MC-PMP-Brussels-2025

Masterclass in the Philosophy of Mathematical Practices 2025

with Michael Barany (Edinburgh)

Situating Mathematical Practices:
Materials, Institutions, and Critical Context

June 18–20, 2025

Vrije Universiteit Brussel

Brussels, Belgium

About

The Centre for Logic and Philosophy of Science (CLPS) of the Vrije Universiteit Brussel (VUB) will host its 7th Masterclass in the Philosophy of Mathematical Practices on June 18–20, 2025 with Michael Barany (University of Edinburgh). We intend the masterclass to be a fully interactive in-person event, with the twofold objective to understand in depth the materials presented in the lectures, and to provide early career researchers (MA students, PhD students and Postdocs) with an opportunity to discuss their ongoing work in a helpful and constructive environment.

The event will start on June 18 at 1pm and finish on the 20th.

We welcome talks on all topics in the Philosophy of Mathematical Practices. Talks connecting to the work of Michael Barany and the theme of this year's iteration "Situating Mathematical Practices: Materials, Institutions, and Critical Context" are particularly encouraged.

Lecture Titles and Abstracts

Materials

The first lecture will examine the materiality of mathematical practices and the sociological and historical methodologies by which one may investigate them. We will consider blackboards, paper, and a range of other technologies of reading, writing, and communication in mathematics, together with their implications for characterizing mathematical knowledge and practice.

Institutions

The second lecture will examine the institutional conditions and processes of mathematical work and identity. We will consider the social, epistemic, and bureaucratic organisation of mathematical workplaces, funders, meetings, societies, and other institutional forms, and will identify their significance for understanding the people, interactions, and knowledges of mathematics.

Critical Contexts

The final lecture will develop a suite of themes and interventions for the critical social and cultural study of mathematical practices based on the imperative of situating mathematical practices in relevant contexts.

Further details of lectures and suggested background readings and preparatory exercises will be distributed to registered participants in June.

Program

Wednesday, June 18: Room P2.0.01

13.00-13.15: Welcome

13.15-15.30: LECTURE 1, Materials

15.30-16.00: Coffee break

16.00-16.30: Erica Meszaros Writing Outside the Lines (in Clay)

16.30-17.00: Tianyi Zheng Unifying Mathematical Explanations: Applicational Use and Analogical Reasoning

17.00-17.30: Coffee break

17.30-18.00: J.P. Ascher Material and Conceptual Supply: The Problem Answered in Henry Oldenburg’s Office

18.00-18.30: David Bakker Newton Versus Descartes on the Exactness of Mathematics

Thursday, June 19: Room P2.0.01

09.00-12.00: LECTURE 2, Institutions

12.00-13.00: Lunch

13.00-13.30: Liis Soon “Aesthetic Considerations” in Mathematics and Formulations of Wigner’s Puzzle

13.30-14.00: Maksim Novokreshchenov Modelling in Practice: An Institutional Ethnography of Avian Flu Mathematical Models

14.00-14.30: Elena Menta Oliva Formal Proof Assistants in the Landscape of Mathematical Rigor

14.30-15.00: Coffee break

15.00-15.30: Aiden Sagerman Modernisms through Practices: A History of Diagrams in Algebraic Topology, 1930s-1970s

15.30-16.00: Karolina Tytko The Diversity of Mental Representations in Mathematical Practice. The Historical Case Study

16.00-16.30: Coffee break

16.30-17.00: Cato Andriessen Meaning and mechanization: Revising Frege and Boole on Logical Calculus. A response

17.00-17.30: Helena Winiger Formal Practices and Interdisciplinary and Transdisciplinary Research in STEM Dominated Collaborations

Friday, June 20: Room P2.0.01

09.00-09.30: Stef Bracke “More than Russell’s Dummy: Couturat’s Logicism in Les Principes des Mathématiques and Manuel de Logistique”

09.30-10.00: Atahan Erbas Formal Diagrams and Informal Proofs

10.00-10.30: Coffee Break

10.30-11.00: Tommaso Peripoli Hypotheses about the Nature of Mathematical Objects of Enrico Giusti: The Objectivation Process of Mathematical Entities and Group Theory as a Case Study

11.00-11.30: Nicola Gianola Theorems and Tautologies: Mathematics Reduced to Units of Measurement

11.30-12.00: Coffee Break

12.00-12.30: Martin Speirs Early Careers in the Early 20th Century –Perspectives from Niels and Harald Bohr

12.30-13.00: Mia Joskowicz Framing Mathematics: Title Pages of Euclid’s Elements in Early Modern Europe

13.00-14.00: Lunch

14.00-17.00: LECTURE 3, Critical Contexts

Abstracts

June 18, 2025

LECTURE 1: Materials

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Erica Meszaros (Brown University), Writing Outside the Lines (in Clay)

Babylonian mathematicians often recorded their work on clay. While wax writing tablets or leather offered these practitioners alternative ways of writing numbers, particularly in the later periods and in alphabetic scripts like Aramaic and Greek, the cuneiform numerals they developed were designed for impressing into clay tablets. Though notoriously durable and “permanent” in a way many biodegradable materials are not, these tablets are often viewed by modern scholars as static and linear. However, a careful examination of mathematical texts showcases how Babylonian mathematicians played with the placement of numbers and writing as part of the mathematical process.

This presentation will give an overview of the materiality of Babylonian mathematical texts, with a focus on achieving “nonlinearity” within the generally standard tradition of cuneiform writing. Erasures and the incorporation of previous breaks in subsequent copying are perhaps the most mundane examples, but recent research has highlighted the role that marginal numbers and numbers otherwise outside of the standard linear flow function as part of the mathematical process. To these examples, I will add instances from later mathematical astronomy of nonlinearity in the presentation and use of named quantities that function like variables, as well as where the compendium nature of the text suggests nonlinear use. My goal with these examples is to highlight how Babylonian practitioners achieved nonlinearity on tablets that often seem staid, and to discuss how this nonlinearity may be a part of a later “algorithmic” culture.

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Tianyi Zheng (KU Leuven, LMU Munich), Unifying Mathematical Explanations: Applicational Use and Analogical Reasoning

In recent research on mathematical explanation, a widespread approach has been to distinguish between intra-mathematical and extra-mathematical explanations. Many philosophers emphasize the explanatory role of extra-mathematical explanations by appealing to Greater Modal Strength (GMS)—the idea that a Distinctive Mathematical Explanation (DME) derives its explanatory power from mathematical inference itself, rather than from any underlying causal structure. However, this reliance on modality faces a serious tension: if we accept GMS, it becomes difficult to explain how mathematical properties could determine physical modal structures (Lange 2021). Yet if we reject GMS, it is equally hard to account for cases that consistently exhibit GMS, as highlighted by Barman (2025), particularly if DMEs are to be treated as genuine modal explanations.

In this talk, I will argue that the tension within DMEs can be illuminated by rejecting the traditional distinction between intra- and extra-mathematical explanations and instead unifying certain mathematical explanations through their applicational use. Much of the current account of DMEs, I contend, overlooks the crucial role that analogical reasoning plays in underwriting these explanations. I will defend this approach by drawing on an inferentialist framework, emphasizing how analogical reasoning between inferential structures operates at the level of mathematical application. To illustrate this, I will discuss two examples: (1) a DME utilizing the Borsuk–Ulam theorem as an instance of external application, and (2) the explanatory use of large-scale geometric properties of groups as an instance of internal application.

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J.P. Ascher (University of Edinburgh), Material and Conceptual Supply: The Problem Answered in Henry Oldenburg’s Office

Supply means to both provide materials and substitute equivalent materials. Considering the materials of mathematics, office suppliers can supply equivalent sheets from several different paper mills, or--in the 20th century--can provide coloured pencils from this or that manufacturer. Yet conceptual supply extends to problem lists, reviews, and copy for publishing. During the 20th century self-described supply houses contended for what was an acceptable substitution, while the new form of the abstracting journal supplied the structural images of mathematics. Interlinked material and conceptual supply can also be seen in Kevin Lambert’s recent work on the materiality of mathematical symbols. He argues that the material supplies used for thinking could be considered a kind of “mindware” and that office supplies can be used to trace systems of exchanging ideas.

Building on Lambert’s and other recent theories relating to supply, my talk outlines what I call the “supply history” of Henry Oldenburg’s office as Secretary for the Royal Society, one of the largest European offices of the seventeenth century. I propose to show how this history sheds light on the concept of supply for seventeenth-century mathematics. In particular, I argue that Oldenburg’s office supplied and was supplied by ephemeral knowledge from around the European view of the globe by strategically provisioning and substituting mathematical materials and concepts. I relate this analytic of supply to one used in 20th-century mathematics and abstracting journals, connecting early-modern and modern concepts of supply.

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David Bakker (Utrecht University), Newton Versus Descartes on the Exactness of Mathematics

It is well-known that Newton brought an unprecedented exactness to the study of mechanics, an exactness that was usually associated with mathematics, but which most thought to be impossible for such a physical subject as mechanics. A vocal advocate for characterizing geometry as exact, in contrast to mechanics, was Descartes. It is also well-known that Newton became increasingly hostile against the Cartesian scientific methodology. In this talk, I will describe Newton’s conception of ‘exactness’ and show how this is the opposite of the Cartesian view. According to Newton, it is not the subject of mechanics that is inherently inexact, it is merely our practice or knowledge of that subject: “[T]he errors do not come from the art but from those who practice the art.” (Principia, Preface). It is such a view of exactness that allowed Newton to bring an unprecedented exactness to mechanics. Newton’s view of the exactness of mechanics is itself part of a broader view about the exactness of mathematics. According to Newton, mechanics is not ‘watered down’ geometry; on the contrary, geometry is a form of mechanics and derives its exactness from the exactness of mechanical practice (and not from the ‘exactness’ of certain clear and distinct ideas).

June 19, 2025

LECTURE 2, Institutions

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Liis Soon (University of Tartu), “Aesthetic Considerations” in Mathematics and Formulations of Wigner’s Puzzle

A common way to phrase the issue of the unreasonable effectiveness of mathematics in the natural sciences is to frame mathematics as something developed primarily with “aesthetic considerations” in mind, while the empirical sciences are about the empirical world and seem to have nothing to do with our sense for the aesthetic. I aim to show that this is not a productive way of formulating Wigner’s puzzle: these assumptions about mathematics and empirical sciences don’t hold their ground.

First, building on Todd’s (2008) skepticism about the sincerely “aesthetic” nature of judgements made about mathematics, there is reason to doubt the “aesthetic” nature of such considerations – aesthetic language is covering up epistemic considerations. Unlike aesthetic judgements, such considerations are in no way “disinterested” in truth and utility. The presence of some form of pleasure in encounters with mathematics does not immediately make these experiences aesthetic.

Second, even if the “aesthetic” nature of such considerations is taken at face value, this formulation of Wigner’s puzzle can still be undermined. It could be shown that aesthetics is not the primary force shaping mathematics. We want our mathematics to be a coherent and consistent thing for epistemic reasons. Moreover, the role of “aesthetic considerations” in empirical sciences might not be so different from mathematics. Empirical evidence can only be interpreted with a theory already in hand, and the move from scientific theory to empirical reality isn’t as straightforward as this formulation of Wigner’s puzzle makes it out to be.

Reference:

Todd, Cain S. (2008). Unmasking the truth beneath the beauty: Why the supposed aesthetic judgements made in science may not be aesthetic at all. International Studies in the Philosophy of Science 22 (1): 61–79.

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Maksim Novokreshchenov (University of Edinburgh), Modelling in Practice: An Institutional Ethnography of Avian Flu Mathematical Models

Mathematical modelling of zoonotic diseases is a prevalent biosecurity tool used to predict and anticipate infectious outbreaks or the development of an ongoing epidemic. The models can have far-reaching consequences for multispecies collectives, as they often ground decisions and regulations about vaccinations, lockdowns, patterns and boundaries of grazing, and cullings. The outcomes of modelling are largely dependent on the ways more-than-human relationships are imagined and operationalised. This task is as complex, as these relationships themselves and the modelling of avian influenza is a prime example of that. Influenza H5N1 virus primarily affects two very different groups of animals — poultry and wild birds — with different needs, contagion patterns and risks, while it is also infamous for its ability to mutate and infect, as the current outbreak in the US shows, cows and other mammals. In my presentation, I will share preliminary findings from an ethnographic study of avian flu modellers in the UK. I will explore how these findings reveal the contingent nature of mathematical modelling practices across disciplines, institutions, and research contexts. Finally, I will reflect on how this study can support a critical examination of the environmental performativity of avian flu models, which, like many other mathematical tools, do not merely describe but shape ecological and epidemiological realities.

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Elena Menta Oliva (University of Barcelona), Formal Proof Assistants in the Landscape of Mathematical Rigor

Formal Proof Assistants (FPAs) are increasingly used in contemporary mathematics, thereby introducing new issues for the Philosophy of Mathematical Practice (PMP) to explain. A question arises as to whether these tools may require a reassessment of established standards of mathematical knowledge and related concepts. This talk will focus on one of these: What does it mean for a proof constructed within an FPA to be considered rigorous?

FPAs have already been mentioned in the literature on mathematical rigor to illustrate and reinforce the standard view —which relates rigor to formal proof. However, alternative accounts remain underexplored in connection with FPA-based practices. These include argumentation and dialogical models of proof, the recipe model of proof, and virtue-theoretic approaches to rigor (Tanswell, 2024).

In this talk, I argue that, while FPAs clearly work towards the standard view's ideal of rigor, they also introduce new interaction dynamics, writing practices, and epistemic values that resonate with the other accounts of rigor as well. For instance, the interface allows for interactions that resemble a strict audience scrutinizing a proof; the imperative programming paradigm reflects features emphasized by the recipe model; and the work in FPAs is motivated by virtues such as meticulousness.

Engaging more systematically with FPAs from within the PMP allows for a richer philosophically-informed description of these computer-assisted practices, bringing to light dimensions often overlooked when FPAs are viewed primarily as tools for formalization and verification. It also prompts a reassessment and refinement of existing theories to better accommodate case studies involving emerging tools.

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Aiden Sagerman (Cambridge University), Modernisms through Practices: A History of Diagrams in Algebraic Topology, 1930s-1970s

This talk investigates the shifting diagrammatic practices of algebraic topologists in the mid twentieth century. Beginning with Herbert Mehrtens, historians of mathematics have long been interested in a “modernist transformation” in European mathematics in the 1890s-1930s characterized by growing abstraction, formalism, and an anxious relationship with external referents. As Alma Steingart has argued, algebraic topology (and American mathematics at large) subsequently underwent a mid-twentieth-century “high modernist transformation” that centered on axiomatization and generalization as ends in themselves—repeatedly “invert[ing] the relationship between structure and content” within the subject. However, even as the subject’s language and conceptual structure radically changed, many of the practices of algebraic topology—particularly surrounding diagrams—were conserved. Drawing on a mixture of pedagogical texts and the archive of the Cambridge algebraic topologist Frank Adams, I consider three episodes in the history of topology: the use of geometric diagrams in the topology of the 1930s, the replacement of geometric diagrams by commutative diagrams in the 1940s-50s, and the hybridization of the two types of diagrams in the 1960s-70s. As topologists developed new practices and hybridized them with old ones, I argue, they simultaneously (re)constructed mathematical modernity and the boundaries between algebra and geometry. The high modernist transformation should thus be understood as a process of resignification and of shifting practices across contexts rather than a wholesale remaking of mathematical practice.

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Karolina Tytko (Independent Scholar), The Diversity of Mental Representations in Mathematical Practice. The Historical Case Study

The main idea of my proposal is described in (Tytko et al. 2023). The basic question is whether the diversity of representations (imagistic and linguistic) can be a positive factor, supporting solving various problems in a more effective way. And how this diversity can be shaped at the mental level in a social context, especially in the era of AI.

These considerations come from the analysis of the mathematical practice of Georg Cantor and Richard Dedekind (Cantor 1887; Błaszczyk, Fila 2020; Dedekind 1872; Błaszczyk 2005), two nineteenth-century mathematicians. There I noticed that the effectiveness of problem solving may be related to the use of the diversity of mental representations associated with specific concepts. It was also correlated with various factors, such as the influence of the scientific environment. I have explored this relationship using especially the example of Dedekind’s construction of real numbers (Tytko 2023). On the other hand, strong, erroneous beliefs may result from unconscious mental representations associated with a specific concept - i.e. from the lack of conscious reference to the diversity of these representations. I have analysed this relationship using the example of Cantor's proof against infinitesimals (paper under review for Annales of Mathematics and Philosophy, Special Issue: On Proofs).

The perspective of these considerations is related to semiotic embodied mathematics (Giardino 2018), using the notion of perceptual mental imagery (Nanay 2023) and is related to the problem of mathematical intuition (D’Alessandro, Stevens 2024) or the alignment problem (AP) in the context of AI (Pérez‑Escobar, Sarikaya 2024).

Błaszczyk, Piotr. 2005. „On the Mode of Existence of the Real Numbers”. Logos of Phenomenology and Phenomenology of the Logos, A.T. Tymieniecka (ed.), 88:137–55.

Analecta Husserliana. Dordrecht: Springer. https://doi.org/10.1007/1-4020-3680-9_8

Błaszczyk, Piotr, Marlena Fila. 2020. „Cantor on infinitesimals. Historical and modern perspective”. Bulletin of the Section of Logic 49 (2): 149–79. http://dx.doi.org/10.18778/0138-0680.2020.09

Cantor, Georg. 1887. „Mitteilungen zur Lehre vom Transfiniten. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, E. Zermelo (ed.), 378–439. Berlin: Verlag von Julius Springer. https://gdz.sub.uni-goettingen.de/download/pdf/PPN237853094/PPN237853094.pdf

D’Alessandro, William, Irma Stevens. 2024. „Mature intuition and mathematical understanding”. The Journal of Mathematical Behavior 76, 101-203. https://doi.org/10.1016/j.jmathb.2024.101203

Dedekind, Richard. 1872. „Stetigkeit und irrationale Zahlen”. Gesammelte mathematische Werke, Robert Fricke, Emmy Noether, Oystein Ore, 315–34. Braunschweig: Verlag von Friedr. Vieweg&Sohn Akt.-Ges. https://gdz.sub.uni-goettingen.de/download/pdf/PPN23569441X/PPN23569441X.pdf

Giardino, Valeria. 2018. „Manipulative imagination: How to move things around in mathematics”. THEORIA. An International Journal for Theory, History and Foundations of Science 33. https://doi.org/10.1387/theoria.17871

Nanay, B. 2023. Mental Imagery: Philosophy, Psychology, Neuroscience. 1. ed. Oxford University PressOxford. https://doi.org/10.1093/oso/9780198809500.001.0001

Pérez‑Escobar, J.A., D. Sarikaya. 2024. „Philosophical Investigations into AI Alignment: A Wittgensteinian Framework”. Philosophy & Technology 80 (37). https://doi.org/10.1007/s13347-024-00761-9

Tytko, Karolina. 2023. „Visual thinking and the socio-historical aspects in the introduction of number systems by Dedekind”. Filozofia Nauki 31 (2): 1–26. https://doi.org/10.14394/filnau.2023.0012

Tytko, Karolina, Franci Mangraviti, Natalia Rylko, Nurtazina Karlygash B. 2023. „New epistemology of mathematics and formal sciences in the age of AI. Critical concept kinds and diversity of mental representations”. Edukacja Biologiczna i Środowiskowa 2 (80). https://doi.org/10.24131/3247.230207

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Cato Andriessen (Ghent University), Meaning and mechanization: Revising Frege and Boole on Logical Calculus. A response

Building on Van Heijenoort’s (1967) distinction between “logic as calculus” and “logic as language”, historians of logic have revisited its roots in the 19th-century polemic between Frege and Schröder. With Schröder explicitly comparing Frege’s logic to Boole’s, the polemic reveals surprising similarities and diﬀerences between Fregean and Boolean logic, particularly in their symbolic notations. This paper elaborates on a recent interpretation by Waszek and Schlimm (2021), focusing on the concepts of calculus and mechanisation in Frege and Boole. Waszek and Schlimm argue that (1) Frege and Boole put forward very diﬀerent conceptions of a logical calculus, and (2) that Boole’s conception of a calculus as a symbolic method for problem-solving should be revived in light of a dominant Fregean outlook on the role of logical symbolisms. In line with (1), I examine how each thinker responded to diﬀerent conceptions of ‘mechanisation’ within their respective logical frameworks, and discuss the historical and institutional context underlying these claims. I deviate from Waszek and Schlimm, however, in arguing against (2). By emphasizing the connection between their concepts of mechanisation and formalisation, I suggest that Frege’s construction of an anti--mechanical (philosophy of) logic brings with it an implicit philosophical stance on the nature of language and of signs in particular – a stance foreign both to Boole and to contemporary philosophical accounts of logical symbolism. This Fregean perspective oﬀers valuable insights for current studies on the role of written symbols in logic and mathematics.

Sources cited

Van Heijenoort, J. (1967). Logic as Calculus and Logic as Language. Synthese, 17(3), 324–330.

Waszek, D., & Schlimm, D. (2021). Calculus as method or calculus as rules? Boole and Frege on the aims of a logical calculus. Synthese, 199(5–6), 11913–11943.

h\ps://doi.org/10.1007/s11229-021-03318-x

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Helena Winiger (ETH, Zurich), Formal Practices and Interdisciplinary and Transdisciplinary Research in STEM Dominated Collaborations

From a societal and scientific perspective, many problems increased in complexity over the past decades. Such complex problems are asking for collaborative solutions across fields, disciplines, and societal actor groups. To address them collaboratively and accordingly, inter- and transdisciplinary research (IDR/TDR) are fostered through policy and funding schemes, since their practices are known for diverse collaboration and for a science-society dialogue sensitive to societal needs. However, in being institutionalized, IDR/TDR can face barriers to responsively address such complex problems across fields and societal actor groups.

The formalization of research practices can be thought of as a relevant factor in institutional boundary-setting to IDR/TDR collaboration. In many cases of science, technology, engineering, and mathematics (STEM) fields and of STEM-dominated IDR/TDR collaborations, practices are increasingly abstracted, formalized, standardized, and automated. They can, therefore, seem detached from the particular, less formalized, changing, and lived realities in focus of the practices in arts, humanities, or social sciences (AHSS). Also, they can seem closed to a dialogue with society, since their practices claim to be objective, and hence refuse to an interdependence with societal practices.

This study is anchored in the project “Investigating interdisciplinarity and transdisciplinarity: intersections of practices, culture(s) and policy in collaborative knowledge production (INTERSECTIONS)». I approach questions related to philosophy of formal practices and IDR/TDR from a meta-research perspective. In this study, qualitative methods are used. Following a literature review, empirical data is gathered through multi-sited ethnographic research in large research consortia in Switzerland conducting IDR/TDR in STEM-dominated fields and collaborations.

June 20, 2025

Stef Bracke (Ghent University), “More than Russell’s Dummy: Couturat’s Logicism in Les Principes des Mathématiques and Manuel de Logistique”

Louis Couturat’s logicism is often considered to be a mere oversimplification of Bertrand Russell’s logicism. Couturat’s Les Principes des Mathématiques, scholars argue, is nothing more than an oversimplification of Russell’s The Principles of Mathematics. In this paper, however, I will argue that Couturat was an original philosopher in the logicist tradition, and I will show what his originality consists of. Using Couturat’s recently recovered Manuel de Logistique—which was written contemporaneously to his Principes—to shed new light on the Principes, I will emphasize two crucial differences of Couturat’s logicism to Russell’s logicism.

The first difference concerns their respective conceptions of logic. Couturat’s logical system, it will be argued, consists of an original synthesis of, on the one hand, Russell’s ideography and, on the other hand, logical operations in line with the algebra of logic tradition—the last of which is explicitly rejected by Russell. The second difference concerns their conceptions of the relationship between logic and mathematics. Whereas Russell reduces mathematics to logic using a set of logical axioms, Couturat explicitly rejects any use of axioms. Instead, he proposes to solely use definitions. An exploration of Couturat’s position on this issue culminates in a discussion of his striking characterization of the logical analysis of mathematics as analytic a posteriori. Taking Couturat’s conception of logic and its relation to mathematics together, I will conclude that Couturat’s logicism, far from being an oversimplification of Russell’s, is highly original.

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Atahan Erbas (KU Leuven), Formal Diagrams and Informal Proofs

In the current literature on the philosophy of mathematical practice, it is claimed that a proof is not necessarily formal in order to be mathematical (Tanswell, 2024). In other words, informal proofs can also be mathematical proofs. In contrast, it is argued that for diagrams to be mathematical (or logical), they must be formal and systematic (De Toffoli, 2022). The motivation behind this idea is to show that diagrams also play an important role in mathematical proofs. But why can mathematical proofs be informal, but mathematical diagrams used in mathematical proofs must be formal? This talk seeks to answer this question and is divided into three parts.

First, we give a new definition of formality, based on Novaes’ book Formal Languages in Logic (2012). We argue that a necessary condition for being formal is being decomposable. By decomposability we mean that the steps of an inference can be clearly distinguished from each other after the inference has been made. From this we argue, second, that proofs have a formal structure no matter how informal they are called. Third, we argue, on the contrary, that diagrams have an informal structure no matter how formal they are. Accordingly, we go one step further and conclude with a new twist on the increasingly blurred distinction between diagrammatic and non-diagrammatic notations.

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Tommaso Peripoli (University Statale di Milano and the School for Advanced Studies IUSS Pavia), Hypotheses about the Nature of Mathematical Objects of Enrico Giusti: The Objectivation Process of Mathematical Entities and Group Theory as a Case Study

Hypotheses about the Nature of Mathematical Objects is a small booklet published by Giusti in 1999. Although it primarily addresses the history of mathematics, it also presents several philosophical theses. The book's main argument is that mathematics is a discipline that deals with entities undergoing a specific process of objectification. Initially, mathematicians invent tools to solve specific problems. Over time, these tools become associated with specific problems, and eventually, they are recognized as independent objects, detached from their original practical contexts. Giusti illustrates this thesis with the aid of numerous historical examples. He takes into consideration various types of mathematical objects and analyzes their relationships to different phases of the objectivation process.

Among the many case studies presented, the most significant one is that of groups. Galois viewed groups as an innovative method for solving algebraic equations, but it was only later that groups emerged as autonomous entities within the Platonic realm.

I intend to develop more the case study of groups to explore Giusti's ideas further. Ultimately, I will argue that while Giusti's book is not a philosophical essay, it aligns well with the contemporary philosophical sensibility surrounding mathematical practice; it offers a philosophical reflection that starts with the historical development of mathematics.

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Nicola Gianola (University of Rome Tor Vergata), Theorems and Tautologies: Mathematics Reduced to Units of Measurement

My proposal for the Masterclass examines Ludwig Wittgenstein’s critique of the logicist conception of mathematics, which holds that mathematical propositions are mere tautologies devoid of informational content. As pointed out by Hans Hahn, when taken to its extreme, this view fails to justify the effectiveness of the application of mathematics to the natural sciences. The aim of this contribution is to show how Wittgenstein, while maintaining that mathematical propositions are true by virtue of the meaning of their terms, partially recovers the Kantian insight of mathematics as a synthetic a priori science. Through two elementary examples – an arithmetic operation (3 + 4 = 7) and a Euclidean geometry theorem (the congruence of the angles opposite the equal sides of an isosceles triangle) – I aim to illustrate how, according to Wittgenstein, mathematical propositions cannot be reduced to tautologies but are instead 'grammatical propositions' that define and enrich the meaning of the mathematical concepts involved. Mathematical proofs are interpreted as a paradigmatic procedure that establishes new rules of inference (theorems), which can be applied to experience in order to gain innovative information. In this sense, mathematics not only retains its universal, necessary, and a priori character, but also performs a transcendental function, making it possible to obtain new knowledge about the world. This proposal seeks to contribute to the re-evaluation of the epistemic role of mathematics by overcoming the dichotomy between analyticity and syntheticity.

References:

Hahn, Hans (1959), “Logic, mathematics and knowledge of nature”, in Logical positivism, The Free Press;

Schroeder, Severin (2021), Wittgenstein on Mathematics, Routledge, New York;

Skovsmose, Ole (2024), “Mathematics as Logical Tautologies”, in Critical Philosophy of Mathematics, Springer;

Steiner, Mark, (1998), The Applicability of Mathematics as a Philosophical Problem, Harvard University Press;

Wittgenstein, Ludwig (1922) (1969), Tractatus Logico-Philosophicus, Routledge & Kegan Paul, London;

Wittgenstein, Ludwig (1974), Philosophical Grammar, ed. R. Rhees, trans. A. J. P. Kenny, Blackwell, Oxford;

Wittgenstein, Ludwig (1975), Philosophical Remarks, Blackwell, Oxford, 2nd edn;

Wittgenstein, Ludwig (1976), Wittgenstein’s Lectures on the Foundations of Mathematics, Cornell University Press, New York;

Wittgenstein, Ludwig (1978), Remarks on the Foundations of Mathematics, Blackwell, Oxford, 3rd edn;

Wittgenstein, Ludwig (1979), Wittgenstein and the Vienna Circle. Conversations recorded by Friedrich Waismann, Basil Blackwell, New York;

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Martin Speirs (University of Copenhagen), Early Careers in the Early 20th Century –Perspectives from Niels and Harald Bohr

The brothers Niels Bohr and Harald Bohr wrote many letters to each other between 1906 and 1950. The letters from the period 1909-1916 provide a glimpse into the early stages of their scientific careers, Harald as a mathematician and Niels as a physicist. This project focuses on their careers during this period, with a special focus on publication practices. The letters display a remarkable sense of publication imperatives, and publication strategies: Where to publish? Which languages to choose? Which types of results to publish? Why publish?

In the 21st century it is common sense for an early career mathematician or physicist to want to publish many scientific papers. This is the only way to “build a career” within the modern institutionalization of science. At the beginning of the 20th century this was not as clear cut. Indeed, at the time academic credit could also be built using other forms of criteria, for example teaching qualifications, nationality, and institutional seniority. By studying the correspondence between the two brothers, this project aims to contribute to an understanding of the interrelation between publication practices and mathematical knowledge and practices.

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Mia Joskowicz (Tel Aviv University), Framing Mathematics: Title Pages of Euclid’s Elements in Early Modern Europe

Euclid’s Elements stands as one of the most influential mathematical works since antiquity, with numerous translations and adaptations published across early modern Europe. This presentation examines the title pages of these editions as liminal spaces that mediate between the text, its producers, and its readers. Featuring the title, content descriptions, contributors, and sometimes illustrations, title pages serve multiple roles: They designate the book's identity, advertise its contents, and reflect contemporary aesthetic and intellectual conventions. These elements reveal pedagogical and epistemological frameworks, offering insight into the circulation of the text and the social, intellectual, and commercial forces shaping its transmission.

This study analyzes over 300 title pages from Elements editions and related texts printed before 1700. The title pages are analyzed through a paratextual lens, categorizing each of their segments to reveal how the mathematical and non-mathematical content of the text was selectively emphasized, reinterpreted, or omitted based on the intended readership, authorship, and social context. This approach highlights how authors, printers, and publishers adapted and framed Elements for a wide range of audiences.

Through the examination of its title pages, the study exposes the active role of the text in contemporary debates. It highlights tensions between competing intellectual priorities—fidelity to Greek sources, philological rigor, mathematical innovation, pedagogical concerns, and practical utility. These discourses and tensions, rather than signaling a linear progression, shed light on broader mathematical shifts during the period and the connections between them and the changing, diverse practitioners who engaged with the text.

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LECTURE 3, Critical Contexts

Call for abstracts

We invite interested early career researchers to send us an abstract of at most 250 words by May 2. Please submit your abstract, including your affiliation and status (bachelor’s student, master’s student, PhD student, postdoc, other) by sending it to the following email address: zhao.fan@vub.be .

The talks will consist of a 20 minute presentation followed by 10 minutes for discussion. Notification of acceptance will be sent out by May 15. Notice that submitting an abstract is not mandatory for attending the Masterclass.

Registration

Registration for the event is closed. Attendance is free and open to anyone interested, but registration is required by sending an email to zhao.fan@vub.be including your affiliation and status (bachelor’s student, master’s student, PhD student, postdoc, other).

Travel Grants

We hope to be able to provide travel grants (up to 300 Euro each) for those who do not have other sources of funding to attend the event. To apply, please send a short description of your situation to zhao.fan@vub.be . Priority will be given to speakers, but all attendees may apply for a travel grant. Deadline was May 2.

Organizers

The masterclass is organized by Line Edslev Andersen (VUB), Zhao Fan (VUB), Thomas Glasman (VUB), Sander Pouliart (VUB), Colin Jakob Rittberg (VUB), Deniz Sarikaya (VUB & Universität zu Lübeck)

The Masterclass honors Joachim Frans (1989-2023) who co-organized the Masterclass for many years.

For any questions write an email to zhao.fan@vub.be

Acknowledgements and Supports

The masterclass is supported by the Centre for Logic & Philosophy of Science (CLPS) of the Vrije Universiteit Brussel (VUB), UK Research and Innovation (UKRI), UK government’s Horizon Europe funding guarantee [grant number EP/X033961/1], the Belgian Society for Logic and Philosophy of Science (BSLPS) // National Centre for Research in Logic (NCNL/CNRL), and the Doctoral School of Human Sciences (DSh) of the VUB.This event is endorsed by DMRCP: Diversity of Mathematical Research Cultures and Practices and the APMP: The Association for Philosophy of Mathematical Practice.

This event is endorsed by:

The Association for the Philosophy of Mathematical Practice and the CIPSH Chair: Diversity of Mathematical Research Cultrues and Practice

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