16 Nov The laws of truth and half of truth. | Part B’ (LEONARD MLODINOW)
THE ROMANS generally scorned mathematics, at least the mathematics of the Greeks. In the words of the Roman statesman Cicero, who lived from 106 to 43 B.C., “The Greeks held the geometer in the highest honor; accordingly, nothing made more brilliant progress among them than mathematics. But we have established as the limit of this art its usefulness in measuring and counting.” Indeed, whereas one might imagine a Greek textbook focused on the proof of congruences among abstract triangles, a typical Roman text focused on such issues as how to determine the width of a river when the enemy is occupying the other bank. With such mathematical priorities, it is not surprising that while the Greeks produced mathematical luminaries like Archimedes, Diophantus, Euclid, Eudoxus, Pythagoras, and Thales; the Romans did not produce even one mathematician. In Roman culture it was comfort and war, not truth and beauty, that occupied center stage. And yet precisely because they focused on the practical, the Romans saw value in understanding probability. So while finding little value in abstract geometry,
Cicero wrote that “probability is the very guide of life.” Cicero was perhaps the greatest ancient champion of probability. He employed it to argue against the common interpretation of gambling success as due to divine intervention, writing that the “man who plays often will at some time or other make a Venus cast: now and then indeed he will make it twice and even thrice in succession. Are we going to be so feeble-minded then as to affirm that such a thing happened by the personal intervention of Venus rather than by pure luck?” Cicero believed that an event could be anticipated and predicted even though its occurrence would be a result of blind chance. He even used a statistical argument to ridicule the belief in astrology. Annoyed that although outlawed in Rome, astrology was nevertheless alive and well, Cicero noted that at Cannae in 216 B.C., Hannibal, leading about 50,000 Carthaginian and allied troops, crushed the much larger Roman army, slaughtering more than 60,000 of its 80,000 soldiers. “Did all the Romans who fell at Cannae have the same horoscope?” Cicero asked. “Yet all had one and the same end.” Cicero might have been encouraged to know that a couple of thousand years later in the journal Nature a scientific study of the validity of astrological predictions agreed with his conclusion. The New York Post, on the other hand, advises today that as a Sagittarius, I must look at criticisms objectively and make whatever changes seem necessary.
In the end, Cicero’s principal legacy in the field of randomness is the term he used, probabilis, which is the origin of the term we employ today. But it is one part of the Roman code of law, the Digest, compiled by Emperor Justinian in the sixth century, that is the first document in which probability appears as an everyday term of art. To appreciate the Roman applications of mathematical thinking to legal theory, one must understand the context: Roman law in the Dark Ages was based on the practice of the Germanic tribes. It wasn’t pretty. Take, for example, the rules of testimony. The veracity of, say, a husband denying an affair with his wife’s toga maker would be determined not by hubby’s ability to withstand a grilling by prickly opposing counsel but by whether he’d stick to his story even after being pricked—by a red-hot iron. (Bring back that custom and you’ll see a lot more divorce cases settled out of court.)
In replacing, or at least supplementing, the practice of trial by battle, the Romans sought in mathematical precision a cure for the deficiencies of their old, arbitrary system. Seen in this context, the Roman idea of justice employed advanced intellectual concepts. Recognizing that evidence and testimony often conflicted and that the best way to resolve such conflicts was to quantify the inevitable uncertainty, the Romans created the concept of half proof, which applied in cases in which there was no compelling reason to believe or disbelieve evidence or testimony. In some cases the Roman doctrine of evidence included even finer degrees of proof, as in the church decree that “a bishop should not be condemned except with seventy-two witnesses…a cardinal priest should not be condemned except with forty-four witnesses, a cardinal deacon of the city of Rome without thirty-six witnesses, a subdeacon, acolyte, exorcist, lector, or doorkeeper except with seven witnesses.” To be convicted under those rules, you’d have to have not only committed the crime but also sold tickets. Still, the recognition that the probability of truth in testimony can vary and that rules for combining such probabilities are necessary was a start. And so it was in the unlikely venue of ancient Rome that a systematic set of rules based on probability first arose.
Unfortunately it is hard to achieve quantitative dexterity when you’re juggling VIIIs and XIVs. In the end, though Roman law had a certain legal rationality and coherence, it fell short of mathematical validity. In Roman law, for example, two half proofs constituted a complete proof. That might sound reasonable to a mind unaccustomed to quantitative thought, but with today’s familiarity with fractions it invites the question, if two half proofs equal a complete certainty, what do three half proofs make? According to the correct manner of compounding probabilities, not only do two half proofs yield less than a whole certainty, but no finite number of partial proofs will ever add up to a certainty because to compound probabilities, you don’t add them; you multiply.
That brings us to our next law, the rule for compounding probabilities: If two possible events, A and B, are independent, then the probability that both A and B will occur is equal to the product of their individual probabilities. Suppose a married person has on average roughly a 1 in 50 chance of getting divorced each year. On the other hand, a police officer has about a 1 in 5,000 chance each year of being killed on the job. What are the chances that a married police officer will be divorced and killed in the same year? According to the above principle, if those events were independent, the chances would be roughly 1 /50 × 1 /5,000 , which equals 1 /250,000 . Of course the events are not independent; they are linked: once you die, darn it, you can no longer get divorced. And so the chance of that much bad luck is actually a little less than 1 in 250,000.
Why multiply rather than add? Suppose you make a pack of trading cards out of the pictures of those 100 guys you’ve met so far through your Internet dating service, those men who in their Web site photos often look like Tom Cruise but in person more often resemble Danny DeVito. Suppose also that on the back of each card you list certain data about the men, such as honest (yes or no) and attractive (yes or no). Finally, suppose that 1 in 10 of the prospective soul mates rates a yes in each case. How many in your pack of 100 will pass the test on both counts? Let’s take honest as the first trait (we could equally well have taken attractive).
Since 1 in 10 cards lists a yes under honest, 10 of the 100 cards will qualify. Of those 10, how many are attractive? Again, 1 in 10, so now you are left with 1 card. The first 1 in 10 cuts the possibilities down by V 10 , and so does the next 1 in 10, making the result 1 in 100. That’s why you multiply. And if you have more requirements than just honest and attractive, you have to keep multiplying, so… well, good luck.
Before we move on, it is worth paying attention to an important detail: the clause that reads if two possible events, A and B, are independent.
It is important to remember that you get the compound probability from the simple ones by multiplying only if the events are in no way contingent on each other.
There are situations in which probabilities should be added, and that is our next law. It arises when we want to know the chances of either one event or another occurring, as opposed to the earlier situation, in which we wanted to know the chance of one event and another event both happening. The law is this: If an event can have a number of different and distinct possible outcomes, A, B, C, and so on, then the probability that either A or B will occur is equal to the sum of the individual probabilities of A and B, and the sum of the probabilities of all the possible outcomes (A, B, C, and so on) is 1 (that is, 100 percent). When you want to know the chances that two independent events, A and B, will both occur, you multiply; if you want to know the chances that either of two mutually exclusive events, A or B, will occur, you add.
These three laws, (together with the first law of probability, which is one of the most basic of all: The probability that two events will both occur can never be greater than the probability that each will occur individually. Why not? Simple arithmetic: the chances that event A will occur = the chances that events A and B will occur + the chance that event A will occur and event B will not occur) simple as they are, form much of the basis of probability theory. Properly applied, they can give us much insight into the workings of nature and the everyday world. We employ them in our everyday decision making all the time. But like the Roman lawmakers, we don’t always use them correctly.
The Drunkard’s Walk