Is God a Mathematician? (17 page)

We all use probabilities and statistics in almost every decision we make, sometimes subconsciously. For instance, you probably don’t know that the number of fatalities from automobile accidents in the U.S. was 42,636 in 2004. However, had that number been, say, 3 million, I’m sure you would have known about it. Furthermore, this knowledge would have probably caused you to think twice before getting into the car in the morning. Why do these precise data on road fatalities give us some confidence in our decision to drive? As we shall see shortly, a key ingredient to their reliability is the fact that they are based on very large numbers. The number of fatalities in Frio Town, Texas, with a population of forty-nine in 1969 would hardly have been equally convincing. Probability and statistics are among the most important arrows for the bows of economists, political consultants, geneticists, insurance companies, and anybody trying to distill meaningful conclusions from vast amounts of data. When we talk about mathematics permeating even disciplines that were not originally under the umbrella of the exact sciences, it is often through the windows opened by probability theory and statistics. How did these fruitful fields emerge?

Statistics—a term derived from the Italian
stato
(state) and
statista
(a person dealing with state affairs)—first referred to the simple collection of facts by government officials. The first important work on statistics in the modern sense was carried out by an unlikely researcher—a shopkeeper in seventeenth century London. John Graunt (1620–74) was trained to sell buttons, needles, and drapes. Since his job afforded him a considerable amount of free time, Graunt studied Latin and French on his own and started to take interest in the Bills of Mortality—weekly numbers of deaths parish by parish—that had been published in London since 1604. The process of issuing these reports was established mainly in order to provide an early warning signal for
devastating epidemics. Using those crude numbers, Graunt started to make interesting observations that he eventually published in a small, eighty-five-page book entitled
Natural and Political Observations Mentioned in a Following Index, and Made upon the Bills of Mortality.
Figure 32 presents an example of a table from Graunt’s book, where no fewer than sixty-three diseases and casualties were listed alphabetically. In a dedication to the president of the Royal Society, Graunt points out that since his work concerns “the Air, Countries, Seasons, Fruitfulness, Health, Diseases, Longevity, and the proportion between the Sex and Ages of Mankind,” it is really a treatise in
natural history. Indeed, Graunt did much more than merely collect and present the data. By examining, for instance, the average numbers of christenings and burials for males and females in London and in the country parish Romsey in Hampshire, he demonstrated for the first time the stability of the sex ratio at birth. Specifically, he found that in London there were thirteen females born for every fourteen males and in Romsey fifteen females for sixteen males. Remarkably, Graunt had the foresight to express the wish that “travellers would enquire whether it be the same in other countries.” He also noted that “it is a blessing to Man-kind, that by this overplus of
Males
there is this natural Bar to
Polygamy
: for in such a state Women could not live in that parity, and equality of expence with their Husbands, as now, and here they do.” Today, the commonly assumed ratio between boys and girls at birth is about 1.05. Traditionally the explanation for this excess of males is that Mother Nature stacks the deck in favor of male births because of the somewhat greater fragility of male fetuses and babies. Incidentally, for reasons that are not entirely clear, in both the United States and Japan the proportion of baby boys has fallen slightly each year since the 1970s.

Figure 32

Another pioneering effort by Graunt was his attempt to construct an age distribution, or a “life table,” for the living population, using the data on the number of deaths according to cause. This was clearly of great political importance, since it had implications for the number of fighting men—men between sixteen and fifty-six years of age—in the population. Strictly speaking, Graunt did not have sufficient information to deduce the age distribution. This is precisely where, however, he demonstrated ingenuity and creative thinking. Here is how he describes his estimate of childhood mortality:

In other words, Graunt estimated the mortality before age six to be (71,124 + 6,105) ÷ (229,250–16,000) = 0.36. Using similar arguments and educated guesses, Graunt was able to estimate the old-age mortality. Finally, he filled the gap between ages six and seventy-six by a mathematical assumption about the behavior of the mortality rate with age. While many of Graunt’s conclusions were not particularly sound, his study launched the science of statistics as we know it. His observation that the percentages of certain events previously considered purely a matter of chance or fate (such as deaths caused by various diseases) in fact showed an extremely robust regularity, introduced scientific, quantitative thinking into the social sciences.

The researchers who followed Graunt adopted some aspects of his methodology, but also developed a better mathematical understanding of the use of statistics. Surprisingly perhaps, the person who made the most significant improvements to Graunt’s life table was the astronomer Edmond Halley—the same person who persuaded Newton to publish his
Principia.
Why was everybody so interested in life tables? Partly because this was, and still is, the basis for life insurance. Life insurance companies (and indeed gold diggers who marry for money!) are interested in such questions as: If a person lived to be sixty, what is the probability that he or she would also live to be eighty?

To construct his life table, Halley used detailed records that were kept at the city of Breslau in Silesia since the end of the sixteenth century. A local pastor in Breslau, Dr. Caspar Neumann, was using those lists to suppress superstitions in his parish that health is affected by the phases of the Moon or by ages that are divisible by seven and nine. Eventually, Halley’s paper, which had the rather long title of “An Estimate of the Degrees of the Mortality of Mankind, drawn from curious Tables of the Births and Funerals at the City of Breslaw; with an Attempt to ascertain
the Price of Annuities upon Lives,” became the basis for the mathematics of life insurance. To get an idea of how insurance companies may assess their odds, examine Halley’s life table below:

Halley’s Life Table

The table shows, for instance, that of 710 people alive at age six, 346 were still alive at age fifty. One could then take the ratio of 346/710 or 0.49 as an estimate of the probability that a person of age six would live to be fifty. Similarly, of 242 at age sixty, 41 were alive at age eighty. The probability of making it from sixty to eighty could then be estimated to be 41/242, or about 0.17. The rationale behind this procedure is simple. It relies on past experience to determine the probability of various future events. If the sample on which the experience is predicated is sufficiently large (Halley’s table was based on a population of about 34,000), and if certain assumptions hold (such as that the mortality rate is constant over time), then the calculated probabilities are fairly reliable. Here is how Jakob Bernoulli described the same problem:

After concluding that this and similar forecasts “depend on factors that are completely obscure, and which constantly deceive our senses by the endless complexity of their interrelationships,” Bernoulli also suggested a statistical/probabilistic approach:

Halley followed his mathematical articles on mortality with an interesting note that had more philosophical overtones. One of the passages is particularly moving:

While the situation in much of the modern world has improved significantly compared to Halley’s sad statistics, this is unfortunately not true for all countries. In Zambia, for instance, the mortality for ages five and under in 2006 has been estimated at a staggering 182 deaths per 1,000 live births. The life expectancy in Zambia remains at a heartbreaking low of thirty-seven years.

Statistics, however, are not concerned only with death. They penetrate into every aspect of human life, from mere physical traits to intellectual products. One of the first to recognize the power of statistics to potentially produce “laws” for the social sciences was the Belgian polymath Lambert-Adolphe-Jacques Quetelet (1796–1874). He, more than anyone else, was responsible for the introduction of the common statistical concept of the “average man,” or what we would refer to today as the “average person.”

The Average Person

Adolphe Quetelet was born on February 22, 1796, in the ancient Belgian town of Ghent. His father, a municipal officer, died when Adolphe was seven years old. Compelled to support himself early in life, Quetelet started to teach mathematics at the young age of seventeen. When not on duty as an instructor, he composed poetry, wrote the libretto for an opera, participated in the writing of two dramas, and translated a few literary works. Still, his favorite subject remained mathematics, and he was the first to graduate with the degree of doctor of science from the University of Ghent. In 1820, Quetelet was elected as a member of the Royal Academy of Sciences in Brussels, and within a short time he became the academy’s most active participant. The next few years were devoted mostly to teaching and to the publication of a few treatises on mathematics, physics, and astronomy.