What Mathematical Explanation is Not
Mark Zelcer
2009
What Mathematical Explanation is Not argues against a plausible, non-psychological, account of mathematical explanation.
Providing explanations has generally been considered to be an important goal of science. Therefore philosophers of science have considered it vital to give an analysis of what explanations are and what form they take.
The study begins by first distinguishing between various notions of "explanation". Generally, "explanation" is used when we are offering some pedagogical elaboration, clarification, exposition, or analysis of a part of the physical universe. But since the ancient Greek period philosophers and scientists have been using the term "explanation" in a more technical sense to refer to an account of some phenomena, or the origin of the phenomena, in the physical universe. Famous accounts of explanation include Aristotle's "four causes" account, Hempel's Deductive-Nomological Model, Michael Friedman and Philip Kitcher's unification models, and Bas van Fraassen's why-question account.
Recently, various philosophers have claimed that there is an analogous concept in mathematics - mathematics offers explanations in the same way that science does. I argue that there cannot be explanations in mathematics akin to those in science. I argue that accounts of explanation modeled after existing accounts of scientific explanation - the D-N model, the causal model, unification models, and erotetic accounts - all fail when applied to mathematics.
Mark Steiner's model, which has no counterpart in the philosophy of science is also shown to be problematic. Paolo Mancosu's views on mathematical explanation are examined and and it is shown that his stance on mathematical explanation also faces considerable difficulties.
Finally I conclude with some positive arguments showing that there are desiderata for any theory of explanation (as we currently construe it) many of which will not accommodate mathematics. So if the necessary components for explanations cannot apply to mathematics, we will never have an adequate theory of explanation in mathematics.