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

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

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.

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