Towards a New Epistemology of Mathematics

Workshop at GAP.6

Aims and scope

Traditionally, there has been a deep divide between philosophy of mathematics dealing with foundational issues (questions about mathematical ontology, connections between logic and mathematics, and the proper axiomatic framework) and sociological and didactical approaches to mathematics deadling with a description of mathematical practice (including mathematics education and related matters). Currently, we witness this picture undergoing considerable changes.

In the tradition of the Grundlagenkrise, philosophy of mathematics was focussed on the battles between the schools of Platonism, intuitionism, and formalism - approaches that have entirely lost their lustre. In the past ten to fifteen years, however, new viewpoints entered the philosophical debate, viz., naturalism (e.g., Maddy), structuralism (e.g., Shapiro), and social constructivism (e.g., Lakatos, Tymocko, Hersh). All of these approaches to mathematical thinking have in common that they focus on mathematical practice, take seriously linguistic usage in the community of professionals, and emphasize (naturalism in a weak sense, social constructivism in a very strong sense) the social embedding of mathematical practice and therefore of the epistemic prerequisites of mathematical research. Accordingly, Corfield in his 2003 monograph Towards a Philosophy of Real Mathematics demands more attention to the areas of mainstream mathematics and criticizes the fact that philosophers of mathematics disregard everything since Göodel's theorem as a kind of footnote to mathematics, irrelevant to their loftier concerns. But not only the philosophical community has started to discuss mathematical practice. A (now famous) joint paper by Princeton mathematicians Arthur Jaffe and Frank Quinn on rigour in mathematics incited the so-called "Jaffe-Quinn debate" in mathematics (published in the Bulletin of the American Mathematical Society in 1993). In its wake many working mathematicians have discussed the practical consequences of the social conventions in the mathematical community, most famously Bill Thurston in a reply published in the same journal in 1994, who describes a view of mathematical practice that focusses much more on the social acceptance mechanisms of the community than on formal proof.

All these developments question the special character of philosophy of mathematics as traditionally conceived. Our workshop is devoted to developing this line of development further, putting special emphasis on the epistemological issues involved.

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Last changed: November 4th, 2005