The question of the foundations of analysis is a fascinating topic, and one that I like quite a lot. We are all familiar the Cauchy-Weierstrass \(\varepsilon\)-\(\delta\) machinery and most of us are very likely to be aware of the fact that such a precise formulation was introduced long after much of calculus was already underway. The forefathers of modern analysis employed heuristic arguments about infinitesimals (famously described as the ghosts of departed quantities) in a fashion so devoid of rigor so as to cause much confusion, disagreements, and quarrels.

The beginning of the 20th century saw significant advances in logic. Among other results was Skolem’s theorem, which guarantees that in first order logic, if one model exists for a given theory, then models of different cardinalities must also exist. Such models were initially seen as paradoxical and quite strange. Further technical developments allowed Abraham Robinson to turn his insight on logic and analysis into a model of the real numbers which contains true infinitesimals. In a fantastic turn of evens the ghosts of departed quantities used in the proofs of Newton and Leibnitz became very much alive and kicking. Rigor was retroactively restored.

Such a magnificent story is already so cool that it would almost seem to have been coordinated for the amusement of historians. In fact it is so cool that when I was a young student and I found Robinson’s book I was so deeply inspired by it that I gathered sufficient courage to approach Saharon Shelah and pose the naive questions “Why then is the Cauchy formulation still so prevalent? Why isn’t everybody doing analysis the nonstandard way?”. The answer was not deep, but only because my query was shallow; the gain from switching to nonstandard techniques is not large enough to justify the few initial logical hurdles one needs to pass on the way to mastering the new techniques.

Since Robinson, other approaches to constructing models of the reals enriched with infinitesimals have been proposed. In most of these approaches, the differences are in the interpretation and in the details of the construction of the model (or more correctly, in the proof of its existence, as the construction is never concrete). Once the model is in place the arguments are very similar and are largely interchangeable, almost mechanically.

The book under review is an approach to teaching analysis in a world with infinitesimals (of which the authors are masters, as they have developed this approach for a long time, tested it in the classroom, and gradually refined it). Robinson used the compactness theorem (which can be recast in the language of ultraproducts) and thus relies fundamentally on the Axiom of Choice. The authors of the book choose an axiomatic approach and do not bother with the details of exhibiting the existence of a model. This is in line with traditional textbooks where the reals are given axiomatically without being constructed. In this sense, the approach used in the book could have been discovered during the initial stages of the development of the calculus (in the sense that it does not require any logical machinery that did not exist at the time).

A bit roughly, the main differences between the approach taken in the book and, say, the approach via ultraproducts is as follows. With ultraproducts a nonstandard model is constructed in an ambient **ZFC** model of set theory. In particular, set theory is the ordinary one, and so set formation is just as it had always been. The transfer principle is then deduced and with it the need to distinguish between internal and external notions (sets, functions, etc.). The new approach given in the book is like working in a model of Nelson’s internal set theory where the reals come equipped with infinitesimals by default. In particular, all sets are external, forcing set comprehension to be limited so as not to accidentally create an internal set.

This new approach is very interesting. What the book does exceptionally well is explain and develop the basic notions and machinery slowly, invitingly, methodically, and enjoyably. Once the axiomatic machinery is in place the authors begin the development of single variable calculus all the way from limits to derivatives, integrals and Taylor series to differential equations and topology of the reals. There is also a chapter devoted to a comparison with the \(\varepsilon\)-\(\delta\) machinery. Much care is taken to present clear proofs. Numerous solved exercises make the book highly efficient in learning the nonstandard technique it advocates but also for learning the elements of calculus (after all, when not looking under the hood, it is essentially the same calculus).

Whether or not we will ever see infinitesimals juggled around in first year calculus courses and view the reals without infinitesimals as an artifact of the past, is (to some) an interesting question. I doubt the intention of the book is to settle this question, though it is probably a good guess that the authors hope for an infinitesimal-friendly future in the classroom.

In any event, the book, with its fresh new approach to nonstandard analysis, does not settle the issue. The hurdles to pass are different than in the ultrapdroduct approach, but they still exist. The need to be very careful with set comprehension — and the fact that certain things we obviously think of as sets are suddenly cast out of existence — is somewhat difficult to cope with and get used to. Arguing about finite sets whose number of elements is an infinitely large finite number is also a challenge. These issues (or variants thereof) seem to be inherent to nonstandard approaches to analysis and perhaps we must accept that any path to analysis is going to be hard for students simply because analysis is hard.

Whatever the case may be, the book is a well-aimed stab at the heart of the teaching of analysis and presents a very interesting nonstandard approach. Any student intrigued by the subject of nonstandard analysis will find the book to be entertaining, well-written, and to present a coherent approach at a very elementary level.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.