The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)
Part II. Hope and Monsters
Chapter 22. Patterns Made with Ideas
For the Greeks, the word mathematics had vastly different associations than it does for most of us. Mathematics had almost nothing to do with adding up columns of numbers or figuring out how long it would take Bob and Tom working together to paint a room. The aim of mathematics was to find eternal truths—insights into the abstract architecture of the world—and then to prove their validity. “A mathematician, like a painter or poet, is a maker of patterns,” wrote G. H. Hardy, an acclaimed twentieth-century mathematician and an ardent proponent of the Greek view. “If his patterns are more permanent than theirs, it is because they are made with ideas.”
Let’s take a few minutes to look at the kind of thing Greek mathematicians accomplished, because it was their example—and the way they interpreted their success—that inspired their intellectual descendants in the seventeenth century. (One of Newton’s assistants could recall only one occasion when he had seen Newton laugh. Someone had made the mistake of asking Newton what use it was to study Euclid, “upon which Sir Isaac was very merry.”) The Greeks had looked for their “permanent patterns” in the world of mathematics. Seventeenth-century scientists set out with the same goal except that they expanded their quest to the world at large.
They found mathematics on all sides. When Isaac Newton directed a beam of light through a prism, he marveled at the rainbow on his wall. No one could miss either the beauty or the order in that familiar spectacle, but it was the interplay between the two that so intrigued Newton. “A naturalist would scarce expect to see ye science of those colours become mathematicall,” he wrote, “and yet I dare affirm that there is as much certainty in it as in any other part of Opticks.”
For the Greeks the notion of “proof”—not a claim or a likelihood but actual proof beyond a doubt—was fundamental. A proof in mathematics is a demonstration or an argument. It starts with assumptions and moves, step by step, to a conclusion. But unlike ordinary arguments—who was the greatest president? who makes the best pizza in Brooklyn?—mathematical arguments yield irrefutable, permanent, universally acknowledged truths. Of all the shapes you can make with a piece of string, a circle encloses the biggest area. The list of prime numbers never ends.29 If three points aren’t in a straight line, there is a circle that passes through all three. Everyone who can follow the argument sees that it must be so.
Like other arguments, proofs come in many varieties. Mathematicians have individual, recognizable styles, just as composers and painters and tennis players do. Some think in pictures, others in numbers and symbols. The Greeks preferred to think pictorially. Take the Pythagorean theorem, for instance, perhaps the most famous theorem of them all. The theorem involves a right triangle—a triangle where one angle is 90 degrees—and relates the lengths of the various sides. In the simplest right triangle, one side is 3, another 4, and the longest 5. Many centuries before the birth of Christ, some unknown genius stared at those numbers—3, 4, 5—and saw something that astonished him.
It’s easy to draw a triangle with a side 3 inches long and a side 4 inches long and a third side that’s short (at left, below), or a triangle with a side 3 inches long and a side 4 inches long and a third side that’s long (at right, below). But if the angle between the 3-inch side and the 4-inch one is not just any angle but 90 degrees, then the length of the third side turns out to be precisely 5. So the puzzle pieces that our unknown genius turned over and over in his mind were these: 3, 4, 5, 90 degrees. What tied those numbers together?
No doubt he drew endless right triangles and measured the sides. Nearly always the longest side would be a seemingly random number, no matter how carefully the two short sides were chosen. Even in the simplest case—a triangle where the two short sides were both 1 inch long—the third side didn’t look simple at all. A shade more than 1 ⅜ inches, not even anything that lined up with the divisions on a ruler.
Perhaps he stuck with his experiments long enough to draw the right triangle with short sides 5 and 12. Set a ruler in place to draw the third side and then measure it. Success at last—the long side is precisely 13 inches long, so here is another right triangle with all three sides respectable whole numbers.
Two right triangles, two sets of numbers, like two words from a coded message. First set: 3, 4, 5. Second set: 5, 12, 13. What do the two triplets have in common?
For two thousand years, we have called the answer the Pythagorean theorem—the length of one short side, squared, plus the other short side, squared, equals the long side, squared. 32 + 42 = 52. For the second triangle, 52 + 122 = 132.30 More to the point, the relationship holds for every right triangle whatsoever, whether it is scratched in the sand or splayed across the heavens.31
In modern terms, the theorem is usually written as a2 + b2 = c2. In the pictorial terms the Greeks preferred, the theorem is about squares, not numbers, and Pythagoras’s claim is that the area of one small square added to the area of the other small square is exactly the same as the area of the large square. (See drawing below.) The two approaches, numerical and pictorial, are exactly equivalent. The choice between them is purely a matter of taste, like the choice between an architectural drawing and a scale model.
Pythagoras’s theorem says that the area of one small square plus the area of the other small square is exactly equal to the area of the large square.