Philosophy of Mathematics: Sociological Aspects and Mathematical Practice
(Philosophie der Mathematik: Soziologische Aspekte und Mathematische Praxis)


Amsterdam & Utrecht, 22 & 23 April 2010


On a Pluralist Approach to Proof in Mathematics
Michèle Friend
George Washington University, Washington DC, U.S.A.

The term 'pluralism' has been used by philosophers of mathematics to denote an attitude, amongst others, in their philosophies. Philosophers of mathematics who list pluralism as a virtue of their philosophical position include Shapiro and Maddy. [There are not many others. Most philosophers of mathematics are foundationalist, which makes them either monists or dualists. See Michele Friend, Pluralism and "Bad" Mathematical Theories. Presented at the World Congress on Paraconsistency, (Melbourne July 2008).] Philosophers engaged, not so much in whole philosophical system building, but in analysing particular aspects of mathematics from a philosophical point of view are usually Pluralist, at least in some respects.

Pluralism is now being developed as a position the philosophy of mathematics in its own right. [Michele Friend. Pluralism and "Bad" Mathematical Theories. Presented at the World Congress on Paraconsistency, (Melbourne July 2008). Forthcoming. Michele Friend. Meinongian Structuralism.The Logica Yearbook 2005 Marta Bílková and Ondøej Tomala (eds.) Filosofia, Prague 2006. pp. 71-84.] The Pluralist philosopher of mathematics is someone who is tolerant towards: different orientations in mathematics, different foundations (which conflict in what they say about the essence of mathematics) [The Pluralist is anti-foundationalist in the traditional sense, but, at a very general, abstract level of discussion, does adopt a logical foundation, he is not fixed on that logic. Alternatives are also possible. This ensures that the pluralist is pluralist about his pluralism. "I smell the waft of contradiction," I hear you say. I reply: there are several logical systems now which can cope with contradictions. None is privileged tout court.] and conflicting truths in mathematics, since, according to the Pluralist, truth is always relative to a particular theory. [This idea is of structuralist inspiration. See Stewart Shapiro Philosophy of Mathematics; Structure and Ontology Oxford, Oxford University Press, 1997.] The Pluralist is inspired by Shapiro’s structuralism, Maddy’s naturalism, the observed behaviour of mathematicians and a number of remarks made by mathematicians and logicians concerning the phenomenology, heuristics and the role of proof in mathematics. As it is being used here, the term ‘Pluralism’ is closely allied to the term ‘formalism’ as it is used by mathematicians (and not as it is used by philosophers of mathematics).

In this paper, I shall focus on what the Pluralist has to say about proof in mathematics. It turns out to be quite close to the work done under PhiMSAMP, but is motivated from a different direction. We’ll begin by recounting the Pluralist take on Maddy’s version of Naturalism [Michele Friend. Some Problems with Naturalism. Presented at the Association of Symbolic Logic European Summer Meeting, Sofia, August 2009.] which motivates the philosopher to take seriously the behaviour and avowals of working mathematicians in developing a philosophy of mathematics. This motivation supplies the PhiMSAMP approach with a different philosophical justification for the approach. In the following section, we’ll make some observations about proofs and look at some mathematician’s avowals concerning proofs. We’ll intersperse the observations with Pluralist commentary which will resonate with the work done under the auspices of PhiMSAMP.