Drunkard's Walk - Leonard Mlodinow [17]
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.17 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.) And if the defendant says the chariot never tried to stop but the expert witness says the hoof prints show that the brakes were applied, Germanic doctrine offered a simple prescription: “Let one man be chosen from each group to fight it out with shields and spears. Whoever loses is a perjurer and must lose his right hand.”18
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.”19 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