Introduction

Even though he is arguably one of the most influential philosophers of the 20th century, Ludwig Wittgenstein is often perceived as confusing and misunderstood. Furthermore, in spite of his conviction that his most important work was in the philosophy of mathematics, his contributions in this field are generally lesser known than his other writings. Given the significance of Wittgenstein's work in areas like the philosophy of mathematical practice (PMP), a special issue on the subject felt warranted. I thank the Editor-in-Chief of Philosophical Investigations (PI) for the opportunity to edit this volume and am grateful to the authors for their fine contributions, as well as to the anonymous reviewers and the production staff at Wiley. Without them, this volume would not have been possible. In slightly different worlds, the title of this special issue on ‘Wittgenstein and the Philosophy of Mathematics’ might also have been different. In so far as the later Wittgenstein's emphasis on the role of practices resonates closely with my own work, the volume could have been titled ‘Wittgenstein and the Philosophy of Mathematical Practice’, and indeed, many contributions are from authors who have also published in PMP. Alternatively, a more exegetical approach might have led us to ‘Wittgenstein's Philosophy of Mathematics’ or ‘Wittgenstein and the Philosophy of Mathematic(s|al Practice)’ However, the final title prevailed for three reasons: (1) it is the most inclusive, (2) it is less biased with respect to the relation between philosophy of mathematics and PMP, and (3) the absence of a purely exegetical focus should motivate the reader to gain Wittgensteinian insights even from articles about philosophy of mathematics that may not necessarily represent Wittgenstein's own views, and quite independently of whether that reader has any interest in Wittgenstein as a person. The nine articles in this collection cover a wide range of topics, all of which are relevant to Wittgenstein's philosophy of mathematics and PMP. Frank Scheppers, in ‘Wittgenstein and the Philosophy of Mathematical Practice’, explores how Wittgenstein's later philosophy, particularly his practice-based approach to meaning, foreshadows contemporary PMP. Scheppers argues that Wittgenstein's emphasis on the meaningfulness of mathematics being rooted in its everyday applications, and his insistence on the variability and contingency of mathematical practices, makes him a precursor to modern PMP. Wittgenstein's approach, however, is more radically practice-based than most current PMP work. On the other hand, some of Wittgenstein's ideas on key dimensions crucial to the contemporary domain of PMP, such as ethics, aesthetics and culture, are not shared by most scholars in today's PMP landscape. We then have three articles on Wittgenstein and set theory by Felix Mühlhölzer, Penelope Maddy and Karim Zahidi, respectively. This is an interesting topic, especially since Wittgenstein frequently uses rather basic examples (2 + 2 = 4 or 12 × 12 = 144), and his philosophy of arithmetic and the rootedness of mathematics in actual human practice are importantly connected. Although Wittgenstein's remarks on higher mathematics were at times dismissive, he practised seriously and adopted a ‘philosophy comes second’ position. This means that there is some tension if we read Wittgenstein as being simply dismissive of higher mathematics. We see in these articles that his unpublished work contains remarks on set theory (and the diagonal argument), which reveal a more nuanced view, and that there is a point to be made about the applicability of Wittgenstein's methods and his hesitation to use them for more advanced mathematics. Felix Mühlhölzer, in ‘Wittgenstein and Set Theory’, provides an insightful examination of Wittgenstein's complex relation to set theory, highlighting both his strong criticism of it and his acknowledgement of its philosophical significance. Wittgenstein famously dismissed set theory as ‘hocus pocus’, particularly criticizing the abstraction involved in treating sets as if they existed in their own independent world, separate from the actual mathematical practices that give rise to them. Mühlhölzer clarifies that Wittgenstein's disdain is primarily directed at the metaphysical interpretation of sets, where mathematicians lose sight of the concrete mathematical operations that generate these abstract concepts. Despite his hostility, Wittgenstein also recognized the ‘tremendous’ philosophical depth that set theory reveals. The notion of depth also gets further attention in this article. In ‘Wittgenstein on Mathematics’, Penelope Maddy investigates Wittgenstein's evolving views on mathematics, particularly his struggles with arithmetic and higher mathematics in the period leading up to PI and Remarks on the Foundations of Mathematics (RFM). Maddy argues that Wittgenstein's new method of analysis, developed during this time, can be effectively applied to arithmetic and set theory. Although Wittgenstein himself had abandoned this line of inquiry by 1945, Maddy contends that his method offers an innovative and uniquely Wittgensteinian perspective on mathematics that remains valuable. The article highlights the discrepancy between Wittgenstein's groundbreaking work on sense and logic, which matured into the methods displayed in PI and RFM I, and his more speculative, unfinished explorations into mathematics. She notes that Wittgenstein's critical approach to mathematics—particularly his scepticism of set theory—stemmed from concerns about foundationalism and abstract metaphysical claims, which he found problematic. Despite these reservations, Maddy suggests that Wittgenstein's method could yield fruitful insights into mathematics, especially in understanding how mathematical practices are grounded in human activities and hence contingent on our natural reactions and societal practices. Karim Zahidi, in ‘Wittgenstein in Cantor's Paradise’, evaluates Wittgenstein's critique of Cantorian set theory in order to shed light on the latter's broader philosophical stance on mathematics. Zahidi argues that Wittgenstein's critique is grounded in the belief that mathematical theories are human constructions rather than discoveries of objective mathematical realities. By engaging with Cantorian set theory, Wittgenstein highlights the importance of human agency in the development of mathematical concepts and proofs. This critique encourages a reflective approach to mathematics, urging a reconsideration of its foundations and the role of mathematical practice. Zahidi suggests that Wittgenstein's perspective aligns with PMP, which emphasizes the significance of the human element in mathematical development and understanding. The later Wittgenstein conceives mathematics as rule-following practices exhibiting many parallels with ordinary language. He is explicit about the latter, for instance, in the first pages of the Lectures on the Foundations of Mathematics (LFM). This view has often been described as conventionalist—albeit in different senses, given that rule-following practices are human conventions. This dimension is directly addressed in the articles by Berg, Büttner and Glock, and Pérez-Escobar, the latter considering the role of paradigmatic considerations in the context of applications. In ‘Wittgenstein and Mathematical Facts’, Ásgeir Berg makes the point that the later Wittgenstein did not deny that there were mathematical facts. The author reads the later Wittgenstein as a radical conventionalist and argues that such conventionalism about mathematics is compatible with the existence of mathematical facts: These facts are about what we find to be natural with respect to new cases in mathematics and our agreement concerning the constitution of concepts. Using an example from chess, the author reformulates the role of proofs in mathematics to show that these still play a role under a radical conventionalist perspective (in a way, against Dummett's criticism). Overall, the role of the proof would be to ‘show us how these concepts cannot fit together, given how we actually would find it natural to project them into new cases’, thus establishing a mathematical fact and explanatory value. Hans Johann Glock and Kai Michael Büttner, in ‘Logic and Conventions’, argue that both Wittgenstein and the Logical Positivists' conventionalism resist many common objections, including those famously advanced by Quine and Dummett. These critics, the authors argue, mischaracterize the conventionalist's construal of logical necessity in terms of linguistic conventions. Glock and Büttner discuss several of these misconstruals of Wittgenstein and the Logical Positivists' views on logical necessity, for example ‘to call a sentence true by convention is to claim that what the sentence says is made true by the existence of certain conventions’. Instead, these conventionalisms should be interpreted as saying that ‘given certain linguistic conventions, some sentences are unconditionally true, i.e. true irrespective of how the world happens to be’. Understood in this way, conventionalism is a much more viable position. In ‘The role of pragmatic considerations during mathematical derivation in the applicability of mathematics’, José Antonio Pérez-Escobar argues that the later Wittgenstein's philosophy of mathematics can further inform the ‘unreasonable’ applicability of mathematics. He contends that previous accounts of the applicability of mathematics have treated mathematics as rigid structures, where mathematical derivation is determined by mathematical structure itself. However, Wittgenstein's philosophy shows that mathematical derivation, instead of being determined by mathematical structure, is subject to rule-following considerations. The author shows this by discussing both Dirac's delta function and a short case study about geometry. The last two articles engage with cases that are relevant to PMP, namely diagrams and concrete notational systems (Mumma) and the role of contradictions (Vanrie). There are further interesting anthropological dimensions at play here. Note, for example, how the case of diagrams relates to a practice not infrequently ignored by an overly algebraic zeitgeist, or the case of Alan Turing, who was both an actual practitioner and highly valued by Wittgenstein. John Mumma, in ‘Geometric diagrams as an effective notation’, makes the case that geometric diagrams are distinctively effective in proofs, due to their computational properties. Furthermore, he argues that the later Wittgenstein agreed that the distinctive visual properties of proofs were essential for them to work as proofs. Wittgenstein's overall discussion of proofs in RFM thus supports the author's conclusion and further illuminates our understanding of proofs. In ‘Not a difference of opinion: Wittgenstein and Turing on contradictions in mathematics’, Wim Vanrie argues that, contrary to common readings of Turing and Wittgenstein's debate about contradictions in LFM, there is no strong difference of opinion between the two. This is because Wittgenstein accepts that, in the presence of a contradiction, things can indeed go wrong. In his view, however, the focus of the debate should be on the reasons why Turing thinks that contradictions are harmful and the calculi in which they feature (even if hidden) should be avoided: Turing thinks this is a mathematical defect, while Wittgenstein thinks it is not. For Wittgenstein, if things go wrong, then this must be for other reasons. All in all, this volume may be seen as a collection of papers that helps us to (1) better understand mathematical practice, (2) deepen our discourse about Wittgenstein's philosophy of mathematics and (3) familiarize the wider community of philosophers of mathematical practice with Wittgensteinian terminology, hopefully empowering them to use Wittgensteinian ideas and encouraging Wittgenstein scholars to move towards the first goal listed above.