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!
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N J Sheikh
Really enjoyed reading this book - I only have an A ...
Reviewed in the United Kingdom on August 20, 2015