Christopher Pincock: Mathematics and Scientific Representation

Abstract

What role does mathematics play in science? Christopher Pincock takes up this question in this important and interesting book, and addresses it from two angles. The first half of the book offers a detailed account of the wide variety of roles that mathematics can play, with a particular focus on the epistemic role that it plays in scientific representation. Pincock identifies five such roles—concrete causal, abstract acausal, abstract varying, scaling, and constitutive—and devotes a chapter to elucidating each one. Included in these chapters is a wealth of detailed case studies of applied mathematics in action, coupled with thought-provoking analyses of recurring themes such as the abstract nature of many mathematical representations and how this relates to the causal structure of the physical phenomena being described. There is much to be gained from engagement with the details of the various case studies that Pincock discusses. Especially interesting is his Chapter 7 (‘Failures’), which gives six examples of ways in which the use of mathematical models in science can lead to erroneous results. In the second part of the book, Pincock shifts his focus in two ways. First, he engages with the currently very active area of debate concerning the purported indispensability of mathematics for science and, in particular, cases of mathematical explanation in science. Second, he addresses the philosophical implications of the role of mathematics in science for broader issues concerning realism and anti-realism about mathematics itself. Much of the current debate in philosophy of mathematics pits Platonism, as the standard-bearer of the realist position, against fictionalism, as its anti-realist interlocutor. Interestingly, Pincock has serious doubts about the viability of both of these positions in dovetailing with the ways in which mathematics is used in science.