# Edited Volumes

*Knowledge, Number, and Reality*

Encounters with the Work of Keith Hossack

Encounters with the Work of Keith Hossack

Forthcoming, with Nils Kürbis and Jonathan Nassim (eds.); Bloomsbury Academic, London.

https://www.bloomsbury.com/uk/knowledge-number-and-reality-9781350186439/

# Papers

## The insubstantiality of mathematical objects as positions in structures

*Inquiry: An Interdisciplinary Journal of Philosophy*

**Abstract**. The realist versions of mathematical structuralism are often characterized by what I call ‘the insubstantiality thesis’, according to which mathematical objects, being positions in structures, have no non-structural properties: they are purely structural objects. The thesis has been criticized for being inconsistent or descriptively inadequate. In this paper, by implementing the resources of a real-definitional account of essence in the context of Fregean abstraction principles, I offer a version of structuralism – essentialist structuralism – which validates a weaker version of the insubstantiality thesis: mathematical objects have no non-structural essential properties. Next, I show how this rendition of structuralism alleviates a Fregean worry against insubstantiality, which is directed at the explanation of the applicability of mathematics from the structuralist perspective.

## Mathematical structures, universals, and singular terms

Forthcoming (2022) in *Knowledge, Number, and Reality; Encounters with the Work of Keith Hossack*; edited by N. Kürbis, B. Assadian, and J. Nassim; Bloomsbury Academic, London.

## Referential deflationism *vs*. referential indeterminacy

2021; in* Mathematics, Logic and their Philosophies*; edited by M. Mojathedi, S. Rahman, and M. S. Zarepour; Dordrecht: Springer.

**Abstract.** Indeterminacy of reference appears to be incompatible with the deflationist conceptions of reference: in deflationism, the singular term ‘*a*’ refers to *a*, if it exists, and to nothing else, whereas if the term is referentially indeterminate, it has a variety of equally permissible reference-candidates: referential indeterminacy and deflationism cannot both be maintained. In this paper, I discuss the incompatibility thesis, critically examine the arguments leading to it, and thereby point towards ways in which the deflationist can explain referential indeterminacy.

## Formal logic in philosophy

Survey article in* Introduction to Philosophy: Logic*; open textbook; edited by Benjamin Martin; Rebus Community, 2020.

**Abstract**. This chapter discusses some philosophical issues concerning the nature of formal logic. Particular attention will be given to the concept of logical form, the goal of formal logic in capturing logical form, and the explanation of validity in terms of logical form. We shall see how this understanding of the notion of validity allows us to identify what we call formal fallacies, which are mistakes in an argument due to its logical form. We shall also discuss some philosophical problems about the nature of logical forms. For the sake of simplicity, our focus will be on propositional logic. But many of the results to be discussed do not depend on this choice, and are applicable to more advanced logical systems.

## Indeterminacy and failure of grounding

2019 (with Jonathan Nassim); *Theoria*. 85 (4): 276–288

**Abstract**. Cases of grounding failure present a puzzle for fundamental metaphysics. Typically, solutions are thought to lie either in adding ontology such as haecceities or in re‐describing the cases by means of the ideology of metaphysical indeterminacy. The controversial status of haecceities has led some to favour metaphysical indeterminacy as the way to solve the puzzle. We consider two further treatments of grounding failure each of which, we argue, is a more plausible alternative. As such, the initial dichotomy is a false one, and these alternative options deserve consideration before resorting to the heavyweight machinery of metaphysical indeterminacy.

## Abstractionism and mathematical singular reference

2019; *Philosophia Mathematica*, 27 (2): 177–198.

**Abstract**. Is it possible to effect singular reference to mathematical objects in the abstractionist framework? I will argue that even if mathematical expressions pass the relevant syntactic and inferential tests to qualify as singular terms, that does not mean that their semantic function is to refer to a particular object. I will defend two arguments leading to this claim: the permutation argument for the referential indeterminacy of mathematical terms, and the argument from the semantic idleness of the terms introduced by abstraction principles.

## In defense of utterly indiscernible entities

2018; *Philosophical* *Studies*. 176 (10): 2551–2561.

**Abstract**. Are there entities which are just distinct, with no discerning property or relation? Although the existence of such utterly indiscernible entities is ensured by mathematical and scientific practice, their legitimacy faces important philosophical challenges. I will discuss the most fundamental objections that have been levelled against utter indiscernibles, argue for the inadequacy of the extant arguments to allay perplexity about them, and put forward a novel defence of these entities against those objections.

## Are the natural numbers fundamentally ordinals?

2019 (with Stefan Buijsman); *Philosophy and Phenomenological Research*. 99 (3): 564–580.

**Abstract**. There are two ways of thinking about the natural numbers: as ordinal numbers or as cardinal numbers. It is, moreover, well‐known that the cardinal numbers can be defined in terms of the ordinal numbers. Some philosophies of mathematics have taken this as a reason to hold the ordinal numbers as (metaphysically) fundamental. By discussing structuralism and neo‐logicism we argue that one can empirically distinguish between accounts that endorse this fundamentality claim and those that do not. In particular, we argue that if the ordinal numbers are metaphysically fundamental then it follows that one cannot acquire cardinal number concepts without appeal to ordinal notions. On the other hand, without this fundamentality thesis that would be possible. This allows for an empirical test to see which account best describes our actual mathematical practices. We then, finally, discuss some empirical data that suggests that we can acquire cardinal number concepts without using ordinal notions. However, there are some important gaps left open by this data that we point to as areas for future empirical research.

## The semantic plights of the ante-rem structuralist

2017; *Philosophical Studies,* 175 (12): 3195–3215.

**Abstract**. A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante-rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist fails to explain reference in a way that makes her account different to, and privileged over, that of her eliminativist rivals. Both problems undercut the motivation behind ante-rem structuralism.

*Displacing numbers: on the metaphysics of mathematical structuralism*

2016, Ph.D. Dissertation, Birkbeck College, University of London

# Book Reviews

*Abstractionism: Essays in Philosophy of Mathematics*, edited by Philip A. Ebert and Marcus Rossberg (OUP, 2016); published in *Analysis* 78, Issue 1, 1 2018: 188–191.

*The Metaphysics of Relations*, edited by Anna Marmodoro and David Yates (OUP, 2016); published in *Philosophical Quarterly* 67 (269), 2017: 871–874.

*Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale*, edited by Ivette Fred-Rivera and Jessica Leech. Forthcoming in *Philosophical Quarterly.*