Drunkard's Walk - Leonard Mlodinow [47]
Bernoulli saw the answer in the same terms that Jagger later would: instead of depending on probabilities being handed to us, we should discern them through observation. Being a mathematician, he sought to make the idea precise. Given that you view a certain number of roulette spins, how closely can you nail down the underlying probabilities, and with what level of confidence? We’ll return to those questions in the next chapter, but they are not quite the questions Bernoulli was able to answer. Instead, he answered a closely related question: how well are underlying probabilities reflected in actual results? Bernoulli considered it obvious that we are justified in expecting that as we increase the number of trials, the observed frequencies will reflect—more and more accurately—their underlying probabilities. He certainly wasn’t the first to believe that. But he was the first to give the issue a formal treatment, to turn the idea into a proof, and to quantify it, asking how many trials are necessary, and how sure can we be. He was also among the first to appreciate the importance of the new subject of calculus in addressing these issues.
THE YEAR Bernoulli was named professor in Basel proved to be a milestone year in the history of mathematics: it was the year in which Gottfried Leibniz published his revolutionary paper laying out the principles of integral calculus, the complement to his 1684 paper on differential calculus. Newton would publish his own version of the subject in 1687, in his Philosophiae Naturalis Principia Mathematica, or Mathematical Principles of Natural Philosophy, often referred to simply as Principia. These advances would hold the key to Bernoulli’s work on randomness.
By the time they published, both Leibniz and Newton had worked on the subject for years, but their almost simultaneous publications begged for controversy over who should be credited for the idea. The great mathematician Karl Pearson (whom we shall encounter again in chapter 8) said that the reputation of mathematicians “stands for posterity largely not on what they did, but on what their contemporaries attributed to them.”11 Perhaps Newton and Leibniz would have agreed with that. In any case neither was above a good fight, and the one that ensued was famously bitter. At the time the outcome was mixed. The Germans and Swiss learned their calculus from Leibniz’s work, and the English and many of the French from Newton’s. From the modern standpoint there is very little difference between the two, but in the long run Newton’s contribution is often emphasized because he appears to have truly had the idea earlier and because in Principia he employed his invention in the creation of modern physics, making Principia probably the greatest scientific book ever written. Leibniz, though, had developed a better notation, and it is his symbols that are often used in calculus today.
Neither man’s publications were easy to follow. In addition to being the greatest book on science, Newton’s Principia has also been called “one of the most inaccessible books ever written.”12 And Leibniz’s work, according to one of Jakob Bernoulli’s biographers, was “understood by no one” it was not only unclear but also full of misprints. Jakob’s brother Johann called it “an enigma rather than an explanation.”13 In fact, so incomprehensible were both works that scholars have speculated that both authors might have intentionally made their works difficult to understand to keep amateurs from dabbling. This enigmatic quality was an advantage for Jakob Bernoulli, though, for it did separate the wheat from the chaff, and his