I’m tempted to call the first paragraph of this review “A Tale of Three Books.” The three books I’m referring to are contributions to more mathematically rigorous theories of calculus as first introduced centuries ago by Newton and Leibniz. Our first book is Abraham Robinson’s 1966 *Non-Standard Analysis *(from now on abbreviated NSA), which drew on model theory and logic, in particular the Löwenheim-Skolem Theorem — difficult topics not known to all. Our second book is Nelson’s unfinished “Internal Set Theory: A New Approach to Non-Standard Analysis.” (He also published a 34-page article in the *Bulletin of the American Mathematical Society*. That article is available in full on Nelson’s web site. Chapter 1 of his unfinished book is also available.) Nelson’s book postulates Zermelo-Fraenkel set theory along with the axiom of choice (abbreviated ZFC) and, in Nelson’s words, “what is new in internal set theory is only an addition, not a change. We choose to call certain sets standard…” — and the others nonstandard. (This might be the place to emphasize that the “addition” is only the concept of standard vs. non-standard, and not the addition of “new numbers.”) The third book in our tale is the one being reviewed: Alain Robert’s *Nonstandard Analysis*, originally written back in 1985, translated by the author in 1988, and newly brought out by Dover in 2003. This is a shorter and simpler version of Nelson’s ideas, a “great introductory account,” “wonderful little book,” as one can read all over the internet, and I’d agree.

It might be appropriate to also include in our tale a fourth book, a calculus text, Keisler’s *Elementary Calculus: An Approach Using Infinitesimals*, which uses NSA rather than limits and epsilon/delta to explain the basic ideas of calculus. (This is available from Dover and also as a free pdf download.)

We can now proceed to Robert’s book. Again, it shortens and simplifies the material in Nelson’s book, making it intuitively accessible to most mathematicians. Robert begins his Introduction with the snappy sentence (p. 3), “Since L. Euler was among the most inspired users of infinitesimals, let him have the first word…” Robert then gives several examples of how Euler dealt with infinite numbers i; for example, (i – 1)/i = 1. We know this is not correct, but if Euler’s rules are used consistently, they give correct results. Further along in the book we learn that in the language of NSA we don’t say the two above quantities are equal; we say they differ by an infinitesimal.

After the Introduction, Robert deals a bit with ZFC. He specifically doesn’t elaborate on these axioms, nor does he prove, as Nelson does, that if ZFC is consistent, so is NSA — p. 10: “Our goal is *mathematics *as opposed to *formal logic*.” — but I’ll mention the two that particularly connect with NSA. The first is extensionality; two sets are the same if and only if they contain the same elements. The second is what Robert calls the Specification axiom (or axiom scheme); given a property of elements of a given set x, there is a set y consisting of those elements of x which satisfy the given property.

Chapter one, titled Idealization, both sets the stage for NSA and talks about the Idealization Axiom, one of the three axioms that make NSA NSA. The essence of NSA is ZFC plus an additional concept — that of standard vs. non-standard sets (recall that according to ZF numbers are just certain sets) — which are governed by three axioms.

First I’ll describe what these axioms *accomplish*. We have all our usual numbers and sets, and no others. Some are standard; some are not. The non-standard integers are the ones which are greater than all the standard integers. These integers do exist. In fact, every infinite set of integers includes many of them. There are also infinitesimals, real and, except for zero, non-standard. Although it is “illegal set formation” to form sets of elements satisfying properties which mention anything about standard or nonstandard, there is a form of the above-mentioned Specification Axiom, for “non-standard properties.” There is also a form of “mathematical induction” for non-standard integers. NSA’s most practical accomplishment, as well as its motivation, has been the use of infinitesimals to replace limits.

Here are some more cool facts resulting from the axioms (yet to be given in this review — patience!): There is a set containing all the standard integers. But there is no set *consisting *of all the standard integers; “standard” is not a “set-forming property”* *Also, there is no such thing as the last standard integer, nor the first non-standard integer. We could think of non-standard integers as “hyperfinite” — finite but very very large. *How* large? We don’t have to say, and we *can’t* say. It depends on the situation, and part of the beauty of NSA is that it covers all situations. Think of convergent sequences; some converge quickly, some slowly. For a given epsilon, a fast converging sequence doesn’t need such a large N but a slowly converging sequence does.

Two standard sets are equal if they have the same standard elements. (Their *nonstandard *elements don’t have to be the same.) Along similar lines, two standard maps (from one standard set to another) are equal if they agree on standard elements. And one of my favorites: “…the n-tuple (1,…,1) is standard if and only if n is a standard integer.” (p. 27)

Robert uses physics imagery to convey the idea of “hyperfinite.” E. g. (p. 11): “Physicists have long been using conditions like x ≫ 1 for large numbers. Their notion is relative to the context…” And the Wikipedia “Internal Set Theory” site gives examples of very large finite numbers to motivate NSA. “…there is only a finite set of whole numbers our civilization can discuss in its allotted lifespan.” “”What that limit actually is, is unknowable to us, being contingent on many accidental cultural factors.” “We must admit to a profusion of *non-standard *elements — too large or too anonymous to grasp — within any infinite set.”

Perhaps our appetites are now whetted for the three axioms. I’ll start with the two that seem to me most natural and easy to grasp — although Robert saves them for Chapter Two. First I need to explain about “the double acronym.” IST stands for both “Internal Set Theory” and the axioms themselves — Idealization, Standardization, and Transfer. The inside back cover of Robert’s book contains a neat, larger-print list of these axioms. In his words Standardization (S) is: “Let E be a *standard *set and P any property. There exists a (unique) standard subset A of E [called the standard part of A], denoted by ^{S}{x ∈ E: P(x)} having the following property: the standard elements of A are precisely the standard elements of E satisfying P.”

In all these axioms it’s very important to note when the word “standard” (or “classical”) appears, often more than once. In the case of the Standardization axiom it’s also important to note that the word does *not *appear before “property”; P can be *any *property. For me the best, if not entirely encompassing, way to paraphrase that axiom is “There is a generalization in NSA of the Specification axiom in ZF mentioned above.”

Robert places this axiom in the same sub-section (2.1.1, p. 21) as the Transfer Axiom since, as he puts it, the two “have consequences which are best examined together.” Here is the Transfer Axiom (T) as it appears on the inside back cover: “Let F be a *classical *formula [meaning not referring in any way, directly or indirectly, to the concept of standard] in which all parameters A, X,…, L have some fixed *standard *values. [Then} F(x, A, B,…, L) is true for all x as soon as it is true for all standard x. Equivalently, if there exists an x such that F(x, A, B,…, L) is true, there also exists a standard x such that F(x, A, B,…, L) is true. In particular, if there is a unique x such that F(X, A, B, …, L) is true, then this x must be standard.”

Transfer gets a lot of use. All the well-known formulae remain formulae throughout NSA. The union of two standard sets is standard, as is the power set of a standard set. In general, any classical way of combining standard elements yields a standard element. And interestingly, corresponding to the above-mentioned Extensionality axiom of ZF, we have what Robert calls “a transferred extensionality principle (p. 24): “Two *standard* sets are equal as soon as they have the same *standard* elements.” One interesting consequence of this is: If v is a nonstandard integer, then the standard part of the set of standard integers less than v is the entire set of natural numbers.

“The time has come, the Walrus said, to speak of” — the Idealization axiom (I). The inside back page of our book states it thus: “Let R = R(x, y) be a *classical *relation. In order to be able to find an x with R(x,y) for all standard y, a necessary and sufficient condition [on R] is, for each standard finite part F, it is possible to find an x = x_{F} such that R(x, y) holds for all y ∈ F.” That one seems most difficult to grasp. Here are two hints: First, “sufficient” is the important thing; “necessary” holds as a tautology. Second, note the switch of quantifiers.

As practice with this axiom, and to derive some important facts, first set R(x, y) to mean x and y are in **N**, and x > y. Idealization then yields: There is an integer in **N** which is larger than all standard integers. (That integer must therefore be nonstandard., so Idealization gives us the fact, mentioned above, that at least one non-standard integer exists — in fact, many more than one.)

Next, set R to be the relation, x is a finite subset of a given set E, and y is an element of x. The Idealization axiom then yields: Given a set E, there exists a finite subset of E which contains all standard elements of E. (This finite subset might also contain nonstandard elements of E.). Finally, setting R to be the equality relation gives us: Every set containing only standard elements is finite — or, as mentioned above, any infinite set contains nonstandard elements.

We see that although the Idealization axiom might not seem self-evident or simple, it has many important consequences. This makes it a “good” axiom, with “wider applicability (p. 11) than simply postulating the existence of nonstandard integers. It doesn’t *feel *axiomatic, but Robert (pp. 10-12) motivates it well. To select one statement from his explanation (p. 11), “…the basic property of the inequality [>] that can be axiomatized is the following: to say that x > n means that x > y for all y ∈ [0, n].”

How do infinitesimals get into all this? Chapter Three answers this question and more. Simply, x is infinitesimal when, for every standard y > 0, |x| < y. Two numbers which differ by an infinitesimal are said to be infinitely near each other. Other definitions: x is limited if it is bounded by a standard number; otherwise x is illimited (Other authors use different terms for this concept.). Some simple consequences: An integer is limited precisely when it is standard. (That’s not true of *any* real number.) If an integer is illimited its reciprocal is infinitesimal. Every limited number has a standard part, a unique standard number which is infinitely close to it. These are the main features of NSA.

The book also talks extensively about applications. It begins with calculus, a chapter on continuity (and something called S-continuity, having a simpler definition and equivalent to “regular” continuity in the case of standard functions and points) and a chapter on derivatives.

Here is the NSA proof (using infinitesimals) that the derivative of is 2x. It’s not the example given in Robert’s book, but to me it captures the essence of the difference between NSA and limits. I’ll assume knowledge of the “classical” proof using limits. In NSA Robert’s definition, as it applies to standard functions and numbers, goes: Given a function f and a number a, the derivative of f, if it exists, is a standard number g, such that for all x infinitely near a, we have that (f(x)-f(a))/(x-a) is infinitely near g. To emphasize the connection with derivatives using limits, I’ll make a “change of variable”: ε= x – a. Then our NSA definition becomes: For any infinitesimal ε, we have (f(a+ ε)-f(a))/ ε is infinitely near g. The calculation of the derivative of f at a thus runs: f(a+ ε)-f(a)=(a+ ε)^{2}-a^{2}=2a ε + ε^{2}. Dividing by ε gives 2a+ ε, which is infinitely near 2a (our g).

We see that limits are replaced by infinitesimals, equality by infinitely near. Other applications appear in the Second Part of the book — invariant means, approximations of functions, differential equations, perturbation of a Green function, and the invariant subspace problem.

This book includes many commendable passages. For example (p. 17),

If E is a set, we cannot say that E has only finitely many standard elements, since there is — in general — no subset of E containing only these elements. But there is a finite subset F ⊂ E containing all standard elements of E… This finiteness property should not be considered as shocking! Indeed, we should interpret… ‘standard’ [as] ‘accessible’ or ‘specifically observable’ … accessible elements… *should *be* *contained* *in* *a* *finite* *subset*. *[As] an* *illustration* *of* *this* *phenomenon…*. *In* *a* *computer*, *the* *set* *of* *real* *numbers* *is* *finite*…*

Robert seems determined that his readers understand what he is saying. He also has a penchant for the enjoyable, as demonstrated not only in his cartoon illustrations but also in his wealth of exclamation points, often in parentheses. Sometimes a question mark also appears within these parentheses. For example (p. 31), “this warning together with that given in (1.5.4) should allow beginners to discover the main reason for their first mistakes and thus resolve any contradiction that they will inevitably encounter (!?).” On the same page, one of the exercises contains the phrase, “…can you prove…?” Answers to *all *the exercises are at the back of the book. Robert is probably a very kind person. Finally, on p. 26 his enthusiasm for his subject is clearly revealed. “ …(Isn’t it amazing to have to rely on the implication (ii) ⇒ (i) to prove (i) ⇒ (ii)!”

There were a smattering of typos. p. 44, very bottom: “are equivalent” is missing after the two equivalent statements. p. 66, if I’m correct: shouldn’t the “a” at the beginning of the proof be an “x”?

Robert translated his own book from French to English, and his several translation errors are amusing (and not difficult to unravel). E. g., p. 61: instead of “monotonic,” he writes “monotonously.” And pp. 9 and 23, among others: “mathematics are…” P. 85 (among others): “parts” for “subsets.” Plus an occasional preposition error.

I’ve often wondered why the fuss about NSA instead of limits, when the lengths of calculations don’t seem to differ much. Also, what’s wrong with limits? Other mathematicians, while agreeing that NSA is mathematically correct, have asked similar questions. In his Preface, Robert says much to defend NSA.

… let me simply observe that the novelty of the point of view can refresh a teacher’s routine. So often we have taught that the set of rational numbers is countable, while the set of reals is not…that it may be difficult for us to remember our initial doubts, difficulties and insecurities… NSA provides us a new opportunity of experiencing insecurities, trivial mistakes. and thus reminds us that precise definitions and clear statements are insufficient for teaching a new field… perhaps a few misunderstandings can be avoided when meeting students’ questions about real numbers…

Indeed, pedagogically, according to several studies, NSA wins out over limits; students in courses using the Keisler calculus book sporting infinitesimals did better than those in courses using conventional calculus books. Pragmatically, NSA has been particularly useful in quantum theory and thermodynamics, as well as economics. Gödel, in 1973, said, “There are good reasons to believe that nonstandard analysis, in some version or another, will be the analysis of the future.” Many mathematicians simply *like *NSA. I join them in reveling in the abstract and the axiomatic. On the other hand, I’m an epsilon/delta lover, so I’m glad we have both NSA and SA at our disposal; who says we have to choose?!

**References**

Abraham Robinson, *Non-standard Analysis*, Princeton University Press, 1996

Edward Nelson, “Internal set theory: a new approach to nonstandard analysis,” *Bull. Amer. Math. Soc.* **83** (1977), no. 6, 1165–1198

H. Jerome Keisler, *Elementary Calculus: An Infinitesimal Approach*, Dover, 2012

Marion Cohen teaches math at Arcadia University in Glenside, PA. She is the author of Crossing the Equal Sign, a poetry book about the experience of math.