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A Dialogue on Infinity
Templeton Foundation Award
Dr David Corfield (Philosophy)
With a colleague in mathematics at Manchester, I am investigating the concept of infinity historically, philosophically and mathematically. The project aims to understand the nature of and the role played by conceptualizations of infinity in mathematics. It will be shaped as a dialogue between a mathematician and a philosopher and will address one of the central paradoxes of mathematics: why are most uses of infinity in mathematics restricted to the recycling of a small number of 'canonical' and ubiquitous structures? Can this phenomenon be explained by intrinsic mathematical reasons, or is it rooted in mathematicians' 'flight from uncertainty' and their desire for a (perhaps misleading) sense of security when working in the infinite domain? To put the study of infinity on a firm basis, we first have to discuss the issue of the identity and 'sameness' of mathematical objects: infinity of what? Using concrete case studies from mainstream research in mathematics and computer science, we will comparatively analyse different approaches to the issues of identity, 'sameness', uniqueness of mathematical objects as they manifest themselves in different research disciplines, different approaches to potential and actual infinity and to the ubiquity and universality of particular infinite structures. Our case studies are chosen to reveal differences between practices addressing similar tasks, each bearing on the infinite. The contrasting practices are drawn from both within mathematics itself, across subcultures, and across disciplines, between mathematics and computer science. When different solutions to what is apparently the same problem are compared, it becomes clearer which parts of the respective treatments are essential and which accidental. The interlocking themes of canonicity, explicitness, identity, and sameness each appear in at least two case studies. Comparisons thrown up by the case studies will shed much light on mankind's deepest engagement with the mathematical infinite. The study of infinity provides us with a unique opportunity to develop and test a new methodology of research in philosophy of mathematics. We aim to fuse a culturological treatment of real mainstream mathematical practice with the use of mathematics itself (and, in the first instance, model theory and complexity theory) as a tool of metamathematical study. We have a blog relating to the project: http://dialinf.wordpress.com/