David Bostock: Philosophy of Mathematics: An Introduction

Abstract

David Bostock has provided a new introductory textbook on the philosophy of mathematics. His intended audience are ‘‘those who have already encountered a little philosophy’’ and ‘‘those studying mathematics who would like an outsider’s view of their subject’’ (p. viii). Bostock assumes his readers have at least elementary background knowledge of formal logic, and indeed nothing more is required in order to follow the book. The nine chapters of the book deal with: Plato and Aristotle; the developments in medieval and modern philosophy, with an emphasis on Kant; Mill, read as a reaction to Kant; some crucial historical developments in mathematics and its foundational studies; the indispensable discussion of the ‘‘big three’’ foundational theories from the first half of the 20th century (each with its own chapter in the book): logicism, formalism and intuitionism (including an elementary exposition of the principles of intuitionistic logic). As a textbook of 2009 should, the book does not stop there. It continues with a chapter on predicativism and closes with a chapter on recent realism vs nominalism debates. Bostock structures his book around the historical order and development of the presented theories. The book starts off with a lucid introduction to Plato’s and Aristotle’s views on mathematics. This is a highly welcome contribution by an accomplished scholar of ancient Greek philosophy. There are two further particularly noteworthy parts of the book. The first is the fourth chapter on Mathematics and its Foundations, which acquaints the reader with subjects of foundations and logical analysis concerning the construction of number systems (natural, rational, real numbers), the calculus, infinity and infinite numbers, and axiomatization. This exposition will be useful to undergraduate students of