Mary Leng

Nicole Vincent makes the argument that the gender incongruence thesis is ‘conceptually incoherent, insidiously regressive, and hostile to diversity’ (Vincent 2023: 215). She holds that the idea that there is such thing as congruence or incongruence between experienced gender (or gender identity) and sex is regressive and hostile to diversity on the grounds that it rests on a regressive normative assumption that experienced gender and sex ought to align. She holds further that the gender incongruence thesis is conceptually incoherent on the grounds that sex and experienced gender are conceptually distinct and can vary independently of each other. If this is true, then combinations of sex and experienced gender can be more or less common, but not congruent or incongruent. Finally she proposes an alternative diagnostic assessment of transgender people suffering gender dysphoria, holding that gender dysphoria should be understood as involving a feeling of incongruence between sex and experienced gender, and comments on the implications of this understanding for medical treatment of this condition.We argue that Vincent is wrong in thinking that the incongruence thesis is conceptually incoherent. Nevertheless, we argue that on at least one plausible analysis of incongruence, incongruence itself, as opposed to an individual’s distress at the presence of incongruence, is not something in need of treatment, including medical interventions. Thus, we conclude, Vincent’s proposals concerning medical treatment receive some support, even if the gender incongruence thesis holds.

On the face of it, the same motivations that lead some philosophers to adopt mathematical fictionalism seem also to push in the direction of moral fictionalism. In particular, to the extent that mathematical fictionalists are motivated by epistemological concerns about our ability to know truths about abstract mathematical objects, one might expect them to be similarly worried about our ability to know “queer” moral facts. However, the author argues that existing versions of moral fictionalism fare less well than existing versions of mathematical fictionalism in answering two key questions for fictionalists about a domain _D_, (1) _what_ is it that the _D_-discourse is being used to do, and (2) _why_ should we expect _D_-talk to be useful in this way if we do not believe our _D_-claims to be true? The author closes by presenting considerations that suggest that the combination of mathematical fictionalism with moral realism might be motivated on naturalist grounds.

According to an argument developed by Enoch in his book _Taking Morality Seriously_ (OUP 2011) we have reason to believe in the existence of irreducible normative truths based on the indispensability of such truths to the project of practical deliberation. This argument parallels a more venerable indispensability argument in the philosophy of mathematics, which takes the indispensability of mathematical objects to the practice of scientific theorizing to be grounds for belief in the former. This chapter explores the parallels between the two types of argument, suggesting that Enoch’s more recent argument is rendered problematic insofar as it involves an explicitly normative premise.

Øystein Linnebo (2018) offers an account of mathematical objects that promises to bypass some of the features that can make standard Platonism seem unpalatable. Although Linnebo’s mathematical objects are abstract and indeed exist of necessity, their existence is lightweight: they make little by way of demands on the world. As such the costs of realism about mathematical objects are, in Linnebo’s account, much lower than is standardly assumed. Worries about how we can know about mathematical objects, or indeed refer to them, given what Linnebo calls their ‘shallow nature’, fall away. Why then persist, as the mathematical fictionalist does, with nominalist scruples about accepting the existence of abstract mathematical objects, if the existence of objects turns out to be so metaphysically undemanding?

‘If-thenism’ sees mathematical practice as a matter of working out what follows from our mathematical hypotheses without regard to whether those hypotheses are true. Enhanced if-thenism, as Maddy envisages it, adds a further component to mathematical practice: the identification of mathematically valuable hypotheses to investigate (where axioms as hypotheses are “chosen with an eye to facilitating important mathematical goals”, and only some amongst all the logically possible collections of axioms are supported as tracking “underlying contours of mathematical depth”). While accepting that Maddy is right that the identification of mathematically interesting axioms is a sufficiently important component of mathematical practice to warrant a departure from ‘if-thenism’ in its purest form (according to which any consistent axioms would be fair game for mathematical investigation), this chapter argues that empirical science can also provide support for mathematical axioms. Indeed, a plausible reading of Hilary Putnam’s rather elusive indispensability argument is as an argument against pure ‘if-thenism’ and in favour of an enhanced if-thenism where axioms are chosen for scientific reasons.

The fictionalist aims to avoid commitment to mathematical objects by replacing mathematical truth with fictional correctness: truth-in-the-story (of standard mathematics). For an axiomatically stated mathematical theory T, a sentence S is true-in-the-T-story if and only if it follows logically from the axioms of T. The formalist, on the other hand, seeks to avoid commitment to mathematical objects by replacing mathematical truth with formal derivability: a sentence S is true in a mathematical theory T if there is a derivation of S from the axioms of T. For first-order theories, and if the existence of a derivation is understood to mean the in principle possibility of deriving S from T, formalism and fictionalism would seem to coincide. However, in practice the two views typically diverge, with the divergence being due to different attitudes to modality. For fictionalists such as myself who embrace primitive modality, a modal notion of logical consequence allows for sentences in second order arithmetic to have objective ‘truth-in-the-story’ values even if they are not derivable. On the other hand as the most prominent contemporary advocate of formalism, Alan Weir wishes to develop an austere form of formalism without modal commitments, so that mathematical truth is not grounded in the in principle possibility of a derivation, but only in the existence of actual derivations. In this chapter I continue a long-standing debate between myself and Weir over the nominalistic acceptability of primitive modality, arguing that Weir’s account of informal mathematics involves him in modal commitments whether he likes it or not.

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.

For the past few years, I have been fortunate enough to teach, annually, a third year undergraduate module in the philosophy of mathematics. It is a testimony to Paul Benacerraf’s great influence on the discipline that the module is structured very naturally in two halves, which could quite easily be subtitled “Before Benacerraf” (BB) and “After Benacerraf” (AB). The story I tell starts at the end of the 19th century, with Cantor’s development of the new infinitary set theory, and mathematicians’ and philosophers’ concerns about how (or whether) we can make sense of this new mathematics that is not grounded in Kantian intuition of space and time.

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.

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.

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.

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.

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.