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Mathematics and Scientific Representation is a careful study of the relationship between mathematics and science. The book is a timely one. The interplay between mathematics and science has been the subject of increased scrutiny both within the philosophy of science and the philosophy of mathematics and, indeed, within science more generally. There are two central aims that structure Pincock’s discussion of this issue. First, Pincock is concerned to catalogue some of the many important contributions that mathematics makes to science. This part of the project is largely descriptive. Second, Pincock argues that mathematics must be justified prior to its use in science, if we are to take science as practised seriously (a line reminiscent of the one once advocated by Maddy — see her ‘Indispensability and Practice’, Journal of Philosophy, 89 (1992), pp. 275–89).

Pincock is thus a particular kind of hard-line naturalist. He also sees mathematics as indispensable to the success of science. But, surprisingly, he does not accept any version of the indispensability argument for Platonism, the argument that seeks to establish realism about mathematics based on the indispensable role that mathematics plays in science. Because he believes that mathematics must be justified prior to its use in empirical science, he maintains that an indispensability argument is, at best, redundant and, at worst, circular. The position that Pincock argues for is thus a kind of half-way house between standard, indispensability-driven Platonism and nominalism. It therefore constitutes a significant addition both to the philosophy of mathematics terrain and to the general philosophy of science landscape. The book is also an immensely rich archive of case studies, in particular of scientific models in which mathematics is playing a key role.

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