Drunkard's Walk - Leonard Mlodinow [14]
IN THE STORY of mathematics the ancient Greeks stand out as the inventors of the manner in which modern mathematics is carried out: through axioms, proofs, theorems, more proofs, more theorems, and so on. In the 1930s, however, the Czech American mathematician Kurt Gödel—a friend of Einstein’s—showed this approach to be somewhat deficient: most of mathematics, he demonstrated, must be inconsistent or else must contain truths that cannot be proved. Still, the march of mathematics has continued unabated in the Greek style, the style of Euclid. The Greeks, geniuses in geometry, created a small set of axioms, statements to be accepted without proof, and proceeded from there to prove many beautiful theorems detailing the properties of lines, planes, triangles, and other geometric forms. From this knowledge they discerned, for example, that the earth is a sphere and even calculated its radius. One must wonder why a civilization that could produce a theorem such as proposition 29 of book 1 of Euclid’s Elements—“a straight line falling on two parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles”—did not create a theory showing that if you throw two dice, it would be unwise to bet your Corvette on their both coming up a 6.
Actually, not only didn’t the Greeks have Corvettes, but they also didn’t have dice. They did have gambling addictions, however. They also had plenty of animal carcasses, and so what they tossed were astragali, made from heel bones. An astragalus has six sides, but only four are stable enough to allow the bone to come to rest on them. Modern scholars note that because of the bone’s construction, the chances of its landing on each of the four sides are not equal: they are about 10 percent for two of the sides and 40 percent for the other two. A common game involved tossing four astragali. The outcome considered best was a rare one, but not the rarest: the case in which all four astragali came up different. This was called a Venus throw. The Venus throw has a probability of about 384 out of 10,000, but the Greeks, lacking a theory of randomness, didn’t know that.
The Greeks also employed astragali when making inquiries of their oracles. From their oracles, questioners could receive answers that were said to be the words of the gods. Many important choices made by prominent Greeks were based on the advice of oracles, as evidenced by the accounts of the historian Herodotus, and writers like Homer, Aeschylus, and Sophocles. But despite the importance of astragali tosses in both gambling and religion, the Greeks made no effort to understand the regularities of astragali throws.
Why didn’t the Greeks develop a theory of probability? One answer is that many Greeks believed that the future unfolded according to the will of the gods. If the result of an astragalus toss meant “marry the stocky Spartan girl who pinned you in that wrestling match behind the school barracks,” a Greek boy wouldn’t view the toss as the lucky (or unlucky) result of a random process; he would view it as the gods’ will. Given such a view, an understanding of randomness would have been irrelevant. Thus a mathematical prediction of randomness would have seemed impossible. Another answer may lie in the very philosophy that made the Greeks such great mathematicians: they insisted on absolute truth, proved by logic and axioms, and frowned on uncertain pronouncements. In Plato’s Phaedo, for example, Simmias tells Socrates that “arguments from probabilities are impostors” and anticipates the work of Kahneman and Tversky by pointing out that “unless great caution is observed in the use of them they are apt to be deceptive—in geometry, and in other things too.”4 And in Theaetetus, Socrates says that