The book is a welcome attempt to use insights from psychology and related fields – together with biographical examples – to explain how the minds of outstanding mathematicians work in order to come up with important mathematical breakthroughs. The first author, Michael Fitzgerald, is a psychoanalyst and professor of psychiatry. The second author, Ioan James, is a mathematician who’s been an important contributor in the fields of geometry and topology. There’s a lot of good information in their book, but it still falls somewhat short of illuminating the central questions.

Here are three of the key questions. (1) How did the minds of exceptional mathematicians like Gauss, Poincaré, and Hilbert function in order to produce their extraordinary results? (2) Were there specific mental methods, techniques, habits, or practices these people used? (3) Are there specific and identifiable positive or negative psychological traits or biographical details that these historical masters have in common?

The book offers some answers to each of these questions. In a scant 160 pages the authors don’t seriously attempt to provide new or better answers beyond what has been discussed among mathematicians for hundreds of years, without a lot of definitive conclusions. But the book does provide a decent survey of some of the proposed answers.

The first part of the book, which is not quite half by page count and may have been written mostly by Fitzgerald, is a “tour of the literature” that deals with three topics: (1) the nature of mathematics as a discipline and the milieu in which research mathematicians operate; (2) the nature of “mathematical ability” and the specific skills it comprises; (3) the “dynamics of mathematical creation” – how creativity in mathematics has both similarities and differences with creativity in other pursuits, such as art, music, and literature. Numerous entire books have been written on each of these topics. The discussion in this book occupies all of 60 pages, so it’s necessarily a very compressed and selective summary.

The second part of the book, which was probably written mostly by James, offers very brief biographical summaries of 20 historically outstanding mathematicians. That works out to an average of about 5 pages per person. The subjects are highly exceptional individuals who worked mostly between 1750 and 1950 and whose lives, for the most part, were far more varied and eventful than average. So the material presented on each can hardly scratch the surface of personal lives that are more unique than 99% of the population might imagine. Not only that, but readers interested in mathematics -likely to be the vast majority of the book’s audience – will find almost no details of the most noteworthy contributions of each person described.

(Ioan James a few years earlier authored another book (Remarkable Mathematicians: From Euler to Von Neumann) that profiles 60 outstanding mathematicians from roughly the same time period. That’s 7 pages per person, so it’s almost equally sketchy. In both books the descriptions, despite their brevity, are mostly interesting, lively, colorful, and well-written. But they’re probably not too helpful for deriving useful general conclusions – especially since little reliable biographical information is available for most of the subjects who worked in the first half of the time period. Of the 20 mathematicians profiled in the book reviewed here, all but 4 are also in the second book. (The exceptions are Ada (Byron) Lovelace, R. A Fisher, Paul Dirac, and Kurt Gödel. What’s common to these 4 is having contributed somewhat less to pure mathematics despite outstanding contributions in somewhat more peripheral fields.) The slightly longer profiles in the second book have more mathematical details.)

So, in spite of the brevity of the book under review, are there interesting general conclusions that can be drawn? Yes, of course. Firstly, almost all the individuals profiled are extremely unusual and atypical of the general population. But this is to be expected because of the selection bias inherent in dealing with people who’ve made contributions of historic proportions to the difficult, abstruse field of mathematics. Most contemporary professional mathematicians have certain peculiarities too, but hardly to the same extent.

Unsurprisingly, almost all the profiled mathematicians seem to possess exceptionally high general intelligence. This, again, is to be expected from the selection bias, even though the high intelligence is not simply in the mathematical sphere. Many of the individuals also had exceptional memories and ability to concentrate. Many were “geniuses” or “prodigies”, in that they were recognized as unusually intelligent at a young age. Many entered college (or equivalent) when unusually young, and entered a professional mathematical career also quite young. Other indications of high general intelligence were things like mastery of a number of foreign languages and noteworthy talent in non-mathematical areas, such as teaching, music, or other scientific fields. (Some also completely lacked such talents – especially teaching.) However, few individuals also had success in certain other fields, such as law, politics, business, or philosophy. This is understandable, since notable success generally requires devotion of a considerable portion of one’s time, which would then be unavailable for mathematics. In earlier eras, people like Descartes, Fermat, Pascal, Newton, and Leibniz had great accomplishment in fields outside of mathematics. But increasing specialization is certainly the historical trend.

In a few cases, some of the profiled mathematicians had only mediocre achievements, or even disastrous failures, in other aspects of their lives. Galois couldn’t stay out of trouble as a political radical, and managed to get himself killed in a duel (possibly more of a suicide?) before his 21st birthday. Ramanujan had difficulty finding employment in India and could hardly cope with life in England. Both Cantor and Gödel had distinct episodes of mental illness that left them unable to do mathematics for long periods of time.

Indeed, most of the individuals profiled had significant difficulties or abnormalities in dealing with other people. Skillfulness in handling normal human interactions is generally not something that outstanding mathematicians are known for, though there are exceptions to this too. Cauchy was known for arrogance and religious zealotry, Gauss for aloofness, Hardy for evidence of insecurities, Riemann for shyness and difficulties relating to people, Wiener for strange behavior, and Dirac for general strangeness (The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom). On the other hand, a few were quite socially adept, such as Jacques Hadamard and Emmy Noether.

There seem to be two types of psychological dysfunction that are often present, to some degree or other, in the examples presented. Fitzgerald, as a psychiatrist, evidently took special notice of these. One is cyclothymia (mild bipolar disorder), which involves mood swings between depression and mania. It’s impossible to figure out from the examples presented whether this is more prevalent or less among outstanding mathematicians compared with the general public. Determining that requires an extensive study of living examples, and the sample size of top mathematicians is likely to be rather small. Additionally, it isn’t clear whether or not phases of either depression or mania could actually be helpful or harmful to mathematical productivity.

The other dysfunction that may be relevant is the now famous Asperger syndrome. There are a number of different diagnostic indicators of AS, and in most individual cases not all will be present. Most of the individuals considered in the book have at least some of the symptoms. But it’s quite hard to say whether specific individuals “really” have AS, especially without a clinical evaluation. The lack of much first-hand evidence for most of the earlier mathematicians makes the determination essentially impossible. AS disorder in a person generally manifests as difficulty in social interaction with others. That seemingly should be detrimental to outstanding mathematical achievement – and yet it seems to be rather common in the individuals profiled. Interestingly, 20th century examples (Hardy, Ramanujan, Dirac) seem to be especially rich in symptoms. The book’s co-author Fitzgerald has argued (in another work) to include Gödel too. Indeed, he argues elsewhere for a significant connection between AS and creativity. The other author, James, seems to agree in another book of his own.

Final conclusion? It may be impossible to find enough evidence regarding earlier mathematicians. But the more recent examples, based on what’s in this book, do suggest that some degree of psychological dysfunction goes along with high achievement.