Einstein said he went to his office at the Institute for Advanced Study “just to have the privilege of walking home with Kurt Gödel.” Photograph from Getty Images

In 1933, with his great scientific discoveries behind him, Albert Einstein came to America. He spent the last twenty-two years of his life in Princeton, New Jersey, where he had been recruited as the star member of the Institute for Advanced Study. Einstein was reasonably content with his new milieu, taking its pretensions in stride. “Princeton is a wonderful piece of earth, and at the same time an exceedingly amusing ceremonial backwater of tiny spindle-shanked demigods,” he observed. His daily routine began with a leisurely walk from his house, at 115 Mercer Street, to his office at the institute. He was by then one of the most famous and, with his distinctive appearance—the whirl of pillow-combed hair, the baggy pants held up by suspenders—most recognizable people in the world.

A decade after arriving in Princeton, Einstein acquired a walking companion, a much younger man who, next to the rumpled Einstein, cut a dapper figure in a white linen suit and matching fedora. The two would talk animatedly in German on their morning amble to the institute and again, later in the day, on their way homeward. The man in the suit may not have been recognized by many townspeople, but Einstein addressed him as a peer, someone who, like him, had single-handedly launched a conceptual revolution. If Einstein had upended our everyday notions about the physical world with his theory of relativity, the younger man, Kurt Gödel, had had a similarly subversive effect on our understanding of the abstract world of mathematics.

Gödel, who has often been called the greatest logician since Aristotle, was a strange and ultimately tragic man. Whereas Einstein was gregarious and full of laughter, Gödel was solemn, solitary, and pessimistic. Einstein, a passionate amateur violinist, loved Beethoven and Mozart. Gödel’s taste ran in another direction: his favorite movie was Walt Disney’s “Snow White and the Seven Dwarfs,” and when his wife put a pink flamingo in their front yard he pronounced it furchtbar herzig—“awfully charming.” Einstein freely indulged his appetite for heavy German cooking; Gödel subsisted on a valetudinarian’s diet of butter, baby food, and laxatives. Although Einstein’s private life was not without its complications, outwardly he was jolly and at home in the world. Gödel, by contrast, had a tendency toward paranoia. He believed in ghosts; he had a morbid dread of being poisoned by refrigerator gases; he refused to go out when certain distinguished mathematicians were in town, apparently out of concern that they might try to kill him. “Every chaos is a wrong appearance,” he insisted—the paranoiac’s first axiom.

Although other members of the institute found the gloomy logician baffling and unapproachable, Einstein told people that he went to his office “just to have the privilege of walking home with Kurt Gödel.” Part of the reason, it seems, was that Gödel was undaunted by Einstein’s reputation and did not hesitate to challenge his ideas. As another member of the institute, the physicist Freeman Dyson, observed, “Gödel was . . . the only one of our colleagues who walked and talked on equal terms with Einstein.” But if Einstein and Gödel seemed to exist on a higher plane than the rest of humanity, it was also true that they had become, in Einstein’s words, “museum pieces.” Einstein never accepted the quantum theory of Niels Bohr and Werner Heisenberg. Gödel believed that mathematical abstractions were every bit as real as tables and chairs, a view that philosophers had come to regard as laughably naïve. Both Gödel and Einstein insisted that the world is independent of our minds, yet rationally organized and open to human understanding. United by a shared sense of intellectual isolation, they found solace in their companionship. “They didn’t want to speak to anybody else,” another member of the institute said. “They only wanted to speak to each other.”

People wondered what they spoke about. Politics was presumably one theme. (Einstein, who supported Adlai Stevenson, was exasperated when Gödel chose to vote for Dwight Eisenhower in 1952.) Physics was no doubt another. Gödel was well versed in the subject; he shared Einstein’s mistrust of the quantum theory, but he was also skeptical of the older physicist’s ambition to supersede it with a “unified field theory” that would encompass all known forces in a deterministic framework. Both were attracted to problems that were, in Einstein’s words, of “genuine importance,” problems pertaining to the most basic elements of reality. Gödel was especially preoccupied by the nature of time, which, he told a friend, was the philosophical question. How could such a “mysterious and seemingly self-contradictory” thing, he wondered, “form the basis of the world’s and our own existence”? That was a matter in which Einstein had shown some expertise.

A century ago, in 1905, Einstein proved that time, as it had been understood by scientist and layman alike, was a fiction. And this was scarcely his only achievement that year, which John S. Rigden skillfully chronicles, month by month, in “Einstein 1905: The Standard of Greatness” (Harvard; $21.95). As it began, Einstein, twenty-five years old, was employed as an inspector in a patent office in Bern, Switzerland. Having earlier failed to get his doctorate in physics, he had temporarily given up on the idea of an academic career, telling a friend that “the whole comedy has become boring.” He had recently read a book by Henri Poincaré, a French mathematician of enormous reputation, which identified three fundamental unsolved problems in science. The first concerned the “photoelectric effect”: how did ultraviolet light knock electrons off the surface of a piece of metal? The second concerned “Brownian motion”: why did pollen particles suspended in water move about in a random zigzag pattern? The third concerned the “luminiferous ether” that was supposed to fill all of space and serve as the medium through which light waves moved, the way sound waves move through air, or ocean waves through water: why had experiments failed to detect the earth’s motion through this ether?

Each of these problems had the potential to reveal what Einstein held to be the underlying simplicity of nature. Working alone, apart from the scientific community, the unknown junior clerk rapidly managed to dispatch all three. His solutions were presented in four papers, written in the months of March, April, May, and June of 1905. In his March paper, on the photoelectric effect, he deduced that light came in discrete particles, which were later dubbed “photons.” In his April and May papers, he established once and for all the reality of atoms, giving a theoretical estimate of their size and showing how their bumping around caused Brownian motion. In his June paper, on the ether problem, he unveiled his theory of relativity. Then, as a sort of encore, he published a three-page note in September containing the most famous equation of all time: E = mc2.

All of these papers had a touch of magic about them, and upset deeply held convictions in the physics community. Yet, for scope and audacity, Einstein’s June paper stood out. In thirty succinct pages, he completely rewrote the laws of physics, beginning with two stark principles. First, the laws of physics are absolute: the same laws must be valid for all observers. Second, the speed of light is absolute; it, too, is the same for all observers. The second principle, though less obvious, had the same sort of logic to recommend it. Since light is an electromagnetic wave (this had been known since the nineteenth century), its speed is fixed by the laws of electromagnetism; those laws ought to be the same for all observers; and therefore everyone should see light moving at the same speed, regardless of the frame of reference. Still, it was bold of Einstein to embrace the light principle, for its consequences seemed downright absurd.