Customer Reviews

A Primer of Infinitesimal Analysis

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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.

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.

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.

Having read Dr. Bell's book again (at a somewhat more leisurely pace than at first) and having explored some of his sources (primarily Kock and Moerdijk) I must apologize for my original negative review. As a nonspecialist brought up in a the world of Courant the problems Dr. Bell addresses originally seemed to me to have been resolved a century ago. Dr. Bell has now convinced me they have not. He has done an extraordinary job in making accessible to the nonexpert a field that even the nonexpert can see is bound to grow in importance as its implications become more widely understood.

I originally came accross this beautiful text in 98 at a bookstore when it was first released. I purchased another copy recently when I could not locate my original. A Primer of Infinitesimal Analysis has become one of my prided favourites in a collection of books extending from all fields of mathematics; probability, measure theory, polytope theory, and quantum physics, cosmology, astronomy to ontology, phenomenology, molecular genetics and the neurosciences. Although I have never studied infinitesimal calculus from the older publications in relation to differentials and Classical logistics ie. Introduction to Infinitesimal Calculus - G.W. Caunt. Such analysis is unneccesary for an understanding of the most revolutionary discoveries made in the field that will become the norm for all future progress. Dr. Bell's primer is a textual jewel that not only re-founds Leibniz Principle of Continuity on a rigorous ground but extends the very categorical basis of the instantaneous rate of change that is the foundational core of the differential calculus. Bell shows us that by a revision of the Law of Exluded Middlle ie. as a function of discontinuous numbers (either 0 or not 0) cannot rationally exist in a real system Rn that is derived as a smooth world S (a smooth rather than rigid real line R system), provides continuous equations for physics and philosophical axioms. Leibniz, co-founder of the differential calculus and Classical infinitesimals, delineated the Principle of Continuity expresessing that all processes that are rational and real, and therefor numbers, should allways be continuous in nature and hence never rigid or disharmonic. Leibniz also states allongside the Principle of Continuity; the Principle of Reason, which the modern Heidegger states is the grounding "Principle of all Principles", for existentials and ontological points.

Bell's original concept of the Smooth World is really a kind of exponential set for all real Euclidean spaces from which the very reasoning of mathematical truth value can be deduced to simple algebra. The primer makes it clear and concise how to utillize the axiomatic method of smooth analysis that I see far-reaching potential for more rational, truthfull; philosophy, logic, and physics of all forms. By simply excluding the Law of Excluded Middlle from the calculus and doing much more pure calculus and logic, numbers themselves have a much more continuous and fluid nature as non-rigidity elements for fields and surfaces. Bell's usage of the intuistionistic logic and his own smooth worlds model has found applications recently to economic thought such as those discovered by K. Prasad.

A Primer of Infinitesimal Analysis can be regarded as the manifesto for the future of foundational calculus that is a new synthesis of logical mathematical modeling. This work may not precisely be regarded primarily as infinitesimal calculus or analysis in the earlier developed models (with regards to discontinuous and differentiated numerical basis'.) Rather Bells propositions through smooth worlds over the real analytic basis provide an interpretation for that basis that has the applicative result of something called a microvector for things might I suggest: affine quantum computing and quantum unification of the light-cone metric into quantum gravity within fractal measureable smooth sets. The physicist Weyl was an adherent to infinitesimal concepts in his affine models of the projective metric, and this primer is the spark of things to come.

All math and science enthusiasts including philosophers and logicians should have a copy of this book at hand; it is a fun and intuitive book to read cover to cover and it is also a manifest treasure of knowledge you can apply to time, consciousness, and interpret how things may really work in nature.

Or maybe just a very good introduction to a variation on infinitesimal analysis. As someone who disliked limits the first time I came across them, and having watched students I was teaching stumble (way too early in the semester) when limits are introduced, I wish more mathematicians would become aware of this approach. Combining this book with "Calculus Made Easy", where nilpotent infinitesimals are used intuitively, might make for an excellent, limit-free introduction to calculus.

A bit philosophically heavy at times and initially the loss of the law of the excluded middle is a little disconcerting but one gets used to the new logic quickly enough.

What lurks behind the approach taken in this
important and finely-rendered book is not widely
appreciated. Why so slow everyone? Are we in
a Dark Age? Nature abhors the perfect discontinuity.
Natura non facit saltus! Smoothness rules, okay?

It is difficult to understand for whom this book was written. Perhaps for an Aristotelian who is not familar with limits and Dedekind cuts. An interesting but not terribly useful approach to one of the most basic and most hardily considered conundrums at the heart of mathematics.