
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. 7184.] 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
antifoundationalist 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.
