Mathematicians are uncomfortably aware that theirs is a “cold” subject. Though full of wonders and delights, it has little appeal to the tender side of human nature, little connection with the clayey appetites and longings of our everyday lives. There is a story about the great German mathematician David Hilbert. Noticing that one of his students had been absent from class for some days, Hilbert inquired the reason. He was told that the absentee had abandoned his studies in order to become a poet. Hilbert: “I can’t say I’m surprised. I never thought he had enough imagination to be a mathematician.” We laugh, but without much conviction. No doubt poetry and mathematics both require great resources of imagination, but we can’t help thinking that these are different kinds of imagination: the one delving inward to the stuff we are made of, the other flying out, away from human things, to an aery realm of frigid abstraction.
The author of a pop-math book has to take some consistent attitude towards this unhappy fact. The school of thought which I favor, and which I suppose could be called the “hard” approach to the popularization of math, mainly just ignores the issue. We present the reader with the truths of mathematics as they are, in the plainest language we can muster, link them to the human world as best we can with references to the lives or historical backgrounds of the mathematicians concerned, and season the mix with occasional snippets of personal anecdote or opinion. Our model is the 1940 pop-math classic Mathematics and the Imagination, by Edward Kasner and James Newman. (This was, incidentally, the book that gave our language the word “googol,” which the authors claimed to have got from working with a New York kindergarten class. A googol is ten thousand trillion trillion trillion trillion trillion trillion trillion trillion.)
Robert Kaplan follows a different tack. He seeks to humanize math for the general reader by cramming into his text as much of the Humanities as it will bear. Kaplan first hoist this banner three years ago with The Nothing That Is, “a natural history of zero,” whose very title, as if in defiance of Hilbert, is taken from a poem by Wallace Stevens, and in which, opening it at random (honestly), I light on a sentence beginning: “Shall we scribble across our canvas, as da Vinci did again and again in old age, ‘Di mi se maifu fatta alcuna cosa …’?” Let’s not, I murmur; let’s just cut to the mathematical chase, could we, please?
I don’t mean to be unkind. These things are matters of taste, and The Nothing That Is is actually a useful and informative book. It enjoyed very good sales, too; so whatever I may think of the soft-pop style of math exposition, there is obviously a healthy market for it. Here Robert Kaplan, this time in collaboration with his wife Ellen, has cast his net much wider, seeking to explain nine large mathematical topics to a general readership. The style is, if anything, even more strenuously multidisciplinary than before. By page ten of The Art of the Infinite I had logged references to Blake, Baudelaire, Robert Louis Stevenson, A.E. Housman, Shakespeare, Proust, Tom Paine, Michelangelo, Sir Francis Bacon and Heraclitus, and had struck up a nodding acquaintance with Mixtecs, Sanskrit love lyrics, Australian aborigines, and “the Oksapmin of Papua New Guinea,” whoever they may be. There is no arguing with success, I suppose; but I could not help thinking that the author might have benefited from the attentions of Daniel Webster. Invited to edit William Henry Harrison’s inaugural address, Webster found it so overloaded with classical allusions he later boasted that in the course of his editing he had killed “seventeen Roman proconsuls as dead as smelts, every one of them.”
The nine topics presented here, one per chapter, are as follows: numbers; “foundations” (that is, the attempt to find fundamental axioms from which math can be deduced); the distribution of primes; series; triangles; fields; complex numbers; projective geometry; and transfinite arithmetic. The authors attempt to link these topics using the concept of infinity, though this concept is obviously nearer the surface in some chapters than others. After every second or third chapter comes a one-page “interlude,” adding nothing much in the way of information, but bringing our attention back to the notion of the infinite, while testing to the limit my tolerance for lyricism in math texts. Sample sentence: “The infinite disguised as the indefinite is our onlie begetter.” (And I note that in the first of those interludes the authors recycle that da Vinci quote from The Nothing That Is.)
If you can just put Baudelaire and Leonardo da Vinci out of your mind, the math content of the book is really very good. I was glad to be reminded of the endless fascination contained in a triangle, for example. The nine-point circle and the Euler line are great but little-known wonders. It is a shame the authors had not enough space to go into Simson lines, Morley triangles, Malfatti circles or perimeter bisectors. They have got me thinking that there is a good pop-math book waiting to be written just about the humble triangle. Hmmm … Projective geometry, too, which was taught to high school honors classes in my own time, but which is now unjustly neglected, offers many beautiful results to the casual enquirer, and I commend the Kaplans for giving over a chapter to this unfashionable topic. I only wish they had dug up the connection between Pappus’s Theorem and the traditional forms of Chinese poetry, which someone once explained to me, but the details of which I have since forgotten.
The pons asinorum for lay readers is of course the complex numbers, which even willing and well-prepared people often just cannot “get.” Here I am not sure the Kaplans have found the right way to present this challenging topic. They come at it via the so-called “fundamental theorem of algebra,” and arithmetic in the Argand diagram, then proceed rather laboriously to de Moivre’s formula. I believe a more strictly numerical approach would be better here; though since nothing works reliably, I suppose anything is worth trying. Gauss is supposed to have said that if it is not immediately obvious to you on being told that e πi = −1, you will never be a mathematician. That cuts out rather a lot of the human race; but the thing is not beyond explaining to any intelligent adult, and the Kaplans deserve credit for trying. I am sorry to have been sarcastic about their style, which just isn’t to my taste. There has been — and, judging by how successful The Nothing That Is was, is — a good demand for books explaining math topics to readers with a decent liberal, but non-mathematical, education. The Art of the Infinite is a welcome contribution.