The Mathematician's Brain: A Personal Tour Through the Essentials of Mathematics and Some of the Great Minds Behind Them

by David Ruelle

The Mathematician's Brain poses a provocative question about the world's most brilliant yet eccentric mathematical minds: were they brilliant because of their eccentricities or in spite of them? In this thought-provoking and entertaining book, David Ruelle, the well-known mathematical physicist who helped create chaos theory, gives us a rare insider's account of the celebrated mathematicians he has known-their quirks, oddities, personal tragedies, bad behavior, descents into madness, tragic ends, and the sublime, inexpressible beauty of their most breathtaking mathematical discoveries.


Consider the case of British mathematician Alan Turing. Credited with cracking the German Enigma code during World War II and conceiving of the modern computer, he was convicted of "gross indecency" for a homosexual affair and died in 1954 after eating a cyanide-laced apple--his death was ruled a suicide, though rumors of assassination still linger. Ruelle holds nothing back in his revealing and deeply personal reflections on Turing and other fellow mathematicians, including Alexander Grothendieck, René Thom, Bernhard Riemann, and Felix Klein. But this book is more than a mathematical tell-all. Each chapter examines an important mathematical idea and the visionary minds behind it. Ruelle meaningfully explores the philosophical issues raised by each, offering insights into the truly unique and creative ways mathematicians think and showing how the mathematical setting is most favorable for asking philosophical questions about meaning, beauty, and the nature of reality.



The Mathematician's Brain takes you inside the world--and heads--of mathematicians. It's a journey you won't soon forget.

Genres: Mathematics, Psychology, Science, Popular Science, Philosophy, Nonfiction

176 pages, Hardcover

First published July 16, 2007

About the author

David Pierre Ruelle is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term strange attractor, and developed a new theory of turbulence.

Reviews

[Charles Daney · Rating 5 out of 5]

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 Theoreme Vivant (reviewed here) provides excruciating details of one example of this process.

[Alejandro (IG: todoporleer) · Rating 3 out of 5]

Si eres matemático te vas a aburrir.
Si eres lego en la materia no leas este libro.
Si tienes conocimientos y lo quieres leer para motivarte a estudiar matemáticas, no lo va a conseguir.
Lee este libro por el puro placer de quedarte atónito de cómo se construye una teoría matemática, hay alguna pequeña demostración de ella, trivial para el autor pero ni mucho menos.

El libro presenta capítulos muy interesantes y otros algo repetitivos, se supone que va a entrar en el cerebro de los matemáticos pero eso es una empresa difícil por lo que se queda en la superficie.

Se trata de ideas, conceptos, creatividad, formalismos y alguna que otra curiosidad interesante.

Lo he disfrutado a ratos, así que o te gustan las matemáticas o huye de este libro.

[William Bies · Rating 2 out of 5]

The French mathematician David Ruelle is well known to physicists for his painstakingly rigorous work on the foundations of statistical mechanics and chaos theory, in particular for an innovative theory of turbulence as originating in a strange attractor (which, though speculative when proposed, has subsequently received striking experimental support). In the present volume issued in 2007 by the Princeton University Press, The Mathematician’s Brain, he offers us not just another routine exposition of popular mathematics, but the reflections of a lifetime on the nature of creativity in the field of mathematics – a topic on which, however exciting, few with the requisite qualifications have ventured to publish. Thus, the curious reader must be alert to the prima facie promise when a distinguished mathematician takes up his pen on this subject of perennial interest to any cultured mind.

A glance at the table of contents indicates that Ruelle intends to approach his theme with broad brush strokes. In twenty-three short chapters, typically five to ten pages in length each, he broaches questions such as the nature of mathematics itself, the relation between mathematics and ideology, the unity of mathematics, algebraic geometry and arithmetic from the point of view of Alexander Grothendieck’s revolutionary program, mathematical structures and texts, artificial computers versus the brain, infinity, set-theoretical foundations, psychology and aesthetics, the infinite-dimensional labyrinth, mistakes, the construction of mathematical theories and the respective roles of tinkering versus strategy, mathematical physics and emergent behavior and, lastly, the beauty of mathematics itself. Ruelle cannot be faulted for want of ambition in attempting so much!

Unfortunately, his execution fails to live up to the promise, perhaps because the short-chapter format does not permit very much in the way of systematic development of connected ideas. What we get instead is an array of occasional stray comments that may pique one’s interest, coming as they do not from a crank but someone who has worked with many of the first-class mathematicians of the past century. For instance, Ruelle observes somewhere that about twice as many mathematicians happen to be religiously inclined than physicists yet neglects to follow up with any grounded account of why this disparity across intellectual disciplines should hold sway. In fact, he usually doesn’t enter very deeply into why he thinks his remarks can be justified in terms of an underlying philosophy of mathematics or psychology of invention. Hence, disorganized, rhapsodic, marred in a few places by irrelevant personal or political commentary (on Grothendieck, Turing).

The author moreover cheerfully commits a fault common to those who are versed in a technical field when writing for a popular audience, namely, that of quoting detailed results without any context (for instance, in chapter 17 Lee and Yang’s rather technical circle theorem which originates in a problem in statistical mechanics, or the discussion of the Ising model in chapter 22). It may be all right to bring up such things as illustration either if part of a coherent exposition or if they offer particular insight into a question under discussion (cf. Lessing’s use of the Laocoon sculpture group in his thematic discussion of the relation between the plastic and literary arts), but neither is the case here with Ruelle. His reason for quoting this result seems to be to lend support to his claim that mathematical discovery typically proceeds by starting from a shrewd guess as to what ought to be the case, then casting about more or less indiscriminately amongst what one already knows to see whether a concatenation of steps could lead from there to a justification of one’s conjecture:

I have just described human mathematics as a labyrinth of ideas, through which the mathematician wanders, in search of the proof of a theorem. The ideas are human, and they belong to a human mathematical culture, but they are also very much constrained by the logical structure of the subject. The infinite labyrinth of mathematics has thus the dual character of human construction and logical necessity. And this endows the labyrinth with a strange beauty. It reflects the internal structure of mathematics and is, in fact, the only thing we know about this internal structure. But only through a long search of the labyrinth do we come to appreciate its beauty; only through long study do we come to taste fully the subtle and powerful aesthetical appeal of mathematical theories. (p. 96)

Every productive mathematician has of course his own style, which works for him – one cannot contest it if this is how Ruelle’s mind is disposed. But it scarcely seems plausible that this represents the summit of achievement in the great works of mathematical genius. For indeed, what Ruelle describes does pretty much correspond, say, to how the present reviewer tackled the problem sets when an undergraduate: after reading the chapter, look at the statement of the homework exercises, formulate a guess about how to solve them and flip through the preceding sections in search of theorems or techniques that might get one there – often enough, such a procedure will succeed, but the real learning occurs precisely when it does not: then one has to root around for a long time in order to uncover what concept one is missing or applying faultily. If Ruelle were right and something like this were all the professional mathematician has at hand in his arsenal, it hardly seems that any deep structural results would ever have come to light. In textbook presentations (which, needless to say, misconstrue the true order of discovery), the initial phase in the exposition of a new branch of mathematics may appear to proceed somewhat along the lines Ruelle portrays – once one has the appropriate concepts at one’s disposal! – glance, say at the beginning chapters in group theory or in the theory of field extensions or modules over a ring, but what makes the subject interesting and rewarding is when the preliminaries have been concluded and more profound ideas come into play. The great discoverer seems to be able to call upon a powerful intuition, based in an obscure manner on the peculiarities of his mathematical education, that guides him according to a vision of what ought to be the case. Consider, for instance, Riemann and Weierstrass’ contrasting approaches to the theory of functions of a complex variable, or Ramanujan’s method of conjecturing surprising formulae in number theory. Yet Ruelle’s breezy account does not allow for anything like this.

Since, as already indicated, one cannot look anywhere in the present work for a coherent overarching thesis, let us merely comment on a handful of salient points suggested by a perusal of Ruelle’s exposition:

1) Does Ruelle motivate his anti-Platonism at all? Not really, he merely weasels out by suggesting something along the lines of ‘fine if it works for you, but I don’t need it’ – perhaps if like Ruelle one is a technician one can very well leave the contemplation of ideas to others more endued with inventiveness and get by with what these intrepid explorers report about them: philosophical judgments like aesthetic judgments although to a certain extent matters of taste ought to be objective in the sense that they are supported by reasons. The only contention Ruelle advances in the two paragraphs he devotes to the question is the following: ‘The consideration of the mind may be irrelevant when we discuss the formal aspects of mathematics but not when we discuss conceptual aspects. Mathematical concepts indeed are a production of the human mind and may reflect its idiosyncrasies’ (p. 45). Hence the ideas are mind-dependent. But doesn’t he confuse the order of knowledge with the order of discovery? Supposing there were pure eternal ideas; wouldn’t we come to know them (to the extent that we do) through the exercise of our minds? Ruelle’s condensed argument is neither here nor there.

2) Ruelle’s treatment of intuition and meaning in mathematics. He remarks, the ‘intellectual landscape has changed since Hadamard wrote his wonderful little book’ (p. 87). So: we have now a better understanding of short and long-term memory and, after the advent of digital computers, look differently upon the process of thought in general. Despite the fact that ‘good mathematical ideas are selected on on aesthetic basis’ (paraphrase of Hadamard):

Many scientists try their literary talent on an autobiography, other paint or play an instrument. The results are often not bad, but rarely great. And in many cases really good scientists achieve truly mediocre artistic results. (p. 88)

Mathematical good taste, then, consists of using intelligently the concepts and results available in the ambient mathematical culture for the solution of new problems (p. 89)

We have seen that the presentation of a mathematical theory in a technical article is somewhat removed from what the author had originally in mind. Intuitive ideas and nonverbal concepts have to be dressed up and couched in professional jargon….But the intuitive meaning of mathematics is rooted in formalism. If one were to abandon the formalism and keep only the intuitive meaning, mathematics would soon be a matter of opinion rather than of knowledge. Its progress would then rapidly grind to a stop. (pp. 117-118)

Comment: we are faced at this juncture with a mystery. For, on the one hand, mathematics is characterized by precision, the utmost attainable in any human endeavor. Yet, on the other hand, mathematicians want to entertain constructs far beyond the capacity of language to pin down: for instance, an arbitrary function on some space, or any of the vastly more abstract entities of higher algebra. A verbal description can do little more than to hint at what we see via our imagination. How do these conflicting tendencies interact with each other in the process of mathematical derivation? For it is not enough to have a vague impression: one wants to make a definite statement and then to justify it in all rigor. So, Ruelle barely scratches the surface here.

3) In chapter 19, Ruelle reprises Freud’s psychoanalysis of Leonardo da Vinci’s genius in terms of sublimation (and by implication, Newton’s etc.) – unfortunately a little sketchily; he confines himself to the observation that while sublimation may help us to understand the drive to pursue discoveries in mathematics, it cannot explain any of its content, for mathematical thinking qua thinking is separable from the subjective motivation to undertake it in the first place: true enough! Yet one wishes Ruelle had entered more deeply into the problem of aesthetics in mathematics, for the hallmark of the aesthetic lies in the appeal of the object of contemplation to the will in virtue of the disinterested pleasure it affords, which is a response to perceived good. Thus, it seems to pose a delicate challenge to disentangle the intellectual from the emotional, and no doubt the keen psychologist (whether Freudian or not) could have something valuable to contribute to an improved understanding of specifically mathematical ideation.

4) Does Ruelle have anything important to say about mathematical beauty in the final chapter? For him, the sense of beauty is not ruled by logic alone. To a frail human being mathematical truth is consoling because absolutely certain in a world of uncertainty. Despite the usefulness of mathematics in its applications, good mathematicians rarely are motivated by duty. In answer to the question: what make mathematics beautiful? ‘The beauty of mathematics lies in uncovering the hidden simplicity and complexity that coexist in the rigid logical framework that the subject imposes’ (p. 129). Unlike works in the fine arts and literature, then, which to a large extent are free products of the artist's imagination, the logical framework of mathematics is not a human construction but intrinsic. An indication of this can be found in the possibly surprising circumstance that ‘a short mathematical statement may need a very long proof’ (p. 130).

And as we come to the end of our journey, let me say one more thing: it is while doing mathematical research that one truly comes to see the beauty of mathematics. It faces you in those moments when the underlying simplicity of a question appears and its meaningless complications can be forgotten. In those moments a piece of a colossal logical structure is illuminated, and some of the meaning hidden in the nature of things is finally revealed. (p. 130)

Thus, Ruelle’s take on mathematical beauty may not be very grounded but will do as a prompt to one’s own meditations. Let us seek to consolidate what Ruelle has just said in philosophical terms: the Kantian Zweckmäßigkeit perceived to lie at the root of mathematical beauty consists in a balance of two terms: simplicity and sweep versus content and depth (somewhat like minimization of free energy in thermodynamics, which involves the difference between an energetic and an entropic term and thus in equilibrium a balance between competing tendencies).

All around: stimulating, worth the read but too unsystematic to engage the reader in a protracted reflection on the material Ruelle himself puts forward, a disappointing outcome given how much intrinsic interest his topic enjoys: 2 stars.

Why Hadamard’s work broaches subjects with sustained intellectual interest while, in the main, Ruelle’s fails to: the latter seems to view his task as one of spinning out a few stray observations based on his life’s experience as a working mathematician whereas the former understands himself as a participant in a learned discourse with interlocutors among psychologists, artist and poets etc. and thus has at his disposal many points to offer to us which he evidently has distilled from years of reflection on his problem of particular interest, viz that of invention in the mathematical field, undertaken with the wider context constantly in mind. Hence, Ruelle never takes his own aperçus very far, even when they might promise to yield an avenue into a fruitful theme for investigation. As a case in point, he mentions (briefly as is his wont) a possibly paradoxical-seeming occurrence: very often, faulty steps in the proof of a major new theorem crop up during the process of peer review, but almost always it turns out to be possible to repair them. Why should this be so? Evidently, as Ruelle himself comments, a good mathematician possesses a powerful sense of intuition, but then he declines to delve any more deeply into the reasons behind this curious phenomenon. If he were instead Hadamard, a man of broad intellectual culture, he would have pondered over it for years and have arrived at a few provisional conclusions which he could state in simple terms.

[Bablu · Rating 4 out of 5]

good rapid fire natural language explanations of some math interspersed with historical anecdotes, reflections, and process theories on how mathematical work is done by one particular guy. the shortness of each chapter (about 5-10 pp per) was very helpful in reading through quickly and is something i'll consider adopting in future writing

[mono · Rating 4 out of 5]

Ω(»,Φ,I,β) - I doubt this will make any pedestrian a mathematician, but I enjoyed it more than 📚The Joy of X & 📚Love & Math. It is a metamath book, but there are glimpses of twilight behind the veil.

⇝ - Barry Simon's Case Against Codes, 📚A Course in Arithmetic, Stephen Smale, Erlangen Program, Samuel Eilenberg, Saunders Mac Lane, Laurent Schwartz, ZFC, Jacques Hadamard

[Peter Flom · Rating 3 out of 5]

In the Preface, Ruelle writes: "My ambition is to present here a view of mathematics and mathematicians that will interest those without training in mathematics, as well as many who are mathematically literate." I don't think he really succeeds. I like mathematics, I've read a good deal of "popular math", I worked as a statistician, and I took some math in college. But there was much here that I could not follow.

Nevertheless, what I could follow was fairly interesting, but also quite scattered.

The subtitle of the book is also somewhat misleading. This book doesn't cover what I would regard as "the essentials of mathematics" and, while it does talk about some great mathematicians (Grothendieck, Turing, a bit of Newton) I wouldn't say this is a sampling of the great mathematicians.

I think I was expecting something of an updating and blend of books such as *What is Mathematics?" and "Men of Mathematics". That isn't what I got.

[Fernando Pestana da Costa · Rating 3 out of 5]

This book, by a world famous dynamicist and mathematical physicist, is a kind of digression about mathematics, mathematicians, ethics, politics, philosophy, and more. His discussions of mathematics, in the chapters "What is Mathematics?", "The Erlanger Program", "Mathematics and Ideologies", and "The Unity of Mathematics", are extremely well done, even enlightening. Some other chapters are less compelling. Overall: a book that certainly deserves to be read.

[Pablo · Rating 4 out of 5]

Sobre todo me fascina la vida de Alexander Grothendieck. Un libro fundamental para entender cómo pensaban algunos de los matemáticos más importantes del siglo XX.

[Tim · Rating 4 out of 5]

Ruelle, a mathematical physicist, gives us a careful collection of accounts, reminiscences, results, and arguments from his professional experience and from his broad reading in mathematics and science. As a result the book is arguably about much more than Ruelle's own brain. Among the many themes he revisits, he sympathetically touches on a Platonic view of mathematics - the language of nature as Galileo and others termed it - but foresees the day when logic machines - computers - may supplant entirely human contributions to the development of the language. Others (cf. psychologist George Lakoff) have argued that mathematics as humans know it - and perhaps as humans can program machines to fashion it - is always a construction of our brains. Ruelle recognizes the tensions in human mathematics, particularly that between cultural sufficiency and logical necessity. But the connection to the book of nature out there or the book of nature written in the psyches of individual mathematicians retains enduring allure. As a companion to this volume, Ruelle's earlier account of 'Chaos and Chance' is a wonderful introduction to what chance could mean in the world we inhabit.

[David · Rating 3 out of 5]

This book is not at all what I expected from the blurb. While Grothendieck and Turing are certainly mentioned, this book is not primarily about mathematical characters and their quirks. Rather, it is about the character of mathematicians, and about how research mathematicians look at the world.

The book is organized as a collection of short, essay-like chapters on topics like the story of Turing, Freud's description of Da Vinci and it's possible bearing on mathematical creativity, what it means to write a proof, and some ideas about mathematical physics. I read it a chapter or two per sitting, so I may have an exaggerated view of how loose the organization is.

On reading many chapters, I found myself thinking that I ought to be doing mathematics instead! In part, I think this is because of the flavor of the discussion. In part, too, I think Ruelle's writing style sounds a little too much like my own writing for me to find it completely comfortable.

[Tom · Rating 5 out of 5]

Only problem with this book is that it's too short.

[Brian Page · Rating 3 out of 5]

Inspiring but still a bit disjointed.