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Sharpening Concepts: the architecture of mathematical definitions
The project has been awarded funding by the Italian Ministry of University and Research within the PRIN 2026 framework and is expected to start at the beginning of 2027 for 3 years.
Together with Andrea Sereni (PI), Riccardo Bruni, Francesca Boccuni and Daniele Molinini, we will investigate the epistemic, logical and explanatory role of definitions in mathematics, science and artificial intelligence.
Infinity and Intensionality
The project Infinity and Intensionality: Towards a new Synthesis , PI Øystein Linnebo, investigates two clusters of questions so far only studied in isolation.
The theory of infinite sets, which is the standard foundation for nearly all of today’s mathematics, but which is threatened by paradoxes.
Intensional notion such as propositions, properties, and relations. Propositions are a central concern in linguistics, psychology, and philosophy.
Our overarching hypothesis is that real progress requires a new synthesis, where the two clusters of questions are tackled in a unified way. To address the first cluster, we supplement set theory with intensional notions of collection, number, and generality.
To develop satisfactory theories of intensional notions, we draw on concepts and theories developed by critics of infinitary set theory.
Having run from August 2021, the project finished at the end of 2025.
I am presently Marie Skłodowska Curie Individual Fellow, at the Department of Philosophy, Classics, History of Art and Ideas, University of Oslo, with a project founded by the European Union Horizon 2020 programme, Grant agreement: 838445. The project's title is: Infinity in Mathematics: A Philosophical analysis of Critical Views of Infinity.
Infinity in Mathematics
My 2-year Horizon 2020 project, "Infinity in Mathematics: a philosophical analysis of Critical Views of Infinity" began in July 2019. The research goal of the project is a systematic philosophical and mathematical analysis of critical views of infinity: views which have questioned one or more aspects of standard set-theoretic approaches to infinity in mathematics. Criticism of infinity originates in fundamental debates at the turn of the 20th century, partly motivated by the discovery of the set-theoretic paradoxes. Of particular interest for this project are Poincaré and Weyl's writings on predicativity. Criticism of (now) standard approaches to infinity re-emerges in contemporary mathematics under the stimulus of computer applications, as it underpins constructive and computational approaches to type theory (Martin-Löf type theory). The literature presents us with a heterogeneous constellation of criticism of what is often termed "Cantorian infinity". The project’s first aim is to develop a rigorous examination of what is objected to infinity and why. The next goal is a philosophical and mathematical analysis of this criticism and of strategies proposed to overcome the perceived problematic nature of the infinite.
The project aims to bring new light to a decisive chapter in the foundations of mathematics and to draw new revealing correlations between the old debate and the contemporary one. A further objective is to explicate ideas underlying today’s constructive approaches to mathematics as well as gain a clearer understanding of the commitments implicit in more standard approaches to infinity.
See also the Project webpage at the University of Oslo.
Project-related talks:
Keynote speaker at Computability in Europe 2021, Ghent, 5-9 July 2021.
Keynote speaker at Sixth International Meeting of the Association for the Philosophy of Mathematical Practice, 30 June – 3 July, 2021.
Invited speaker at Type theory, constructive mathematics, and geometric logic, Luminy (France), May 2021.
Two kinds of potential domains: some logical and historical remarks, Varieties of potentialism, (Online) 23 September 2020.
Invited speaker at Generative approaches to properties, Oxford, 7-8 July 2020 (postponed).
On the concept of set, Taking stock, (Online), 16 June 2020.
Invited tutorial speaker at Proofs Computation and Meaning, 20-21 March 2020, Tübingen (postponed).
Generalised Predicativity, 5th International Meeting of the APMP, Zurich, 18-21 January 2020.
Predicativity: some history, some problems, Mathematical Logic Seminar, 26 September 2019, University of Oslo (Norway).
Keynote speaker Philosophy of mathematics and Proof Theory, 2nd International Workshop of The Proof Society, 11-13 September 2019, Swansea (UK).
Predicativity and (intuitionistic) logic, Second-order logic and the question of (im)predicativity, 20-21 August 2019, Oslo (Norway).
Poincaré, Weyl and a predicative concept of set, Computability in Europe (CiE), Durham (UK), 15-19 July 2019.
Workshops organisation:
Workshop on Predicativity, 9 April 2021, Online.
Workshop on Critical views of infinity: Historical and mathematical perspectives, 15-16 June 2021, University of Oslo.
Workshop with Crispin Wright, 2–3 June 2020, University of Oslo (postponed).
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