Mathematics for Poets: A Poet's VersionApril 22, 1999
From The Number Devil: A Mathematical Adventure
The Number Devil: A Mathematical Adventure. By Hans Magnus Enzensberger, Metropolitan Books, Henry Holt, New York, 1998, 260 pages, Illustrated, $22.00.
I suppose there is hardly a college in the country, junior or senior, that doesn't have in its catalog a first-year mathematics course designed for non-science concentrators. Often referred to facetiously as "Math for Poets," such courses may be listed more prosaically as Math 1 or something similar. Their contents are very idiosyncratic---shaped by what happens to appeal to the instructor at the moment.
The instructors of such courses, rarely poets or artists or writers, are more often than not roped in by the department chair: "You teach it this semester. It'll be good for you." Some have brought enthusiasm to the opportunity, and this group has produced easily half a hundred "Math for Poets" textbooks (not to mention abundant Web material), again quite idiosyncratic.
Let us now see what a real poet has come up with. First, who is the poet? Although my university library has 60 of his works on its shelves, Hans Magnus Enzensberger (b. 1929; subsequently referred to here as HME) is not widely known in the United States. In Germany, he has a great reputation as an essayist, a journalist, and one of the country's most prominent poets. In recent months he has written about immigrants in Germany and about the world full of civil wars in which we live. His latest book (February 1999) is Zig Zag: The Politics of Culture and Vice Versa. HME's political writings are controversial; academic conferences have been based on them. Having now read in translation a number of his books, I consider him a fine writer, and one who is much more easily understood than most of the academics who have sought to explicate him.
I first encountered HME last summer in Berlin, at the International Congress of Mathematicians, where he had been invited by the German Academy of Sciences to give a Urania Talk, i.e., one of a series of talks, dedicated to "high culture" at a popular level, given in the Urania Theater. The talk was immediately published in the high-brow Frankfurter Allgemeine Zeitung.
If, as he has stated, he is an "outsider to mathematics," what are HME's qualifications for writing about mathematics? When I sent him this question, he wrote back to me:"My mathematical training is negligible. I have been following, to the extent that they are accessible to a layman, the developments in mathematics, as a sort of intellectual fitness training. I had the good fortune to be exposed to math teaching at school by an extraordinary teacher when I was sixteen. This man was not a professional tea-cher but a scientist whose research institute had ceased operating because of the war. He taught as a sort of hobby or pastime until he could resume his career, and fortunately for me, he created his own curriculum. I thus escaped the boredom of run-of-the-mill arithmetics teaching."
I really shouldn't have referred to "qualifications"---as though one has to be certified by a mathematics organization before writing a popular math book. Mathematics doesn't "belong" to anyone, let alone to the mathematicians.
That vast populations are afflicted with mathophobia, despite the increasing mathematization of civilization, has been well known for a long time. This is the paradox that HME addresses in his Urania talk. Where to place the blame? What can be done to alleviate the affliction?
The obvious targets for blame are the mathematicians themselves, the teachers of mathematics and the curricula they follow, particularly at the elementary grades, and the expositors of mathematics, including the PR sections of the mathematical societies and science journalists. There is, of course, overlap in these groups. That the material itself, of its own nature, may be difficult, is less often charged.
HME finds the mathematicians guilty of elitism; the teachers and their curricula are guilty of deadly dullness; the expositors, if professional mathematicians, are guilty of stiff-neckedness and an inability to lessen the precision of their statements in order to achieve increased comprehensibility. If professional journalists, they do the best they can with the official news releases and try to talk the public into excitement over whatever it is that the math establishment deems the latest blockbuster.
HME's solution? Convey the poetic, imaginative stuff of mathematics; leave the rote behind, whether in arithmetic or in more advanced topics. Tedious rote does not reveal the real substance of mathematics, even though the public may think it does. And leave technical reasoning behind.
While admitting the wonder or the miracle that "something akin to l'art pour l'art should be so capable of explaining and manipulating the real world around us," in the tension between the esthetic and the utilitarian aspects of mathematics, HME definitely leans toward the former. He is essentially preaching a Hardyite message.
The Number Devil (translated from the original, Der Zahlenteufel) preceded HME's talk by at least a year, and this book, together with his stature as an essayist and his considerable visibility, made him an excellent choice as a Urania speaker at the very media-conscious ICM 98. That he was speaking under the aegis of the ICM, moreover, lent considerable cachet to his words.
Let's look at the book to see how HME would go about getting young people interested in mathematics. And is he writing only for young people? Well, because the book is charmingly illustrated---featuring a pixie imp, the "Number Devil," and Robert, age 12---I suppose it is intended for young people.
On the other hand, the German subtitle "A Bedtime Book for Everyone who Has Math Anxiety" (which has been totally suppressed for the American edition), leads me to wonder whether the author had in mind a larger public than juveniles or doting relatives who buy young people birthday presents.
The text consists of 12 mathematical dream-dialogs experienced by Robert. In these dream-dialogs, the Number Devil leads Robert to a variety of mathematical ideas experimentally, visually, intuitively: more or less by the Socratic method. (See Plato's "Meno.") Progressions, prime numbers, imaginary numbers, limits, the Sierpinski triangle, Euler's formula for polyhedra. . . . All in all, the book includes about 150 different mathematical terms, ideas, results, all indexed. There are lots of numbers in the book, lots of diagrams, and hardly any algebra. In bygone days, such material might very well have been called "Mathematical Recreations"; today, it could be a resource for "Math Can Be Fun"-type courses.
One more thing. The author has seen fit to replace a number of standard terms by terms of his own invention, thinking, I suppose, that this would ease some of the mathophobia. Thus: "factorial" becomes "vroom!"; "dodecahedron" becomes "pentagon ball"; "Georg Cantor" becomes "Professor Singer."
Now for my personal reactions to HME's material. First, the talk. I agree with a good deal of what HME says, and I disagree with some things. I agree that there is a failure of communication: The drawbridge between the professionals and the laity remains up. I agree that rote repetition at an early age and even later is deadly boring. On the other hand, as the old joke has it, (boring?) practice is necessary to get you to Carnegie Hall. I agree that young people could probably learn much more math and at an earlier age than they are currently doing.
I don't agree with the implication that math can be made fascinating for everyone. The surprising properties of Pascal's triangle may intrigue you and me, but many, finding the topic deadly dull, might ask "So what?"
We know---and HME points out also---that humans have a wide spectrum of interests and skills. I could never have been a football player or a ballet dancer, nor would it make sense to fill up the world with professional footballers, dancers, or mathematicians. The world can be (and is being) mathematized by a relatively few professionals.
What is needed on the part of the public, besides a very minimum of technical skill, is an awareness of the mathematizations that have been put in place in the world and some recognition that as it engages humans the process has both merits and demerits.
Turning now to The Number Devil, it may be inappropriate for me to pontificate on a book targeted for young people. And because the world of mathematical practice has changed so much, reminiscences of my own engagement with math at age 12 may now be irrelevant. Perhaps I should have asked a 12 year old to write a review of The Number Devil. I did ask an adult mathophobe, a poet actually, to look at the book, and she thought the material would engage only those who had been previously engaged.
Be that as it may, where does the poetic imagination of the author show itself? Not in the selection of material, which is simply the author's choice from among fairly standard items. Perhaps, then, in the dialogs, in the careful manner in which HME has worked out the Socratic method. The dialogs between Robert and the Number Devil approach mathematics directly, not as in Alice in Wonderland, where the mathematics is subtle, veiled, and usually missed by the reader. As a sample from HME:"I assume YOU shake hands when you say goodbye."
"Shake hands! We mumble, 'See you'---if you're lucky."
"A pity," said the number devil, "because I wonder how long it would take for each of them [11 children] to shake hands with each of the others."
"You know perfectly well there would be an untold number of handshakes. Eleven vroom! of them, I suppose, since there are eleven of them."
"Wrong," said the number devil.
"Wait a minute," said Robert, "I see. If there were two of them they would need only one handshake. If there were three . . ."
The book is a best-seller in Europe, and it may very well happen that twenty years from now, a number of professional mathematicians will say that they were nurtured on Robert and the Number Devil. I am not a fan of Professor Dry-as-Dust. I know the delights of recreational material treated lightly. And I hope, and I hope the author hopes, that his treatment will somehow inspire the nonprofessional multitudes to go beyond and acquire a deeper appreciation of the part mathematics plays in the civilization that he has described so vividly in his usual author's role.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island. Davis also presented a Urania talk, "The Prospects for Mathematics in a Multi-Media Civilization," at ICM 98.