Bahram Assadian

According to the realist rendering of mathematical structuralism, mathematical structures are ontologically prior to individual mathematical objects such as numbers and sets. Mathematical objects are merely positions in structures: their nature entirely consists in having the properties arising from the structure to which they belong. In this paper, I offer a bundle-theoretic account of this structuralist conception of mathematical objects: what we normally describe as an individual mathematical object is the mereological bundle of its structural properties. An emerging picture is a version of mereological essentialism: the structural properties of a mathematical object, as a bundle, are the mereological parts of the bundle, which are possessed by it essentially.

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.

According to the neo-Fregean abstractionism, numerical expressions of the form ‘the number of Fs’, introduced by Hume’s Principle, should be read as purportedly referential singular terms. I will explore the prospects of a version of abstractionism in which such expressions have presuppositional content, as in Strawson’s account. I will argue that the thesis that ‘the number of Fs’ semantically presupposes the existence of a number is inconsistent with the required ‘modest’ stipulative character of the truth of Hume’s Principle: since Hume’s Principle is true and provably presupposes that numbers exist, what it presupposes is also true; and so numbers exist. This, however, means that numbers are conjured into existence by a direct stipulation.

Reference may be fixed by stipulation through a speech act, just like bets and marriages. An utterance of _Let n refer to an/the F_ is a speech act by means of which, if successful, a speaker institutes a practice of referring, and a hearer coordinates by choosing a referent from the domain of discourse. We articulate a metasemantics for this view. On our view, the interlocutors can select a referent randomly, if necessary, motivated by the incentive to coordinate on the use of a name. Moreover, we argue that reference fixed by a performative speech act is ‘thin’, or ‘undemanding’. Finally, we defend the thesis that co-reference might not determinately obtain despite reference being fixed. Performative reference makes sense of ordinary speakers’ practices, who appear to be very liberal and unimpressed by skepticism toward causally inert or epistemically indiscernible objects.

According to Øystein Linnebo's account of abstractionism, abstraction principles, received as Fregean criteria of identity, can be used to reduce facts about singular reference to objects such as directions and numbers to facts that do not involve such objects. In this article, first I show how the resources of Linnebo's metasemantics successfully handle Dummett's challenge against the referentiality of the singular terms formed by abstraction principles. Then, I argue that Linnebo's metasemantic commitments do not provide us with tools for dispelling the threat of a version of referential indeterminacy, according to which nothing in our use of a singular term, even when it is guided by an associated criterion of identity, could determine which particular object it refers to. I end by examining the bearing of the indeterminacy challenge to Linnebo's treatment of Frege's Caesar Problem: in the absence of an argument against the indeterminacy of reference, it is unclear how numerical expressions could qualify as genuine singular terms.

Indeterminacy of reference appears to be incompatible with the deflationist conceptions of reference: in deflationismDeflationism, 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 indeterminacyReferential indeterminacy and deflationismDeflationism 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 indeterminacyReferential indeterminacy.

Throughout his career, Keith Hossack has made outstanding contributions to the theory of knowledge, metaphysics and the philosophy of mathematics. This collection of previously unpublished papers begins with a focus on Hossack's conception of the nature of knowledge, his metaphysics of facts and his account of the relations between knowledge, agents and facts. Attention moves to Hossack's philosophy of mind and the nature of consciousness, before turning to the notion of necessity and its interaction with a priori knowledge. Hossack's views on the nature of proof, logical truth, conditionals and generality are discussed in depth. In the final chapters, questions about the identity of mathematical objects and our knowledge of them take centre stage, together with questions about the necessity and generality of mathematical and logical truths.

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.

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.

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.

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.

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.