Mathematics and the Modern Mind

January 2, 2002

Book Review
Philip J. Davis

The Modern Mind: An Intellectual History of the 20th Century. By Peter Watson, HarperCollins, New York, 2000, 846 pages (including 54 pages of notes and references), $40.00.

I ordered this book so I could annoy the author and I stayed on to enjoy his words. But as the reader will see shortly, I've not totally abandoned my original intention.

What we have here is the author's selection of the principal intellectual tendencies and events of the 20th century. The book opens with Freud and Arthur Evans (who excavated the Minoan Knossos in Crete) and ends with Neil Postman (media theorist at NYU). Having myself lived through the greater part of the 20th century and experienced its tendencies either directly or through cultural osmosis, I'd say that Watson has done a creditable job: a labor of Hercules, and presumably of love, and an exhibition of incredible perseverance.

The Modern Mind consists essentially of a series of summaries of many, many books woven together in more or less chronological order by threads of history, biography, personal evaluations, and speculations. In the passage from Freud to Postman, the reader meets Bertrand Russell, Enrico Fermi, Eldridge Cleaver, Jacques Derrida, John Maynard Keynes, Stephen Hawking, and some two thousand additional personalities, including Marlene Dietrich, to whom the author says a brief hello. The reader encounters discussions of capitalism, socialism, cubism, Darwinism, pragmatism, phenomenology, the classification of languages, "e-mailism," nucleosynthesis, and electronic music: almost a thousand indexed ideas. Peter Watson is a liberal-minded journalist who has written for such publications in London and New York as the Observer and the Times. Over the years, his books have revealed a penchant for the sensational: The Caravaggio Conspiracy (scams in the art world), The Death of Hitler, and so forth. Watson is good at getting to the pith of complex situations and arguments. He knows how to keep the reader on edge while he retells in a few pages such thrillers as the discovery of King Tut's tomb. What I profited from most in The Modern Mind was Watson's summaries of books that had been published, in some cases with much �clat, and whose essence I had absorbed as part of a wafting zeitgeist, but that I had never read. When an author puts together more than 800 pages on the "modern mind," the reader can rest assured that at least as much has been left out. And that a lot of what is in the book, by commission or omission, isn't exactly correct. The longer a book, the more flak it is likely to attract. One person can't know or tell it all in any depth. Having a good memory, reading thousands of secondary texts, pumping experts for opinions, having several good assistants, each plugged into the Web (I'm just guessing here) all help enormously.

The Modern Mind can be used backward via its built-in search engine: its two indices listing names and ideas. Hmm, let's see what the author has to say about Kemal Ataturk, or about therapeutic nihilism. Do you suppose he's mentioned Scott Joplin? Now, with this inverse approach, let's see how mathematics has fared.

Not very well. And I'm about to use Modern Mind and deduce from it what can be learned about the state of mathematical knowledge among the literate public. First of all, a fair amount of the book is devoted to science and technology. I consider this much coverage good because I happen to agree with C.P. Snow---and I think Watson would concur---that in this century, science and technology have been the principal driving force for all of human life, all its productions and arrangements.

Yet the author is not as deeply immersed in the world of active science as he is in the arts and humanities, and he projects an idealized view of science. "Science thus has a moral authority as well as an intellectual authority," Watson writes. "This is not always accepted." Well, I am one of those who don't accept it, thinking that the pursuit of science was, in the past century, largely amoral (not: immoral).

At the risk of being called a bean counter, I'd like to look at how Watson sums up the last hundred years of mathematics. Of the 2000 people indexed, perhaps two dozen would count as mathematicians. That's pretty generous for the present kind of book. I therefore conclude by a simple calculation of the sort frequently made by those who toss statistics around that Watson sees mathematics constituting about 1.2% of 20th-century intellectuality. This figure misrepresents the reality.

Watson's treatment is inadequate not because it reduces the percentage of significant mathematicians among world shakers to 1.2% when it might easily have been 1.4%, nor because it does not explain mathematics at its gut level. (By comparison, it spends two and a half pages getting into the guts of Samuel Beckett's Waiting for Godot.) Of course, one cannot reasonably expect such mathematical depth in a book that is targeted for a general literate audience. What makes Watson's treatment of mathematics inadequate is that it pretty much limits the mathematics of the 20th century to G�del's theorem (which is irrelevant to 99.9% of mathematical productions), to a passing reference to Hilbert's 23 problems, to chaos, to catastrophe theory, and to a few other mathematical "thrillers." It misses entirely how mathematics has become a standard underpinning of all aspects of today's civilization, and not just of physics and technology (which Watson does allow en passant).

You know the old joke that to a hammer everything looks like a nail? Well, it doesn't take me long to find mathematics everywhere I look. All aspects of our lives are increasingly mathematicized. We are dominated by and we are accommodating to mathematical machines and to the actions they prescribe. (I still view the computer as a mathematical machine and not, as computer theorists would have it, as a communication machine.) Yet through chipification, for example, computer technology has made it possible for the mathematics itself to disappear from view and for the public to remain totally unaware of its buried presence. Public understanding of mathematics and the role it plays in today's world is accordingly abysmal.

The mathematical community at all levels, both research and educational, is aware of this state of affairs and has tried to ameliorate it. Popularizations in the form of books, videos, and Web sites have multiplied. There are college courses called (jocularly) "mathematics for poets." At least one radio talk show specializes in mathematics. Some universities offer courses on scientific journalism, and a few mathematical think tanks, such as the Mathematical Sciences Research Institute in Berkeley, have a stream of journalists-in-residence. Despite all these good things, much needs to be done to elevate the public's understanding and awareness of mathematics.

Is public understanding necessarily abysmal, as some have claimed? Let's face it, mathematics is hard stuff and most people abhor it. When a journalist writes a piece about mathematics, it is usually a spectacular item, such as "new large prime number discovered" or "famous unsolved problem solved." It very rarely deals with the prosaic day-to-day developments and applications that affect people directly. The situation can be compared to classical historiography, which limited itself to kings and wars and neglected the social, economic, and geographical aspects of human life.

In casual references to mathematics, its assigned role is usually that of the classic "handmaiden of the sciences"---the helper and not the initiator. Yet to an increasing extent, mathematics is inserted all over the place by fiat. Civilization got along and brought forth, let us say, Nabokov and his Lolita without the ability to do product striping (a technology laden with mathematics). The methodology of mathematical applications now seems to say: Look around for what can be done profitably with what we have and then do it. The modern mind is mathematical, and one of the results has been the replacement of increasing numbers of personal decisions with decisions churned out by algorithms that cannot accommodate the whole range of possibilities. No algorithm can know it all.

It is probably the case that, despite the claims of educational theorists, the general population uses less hands-on mathematics than at any time in the last several hundred years. More than faint memories of the binomial theorem, the public needs a critical education that brings awareness and judgment of the mathematics that dominates our lives and the ability to react with some force to evaluate, accept, reject, slow down, redirect, reformulate the operation of these abstract symbols that are affecting daily life. Technology is not neutral. It fosters certain kinds of behavior. Nor is mathematics neutral. With some few exceptions, awareness of this has not yet penetrated the educational process.

Peter Watson has done his homework and has read such expository books on mathematics as are available. Since he has derived from them a very parched view of the subject, what can we expect from a less literate public that gets its impressions from a few of years of drilling for standardized tests in grade school? Perhaps the books themselves are parched. I recall that in 1945, when Bertrand Russell's A History of Western Philosophy came out, the ink had hardly dried on its pages when critical reaction set in from the professional philosophers. Though Russell was one of the leading philosophers of his period and had a fine memory, he didn't know it all or get it all right. He was a specialist, the philosophers said, only on Leibnitz. And specialists in other fields jumped on him: This statement isn't quite right; current opinion now asserts that. . . . I, knowing rather little philosophy, was well pleased with Russell's eminently readable treatment and greatly informed by it. Apparently, the reading public was also pleased, for Russell's book sold well and rescued him from financial pressures. My reaction to Watson's book has been similar to the reception of Russell's book: both appreciative and cranky.

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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