Spanning Multiple Worlds

December 21, 2008

Book Review
Philip J. Davis

Mathematics as Metaphor: Selected Essays. By Yuri I. Manin, American Mathematical Society, Providence, Rhode Island, 2007, 236 pages, $49.00.

Over the years, admirers of Yuri Manin have sent me writings either by or about him, so I do not come to this collection unprepared. I have received among other things a review of his booklet Mathematics and Physics, a letter in praise of the teachability of his book A Course in Mathematical Logic, Manin's own summary of the legacy of Georg Cantor, and an offprint of his article "Mathematical Knowledge: Internal, Cultural and Social Aspects." The last two are reprinted in the book under review.

Manin has made fundamental contributions to algebraic and non-commutative geometry and to mathematical physics. Lest my readers should think that Manin's scope is bounded by mathematics and physics, I can point to a section of the present collection on language and consciousness in which he discusses, among other things, psychological archetypes, the occurrence and problems of autism, and C.P. Snow's Two Cultures.

What is clear from the dazzling display of knowledge and opinion in this collection---mere chips, I suspect, off Manin's block---is that Manin is a polymath who speaks from an extremely high altitude. It's enough to make us "laborers in the field" feel miniaturized insofar as we have not yet come up with a once-in-five-hundred-years meta-theory.

High altitude? Yes. In a foreword to the Selected Essays, Freeman Dyson compares Manin to a bird and himself to a "frog that lives in the mud and sees only the flowers that grow nearby." Thinking of Sir Isaiah Berlin's classic metaphor of foxes who know everything as opposed to hedgehogs who know only one big thing, I should call both Dyson and Manin foxes. But enough zoo-oriented psychological taxonomy.

I think that the mathematical community could plug into Manin's writings with considerable stimulation. Yet a sampling of my colleagues revealed that Manin could be better known to them. Therefore, a few biographical details are appropriate.

Manin was born in Simferopol, in the Crimea, in 1937. Growing up in Russia, he did his PhD at Moscow State University under Igor Shafarevitch. Now head of the Max Planck Institute in Bonn and a professor at Northwestern University, Manin is a mathematical star, a presence, a platonist in the philosophy of science, a jungian in psychology, a romantic.

Reviewing Manin's Mathematics and Physics, Dyson writes:

"He shows us physics and mathematics as two neighboring gardens, each growing luxuriantly with trees and flowers in great variety, while the busy physicists and mathematicians fly to and fro like bees carrying pollen for cross-fertilization."

Manin is an elitist, largely an optimist, and---I've been told---a genial and relaxed person. He is also a fine writer, who displays with apodictic certainty knowledge of cultures ranging from Homeric Greece to that of the Winnebago Indians. He has produced so many clippable quotes that a whole book could be profitably fashioned out of them. Here are some samples:

"One cannot tell what mathematics teaches us, in the same way as one cannot tell what exactly we are taught by War and Peace."

"Wisdom lives in connections."

"As always, the visible must be explained in terms of the invisible."

"Self awareness begins when one starts speaking on behalf of one's self."

"An ideal observer of the macroworld cannot change it, but even an ideal observer of the micro-world is bound to change it."

"Despite the democratization of education, such national poets as Dante and Pushkin so far exceeded the average level of language abilities as to become the nation's language teachers for centuries."

"The Romantic revolution of a century and a half ago did not really influence mathematics because there was little place in it for personal whims."

"We are more deeply troubled when we wonder what the author wants to say than when we do not quite see whether what he or she is saying is correct."

"Arrow's ‘Dictator Theorem' illustrates the content of Jean-Jacques Rousseau's idea of a Contrat Social."

Reading Manin, I get the feeling that once he has spoken on anything, from, say, non-commutative geometry to the history of argyle socks, there is very little left for the rest of us to add.

From these essays as well as from an interview, conducted by M. Aigner and V.A. Schmidt and placed at the end of the book, one gleans his views as to what is significant in mathematics and where the subject is going. I read these views as the analytical continuation of his own interests and contributions. His searchlight casts a powerful beam but does not illuminate the wide diversity of the mathematical enterprises.

Mathematics as metaphor? Of course, in that a successful mathematical model mimics the physical world by means of the symbolic world. But there is also mathematics as fiat---when, for example, Congress legislates a tax policy and American citizens dance to its tune.

I believe also that Manin has seriously misjudged the power of the computer juggernaut to change the face of his and our beloved subject. The computer is not a mere mathematical excrescence, useful for technological ends. Rather, I believe that it is a meta-development that might very well change what mathematics is considered to be. Not being a believer in the metaphysical myth of the unity of mathematics, I foresee that in the course of the 21st century, practitioners of mathematics might be split into two groups: the "Old Believers" and the computer-dedicated "Young Turks," with little love lost and with only an occasional bee buzzing between them.

I prefer, though, to dwell not on disagreements, but on the reverse. In the interview, Manin refers to the "two culture controversy," raised most vehemently in 1959 by C.P. Snow, a British scientist, science administrator, and novelist. In brief, Snow asserted that there are two cultures in operation in the intellectual world: the scientific–technological and the humanistic–literary. Science, Snow wrote, has remade and continues to remake the world; the future lies with it. The two cultures exist as discrete and separate groups in mutual incomprehension and hostility.

The first person whose answer to Snow received wide publicity was F.R. Leavis, a professor of English at Cambridge University, a literary critic and philosopher with a reputation for brilliance and mordant wit. Leavis pointed out, among other things, that there is a creation of the human mind more basic than science. It is language, and without language, the scientific enterprise is impossible. Indeed, Manin goes along with this when he writes,

"(Mathematics) is considered a specialized dialect of the natural language and its functioning as a special case of speech."

Leavis asserted further that humanistic values are those that promote consciousness of full human responsibility, and that these values are fostered by the interplay between living language and the genius of literary sensibility; this is untranslatable into logic or mathematical symbols. I agree and venture to think that Manin would also.

Many articles appeared in the brouhaha that followed the Snow/Leavis exchange, some authors taking one side or the other, and some attempting reconciliation. I believe, with Manin, that the split between the scientific and the humanistic attitudes is alive and well, and that ignorance of each other's values is a very dangerous matter.

As regards a reconciliation between the positions of Snow and Leavis, Yuri Manin's life and thoughts, spanning the two cultures in profound and original ways, show that it is indeed possible. Such a conjunction could help to counter the recent widespread reversion to mysticism, medievalism, and fundamentalism. Or, is it just possible that constructive elements in this reversion will create a vibrant tri-valent culture?

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at [email protected].

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