Science: Number-Juggling

A group of sportsmen, having pitched camp, set forth to go bear hunting. They walk 75 miles due south, then 75 miles due east, where they sight a bear. Bagging their game, they return to camp and find that altogether they have traveled 45 miles. What was the color of the bear?

This sounds like a nonsense riddle, but it isn’t. The only place on earth from which you can walk 15 miles south, then 15 miles east, and still be only 15 miles from the starting point, is the North Pole. Hence the bear must have been a polar bear—therefore white.

The riddle of the bear is one of the tidbits in Mathematics and the Imagination* by Professor Edward Kasner, a distinguished but whimsical mathematician of Columbia University, and James Newman, a Manhattan lawyer who once was Kasner’s pupil. Mathematics is one of the hardest of all sciences to popularize, and the Kasner-Newman book is remarkably successful—perhaps because Dr. Kasner has had a lot of practice talking about mathematics to children. He is the man who gave the world the “googol” and the “googolplex” (TIME, Feb. 28, 1938). The googol is the number 1 followed by 100 zeros—a number greater than that of all the atoms in the universe. The googolplex is very much larger: it is the number 1 followed by so many zeros that the number of zeros is a googol. On a piece of paper stretching all the way across the visible universe—a billion light-years—there would not be enough room to write a googolplex in ordinary notation.

Most scientists use jawbreaking words for relatively simple things. A biologist says he has “hypophysectomized” a pigeon when he has removed its pituitary* gland; a psychologist speaks of “tactual-kinesthetic perception” when a blindfolded person indicates a point on his skin which has been stimulated. The opposite is true in mathematics, where ordinary words have fearfully complex meanings—e.g., “fields,” “groups,” “families,” “spaces,” “rings,” “limits,” “domains,” “functions.” In mathematics, a “simple curve” is a closed curve, no matter how elaborate, which does not cross itself—that is, which has one inside and one outside (see cut). An ordinary figure 8 is not a simple curve because it has two insides.

In a few places, especially in showing how the mathematics of infinity developed, and in explaining the calculus, Mathematics and the Imagination plows through heavy seas, and formidable equations loom like reefs and icebergs. But the book is studded with titillating mathematical believe-it-or-nots, puzzles and paradoxes that have bemused men through the ages.

Zeno of Elea (Fifth Century B. c.) declared it would be impossible for Achilles to catch a tortoise, provided the tortoise had a head start. For, said Zeno, Achilles would first have to reach the spot where the tortoise started. At that moment the tortoise would be some distance, however small, ahead. By the time Achilles reached that point, the tortoise would be a little ahead again. And so on to infinity —Achilles would never catch up. Wise & good men wrestled in vain with this prickly paradox until three 19th-Century mathematicians—Weierstrass, Bolzano and Cantor—demolished it by treating the mathematics of infinity realistically instead of mystically. They showed that an infinite class is no greater than some of its parts. The number of geometric points on a line a foot long is infinite. But the number of points on an inch of that line is also infinite—therefore no less. Thus, though the tortoise was ahead at the start, he had to traverse the same number of points (infinity) as Achilles (infinity), so the strong man would speedily overtake him.

When the mathematician Leibniz started to explain the mathematics of the infinitely small to Queen Sophie Charlotte of Prussia, she said she already understood it from watching the behavior of her courtiers.

<footnote>Simon & Schuster ($2.75).</footnote>