Martin Kruskal, a mathematician at Rutgers university in New Jersey, has two brothers, both of whom are also mathematicians. When my older brother’s oldest child was five, Kruskal recalls, he argued with another little boy about whether there was a largest number. No doubt they were talking about the counting numbers--1, 2, 3, and so on. My nephew said there wasn’t, and his friend said there was. The next day my nephew went back to his little friend and said, ‘I asked my father about it, and he’s a mathematician, and he says there isn’t any largest number.’ And the other little boy said, ‘Well, I asked my father about it, and he says there is, and he’s a lawyer.’
Kruskal père and fils are right, of course. (So sue them.) Their assertion is easily demonstrated--whenever you think you’ve found the biggest number, just add one to it, and the new number will be bigger still.
Mathematicians and precocious five-year-olds have long been fascinated by the endlessness of numbers, and they’ve named the endlessness infinity. Infinity isn’t a number like 1, 2, or 3; it’s hard to say what it is, exactly. It’s even harder to imagine what would happen if you tried to manipulate it using the arithmetic operations that work on numbers. For example, what if you divide it in half? What if you multiply it by 2? Is 1 plus infinity greater than, less than, or the same size as infinity plus 1? What happens if you subtract 1 from it? What do you get if you divide the number 1 into infinitely many parts of equal size? And if you multiply that tiny answer by itself, will the result be bigger or smaller?
Two decades ago, using only a few simple rules and concepts, British mathematician John Conway figured out how to wring meaning from apparently nonsensical questions like these. He did it by developing a technique that produced every number you’ve ever imagined, plus untold zillions more--the surreal numbers. Conway didn’t come up with the catchy name. That was the creation of Stanford computer scientist Donald Knuth, now retired, who wrote a kooky little novella about the numbers after hearing Conway describe them in an informal talk. Knuth’s book was the first thing that appeared about these numbers, says Conway. I’m always very lazy about publishing things.
These strange denizens of the number line lurk among its crevices, so enormous or tiny that previous mathematicians never knew they were there. Yet Conway found them, and he found ways to use them in arithmetic just like their ordinary numerical neighbors. Ever since that momentous discovery, Kruskal has been busy extending and refining Conway’s work. Eventually, if all goes well, Kruskal’s efforts will leave telltale marks all over the mathematical landscape.
