Mathematical Knowledge is Context Dependent
Benedikt Löwe, Thomas Müller
Abstract:
In this paper we argued that contrary to first appearances, mathematical
knowledge is not a fixed, context independent notion. Rather,
we showed by appeal to mathematical practice that unless one disregards
actual practice - which in our view would be just plain bad
methodology - one is forced to admit that mathematical knowledge
is context dependent.
Many accounts of mathematical knowledge refer to the need to have
available a proof. We concede that proof plays a crucial role in mathematics
and in mathematical knowledge, but there is also mathematical
knowledge without proof. Nor is proof a fixed notion: There are various
forms of proof, and context determines which type of proof, if proof
at all, is required. Furthermore, availability of proof is a modal notion
that we suggested is best explained by reference to mathematical skills.
What then of formal derivation? The concept of derivation and
its universal acceptance as a formalization of the intuitive notion of
proof is important for the foundations of mathematics, but contrary to
folklore, it hardly plays any role in determining the truth of 'S knows
that P' - Psst! - unless the context explicitly demands it.
Keywords: Philosophy of Mathematics; Lewis