The universe is not entirely deterministic: Not every “effect” is the result of an identifiable “cause.” These ideas began to find favor late in the 19th century. As scientists learned more about nature they were able to identify phenomena—for example, the motion of an individual molecule in a gas or liquid, or the turbu- lent flow of a fluid—for which the information necessary to iden- tify a cause was not simply unknown but perhaps unknowable. Scientists began to look at nature in a new way. They began to develop the concept of a random, or stochastic, process. In this view of nature, scientists can specify the probability of certain outcomes of a process, but this is all they can do. For example, when studying the motion of molecules in a gas they may predict that there is a 75 percent chance that a molecule that is currently in region A will be found in region B after a given amount of time has elapsed. Or they may predict that the velocity of a turbulent fluid at a particular location at a particular time will lie within a particular range of velocities 80 percent of the time. In some instances, at least, these predictions are the best, most accurate predictions possible. For certain applications, at least, prediction in the sense that Laplace understood the term had become a relic of the past.
This kind of understanding of natural phenomena has as much in common with our understanding of games of chance as it has with the deterministic physics of Newton, Euler, and Laplace. The goal of these new scientists, then, was to state the sharpest possi- ble probabilities for a range of outcomes, rather than to predict the unique outcome for a given cause. This was a profound shift
in scientific thinking, and it began with the work of the British botanist Robert Brown (1773–1858).
Brown, like many figures in the history of mathematics, was the son of a minister. He studied medicine at the Universities of Aberdeen and Edinburgh. As a young man he led an adventurous life. He was stationed in Iceland while serving in the British army, and later he served as ship’s naturalist aboard HMS Investigator. It was as a member of Investigator’s crew that he visited Australia. During this visit he collected thousands of specimens, and on his return to England he set to work classifying the collection and writing about what he found. In 1810 he published part of the results of his work as naturalist, but, because sales of the first volume were meager, he never completed the project.
Today, Brown is remembered for his observations of the motion of pollen in water made many years after his return to England. In 1828 he described his discoveries in a little pamphlet with the enormous title “A brief account of microscopical observations
made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies.” In this work Brown describes what he saw when he used a microscope to observe pollen particles that were about 0.0002 inch (0.0056 mm) in diameter immersed in water. He saw the particles occa- sionally turning on their axis and moving randomly about in the water. Prolonged observation indicated to him that the movements were not caused by currents or the evaporation of the water. At first, Brown referred to Buffon: He assumed that the particles moved because of the motion of the “organic molecules” whose existence had been described in Buffon’s Histoire naturelle générale
et particulière. Further research, however, changed his mind. Brown
observed the same phenomenon with particles that could not be alive. He observed 100-year-old pollen. He ground up glass and granite and observed that the particles moved through the water just as the pollen had. He even observed ground-up fragments of the Sphinx. Every sufficiently small particle suspended in water behaved in essentially the same way: (1) Each particle was as likely to move in one direction as in another, (2) future motion was not influenced by past motion, and (3) the motion never stopped.
That the motions might indeed be random was not a popular hypothesis. Scientists of the time believed that these motions would eventually be explained by some yet-to-be-discovered deterministic theory much as planetary orbits had already been explained. Brown, however, continued to gather data. He was remarkably thorough. When it was suggested that the motions were due to mutual attrac- tion between particles, he observed single grains suspended in indi- vidual droplets that were themselves suspended in oil. The oil prevented evaporation of the water, and the continued motion of isolated grains disproved the hypothesis that the motion was caused by forces between particles. Through his experiments Brown gained considerable insight into what did not cause the motion of these grains, but no one at the time had a convincing theory of what did cause their motion. Interest among his contemporaries, never strong to begin with, began to wane. For the next 30 or so years Brown’s experiments, which described the process now known as
Brownian motion, were pushed aside. Scientists were not yet ready to consider fundamentally random events.