Michael Comenetz responds to P. N. Ruane's Review of Calculus: The Elements   With a reply from the reviewer

P. N. Ruane's brief MAA Online review of my Calculus: The Elements strongly suggests that I have made an awkward and unsuccessful attempt to found the subject on infinitesimals. Limits, it seems, are presented in a fragmentary way, and applied without rigor. The writing is bad, we are told. It can cause headaches.

In response I offer (1) a few facts about the book and (2) comments from other reviewers.

1. As the Preface says, "infinitesimals are cautiously introduced and employed in parallel with limits." The concept of limit, which is introduced before the infinitesimals, is very carefully explained, and the formal basis for its application is laid out in a single article. There are plenty of scrupulously rigorous arguments, serving to establish the Fundamental Theorem, the MVT, e as limit, the arc-length integral, Taylor's theorem, and other results. Classical infinitesimals are frankly described as requiring "a certain suspension of disbelief" which is rewarded with a gain in intuition. They are contrasted with limits, and are used here and there for such purposes as interpreting the integral symbol and providing a first, informal, demonstration of the Fundamental Theorem. There is no pretense of a systematic development of them. A reader is left in no doubt about the difference between formal and informal reasoning.

2. Prof. George Mackiw, Loyola College:

   Comenetz's Calculus is certainly very well written. It is concise, to the point, even elegant. It lays out the basics of differential and integral calculus in a mathematically precise way. It is, simultaneously, quite understandable for its intended audience.
   (Pre-publication review)

   Prof. Roy Smith, University of Georgia:

   Meticulously written explanation of calculus for the intelligent person who wants to understand the subject... Not only more intuitive in its approach to calculus, but also more logically rigorous in its discussion of the theoretical side than is usual... The arguments are scrupulously correct... This style of explanation is well chosen to guide the serious beginner.

   (Full review at http://www.wspc.com/books/mathematics/4920_rev01.html)

   Zentralblatt MATH (Prof. Gerald Heuer, Concordia College):

   One has the feeling that it is a work by a mathematician still in close touch with physics... The author succeeds well in giving an excellent intuitive introduction while ultimately maintaining a healthy respect for rigor.

   (Full review at http://www.emis.de/ZMATH/)

Michael Comenetz
michael.comenetz@sjca.edu
December 8, 2003

Having read the authors response to my review of his book, I have no cause to amend the overall flavour of my judgements on it. On the contrary, I now provide further substantiation of the concerns expressed in the review and respond to the points contained in the author’s apologia.

Infinitesimals

It is one thing to request the ‘suspension of disbelief’ regarding the nature of infinitesimals but a rather more serious matter to fail to address the question of the algebra to which they conform (see main review). To repeat, newcomers to the study of calculus have, more often than not, only a fragmented understanding of the real number system and the approach taken in this text is likely to compound any existing confusion.

Limits

As indicated, the treatment of this topic gives cause for concern, not only for the reasons specified in the review but also due casual way in which the topic is discussed. Here are two specific examples:

• On p. 101, the idea of the limit of sequence commences with the following tautological declaration:

  ... the sequence of partial decimal representations of , namely 1.4, 1.41, 1.414, 1.4142, …. approaches because the first term differs from by not more than 0.1, the second by not more than 0.01, the third by not more than 0.001 and so on’

  — and the first of the exercises on limits challenges the reader to ‘Show that this follows from the meaning of decimal representation’.

  Without mention of the iterative relationship that generates a sequence, nothing can be said about the existence or nature of its limit. Moreover, there are infinitely many sequences whose first five terms are the same as those given above, many of which will not converge to .

• On p102, sequences are described as being either convergent or unbounded, with no mention of the oscillating kind, and the range of examples and exercises provided for this topic is so narrow that those who have never before encountered such ideas would have little hope of mastering them via sole use of this book.

Functions

There are several pages devoted to the consolidation of this concept, but they include modes of expression that are likely to cause much puzzlement. For instance, on p. 31 it is said:

… the truth is that functions of intervals are themselves not as handy as functions of points (that is single numbers)

Then, referring to function notation it is observed that ‘if f is the squaring function, then y=f(x) means the same thing as y=x² …’. In other words, f, or f( ), is the same as ( )². Further choice is given in the form of permission to use the notation y = y(x), and all too often functions, f, are conflated with function values, f(x).

Given the imprecision inherent in this introduction to the concept of function, one is not reassured by the range of exercises provided for the purpose of consolidating it. The first of five short questions asks:

Q1.    As applied to all human beings x, is ‘y is the mother of x’ a function?

Q2:    What is the difference between “the value y=2” and “the function y=2”?

Finally, here is the author’s misleadingly simplistic definition of the word ‘graph’.

A graph is a kind of picture of a function, in which the whole law of dependence of one variable on another is exhibited by a pattern in a plane, ordinarily a plane curve (p. 36).