This book offers a defence of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at (...) least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focused on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. This book, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized. (shrink)

This paper responds to John Burgess's ‘Mathematics and _Bleak House_’. While Burgess's rejection of hermeneutic fictionalism is accepted, it is argued that his two main attacks on revolutionary fictionalism fail to meet their target. Firstly, ‘philosophical modesty’ should not prevent philosophers from questioning the truth of claims made within successful practices, provided that the utility of those practices as they stand can be explained. Secondly, Carnapian scepticism concerning the meaningfulness of _metaphysical_ existence claims has no force against a _naturalized_ version (...) of fictionalism, according to which our ordinary standards of scientific evidence may show that we have no reason to believe the mathematical existence claims made within the context of our mathematical and scientific theories. (shrink)

For many philosophers not automatically inclined to Platonism, the indispensability argument for the existence of mathematical objectshas provided the best (and perhaps only) evidence for mathematicalrealism. Recently, however, this argument has been subject to attack, most notably by Penelope Maddy (1992, 1997),on the grounds that its conclusions do not sit well with mathematical practice. I offer a diagnosis of what has gone wrong with the indispensability argument (I claim that mathematics is indispensable in the wrong way), and, taking my cue (...) from Mark Colyvan''s (1998) attempt to provide a Quinean account of unapplied mathematics as `recreational'', suggest that, if one approaches the problem from a Quinean naturalist starting point, one must conclude that all mathematics is recreational in this way. (shrink)

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions.

Defends an account of mathematical knowledge in which mathematical knowledge is a kind of modal knowledge. Leng argues that nominalists should take mathematical knowledge to consist in knowledge of the consistency of mathematical axiomatic systems, and knowledge of what necessarily follows from those axioms. She defends this view against objections that modal knowledge requires knowledge of abstract objects, and argues that we should understand possibility and necessity in a primative way.

This discussion note responds to Mark Colyvan’s claim that there is no easy road to nominalism. While Colyvan is right to note that the existence of mathematical explanations presents a more serious challenge to nominalists than is often thought, it is argued that nominalist accounts do have the resources to account for the existence of mathematical explanations whose explanatory role resides elsewhere than in their nominalistic content.

ABSTRACT‘expressionist’ accounts of applied mathematics seek to avoid the apparent Platonistic commitments of our scientific theories by holding that we ought only to believe their mathematics-free nominalistic content. The notion of ‘nominalistic content’ is, however, notoriously slippery. Yablo's account of non-catastrophic presupposition failure offers a way of pinning down this notion. However, I argue, its reliance on possible worlds machinery begs key questions against Platonism. I propose instead that abstract expressionists follow Geoffrey Hellman's lead in taking the assertoric content of (...) empirical science to be irreducibly modal, using the ‘non-interference’ of mathematical objects as justification for detaching nominalistic consequences. (shrink)

Debunking arguments against both moral and mathematical realism have been pressed, based on the claim that our moral and mathematical beliefs are insensitive to the moral/mathematical facts. In the mathematical case, I argue that the role of Hume’s Principle as a conceptual truth speaks against the debunkers’ claim that it is intelligible to imagine the facts about numbers being otherwise while our evolved responses remain the same. Analogously, I argue, the conceptual supervenience of the moral on the natural speaks presents (...) a difficulty for the debunker’s claim that, had the moral facts been otherwise, our evolved moral beliefs would have remained the same. (shrink)

In her recent paper ‘The Epistemology of Propaganda’ Rachel McKinnon discusses what she refers to as ‘TERF propaganda’. We take issue with three points in her paper. The first is her rejection of the claim that ‘TERF’ is a misogynistic slur. The second is the examples she presents as commitments of so-called ‘TERFs’, in order to establish that radical (and gender critical) feminists rely on a flawed ideology. The third is her claim that standpoint epistemology can be used to establish (...) that such feminists are wrong to worry about a threat of male violence in relation to trans women. In Section 1 we argue that ‘TERF’ is not a merely descriptive term; that to the extent that McKinnon offers considerations in support of the claim that ‘TERF’ is not a slur, these considerations fail; and that ‘TERF’ is a slur according to several prominent accounts in the contemporary literature. In Section 2, we argue that McKinnon misrepresents the position of gender critical feminists, and in doing so fails to establish the claim that the ideology behind these positions is flawed. In Section 3 we argue that McKinnon’s criticism of Stanley fails, and one implication of this is that those she characterizes as ‘positively privileged’ cannot rely on the standpoint-relative knowledge of those she characterizes as ‘negatively privileged’. We also emphasize in this section McKinnon’s failure to understand and account for multiple axes of oppression, of which the cis/trans axis is only one. (shrink)

Are there genuine mathematical explanations of physical phenomena, and if so, how can mathematical theories, which are typically thought to concern abstract mathematical objects, explain contingent empirical matters? The answer, I argue, is in seeing an important range of mathematical explanations as structural explanations, where structural explanations explain a phenomenon by showing it to have been an inevitable consequence of the structural features instantiated in the physical system under consideration. Such explanations are best cast as deductive arguments which, by virtue (...) of their form, establish that, given the mathematical structure instantiated in the physical system under consideration, the explanandum had to occur. Against the claims of platonists such as Alan Baker and Mark Colyvan, I argue that formulating mathematical explanations as structural explanations in this way shows that we can accept that mathematics can play an indispensable explanatory role in empirical science without committing to the existence of any abstract mathematical objects. (shrink)

A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be developed on the basis of these and other studies, (...) locating the substance of mathematics in the various informal argument methods used by mathematicians. (shrink)

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply (...) the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism. (shrink)

What should a Quinean naturalist say about moral and mathematical truth? If Quine’s naturalism is understood as the view that we should look to natural science as the ultimate ‘arbiter of truth’, this leads rather quickly to what Huw Price has called ‘placement problems’ of placing moral and mathematical truth in an empirical scientific world-view. Against this understanding of the demands of naturalism, I argue that a proper understanding of the reasons Quine gives for privileging ‘natural science’ as authoritative when (...) it comes to questions of truth and existence also apply to other stable and considered elements of our inherited world-view, including, arguably, our firmly held mathematical and moral beliefs. If so, then the ‘thin’ mathematical and moral realisms of Penelope Maddy and T. M. Scanlon, respectively, are vindicated. We do not need to shoehorn mathematical and moral truths into the pushings and pullings of our empirical scientific world-view; for the busy sailor adrift on Neurath’s boat, mathematical and moral truths already have their place. (shrink)

In this paper we explore the hypothesis that constrained uses of imagination are crucial to economic modeling. We propose a theoretical framework to develop this thesis through a number of specific hypotheses that we test and refine through six new, representative case studies. Our ultimate goal is to develop a philosophical account that is practice oriented and informed by empirical evidence. To do this, we deploy an abductive reasoning strategy. We start from a robust set of hypotheses and leave space (...) for the generation of further hypotheses and theoretical claims based on the qualitative analysis of new empirical data. (shrink)

Fictionalists about an area of discourse take the view that the value of participating in that discourse does not depend on the truth of the sentences one utters in the course of one’s participation. Thus, in the paradigm case of story-telling, the value the utterance ‘Once upon a time, there were three bears |$\ldots$|’ doesn’t rest on its truth but on something else — perhaps the imaginative opportunities that it invites us to entertain. One attraction of fictionalism is that, by (...) seeking reasons to speak ‘as if’ there are |${\phi}$|s that do not depend on the truth of one’s utterances about |${\phi}$|s (and therefore on the existence of |${\phi}$|s), it opens up possibilities for defending participation in a discourse even if one is sceptical about the literal truth of the sentences one therein utters.Fictionalism, as the authors note in their Preface, is a ‘hot topic’. Indeed, they point out, “use of the term has doubled since 1980 — a defining year for fictionalism — and looks as though it may continue to rise” (p. viii). In the context of the philosophy of mathematics, readers of this journal may recognise the significance of 1980 as the publication year for Hartry Field’s Science Without Numbers [1980]. Field’s book was for after many years out of print; its long-awaited second edition was released in 2016, itself a sign that fictionalism is once more en vogue. Although Field did not, in Science Without Numbers, refer to the position developed there as a ‘fictionalist’ account of mathematics, by 1989’s collection of essays, Realism, Mathematics, and Modality he was willing to present the position of SWN as a fictionalist one, with the value of speaking ‘as if’ there are numbers and sets in the context of empirical science being identified, in Field’s account, as grounded in the conservativeness of mathematics over nominalistically stated versions of our scientific theories, not in its truth. And although there has since been widespread scepticism about the prospects for fully carrying out Field’s programme of nominalizing our best scientific theories, mathematical fictionalism has not gone away. In recent years, fictionalist approaches to the use of mathematics in science (including [Balaguer, 1998] and [Leng, 2010]) have been developed that argue that even indispensable uses of mathematics can be accounted for from a perspective that sees mathematics as a useful fiction rather than a body of truths. (shrink)

FieldHartry. Science Without Numbers: A Defense of Nominalism 2nd ed.Oxford University Press, 2016. ISBN 978-0-19-877792-2. Pp. vi + 56 + vi + 111.

(No abstract is available for this citation).

Stewart Shapiro’s ante rem structuralism recognizes the structural or ‘algebraic’ aspects of mathematical practice while still offering a face-value semantics. Fictionalism, as a purely ‘algebraic’ approach, is held to be at a disadvantage, as compared with Shapiro’s structuralism, in not interpreting mathematics at face value. However, the face-value reading of mathematical singular terms has difficulty explaining how we can use such terms to pick out a unique referent in cases where the relevant mathematical structures admit non-trivial automorphisms. Shapiro offers a (...) solution to this difficulty, but his solution, I argue, evens the score between Shapiro’s structuralism and fictionalism. (shrink)

Guest Editor’s introduction to the Monographic Section.