When physicists write books for the general public, they write about black holes, dark matter, or strings that wriggle like mad. The universe is their subject. Mathematicians write about mathematics and what it all means. Their subject is their subject.
The mathematician David Ruelle is well known for his work on nonlinear dynamics and turbulence, and his new book, “The Mathematician’s Brain” (Princeton University Press, 172 pages, $22.95), is a book about mathematics and what it all means.
If the entomologist studies bugs, and the linguist languages, just what is it the mathematician studies? Sets, numbers, equations—that much is clear. Thereafter, everything solid dissolves into thin air. What is a number? Or a set? Or a shape, for that matter?
If mathematicians cannot say what mathematics is about, neither can they say why its conclusions are certain. “The instability of human knowledge is one of our few certainties,” the journalist Janet Malcolm remarked recently. Yet mathematicians believe that their conclusions are forever.
In answering his own questions, Mr. Ruelle advances two general theses. The first is that “the structure of human science is largely dependent on the special nature and organization of the human brain.” The second is that “the scientific method is a different thing in different disciplines.”
The first claim is empty. We do not know how the brain generates its thoughts. If the brain is simply a physical organ, there is no reason to suppose that it has access to any form of certainty beyond the calculations needed to climb the greasy pole of life. If the brain does have such access, then the structure of human science cannot be largely dependent on its physical organization.
It is quite true that there is no such thing as the scientific method. There are many methods. What counts in mathematics is proof, a systematic way of deriving conclusions from assumptions. Under ordinary circumstances, a mathematical proof is written in the mathematician’s vernacular. Precision is demanded — physicists need not apply — but not logical obsession. Yet mathematicians have in the 20th century learned just how an informal proof may be expressed as the driest of dry structures, a system in which meaning has been stripped from symbols and their manipulation governed by precisely stated rules. The result resembles a text written in an alien language, or a program written in assembly code. No mathematician would dream of presenting his proofs in this way. They would take too long to write and once written, they would never be read.
While Mr. Ruelle is an excellent mathematician, he is no logician. And it shows. “We can in principle,” Mr. Ruelle writes, “give a completely formalized presentation of mathematics.” This is quite true. It hardly follows, as Mr. Ruelle concludes, that “mathematics is the unique human endeavor where the use of a human language is, in principle, not necessary.” If proofs are stripped to their syntactic shell, they have no interest. The human language that Mr. Ruelle is eager to dismiss reasserts itself the moment the mathematician asks about the meaning of the proof.
If “The Mathematician’s Brain” does not answer the questions it poses, this is because no other book has answered these questions either. The book’s value lies in Mr. Ruelle’s description of the curious inner life of mathematicians. Their subject is very difficult. It requires unusual gifts. Physicists may disguise the triviality of their results by bustling about in large research groups. Mathematicians work alone. They are professionally naked.
As a result, many mathematicians have unstable personalities. Alexandre Grothendieck is an extreme example. His is hardly a household name, especially in the English-speaking world. Yet for the 15 years between 1958 and 1973, Mr. Grothendieck dominated the field of algebraic geometry and ruled like a prince over a court comprising some of the most talented mathematicians in the world. His immense treatise on algebraic geometry is, as Mr. Ruelle observes, the last great mathematical oeuvre written in the French language.
To algebraic geometry, Mr. Grothendieck brought an entirely new level of power and abstraction, so much so that his colleague René Thom — a Field medalist and a great mathematician — acknowledged that he left pure mathematics because he was oppressed by Mr. Grothendieck’s “crushing technical superiority.” His technique was only a part of his genius. Mr. Grothendieck was a great mathematical visionary. Like mystics searching for the face of God, he was passionately concerned to see the unity of form behind various mathematical experiences. He did not simply solve isolated problems but, as Mr. Ruelle writes, enveloped them “in a rising tide of very general theories.”
If Mr. Grothendieck was a magnificent mathematician, he was also a political simpleton. Departing the Institut des Hautes Etudes Scientifiques after a pointless dispute about the Institut’s military funding, he found it impossible to obtain a position in France commensurate with his stature. For a time, he taught undergraduate mathematics at a provincial French university. Mr. Ruelle considers Mr. Grothendieck’s internal exile a great injustice. It was nothing of the sort. Just as he was a political simpleton, Mr. Grothendieck was a personal pest, sending endless letters to leading mathematicians accusing them of insufficiently appreciating his genius or stealing his ideas.
Mr. Grothendieck is now said to be living in a shepherd’s hut in the Pyrénées, where he lives on a vegetarian diet and is sustained by meditation. During the 1980s, he wrote an immensely long autobiography entitled “Récoltes et semailles” (Harvest and Sowings). As Mr. Grothendieck devoted a large part of the book to the denunciation of his colleagues, no French publisher accepted it for publication. Sections have appeared on the Internet. They are often interesting in the extent to which they express the nature of mathematical longing.
Mr. Ruelle is far too discreet to tell Mr. Grothendieck’s story well, but it is important that he has told it at all. Of all the arts, mathematics expresses most completely the peculiar human desire for perfect understanding. It is a desire that when fulfilled evokes a form of gratitude, but the terms of the mathematician’s contract are severe, and if gratitude is the mathematician’s reward, a certain personal deformation is the price that all too often he must pay.