by Mario Livio
A few of the first explorers of the new vistas opened by differential equations were members of the legendary Bernoulli family. Between the mid-seventeenth century and the mid-nineteenth century, this family produced no fewer than eight prominent mathematicians. These gifted individuals were almost equally known for their bitter intrafamily feuds as they were for their outstanding mathematics. While the Bernoulli quarrels were always concerned with competition for mathematical supremacy, some of the problems they argued about may not seem today to be of the highest significance. Still, the solution of these intricate puzzles often paved the way for more impressive mathematical breakthroughs. Overall, there is no question that the Bernoullis played an important role in establishing mathematics as the language of a variety of physical processes.
One story can help exemplify the complexity of the minds of two of the brightest Bernoullis—the brothers Jakob (1654–1705) and Johann (1667–1748). Jakob Bernoulli was one of the pioneers of probability theory, and we shall return to him later in the chapter. In 1690, however, Jakob was busy resurrecting a problem first examined by the quintessential Renaissance man, Leonardo da Vinci, two centuries earlier: What is the shape taken by an elastic but inextensible chain suspended from two fixed points (as in figure 31)? Leonardo sketched a few such chains in his notebooks. The problem was also presented to Descartes by his friend Isaac Beeckman, but there is no evidence of Descartes’ trying to solve it. Eventually the problem became known as the problem of the catenary (from the Latin word catena, meaning “a chain”). Galileo thought that the shape would be parabolic but was proven wrong by the French Jesuit Ignatius Pardies (1636–73). Pardies was not up to the task, however, of actually solving mathematically for the correct shape.
Figure 31
Just one year after Jakob Bernoulli posed the problem, his younger brother Johann solved it (by means of a differential equation). Leibniz and the Dutch mathematical physicist Christiaan Huygens (1629–95) also solved it, but Huygens’s solution employed a more obscure geometrical method. The fact that Johann managed to solve a problem that had stymied his brother and teacher continued to be an immense source of satisfaction to the younger Bernoulli, even as late as thirteen years after Jakob’s death. In a letter Johann wrote on September 29, 1718, to the French mathematician Pierre Rémond de Montmort (1678–1719), he could not hide his delight:
You say that my brother proposed this problem; that is true, but does it follow that he had a solution of it then? Not at all. When he proposed this problem at my suggestion (for I was the first to think of it), neither the one nor the other of us was able to solve it; we despaired of it as insoluble, until Mr. Leibniz gave notice to the public in the Leipzig journal of 1690, p. 360, that he had solved the problem but did not publish his solution, so as to give time to other analysts, and it was this that encouraged us, my brother and me, to apply ourselves afresh.
After shamelessly taking ownership of even the suggestion of the problem, Johann continued with unconcealed glee:
The efforts of my brother were without success; for my part, I was more fortunate, for I found the skill (I say it without boasting, why should I conceal the truth?) to solve it in full…It is true that it cost me study that robbed me of rest for an entire night…but the next morning, filled with joy, I ran to my brother, who was still struggling miserably with this Gordian knot without getting anywhere, always thinking like Galileo that the catenary was a parabola. Stop! Stop! I say to him, don’t torture yourself any more to try to prove the identity of the catenary with the parabola, since it is entirely false…But then you astonish me by concluding that my brother found a method of solving this problem…I ask you, do you really think, if my brother had solved the problem in question, he would have been so obliging to me as not to appear among the solvers, just so as to cede me the glory of appearing alone on the stage in the quality of the first solver, along with Messrs. Huygens and Leibniz?
In case you ever needed proof that mathematicians are humans after all, this story amply provides it. The familial rivalry, however, does not take anything away from the accomplishments of the Bernoullis. During the years that followed the catenary episode, Jakob, Johann, and Daniel Bernoulli (1700–1782) went on not only to solve other similar problems of hanging cords, but also to advance the theory of differential equations in general and to solve for the motion of projectiles through a resisting medium.
The tale of the catenary serves to demonstrate another facet of the power of mathematics—even seemingly trivial physical problems have mathematical solutions. The shape of the catenary itself, by the way, continues to delight millions of visitors to the famous Gateway Arch in St. Louis, Missouri. The Finnish-American architect Eero Saarinen (1910–61) and the German-American structural engineer Hannskarl Bandel (1925–93) designed this iconic structure in a shape that is similar to that of an inverted catenary.
The astounding success of the physical sciences in discovering mathematical laws that govern the behavior of the cosmos at large raised the inevitable question of whether or not similar principles might also underlie biological, social, or economical processes. Is mathematics only the language of nature, mathematicians wondered, or is it also the language of human nature? Even if truly universal principles do not exist, could mathematical tools, at the very least, be used to model and subsequently explain social behavior? At first, many mathematicians were quite convinced that “laws” based on some version of calculus would be able to accurately predict all future events, large or small. This was the opinion, for instance, of the great mathematical physicist Pierre-Simon de Laplace (1749–1827). Laplace’s five volumes of Mécanique céleste (Celestial Mechanics) gave the first virtually complete (if approximate) solution to the motions in the solar system. In addition, Laplace was the man who answered a question that puzzled even the giant Newton: Why is the solar system as stable as it is? Newton thought that due to their mutual attractions planets had to fall into the Sun or to fly away into free space, and he invoked God’s hand in keeping the solar system intact. Laplace had rather different views. Instead of relying on God’s handiwork, he simply proved mathematically that the solar system is stable over periods of time that are much longer than those anticipated by Newton. To solve this complex problem, Laplace introduced yet another mathematical formalism known as perturbation theory, which enabled him to calculate the cumulative effect of many small perturbations to each planet’s orbit. Finally, to top it all, Laplace proposed one of the first models for the very origin of the solar system—in his influential nebular hypothesis, the solar system formed from a contracting gaseous nebula.
Given all of these impressive feats, it is perhaps not surprising that in his Philosophical Essay on Probabilities Laplace boldly pronounced:
All events, even those which on account of their insignificance do not seem to follow the great laws of nature, are a result of it just as necessary as the revolutions of the Sun. In ignorance of the ties which unite such events to the entire system of the universe, they have been made to depend upon final causes for or upon hazard…We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situations of the beings who compose it—an intelligence sufficiently vast to submit these data to analysis—it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes. The human mind offers, in the perfection which it has been able to give to astronomy, a feeble idea of this intelligence.
Just in case you wonder, when Laplace talked about this hypothetical supreme “intelligence,” he did not mean God. Unlike Newton and Descartes, Laplace was not a religious person. When he gave a copy of his Celestial Mechanics to Napoleon Bonaparte, the latter, who had heard that there was no r
eference to God in the work, remarked: “M. Laplace, they tell me you have written this huge book on the system of the universe and have never even mentioned its creator.” Laplace immediately replied: “I did not need to make that hypothesis.” The amused Napoleon told the mathematician Joseph-Louis Lagrange about this reply, and the latter exclaimed: “Ah! That is a beautiful hypothesis; it explains many things.” But the story doesn’t end there. When he heard about Lagrange’s reaction, Laplace commented dryly: “This hypothesis, Sir, explains in fact everything, but does not permit to predict anything. As a scholar, I must provide you with works permitting predictions.”
The twentieth century development of quantum mechanics—the theory of the subatomic world—has proven the expectation for a fully deterministic universe to be too optimistic. Modern physics has in fact demonstrated that it is impossible to predict the outcome of every experiment, even in principle. Rather, the theory can only predict the probabilities for different results. The situation in the social sciences is clearly even more complex because of a multiplicity of interrelated elements, many of which are highly uncertain at best. The researchers of the seventeenth century realized soon enough that a search for precise universal social principles of the type of Newton’s law of gravitation was doomed from the start. For a while, it seemed that when the intricacies of human nature are brought into the equation, secure predictions become virtually impossible. The situation appeared to be even more hopeless when the minds of an entire population were involved. Rather than despairing, however, a few ingenious thinkers developed a fresh arsenal of innovative mathematical tools—statistics and probability theory.
The Odds Beyond Death and Taxes
The English novelist Daniel Defoe (1660–1731), best known for his adventure story Robinson Crusoe, also authored a work on the supernatural entitled The Political History of the Devil. In it, Defoe, who saw evidence for the devil’s actions everywhere, wrote: “Things as certain as death and taxes, can be more firmly believed.” Benjamin Franklin (1706–90) seems to have subscribed to the same perspective with respect to certainty. In a letter he wrote at age eighty-three to the French physicist Jean-Baptiste Leroy, he said: “Our Constitution is in actual operation. Everything appears to promise that it will last; but in this world nothing can be said to be certain but death and taxes.” Indeed, the courses of our lives appear to be unpredictable, prone to natural disasters, susceptible to human errors, and affected by pure happenstance. Phrases such as “[- - - -] happens” have been invented precisely to express our vulnerability to the unexpected and our inability to control chance. In spite of these obstacles, and maybe even because of these challenges, mathematicians, social scientists, and biologists have embarked since the sixteenth century on serious attempts to tackle uncertainties methodically. Following the establishment of the field of statistical mechanics, and faced with the realization that the very foundations of physics—in the form of quantum mechanics—are based on uncertainty, physicists of the twentieth and twenty-first centuries have enthusiastically joined the battle. The weapon researchers use to combat the lack of precise determinism is the ability to calculate the odds of a particular outcome. Short of being capable of actually predicting a result, computing the likelihood of different consequences is the next best thing. The tools that have been fashioned to improve on mere guesses and speculations—statistics and probability theory—provide the underpinning of not just much of modern science, but also a wide range of social activities, from economics to sports.
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:
Our first Observation upon the Casualties shall be, that in twenty Years there dying of all diseases and Casualties, 229,250, that 71,124 dyed of the Thrush, Convulsion, Rickets, Teeths, and Worms; and as Abortives, Chrysomes, Infants, Livergrown, and Overlaid; that is to say, that about 1/3 of the whole died of those diseases, which we guess d
id all light upon Children under four or five Years old. There died also of the Small-Pox, Swine-Pox, and Measles, and of Worms without Convulsions, 12,210, of which number we suppose likewise that about 1/2 might be Children under six Years old. Now, if we consider that 16 of the said 229 thousand died of that extraordinary and grand Casualty the Plague, we shall finde that about thirty six percentum of all quick conceptions, died before six years old.”
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?