The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)

Part II. Hope and Monsters

Chapter 23. God’s Strange Cryptography

If you had somehow happened to guess the Pythagorean theorem, how would you prove it? It’s not enough to draw a right triangle, measure the sides, and do the arithmetic. That would only serve to verify the theorem for one example, not for all right triangles. Moreover, not even the most careful measuring could confirm that the sum worked out precisely, to the millionth decimal point and beyond, as it must. But even a dozen successful examples, or a hundred, or a thousand, would still fall short of proof. “True beyond a reasonable doubt” applies in law and in ordinary life—who doubts that the sun will rise tomorrow?—but the Greeks demanded more.

Here is one classic proof, which proceeds in jigsaw-puzzle fashion and almost wordlessly. In math as in chess, crucial moves often look mysterious at the time. Why move a knight there when all the action is here? In this case, the unexpected move that brings the answer in reach is this: take the original triangle and make three identical copies of it, so that you have four triangles all exactly the same.


What’s the gain in dealing with four triangles when we weren’t sure what to do with one? The gain comes in imagining the triangles as cardboard cutouts and then sliding them around on a table in different arrangements. Look at Figure X and Figure Y below. Two different arrangements, both with the same four triangles and some white space. In both cases, the outlines (in bold) look like squares. How do we know they really are squares and not just four-sided, squarish shapes?

Stare at Figure X and Figure Y for a few seconds. All the bold sides are the same length (because each bold side is made up of a long side of the original triangle and a short side). And all the corners are right angles. So the bold shape in Figure X is a square, and so is the bold shape in Figure Y, and both squares are precisely the same size.

Almost done. Each bold square encloses the same area. Each bold square is made up of four identical triangles and some white space. Stare at the pictures again. The large white square in Figure X has to beexactly the same in area as the two smaller white squares in Figure Y. Voilà, we have Pythagoras!


Figure X


Figure Y

Why did the Greeks find that discovery so astonishing? Not for its utility. No Greek would have asked, “What good is it?” What good is a poem or a play? Would a sculpture be more admirable if it could also serve as a doorstop? Mathematics was true and it was beautiful, and that was more than enough. The point was not to find the length of a diagonal across a rectangular field without having to measure it, although the Pythagorean theorem lets you do that. The Greeks had loftier goals.

The Pythagorean theorem thrilled the Greeks for two reasons. First, simply by thinking—without using any tools whatsoever—they had discovered one of nature’s secrets, an eternal and never-before-suspected truth about the structure of the world. Second, they could prove it. Unlike nearly any other valid observation—vinegar is tart, Athens is dusty, Socrates is short—this particular observation was not only true but necessarily true. One of God’s thoughts, finally grasped by man.

Like all the best insights, it is simultaneously inevitable and surprising. But it may also be surprising that the Greeks took for granted that their mathematical theorems were facts about the world rather than man-made creations like houses or songs. Is mathematics invented or discovered? The Greeks came down emphatically in favor of “discovered,” but the question is ancient, like what is justice? and apparently just as difficult to resolve.

On the one hand, what could more plainly be human inventions than the concepts of geometry and algebra? Even the simplest mathematical notion has no tangible existence in the everyday world. Who goes for a walk and trips over a 3? On the other hand, what could be more obvious than that the truths of mathematics are facts about the world, whether or not any humans catch on to them? If two dinosaurs were at a watering hole and two more dinosaurs came along to join them, the science writer Martin Gardner once asked, weren’t there four dinosaurs altogether? Didn’t three stars in the sky form a triangle before the first humans came along to define triangles?32

Newton and the other scientists of the seventeenth century shared the Greek view, and they coupled it with their own fundamental belief that the world was a cosmic code, a riddle designed by God. Their mission, in the words of one prominent writer of the day, was to decode that “strange Cryptography.” The Greeks had harbored similar ambitions, but the new scientists had advantages their predecessors had lacked. First, they had no taboos about studying motion mathematically. Second, they had calculus, a gleaming new weapon in the mathematical arsenal, to study it with.

Just as important, they had complete, unbreakable faith that the riddle had an answer. That was vital. No one would stick with a crossword puzzle if they feared that the areas of order might be mixed with patches of gibberish. Nature presented a greater challenge than any crossword, and only the certain knowledge that God had played fair kept scientists struggling year after year to catch on to His game.

Even so, the task was enormously difficult. Nonmathematicians underestimated the challenge. When Francis Bacon spoke of the mysteries of science, for instance, he made it sound as if God had set up an Easter egg hunt to entertain a pack of toddlers. God “took delight to hide his works, to the end to have them found out.”

Why would God operate in such a roundabout way? If his intent was to proclaim His majesty, why not arrange the stars to spell out BEHOLD in blazing letters? To seventeenth-century thinkers, this was no mystery. God could have put on a display of cosmic fireworks, but that would have been to win us over by shock and fear. When it came to intellectual questions, coercion was the wrong tool. Having created human beings and endowed us with the power of reason, God surely meant for us to exercise our gifts.

The mission of science was to honor God, and the best way to pay Him homage was to discover and proclaim the perfection of His plans.