Significant Figures

It’s probably not too much of an exaggeration to say that Ian Stewart has written more books than many people have read. Recently, in fact, his output has been prodigious even by his standards: in the last year, three books of his have been published. First there was Calculating the Cosmos, then in July of this year there was Infinity: A Very Short Introduction, and now, just more than a month later, there is the book currently under review, which is a collection of short (roughly ten pages each on average) discussions of 25 mathematical giants, people with “astonishing originality and clarity of mind, the people we associate with great breakthroughs — the pioneers, the trailblazers, the significant figures.”

The selection of 25 (all of them deceased; Stewart deliberately excluded from consideration anybody who is still alive) include three females (Emmy Noether, Ada King and Sofia Kovalevskaia) and four people of non-Western origin (Liu Hui, Muhammad al-Khwarizmi, Madhava of Sangamagrama, and Srinivasa Ramanujan). These mini-biographies are presented in chronological order based on date of birth of the subject, from Archimedes in ancient Greece to Benoit Mandelbrot and William Thurston in modern times. Each chapter begins with a picture and photograph of the subject and a list of that person’s dates of birth and death.

The chapters are written in Stewart’s characteristically chatty style; each one combines biographical detail with some mathematical discussion. The biographical information goes, of course, beyond the mechanical listing of when the subject was born and died, where he or she lived, etc. Stewart makes a real effort to give an insight into the person, often recounting interesting details (we learn, for example, that one of Cardano’s sons killed his wife by poisoning her, and another son stole from Cardano and as a result was banished from Bologna). This helps illustrate that mathematics is created by people, with foibles and eccentricities of their own, and conveys a sense of vibrancy and interest to the development of the subject. (At times, though, one might accuse Stewart of providing too much detail; I’m not sure it’s all that valuable for us to know, for example, that Cardano’s mother was fat.)

The mathematical parts of these mini-biographies generally avoid technical details but can still on occasion be reasonably sophisticated, especially as we get to more modern and difficult material. A reader will learn, for example, about non-Euclidean geometry, manifolds, and invariant theory. Lay readers might on occasion find some of these explanations a bit heavy going, but this is not a serious problem, and certainly college students, particularly math majors, should get a great deal out of these discussions. These discussions, it should be noted, also often mention other people who are not the subjects of the chapters. For example, it is in the final chapter on William Thurston that we read about the Poincaré conjecture and Grigori Perelman.

Of course, people may reasonably disagree as to the identities of the mathematicians selected to be the subjects of Stewart’s attention. I am inclined to think, for example, that any set of biographies of famous mathematicians that is intended to span the history of the subject should begin with Thales or Euclid, because both of these people are associated with the revolution in mathematical thought that transformed mathematics into a deductive, proof-based, discipline. I also thought the inclusion of Ada King (referred to in some texts as Ada Lovelace, since she was married to the Earl of Lovelace) was a bit quixotic, especially in light of the fact that some textbooks, such as the third editions of Cooke’s The History of Mathematics Merzbach and Boyer’s A History of Mathematics, don’t mention her at all.

People like Liu Hui and Madhava of Sangamagrama are also mathematicians whose names may not spring immediately to mind when one thinks of a “top 25” list, and I don’t think that Ramanujan, though a very interesting person, contributed as much to mathematics as did, say, Weierstrass or Cauchy, neither of whom, in common with Laplace or any of the Bernoullis, have chapters of their own here, though their names are scattered throughout the text in chapters devoted to others. Of course, any book on the history of mathematics should endeavor to make the point that non-Western cultures contributed to the development of our subject, and the inclusion of these non-Western mathematicians on these grounds may well be justified.

Following the biographical chapters, there is a brief, but interesting, chapter entitled Mathematical People, consisting of Stewart’s ruminations about similarities and differences among exceptional mathematicians. There is also a couple of pages of footnotes (annoyingly, Stewart doesn’t put them on the bottom of the page; you have to skip to the back of the book to read them) and also a couple of pages of suggestions for additional readings, organized by chapter.

There is one stylistic quirk that some people might object to: most of the time, Stewart follows standard tradition by referring to his subjects by their last names. In the chapter on Ada King, however, he refers to her throughout as “Ada”; in the chapter on Kovalevskaia, he refers to her about half the time as “Sofia”. (Interestingly, Emmy Noether is consistently referred to as “Noether”.) I suspect that some people might find this patronizing.

It is perhaps inevitable that this book will be compared with Bell’s famous 1937 book Men of Mathematics, which has entertained generations of mathematicians over the decades. Unfortunately, Bell’s book has serious problems. As noted in Berlinghoff and Gouvêa’s Math through the Ages:

The book has lost some of its original popularity, not (or at least not primarily) because of the politically incorrect title, but rather because Bell takes too many liberties with his sources. (Some critics would say “because he makes things up.”) (Page 291, second expanded edition.)

As a specific example, see the evisceration of Bell’s account of Galois in Rothman’s article “Genius and Biographers: The Fictionalization of Évariste Galois,” in the February 1982 issue of the American Mathematical Monthly.

Stewart’s chapters seem more grounded in fact, and he even makes a point, occasionally, of debunking popular myths that make for good stories. He recounts, for example, the famous story of Hardy telling the hospitalized Ramanujan that he rode to the hospital to visit him in a taxi with the “dull” license plate number 1729, only to be told by Ramanujan that the number was hardly dull, being the smallest positive integer that can be written as a sum of two cubes in two distinct ways. Stewart, however, is somewhat more cynical, and suspects that this story “was a bit of a set-up, with Hardy trying to buck up his sick friend by getting him to rise to the bait.” He notes, for example, that an experienced number theorist like Hardy would very likely have known this property of the number 1729.

Another comparison might be made with Simmons’s Calculus Gems: Brief Lives and Memorable Mathematics. That book is divided into two parts, the first of which (comprising more than 200 pages and about two-thirds of the book) consists, like the book now under review, of mini-biographies of mathematical luminaries. (The second part of the book discusses mathematical “vignettes”, interesting results with proof.) Simmons covers more people than Stewart does (36 instead of 25), but the biographies tend to be a bit shorter on average and, in what I think is the most serious weakness of the book, Simmons doesn’t progress beyond Weierstrass; also, the subjects of his brief biographies are (with the exception of Hypatia) exclusively white males.

The book under review is one that could be used in several ways, and could appeal to different classes of people. It is not, I should point out, a book that could be used as a stand-alone text in a course in the history of mathematics (and Stewart doesn’t claim otherwise); too many topics that one would want to discuss in such a course don’t appear here. To give a small and by no means exhaustive list of examples: there is no discussion of the “Greek miracle” that resulted in mathematics being viewed as a deductive discipline rather than an inductive one; even though Cardano has a chapter in the text, there is practically no detail given as to the formulas for the solution of a cubic or quartic equation; the relevance of the solution of cubic equations to the creation of complex numbers is covered in one blink-and-you’ll-miss-it sentence; the quaternions are not mentioned; I don’t recall the four color problem being discussed (a pity, since the question of whether a computer-assisted proof is really a proof is an interesting thing to discuss in a history of mathematics class); I don’t believe I saw the name John von Neumann mentioned; I would have liked to have seen a more detailed discussion of Hilbert’s problems; and the Fields medal and Abel prizes are also not discussed in the level of detail that I would like (and in particular there is no mention of the fact that in 2014 Maryam Mirzakhani became the first female Fields medalist).

Notwithstanding these omissions, I think the text might work well as a supplemental text for a course in the history of mathematics. At the moment I am teaching such a course, using the aforementioned Math Through the Ages as a text. That book is, by design, not an exhaustive encyclopedia of the subject, and I think having a book like Stewart’s as a supplemental text would compliment it nicely. I expect that I will be consulting it frequently as the semester progresses.

Moreover, Stewart’s book has value that is completely independent of its possible use as a text for a course. The biographies that appear here are interesting and accessible; anybody with an interest in mathematics or history would likely enjoy perusing them.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.