Abstract Structures are ubiquitous in mathematics. But how should they be understood? Modelists claim they are model-theoretic structures. This thesis can be read in two ways: as a claim about what structures refer to, or about how we conceptualize them. Objects-modelism, developed by Button and Walsh, pursues the first; the second leads to concepts-modelism, which remains underexplored. In this paper we develop and defend a version of concepts-modelism, cognitive modelism, drawing on Carey’s theory of conceptual development, and we show how it addresses the challenges Button and Walsh pose for a conceptual account of mathematical structures.

Abstract In non-well-founded set theory, which anti-foundation axiom is philosophically justified, BAFA, FAFA, SAFA, AFA, or some other one? In this paper, we investigate a general approach to answering this question: first, considering which identity condition for sets is justified; second, considering which anti-foundation axiom it justifies. Specifically, we study in detail two plausible identity conditions, $ \text{IC}_{1} $ and $ \text{IC}_{2} $: we show that $ \text{IC}_{2} $ justifies $ \text{FAFA}_{2} $ and $ \text{IC}_{1} $ justifies AFA, and argue for AFA by offering an argument for $ \text{IC}_{1} $.

Abstract This paper considers a range of proofs of Morley’s theorem, and uses this variety to argue for pluralism about explanatory proof. The main alternative to pluralism is Lange’s proposal that an explanatory proof obtains the salient feature of some theorem by exploiting the same kind of feature of the theorem’s setup. This proposal can be amended in a wide variety of different ways to deal with the proofs of Morley’s theorem that I consider. However, it is not clear how to revise Lange’s proposal to reach the right verdict in all of these cases.

Abstract The iterative conception (IC) is arguably the best worked out conception of set available. What is the status of the axiom of choice under this conception? Boolos argues that it is not justified by IC. We show that Boolos’s influential argument overgenerates. For, if cogent, it would imply that none of the axioms of ZFC which Boolos took to be justified by IC is so justified. We furthermore show that, to the extent that they are consequences of a plural formulation of stage theory, all those axioms are justified by IC — axiom of choice included.

Abstract This paper explores the relationship of artificial intelligence to resolving open questions in mathematics. We first argue that limitative results from computability and complexity theory retain their significance in illustrating that proof discovery is an inherently difficult problem. We next consider how applications of automated theorem proving, Sat-solvers, and large language models raise underexplored questions about the nature of mathematical proof — e.g., about the status of brute force and the relationship between logical and discovermental complexity. Nevertheless, we finally suggest that the results obtained thus far by automated methods do not tell against the inherent difficulty of proof discovery.

Abstract We show that structuralism has the very serious defect of having no satisfactory notion of identity which can be associated with its central notion: structure. We also refute the structural thesis about the nature of the natural numbers by showing that there are at least two completely different structures that are entitled to be taken as ‘the structure of the natural numbers’, and any choice between them would arbitrarily favor one of them over the equally legitimate other. Finally, we argue for the high plausibility of the identification of the natural numbers with the finite von Neumann ordinals.

Abstract Mac Lane identified and meticulously described dozens of mathematical ideas in his book Mathematics: Form and Function. At the same time, he acknowledged the difficulty of precisely defining the nature of these ideas, while firmly rejecting any association with platonism. In this paper, I reconstruct Mac Lane’s ontology of ideas by drawing upon Roman Ingarden’s phenomenological ontology. I argue that, contrary to his own assertions, Mac Lane’s ideas belong to the realm of ideal beings, and that his mathematical forms constitute intentional objects in Ingarden’s sense. Furthermore, I present Mac Lane as one of the foremost theorists of ideas.

Abstract How the semantic significance of numerical discourse gets determined is a metasemantic issue par excellence. At the sub-sentential level, the issue is riddled with difficulties on account of the contested metaphysical status of the subject matter of numerical discourse, i.e., numbers and numerical properties and relations. Setting those difficulties aside, I focus instead on the sentential level, specifically, on obvious affinities between whole numerical and non-numerical sentences and how their significance is determined. From such a perspective, Frege’s 1884 construction of number, while famously mathematically untenable, fares better metasemantically than extant alternatives in the philosophy of mathematics.

Abstract We deal with the issue of whether large cardinals are intrinsically justified set-theoretic principles (Intrinsicness Issue). To this end, we review, in a systematic fashion, the abstract principles that have been formulated to motivate them and their mathematical expressions, and assess their intrinsic justifiability. A parallel, but closely linked, issue is whether there exist mathematical principles that yield all large cardinals (Universality Issue), and we also test principles for their ability to respond to this issue. Finally, we discuss Structural Reflection Principles and their responses to Intrinsicness and Universality, and also make some further considerations on their general justifiability.