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.