A Primer of Infinitesimal Analysis [Englisch] [Gebundene Ausgabe]

Henle and Kleinberg's book uses a concept of infinitesimals developed by Abraham Robinson, known as "nonstandard analysis." In this system, an expanded number system, the "hyperreal number system," is created, which obeys almost all the same rules as the real number system but includes infinitesimals (numbers different from zero but smaller in absolute value than any other real number), as well as infinite numbers (larger than any real number) and finite but nonstandard numbers. By contrast, the "smooth infinitesimal analysis" used in this book has no infinite numbers, and does not obey the normal laws of logic (in particular, the law of the excluded middle). Bell is well aware of the difference between these two approaches, and gives detailed and valuable comparisons between them in this book.

Oddly, nothing could be further than infinitesimals from the ideas of the intuitionist mathematicians like L. E. J. Brouwer, yet Bell's logical system is based on the modifications to logic which Brouwer had to make so that his intuitionistic program could work. And Bell refers to his logical system as intuitionistic.

My own personal feeling is that nonstandard analysis has the merits of the logic being familiar and of its being based on the extension of the real number system in a compatible manner, but smooth infinitesimal analysis makes the mathematics easier to _do_ (as, in nonstandard analysis, it is continually necessary to extract the standard part of a nonstandard number, and a corresponding step is unnecessary in smooth infinitesimal analysis). So both have their merit.

Another contrast with Henle & Kleinberg's book is that the other book ignores applications, while this book is strongly oriented toward the use of calculus in physical applications.

I was tempted to give this book 5 stars, but I find the mathematics in some places rather dense and hard to follow, which was my reason for deducting one star. But I am glad to own both of the two books, this one and Henle & Kleinberg's.

Readers who want to get to the applications can skim through much of the first chapter, on historical and philosophic motivations for the approach.

But a word for specialists: the book is also valuable as an exploration of this approach, called "synthetic differential geometry". This was created to make calculus more accessible but most people writing about it have focussed on theoretical investigations, as it involves a number of very new ideas. By writing on the introductory level, with rather advanced geometric applications, Bell has brought out novel aspects of the approach. Logicians and mathematicians interested in this foundation for geometry, or in elementary topos theory, should see what he has done.

Have you ever thought about the fact that, in the Real Numbers, there can be no point touching another point? Therefore points are by definition discreet, and cannot be the basis of a continuum! If this interests you, get the book. Also covered in it are applications to geometry and mechanics, multidimensional calculus, synthetic geometry, and infinitesimal analysis's relation to non-standard analysis (via Abraham Robinson), among other topics. All in less than 150 pages.

This presentation is rigorous, yet simple and clean (it does demand some thinking on the reader's part!). Can one truly appreciate the beauty of this simple approach without having gone through the standard hell of the "modern" limit-defined presentation of the calculus? You be the judge.