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

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

Neologicism (NL) invokes a “syntactic-priority thesis” (SPT) to derive existence of numbers, etc., from abstraction principles. Innumerable counterexamples to the SPT, however, are seen to arise from fiction, e.g., “Pegasus is (entirely) fictive”. Examination of possible defenses of the SPT leads to just one viable option, based on quasi-modal “in-fiction” operators. This, however, applies just as

The International Humanities Council has established a new international research network Diversity of Mathematical Research Cultures & Practices (DMRCP) at the Universität Hamburg. In a tripartite contribution, we outline and discuss the specific philosophical approach that DMRCP seeks to promote for which we use the term ‘empirical philosophy of mathematics’: the contribution is therefore programmatic

Modal potentialism argues that mathematics has a generative nature, and aims to formalise mathematics accordingly using quantified modal logic. This paper shows that Øystein Linnebo’s approach to modal potentialism in his book Thin Objects is incoherent. In particular, he is committed to the legitimacy of introducing a primitive modal predicate of formulae. However, as with the semantic paradoxes,

This paper proposes that mathematicians routinely use inference to the best explanation (IBE) to confirm their conjectures. Mathematicians can justly reason that the ‘best explanation’ of some mathematical evidence they possess would be a proof of it that likewise proves a given conjecture. By IBE, the evidence thereby confirms that such an as-yet-undiscovered proof exists and that the conjecture holds

Free choice sequences play a key role in the Brouwerian continuum. Using recent modal analysis of potential infinity, we can make sense of free choice sequences as potentially infinite sequences of natural numbers without adopting Brouwer’s distinctive idealistic metaphysics. This provides classicists with a means to make sense of intuitionistic ideas from their own classical perspective. I develop

Hume’s Principle (HP) does not determine the truth values of ‘mixed’ identity statements like ‘$ \#F $ = Caesar’. This is the Caesar Problem (CP). Still, neologicists such as Hale and Wright argue that (1) HP is a priori, and (2) HP introduces the pure sortal concept Number. We argue that Neologicism faces a Caesar-problem Problem (CPP): if neologicists solve CP by establishing that ‘$ \#F\neq $ Caesar’

Michael Dummett argues that acceptance of potentially infinite collections requires that we abandon classical logic and restrict ourselves to intuitionistic logic. In this paper we examine whether Dummett is correct. After developing two detailed accounts of what, exactly, it means for a concept to be potentially infinite (based on ideas due to Charles McCarty and Øystein Linnebo, respectively), we

While sets are combinatorial collections, defined by their elements, classes are logical collections, defined by their membership conditions. We develop, in a potentialist setting, a predicative approach to (logical) classes of (combinatorial) sets. Some reasons emerge to adopt a stricter form of potentialism, which insists, not only that each object is generated at some stage of an incompletable process

In ‘Intuition, iteration, induction’, Mark van Atten argues that iteration is an object of intuition for Brouwer and explains the intuitive character of the act of iteration drawing from Husserl’s phenomenology. I find the arguments for this reading of Brouwer unconvincing. In this note I set out some issues with his claim that iteration is an object of intuition and his Husserlian explication of iteration

Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual

In his groundbreaking book, Thin Objects, Øystein Linnebo argues for an account of neo-Fregean abstraction principles and thin existence that does not rely on analyticity or conceptual rules. It instead relies on a metaphysical notion he calls “sufficiency”. In this short discussion, I defend the analytic or conceptual-rule account of thin existence.

According to the iterative conception of set, sets are formed in stages. According to the purely iterative conception of set, sets are formed by iterated application of a set-of operation. The cumulative hierarchy is a mathematical realization of the iterative conception of set. A mathematical realization of the purely iterative conception can be found in Peter Aczel’s type-theoretic model of constructive

The constructive mathematics developed by Bishop in Foundations of Constructive Analysis succeeded in gaining the attention of mathematicians, but discussions of its underlying philosophy are still rare in the literature. Commentators seem to conclude, from Bishop’s rejection of choice sequences and his severe criticism of Brouwerian intuitionism, that he is not an intuitionist–broadly understood as

Non-well-founded set theories allow set-theoretic exotica that standard ZFC will not, such as a set that has itself as its sole member. We distinguish plenitudinous non-well-founded set theories, such as Boffa set theory, that allow infinitely many such sets, from restrictive theories, such as Finsler–Aczel or AFA, that allow exactly one. Plenitudinous non-well-founded set theories face a puzzle: nothing

Informally rigorous mathematical reasoning is relevant. So too should be the premises to the conclusions of formal proofs that regiment it. The rule Ex Falso Quodlibet induces spectacular irrelevance. We therefore drop it. The resulting systems of Core Logic $ \mathbb{C}$ and Classical Core Logic $ \mathbb{C}^{+}$ can formalize all the informally rigorous reasoning in constructive and classical mathematics

Boffa non-well-founded set theory allows for several distinct sets equal to their respective singletons, the so-called ‘Quine atoms’. Rieger contends that this theory cannot be a faithful description of set-theoretic reality. He argues that, even after granting that there are non-well-founded sets, ‘the extensional nature of sets’ precludes numerically distinct Quine atoms. In this paper we uncover

This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a

Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead

Bob Hale has defended a new conception of properties that is broadly Fregean in two key respects. First, like Frege, Hale insists that every property can be defined by an open formula. Second, like Frege, but unlike later definabilists, Hale seeks to justify full impredicative property comprehension. The most innovative part of his defense, we think, is a “definability constraint” that can serve as

I examine the classical idea of ‘algorithm’ as a sequential, step-by-step, deterministic procedure (i.e., the idea of ‘algorithm’ that was already in use by the 1930s), with respect to three themes, its relation to the notion of an ‘effective procedure’, its different roles and uses in logic, computer science, and mathematics (focused on numerical analysis), and its different formal definitions proposed

Using ideas proposed in Aboutness and developed in ‘If-thenism’, Stephen Yablo has tried to improve on classical if-thenism in mathematics, a view initially put forward by Bertrand Russell in his Principles of Mathematics. Yablo’s stated goal is to provide a reading of a sentence like ‘The number of planets is eight’ with a sort of content on which it fails to imply ‘Numbers exist’. After presenting

Who gets to contribute to knowledge production of an epistemic community? Scholarship has focussed on unjustified forms of exclusion. Here I study justified forms of exclusion by investigating the phenomenon of so-called ‘cranks’ in mathematics. I argue that workload-management concerns justify the exclusion of these outsiders from mathematical knowledge-making practices. My discussion reveals three

In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of ‘suggesting’ a conjecture in the psychological sense). The proposed conditions generalize the criteria of Hesse in her influential work on analogical

In their recent article ‘Resolving Frege’s other Puzzle’ Eric Snyder, Richard Samuels, and Stewart Shapiro defend a semantic type-shifting solution to Frege’s other Puzzle and criticize my own cognitive type-shifting solution. In this article I respond to their criticism and in turn point to several problems with their preferred solution. In particular, I argue that they conflate semantic function

The mathematician G.F.C. Griss is known for his program of negationless intuitionistic mathematics. Although Griss’s rejection of negation is regarded as characteristic of his philosophy, this is a consequence of an executability requirement that mental constructions presuppose agents’ executing corresponding mental activity. Restoring Griss’s executability requirement to a central role permits a more

Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature

A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry

In a 2022 paper, Hamami claimed that the orthodox view in mathematics is that a proof is rigorous if it can be translated into a derivation. Hamami then developed a descriptive account that explains how mathematicians check proofs for rigor in this sense and how they develop the capacity to do so. By exploring introductory texts in computability theory, we demonstrate that Hamami’s descriptive account

Gödel argued that the incompleteness theorems entail that the mind is not a machine, or that certain arithmetical propositions are absolutely undecidable. His view was that the mind is not a machine, and that no arithmetical propositions are absolutely undecidable. I argue that his position presupposes that the idealized mathematician has an ability which I call the recursive-ordinal recognition ability

This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in

ABSTRACT Number and Count Sentences like ‘The number of Martian moons is two’ and ‘Mars has two moons’ give rise to a puzzle. How can they be equivalent if only the truth of Number but not that of Count Sentences requires the existence of numbers? Proponents of Linguistic Deflationism seek to resolve this puzzle by arguing that on their correct linguistic analysis the truth of Number Sentences does

AberdeinA., RittbergC.J, and TanswellF.S., eds. Virtue Theory of Mathematical Practices. Topical collection in two double issues of the journal Synthese 199, 2021.

Geoffrey Hellman.**Mathematics and Its Logics: Philosophical Essays.Cambridge University Press, 2021. Pp. viii + 286. ISBN 978-1-108-49418-2 (hbk); 978-1-108-65741-9 (ebk). doi.org/10.1017/9781108657419.

FloydJuliet.**Wittgenstein’s Philosophy of Mathematics. Cambridge Elements of Philosophy of Mathematics. Cambridge University Press, 2021. Pp. iv + 88. ISBN: 978-1-108-45630-2 (pbk); 978-1-108-68712-6 (online). doi/org/10.1017/9781108687126.

We use Gödel’s incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel’s incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel’s incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity

BenciVieri and Di NassoMauro. How to Measure the Infinite: Mathematics with Infinite and Infinitesimal Numbers.Singapore: World Scientific, 2019. Pp. xxviii + 317. ISBN: 978-981-283-737-3 (hbk); 978-981-3276-60-4 (e-book); 978-981-283-638-0 (institutional e-book). doi/org/10.1142/7081.

TrepczyńskiMarcin, ed. Philosophical Approaches to the Foundations of Logic and Mathematics: In Honor of Stanisław Krajewski. Poznań Studies in the Philosophy of the Sciences and the Humanities; 114. Leiden: Brill Rodopi, 2021. Pp. vi + 310. ISBN 978-90-04-44594-9 (hbk); 978-90-04-44595-6 (pdf e-book). doi.org/10.1163/9789004445956.

Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve

Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal

WeingartnerPaul and LeebHans-Peter, eds, Kreisel’s Interests: On the Foundations of Logic and Mathematics. Tributes; 41. London: College Publications, 2020. Pp. viii + 171. ISBN: 978-1-84890-330-2 (pbk).

CentroneStefania, KantDeborah, and SarikayaDeniz, eds, Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory, and General Thoughts. Studies in Epistemology, Logic, Methodology, and Philosophy of Science; 407. Springer, 2019. Pp. xxviii + 494. ISBN: 978-3-030-15654-1 (hbk); 978-3-030-15655-8 (e-book). doi.org/10.1007/978-3-030-15655-8††.

MojtahediMojtaba, RahmanShahid, and ZarepourMohammad Saleh, eds. Mathematics, Logic, and their Philosophies: Essays in Honour of Mohammad Ardeshir. Logic, Epistemology, and the Unity of Science; 49. Springer, 2021. Pp. xviii + 483. ISBN 978-3-030-53653-4 (hbk); 978-3-030-53654-1 (e-book). doi: 10.1007/978-3-030-53654-1.

WilsonMark.**Innovation and Certainty. Cambridge Elements in the Philosophy of Mathematics. Cambridge University Press, 2020. Pp. 74. ISBN: 978-1-108-74229-0 (pbk); 978-1-108-59290-1 (online). doi.org/10.1017/9781108592901.

AbstractDomain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not

RusnockPaul** and ŠebestíkJan. Bernard Bolzano: His Life and His Work.Oxford University Press,2019. Pp. xxxiii + 667. ISBN: 978-0-19-882368-1 (hbk); 978-0-19-255683-7 (pdf); 978-0-19-255684-4 (e-book). doi.org/10.1093/oso/9780198823681.001.0001.

PosyCarl J..**Mathematical Intuitionism. Cambridge Elements in the Philosophy of Mathematics. Cambridge University Press, 2020. Pp. 106. ISBN: 978-1-108-72302-2 (pbk); 978-1-108-67448-5 (e-book). doi.org/10.1017/ 9781108674485.

ABSTRACTSome mathematical proofs explain why the theorems they prove hold. This paper identifies several challenges for any counterfactual account of explanation in mathematics (that is, any account according to which an explanatory proof reveals how the explanandum would have been different, had facts in the explanans been different). The paper presumes that countermathematicals can be nontrivial

FenstadJ.E.. Structures and Algorithms: Mathematics and the Nature of Knowledge. Logic, Argumentation & Reasoning; 15. Springer, 2018. Pp. x + 134. ISBN 978-3-319-72973-2 (hbk); 978-3-030-10294-4 (pbk); 978-3-319-72974-9 (e-book). doi.org/10.1007/978-3-319-72974-9.

ABSTRACTThis paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse

HossackKeith.**Knowledge and the Philosophy of Number: What Numbers Are and How They Are Known.London: Bloomsbury Academic, 2020. Pp. ix + 206. ISBN: 978-1-3501-0290-3 (hbk); 978-1-3502-7796-0 (pbk); 978-1-3501-0292-7 (epub); 978-1-3501-0291-0 (ebk).

ABSTRACTThere is no prospect of discovering measurable cardinals by radio astronomy, but this does not mean that higher set theory is entirely irrelevant to applied mathematics broadly construed. By way of example, the bearing of some celebrated descriptive-set-theoretic consequences of large cardinals on measurable-selection theory, a body of results originating with a key lemma in von Neumann’s work

HaleBob. Essence and Existence: Selected Essays. Jessica Leech,** ed. Oxford University Press, 2020. Pp. xii + 305. ISBN: 978-0-19-885429-6 (hbk); 978-0-19-259622-2 and 978-0-19-259623-9 (e-book). doi.org/10.1093/oso/9780198854296.001.0001.

KennedyJuliette.**Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness.Cambridge University Press, 2021. Pp. xii + 187. ISBN: 978-1-107-01257-8 (hbk); 978-0-511-99839-3 (online); 978-1-00902851-6, 978-1-00902823-3 (e-book). doi.org/10.1017/9780511998393.

PradelleDominique.**Intuition et idéalités. Phénoménologie des objets mathématiques [Intuition and idealities: Phenomenology of mathematical objects.] Collection Épiméthée. Paris: PUF [Presses universitaires de France], 2020. Pp. 550. ISBN: 978-2-13-082237-0 (pbk).

Four philosophical concerns about higher-order logic in general and the specific demands placed on it by the neo-logicist project are distinguished. The paper critically reviews recent responses to these concerns by, respectively, the late Bob Hale, Richard Kimberly Heck, and myself. It is argued that these score some successes. The main aim of the paper, however, is to argue that the most serious

It is currently fashionable to take Putnamian model-theoretic worries seriously for mathematics, but not for discussions of ordinary physical objects and the sciences. However, I will argue that (under certain mild assumptions) merely securing determinate reference to physical possibility suffices to rule out the kind of nonstandard interpretations of our number talk Putnam invokes. So, anyone who