. In brief, the proposal is that before 1817 the meaning of the term ‘continuous’ was determined by a number of ‘platitudes’ which had some (...) special epistemic status. (shrink)

This is my entry on natural language ontology in the Stanford Encyclopedia of Philosophy.

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)

I will argue that the ontological doctrine of physicalism inevitably entails the denial that there is anything conceptual in logic and mathematics. The elements of a formal system, even if they are tagged by suggestive names, are merely meaningless parts of a physically existing machinery, which have nothing to do with concepts, because they have nothing to do with the actual things. The only situation in which they can become meaning-carriers is when they are involved in a physical theory. But (...) in this role they refer to elements of the physical reality, i.e. they represent a physical concept. “Mathematical concepts” are just idols, that philosophy can completely deny and physics can completely ignore. (shrink)

C. S. Jenkins has recently proposed an account of arithmetical knowledge designed to be realist, empiricist, and apriorist: realist in that what’s the case in arithmetic doesn’t rely on us being any particular way; empiricist in that arithmetic knowledge crucially depends on the senses; and apriorist in that it accommodates the time-honored judgment that there is something special about arithmetical knowledge, something we have historically labeled with ‘a priori’. I’m here concerned with the prospects for extending Jenkins’s account beyond arithmetic—in (...) particular, to set theory. After setting out the central elements of Jenkins’s account and entertaining challenges to extending it to set theory, I conclude that a satisfactory such extension is unlikely. (shrink)

Scanlon’s Being Realistic about Reasons (BRR) is a beautiful book – sleek, sophisticated, and programmatic. One of its key aims is to demystify knowledge of normative and mathematical truths. In this article, I develop an epistemological problem that Scanlon fails to explicitly address. I argue that his “metaphysical pluralism” can be understood as a response to that problem. However, it resolves the problem only if it undercuts the objectivity of normative and mathematical inquiry.

It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe. In this paper, I am going to challenge this claim by taking seriously the idea that we can (...) talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched. (shrink)

According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical literature is that sets are somehow constituted by their members. In the first part of the paper, I present a number of problems for this answer, paying special attention to the view that sets are metaphysically dependent upon their members. In the second part of the paper, (...) I outline a different approach, which circumvents these problems by dispensing with the priority or dependence relation altogether. Along the way, I show how this approach enables the mathematical structuralist to defuse an objection recently raised against her view. (shrink)

En esta nota crítica (i) se hace una breve descripción de cada uno de los artículos que componen Orayen: de la forma lógica al significado, (ii) se señalan algunas cuestiones que no están claras en ellos o en las réplicas de Orayen y, (iii) en la medida de lo posible, se indica si los autores desarrollan ulteriormente los problemas abordados en sus artículos. The aim of this critical note is threefold: (i) it briefly describes and comments on each of the (...) articles of Orayen: de la forma lógica al significado; (ii) identifies some issues that may not be clear enough or not fully developed whether in the articles or even in Orayen's replies; (iii) as far as possible, it refers to further studies made by the authors themselves on the same, or quite related, subjects addressed by them in their papers. (shrink)

The Aristotelian Society’s Virtual Issue is a free, online publication, made publically available on the Aristotelian Society website. Each volume is theme-based, collecting together papers from the archives of the Proceedings of the Aristotelian Society and the Proceedings of the Aristotelian Society Supplementary Volume that address the chosen theme. This year's Virtual Issue includes a selection of papers from across the Society’s fourteen decades, each accompanied by a specially commissioned present-day response. The aim of the volume is to aid reflection (...) on how to go about doing moral and political philosophy, by bringing together papers that bear on their epistemology and methodology. In some cases this involves pieces whose declared focus is on one of those areas, as is the case with Margaret MacDonald’s 1941 essay ‘The Language of Political Theory’ and Mary Midgley’s 1974 article ‘The Neutrality of the Moral Philosopher’. In other cases the editor, Ben Colburn, selected papers whose primary focus is elsewhere, but which include some interesting epistemological or methodological reflections or implications along the way. The contributions from our present-day commentators vary similarly in how far their explicit attention is methodological. Nevertheless, there is throughout a concern on how to ‘be an honest reasoner’, to borrow a phrase from the late Dudley Knowles. (shrink)