Lecture Titles and Abstracts

We welcome talks on all topics in the Philosophy of Mathematical Practices. Talks connecting to the work of Silvia De Toffoli are particularly invited. The titles and abstracts will be announced soon on the homepage. 

Program

Day 1

12:00-13:00 Lunch

13:00-16:00 Silvia De Toffoli

16:00-16:30: Coffee

16:30-17:00 Simone Paolo Roca: Understanding Proofs As Models

17:00-17:30 Carlos Falcon: Rule-Governed Semantics in Mathematical Practice Extending Wittgenstein’s Account of Rule-Following

Day 2

9:00-12:00 Silvia De Toffoli

12:00-13:00 Lunch

13:00-13:30 Zhuohan Yang: What the Blueprint Does in the Hybrid Epistemic Network: A Closer Look at Mathematical Formalization

13:30-14:00 Til Eyinck: Mathematical Folklore, Or: Shared Cognition In Mathematics

14:00-14:30 Coffee

14:30-15:00 Yanji Ding: Mathematical Definitions as Pragmatic Acts: Proof, Inquiry, and Local Authority in Mathematical Practice

15:00-15:30 Fleur Hubau: And Then There Was Chaos? The “Puzzling” Gap Before Chaos Theory

15:30-16:00 Coffee

16:00-16:30 Adriana-Georgiana Gustă: Rethinking Statistical Totality through Badiou’s Set Theory

16:30-17:00 Stefanos Jones: Formalization as a piece of mathematical practice

Day 3

10:00-10:30 Fabio Ceravolo: Mathematical justification as community-aligned mathematical belief

10:30-11:00 Paul Hasselkuß: Re-Evaluating Sharability, Transferability, and Formalizability in Mathematical Proofs

11:00-11:30: Coffee

11:30-12:00: Flavia Bruni: Constructing Continuity: A Kantian Interpretation of Dedekind’s Analogy Between Real Numbers and the Line

12:00-13:00: Lunch

13:00-16:00: Silvia De Toffoli

Abstracts

Invited Talks by Siliva De Toffoli

Lecture 1: Fallibilist Epistemology of Mathematics

In my first lecture, I will discuss how to be a fallibilist in mathematics. I will start by clarifying what it means to be a fallibilist about justification and knowledge and why it makes sense to be fallibilist not only about contingent matters but also about necessary truths. I will then survey different forms of mathematical fallibilism, ranging from mundane to more controversial.  Finally, I will zoom in on my preferred form of fallibilism in mathematics, according to which a subject can be justified in believing a mathematical proposition in virtue of a simil-proof, roughly an argument that is put forward as a proof and satisfies the standard of acceptance for proofs of a legitimate mathematical community although it might later turn out to contain a mathematically significant error.

Lecture 2: The Technological Turn in Mathematics (joint work with Fenner Tanswell)

Quickly evolving technologies, such as Interactive Theorem Provers (ITPs), Automated Theorem Provers (ATPs), and Large Language Models (LLMs), all falling under the general heading ‘AI for mathematics,’ are transforming mathematical practice in profound ways. This lecture explores the implications of these innovations, focusing on their impact on how mathematical knowledge is created and shared. It also discusses how they are reshaping the social dimension of mathematics, altering collaboration dynamics, trust relationships, and the collective production of knowledge. For instance, tools like ITPs facilitate large-scale collaborations and make new types of teamwork possible, where trust is not a necessary ingredient. ITPs also help us mitigate our human fallibility, yet they raise questions about the nature of formalization and the relationship between traditional and formal mathematics. Technologies such as LLMs are reshaping the division of epistemic labor between humans and machines and urge philosophers of mathematics to ask questions about the value of their work.

Lecture 3: The Future of Mathematical Practice

Rapid advances in AI for mathematics raise questions not only about how mathematics will be done and how scientific collaborations will be organized, but also about the value and aims of mathematics as a human activity. Traditionally, proofs have been regarded as the primary outputs of mathematical inquiry. However, as AI systems become increasingly capable of producing proofs, mathematicians are realizing that mathematical inquiry is not only about proofs. Mathematicians may be presented with formally correct proofs that they nevertheless find unsatisfying, opaque, or unilluminating.  As Klowden and Tao (2026) put it, machine-generated proofs can seem “aseptic,” whereas human proofs come with a distinctive “smell.”  In this lecture, I will discuss recent developments in AI in mathematics to explore possible futures of mathematical practice. Drawing on Thi Nguyen’s distinction between process-oriented and outcome-oriented activities, I will suggest that mathematics occupies an intermediate position between science and the arts. Mathematicians value the results of inquiry, but they also value understanding, insight, explanation, elegance, and the experience of discovery—goods that are often inseparable from the processes through which mathematical results are obtained. This raises broader questions about the future division of epistemic labor between humans and machines: What should mathematicians do when machines can prove theorems? Which parts of mathematical inquiry should we want to automate? What forms of mathematical understanding might remain distinctively human? I will argue that the rise of AI in mathematics does not merely provide mathematicians with new tools; it gives us an opportunity to reconsider the aims and values of mathematical practice itself.

Contributed Talks

September 16, 2026

Zhuohan Yang (Durham University), What the Blueprint Does in the Hybrid Network: A Closer Look at Mathematical Formalization

De Toffoli and Tanswell (2026) describe mathematical formalization as a hybrid epistemic network: humans, machines, and infrastructure jointly produce and certify proofs. They note that the concept “suggests something more discrete than the case we have examined.” This paper takes a closer look, anchored on one node within the network: the Lean blueprint.

Buzzard, leading the formalization of Fermat’s Last Theorem, calls the blueprint the “crucial ingredient” of large-scale formalization projects. The blueprint is a document that breaks a proof into formalizable lemmas, shows the dependencies among them, coordinates contributors, and tracks progress.

I argue that the blueprint occupies a distinctive position in the network, different in kind from other nodes. Zullip and GitHub are infrastructures for collaboration: they enable mathematical work without themselves being about mathematical content. The blueprint, by contrast, is the organization of mathematical content itself. It is the proof, decomposed and made workable. Producing a blueprint requires mathematical expertise of its own. The author has to decide how to modularize a long argument and what adjustments the paper proof needs to fit a formal system. A blueprint is also revised as the project’s understanding develops. It bridges informal mathematical thought, and the formal proof Lean’s kernel will verify.

This reframes De Toffoli and Tanswell’s concept. The hybrid character of the network is not only that humans and machines work together; it is also that the network’s nodes are of different kinds. Some carry mathematical content; others provide infrastructure. Specifying this internal structure begins that close examination.

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Carlos Falcon (National University of Distance Education), Rule-Governed Semantics in Mathematical Practice: Extending Wittgenstein’s Account of Rule

This work-in-progress explores whether the later Wittgenstein’s account of mathematics as rule-following can be extended from the syntactic level of mathematical operations to the semantic and interpretative level of mathematical practice. Wittgenstein’s remarks on mathematics often emphasize rules, techniques, calculation, and grammar: what counts as following a rule, applying a procedure, or continuing a calculation correctly. However, mathematical practice also involves higher-order interpretations of symbols and operations: saying that a statement is true, that a mathematical object exists, that a symbol represents a quantity, or that a transformation preserves meaning.

The central hypothesis of this paper is that such semantic interpretations need not be understood as references to independently given abstract entities, nor as merely subjective additions to formal manipulation. Rather, they may be understood as rule-governed practices stabilized within mathematical communities. On this view, the operativity of mathematical expressions presupposes not only syntactic rules of transformation, but also a minimal, normative semantics that determines what counts as a correct interpretation, a valid transformation, or an acceptable application.

The paper will examine this hypothesis through elementary cases of arithematical notation and symbolic transformation, asking how signs such as “5”, “+”, and “=” become meaningful and operative within shared practices of counting, calculating, correcting, and justifying. The aim is to sketch a preliminary account of mathematical semantics as a communal and normative practice, without reducing mathematics to pure syntax or appealing to absolutist metaphysics.

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September 17, 2026

Simone Paolo Roca (IUSS Pavia), Understanding Proofs As Models

Our understanding of mathematics mostly comes from our way of constructing, grasping and explaining proofs. However, many of the proofs that the mathematical community currently produces and publishes (we can call them contemporary proofs) are very sophisticated and long. If computer-assisted proofs are also considered, this trend goes to the extreme (Klowden&Tao 2026).

These proofs are informal, in the sense that they contain lots of gaps and implicit elements, often indexed to a certain mathematical field. This makes them very different from the ideally rigorous structure of mechanically checkable formal proofs (Burgess&De Toffoli, 2022).

How can human agents feasibly understand contemporary proofs? What is the most effective and efficient way to carry out this epistemic process? Given our limited cognitive resources, available information, and time, we are not always able to go through all proofs in all their details. Moreover getting lost in an ultra-particular level of granularity might be counterproductive when it comes to understanding them.

I therefore suggest that understanding a proof is best framed as a practice of scientific modeling. Modeling is the practice of representing complex objects through models to make them cognitively accessible (Frigg 2023). An agent (or community) feasibly understands a proof if she constructs a non-factive, defeasible model of the proof’s architecture. This process is driven by identifying crucial elements within the proof that act as shareable and contextual epistemic clues of its architecture.

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Til Eyinck (University of Cologne), Mathematical Folklore, Or: Shared Cognition in Mathematics

There is an ongoing trend towards shared cognition – or, at least, towards shared authorship – in mathematics. This trend is diametrically opposed to what looks like a lasting individualist focus in the epistemology of mathematics. For there seems to be a trend towards wanting to understand central topics in (the philosophy of) mathematics and (in the philosophy of) its practice primarily at the level of individual, cognitive agents. There is no doubt that the individual-level-approach (ILA) is a good and valuable approach. However, the social, cooperative nature of mathematics and mathematical practice is arguably often overlooked in going down (exclusively) the ILA-road. This talk attempts to contribute to this often neglected dimension of mathematics (at least in the philosophy of mathematics, perhaps less so in the considerations of mathematicians regarding their own field). The talk is an attempt to consider not only the ILA, but also a group-level-approach (GLA) as a promising approach in the philosophy of mathematics. The aim is to explore mathematical folklore, that is, cumulative cultural knowledge within mathematical practice and the mathematical community.

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Yanji Ding (Ruhr University Bochum), Mathematical Definitions as Pragmatic Acts: Proof, Inquiry and Local Authority in Mathematical Practice

This project investigates the pragmatic role of mathematical definitions in mathematical inquiry. Starting from Lakatos’s account of proofs and refutations, I examine how definitions emerge, change, and acquire authority within mathematical practice. Lakatos shows that mathematical definitions are often not fixed in advance, but are revised through the process of conjecture, proof-attempt, counterexample, and conceptual adjustment (Lakatos, 1976). His notion of proof-generated definitions is especially important: some definitions are justified because they are required for a valuable proof to work. However, as Werndl argues, proof-generation is only one way in which mathematical definitions are justified; definitions may also be supported by their ability to capture preformal ideas, correspond to important mathematical conditions, or remove redundant assumptions (Werndl, 2009).

Building on this literature, I propose to analyze mathematical definitions as pragmatic acts. A definition does not merely introduce a term or abbreviate a formal condition; it also updates the context of mathematical inquiry. It determines which objects count as relevant cases, which inferential moves are licensed, and which questions become mathematically salient. This allows us to explain a distinctive tension in mathematical practice: definitions are fallible and revisable at the level of inquiry, yet locally authoritative within a well-formed proof. Once accepted in a given context, a definition regulates the use of concepts and structures deductive reasoning. The project therefore aims to connect philosophy of mathematical practice with formal pragmatics.

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Fleur Hubau (Radboud University), And then There Was Chaos? The “Puzzling” Gap Before Chaos Theory

Physics in the 20th century saw the rise of two rapid scientific revolutions: quantum mechanics and relativity. In classical physics, a delayed, third revolution is associated with the phenomenon of “chaos”. Between the late 1960s and the beginning of the 1980s, numerous publications were centred around the idea of sensitivity to initial conditions, a core feature of chaotic phenomena. However, over more than half a century earlier, Henri Poincaré (1854-1912) had already laid the foundations of what is now called “dynamical systems theory” (or more popularly “chaos theory”). This “conspicuous” (Barrow-Green) and “puzzling” (Ruelle) “nontreatment of chaos” (Kellert) has startled commentators. While previous explanations pointed to the influence of computational tools and the emergence of quantum mechanics and relativity, Park has recently argued that empiricist philosophy “exiled chaos from physics”. Using the work of Duhem and Hadamard, Poincaré's contemporaries, Park concludes that sensitive dependence was seen as a “mathematical pathology” rather than a “real physical phenomenon”. In this talk, I will argue that Duhem and Hadamard did not dismiss the idea of unobservable differences. Rather, following Poincaré, they questioned the usefulness of mathematical deduction given practically determined initial conditions. Since the measurement of initial conditions always contains some error, the application of mathematical models to predict physical phenomena may not always be justified. The basis of this argument is the acknowledgement that small variations of initial conditions can produce large variation in the effect, an observation that is very much in line with what was to grow into “chaos” later.

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Adriana-Georgiana Gustă (Alexandru Ioan Cuza University), Rethinking Stastical Totality through Badiou’s Set Theory

As Leibniz recalls in his "Essays of Theodicy", the habit of geometers has always been to forget that „wholes are made of parts that are not so much similar, but put together similarly”. Since then, though, many have argued that this type of obscuration has become the habit of statisticians, with the aid of "abstraction”, proper to mathematical thought, as they tend to break society up into parts, portraying it fully analyzable. We can see a plethora of negative implications in this conception, yet if we see mathematics as a mere tool, we are neglecting important internal constraints. Through a deeper dive into Aristotle, we can see how counting, using aphaeresis, strips entities down, yet we can also see how it creates a levelled plane. We argue that, while the act of counting can be viewed as a "tool" that draws up biased statistics, it can also rethink the relationship between the part and the whole.

Central to this project is the effort to contribute to Alain Badiou’s "Number and Numbers", drawing from modern set theory. By closing in on von Neumann ordinals, we investigate how the statistician transforms inconsistent multiplicity into structured presentation. Furthermore, we utilize Badiou’s concept of the "excess of inclusion over belonging" to demonstrate that mathematical totality is inherently dynamic. By critiquing mathematics being seen as a mere "sterile machine" or "neutral tool," this work seeks to reconcile internal mathematical constraints and the social deployment of number, offering renewed potential to the statistical act.

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Stefanos Jones (University of Chicago), The myth of the "essentially informal" proof. 

The Standard View of proof holds that since fully formal proof is the ideal of proof, an informal proof is correct only to the extent that it can be routinely formalized, and moreover that this is the criterion we rely on in ordinary mathematics. An extensive literature—often from the PMP tradition—attacks the Standard View on the grounds that informal proof is of a fundamentally different nature to formal proof and needs to be explained on its own terms. Without reverting to the Standard View, I argue against this. I try to show that it is distinctive of modern mathematics that its informal practice is tuned to some or another formal system, and that this needs to be a part of any account of the success of how ordinary mathematics ‘works.’ Drawing on the literature on the notion of mathematical style, I discuss both 20th-century and recent examples of this phenomenon. This sheds a certain light on the importance of type-theoretic interactive provers and raises questions about the current trends towards autoformalization. 

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September 18, 2026

Fabio Ceravolo (Sichuan University), Mathematical justification as community-aligned mathematical belief

Call guideline fallibilism the view that one is justified in believing a mathematical claim even when that belief tests on incomplete, unsound or invalid arguments, provided the arguments meet the standards of a mathematical community with specified features (De Toffoli 2021; 2024; 2025). Satisfaction of standards is a counterfactual or even temporal matter: the relevant community would (De Toffoli 2025) or will (Hamami 2019) X the argument, where X is a collective practice such as a group acceptance, understanding, revision, or evaluation.

A question for guideline fallibilist is whether justification necessitates satisfaction of standards. With necessitation, fallibilists permit a reductionist view of justification. Justified belief in (e.g.) Fermat’s theorem based on Wiles’ original incomplete argument just is the fact that the relevant community would X that argument. This reductionism implies that even incompetent or temporarily incapacitated mathematicians would be justified in believing the theorem, provided the community would X the argument (De Toffoli 2025: 26-7).

One way out is to demand for justification individual reliability. On this view, justified belief necessitates the satisfaction of standards, but the two are not identical because reliability must also be present for justification (Ib. 27-8). The path I explore here is less common. Identify justification with satisfaction of standards and then offer an error theory for the intuition that incompetent and incapacitated mathematicians are unjustified when their beliefs rest on arguments the community would X. On this theory justification is a genus property of belief (Gerken 2013; 2022: 57-9) and its species are the properties that the belief issues from differently reliable capacities to reason. The intuition fails, then, simply because we are prone to mistaking non-species for non-genera (when genus boundaries are unknown).

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Paul Hasselkuß (Heinrich Heine University Düsseldorf), Re-Evaluating Sharability, Transferability, and Formalizability in Mathematical Proofs

In the philosophy of mathematical practice, sharability, transferability, and formalizability distinguish types of mathematical justification. However, their precise logical interrelations remain under-theorized. This paper investigates their necessary and sufficient conditions, arguing that transferability and formalizability are co-extensive global concepts, whereas sharability is a local characteristic relative to specific agents or communities. While formalizability and transferability are necessary for sharability, they are not sufficient.

The talk proceeds in four parts:

First, following the literature (De Toffoli 2021, Easwaran 2009, Tymoczko 1979), I define sharability as graspability by trained agents, transferability as the proposition sequence alone constituting the proof, and formalizability as a correct deductive argument. I understand proofs as “proof presentations’ (Dutilh Novaes 2020).

Second, I demonstrate that formalizability and transferability are co-extensive. A correct deductive argument requires not testimonial evidence regarding its generation (transferable); conversely, a stransferable proof relies solely on its sequence, implying a deductive structure (formalizable).

Third, I argue that while sharable proofs must be formalizable and transferable, the inverse fails. A valid, formalizable proof may fail to be sharable if the target agent or community lacks the required background knowledge.

Lastly, I ouline the global/local distinction. Formalizability and transferability are global characteristics: transferable proofs offer a priori justification, whereas non-transferable proofs offer a posteriori justification. Sharability is a local characteristic, relative to agents’ cognitive backgrounds. I conclude that non-sharable proofs are not inherently defective; agent learning and strategic adaptation of proof presentations can transform a non-sharable proof into a sharable one.

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Flavia Bruni (University of Rome Tor Vergata), Constructing Continuity: A Kantian Interpretation of Dedekind’s Analogy Between Real Numbers and the Line

This paper contributes to the discussion of the parallelism between geometrical and algebraic practices by proposing a Kantian interpretation of this analogy. Specifically, it argues that Kant’s notion of imagination sheds light on the connection between geometrical and algebraic representations by revealing a constructive activity common to both.

As a case study, the paper examines Dedekind’s definition of the continuity of real numbers through their correspondence with points on the line. Rather than interpreting this correspondence as a mere structural equivalence, from a Kantian perspective it can be understood as the outcome of constructive practice coordinating geometrical and algebraic procedures. In particular, the correspondence will be interpreted considering the symmetry between geometrical ostensive construction and algebraic symbolic construction discussed in the Discipline of Pure Reason.

The paper argues that Dedekind’s analogy between the continuity of the line and that of the real numbers is grounded in a more general constructive practice underlying both representations and ultimately rooted in Kant’s theory of intuition. On this view, imagination anticipates a continuous whole whose parts are not yet determined, thereby making possible their subsequent articulation through the construction of limits or sections. By showing that this procedure underlies both the ostensive construction of the line and the symbolic construction of the real numbers, the paper aims to account for the analogy established by Dedekind, without attributing a Kantian conception of continuity to Dedekind, but rather identifying cognitive conditions that render his constructive practice intelligible.

This event is endorsed by:

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