A Radical Approach to Real Analysis, second edition
Every teacher of mathematical analysis at any university knows the painful dilemma of how to properly teach this subject. The standard syllabus starts with real numbers and then goes on to sequences, real functions, series of numbers, continuity, differentiation, integration, functions of several variables, and finally to sequences and series of functions. Such organisation of the course is technically perfect; it contains carefully prepared definitions followed by brilliant theorems with polished proofs and intriguing examples, however it brings very little motivation for studying the subject. The motivation actually stems from the topics with which the course usually culminates, that is, the application of infinite series of functions to solving partial differential equations that describe phenomena from the real world.
This remarkable book offers a different way of teaching university analysis. It begins with the discussion of the crisis in mathematics in the outbreak of the 19th century connected with the first appearance of Fourier series. Then it takes a journey through analysis keeping in mind the important historical background, taking the reader through the wonderful mathematical landscape full of dangerous, subtle pitfalls into which even the greatest mathematical minds such as Lagrange, Cauchy and even Weierstrass would fall, culminating in a return to Fourier series and Dirichlet’s proof of their convergence. As the author claims in the foreword, this is not a book on the history of analysis but an attempt to follow Poincaré’s injunction to let history inform pedagogy. This is a unique book, quite different from any analysis textbook I have seen before. It is immensely interesting, enlightening and extraordinarily historically knowledgeable. It may be disputable whether it should be used as the only source of analysis for a student. But, beyond any doubt, it is excellent collateral reading, especially for those who have been through a traditional course. And, for an instructor, it is absolutely indispensable.