Applications of the mathematical theory of probability to prob- lems in science and technology can be very controversial. Although the theory of probability as a mathematical discipline is as rigorous as that of any other branch of mathematics, this rigor is no guarantee that the results obtained from the theory will be “reasonable.” The application of any mathematical theory to a real-world problem rests on certain additional assumptions about
Part of a 1553 text on smallpox (Library of Congress, Prints and Photographs Division)
the relationship between the theory and the application. The mathematically derived conclusions might be logically rigorous consequences of the assumptions, but that is no guarantee that the conclusions themselves will coincide with reality.
The sometimes-tenuous nature of the connections between the mathematical theory of probability and scientific and tech- nological applications accounts for the frequent disputes about the reasonableness of deductions derived with the help of the theory. Some of the results are of a technical nature, but others center on deeper questions about philosophic notions of chance and probability. Historically, one of the first such disputes arose when probability theory was first used to help formulate a government health policy. The issue under discussion was the prevention of smallpox. The discussion, which took place centuries ago, still sounds remarkably modern. Today, the same sorts of issues are sources of concern again. As we will later see, the discussion that began in the 18th century never really ended. It continues to this day.
Smallpox is at least as old as civilization. The ancient Egyptians suffered from smallpox, and so did the Hittites, Greeks, Romans, and Ottomans. Nor was the disease localized to northern Africa and the Mideast. Chinese records from 3,000 years ago describe the disease, and so do ancient Sanskrit texts of India. Slowly, out of these centuries of pain and loss, knowledge about the disease accu- mulated. The Greeks knew that if one survived smallpox, one did not become infected again. This is called acquired immunity. The Islamic doctor ar-Razi, who lived about 11 centuries ago, wrote the historically important Treatise on the Small Pox and Measles. He describes the disease and indicates (correctly) that it is transmitted from person to person. By the time of the Swiss mathematician and scientist Daniel Bernoulli (1700–82), scientists and laypeople alike had discovered something else important about the disease: Resistance to smallpox can be conferred through a process called variolation. To understand the problem that Bernoulli tried to solve it helps to know a little more about smallpox and variolation.
Smallpox is caused by a virus. It is often described as a disease that was fatal to about a third of those who became infected, but
there were different strains of the disease. Some strains killed only a small percentage of those who became infected; other strains killed well over half of those who became ill. Those who survive smallpox are sick for about a month. There is an incubation period of about seven to 17 days, during which the infected person feels fine and is not contagious. The first symptoms are a headache, a severe backache, and generalized flulike symptoms. Next a rash, consisting of small red spots, appears on the tongue and mouth. When these sores break, the person is highly contagious. The rash spreads to the face, arms, legs, and trunk of the body. By the fifth day the bumps become raised and very hard. Fever increases. Scabs begin to form over the bumps. Sometime between the 11th and 14th days the fever begins to drop, and sometime around the third week the scabs begin to fall off. Around the 27th day after the first symptoms appear the scabs have all fallen off and the person is no longer contagious. Numerous pitted scars mark the skin of a person who has recovered from smallpox. The scars remain for life.
Before the discovery of a smallpox vaccine in the last years of the 18th century by the British doctor Edward Jenner, there were only two strategies for dealing with smallpox. One strategy was to do nothing and hope to escape infection. This strategy carried a sig- nificant risk because smallpox was widespread in the 18th century and was a major cause of mor-
tality. Moreover, there was no successful treatment for some- one who had contracted the disease. The other strategy for coping with smallpox was a technique called variolation. This was a primitive method of using live smallpox virus to confer immunity on an other- wise healthy person. Various methods of inoculation were used, but the idea is simple enough: Transfer a milder,
The hands and arms of a person suffering from smallpox (Courtesy of Dr. William Foege/U.S. Centers for Disease Control)
weakened form of the disease from someone who is already infected to an otherwise healthy person. The healthy person will generally become sick, but not sick enough to die. When that person recovers he or she will have acquired immunity against all future infections. In particular, the more virulent strains of the disease will pose no risk. This is how variolation works in theory. In practice, variolation has risks as well as benefits. The most obvious risk was that some of those who were variolated died of the procedure. The problem, then, was to determine whether variolation, on balance, was a better strategy than inaction in the hope of escaping infection. The answer, as it turned out, was by no means obvious.
Enter Daniel Bernoulli. He was the son of the prominent scien- tist and mathematician Johann Bernoulli and nephew of Jacob Bernoulli, author of the law of large numbers. A prominent mathematician in his own right, Daniel attended universities in Heidelberg and Strasbourg, Germany, and Basel, Switzerland. He studied philosophy, logic, and medicine, and he received an M.D. degree. Almost immediately after graduation, however, he began to contribute to the development of mathematics and physics. He soon moved to Saint Petersburg, Russia, where he lived for a number of years and became a member of the Academy of Sciences. Daniel Bernoulli eventually returned to Basel, where he found a position teaching anatomy and botany.
Bernoulli decided to use probability theory to study the effect of variolation on mortality, but to do so he had to phrase the prob- lem in a way that made it susceptible to mathematical analysis. Moreover, the problem had to be more than mathematical; it had to be phrased in a way that would make his results, whatever they turned out to be, relevant to the formulation of public health pol- icy. Suppose, he said, that a large group of infants were variolated. Those babies who survived the procedure could live their life free of the threat of smallpox. Some of the babies, however, would cer- tainly die within a month of being variolated as a result of the pro- cedure itself. On the other hand, if the infants were not variolated, many of them—but probably not all of them—would eventually contract the smallpox, and some of those could be expected to die
of the disease. There were substantial risks associated with either strategy. Which strategy, variolation or no variolation, was more likely to benefit the public health?
In 1760 Bernoulli read the paper, “An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoc- ulation to prevent it,” to the Paris Academy of Sciences. In this paper Bernoulli summarized what evidence was available about the probability of dying of smallpox. He presented his mathematical model and his results. What he discovered is that life expectancy would increase by almost 10 percent among the variolated.
Bernoulli decided that variolation was a valuable tool for pro- tecting the public health. He recommended it, and he was sup- ported in this belief by many scholars and philosophers around Europe. Others disagreed. Some disagreed with his reasoning; others simply disagreed with his conclusions. The French mathe- matician and scientist Jean le Rond d’Alembert scrutinized Bernoulli’s paper and, although he concluded that Bernoulli’s rec- ommendation for variolation was a good one, did not entirely agree with Bernoulli’s analysis. D’Alembert wrote a well-known critique of Bernoulli’s well-known paper. D’Alembert’s response to Bernoulli’s ideas illustrates the difficulty of interpreting real-world problems in the language of probability theory.