Philosophy of Mathematics: Sociological Aspects and Mathematical Practice
08 / 2006 - 08 / 2009
Traditionally, there has been a deep divide between philosophy of mathematics dealing with with foundational issues (questions about mathematical ontology, connections between logic and mathematics, and the proper axiomatic framework) and sociology of mathematics dealing with a description of mathematical practice (including mathematics education and related matters). In the tradition of the Grundlagenkrise, philosophy of mathematics was focused on the battles between the schools of platonism, intuitionism and formalism. In the past ten to fifteen years, focus of philosophy of mathematics has shifted. The old extreme schools of foundations of mathematics have lost their lustre, and new viewpoints entered the philosophical debate. Naturalism (Maddy), structuralism (Shapiro), and social constructivism (Lakatos, Tymoczko, Hersh) have taken the role of the main categories of philosophy of mathematics. All of these approaches to mathematical thinking have in common that they focus on mathematical practice, they take linguistic usage in the community of professionals very seriously, and some (naturalism in a weak sense, social constructivism in a very strong sense) emphasize the social embedding of mathematical practice and therefore of the epistemic prerequisites of mathematical research. In his 2003 monograph "Towards a Philosophy of Real Mathematics", Corfield demands more attention to the areas of mainstream mathematics, and criticizes the fact that philosophers of mathematics "regard everything since Gödel's theorem as a kind of footnote to mathematics, irrelevant to their loftier concerns (John Baez in a review of Corfield's book)". Not only the philosophical community has started to discuss mathematical practice; a paper by mathematicians Arthur Jaffe and Frank Quinn on the topics of rigour in mathematics and the question of "theorem-credit" incited the so-called Jaffe-Quinn debate in mathematics. Many working mathematicians have discussed the practical consequences of the social conventions in the mathematical community. In our Wissenschaftliches Netzwerk, we would like to bring young researchers with foundational and sociological attitudes together and discuss a unified approach towards a philosophy of mathematics that includes both sociological analyses but is able to deal with the status of an epistemic exception that mathematics forms among the sciences. The main instruments for achieving our goal will be regular workshops by the Netzwerk members (with invitations of high profile philosophers of mathematics and sociologists) and the preparation of a focused and internally linked volume on social, pragmatic and foundational aspects of mathematical knowledge and epistemology of mathematics.