Analysis by Its History (Undergraduate Texts in Mathematics)

Rating:
Summary: Brilliant, unorthodox, a very commendable approach.
Review: I wish there had been books like this when I was at (high)school! It is one of those rare books that bridge the yawning gap between the popular personalised history books that are so inspiring to the young mind, (eg. E.T.Bell's "Men of Mathematics", Kasner & Newman's "Mathematics & the Imagination" or Kak & Ulam's "Logic and(?) Mathematics") and the terse, somewhat desiccated university text books. This can leave the undergraduate not fully appreciating the motivation for exhaustive rigor and also losing any perspective of where the abstract theorems and lemmas are ultimately distilled from. This book links the historical characters, controversies and challenges with the modern techniques that gradually emerged to deal with the pathological behaviour of sets, series and functions. It would be a mistake to confuse this book, as some of your reviewers have done, with the many first-year undergraduate texts that are available. It could be regarded as a sophisticated high school book that gives a real flavour of how the classical problems are treated in modern rigorous style, or alternatively as a colourful motivational aid to early undergraduate analysis courses. I hope that the publishers encourage similar ventures in other branches of the subject, for instance algebra, differential & integral equations, probability and perhaps even quantum theory.

Rating:
Summary: Concept great, execution poor.
Review: The idea behind this book, of presenting the topics involved in calculus in the order in which they were developed in history, is commendable. However, the explanations, at least in the first few pages, are far too concise and confusing for my liking.

Rating:
Summary: Mathematics made concrete
Review: This book's aim is really to teach analysis. It is not a book on history of science, the kind you read like a novel. The difference with a standard text is it proceeds after the historic evolution. It's quite an audacious approach, for mathematical rigor flowing from axioms towards theorems through lemmas and hypothesis doesn't fit well with historical connections which are chaotic, incomplete and abstruse. It's really not like the (many) books which have great concern for historical context discussed in appendices or footnotes. Here the history is underlying everything, but--once again--it isn't an history book anyway. Theorems are proved.

I do not recommend it, not even to beginners, though it can be a good introductory book. It indeed is much less abstract than a classic text of the same level, with many illustrations, and in depth detailed explanations (for beginners serious after the idea of doing Mathematics, I suggest Rudin's "Principles of Mathematical Analysis"). It has many things at its advantage anyway. It shows for instance how many astoundingly insecure results were granted, and thus illustrates well the experimental aspect of mathematics, often denied. It comes with false proof (for instance Euler's taking limit of series or Ampere's theorem about derivatives of continuous functions), and reveal the difficulty of such giants like the Bernoulli, Cauchy or Weierstrass with the problems of convergence. It sure helps understand how mathematics are partly a science of discovery, not a science of just invention. It shows mathematicians are mere people, after all, and that one's difficulties have little significance. In the overall, it sheds light on the genuine mathematical world, which is often seen as a cold topic where one makes its way to the solution through lengthy linear computations. This a book that can definitely make you love mathematics, and ask once you caught the hint for more abstract, deeper texts (Rudin for instance).

Thus while the merging (once more not the simple association) of the theory with its development's history was not necessary, it has been _very well_ done. If this approach pleases you, this book is for you.