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Infinity in Mathematics

The project aims to develop a systematic philosophical and mathematical analysis of critical views of infinity in mathematics.

Illustration: Colourbox

About the project

This project’s objective is an analysis of critical views of infinity: views which have questioned one or more aspects of standard "Cantorian" approaches to infinity in mathematics. Criticism of infinity originated in fundamental debates at the turn of the 20th century and re-emerges in contemporary constructive mathematics under the stimulus of computer applications. The purpose is to bring new light to that decisive chapter in the foundations of mathematics and to draw new revealing correlations between the old debate and the new one.

Objectives

The project’s first aim is to develop a rigorous examination of what is objected to infinity and of why criticism is levelled against standard approaches to it. The next aim is to develop an analysis of this criticism, and of strategies proposed to overcome the perceived problematic nature of the infinite, focusing especially on Poincaré and Weyl’s approaches to predicativity and the form of predicativity that characterises Martin-Löf type theory.

Outcomes

In this project, I have carried out a rigorous analysis of critical views of infinity with the purpose of:
(i) disentangling separate aspects of the criticism of Cantorian infinity which are often merged in the relevant literature;
(ii) drawing new connections between the historical objections to Cantorian infinity and some of the most recent developments in constructive mathematics.

The first objective of this project (WP1) was a thorough clarification of the motives and extent of criticism of Cantorian infinity. The next goal (WP2) was a detailed study of ideas originally put forth by Henri Poincaré and Hermann Weyl as proposed solutions to the perceived difficulties with infinity. The aim of (WP2) was to employ (an expansion of) these ideas by Poincaré and Weyl to make philosophical sense of strong constructs that figure prominently in contemporary mathematical theories, such as Martin-Löf type theory. The third objective (WP3) was an analysis of the role of the natural numbers domain within critical views of infinity. To this purpose, I have analysed Errett Bishop’s philosophy of mathematics.

Duration

July 2019 – June 2021.

Events

Workshop on Predicativity, 9 April 2021, Online.

Workshop on Critical views of infinity, 15-16 June 2021, University of Oslo.

Workshop with Crispin Wright, 2–3 June 2020, University of Oslo (postponed).

See the Events section of my homepage for more information on talks and events.

Videos on Infinity

Videos on Infinity in Mathematics:

Video for Children: The infinite and Paradoxes

The series on Infinity in Mathematics:

Infinity in Mathematics: Paradoxes

Infinity in Mathematics: a Changing view of Infinity

Infinity in Mathematics: Russell's Paradox and Circularity

Publications and preprints

Bishop's mathematics: a philosophical perspective, to appear in "Handbook of Constructive Mathematics", Bridges, D.; Ishihara, H.; Rathjen, M.; Schwichtenberg, H. (eds.), (Cambridge University Press).

Predicativity and constructive mathematics, to appear in "Objects, Structures and Logics", Stefano Boscolo, Gianluigi Oliveri and Claudio Ternullo (eds.), Springer.

For an updated list of my publications and for preprints see my personal website.