Customer Reviews

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

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The author, who is a very distinguished mathematician, gives his personal view on how mathematicians think. It is welcome to have books like this written by real mathematicians, as opposed to philosophers who doesn't know that much math. While professional mathematicians might not learn much, students of mathematics can get some very nice insights into how mathematics and mathematicians work.

Unfortunately, some parts of the book that discuss specific mathematics (as opposed to what mathematics is like in general) are not clearly written and should have been edited better. For example, it shakes the confidence of the reader when early on, the pythagorean theorem is stated incorrectly, and then on the next page a statement is asserted to follow from the pythagorean theorem, when it actually follows from the converse of the pythagorean theorem. Most readers of the book will probably know this anyway so it doesn't matter, but later, descriptions of more advanced mathematical concepts are sometimes so brief that they would probably be incomprehensible to someone who does not already know them, and puzzling to someone who does.

Disclosure: I only skimmed this in the bookstore because I didn't feel like paying 20 cents per page for it. I hope that an inexpensive paperback edition will appear, with corrections.

In this small book the author (a distinguished professor of mathematical physics) touches on what mathematicians do, how they do it, how they think and feel about it, and how they relate to the world at large. On such a quick tour there are bound to be some mysterious turns and bumps on the road. More than necessary occur in this book: advanced topics are frequently introduced with unhelpful advice for the novice such as "Just go through it rapidly." Nevertheless I enjoyed learning a new bits of math (now I can define algebraic geometry) and stories of mathematicians. What kept me going was the author's skeptical attitude toward the mathematical establishment of which he is a part, and his genuine compassion for colleagues whose genius can so easily turn to madness.

David Ruelle continues the venerable French tradition of great mathematicians and
scientists writing for the general educated public about their craft, and about
the deeper meanings of it. Especially intriguing are Ruelle's insights into
mathematicians' minds, and his balanced view of platonism vs. the contingency of
history and the human brain.

Ruelle mentions that, with very few exceptions, great scientists are not great writers, and he states Henri Poincare as a notable counterexample. I would add that
Ruelle himself is even a better specimen of a great mathematician and a great writer.

I sense that David Ruelle wrote this book as a labor of love, and I feel priveleged to have been able to read it (as with his wonderful book Chance and Chaos). He provides a fairly penetrating and sophisticated treatment of the nature of mathematics and what it's like to be a research mathematician. His writing style is informal and friendly without sacrificing clarity, precision, and elegance. He doesn't shy away from including some real and nontrivial mathematics (for demonstration purposes), but the book isn't overly technical and he puts the harder stuff in the endnotes. If you've at least dabbled in higher mathematics and have some rudimentary familiarity with set theory, abstract algebra, topology, number theory, Turing machines, etc., you should be able to handle the book (and love it); without that background, it may be tough going.

Perhaps the best way to describe the content of the book is to summarize some of the key points:

(1) A goal of mathematical deduction is to derive nontrivial and interesting results (particularly mathematical theories), not just any or all results which follow from the axioms. Mathematics makes progress because new theorems are built on prior theorems. As it has developed, mathematics has generally become more difficult, though breakthroughs sometimes allow the solution of many problems to be greatly simplified.

(2) Solution of mathematical problems is aided by proper (or clever) classification of problems, imagination, allowing problems to incubate in the unconscious, use of analogy as a heuristic (though not highly reliable), and brute-force use of computers (which is controversial, since such methods have little appeal to our intuition and our desire for insight).

(3) Finding proofs can sometimes be very difficult because the process is like "walking in an infinite-dimensional labyrinth," trying to connect ideas in a sequence which meets the requirements of logic. Even seemingly simple theorems may require very long proofs (eg, Fermat's last theorem).

(4) When errors and gaps in proofs are found, it's often not overly difficult to correct them, so the resulting theorems tend to be fairly stable. In other words, the same destination can often be reached by many paths.

(5) Mathematical papers generally consist of figures, sentences, and formulas. Figures make use of our visual skills, but they're rarely mandatory. Sentences in natural language are indispensible. Formulas are compact and efficient ways of expressing sentences. Putting all of this together well is an art. Formal language could be used in principle but is unworkable in practice.

(6) The conceptual or intuitive aspect of mathematics is related to its natural structures, which are not the same as the formal aspects of mathematics. These structures may reflect human and historically contingent elements, rather than being purely "natural."

(7) The different branches of mathematics are deeply related, sometimes in surprising ways. Set theory (eg, ZFC) is perhaps the most fundamental branch of mathematics. The natural structures of mathematics often guide the development of new branches of mathematics.

(8) "Active research in mathematics gives intellectual rewards different from those enjoyed by a spectator." This research is primarily an individual rather than group activity, but the overall body of mathematics is a collective achievement.

(9) Many (but not all) mathematicians are prone to a "somewhat rigid way of thinking and behaving," mathematicians are twice as likely as physicists to be religious, and, on average, mathematicians don't possess greater artistic ability than the general population. Their special aesthetic sense is therefore of a different kind from that of artists.

(10) Nature is remarkably amenable to mathematical modeling ("unreasonable effectiveness"), especially in physics, and tends to give hints regarding which models to use.

(11) There's a striking contrast between the fallibility of the human mind and the infallibility of mathematical deduction. Unlike science and other intellectual endeavors, mathematics transcends uncertainties and offers a (Platonic) "perfection, purity, and simplicity" which we naturally yearn for, even if we can't be sure how mathematics ultimately relates to us and physical reality. Moreover, "the beauty of mathematics lies in uncovering the hidden simplicity and complexity that coexist in the rigid logical framework that the subject imposes."

(12) Gödel showed that, for a consistent and nontrivial axiomatic system, the system will contain true statements which can't be proven from within the system, including its own consistency. This discovery of incompleteness doesn't overly trouble most mathematicians in their daily work, though I personally find it to be profound and somewhat disturbing, or at least very perplexing ...

If these key points interest you, I urge you not to miss this book. If you find them obvious, I recommend reading the book anyway, since a list of key points doesn't do justice to the richness and charm of Ruelle's discussion. Personally, this book ranks among my favorite mathematics books and I'm a bit saddened to have reached the end of it. Now I just hope that Ruelle will write more books for nonspecialists!

The man who coined the term 'strange attractor' provides a contemporary and a personal look at mathematics in an easy to read fashion. This book is a little bit eclectic which may be considered as a positive point for people outside the world of mathematics, and Ruelle does not adhere to a linear organization, preferring to jump from one subject to another but manages to provide good connections and insights.

Among the mathematicians he writes about, I found the case of Alexander Grothendieck very remarkable, inspiring, sad and hilarious [1]. This is a very interesting part of the history of mathematics which includes important lessons about organizations, politics and power relations.

Ruelle's discussion on some messy parts of math and proof-checking is very good and he poses important questions about proofs getting longer and longer and formalisms required to handle things as rigorously as possible.

The closing chapters are devoted to Ruelle's area of expertise and he writes very strongly on mathematical physics and give very good examples how diverse scientific fields help each other.

1- See my blog entry 'Corporatism in Science and Math: Mathematician Missing - Part 2':

http://ileriseviye.org/blog/?p=2223

The book is philosophically shallow and mathematically unfocused. The author rambles and his writing lacks intellectual vigor. The chapters read like first thoughts dashed off before drifting off to sleep. The endnotes function as next-day supplements to give the book an illusion of depth. At most, these pages may find a respectable place online where they can be skimmed and forgotten. Anyone interested in Alexander Grothendieck, for example, who appears in chapters six and seven, can find through a simple online search, narrative portraits of more substance and value than what Ruelle offers.