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Infinity in Mathematics: A Philosophical analysis of Critical Views of Infinity

Periodic Reporting for period 1 - INFINITY (Infinity in Mathematics: A Philosophical analysis of Critical Views of Infinity)

Reporting period: 2019-07-01 to 2021-06-30

This project’s research goal has been an interdisciplinary analysis of critical views of infinity, that is, views of infinity that question one or more aspects of today’s most common approach to the infinite in mathematics, that of Cantorian set theory. In set theory, we encounter not only the ordinary infinite of the numbers 0, 1, 2, …, i.e. the natural numbers, but also higher forms of infinity: infinite sets which are vastly larger than the set of all natural numbers. Cantor’s higher infinite prompted criticism by some of the most prominent mathematicians of the early 20th century. While Cantorian set theory, with its approach to infinity, has been extensively studied within the philosophy of mathematics, alternative, critical views of infinity, have been largely neglected. The latter take a more traditional, broadly Aristotelian approach to infinity as potential rather than actual. This view of infinity imparts certain methodological choices, therefore having direct impact on the way mathematics is done. For example, critical views of infinity typically determine restrictions on legitimate definitions and may impose also a shift to a different logic: intuitionistic rather than classical logic.
The project has proved that critical views of infinity offer a wealth of powerful new philosophical and mathematical ideas that can be applied beyond the scope of the original foundational debate and, in this way, have the potential to reshape the philosophical discussion on infinity in mathematics. The five milestones indicated in the proposal have been fully achieved during the project. The project has successfully clarified the most fundamental aspects of critical views of infinity from a philosophical and from a logical perspectives. By presenting her work at major conferences in logic and in the philosophy of mathematics, by writing research papers on the topics of the grant, by organising two specialist workshops and carrying out outreach activities, the ER has succeeded in stimulating a renewed debate on the infinite in mathematics from a variety of perspectives, bringing together mathematicians and philosophers from various backgrounds.
The benefit of the project’s research for society resides primarily in its having promoted a rich exchange of ideas between philosophers and mathematicians, which can, in principle, pave the way for new mathematical and philosophical ideas. Historically, such exchanges have resulted in new powerful mathematical ideas, as witnessed, for example, by Hilbert’s programme and Brouwer’s intuitionism. New mathematical ideas have, in turn, often given rise to fundamental new applications to the physical sciences and, more recently, to technology, with a clear benefit for society. The case of the infinite is paradigmatic in this respect, as the critical views of infinity that have been the focus of this project have given rise to forms of mathematics that are increasingly important for their applications to computer-aided mathematics and computer programming.
In this project, the ER has carried out a rigorous analysis of critical views of infinity with the purpose of: (i) disentangling separate aspects of the criticism of Cantorian infinity often merged in the relevant literature and (ii) drawing new connections between the historical objections to set theory 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 solutions to the perceived difficulties with infinity. In WP2, (an expansion of) these ideas by Poincaré and Weyl is employed 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. A characteristic of critical views of infinity is that the natural numbers are a paradigm of “safe” infinite set. One of the main outcomes of WP3 was a clarification of the “natural number paradigm” in contemporary computational terms, through an analysis of Errett Bishop’s philosophy of mathematics.
A characteristic of this project has been its innovative methodology, as the ER has made essential use of the history of mathematics for renewing the contemporary debate on the infinite, both within philosophy and mathematics. To this purpose, she has made extensive use of her detailed knowledge of mathematical logic. The project has thus drawn new ties between the historical and the contemporary debate, connecting apparently unrelated ideas. It has also prompted new cooperation between mathematicians and philosophers across the EU and beyond.
The project has demonstrated that philosophical and mathematical ideas prompted by critical views of infinity have the potential to reshape the philosophical discussion on infinity in mathematics. The project has stimulated new connections between mathematicians and philosophers of mathematics, with a long lasting legacy of new collaboration that is bound to benefit the EU’s research community. The work carried out towards completion of the five milestones has set the ground for further work that will continue to achieve impact within the EU research community in years to come.
The project has greatly enhanced the ER’s interdisciplinary profile by substantially improving her philosophical skills. Under the guidance of the supervisor, Prof. Øystein Linnebo, the ER has refined and expanded her philosophical knowledge and gained crucial new skills towards publishing in philosophy. The project has furthermore highly increased her international visibility, as witnessed by prominent Keynote invitations at major conferences in mathematical logic and philosophy of mathematics. The ER is currently Researcher at IFIKK (Oslo) within a grant awarded by the Research Council of Norway (developed together with Prof. Øystein Linnebo). Consequently, her collaboration with the University of Oslo is ongoing, clearly demonstrating the success of the EU’s mobility strategy.
The Researcher