How Mathematicians Think:
Using Ambiguity, Contradiction, and Paradox to Create Mathematics

by William Byers
Princeton, NJ: Princeton University Press, 2007 425 pp.; $35.00 (cloth)

William Byers, a working mathematician and professor of mathematics and statistics at Concordia University in Montreal, has written a passionate defense of the uniquely human aspect of mathematics. He is a Luddite in the noblest sense of the word, advocating, in the teaching of math, a respect for its deep ideas as against the dead algorithmic recitations that have been reinforced and ramified by the spread of computer culture. He acknowledges the inspiration of Zen teacher Albert Low in his development of a view of math history, theory, and methodology based on a central idea he calls “ambiguity.”

Indeed, Byers’s treatment of “ambiguity” is reminiscent of a koan of Mumon’s: “Keichu had a cart. . . . If you take away the wheels and axles, what have you got?” Like the koan, Byers’s work pays special attention to the ideas of zero and infinity as wonderfully fertile exactly because of the contradictions they contain, words denoting in the one case nothing, and in the other something never finally to be exhausted in a word—or so we intuit. Out of this intentional puzzlement, as from the traditional handling of the koans, comes dynamic engagement with a deep dilemma, ultimately producing insight.

The book is not without its difficulties. Parts of it read like a book proposal, parts like a rabid manifesto, and the word ambiguity is here and there made to hold far more weight than it can really bear. Byers occasionally uses the word in different senses, sometimes to mean “unclear” or “existentially indeterminate” and other times (as he mainly wants to define it) the play of mutually exclusive but self-consistent interpretations of a single phenomenon.

This is a characteristic flaw in books with spiritual pretensions. Robert Pirsig’s very popularZen and the Art of Motorcycle Maintenance, for example, though sympathetic in many ways, is sophomoric in its philosophical meanderings because of its misuse of the word “quality,” something critical to what Pirsig thinks of as his central insight. Pirsig gives the word two utterly distinct meanings: essence or “whatness,” on the one hand, and virtue or positive value on the other. This conflation renders his reasoning nonsensical. A similarly false conflation of terms from different contexts has also been attributed to Fritjof Capra (by the physicist Jeremy Bernstein in The American Scholar, for example) as one of the means by which he limns his mystically informed (deformed?) views of physics and ecology. The urge to mingle scientific and mystical thinking is not necessarily fatal or even necessarily unhelpful, however. For example, the great mathematician Georg Cantor’s groundbreaking researches into the nature of infinity, which Byers examines, were motivated (according to Cantor) by his desire to understand God.

Happily, the confusion of terms does not overwhelm Byers’s book but merely represents an irritating distraction that crops up when Byers wants to transcend his field of expertise and talk about humanity generally. When he sticks to math, Byers knows what he’s doing. There is rigor. This is consoling, because it is too often the case that scientists who invoke spiritual principles and mystics who invoke scientific principles make a godawful mess. Consider the monumental inaptitude, if not downright quackery, of the invocation of particle physics in “quantum” healing, or the evidence-resistant Hundredth Monkey fantasies of Rupert Sheldrake. Or the vitalist atavism of Capra, who, despite his noble purpose, cherry-picks data to support a romantic worldview obnoxious to working biologists, ecologists, and geneticists: scientists constrained, as Capra apparently is not, by observed facts.

Not long ago I attended a class taught by a mystically inclined dancer who, after observing that we tended to pirouette clockwise, attributed this “phenomenon” to the Coriolis Effect, i.e., to the distortion of the paths of moving objects that is caused by the earth’s rotation. Sounds deep, but it’s a palpable absurdity for anyone who’s seen the equations for the forces involved, forces so miniscule that they are observable only in gigantic tidal movements of water or air.

Science and the appearance of rigor seem to have an irresistible appeal to a segment of the spiritual community. Consider astrology, with its intricate calculations—could something so sophisticated actually be based, as statistical studies of its predictions consistently have shown, on absolutely nothing? (Even Carl Jung’s famous study in his essay on synchronicity actually proved that there was no significant correlation between a canonical prediction of astrology and observed facts, despite the fact that astrologers often invoke the study as confirmatory.) Think of the calculations of homeopathy, succussions and dilutions precisely recorded, even though all the numbers have, in terms of physical reality, whether referring to concentration or to provable potency, the identical referent: zero. There’s Theosophy, of course, with the erudite Madame Blavatsky and the romantic Annie Besant penning whole libraries full of books establishing a science to which no reality, as it happens, corresponds.

Consider Buddhism’s lists of numbers of things, useful if one doesn’t enshrine the numbers as absolute. Even the Four Noble Truths: do we really imagine that there must be four? Take a look at the Hebrew Bible and see if you can delineate ten distinct commandments in the so-called Decalogue. Jewish mystics practicing the sacred art of gematria dissect the Pentateuch, reading its alphabet as a system of numbers to divine the Author’s intention. Mandala makers in India and Tibet express themselves geometrically and in strict symmetries. The number of grains in the River Ganges and the number of years in an asamkhya of kalpas are quantities of interest in some sacred scriptures, whether or not they are poetically intended. Number mysticism is ubiquitous, as occultists grope for a longed-for but elusive rigor. All too human. To a degree, albeit harmlessly in his case, Byers, inspired by his friend Albert Low, buys into the conflation of mystical and scientific discourse.

I knew Albert Low a little bit at the Rochester Zen Center. At Albert’s “dharma dialogue,” a student ritual in which, after giving a short talk, he was questioned by the community of meditators, Dane Gifford (later “Zenson”) picked up a copy of Albert’s new book, Zen and Creative Management, and hurled it at him across the Zendo. What had this book to do with the vital matter of Zen? Said Albert: “I am most terrifically sorry.“

There is a sort of anti-intellectualism in Zen, of course—or the appearance of it. Around the same time as Albert’s biblio-projectile encounter, a young woman at the Zen center was preparing to marry a senior student there. Her Quaker father, like many Quakers, was a religiously inspired political activist, and he was outraged to learn that center’s founder, Philip Kapleau, counseled students (at a certain stage of their practice) to forego newspapers, magazines, and books unrelated to Zen practice. I remember the groom closing his eyes in soulful concentration as he tried to explain the reason to the father of the bride.

It didn’t work.

Interesting then that Albert, a teacher now, should have inspired a work of such complexity and discursive rigor as this book. After all, complexity of thought—in its place—is not inimical to Zen practice. In fact, if anybody can beat Pirsig’s teacher, Katagiri Roshi, for complexity of thought, I’ve not yet read or heard him. I once had the job of transcribing some of Katagiri Roshi’s talks, and, believe me, placing commas and parsing clauses were, though Stygian labors in themselves, the least of my worries.

The last time I ever spoke with Philip Kapleau—Albert Low’s teacher and mine—he characterized true Zen understanding as a simultaneous affirmation of the is and the is not.This is very much the understanding put forward by Byers as crucial to mathematical thinking. He delineates various crises in the history of math from Pythagoras, bedeviled by the irrationality of the square root of two, through the Greeks, who could countenance neither zero nor infinity as proper entities, through the wars over transfinite numbers, the infinity of infinities with which Georg Cantor enriched modern mathematics, and the reaction against that explosion of forms: intuitionism, the movement to restrict math to ideas that can be constructed in a naively plausible way.

The final skirmish, and the one which seems to be fueling Byers’s current project, involvesformalism, whose agenda is to reduce math to the systematic manipulation of symbols. Byers presents the best lay account I have seen of Kurt Gödel’s famous counterblast to the formalist agenda, his astonishing metamathematical proof that no formal system can ever be complete. Given any system, humans will always be able to construct theorems that we can see to be true, but that are unprovable in that system.

It is a great strength of Byers’s view, as it is of Mumon’s, that he is grateful to formalism for the necessary creative role it played in pricking the issue that generated this amazing result. Indeed, the genesis of mathematical insight in Byers’s account sounds a lot like the way koans are supposed to work in Zen.

In Zen work, an existential contradiction, Mumon’s “red hot coal,” sticks in the student’s throat; the inability “to swallow it or spit it out” precipitates a crisis to be resolved through an insight that is simultaneously an existential gesture. “If I am whole and complete as I am, why do I feel ignorant and incomplete?” might be one formulation of the conundrum, though encoded in a ritual question like “What is the sound of one hand?” The greater the contradiction, the greater the tension—”doubt mass”—and the greater the breakthrough, according to Zen tradition.

In mathematics, as Byers shows us, the breakthroughs, the “Great Ideas,” have also come about as the result of the tension generated by great contradictions: infinity as a completed quantity, lengths that cannot be measured, even the pivotal idea of equality, since one must first distinguish as different the quantities or representations that are then held to be the same.

A novel and exciting discovery by Byers is that the fecund thematic ambiguities in the historical development of mathematics manifest today as difficulties in understanding by students. These are just the right difficulties for them to have!

Although he is at his best when discussing the inner workings of the mathematician’s mind or advancing the creative, “ambiguous,” idea-centered view of math, Byers’s seminal impulse, judiciously treated, is universal in its implications. Despite occasional excesses, Byers really does manage to demonstrate that the insights of mathematicians come about through a discipline that, in its treatment of dynamic tension, has something in common with Zen practice. First, there is the positive use of difficulty: “the paradox has the enormous value of highlighting a fertile area of thought.” Then the breakthrough: “An idea emerges in response to the tension that results from the conflict inherent in ambiguity.” These sentences from Byers’s book apply equally to scientific and spiritual work.

Contributing editor Eliot Fintushel lives in Santa Rosa, California.

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