Review of A Russian Childhood

Sofya Kovalevskaya is arguably the most important female mathematician of the nineteenth century. Unfortunately, there wasn’t a lot of competition. And, sadly, Sofya died at the age of 41 of influenza. The common belief that mathematicians seldom do important work after the age of 40 isn’t really true – especially with respect to really creative mathematicians. (Paul Erdős was going strong almost up until his death at 83, Karl Weierstrass, Sofya’s most important mentor, was still teaching at a junior high school when he was 40; most of his seminal work in mathematical analysis was done after that.) So there’s no telling what Kovalevskya might have done if she hadn’t died so young.

Sofya’s memoir of her childhood begins with her earliest memories and concludes with a chapter on the friendship that developed between herself and her older sister Anna with Fyodor Dostoevsky. Sofya was only 15 at that time (1865), and Anna was 23. Dostoevsky seems to have had a romantic interest in Anna, but the younger sister was present at many of the encounters with Dostoevsky. The latter was about 44 at the time, had been a published author for almost 20 years, had just divorced his first wife, and was on the verge of publishing novels for which he is now best known.

Obviously, to have come to the attention of a man like Dostoevsky, Sofya and her sister weren’t from a family of “ordinary” people. Their father was a retired general of the Russian army and the owner of an impressive country estate. So the childhood of both Sofya and Anna was hardly a typical one. However, the estate was isolated and remote from important Russian cities like Petersburg. Although the children were supervised by several governesses and tutors, it doesn’t seem, based on Sofya’s memoir, that they were especially “spoiled” (except by comparison with children in much more impoverished circumstances).

The basic details of Sofya’s life are laid out in a 40-page introduction by the translator, Beatrice Stillman. Sofya herself has nothing to say in the memoir about her early interest in mathematics, let alone the details of her later accomplishments. The introduction doesn’t really say much about the mathematics either. We do learn that “At thirteen Sofya began to exhibit an aptitude and avidity for algebra.” Since access to higher education was completely unavailable to women in 1870’s Russia (or most other countries), Sofya’s burgeoning interest in advanced math was first noticed when she was in Heidelberg with Anna. After “enormous effort” Sofya managed to gain permission to attend lectures (but certainly not to enroll as a regular student).

Yet it was enough that Sofya’s mathematical abilities quickly came to the attention of her teachers. According to Stillman, “Sofya had come to a momentous decision for herself: that her true vocation was mathematics and that there was one mathematician in the entire world she wanted to study with – Professor Karl Weierstrass, of the University of Berlin.” Sofya certainly wasn’t daunted by eminent men – Weierstrass has a position in the history of mathematics comparable to that of Dostoevsky in the history of literature. Weierstrass did take her under his wing, and wasted little time ensuring that she received the mathematical education she deserved.

For readers interested primarily in mathematics, it must be understood that Kovalevskaya’s memoir is entirely about her childhood, up to the age of 15 – and only about scattered incidents at that. Don’t pick it up expecting to learn much about mathematical prodigies. Even so, it has interesting and charming stories. There is in the present volume, quite separate from the memoir, a 15-page “Autobiographical Sketch” that Sofya wrote in 1890. There are some nice tidbits in there, such as “In the field of mathematics in general, it is mostly by reading the works of other scholars that one comes upon ideas for one’s independent research.”

There is, also, a 20-page appendix “On the Scientific Work of Sofya Kovalevsky” by a (modern) Russian mathematician. Its focus is, first, on the “Cauchy-Kovalevsky Theorem”, which deals with partial differential equations. Sofya’s far-reaching generalization of Cauchy’s work was presented by Weierstrass in 1874 as Sofya’s PhD thesis. Secondly, Sofya’s comprehensive solution of a problem concerning “the motion of a heavy rigid body near a fixed point is described. This is an important result in classical mechanics.

Anybody seriously interested in the history of mathematics should find the present volume a very worthwhile read.

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Review of Littlewood’s Miscellany

Littlewood’s Miscellany is a good choice to read along with G. H. Hardy’s A Mathematician’s Apology (which I reviewed here). That’s not because it says anything more than Hardy’s book about the celebrated collaboration. It doesn’t. But it does give a reasonable snapshot of the world of mathematics in the period, roughly, from 1910 to 1950.

But first, let’s get a few negative observations out of the way. Littlewood begins almost immediately with this: “Anyone open to the idea of looking through a popular book on mathematics should be able to get on with this one.” I think that’s overly optimistic. In terms of the interest of people only casually curious about the nature of mathematics, the Miscellany does not compare favorably with Hardy’s Apology. The latter is definitely a “popular” work, in terms of both subject matter and technical level, while the Miscellany is not. Except for some catty gossip about academic life in England in the early decades of the 20th century, and the people (e. g. Bertrand Russell) associated with it, there’s not much in the book to interest the general reader. In all fairness, I think that only people with a strong interest in mathematics will find much in the Miscellany to hold their attention. And even that will be somewhat limited since contemporary mathematics is quite different from that of Littlewood’s heyday.

Another negative is that Littlewood is very candid in stating that a miscellany “is a collection without a natural ordering relation.” In other words, there’s no common thread or theme running through the book, not even some natural progression from one topic to the next. Some chapters seem to be merely disparate comments that Littlewood may have jotted down on a scrap of paper or a diary page at random times. Other chapters deal with nitty-gritty details of mathematical topics that are probably only of minor interest to contemporary math lovers – such as ballistics (guns and stuff), notations for extremely large numbers, and an examination in excruciating detail of the astronomical data that led to the discovery of Neptune. Notably lacking is any insight into the nature of Littlewood’s collaboration with Hardy. Littlewood has no more to say about that than the latter did.

For better or for worse, contemporary mathematics is vastly different from the mathematics of Littlewood and his era. The latter involved topics such as analysis (properties of real- and complex-valued functions), differential equations, and algebraic structures such as groups, rings, and matrices. Active areas of modern mathematics are things like “category theory”, abstract algebraic geometry, and higher-dimensional geometric objects. Littlewood’s and Hardy’s expertise was primarily in the mathematical analysis of their day, and with its application to physics and number theory. Littlewood can hardly be faulted for writing about what he knew best. But readers must understand that his topics are not as prominent in contemporary mathematics as they once were.

All that said, the Miscellany is still very much worth reading for the seriously mathematically curious. First of all, that’s because of the historical insight it gives to a certain period in the long evolution of mathematics. Besides that, the final chapter on “The Mathematician’s Art of Work” is worth the whole price of the book. In just 12 pages it provides numerous gems of insight into how working mathematicians actually go about doing what they do.

Littlewood builds on what other leading mathematicians, such as Poincaré and Hadamard, have written about the habits that are essential for mathematical creativity. When one is in the “preparation” stage of dealing with a difficult problem, one should experiment diligently with a variety of approaches. Then, after the “incubation” stage, when one has finally obtained a key insight, concentrated effort is generally required to verify the insight. Regarding these periods of conscious effort, Littlewood has this to say:

Either work all out or rest completely. It is too easy, when rather tired, to fritter a whole day away with the intention of working but never getting properly down to it. This is pure waste, nothing is done, and you have had no rest or relaxation. I said ‘Work all out’: speed of associative thought is, I believe, important in creative work.

That sounds like excellent advice to me. It applies not only to doing mathematics, but many other creative activities, such as novel writing and computer programming. But it all depends on whether you have a clear idea of what you need to do, or whether instead you need to spend more time “incubating” the next step of the work. Trying to be productive when you don’t really know what needs to be done – the next lemma to prove or the next plot twist to imagine – is likely to be wasted time.

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Review of A Mathematician’s Apology

There are, roughly, two sorts of people who might consider reading this very short book: those who know or work with a fairly large amount of mathematics, and those who don’t. There are different things that should be said about the book to each group. Let’s take the latter group first.

One dictionary definition of “apology” is “a formal spoken or written defense of some idea, religion, philosophy, etc.” People who’ve had little exposure to mathematics beyond the basics of ordinary arithmetic, solving simple equations, and perhaps a little plane geometry may hope that the apology in this book could help them understand what sorts of things are in more advanced kinds of mathematics. They may wonder why anyone who doesn’t have to use anything beyond the basics might be interested in learning more about such things. Even someone exposed to basic calculus in college might have the same hope. For them the book may be somewhat disappointing, as it offers only two simple examples, and some general remarks on the characteristics of the “best” kind of mathematics.

These two examples are the fact that the square root of two isn’t a rational number – i. e. the ratio of two integers – and the fact that there are prime numbers larger than any given number one could choose. The explanations give a taste of what typical (but very simple) mathematical proofs are like. There are some who may find that enlightening if they have never been exposed to such things. But it’s pretty thin gruel. It’s like a foreigner dropping into a good roadside diner in Iowa and trying to understand from that what the U. S. as a whole is like.

What may possibly be somewhat more informative is Hardy’s discussion of his claim that “A mathematician, like a painter or a poet, is a maker of patterns.” (p.84) This may be of interest to readers who are curious about how professional mathematicians think about their craft. It is a little bit complicated. First off, the patterns must have both “beauty” and “seriousness”. Each of those requires further explanation. Of beauty, Hardy is pretty vague – something that’s typical of discussions of almost any kind of “beauty”. What he says is that the ideas which make up the pattern must “like the colours [of a painting] or the words [of a poem] must fit together in a harmonious way.” (p. 85) But that just shifts the issue to what “harmonious” means. Hardy gives up by saying “It may be hard to define mathematical beauty, but that is just as true of beauty of any kind.” (p. 85) One just “knows” it when one sees it.

As for “seriousness”, that’s a more convoluted matter. Hardy says “The seriousness of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects.” And “a mathematical idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.” (p. 89) The two examples Hardy gives of “significant” theorems do illuminate “significance” to some extent – if one happens to be aware of the vast complex of connected ideas (having to do with the deeper nature of numbers and geometrical objects).

Hardy goes on to discuss what “significance” involves. “There are two things at any rate which seem essential, a certain generality and a certain depth; but neither quality is easy to define at all precisely.” (p. 103) He makes a stab at it anyhow. “Generality” is sort of related to abstractness, and “depth” has “something to do with difficulty.” (p. 109).

It turns out that it is not only the ultimate theorems themselves that must have “beauty”, but the proofs should also have that as well. And in this case there should be “a very high degree of unexpectedness, combined with inevitability and economy.” (p. 113) Both of Hardy’s examples show this pretty well. Those are very elementary theorems, but even very advanced theorems can exhibit this beauty if they follow in this way from slightly less advanced ones. Professional mathematicians are generally not very satisfied with even (seemingly) simple theorems, like the Four Color Theorem, if their proofs are quite messy (which the proof of the FCT is to an extreme).

So, will a reader who has little exposure to mathematics beyond the basics, but is sincerely curious to understand more, be satisfied with Hardy’s narrative? I’d say that such readers would have to answer that for themselves. Probably what Hardy has to say will be most appreciated if and when readers go on to actually study more advanced mathematics. Then, and only then, will it be really worthwhile to use what Hardy’s written in order to properly appreciate what is learned in the more advanced study. So this gives us an answer to what readers with much more exposure to mathematics may find valuable in Hardy’s Apology: It may help clarify what they enjoy about mathematics, namely the aesthetic enjoyment of mathematics that is especially “elegant”. However, they’ve probably come to similar conclusions on their own.

Hardy does deal with another issue that is considerably more controversial: that is, whether it’s important for the “best” mathematics to have – or not have – practical utility and applicability. Hardy, a confirmed pacifist, insisted that “Real mathematics has no effect on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.” (p. 140) Although counterexamples have appeared since that was written, they are quite insignificant in comparison to the beauty and (can I say?) grandeur of both theories. (“Grandeur” is a property a related collection of theorems on the same topic may possess. Hardy doesn’t mention it.)

It’s a waste of time to argue over the relative merits of “pure” vs. “applied” mathematics. If a particular professional mathematician prefers one over the other, it’s just a matter of individual taste. If a given body of mathematics happens to have important applications, that fact needn’t either enhance nor detract from whether the mathematics has significant value to those who work on it. And so neither case requires more or less “apology” (in the sense of an excuse or justification) that needs to be given for working on it.

Hardy offers very little of a biographical nature in his Apology, although there are a few pages along such lines at the end. This is rather unfortunate. One would have hoped that such an eminent mathematician as Hardy might have offered a rather more detailed account of his professional working habits and how he arrived at his most noteworthy accomplishments. He wrote some great textbooks (alone or in collaboration), but his best theoretical work was in a legendary collaboration with J. E. Littlewood.

The few biographical remarks that Hardy offers are somewhat downbeat, as the Apology was written late in his life, when he felt most of his creative energy had faded. Too, he seemed to have had for much of his life somewhat of an inferiority complex. Hardy was eight years older than Littlewood, so he may have always been at a disadvantage to the vigor of his collaborator. Littlewood also published one notable book for a general audience (Littlewood’s Miscellany), which is quite different from the Apology, and has its own idiosyncratic strengths and weaknesses. Unfortunately, neither man had almost anything to say about their unique collaboration.

The reprinted (1967) edition of the Apology, which came out 20 years after Hardy’s death, has a long foreword by Hardy’s friend C. P. Snow. As Snow wasn’t a mathematician, it has little to say about mathematics, but contains some very useful biographical details, including much about Hardy’s other great collaboration, with Srinivasa Ramanujan. Professional mathematicians may find this foreword the most interesting part of the book.

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Review of The Mind of the Mathematician

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.

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Review of The Best Writing on Mathematics 2012

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In describing the essays in this volume as the “best” writing on mathematics, the word “best” can’t be taken literally. For one thing a mathematician would naturally point out that there is no simple, obvious linear ordering on the set of writings about mathematics that reflects “quality”. There is certainly no metric to quantify quality for this type (or any other type) of writing. Specifying a particular audience that might have useful opinions on quality would help, but still be inadequate. That said, let’s assume the audience is at least people who understand some mathematics and who appreciate it and value it. That’s still a number of audiences, since it includes professional mathematicians, teachers of mathematics at all levels, and users of mathematics in fields like physics, statistics, economics, etc.

There’s a little something in this volume for many of these audiences, but probably not enough to satisfy most of them. The editor of this collection is a specialist in mathematics education, so that category is over-represented. People who apply mathematics in their own special fields of expertise as well as research mathematicians will probably find the material on mathematics education to be of minor interest.

Many of the other topics tend to be treated in a way that would appeal to that fairly small part of the “general public” that actually has any interest at all in mathematics. Check out the table of contents to see if any topic of that sort interests you. Since this is a personal evaluation, I’ll just mention articles that seemed interesting to me.

Perhaps unsurprisingly, the two best articles in the volume were written by research mathematicians who’ve won Fields Medals: Terrence Tao and Timothy Gowers. Tao writes about a topic he knows extremely well (of which there are quite a few): the distribution of prime numbers. The question is whether there are any patterns in how prime numbers are distributed, or whether the distribution is essentially random. Gowers has two contributions, but the interesting one is the question of whether mathematical truths are “invented” or “discovered”. This can be treated as either an empirical question or a philosophical one. Gowers focuses on the empirical question, and the answer is “both”.

There are several good articles on the history of mathematics. One (by Peter Rowlett) offers a number of examples where solutions to practical problems were enabled by theoretical work that may have occurred centuries earlier. An article by John Baez and John Huerta discusses unfamiliar sorts of “numbers”, including quaternions and octonions, and how they are relevant to symmetry, especially in string theory (as in theoretical physics). Charlotte Simmons writes about the influence of the logician Augustus De Morgan on other notables, such as William Hamilton (discoverer of quaternions) and George Boole (inventor/discoverer of Boolean algebra). Fernando Gouvea shows how the modern theory of real numbers and of sets arose from interactions over many years between George Cantor and his teacher, Richard Dedekind. A satisfactory theory was discovered only after various promising but false starts.

Only three other articles seem worthy of mention. Brian Hayes explains the remarkable topological fact that the volume of a unit n-ball (n-dimensional solid sphere of radius 1) approaches 0 as n tends to infinity. Richard Elwes writes about the very esoteric theory of infinite cardinal numbers (as originally developed by George Cantor). Some of the most recent work has been done by Hugh Woodin, and it suggests that even Kurt Gödel and Paul Cohen didn’t have the last word on Cantor’s Continuum Hypothesis.

Lastly, philosopher Ian Hacking asks the question, “Why is there a philosophy of mathematics at all?” As a philosopher, he attempts to present several plausible justifications. This discussion concerns mathematical Platonism and the “invented” vs. “discovered” issue. Professional mathematicians, though, aren’t that concerned with the philosophy, since for them the really important thing is proving theorems, while letting philosophers worry about what that “means”.

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Review of The Mathematician’s Brain

Birth of a Theorem

The title is slightly misleading, as it might lead one to expect an analysis of how mathematicians work based on psychology or neuroscience. In fact, there is very little of that, especially if one discounts a short chapter on Freud’s views (which don’t really have much of value on the subject). So it would be justifiable to be disappointed with the book if one had that expectation.

While it would be great if someday scientific study of the brain could reveal the mechanisms of creativity in mathematics or in general, that’s not the situation now. Ruelle does report the ideas on mathematical creativity of Poincaré and Hadamard, but there hasn’t been much to add in the last 100 years. Nevertheless, the present book is an excellent follow-on to Ruelle’s Chance and Chaos (reviewed here). These two have much in common: they are relatively short, comprise a number of brief chapters on diverse topics, and are quick reads. Both books cover a lot more ground than their titles would suggest. But, happily, there’s little overlap between them.

In one respect the present book is even shorter than its 130 pages (exclusive of the excellent notes at the end) would suggest, because it’s really two books in one. The first part, about 60 pages, is mostly nontechnical and quite accessible to general readers. It gives, in that short space, a clear picture of the general subject matter of concern to mathematicians since the ancient Greeks. (Which is far more than just arithmetic, geometry, algebra, and calculus.) The remainder of the book deals with more technical topics. There are some more detailed explanations in the notes. However, a reader without some college-level math courses would need to be satisfied to accept the technical terminology without much explication.

But the reward from reading the later chapters, for readers with any amount of mathematical background, is a clear picture of how professional mathematicians actually work. I’ve given the book a top rating for its coverage of topics such as sketches of one of the 20th century’s most original mathematicians, Alexander Grothendieck (whom Ruelle knew personally), set theory as a foundation for mathematics, Gödel’s incompleteness theorem (very succinctly explained), and the Riemann Hypothesis (and the intriguing idea that if RH could be proven to be undecidable within standard arithmetic, it should be provable in a broader theory).

There’s a lot more. The general idea of mathematical structures is explained, and how new concepts are created. This leads to a discussion of whether mathematical concepts are “created” or “discovered” – a very old debate. (There seems to be consensus now that the truth is “some of both”.) There’s a discussion of a theorem, the “Circle Theorem”, which was actually discovered by physicists T. D. Lee and C. N. Yang (best known as winners of a Nobel Prize for discovering CP parity nonconservation). The Circle Theorem, which isn’t well-known to mathematicians, is surprising but simple to state, and has a relatively simple proof.

Finally, there is a discussion of the nitty-gritty details of proving some of the complicated theorems of modern mathematics. In general, what makes this activity difficult is the huge number of possible approaches to finding proofs. It’s like finding one’s way through a high-dimensional labyrinth to a valid proof. There are a huge number of possible paths to explore, almost all of which lead nowhere. Yet, somehow, a process of enlightened “tinkering” with the details of a proof can succeed. This involves proving just the right set of lemmas with a variety of slightly different hypotheses and conclusions in order to reach the desired result. The chapter on “The Strategy of Mathematical Invention” gives a sketch of how this is done, but there are no explicit examples. Fortunately, for anyone who’s curious about this, Cédric Villani’s Birth of a Theorem (reviewed here) provides excruciating details of one example of this process.

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Review of My Brain Is Open

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It’s probably fair to say that a large majority of the general public in the U. S. could not name a single important mathematician who was active in the period 1933 to 1996 (the years of Paul Erdős’ adult life). A few people might think of Andrew Wiles, John von Neumann, or perhaps Kurt Gödel. It would be surprising if more than a percent or two could mention Paul Erdős or even recognize the name. Yet Erdős is probably one of the dozen or so most important mathematicians of that period. For instance, nobody in the history of the world published more new mathematical results (almost 1500), except for Leonhard Euler (who died in 1783).

Erdős created himself or was among the creators of a number of new mathematical specialties, including combinatorics, Ramsey theory, probabalistic number theory, and combinatorial geometry. He also made ground-breaking advances in the early stages of set theory and graph theory. And he added very substantially to classical number theory (one of Euler’s specialties). In addition to his astonishing productivity, he remained active and prolific until he died at the age of 83 – something almost unheard of among mathematicians. Nevertheless he had little or no interest in many other branches of modern mathematics, such as topology, abstract algebra, or mathematical physics.

Schechter’s book is very good at explaining in general terms what each of those mathematical topics are about and the contribution made by Erdős. Unlike most modern mathematics, the things that interested Erdős the most were “simple” things, like numbers and geometrical figures that are familiar to most people. So it’s easy enough to understand most of what Erdős worked on. It would be nearly impossible, however, to explain to non-mathematicians the techniques by which the results were obtained. (After all, his results had eluded all earlier mathematicians.) And even mathematicians are mystified by the mental processes that led to the results.

Erdős was quite an unusual person in other respects as well. He never married or had any apparent sexual interests. During most of his adult life he had no permanent home and almost no possessions other than a couple of suitcases, some notebooks, and a few changes of clothes. He was almost perpetually on the go from one place to another after his welcome at one host’s abode started to wear thin. And his memory was phenomenal – though only for details that were important to him, namely anything relevant to mathematics he cared about and his vast network of mathematical collaborators.

He co-authored papers with almost 500 other mathematicians, and personally discussed mathematics with hundreds of others. He generally knew the phone numbers and other personal details of most in his network. (Yet he was often unable to associate their names and details with their faces when he encountered them at meetings.) He was also able to recall technical details and publication information of thousands of mathematical papers, which may have been published decades previously. This fact probably helps explain his ability in many cases to solve new problems within minutes, because he could recall such a vast number of earlier results and techniques.

Schechter’s biography is generally quite good. The author has a PhD in physics and is obviously conversant with the mathematics that interested his subject. There are just a few minor issues. Although Schechter never actually met Erdős, many personal anecdotes are reported. Those must have come from conversations with associates of Erdős or articles about him, but there’s only a two-page “Note on Sources” instead of footnotes with specific details. There is a good bibliography, however.

The book is only about 200 pages, so it’s a quick read. However, if it had been a little longer, it could have gone into somewhat more detail about Erdős’ mathematics – such as set theory and probability theory. Another Erdős biography (The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth) was published about the same time, less than 2 years after the subject’s death. (I’ve reviewed that one too.) So both biographers most likely wanted to get published as quickly as possible. There are clear signs of haste in both cases. In the present book there are few things that a proofreading should have caught, but only one more serious error: in Chapter 4 there are incorrect references to two of the graph diagrams.

All in all, this biography can be highly recommended for an overview of Erdős’ mathematics, a fine portrait of a very unique and colorful individual, and the opportunity to gain a little understanding of the social process in which mathematics of the highest caliber is actually created.

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