The German Genius

However, the person inside the train will see the light beam hitting the back of the carriage at the same time as it hits the front of the carriage. Thus the time the light beam takes to reach the back of the carriage is different for the two observers. The discrepancy, Einstein said, can only be explained by assuming that the perception is relative to the observer and that, because the speed of light is constant, time must change according to circumstance. His most famous prediction was that clocks would move more slowly when traveling at high speeds. This anti-commonsense notion was actually borne out by experiment many years later. Physics was transformed. 30
    T HE S ECRET OF C ONTINUITY AND THE M EANING OF “B ETWEEN”
     
    In the late nineteenth century, Germany bred an extraordinary generation of “pure” mathematicians who were very concerned with ideas that, though extremely theoretical to begin with, would eventually prove fundamental and practical in equal measure. 31 Together with Planck, but operating from a very different starting point, they conceived the basis of what would, in the future, become the digital revolution.
    As we have seen, Carl Friedrich Gauss, Bernhard Riemann, and Felix Klein had helped established Göttingen as the world capital of mathematics, though other German university towns—Heidelberg, Halle, and Jena—were close seconds. In these small, remote, and self-contained worlds, away from the teeming metropolises, mathematicians’ minds were free and cleared to explore basic issues. And for many, number theory was the ultimate abstraction.
    Richard Dedekind, born in 1831, was one of Gauss’s last students at Göttingen and a pallbearer at his funeral. 32 Dedekind was a dedicated academic who never married and spent much of his time editing for publication one or two contributions by Gauss and the bulk of the papers by his other great teacher, Peter Lejeune Dirichlet, on differentiable functions and trigonometric series (Dirichlet, he liked to say, had made “a new man” of him). 33 This exercise had given Dedekind some ideas of his own, and though it was no more than a pamphlet when it was published in 1872, Continuity and Irrational Numbers soon became a classic, the best description to date of what mathematicians call the “numerical continuum” or the secret of continuity.
    The “secret” of continuity is one of those issues that troubles mathematicians if no one else (though of course it was theoretically linked to quantum theory, which advocated that energy was not continuously emitted, as Newton had said). The problem of continuity becomes apparent once you try to grasp what it means to be “between.” As early as the sixth century B.C., Pythagoras knew that fractions lay “between” whole numbers. Then irrational numbers quashed this thinking, their decimal representation just went on and on and on. The problem of “between” now had to be restated. If irrational numbers lay between whole numbers and rational fractions, how many numbers were there between, say, 0 and 1? More perplexing still, there seemed to be just as many numbers between 0 and 1 as there were between 1 and 1,000. How could that be? 34
    Dedekind’s solution was as simple as it was elegant. In mathematical terms, he wrote, “if one could choose one and only one number, a , which divided all the others in the interval into two classes, A and B , such that all numbers in A were less than a , all in B were greater than a , while a itself could be assigned to either class, then the interval was continuous by definition.” Dedekind had defined (the numerical meaning of) continuity by removing the concept of “between.”
    The concept of “between” borders on the philosophical, and brings to mind such concepts as “before” and “beyond,” which had troubled Kant. This, in a sense, is what interested Dedekind’s colleague, Georg Cantor. Just as Dedekind had studied with Gauss, so Cantor had been a pupil of Karl Weierstrass in Berlin at least until 1866. Born in 1845 into a devout Lutheran family, Cantor was very interested in metaphysics and believed that when he made his major discovery—infinite cardinal numbers—they had been revealed to him by God. 35 A manic-depressive, he ended his days in an asylum, but between 1872 and 1897 he created the theory of sets and the arithmetic of infinite numbers. 36
    The paper that started it was entitled “On the Consequences of a Theorem in