15 Nov The laws of truth and half of truth. | Part A’ (LEONARD MLODINOW)
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 Godel — 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.” And in Theaetetus, Socrates says that any mathematician “who argued from probabilities and likelihoods in geometry would not be worth an ace.” 5 But even Greeks who believed that probabilists were worth an ace night have had difficulty working out a consistent theory in those days before extensive record keeping because people have notoriously poor memories when it comes to estimating the frequency — and hence the probability — of past occurrences.
Which is greater: the number of six-letter English words having n as their fifth letter or the number of six-letter English words ending in ing?” Most people choose the group of words ending in -ing. Why? Because words ending in -ing are easier to think of than generic six letter words having n as their fifth letter. But you don’t have to survey the Oxford English Dictionary—or even know how to count—to prove that guess wrong: the group of six-letter words having n as their fifth letter words includes all six-letter words ending in -ing. Psychologists call that type of mistake the availability bias because in reconstructing the past, we give unwarranted importance to memories that are most vivid and hence most available for retrieval.
The nasty thing about the availability bias is that it insidiously distorts our view of the world by distorting our perception of past events and our environment. For example, people tend to overestimate the fraction of homeless people who are mentally ill because when they encounter a homeless person who is not behaving oddly, they don’t take notice and tell all their friends about that unremarkable homeless person they ran into. But when they encounter a homeless person stomping down the street and waving his arms at an imaginary companion while singing “When the Saints Go Marching In,” they do tend to remember the incident. 7 How probable is it that of the five lines at the grocery-store checkout you will choose the one that takes the longest? Unless you’ve been cursed by a practitioner of the black arts, the answer is around 1 in 5. So why, when you look back, do you get the feeling you have a supernatural knack for choosing the longest line? Because you have more important things to focus on when things go right, but it makes an impression when the lady in front of you with a single item in her cart decides to argue about why her chicken is priced at $1.50 a pound when she is certain the sign at the meat counter said $ 1 .49.
By distorting our view of the past, the availability bias complicates any attempt to make sense of it. That was true for the ancient Greeks just as it is true for us. But there was one other major obstacle to an early theory of randomness, a very practical one: although basic probability requires only knowledge of arithmetic, the Greeks did not know arithmetic, at least not in a form that is easy to work with. In Athens in the fifth century B.C., for instance, at the height of Greek civilization, a person who wanted to write down a number used a kind of alphabetic code. The first nine of the twenty- four letters in the Greek alphabet stood for the numbers we call 1 through 9. The next nine letters stood for the numbers we call 10, 20, 30, and so on.
The first nine of the twenty-four letters in the Greek alphabet stood for the numbers we call 1 through 9. The next nine letters stood for the numbers we call 10, 20, 30, and so on. And the last six letters plus three additional symbols stood for the first nine hundreds (100, 200, and so on, to 900). If you think you have trouble with arithmetic now, imagine trying to subtract ΔΓΘ from ΩΨΠ! To make matters worse, the order in which the ones, tens, and hundreds were written didn’t really matter: sometimes the hundreds were written first, sometimes last, and sometimes all order was ignored. Finally, the Greeks had no zero.
The concept of zero came to Greece when Alexander invaded the Babylonian Empire in 331 B.C. Even then, although the Alexandrians began to use the zero to denote the absence of a number, it wasn’t employed as a number in its own right. In modern mathematics the number 0 has two key properties: in addition it is the number that, when added to any other number, leaves the other number unchanged, and in multiplication it is the number that, when multiplied by any other number, is itself unchanged. This concept wasn’t introduced until the ninth century, by the Indian mathematician Mahāvīra.
Even after the development of a usable number system it would be many more centuries before people came to recognize addition, subtraction, multiplication, and division as the fundamental arithmetic operations — and slowly realized that convenient symbols would make their manipulation far easier. And so it wasn’t until the sixteenth century that the Western world was truly poised to develop a theory of probability. Still, despite the handicap of an awkward system of calculation, it was the civilization that conquered the Greeks — the Romans — who made the first progress in understanding randomness.
The Drunkard’s Walk
LEONARD MLODINOW
