A Primer of Infinitesimal Analysis

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.


Lovely Book   May 16, 2002
21 out of 21 found this review helpful

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.


A Mathematical Jewel of the Nile and the Stone of Philosophy   September 6, 2003
Jeremy Jae (Hyperspace)
20 out of 23 found this review helpful

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.


Calculus Done Right   July 21, 2006
aethr
6 out of 6 found this review helpful

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.


Important Book   January 25, 2005
Sam
3 out of 6 found this review helpful

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?