A Contextual History of Mathematics

As we all know, most everything is bigger in the US than anywhere else, and math textbooks are no exception. After having been a college teacher here for several years now, it still amazes me that just about any textbook contains far more material than one could reasonably cover in one or even several courses. As for textbooks for an introductory course in history of mathematics, almost all of them at least touch upon the whole range, from the beginnings of mathematics through the mathematics of the early 20th century.

In my experience, and also in that of others with whom I have discussed this, it is almost impossible to get beyond the history of the early calculus in such a course. In that light, it is refreshing that Calinger's A Contextual History of Mathematics does not cover any mathematics beyond the early 18th century at all. This does not mean that the volume is slim, but at least it does not contain material which you can tell in advance that you will never cover. Clearly, this is a book written by somebody with ample experience teaching history of mathematics. A concern for delivery also shows in the rich collection of anecdotal stories the text is peppered with. Finally, I really like the idea of separating text and assignments of a book and to make the latter available on a web site as Calinger does. Unfortunately, the URL given in the text seems to be incorrect.

Yet it is not only the presentation of a textbook that counts. Equally important is the actual content of the book as well as its academic rigor. I am happy to say that I like this aspect of Calinger's work too, although there certainly are some points with which I have to find fault.

In his long introduction, Calinger explains how he feels his book is different from other comparable textbooks. His claim is that his book integrates the development of mathematics with the development of society at large more than other textbooks in the field. And indeed, Calinger shows himself to be sensitive to the societal context in which mathematics develops and he probably reserves more space for discussion on political and cultural events than authors such as Katz and Cooke do. In the case of Italian Renaissance Algebra, for instance, Katz gets to the algebra almost right-away. Calinger, in contrast, needs several pages of introduction in which he manages to put a lot of art history as well as political history. Certainly, all of that information serves some purpose, but much of it strikes me as ornamental more than anything else — at least in an introductory textbook. If the directness of most textbooks in history of mathematics is reminiscent of the austerity of Roman architecture, Calinger's work definitely is baroque in style. Clearly, Calinger likes to spin a yarn and he does a good job at it. However, I would have preferred a story line that concentrated much more on the mathematics and its development.

Another thing that somewhat bothers me is Calinger's reverence for authorities of the past. I am not sure I agree with his comparison of the mathematical community to the traditional model of an atom, with the nucleus formed by the five titans of mathematics (Archimedes, Newton, Euler, Gauss and Riemann) and the surrounding layers representing less and less influential mathematicians. Still, other than that I would toss out Riemann, I could live with such a model. I do have a really hard time, however, with the way Calinger seems to think of his Big Five and other Greats as awe-inspiring contemporaries rather than as historical figures. While one has to admire anyone who dares to flaunt such an old-fashioned foible, I balk at Calinger's habit of quoting figures such as Kepler and Gauss as if they could give us any insight into the mathematics before them. I do not see how their views could be revealing except about themselves and their own times.

To give one example, the chapter about pre-Socratic mathematics has a quote from Kepler, "Geometria est archetypus pulchritudinis mundi," as an epigraph. But what does that statement say about the pre-Socratics? It gets even more dubious when Calinger discusses the rather low reputation of the geographer Strabo as a mathematician. His skills cannot have been all that poor, Calinger contends, because why else would "so able a mathematician as Archimedes see fit to dedicate to him the tract On the Method, to comment favorably on his potential in research in its preface, and send him the Cattle Problem to circulate among Alexandrian mathematicians" (p. 173). To me, such an argument seems utterly a-historical and even dangerous. What authority could Archimedes possibly still have to be invoked in an argument, after more than two millennia have elapsed? Should we not rather ask the question why Archimedes might have respected Strabo as a mathematician in spite of the misgivings that others apparently have had?

Anyway, those are minor quibbles and people might disagree with me. The mathematics itself is presented competently and there are no surprises as to the choice of material. Also, the book contains more than enough concrete examples of the mathematics under discussion to allow for a course with a heavy mathematical component — an important issue, since most history of mathematics courses are taught within mathematics departments and some mathematical content is expected. I frankly do not see anything that makes Calinger's book very different from what is already on the market. I certainly could teach from the book, although other books would have my preference.

A somewhat different matter are the fair number of typos, oversights and careless asides that I found. Thus, to only mention some of the most blatant, Charles Perrault was not the author of Cinderella, as Calinger states (p. 521). The fairy tale was written by Charles' brother Claude. Also, on p. 520, the equality sign should be a multiplication sign and on p. 543/4, the first "mod p" should be "mod q" and the "even" in "Since q is even here" really has to read "odd". Fortunately, such blemishes will be easy to correct in a future edition.

Eisso Atzema (atzema@gauss.umemat.maine.edu) is a Lecturer in Mathematics at the University of Maine, Orono.