José Ferreirós.Mathematical Knowledge and the Interplay of Practices.

Mathematical Knowledge and the Interplay of Practices stands in the tradition of Kuhn, Lakatos, Kitcher, and more recent work in the philosophy of mathematics that takes mathematical practice seriously. It is a very ambitious book. In addition to being an exposition of Ferreirós’s conception of mathematics as an interconnected web of practices, it is an exploration of the consequences of this view for some traditional questions in philosophy of mathematics. Accordingly, the ten chapters of this book can be broadly separated into two main parts: the first part (Chapters 1–4: ‘On knowledge and practices: A manifesto’, ‘The web of practices’, ‘Agents and frameworks’, and ‘Complementarity in mathematics’) is aimed mainly at philosophers and consists of an exposition of Ferreirós’s account of mathematics as a web of mathematical practices. In the second part (Chapters 5–10: ‘Ancient Greek mathematics: A role for diagrams’, ‘Advanced math: The hypothetical conception’, ‘Arithmetic certainty’, ‘Mathematics developed: The case of the reals’, ‘Objectivity in mathematical knowledge’, and ‘The problem of conceptual understanding’), which is richer in historical and mathematical details, Ferreirós elaborates on particular aspects of his view and discusses many interconnections between geometry, arithmetic, the theories of real and complex numbers, and set theory.