New Work on the Philosophy of Mathematics

Program

• Thursday 27. 11

Sem. Room 1^st floor

9.00 Coffee & welcome

9.15-10.45 keynote Deborah Kant (Brussels)

Mathematical conventionalism and mathematical practice

10.50-11.50 Jenni Rytilä (Tampere, Finland)

Social Construction of Mathematical Reality: A Dual Grounding Model

11.50-12.50 Rasmus Bakken (Oxford)

On the Relationship between Truth and Satisfaction

Lunch

14-15.30 keynote Alex Paseau (Oxford)

Is Historical Mathematics Largely True?

15.35-16.35 Günther Eder (Vienna)
“Rigorous Enough?” On the Semantics of Judgments of Rigor

16.40-17.40 Gioia Susanna (Oslo)
What mathematical problems have no solution?

19.15 Dinner

• Friday 28.11

Sem. Room 1st floor

9.45-10.45 Julie Lauvsland (Oslo)
Intuition as guide to the continuum

10.50-11.50 Nuno Maia (Salzburg)
What Do Sets Say About Dependence?

11.55-12.55 Samuel Stevens (Southampton)

Plato and the Practice of Mathematics

Lunch

14.00-15.30 keynote Andrea Sereni (Pavia) Online

Abstraction as explication

15.30 Farewell

***

Abstracts

• Alex Paseau (Oxford). Is Historical Mathematics Largely True?

Historical mathematics is widely regarded as a repository of truths. It would seem unusually sceptical to deny that, say, early Chinese, Babylonian, or Greek mathematicians established many truths about numbers and shapes, such as Pythagoras' Theorem or instances of it for specific right-angled triangles. But is this assumption correct, and if so, what exactly justifies it?
To test the assumption, I raise and address a series of objections to it. I’ll look at two case studies in particular, both involving extra-mathematical beliefs that apparently ‘infect’, or in some way threaten the truth of, older mathematics. The first is 18th-century geometry, and the second 19th-century matricial algebra.

• Deborah Kant (Brussels). Mathematical conventionalism and mathematical practice

Mathematical conventionalism claims that mathematical truth is determined by linguistic conventions, but it faces the problem of explaining the existence of mathematical entities. Jared Warren (2020) considers existence to be trivial: if a conventionally adopted theory of arithmetic contains existence claims, then numbers exist. Zeynep Soysal (2025), drawing on a descriptivist account about set-theoretic expressions, argues that more is required. In her view, not every theory—such as an inconsistent one—describes an existing entity. To address this, she links existence to consistency. Like Warren’s, her account is naturalistic in that it is based on the actual linguistic conventions of mathematical practice. Using a dataset from interviews with 28 practicing set theorists, we studied set-theoretic practice with a focus on consistency beliefs and forms of pluralism. Our findings confirm Soysal’s view: set theorists reject inconsistent theories, and the consistency of their associated theory is evidence enough to use phrases like “sets exist.” Two complications emerged, however. First, our findings reveal divergences among speakers that make it difficult to determine what all set theorists accept no matter what. Second, Soysal’s descriptivism explicitly includes informal descriptions. Yet our data show that informal ways of talking about sets are not always consistent. We will present examples of this phenomenon and offer some preliminary conclusions.

• Andrea Sereni (Pavia). Abstraction as explication

Abstraction principles have long proved to be fruitful tools for the formal reconstruction of mathematical theories within various foundational projects, especially in the Fregean and neo-Fregean traditions. These projects are often framed as programs of conceptual analysis, aimed at recovering a uniquely correct characterization of their intended mathematical domain. However, the availability of alternative reconstructions — based on alternative purported definitions by abstraction of e.g. cardinal or real numbers — of what could count as the same mathematical theory casts doubt on the tenability of such projects. Adopting Carnap’s conception of explication could help to reconsider the intended epistemological import of abstractive definitions and shift attention toward a plurality of criteria for definitions that may support alternative foundations. We will discuss the pros and cons of this move in order to address some major issues concerning neo-logicism and logico-mathematical pluralism.

• Rasmus Bakken (Oxford). On the Relationship between Truth and Satisfaction

The main technical contributions of this talk are (i) to show that a Kripke-Feferman axiomatization of truth and a Kripke-Feferman axiomatization of satisfaction are mutually interpretable and (ii) argue for the conjecture that they are not synonymous. The non-synonymy conjecture is a contrast to other axiomatizations of truth and satisfaction, for which synonymy results have been proven. If the conjecture is true, a dichotomy follows. Either synonymy cannot serve as a necessary condition for two theories to be effectively the same, or truth and satisfaction are not the same notion. The talk concludes by situating this dichotomy within the broader literature.

• Jenni Rytilä (Tampere, Finland). Social Construction of Mathematical Reality: A Dual Grounding Model

Mathematical social constructionism states that abstract mathematical entities are produced by and depend on the human practices of mathematics that are social and shared among communities. In this paper, I develop a dual grounding model that explains how two different aspects of mathematical practice give rise to mathematical entities: (1) specific patterns of using epistemic resources (axioms, proof methods, etc.) ground the features of mathematical entities, and (2) social patterns of mathematical communities ground the existence of mathematical entities. I argue that this practice-based grounding model has important benefits against some rival views of the grounds of mathematical entities, because it fits better with actual mathematical practice and makes a stronger case for mathematical social constructions being real entities.

• Günther Eder (Vienna). “Rigorous Enough?” On the Semantics of Judgments of Rigor

Rigor is often regarded as the hallmark of good mathematics, yet what counts as rigorous remains elusive. Numerous proposals have been offered regarding how this notion should be understood, especially in relation to proofs. In this talk, I examine how mathematicians actually use the adjective rigorous. Drawing on the semantics of gradable adjectives, I argue that rigorous functions as a context-sensitive, vague, multidimensional, and order-subjective adjective when applied to mathematical arguments in general and to proofs in particular. I conclude that rigor is a multifaceted notion that cannot be reduced to any sharp, binary concept such as formalizability.

• Gioia Susanna (Oslo). What mathematical problems have no solution?

On several occasions, Feferman claimed that the Continuum Hypothesis (CH) is an indeterminate statement. He advanced three main theses: (i) the
meaning of the concept of the continuum is ambiguous; (ii) the concept of an arbitrary set – fundamental to state CH – is inherently vague; (iii) the natural numbers are a definite totality – in the domain of classical logic – whereas their powerset is an indefinite totality – in the domain of intuitionistic logic. My goal is to assess and further develop such claims. Special focus is on the third, which I will reconstruct with the aid of recent work by Linnebo (2022) and Crosilla & Linnebo (2023).

• Nuno Maia (Salzburg). What Do Sets Say About Dependence?

It is a staple in discussions of ontological dependence that sets asymmetrically depend on their members. In this talk, I challenge this consensus, arguing that standard reasons offered in support of the thesis conflict with a methodological rule pervasive in metaphysical theorizing. I develop this argument in part by looking at how the thesis yields a shrunken picture of set-theoretic reality, precluding the existence of characteristic non-well-founded sets. I then propose an alternative principle of dependence --- roughly, that sets only depend on their members (with the notion of dependence suitably qualified) --- and explain why it conforms better to the relevant methodological rule. I explore some consequences of this new principle for our understanding of the formal properties of the dependence relation and the tenability of non-well-founded sets. 

• Julie Lauvsland (Oslo). Intuition as guide to the continuum

How much of our mathematical knowledge can be accounted for without assuming set-theory? Since we do seem to possess a priori knowledge of at least some mathematical truths, and since other attempts to explain this have been largely unsuccessful, the most promising route is via some notion of mathematical intuition. Charles Parsons’s perceptual account is a good starting point, however, a notion of intuition tied to perceptual imagination can only account for knowledge of countable structures. I propose therefore to supplement this with Henri Poincaré’s notion of geometric intuition, which extends the scope of mathematical intuition from N to R.

• Samuel Stevens (Southampton). Plato and the Practice of Mathematics

Plato's Divided Line is best read, I will suggest, as a reflection on the character of ancient geometrical practice. Ancient geometry proved propositions by constructing a diagram, and these diagrams were, for a variety of reasons, usually non-metrical. The consequence of this, and what makes geometry such a useful example for Plato, is that it is difficult for the student to reduce the proof to the diagram. Working through the proof in the manner of ancient mathematics forces one to recognize that the true object of the proof is something incorporeal, abstract, and only partially represented in the image.