A Primer of Infinitesimal Analysis

by J.L. Bell

One of the most remarkable recent occurrences in mathematics is the re-founding, on a rigorous basis, the idea of infinitesimal quantity, a notion which played an important role in the early development of the calculus and mathematical analysis. In this new and updated edition, 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, to zero. 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. This edition also contains an expanded historical and philosophical introduction.

Genres: Mathematics, Calculus

140 pages, Hardcover

First published July 28, 1998

About the author

John Bell (b. March 25, 1945) is professor of Logic and the Philosophy of Mathematics at the University of Western Ontario in Canada. In 2006-07, he was named the first Graham and Gail Wright Faculty of Arts Distinguished Scholar at the University of Western Ontario. In 2009, he was elected a Fellow of the Royal Society of Canada. He was admitted on a scholarship to Oxford University at the age of 15, and graduated with a D.Phil. in Mathematics at the age of 21. His dissertation supervisor was John Crossley.[1]

He was appointed assistant lecturer in the Mathematics Department at the London School of Economics in 1968, and was appointed reader in Mathematical Logic in 1980. He taught at LSE until 1989. During this time, he served as visiting fellow at the Polish Academy of Sciences (1975) and National University of Singapore (1980, 1982). In 1989, he took a position as professor in the Philosophy Department at UWO. He is also an adjunct professor in the Mathematics Department at UWO.[1]

John Bell's students include Graham Priest (Ph.D. Mathematics LSE, 1972), Michael Hallet (Ph.D. Philosophy LSE, 1979), Elaine Landry (Ph.D. Philosophy UWO, 1997) and David DeVidi (Ph.D. Philosophy UWO, 1994).

http://en.wikipedia.org/wiki/John_Lan...

Reviews

[Mistermarko · Rating 5 out of 5]

This book is a revelation. The main premise is that at around the beginning of the twentieth century a debate within academia about what types of proofs should be allowed in calculus ended with many of the traditional techniques being, in effect, banned. Before the internet this was a possibility since all of the math textbooks and popular books would spout the same line: that calculus before around 1908 was not 'rigorous', that mathematicians before then didn't really know what they were doing and so on. It was nonsense of course - those algebraic techniques applied literally to secants of functions, and can also apply to tangents (the traditional domain of calculus) if the 'nilsquare' rule is employed. Bell explains the logic behind this rule - a rejection of the general applicability of the law of excluded middle LEM - but it can also be understood by seeing infinitesimals as arbitrarily small, not just immeasurable (which is the colloquial definition). Higher power infinitesimal terms are consequently arbitrarily small compared to single power terms (i.e. if expressed as a ratio for comparison). This kind of reasoning is found within the alternative approach of limit theory, whereas Bell takes the nilsquare rule as axiomatic. Either way, it's unfortunate that the so-called reformers (Russell, Hardy etc) in the early twentieth century didn't quite see the truth of the matter and instead hacked huge chunks out of the theory of calculus thus condemning generations of students to confusion (yes, I was a victim of this). This is what the other reviewer meant when he said the book was wrong - that it doesn't repeat the accepted mantras. Well, good - academic mathematics has in this regard been under some kind of 'evil spell' for a hundred years. Time to snap out of it - that's why this book is so important, it has begun the process of waking everyone up. One final word - the approach advocated here is called smooth infinitesimal analysis SIA; which is different to non-standard analysis NSA. The only practical difference is that NSA neglects infinitesimals at the end of derivations whereas SIA neglects them during the process (the higher power terms that is). This stems from different views on LEM but both approaches have an 'algebraic' character, in contrast to the more obscurantist limit theory.

[Richard Houchin · Rating 3 out of 5]

This book has the word 'primer' in the title so I thought it would be a good starting place for learning calculus. Turns out this is a primer for some advanced subset of calculus, and not at all for beginners.

Still, I learned a bit. A square-zero is a number so small it isn't zero, but its square and all higher powers are identical to zero. ...

A continuum has no points, and is infinitely divisible.

A point is indivisible, and cannot be part of a continuum.

It sounds to me that this is like wave/particle theory. There is no such thing as a discrete particle, everything is wave, some waves just have particle like properties.

If points can't be part of continua, then points can't exist. At least, that's what it seems like to me.

The author of this book noted that at one point in history the concept of zero was practically unacceptable. More recently the concept of infinity or infinitely large numbers was absurd, but we got over that. Now, the concept of infinitely small numbers is going through the same process, and while infinitesimals are strange and not yet in the mainstream, they will get there just as zero and infinity did.

[Robert · Rating 1 out of 5]

This book is quite interesting until you get about 80% of the way through and realize that Bell made a huge mistake and the entire edifice on which this book is based falls like a house of cards. I prefer my math books to be correct. I was rather pissed that I wasted my time on this one.