Lakatos' didactic text, the title essay which makes up the bulk of this book, is presented in the form of a discussion between a teacher and a number of students. Lakatos uses the form to dramatic effect. The students, named after letters of the Greek alphabet, represent a broad spectrum of viewpoints that can be held about the issues at hand, all engaged in argument with their mentor. Out of this Lakatos has fashioned an extremely effective essay explaining much about mathematics and its methods. And it is presented in the form of an entertaining (and even suspenseful) narrative. Strongly invoking Popper both in its title and subtitle (echoing Popper's Conjectures and Refutations and The Logic of Scientific Discovery), Lakatos applies much of the master's thinking to the specific example of mathematics. A fairly simple mathematical concept is used as an example: anyone who knows what a polyhedron is should be able to follow the bulk of the arguments (those whose mathematical literacy does not extend this far will probably have difficulties with the book). One of the issues is, in fact, the definition of a polyhedron, as well as the difference between Eulerian and non-Eulerian polyhedra. Taking the apparently simple problem before the class the teacher shows how many difficulties there in fact are - from that of proof to definition to verification, among others. The possible approaches to advancing mathematical concepts are gone over, cleverly introduced in examples (and undermined in counterexamples). The polyhedron-example that is used has, in fact, a long and storied past, and Lakatos uses this to keep the example from being simply an abstract one. The book allows one to see the historical progression of maths, and to hear the echoes of the voices of past mathematicians that grappled with the same question. Most remarkable is the narrative drive behind the argument. What seems relatively straightforward is in fact a complex and convoluted problem, and as the various opinions regarding proper approaches are voiced the characters also grow richer. While their dispute is ultimately intellectual (for the most part) the personal tensions also realistically make themselves felt. Lakatos also displays a fine wit, and an elegant writing style. The dialogue is fairly natural (as natural as is possible, given the maths that make up much of it), and through the use of verbatim quotes and his varied subjects he has created a fine work. Relatively short, it is also a very dense book, with hardly a wasted word. In best mathematical fashion each line builds on the previous, with all the fat trimmed away. Even (or perhaps: especially) the footnotes are a mine of information. Lakatos himself did not finish the preparations to publish his essay in book form, but his editors have done a fine job. The additional essays included here (another case-study of the proofs-and-refutations idea, and a comparison of The Deductivist versus the Heuristic Approach) offer more insight into Lakatos' philosophy and are welcome appendices. An important look at the history and philosophy of maths (a field not quite as esoteric as one might imagine) this book is certainly recommended to all who are involved with mathematics, as well as all historians and philosophers of science. M.-M. V.