Bahram Assadian

This paper explores the abstractionist variant of ‘easy ontology’ and its reliance on the Syntactic Priority Thesis (SPT), according to which the occurrence of singular terms in true atomic sentences, embedded in extensional contexts, ensures reference to a range of objects. Recent objections by Geoffrey Hellman (2025), however, suggest that SPT generates an unwanted ontology of fictional and mythological entities, thereby undermining its credibility and obstructing the abstractionist route to mathematical platonism. I critically examine Hellman’s arguments and advance strategies to secure SPT. Furthermore, on the reading of Frege’s context principle that I develop, I contend that – contrary to Hellman – an assessment of the abstractionist route to mathematical platonism cannot proceed solely in terms of SPT, bypassing abstraction principles that contextually define the relevant singular terms. In light of this, I revisit the platonist commitments of abstractionism.

According to the realist or ‘non-eliminative’ renderings of mathematical structuralism, mathematical objects are merely positions in structures, ontologically dependent on the structure to which they belong. The purely structural character of mathematical objects leads to various treatments of identity statements linking positions from distinct structures, such as ‘the natural number 2 is identical to the real number 2’. I develop a novel, Aristotelian (in re) account of ontological dependence to argue that cross-structural identities are strictly false, while the expressions flanking the identity sign are referentially indeterminate. This approach avoids reliance on the contentious metaphysical principles that underlie other approaches to the falsity of such statements. I also argue for the referential indeterminacy of mathematical expressions even if isomorphism is taken to be sufficient for the numerical identity of structures. Along the way, I provide more precise accounts of cross-structural identities and critically examine some of the extant arguments.

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.

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.

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.

A central thesis of neo-Fregean abstractionism is that numerical expressions of the form ‘the number of Fs’, introduced by Hume’s Principle, should be read as genuine singular terms whose semantic function is to refer to particular objects. This paper explores the prospects of a variant of abstractionism in which such expressions have existential assertoric content, as in Russell’s analysis of definite descriptions. The neo-Russellian abstractionist faces three initial challenges: (i) the Russellian rendering of Hume’s Principle does not retain the ontological modesty that any admissible abstraction principle must respect. (ii) It defuses the objectual character of natural numbers, and thereby fails to ground their infinity. And (iii) it does not involve the means for engaging in identifying reference to numbers as particular objects, thereby constituting a derogation of arithmetical platonism. I shall investigate these challenges and propose solutions to address them.

Radical indeterminacy of reference is the thesis that there is no fact of the matter as to which objects singular terms refer to, and which sets of objects are in the extensions of predicates. For instance, it is indeterminate as to whether ‘London’ refers to a city in England or instead to a dormant volcano in Africa. This paper addresses a largely unexplored challenge against radical indeterminacy of reference: the claim that it is self-refuting, rendering its own thesis ineffable. I develop several versions of this challenge and argue that they mischaracterize referential indeterminacy – either by depicting it as reliant on the determinacy of reference and singular thought, or by tying it to eliminativism about reference.

Abstractionism is the view that Fregean abstraction principles underlie our knowledge of the existence of mathematical objects. It is often assumed that the abstractionist proof for the existence of such objects requires ‘negative free logic’ in which all atomic sentences with empty terms are false. I argue that while negative free logic is not indispensably needed for the proof of abstract existence, there is a motivation for it—along broadly Fregean lines. The standard motivation for negative semantics rests on the explanation of truth in terms of reference. This line of reasoning, however, is not available in a context in which the reference of abstract terms must be proved, and not presupposed. I reverse the direction of explanation, thereby offering a novel motivation, Truth Priority, for the use of negative semantics. Some of the implications of Truth Priority for the abstractionist conception of ontology and reference will also be explored.

Against mathematical platonism, it is sometimes objected that mathematical objects are mysterious. One possible elaboration of this objection is that the individuation of mathematical objects cannot be adequately explained. This suggests that facts about the numerical identity and distinctness of mathematical objects require an explanation, but that their supposed nature precludes us from providing one. In this paper, we evaluate this nominalist objection by exploring three ways in which mathematical objects may be individuated: by the intrinsic properties they possess, by the relations they stand in, and by their underlying ‘substance’. We argue that only the third mode of individuation raises metaphysical problems that could substantiate the claim that mathematical objects are somehow mysterious. Since the platonist is under no obligation to accept this thesis over the alternatives, we conclude that, at least as far as individuation is concerned, the nominalist objection has no bite.

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.

How do we acquire the notions of cardinality and cardinal number? In the (neo-)Fregean approach, they are derived from the notion of equinumerosity. According to some alternative approaches, defended and developed by Husserl and Parsons among others, the order of explanation is reversed: equinumerosity is explained in terms of cardinality, which, in turn, is explained in terms of our ordinary practices of counting. In their paper, ‘Cardinality, Counting, and Equinumerosity’, Richard Kimberly Heck proposes that instead of equinumerosity or counting, cardinality is derived from a cognitively earlier notion of just as many. In this paper, we assess Heck’s proposal in terms of contemporary theories of number concept acquisition. Focusing on bootstrapping theories, we argue that there is no evidence that the notion of just as many is cognitively primary. Furthermore, since the acquisition of cardinality is an enculturated process, the cognitive primariness of these notions, possibly including just as many, depends on various external cultural factors. Therefore, being possibly a cultural construction, just as many could be one among several notions used in the acquisition of cardinality and cardinal number concepts. This paper thus challenges those accounts which seek for a fundamental concept underlying all aspects of numerical cognition.

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.

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.

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.