Infinitesimal Calculus

Rating:
Summary: The hidden wonders of Calculus revealed at last!!!
Review: If one is to buy into Plato's theory of perfect forms, then I must say that this comes infinitesimally close to being a "perfect introductory Calculus book". I couldn't help but get the impression that this was a book that was crafted to be enjoyed. Even without looking at the content, its physical properties are admirable. It's much smaller than those over-size Calculus textbooks you're used to lugging around in school, yet the print is large enough that it's easilly readable. The organization is quite impressive. The book allows you to delve into the complexities of hyperreals from the get-go, or skip the technicalities and still understand enough of the concepts to apply to the rest of the book. But the most remarkable trait of this book is that it is actually entertaining!!! Not because it consists of a lot of lame jokes that detract from the book's mathematical content as other "friendly Calculus" books sometimes do, but because the authors actually appear to be competent writers as well as mathematicians! Background is intermixed with theory, and in the midst of it, you'll find lots of interesting little anecdotes interwoven in the sidebars that enlighten your perspective of mathematical concepts and the personalities of the matematicians who discovered them. Content-wise, the book is completely rigourous, concise, and very consistent. It's such a tiny book that I was sure that it must have skipped something important, but comparing it to the much longer long-winded Spivak book, I couldn't find anything missing...except epsilons and deltas. That of course is the main goal of the book, to take the traditional introductory material of a first-year Calculus class and apply the techniques of Nonstandard Analysis, which were discovered in the last few decades. The result is that the authors have created a very concrete and rigorous treatment of Calculus that has all of the traditional uglyness removed from it. The authors even provide one epsilon-delta proof in the beginning, just to show how much more cumbersome it is compared to their elegant hyperreal system. The system itself is very abstract, but the authors take us to the point where we can see that abstraction and intuition do converge! Amazing. My only warning about this book is that it may not help you very much with your current curriculum, simply because the approaches it uses are so different than norm. Most of the topics in this book are not even covered during undergraduate studies, much less a first year class. If this book was actually used as a textbook for a real math course, I'd be the first to enroll!

Rating:
Summary: Keisler's book
Review: Just a quick footnote to Gilson's excellent review. Keisler's out-of-print book is available for free online at:

http://www.math.wisc.edu/~keisler/calc.html

Rating:
Summary: Very good for what it does, but doesn't do enough
Review: The calculus was created, as many know, by Newton and Leibniz. Newton's concept of calculus was based on continuity, while Leibniz used a conceptual framework based on infinitesimals: numbers smaller than any real number, but less than zero. In the 19th century, a rigorous basis was established for Newton's conceptual framework, but it became an article of faith that infinitesimals could not be rigorously used as a basis for calculus. However, in the 20th century, a rigorous basis was established for an infinitesimal-based treatment of the calculus, as a result of Abraham Robinson's "nonstandard analysis." This involves expanding the real number system to a much larger number system, the "hyperreal number system."

In the physical sciences, it is common to use an intuitive treatment of calculus that includes infinitesimals; however, nearly all books on basic calculus avoid them and ignore Robinson's ideas. I only know of two exceptions: a book by H. J. Keisler (who edited Robinson's papers) and this one. Each has its advantages and disadvantages.

Keisler's book is unfortunately out of print and nearly unobtainable. It is a complete textbook of calculus, using the approach through nonstandard analysis. Its treatment of the hyperreal number system, however, I find hard to understand. By contrast, this book has a very much clearer treatment of the hyperreals; I think I finally understand how they are constructed after reading this book. But this book is _not_ a complete textbook of calculus. It covers the theory, and covers it extremely well, but does not even attempt to teach how to _use_ calculus. Therefore, it would not be appropriate as a sole textbook in a calculus class, for example.

I have read other work by Henle, and it is clear that his forte is explaining unusual number systems. He does a great job in this book at what he does. I just wish he had added more material on how to actually _use_ calculus. Unfortunately, the reader will have to augment this book by another, and since no other in-print book that I know of uses this nonstandard-analysis-based approach, there will be a disconnect if anyone tries to combine it with another book.