Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)
CHAPTER V. THE MATHEMATICS OF RANDOMNESS
38. RAPID DEVELOPMENT
The analysis of the unfinished-game problem presented by Fermat and Pascal came just at the time when the need for mathematics that would describe randomness was growing. Possible uses of the concept of randomness led to a growing number of mathematicians involving themselves in the subject, and the theoretical aspects proposed by Fermat and Pascal played a central role in the rapid development of the mathematics of randomness.
There were three main reasons for the increased interest in this area. One, mentioned above, was the spread of gambling and games of chance, with quite a number of scientists and mathematicians among the participants.
The second reason was related to the bankruptcies of many European cities as a result of liabilities for payments of pensions and allowances they had taken upon themselves. Governments and municipalities in Europe, mainly in Holland and England, had for a long time followed the practice of financing their expenditures by taking loans from their citizens, in exchange for which they undertook to pay the lenders fixed amounts as long as they lived. The problem was that the mathematics required to calculate the proper amount of the repayments, and in particular to avoid incurring liabilities that the borrowers would not be able to meet, did not exist at that time. The amount of the regular repayments was determined intuitively or by using calculations that were biased by the borrowers’ desire to raise more and more money. The government of England, for example, issued an ordinance in 1540 that the amount of the annual repayments on a loan had to be such a sum equivalent to the entire amount of the loan that would be repaid in seven years, whereas the annual payments continued until the death of the lender. The age of the lender and the life expectancy of potential lenders was not taken into account. Newton included in his lectures in 1675, and later published in a short paper, a description of how to use the binomial formula to calculate the current value of an interest-bearing account, what is today known as capitalization. He did not even hint at the need for anyone guaranteeing the payments to relate to the average length of life of the recipients of the payments. The result of this practice was that more and more municipalities went bankrupt. The relevant branch of mathematics, unknown at that time, is of course what the actuarial profession deals with today, namely, calculating the life expectancy of the population of lenders, and hence the expected liability of future payments, and comparing those liabilities with the expected tax revenue. The data required for these calculations, such as mortality tables for different towns, already existed. These had been compiled for years out of curiosity and interest, and especially to understand the effects on mortality rates in different places of the various plagues that struck Europe. The basic concepts of probability and expectation, however, were not known. When municipality after municipality encountered financial difficulties and even bankruptcy, it created pressure to find a solution to the problem, and the development of the mathematics of probability and statistics was given a boost.
The third reason for the development of probability concepts was the development of jurisprudence. The awareness of the validity of legal arguments was increasing in Europe in those days, as was the recognition that legal proof that leads to conviction or acquittal beyond all reasonable doubt was almost impossible to obtain. Probability considerations in legal arguments had already appeared in ancient times, but with social progress the requirement arose that litigants base their case on quantitative assessments of the probability that their case was correct. In any event there was a growing need to develop terminology and methods of analysis that would help in such quantification.
The questions that arise in an analysis of chances and risks in ensuring payments are totally different from those that arise in, say, legal claims. Claims about betting, insurance, public-opinion surveys, and so on, relate to a random occurrence among many possibilities or random sampling from a large population, namely, statistics. In contrast, legal claims generally relate to a single, non-repeated event or to the degree of belief in the correctness of a certain argument. There is no apparent reason to expect the same theory and the same mathematical methods to apply to both types of situation, but nevertheless the same terminology of probability theory is used in the analysis of repeated and one-off incidents. This is the duality inherent in probability theory and statistics, and the pioneers of the theory were aware of it. We will discuss this in more detail later on.
A few years after the Pascal-Fermat correspondence, in 1657, Christiaan Huygens (1629–1695) published a book that summarized his own work on the theory of probability and the knowledge on the subject that had been accumulated until then. He was encouraged to publish it by Artus Gouffier de Boissy, Duke of Roannez, who was also a patron of Pascal. Despite his efforts, Huygens never met Pascal face-to-face, as Pascal was at that time more involved in theology than mathematics, but he was well aware of Pascal's work. Huygens's book was the first published on the subject of probability and was entirely devoted to an analysis of the randomness related to games of chance, repayments of loans, and so on, in other words, its statistical aspects. Among other things, he wrote about the methods of calculation developed over many years by mathematicians related to random draws, counting methods, and the like. Huygens, who was Dutch, was one of the most respected and esteemed mathematicians, physicists, and astronomers of the period. Apart from his contribution in various other fields, he was famous for his explanation of the propagation of waves before the wave equation had been formulated. He traveled widely on the Continent and to Britain, and in 1663 he was elected to the Royal Society and later to the French Academy.
Huygens's book was also the first to discuss the idea of an average and of expected value. He coined the term in Latin expactatio (“expectation”). Today, engulfed as we are by statistical data of all sorts on all subjects and in all locations, it is difficult to imagine that until the middle of the seventeenth century the concept of the average was not in widespread use as a statistical quantity. At that time averages were calculated by physicists to obtain a good estimate of inaccurate measurements, for instance the paths of the celestial bodies, but not for statistical analyses. The mathematical analysis of games of chance and expected future payments led naturally to the development and use of the concept of the average. The first well-ordered presentation of that concept appeared in Huygens's book.
The discussion was incisive. To illustrate: Christiaan's brother Ludwig used the existing tables of mortalities and found that the average lifetime of someone born in London was eighteen years. “Does that number have any significance?” Ludwig asked his brother. It is known, he claimed, that infant mortality is very high, and many children die before they are six years old, whereas those who do survive live to the age of fifty or longer. Indeed, a good question, and in our days the answer would be that different indices are used for different purposes. Christiaan Huygens did not deal with these situations but focused on the subject of gambling, and he stuck to his opinion that expectation was the right measure for estimating the value of bets, with the value being the expected payments weighted according to the chances of winning.
Here we will repeat the precise mathematical definition of expectation. If the monetary payments A1, A2,…, An in a draw are won with probabilities p1, p2,…, pn, respectively, the expectation of the draw is p1A1 + p2A2 +…+ pnAn. Just as with the concept of the average, questions about the justification and interpretation of the idea of expectation were asked, forcefully, because the idea of probability had not been sufficiently clarified. It was recognized only in the context of relative frequency, in other words, p1 in the definition is the approximate proportion in which the win of A1will occur if the draw is repeated many times.
One basic question that was asked was where do these probabilities come from? Also, how are they calculated? In throws of a die, for example, as there is no reason that one face should be on top more than any other, the probability of each face coming on top can be calculated; but how, Huygens had already asked himself, can the probability of catching a particular disease or being injured in an accident be calculated?
Despite the difficulty in establishing the concepts, the relevance to what they could be used for was clear. The first use was statistical analyses. The word statistics is derived from the word state, in the sense of country, and statistics did in fact deal mainly with subjects related to the management of a country. The most rapid progress was made in Holland, in relation to the pricing of the repayment of loans. This was because both the politician responsible for that subject, Johan de Witt, who was a leading figure in Dutch political life, and the burgomaster (mayor) of Amsterdam, Johannes Hudde, a town that was hard hit because of promises of unrealistic payments, were good mathematicians. They both engaged in and contributed to Descartes's Cartesian geometry. They consulted Huygens himself, who was then in Amsterdam, and in 1671 de Witt published a book in which he set out the theory as well as the practice of calculating loan repayments under various conditions with detailed exemplary calculations. The book came with a confirmation by Hudde that the calculations were correct. The method soon spread across all of Europe. Of all countries it was England that was slow in adopting this use of the theory, and even a hundred years later the government authorities there were still selling pensions at prices that were not based on proper mathematical costings.
Alongside the development of the practical statistical use of the mathematics of randomness, progress was also made in theory regarding the connection between probability and legal evidence. Leibniz was the leader in this field, and in 1665 he published a paper on probability and law, with a more detailed version published in 1672. Leibniz's interpretation of probability was similar to the sense in which Aristotle viewed it, that is, the likelihood of an event in light of partial information. Leibniz, who came from a family of lawyers, tried to present quantitative measurements for the correctness of legal claims. After a visit to Paris and after becoming familiar with the Fermat-Pascal correspondence as well as Pascal's wager, Leibniz realized that a similar analysis could also be used in cases in which one had to assess the probability that a claim or certain evidence, even if it is one-off, that is, non-repeated, is correct or not. He analyzed the logic underlying the information brought before a judge and proposed measuring the likelihood of a conclusion, giving it a value of between zero (in the event that the conclusion is clearly wrong) and one (in the case when it is without doubt correct). Thus Leibniz laid the foundation of the analogy between the likelihood of an occurrence and the mathematics of situations that are repeated randomly. Herein apparently lies the secret of the great impact of the Fermat-Pascal correspondence and Pascal's wager. They used the same mathematical tools in discussing events that can be repeated, such as a game of chance stopped in the middle, as well as non-repeated occurrences, such as the question of the existence of God. Yet neither Leibniz nor others who dealt with the concepts of likelihood and probability reached an understanding or consensus of what these probabilities were derived from or where they were formed.