Mariana Cook opens her preface to Mathematicians with these sentences: “They are not like the rest of us. They may look like the rest of us, but they are not the same.” One is reminded of how F. Scott Fitzgerald began a short story: “Let me tell you about the very rich. They are different from you and me.” To which Ernest Hemingway famously responded, “Yes, they have more money.” One is tempted to make a similar reply regarding Ms. Cook’s mathematicians—“Yes, they know more about math.”
Mathematicians is a huge beautiful book, ten inches by a foot. On the left side of each double page is a crisp autobiography by a world-famous mathematician. On the right is a superb black-and-white photograph of the mathematician taken by Ms. Cook, also the book’s editor. The volume follows the format of Ms. Cook’s earlier book, Faces of Science, about eminent scientists. She was a protegé of Ansel Adams, and her photographs hang in art musuems and galleries throughout the world. “A Couple in Chicago,” her recent portfolio featuring President Obama and his wife, is part of her ongoing work on portraits of famous couples.
Ninety-two mathematicians appear in the book under review. Over half are white male professors now in their elderly years, six are Afro-American men, and there is one black woman. Five are Asian men, four Asian women, two men are from India, and there is one handsome woman from Iran. Obviously, it is not possible to comment on all ninety-two mathematicians. I will, therefore, limit my attention to the few I think will be of most interest to New Criterion readers.
Persi Diaconis, Stanford University’s famous statistician, writes about his strange double life as both mathematician and magician. Persi has been a friend of mine since his undergraduate days at Manhattan’s City College. I actually played a role in getting him into Harvard for graduate work towards a doctorate. Persi is one of my few magic friends who can do a perfect Zarrow shuffle. Named after its inventor, Herb Zarrow, it is neither a push-through nor a pull-out false shuffle, yet it leaves the deck unaltered. In Persi’s hands, it is undetectable from a genuine riffle shuffle.
Persi is best known for having proved that seven riffle shuffles are necessary and sufficient to turn an ordered deck of fifty-two cards into a random order. More recently he was able to show that when an average person flips a coin the probability is a trifle over 1/2 that it will fall with its up side the same as it was before the flip. The reason is that when a coin is flipped, say, a hundred times, there are likely to be a few times when the coin wobbles and keeps the same side up. Persi is one of a few magicians who can flip a coin so it wobbles but looks like it is rotating. His technical papers are, of course, more significant.
Sir Roger Penrose, England’s great mathematical physicist, is in the book mainly because of his discovery of what are called Penrose tiles. These are two simple shapes, named darts and kites, that have the amazing property of tiling the plane only in what is called an aperiodic manner. Periodic tiling has a region that repeats itself to infinity like patterned wallpaper. An infinite number of tiles, including all triangles and rectangles, can be arranged to tile the plane aperiodically, but they also will tile periodically. Penrose tiles cover the plane only aperiodically.
I had the enormous pleasure of introducing Penrose tiles to the general public in my column on recreational math in Scientific American. After my first column on the tiles appeared, an amazing thing happened. Crystals were discovered with an aperiodical structure that was a three-dimensional analog of Penrose tiling! Such crystals had not been believed to be possible. Known as quasi-crystals, hundreds of papers have now been devoted to them. They are wonderful examples of how a mathematical discovery, made with no inkling of its applications, can turn out to have long been familiar to Mother Nature! In his autobiographical sketch, Penrose includes his drawing of a tiling by darts and kites in which the pieces have been distorted to resemble chickens.
The essay by John Conway, arguably the world’s most creative mathematician, is my favorite. I suspect that almost every mathematician in the book is a Platonic realist, one who believes that mathematical theorems are forever true in all possible worlds and are independent of human culture. Put another way, they believe that if all sentient creatures vanished from the universe, planets would still travel in ellipses, spiral galaxies would still spiral, all primes would remain prime, and two plus two would still be four. Here is how the great Conway puts it at the end of his essay:
It’s quite astonishing and I still don’t understand it, having been a mathematician all my life. How can things be there without actually being there? There’s no doubt that 2 is there or 3 or the square root of omega. They’re very real things. I still don’t know the sense in which mathematical objects exist, but they do. Of course, it’s hard to say in what sense a cat is there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can’t push a cat in a direction it doesn’t want to go. You can’t do it with a number either. I’m only using the word number because you’ll have a vague idea in your head as to what I mean. The objects that a mathematician studies are more abstract than numbers but very real.
I often think of cats. I think of trees. I think of dogs occasionally but I don’t think of them all that much because dogs are agreeable. They do what you want them to do to some extent. Some people believe that mathematics is what we think it is and it’s created by our thoughts. I don’t. I’m a Platonist at heart, although I know there are very great difficulties in that view.
Benoît Mandelbrot, the father of fractals, is in the book as he surely deserves to be. He reproduces a picture of the famous Mandelbrot set, a pattern generated on a computer screen by a ridiculously simple formula. It is said to be the most complicated of all geometrical patterns. Penrose, an unashamed Platonist, likens exploring the Mandelbrot set to someone exploring the alps or an African jungle. The set is “out there,” independent of one’s brain as much as a cat is.
To listen to the rhetoric of a small but noisy group of anti-realists, you would think that Penrose and Conway, along with Hardy and Hilbert and Gödel—indeed almost every great mathematician past or present—is an idiot unable to grasp the great truth that the locus of mathematical reality is inside human culture, like fashions in clothes, traffic regulations, and trends in art and music. I also suspect that not every mathematician in the book is an atheist, but only one, Arlie Petters, had the courage to mention God. Petters is a brilliant Afro-American from, of all places, a little known town in Central America called Dangruiga, in Belize. He is now a professor at Duke.
In his first paragraph Petters asks, “Is there a God?” He answers in his last paragraph. “God, love, meditation, and prayer are an integral part of my day-to-day life.” The scientific method, he continues, “has practical relevance only to a restricted part of the human condition. Indeed, the moment I begin to act as if science can access and resolve all the deep mysteries of existence, pinch me! And as I awaken, lovingly and forgivingly, remind me that I am only human.”
Welcome, Arlie, from those of us who like to call ourselves mysterians.