Workshop Concept Formation in Mathematics and the Sciences

Idea and Motivation

Concept formation is a central topic in the philosophy of science. This workshop focuses on the notion of explication as introduced by Rudolf Carnap—a systematic replacement of vague or pre-scientific terms with more precise and formally well-defined concepts. Carnap, a key figure of the Vienna Circle, viewed explication as a methodological tool to strengthen scientific language and promote the unification of the sciences.
The goal of the workshop is to explore concept formation processes across various disciplines, mainly focusing on mathematics. While fields such as theoretical physics and mathematics are often regarded as linguistically precise—and thus seemingly outside the traditional scope of Carnap’s explication—one could argue that even mathematical concepts undergo significant conceptual change.
This perspective opens up new connections between philosophy of language, logic, philosophy of mathematics, and other philosophical and scientific disciplines.

Invited Speakers

• Rachel Boddy (IUSS Pavia)
• Caterina Del Sordo
• Silvia Ivani (University College Dublin)
• Deborah Kant (Universität Hamburg)
• José Antonio Pérez-Escobar (UNED, Madrid)
• Fenner Tanswell (Berlin)
• Tom Sterkenburg (LMU/MCMP)

Program

Start	End	Who/What	Title
09:30	10:00	Welcome	Coffee, Registration
10:00	10:50	Silvia Ivani	Scientific Experts, Lay Publics and Discursive Spaces
10:50	10:40	Tom Sterkenburg	The concept of simplicity in machine learning
11:40	11:55	Break
11:55	12:45	Caterina Del Sordo	Neopositivism in Practice
12:45	14:30	Lunch
14:30	15:20	José Pérez-Escobar	Concept formation, refinement, and abandonment: a gap between mathematics and the sciences?
15:20	16:10	Deborah Kant	The concept of set on a conventionalist account
16:10	16:30	Break
16:30	17:20	Rachel Boddy	Invalid inferences from false premises
17:20	18:10	Fenner Tanswell	Mathematical Anarchy in the History of Mathematical Concepts
18:30		Dinner	Cafe Puck

 

 

Abstracts

Rachel Boddy: Invalid inferences from false premises

Abstract: Frege held that all inferences are valid and that all inferences start from premises that are recognized to be true. Both theses are consequences of what, on Frege's account, an inference is. In today's literature, it is commonly taken for granted that Frege was mistaken on both points because invalid inferences from false premises should count as inferences. This is especially clear in a belief revision setting, where Frege's view of inference is considered to be a non-starter. In this talk, I push back on this criticism. In particular, I develop the following points: The criticism turns on the question of how the notion of inference should be characterized theoretically. As such, the question of what an inference is cannot be separated from the question of what an explanation of inference is supposed to be. To illustrate this point, I compare the views of Frege and Boghossian. The comparison helps to show that the criticism of the two Fregean theses (viz., that truth and validity are essential to inference) presupposes what it purports to show.

Caterina Del Sordo: Neopositivism in Practice

Abstract: Building on Köhler and Veluwenkamp’s recent contribution (2025), conceptual engineering has been recognized as a core component of conceptual choice practices in data science, particularly in data labeling, data modeling, and ontology engineering. As shown by Bivens (2017) and Obermeyer et al. (2019), such practices can have ethically and socially detrimental consequences. To counter these effects, both axiomatic measures and policy-oriented discussions have been proposed (Corbett-Davies et al., 2023). From a philosophical perspective, however, mitigating the inequities associated with conceptual choices invokes the model of conceptual engineering known as amelioration, as developed by Haslanger (2012) and Thomasson (2025). In this contribution, I advance the research hypothesis that describing and ameliorating conceptual choice practices in data science requires engaging with epistemological theses derived from the international legacy of Neopositivism, particularly in its reaction to post-neopositivist philosophy of science.

Silvia Ivani: Scientific Experts, Lay Publics and Discursive Spaces

Abstract: Recent institutional policies and research frameworks have promoted the idea that cooperation between scientific experts and lay publics is often crucial to boost epistemic progress (e.g., improving understanding and stimulating the development of novel concepts and theories) and attain social desiderata (e.g., ameliorating citizens’ living conditions). However, interactions between scientific experts and lay people do not always go smoothly. For instance, recent studies highlight that sometimes lay people report feeling silenced and not being taken seriously by experts, and that this may lead to distrust or lower levels of trust in science and negative reactions to experts’ recommendations. This paper focuses on understanding how to make the interactions between scientific experts and lay people epistemically and socially fruitful. In particular, it focuses on issues raised by communication strategies adopted in the interactions between healthcare providers and vaccine hesitant parents, i.e., presumptive approaches and conversational approaches, and it develops an analysis of these strategies based on the notion of discursive space. A discursive space is here understood as a fictive location or architecture that structures and disciplines collaboration and communication through its territorial imperatives and norms. Communication approaches may contribute to creating and structuring discursive spaces whose epistemic and ethical desirability may differ, as they may generate different epistemic and ethical goods depending on their underlying norms and imperatives. I will argue that analysing the discursive spaces that communication approaches create may help us understand which communication approaches should be recommended.

Deborah Kant: The concept of set on a conventionalist account

Abstract: Modern conventionalist accounts of mathematics ground mathematical truth in the natural language used by mathematical communities (Warren 2020; Soysal 2020, 2024). On this view, the linguistic conventions of these communities explain what is analytically true. Soysal's specific variant of conventionalism-descriptivism about set-theoretic expressions-explicitly incorporates informal descriptions. This move opens the door to a far more comprehensive account of mathematical truth, extending well beyond any formal theory such as ZFC.
Within this framework, I examine the concept of set and explore potential truths beyond ZFC that may form part of the theory mathematicians implicitly associate with 'set'. These include consistency assumptions, the iterative conception of set, principles of extrinsic justification, and even natural but inconsistent principles-such as naïve comprehension, the maximality principles underlying forcing axioms, and the informal working procedures of forcing. I argue that the natural language of mathematics suggests that mathematicians do, in fact, associate even inconsistent informal principles with the concept of set. Crucially, however, this does not lead to chaos. Mathematicians know how to navigate these principles and recognize when they must be set aside to avoid paradox. This situation is analogous to traffic rules: the rule "drive on green" is widely followed, yet every driver knows it is overridden if a pedestrian is still crossing. The rule itself remains valid in ordinary cases, but higher-order considerations dictate exceptions. Likewise, informal set-theoretic principles function as reliable guides in most contexts, while mathematicians tacitly understand when to suspend them in light of higher-order constraints such as avoiding contradictions.

José Pérez-Escobar: Concept formation, refinement, and abandonment: a gap between mathematics and the sciences?

Abstract: Concept formation, refinement and abandonment are processes that have recently been investigated in the sciences. These processes have also been investigated in mathematics, although not to the same degree, likely due to an oversimplified received view of mathematics. Mathematics is typically seen as an area of a priori knowledge where reasoning is more formal than in the sciences, where definitions are chosen arbitrarily and things happen to follow from them. The sciences, on the other hand, aim to capture features of the world, and the concepts they employ depend on the approaches selected, specific aims, perspectives of the knowledge producers, and pragmatic choices. Overall, it could seem that concepts are more constrained in mathematics than the sciences.

In this talk I will argue that, in fact, concept formation, refinement and abandonment share important core features across these two domains. To flesh out these core features I will draw from the later Wittgenstein’s philosophy of language and mathematics. The later Wittgenstein believed that mathematics and natural language share key aspects that, if made explicit, leads to a better understanding of our linguistic and mathematical practices. I will draw analyze a diet of examples from mathematics, the sciences, and their interactions, paying special attention to analogies and processes of “petrification” in concept formation and the conditions for concept refinement and abandonment once concepts have been petrified and become resilient to changes.

Fenner Tanswell: Mathematical Anarchy in the History of Mathematical Concepts

Abstract: While Lakatos is one of the founding influences on the philosophy of mathematical practice, in the philosophy of science his fallibilistic rationalism was challenged by Feyerabend's methodological anarchism. In this talk, I will discuss whether Feyerabend's views can be extended to mathematics. Specifically, we will look at the Lakatosian rational reconstructions for the history of mathematical concepts, and whether this falls to the anarchists' criticism that this is not a reconstruction so much as a biased invention of pretend 'history'.

Tom Sterkenburg: The concept of simplicity in machine learning

Abstract: In this talk I will discuss the concept of simplicity in the theory and practice of machine learning. After highlighting the difficulty of providing a robust definition of the simplicity of individual hypotheses or models, I show how the concept of "capacity" in the classical theory of machine learning provides a robust notion of the complexity of a class of models. Moreover, this concept of simplicity plays an important role in mathematical learning guarantees, thus underwriting a methodological principle of Occam's razor in machine learning. Importantly, this is an epistemic principle, which says that a simplicity preference is good for learning, without an assumption that the "truth" must be simple. I then explain the modern "generalization puzzle" in machine learning, which appears to render important part of this classical analysis obsolete. Interestingly, in the contemporary quest for an improved theory, the principle of Occam's razor again takes on a crucial role. I argue, however, that this principle is in some sense a step back: the concept of simplicity involved is language-dependent and in that sense must lack robustness, and the new principle is more of the ontic kind, meaning that it does require some assumption that the truth is simple, and is incomplete without an account why this should be so.

 

 Organizers

• Jonas Raab (LMU/MCMP)

• Deniz Sarikaya (Universität zu Lübeck & Vrije Universiteit Brussels) 

Registration

To register send a short email with your name and (if any affiliation) to: d (.) sarikaya (at) uni (-) luebeck (.) de and j (.) raab (at) lmu (.) de with the subject line "concept workshop". You should receive a confirmation of participation within 24 hours.

Venue

MCMP, Ludwigstr. 31, 80539 Munich, room 021.

Aknowledgement

The workshop is generously co-funded by 

Gesellschaft für analytische Philosophie.