Is God a Mathematician?

by Mario Livio

  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:

  What mortal, I ask, could ascertain the number of diseases, counting all possible cases, that afflict the human body in every one of its many parts and at every age, and say how much more likely one disease is to be fatal than another…and on that basis make a prediction about the relationship between life and death in future generations?

  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:

  There is, however, another way that will lead us to what we are looking for and enable us at least to ascertain a posteriori what we cannot determine a priori, that is, to ascertain it from the results observed in numerous similar instances. It must be assumed in this connection that, under similar conditions, the occurrence (or nonoccurrence) of an event in the future will follow the same pattern as was observed for like events in the past. For example, if we have observed that out of 300 persons of the same age and with the same constitution as a certain Titius, 200 died within ten years while the rest survived, we can with reasonable certainty conclude that there are twice as many chances that Titius also will have to pay his debt to nature within the ensuing decade as there are chances that he will live beyond that time.

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

  Besides the uses mentioned in my former, it may perhaps not be an unacceptable thing to infer from the same Tables, how unjustly we repine at the shortness of our lives, and think our selves wronged if we attain not Old Age; whereas it appears hereby, that the one half of those that are born are dead in Seventeen years time, 1238 being in that time reduced to 616. So that instead of murmuring at what we call an untimely Death, we ought with Patience and unconcern to submit to that Dissolution which is the necessary Condition of our perishable Materials, and of our nice and frail Structure and Composition: And to account it as Blessing that we have survived, perhaps by many Years, that Period of Life, whereat the one half of the whole Race of Mankind does not arrive.

  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.

  Quetelet used to open his course on the history of science with the following insightful observation: “The more advanced the sciences become, the more they have tended to enter the domain of mathematics, which is a sort of center towards which they converge. We can judge of the perfection to which a science has come by the facility, more or less great, with which it may be approached by calculation.”

  In December of 1823, Quetelet was sent to Paris at the state’s expense, mostly to study observational techniques in astronomy. As it turned out, however, this three-month visit to the then mathematical capital of the world veered Quetelet in an entirely different direction—the theory of probability. The person who was mostly responsible for igniting Quetelet’s enthusiastic interest in this subject was Laplace himself. Quetelet later summarized his experience with statistics and probability:

  Chance, that mysterious, much abused word, should be considered only a veil for our ignorance; it is a phantom which exercises the most absolute empire over the common mind, accustomed to consider events only as isolated, but which is reduced to naught before the philosopher, whose eye embraces a long series of events and whose penetration is not led astray by variations, which disappear when he gives himself sufficient perspective to seize the laws of nature.

  The importance of this conclusion cannot be overemphasized. Quetelet essentially denied the role of chance and replaced it with the bold (even though not entirely proven) inference that even social phenomena have causes, and that the regularities exhibited by statistical results can be used to uncover the rules underlying social order.

  In an attempt to put his statistical approach to the test, Quetelet started an ambitious project of collecting thousands of measurements related to the human body. For instance, he studied the distributions of the chest measurements of 5,738 Scottish soldiers and of the heights of 100,000 French conscripts by plotting separately the frequency with which each human trait occurred. In other words, he represented graphically how many conscripts had heights between, say, five feet and five feet two inches, and then between five feet two inches and five feet four inches, and so on. He later constructed similar curves even for what he called “moral” traits for which he had sufficient data. The latter qualities included suicides, marriages, and the propensity to crime. To his surprise, Quetelet discovered that all the human characteristics followed what is now known as the normal (or Gaussian, named somewhat unjustifiably after the “prince of mathematics” Carl Friedrich Gauss), bell-shaped frequency distribution (figure 33). Whether it was heights, weights, measurements of limb l
engths, or even intellectual qualities determined by what were then pioneering psychological tests, the same type of curve appeared again and again. The curve itself was not new to Quetelet—mathematicians and physicists recognized it from the mid-eighteenth century, and Quetelet was familiar with it from his astronomical work—it was just the association of this curve with human characteristics that came as somewhat of a shock. Previously, this curve had been known as the error curve, because of its appearance in any type of errors in measurements.

  Figure 33

  Imagine, for instance, that you are interested in measuring very accurately the temperature of a liquid in a vessel. You can use a high-precision thermometer and over a period of one hour take one thousand consecutive readings. You will find that due to random errors and possibly some fluctuations in the temperature, not all measurements will give precisely the same value. Rather, the measurements would tend to cluster around a central value, with some measurements giving temperatures that are higher and others that are lower. If you plot the number of times that each measurement occurred against the value of the temperature, you will obtain the same type of bell-shaped curve that Quetelet found for the human characteristics. In fact, the larger the number of measurements performed on any physical quantity, the closer will the obtained frequency distribution approximate the normal curve. The immediate implication of this fact for the question of the unreasonable effectiveness of mathematics is quite dramatic in itself—even human errors obey some strict mathematical rules.

  Quetelet thought that the conclusions were even more far-reaching. He regarded the finding that human characteristics followed the error curve as an indication that the “average man” was in fact a type that nature was trying to produce. According to Quetelet, just as manufacturing errors would create a distribution of lengths around the average (correct) length of a nail, nature’s errors were distributed around a preferred biological type. He declared that the people of a nation were clustered about their average “as if they were the results of measurements made on one and the same person, but with instruments clumsy enough to justify the size of the variation.”

  Clearly, Quetelet’s speculations went a bit too far. While his discovery that biological characteristics (whether physical or mental) are distributed according to the normal frequency curve was extremely important, this could neither be taken as proof for nature’s intentions nor could individual variations be treated as mere mistakes. For instance, Quetelet found the average height of the French conscripts to be five feet four inches. At the low end, however, he found a man of one foot five inches. Obviously one could not make an error of almost four feet in measuring the height of a man five feet four inches tall.

  Even if we ignore Quetelet’s notion of “laws” that fashion humans in a single mold, the fact that the distributions of a variety of traits ranging from weights to IQ levels all follow the normal curve is in itself pretty remarkable. And if that is not enough, even the distribution of major-league batting averages in baseball is reasonably normal, as is the annual rate of return on stock indexes (which are composed of many individual stocks). Indeed, distributions that deviate from the normal curve sometimes call for a careful examination. For instance, if the distribution of the grades in English in some school were found not to be normal, this could provoke an investigation into the grading practices of that school. This is not to say that all distributions are normal. The distribution of the lengths of words that Shakespeare used in his plays is not normal. He used many more words of three and four letters than words of eleven or twelve letters. The annual household income in the United States is also represented by a non-normal distribution. In 2006, for instance, the top 6.37% of households earned roughly one third of all income. This fact raises an interesting question in itself: If both the physical and the intellectual characteristics of humans (which presumably determine the potential for income) are normally distributed, why isn’t the income? The answer to such socioeconomic questions is, however, beyond the scope of the present book. From our present limited perspective, the amazing fact is that essentially all the physically measurable particulars of humans, or of animals and plants (of any given variety) are distributed according to just one type of mathematical function.

  Human characteristics served historically not only as the basis for the study of the statistical frequency distributions, but also for the establishment of the mathematical concept of correlation. The correlation measures the degree to which changes in the value of one variable are accompanied by changes in another. For instance, taller women may be expected to wear larger shoes. Similarly, psychologists found a correlation between the intelligence of parents and the degree to which their children succeed in school.

  The concept of a correlation becomes particularly useful in those situations in which there is no precise functional dependence between the two variables. Imagine, for example, that one variable is the maximal daytime temperature in southern Arizona and the other is the number of forest fires in that region. For a given value of the temperature, one cannot predict precisely the number of forest fires that will break out, since the latter depends on other variables such as the humidity and the number of fires started by people. In other words, for any value of the temperature, there could be many corresponding numbers of forest fires and vice versa. Still, the mathematical concept known as the correlation coefficient allows us to measure quantitatively the strength of the relationship between two such variables.

  The person who first introduced the tool of the correlation coefficient was the Victorian geographer, meteorologist, anthropologist, and statistician Sir Francis Galton (1822–1911). Galton—who was, by the way, the half-cousin of Charles Darwin—was not a professional mathematician. Being an extraordinarily practical man, he usually left the mathematical refinements of his innovative concepts to other mathematicians, in particular to the statistician Karl Pearson (1857–1936). Here is how Galton explained the concept of correlation:

  The length of the cubit [the forearm] is correlated with the stature, because a long cubit usually implies a tall man. If the correlation between them is very close, a very long cubit would usually imply a very tall stature, but if it were not very close, a very long cubit would be on the average associated with only a tall stature, and not a very tall one; while, if it were nil, a very long cubit would be associated with no especial stature, and therefore, on the average, with mediocrity.

  Pearson eventually gave a precise mathematical definition of the correlation coefficient. The coefficient is defined in such a way that when the correlation is very high—that is, when one variable closely follows the up-and-down trends of the other—the coefficient takes the value of 1. When two quantities are anticorrelated, meaning that when one increases the other decreases and vice versa, the coefficient is equal to–1. Two variables that each behave as if the other didn’t even exist have a correlation coefficient of 0. (For instance, the behavior of some governments unfortunately shows almost zero correlation with the wishes of the people whom they supposedly represent.)

  Modern medical research and economic forecasting depend crucially on identifying and calculating correlations. The links between smoking and lung cancer, and between exposure to the Sun and skin cancer, for instance, were established initially by discovering and evaluating correlations. Stock market analysts are constantly trying to find and quantify correlations between market behavior and other variables; any such discovery can be enormously profitable.

  As some of the early statisticians readily realized, both the collection of statistical data and their interpretation can be very tricky and should be handled with the utmost care. A fisherman who uses a net with holes that are ten inches on a side might be tempted to conclude that all fish are larger than ten inches, simply because the smaller ones would escape from his net. This is an example of selection effects—biases introduced in the results due to either the apparatus used for collecting the data or the methodology used to analyze the
m. Sampling presents another problem. For instance, modern opinion polls usually interview no more than a few thousand people. How can the pollsters be sure that the views expressed by members of this sample correctly represent the opinions of hundreds of millions? Another point to realize is that correlation does not necessarily imply causation. The sales of new toasters may be on the rise at the same time that audiences at concerts of classical music increase, but this does not mean that the presence of a new toaster at home enhances musical appreciation. Rather, both effects may be caused by an improvement in the economy.

  In spite of these important caveats, statistics have become one of the most effective instruments in modern society, literally putting the “science” into the social sciences. But why do statistics work at all? The answer is given by the mathematics of probability, which reigns over many facets of modern life. Engineers trying to decide which safety mechanisms to install into the Crew Exploration Vehicle for astronauts, particle physicists analyzing results of accelerator experiments, psychologists rating children in IQ tests, drug companies evaluating the efficacy of new medications, and geneticists studying human heredity all have to use the mathematical theory of probability.

  Games of Chance

  The serious study of probability started from very modest beginnings—attempts by gamblers to adjust their bets to the odds of success. In particular, in the middle of the seventeenth century, a French nobleman—the Chevalier de Méré—who was also a reputed gamester, addressed a series of questions about gambling to the famous French mathematician and philosopher Blaise Pascal (1623–62). The latter conducted in 1654 an extensive correspondence about these questions with the other great French mathematician of the time, Pierre de Fermat (1601–65). The theory of probability was essentially born in this correspondence.