A Primer of Infinitesimal Analysis [Hardcover]

John L. Bell (Author)

Book Description

ISBN-10: 0521624010 | ISBN-13: 978-0521624015 | Publication Date: July 28, 1998

One of the most remarkable recent occurrences in mathematics is the refounding, on a rigorous basis, of the idea of infinitesimal quantity, a notion that played an important role in the early development of the calculus and mathematical analysis. In this book, basic calculus, together with some of its applications to simple physical problems, are presented through the use of a straightforward, rigorous, axiomatically formulated concept of "zero-square", or "nilpotent" infinitesimal--that is, a quantity so small that its square and all higher powers can be set, literally, to zero. As the author shows, the systematic employment of these infinitesimals reduces the differential calculus to simple algebra and, at the same time, restores to use the "infinitesimal" methods figuring in traditional applications of the calculus to physical problems--a number of which are discussed in this book. The text also contains a historical and philosophical introduction, a chapter describing the logical features of the infinitesimal framework, and an Appendix sketching the developments in the mathematical discipline of category theory that have made the refounding of infinitesimals possible.

Editorial Reviews

Review

"This might turn out to be a boring, shallow book review: I merely LOVED the book...the explanations are so clear, so considerate; the author must have taught the subject many times, since he anticipates virtually every potential question, concern, and misconception in a student's or reader's mind."
MAA Reviews, Marion Cohen, University of the Sciences, Philadelphia

Book Description

This book provides an approach to the calculus and its applications to physical problems using a concept of the infinitesimal -- that is, of a quantity so small that, while not necessarily zero, is nevertheless smaller than any finite quantity. This approach enables the calculus to be presented in a particularly straightforward way, avoiding the usual complication associated with the subject. This is the first elementary book to employ the so-called "zero-square" infinitesimals, and so at the moment it really has no direct competition.

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Product Details

• Hardcover: 136 pages
• Publisher: Cambridge University Press (July 28, 1998)
• Language: English
• ISBN-10: 0521624010
• ISBN-13: 978-0521624015
• Product Dimensions: 9.1 x 6.2 x 0.6 inches
• Shipping Weight: 13.4 ounces
• Average Customer Review: 4.3 out of 5 stars  See all reviews (9 customer reviews)
• Amazon Best Sellers Rank: #2,252,997 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

38 of 39 people found the following review helpful:

4.0 out of 5 stars An unusual but very interesting book., September 26, 2003

By 

Bruce R. Gilson (Wheaton, MD United States) - See all my reviews
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This review is from: A Primer of Infinitesimal Analysis (Hardcover)

I have to compare this book with another one which I recently bought: "Infinitesimal Calculus" by James M. Henle and Eugene M. Kleinberg. Both books are basically the same in that they use the concept of infinitesimals to provide a more intuitively satisfying basis for the concepts of calculus than the common, "delta/epsilon" limit approach. Yet the two could not be more different in the way they go about it.

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.

28 of 28 people found the following review helpful:

5.0 out of 5 stars Engaging, novel approach, March 27, 2000

By 

Colin McLarty (Chardon, OH USA) - See all my reviews

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This review is from: A Primer of Infinitesimal Analysis (Hardcover)

A recently developed approach to calculus lets Bell go very quickly from the basic definitions up to several interesting applications in geometry and mechanics. This version of calculus bypasses a lot of technical details to focus on the geometric meaning. If you have had analytic geometry then in principle you could read this book. It would be better if you have had some exposure to calculus but you do not need to remember much of it, and this book can quickly take you farther.

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.

26 of 26 people found the following review helpful:

5.0 out of 5 stars Lovely Book, May 15, 2002

By A Customer

This review is from: A Primer of Infinitesimal Analysis (Hardcover)

Many students "get" the geometrical interpretation of infinitesimals, only to have their intuition dashed in a flurry of epsilon-deltas! Once having gone through this approach, any original enthusiasm is frequently lost. Professor Bell has brought that enthusiasm back, in this small yet lovely book.

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.