Wittgenstein and the Philosophy of Mathematical Practice

In 1944 Ludwig Wittgenstein wrote that his “chief contribution ha[d] been in the philosophy of mathematics” (Monk, 1990,p. 466). Oddly enough, however, as opposed to his other philosophical contributions, his writings and lectures on mathematics have remained largely uncharted if not misconceived. The peculiarity of the matter becomes even more apparent, when we realize that the bulk of his oeuvre past 1929 was devoted to the topic.

What comes to the front, when examining these remarks on mathematics, is Wittgenstein’s insistence on understanding mathematics as a ‘human invention’ or ‘anthropological phenomenon’ set within and driven by a certain community. Mathematics is an activity; a practice that constructs “various ways of seeing conceptual possibilities and empirical situations, proof and logical methods central to its progress”. (Floyd, 2021)

Interestingly, in recent years, a subset of philosophers of mathematics have concerned themselves with the practice of mathematicians, instigating what is now known as the ‘philosophy of mathematical practice’ (PMP).

Among these philosophers, Wilder (1950, 1981), Pólya (1945, 1954, 1962), Lakatos (1976) and Kitcher (1984) are often mentioned as initiators of this movement, although PMP has mainly by driven in the last 20 years by the influential monographs and collected by the likes of Van Kerkhove (2007, 2009) and Van Bendegem (2007), Löwe and Muller (2010), Mancosu (2008), Ferreirós and Gray (2006), Giaquinto (2007) and Macbeth (2014), among many others.

The aim of this conference is to investigate the historical, present and future significance and influence of Ludwig Wittgenstein to the Philosophy of Mathematical Practice, bridging the gap between both scholarships.

Senior organizers: Bart Van Kerkhove and Erik Weber;
Junior organizers: Deniz Sarikaya and Vincent M. P. Vincke

We envision this conference mainly as a activity with in person exchange, so we strongly motivate contributed speakers to join us in person.

There is no participation fee for online participants,
There will be at most a small fee for in person participants.
We might need ask you to cover your own food at the conference venue (around 20 EUR).
We have no funds to cover travel or accommodation.

Please register / apply via this google form .

Thursday, 30. March 2023

10.00 – 10.30: Registration
10.30 – 10.45: Welcome
10.45 – 11.30: Invited talk 1: Sorin Bangu
11.30 – 12.00: Coffee break
12.00 – 13.00: Contributed Talks 1 & 2 : Wei Zeng*   //   Jack Verschoyle
13.00 – 14.30: Lunch
14.30 – 15.30: Contributed Talks 3 & 4:  Ásgeir Berg Matthíasson   // Deniz Sarikaya & José  A. Perez Escobar
15.30 – 15.45: Coffee break
15.45 – 16.45: Contributed Talks 5 & 6   David Chandler  //   Johannes Lenhard
16.45 – 17.30: Invited talk 2:  José A. Perez Escobar

19:30 - open end Conference dinner at Chez Léon

Friday, 31. March 2023

10.30 – 10.45: Welcome
10.45 – 11.30: Invited talk 3: Wim Vanrie
11.30 – 12.00: Coffee break
12.00 – 12.45: Invited talk 4:  Jordi Fairhurst & José  A. Perez Escobar
12.45 – 14.15: Lunch
14.15 – 15.15: Contributed Talks 7 & 8: Matteo de Ceglie  //   Paolo Degiorgi*
15.15 – 15.30: Coffee break
15.30 – 16.15: Invited talk 5:  Karim Zahidi
16:15 – 16:30: Closing Statement

Wittgenstein on the Benacerraf Dilemma
by Sorin Bangu

ABSTRACT: The Benacerraf Dilemma is one of the most discussed issues in the philosophy of mathematics today. In this talk, I shall explore a (later) Wittgensteinean take on it. I shall present the problem that Benacerraf formulated in his famous 1973 paper “Mathematical Truth,” and I shall also offer a very rough outline of Wittgenstein’s views on the philosophy of mathematics (as extracted from his Remarks on the Foundations of Mathematics [1937-1938] and his Lectures on the Foundations of Mathematics [1939]). Based on the distinction between solving and dissolving a problem (and implicitly, between a genuine problem and a pseudo-problem), the gist of the talk will be to argue that, instead of proposing (yet) another solution to the problem (as several philosophers have tried since Benacerraf articulated it), a Wittgensteinean way to deal with it would be to propose a dissolution of it.

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Mathematical practice and set-theoretic pluralism
by Matteo de Ceglie

ABSTRACT: In this paper, I argue that set-theoretic pluralism -- i.e. the position that there is no intended model of set theory, but a multiverse of them -- is a better position than universism -- the position that there exists only one set-theoretic universe, namely the cumulative hierarchy V -- regarding mathematical practice. To do so, I will focus on one particular part of mathematical practice: the correct formulation of the problem at hand. That this is a fundamental part of what the actual practice of mathematics entails can be already seen in Wittgenstein, that remarks in his Nachlass that ``[...] the correct formulation of the question is already the answer.'' (Wittgenstein Nachlass Ms-305,1[4]). Moreover, such a position can also be seen in the debate on the definiteness of CH (see e.g. Feferman (2014)), in which the interest of finding a solution to CH is challenged on the grounds that it is not a correctly formulated mathematical problem. Against this background, I argue that set-theoretic pluralism gives us the tools and methods to correctly formulate such problems. Following Hamkins (2012), the correct formulation of mathematical problems for pluralism involves exploring the multiverse, finding the conditions on which a statement can be made true or false with forcing or other set-theoretic methods. In particular, the CH can be seen, from the pluralist perspective, at least as a definite logical problem (in Feferman's sense, i.e. a problem involving reference to axiomatic theories and consistency), if not even a definite ordinary mathematical problem. 

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Should we investigate the significance and influence of Wittgenstein on the philosophy of mathematical practice?
by David Chandler 

ABSTRACT: One of the more interesting aspects of Wittgenstein’s later writings, those which come after his return to both Cambridge and philosophy in general in 1929, is his promise to leave mathematics as it is, a commitment which he, no doubt, routinely violates on many occasions within the 'Philosophical Investigations' such as in his dismissive treatment of set theory as illegitimate. This sentiment extends from his overwhelmingly negative attitude towards Cantor’s diagonal argument in the earlier 'Remarks on the Foundations of Mathematics'. Nevertheless, the intention behind this assertion, as mentioned above, is almost entirely clear for all who possess any familiarity with his thought: philosophy is said to leave everything as it is, including mathematics, and no mathematical discovery, no matter how theoretically revelatory, can contribute to new solutions to genuine philosophical problems. (If there is, in fact, such a thing).

Therefore, if we wish to provide a resolute reading, the question is: should we investigate the significance and influence of Wittgenstein on the philosophy of mathematical practice? After all, we must ask ourselves what we want to come about through this investigation. Is the intention for mathematicians to understand Wittgenstein’s suggestions, and our observations regarding them, in order to incorporate them into their future work? Or rather, are we to avoid sharing these findings in some form of an interdisciplinary project? It is these normative issues that I will grapple with in this paper, alongside establishing a historical and philosophical background to the subject.

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The varieties of surprises in the Tractatus
by Paolo Degiorgi 

ABSTRACT: I draw on Wittgenstein’s later reflections on the different roles of the surprising in mathematical practice to shed light on two exegetical issues regarding the Tractatus. My point of departure is the second appendix to book I of the Remarks on the Foundations of Mathematics, where Wittgenstein notoriously distinguishes between two roles that the surprising might play in mathematical practice. Surprise of the first kind is essential to fruitfully engage with mathematics; it vanishes once the solution to the problem is thoroughly understood. The second form stems from erroneously thinking that mathematicians investigate ‘depths’ of which we have ‘no inkling of,’ as well as confusing calculations and proofs with experiments. Wittgenstein saw the predominance of this form of surprise to be detrimental to a genuine understanding of mathematics, because of its tendency to permanently enshroud mathematical results with an aura of mystery.

I argue that several passages from the Tractatus, such as 6.125 and 6.1261, are best understood as employing the second sense of 'surprise' discussed in RFM. I thereby apply Wittgenstein’s later remarks on the surprising in mathematics to elucidate the Tractarian distinction between the sinnvolle Sätze of natural science and the sinnlose Sätze of logic, as well as the Scheinsätze of mathematics. Then I argue that we can employ the discussion of the first role of surprise in RFM to shed light on the Tractarian distinction between significant propositions and Wittgenstein’s own elucidatory propositions (discussed in the 6.5s).

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A unified hinge account of testimony and regularities in mathematics

by Jordi Fairhurst

 Abstract: Recent literature has been concerned with hinge accounts of testimony (esp. prejudices) and mathematical propositions individually. However, it is not clear what the relationship between these systems of hinges is. In this talk, we will present a novel account that characterizes the relationship between these two systems of hinges in the mathematical practice. Furthermore, in the light of this unified hinge account of testimony in mathematics, we will explore how one can minimize or avoid epistemic injustice.

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The Most Important Thing: Wittgenstein, Engineering, and the Foundations of Mathematics
by Johannes Lenhard

ABSTRACT: In his “Remarks on the Foundations of Mathematics” (RFM), Ludwig Wittgenstein argues that contradictions might happen in mathematics without harm to the foundations and without harm to using mathematics. This liberal view caused a scandal and counts as untenable. My contribution revisits Wittgenstein’s claims from the perspective of mathematics as a tool and shows that it is much better supported than usually assumed.

The talk combines historical with systematic argument. I provide little known background about the situation at TU Charlottenburg (Berlin) when Wittgenstein set out to study engineering there. Alois Riedler, then the leading engineer at Charlottenburg, pushed for a new conception of mathematization in line with Wittgenstein’s later claims. Additionally, two examples of inconsistent, but widely used mathematical tools are sketched. Against this background, I (re-)interpret seminal passages of Wittgenstein’s RFM and also his exchange with Alan Turing during his 1939 lectures. There, Wittgenstein highlights that basic laws of thought are at issue and that reflecting on them would be “the most important thing” he has talked about.

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Was Wittgenstein a radical conventionalist?
by Ásgeir Berg Matthíasson 

ABSTRACT: According to Michael Dummett’s infamous reading of Wittgenstein’s philosophy of mathematics, Wittgenstein was a radical conventionalist who held that “the logical necessity of any statement is a direct expression of a linguistic convention” and not just “consequences of conventions, but individually conventional”. According to the radical conventionalist, then, our mathematical practices determine directly, however that is specified, for each mathematical proposition individually, that it is true or false.

In this paper, I defend Dummett’s reading of Wittgenstein as a radical conventionalist and argue that there, Wittgenstein’s view can best be described as one where our agreement about the particular case is constitutive of our mathematical concepts, and hence that each true statement of mathematics is directly conventional, and not a consequence of a prior adoption of a convention or rules.

I will argue that his view is actually able to withstand some of the most difficult objections that have been brought forward against the view, including those of Dummett himself. My goal is therefore not merely exegetical.

In the last part of the paper, I argue that if we read Wittgenstein as a radical conventionalist, interesting connections with work on mathematical practice reveal themselves, for example to mathematics as concept-formation (Lakatos 1976, Tanswell 2018), the relation of elementary mathematics to the actual world (Kitcher 1984, Ferreirós 2016), explaining “the unreasonable effectiveness of mathematics” (picking up on a suggestion from Pérez-Escobar 2019) and finally, that radical conventionalism can serve as a foundational theory for the philosophy of mathematical practice. 

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Meaning as use, rule-bending and AI safety
José Antonio Pérez Escobar and Deniz Sarikaya

ABSTRACT: TBA

The relevance of Wittgenstein’s philosophy of mathematics in measuring and modeling in biology
by José A. Perez Escobar

ABSTRACT: Biologists attempt to control the historicity, variability and “messiness” of biology via measurement practices to develop a rigorous, quantitative biology and improve the reproducibility of experiments. However, informed by the later Wittgenstein’s philosophy of mathematics, I will argue that the mathematical models produced are not mere representations of empirical structures (mathematics as descriptions) but can be used to guide measurement so that empirical phenomena conform to quantitative expectations (mathematics as rules of description). The first use can lead to the revision of the model if expectations are not met, while the second guides how measurement should be performed and adjusted. I characterize how each use of mathematical models makes sense depending on contextual nuances of the scientific practice. I will also raise caution on how scientists, in their quest to control the “messiness” of biology, may inadvertently exploit mathematical models in certain situations.

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Contradiction in Wittgenstein's 1939 Cambridge Lectures on the Foundations of Mathematics
by Wim Vanrie 

ABSTRACT: In the 1939 Cambridge Lectures, Wittgenstein characterizes the law of contradiction as "a result of continuing in a particular way the technique which we have in dealing with propositions" (p. 231). The aim of this paper, in a nutshell, is to seek an understanding of this remark — and others like it, which are spread throughout the lectures. This requires, among other things, achieving clarity about how Wittgenstein wishes to regard logic and mathematics as intrinsically connected to certain notions of practice and technique, and what role our 'fear of contradiction' plays in these practices.

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Application and a Wittgensteinian Objection to Formalism
By Jack Verschoyle

ABSTRACT: The aim of my talk is to suggest that a radical account of application attributed to the later Wittgenstein by Perez-Escobar and Sarikaya makes it hard to grasp what a Wittgensteinian criticism of formalism could be (If there is one – there are mixed opinions about this). If Wittgenstein does have a criticism of formalism it appears to be that formalism fails to appreciate the applicability of mathematics, in the sense of forming rules for descriptions. But this won’t do if Perez-Escobar and Sarikaya are right about Wittgenstein’s understanding of application because a formalist could then tell a story of the application/use of empty signs in this sense (imagine if the wallpaper people of the Lectures applied their designs).

In order to draw out this problem fully, I will suggest that – contrary to popular opinion – there are some philosophically significant points of agreement between the later Wittgenstein and Frege. Both Wittgenstein and Frege share in three interdependent aspects of their philosophy of mathematics: their theory of the nature of mathematical propositions; the importance of applicability to mathematics; and their objections to (at least complete) formalism. Wittgenstein’s accounts of the former two mean that he cannot use Frege’s criticism of the formalist (that it does not allow for the instantiation of mathematical concepts with other concepts (e.g., physical spacial concepts)).

I find that this discussion leads us back to the problem of correspondence between mathematics and the world, which has been discussed in recent work.

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Wittgenstein in Cantor's paradise
by Karim Zahidi

Abstract: Throughout Wittgenstein's later writings on mathematics there is a consistent negative appraisal of Cantorian set theory. In this talk I want to explore what Wittgenstein's critique amounts to. I will focus on his critique of Cantor's diagonal proof of the uncountability of the real numbers and will explore whether this critique is also applicable to Cantor's first proof of uncountability of the reals. This case study will then be used to explore Wittgenstein's view on the role of mathematical proof in concept formation.

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Wittgenstein and Conceptual Engineering in Mathematics
By Wei Zeng 

ABSTRACT: Although revising and replaying concepts is not a new method in the history of philosophy, it is only recently that conceptual engineering became a popular independent field in philosophy. In the literature on conceptual engineering, in addition to the general discussion of the conceptual engineering approach (Cappelen 2018), there are also discussions about projects of revising or replacing specific concepts, such as the concept of “woman” (Haslanger 2000), “truth” (Scharp, 2013). However, there is still little literature on whether mathematical concepts can be engineered. Tanswell (2018) argued that mathematical concepts are open-textured and, therefore, could be engineered, but she didn’t fully explore the practical issues of how to engineer mathematical concepts. Following Tanswell, I examine the possibility of applying the conceptual engineering method to mathematics from a practice-oriented perspective. Moreover, I investigate Wittgenstein’s view on mathematical concepts and conceptual change in mathematics and argue that Wittgenstein provides an account that makes engineering mathematical concepts practically possible. In particular, I argue that Wittgenstein’s account solves the “lack of control” problem that Cappelen takes as a challenge for the conduction of conceptual engineering projects.

The conference would not have been possible, without the generous support of the Royal Flemish Academy of Belgium for Science and the Arts, the CLPS: Center for Logic & Philosophy of Science of the Vrije Universiteit Brussel (VUB) and the Doctoral School of Human Sciences (DSh) of the VUB.  This event is endorsed by DMRCP: Diversity of Mathematical Research Cultrues and Practices.