Double-slit experiment: a method for analyzing light by diffracting light beams through two slits and observing the resulting patterns on a screen
Exclusion principle: Pauli's discovery, expressed as a law, that no two particles occupy the same space at the same time
Fermions: see Bosons
Field: the extension of a physical quality throughout space (as the electromagnetic field); in classical or quantum physics, field theory describes the dynamics and effects within the field.
General Relativity: Einstein's generalization of his theory of special relativity to include gravity. It reconceived Newton by showing that apples fall to the ground because the earth's mass curves the adjacent space-time, forcing apples to move in a special way, that is, towards the surface of the earth. It has proven extremely difficult to unify general relativity with quantum mechanics. String theory is currently the best hope.
Gödel numbering: the assignment of numbers to mathematical symbols and formulas, to allow for mathematical statements about mathematics (that is, metamathematical statements)
Incompleteness theorems: Gödel's demonstrations that in any formal system, there are statements that are true, but are not provable using the axioms of that system. The paper containing the theorems was entitled “On Formally Undecidable Propositions of Principia Mathematica and Related Systems,” a reference to the work by Russell and Whitehead.
Induction: reasoning that begins with information from particular instances and leads to general propositions
Locality: in classical physics, the idea that two objects in separate places are independent and cannot interact, a restriction seemingly disproved by recent experiments demonstrating quantum particles to be “entangled” no matter what the distance between them
Logicism: an approach to philosophy in which mathematics is subsumed into logic
Matrix mechanics: the first complete definition of the laws and properties of subatomic particles using matrices to describe their properties
Maxwell's equations: four equations that describe electric and magnetic fields and their interaction with matter
Neutrino: an extremely light particle, the existence of which was hypothesized by Pauli
Paradox: in logic, reasoning that leads to contradiction, revealing false assumptions or faulty processes
Particle physics: the study of the basic subatomic elements and the forces acting upon and among them
Photoelectric effect: the emission of electrons when exposed to electromagnetic radiation and the subject of Einstein's first 1905 paper, in which he proposed that rather than waves, light was made of quanta (later called photons)
Photon: the elementary particle that forms light
Planck's constant: the ratio of a photon’s energy to its frequency
Quantum mechanics: a theory of subatomic systems that derives information about particles through an application of statistics, conceives of electrons and protons as both particle and wave in their behavior, and acknowledges uncertainty in measuring both movement and position simultaneously
Quantum physics: the entire body of modern, postclassical physics incorporating quantum mechanics and treating both large and small scale forces
Russell's paradox: a conundrum discovered by Russell in 1901. The problem occurs when, in attempting to account for and classify all sets, one imagines a “set of all sets that are not members of themselves.” A set can be a member of itself: Imagine the set of all objects that are not cars. Since the set of all noncars is not a car, it can be a member of itself. Now, we must move up a rung in abstraction, because set theory conceives of “sets of sets.” The set of all sets that are members of themselves is, indeed, possible: e.g., the set of all sets of cars can be a member of itself. But the set of all sets that are not members of themselves is a paradox. Is it a member of itself? It must be, since it is a set that includes just that: sets that are not members of themselves; and yet it cannot be, since by being a member of the set of all sets that are not members of themselves, it would become a member of itself.
Special relativity: Einstein's explanation of the relationships among light, energy, and matter, introducing the concept of space-time and defining the speed of light as the one constant not dependent on the observer
Spin: in quantum mechanics, a particle's angular momentum—either half-integer (fermions) or integer (bosons)
String theory: a theory (as yet unproven) that attempts to unify all of the known forces (weak, strong, electromagnetic, and gravitational). In this theory, matter is made not of particles that behave like waves, but of strings that vibrate.
Strong and weak nuclear forces: within the atom, the strong force holds neutrons and protons in the nucleus; the weak force allows beta decay, or radiation, to occur.
Ultraviolet catastrophe: a false prediction, based on classical physics, that when black-body radiation reaches equilibrium, it will emit infinite heat
Uncertainty principle: a cornerstone of quantum mechanics; Heisenberg's assertion that it is impossible to measure and determine both the position and the momentum of a particle simultaneously
Unified theory: a theory of everything, that is, all of the elemental forces (weak, strong, electromagnetic, and gravitational)
Wave mechanics: Erwin Schrödinger's explanation of quantum behavior, in which electrons move like waves around the nucleus
Wave/particle duality: in quantum mechanics, the idea that all objects exhibit the properties of both waves and particles
PART 1
THE PATHOS OF
SCIENCE
PRINCETON, WINTER 1943–44
IN PRINCETON, DURING THE COLD, wartime winter of 1943–44, four men—scientists and luminaries all, with common interests and uncommon theories—met once a week over the course of several months. These casual meetings took place far from the horrific battlefields of the World War and far from Los Alamos, the (then) secret lair of experimental atomic physicists.
They were extraordinary meetings, although they probably did not contribute to the advancement of the sciences. The four participants were uniquely matched within the adversarial and communal culture of a pure and disinterested science: Albert Einstein (at whose 112 Mercer Street house the men met); Bertrand Russell, the British logician, philosopher, and gadfly; Wolfgang Pauli, the boy wonder of quantum physics, who formulated the “exclusion principle” in 1925 and postulated the existence of the neutrino in 1930; and Kurt Gödel, whose “incompleteness” theory of 1931 shattered the link between logic and mathematics that Russell's monumental work Principia Mathematica had attempted to forge.
We know of these meetings only from passing remarks in Russell's Autobiography:
While in Princeton, I came to know Einstein fairly well. I used to go to his house once a week to discuss with him and Gödel and Pauli. These discussions were in some ways disappointing, for, although all three of them were Jews and exiles and, in intention, cosmopolitans, I found that they all had a German bias toward metaphysics, and in spite of our utmost endeavour we never arrived at common premises from which to argue.1
Eventually, the conversations seem to have sputtered out. Russell gives us no other details about what was said, and perhaps nothing worth reporting was said.
That such exemplars of our scientific age had occasion to chat in the cloistered world of Princeton's Institute for Advanced Study might seem fodder for yet another London stage play set in the world of physics—Heisenberg and Bohr had met in Copenhagen in 1941, and from that meeting, with its uncertainties and relative perspectives, can be seen, in retrospect, angled perspectives into the collision of particles that is war.
Indeed, it was war and Hitler that had brought all four to Princeton. Einstein, Pauli, and Gödel, having fled the chaos of Europe, found refuge at the Institute (Einstein in 1933, Pauli and Gödel in 1940). Russell was in temporary exile from England, yearning to return, but as yet unable, owing to travel restrictions. Solidly entrenched in the small-town atmo
sphere, the four men spent their days thinking, writing, and, periodically, lecturing, either within the Institute or elsewhere at professional meetings.
For years, Einstein had enjoyed world fame. Princeton was no different. Shop owners hoarded his signed checks, children pleaded for help with homework, strangers approached him on the street or in museums. He and his second wife, Elsa, had moved to the modest white house at 112 Mercer Street in 1935, just a year before Elsa's death. During Elsa's illness and after her death, Elsa's daughter, Margot, and Helen Dukas, Einstein's secretary, who had been part of the family since 1928, ran the household. They were joined by Maja, Einstein's gifted and beloved sister, in 1939. By 1943, then, Einstein's household consisted of himself and three exceedingly intelligent women. One of the three would surely have served tea to the guests.
The colonial-style house (which Einstein paid for by selling a manuscript to the Morgan Library) was filled with solid German furniture rescued from the couple's apartment in Berlin. Typically Biedermeier, a style associated with the pretentious nineteenth-century German bourgeoisie, the cumbersome, clumsy, outmoded furniture was relegated to the first floor at 112 Mercer—Elsa's domain. His own study, on the upper floor, was furnished in much plainer fashion, with tall bookshelves and a paper-strewn table.2 It was there that Einstein often entertained colleagues, and there, presumably, that the four men met.
Clearly, the meetings did not make history. But they certainly embodied it. Einstein's special and general theories of relativity had reshaped modern physics; Pauli's exclusion principle helped launch the revolution in quantum physics; Russell's early eminence as a logician resulted in the towering Principia Mathematica (written with Alfred Whitehead), which laid out the foundations of symbolic logic; Gödel's incompleteness theorem quashed any hope (including Russell's) of mathematics as a universal, consistent, and complete system.
A more illustrious scientific group of friends probably never gathered, at least not in such a relaxed and intimate setting as Einstein's study. After all, Einstein was Einstein, and Gödel was considered the most important logician since Aristotle. In terms of stature, their only equivalents would be Newton and Leibniz in the seventeenth century, two geniuses who never met. Of course, conferences and congresses were typical meeting places, with their rituals of podium, prepared papers, and hallway talk. But the conversations Russell alludes to would have had nothing in common with academic conferences.
It was an intimate little group. Gödel and Pauli were among Einstein's closest friends in Princeton. Einstein was fond of Russell, who, as the only nonmember of the Institute for Advanced Study, might have seemed a bit of an interloper. Later, Russell would develop a friendship of sorts with Wolfgang Pauli.
It was also a pot simmering with outsized personalities. Pauli, then aged forty-three, was famously intimidating, given his brilliance and rough sarcasm—and he looked the part: florid and barrel shaped, with strangely slanted eyes in a moonface. (George Gamow, the mathematician, once sketched Pauli as a plump devil, an allusion to his reputation as an intense critic, as well as to his devilish wit.) Gödel, aged thirty-seven, a mathematical Platonist happily married to a former cabaret dancer from Vienna, was a notorious recluse, shadowed by bad health and nervous breakdowns. He was thin and intense, his eyes shielded by horn-rim glasses with tinted lenses—he might be staring at you, but it was hard to tell. The seventy-two-year-old Russell, small and bony-featured, was famous for his lightning mind, malicious wit, and boundless energy; he was now the third Earl Russell and one of the world's leading philosophers, but he was also impoverished and had been in America since 1938 trying (vainly) to find a paying job. Then there was Einstein, his hair wild as ever, wearing a sweatshirt with a handy pen clipped to its neckline.
German accents mingled with upper-class British tones: Einstein talked very softly, Russell in a snapping high pitch, Pauli in a growl, and Gödel quietly and precisely. It would have been a smoke-filled room as well: Einstein, Russell, and Pauli were incessant pipe smokers.
One fascinating topic probably never came up. Two of the four scientists dabbled in the realm of the irrational. Pauli the physicist was a scourge of any idea that did not meet the most rigorous, objective standards of proof. But privately he subscribed to Carl Jung's psychology, with its archetypes and archaic myths. Indeed, he had been Jung's patient in the early 1930s, and Jung had mined his client's rich dream life in several lectures and articles, suppressing Pauli's name, of course. Among Jung's writings on Pauli's dreams is the chapter “Individual Dream Symbols in Relation to Alchemy,” which alludes to another of Pauli's extrarational interests: medieval alchemy. Pauli found in alchemy a model for how matter could be joined to spirit—a task he felt urgent for the modern mind. As for Gödel, the astringent logician was highly sympathetic to telepathy, reincarnation, and the existence of ghosts—not despite being a logician, but because of it: He thought such phenomena rationally justifiable. The young Gödel may have attended a séance; certainly he read widely on parapsychology throughout his life.
Had they been aired, Pauli's Jungian affinities and Gödel's excursions into the paranormal would have startled the two elder scientists. Einstein and Russell both adhered to more conventional metaphysical views. Einstein, though close to both Pauli and Gödel, seems to have been unaware of their otherworldly interests. Perhaps it is no wonder, as Russell complained, that the group could not find common premises, at least when it came to philosophizing about science.
AGING GENIUS
Besides tangled friendships and rivalries, the four men who met in Einstein's living room shared something else: the common fate of being past their prime. By 1944 each had already done his important scientific work. Einstein had already spent the last twenty years pursuing his hope of a grand unified theory for physics—a pursuit destined for failure. Russell's creative years in logic lay more than thirty years in the past; now, he was busily churning out popularizing books to earn badly needed money. (The History of Western Philosophy, which he was finishing up, surprised him by becoming a runaway best seller the following year.) Pauli's many achievements included his 1925 discovery of the exclusion principle, one of the great clarifying insights in physics, and his daring prediction in 1930 of the neutrino; but there were no more such triumphs ahead. Gödel's startling theorem of 1931 proved that mathematics must remain incomplete—the “most significant mathematical truth of this century,” as his honorary degree from Harvard later put it; but by 1944, he had shifted from logic to make a new start in philosophy.
The history of science is a long procession of figures, some famous, many forgotten, who come forward, work their special wonders, make their mark, lose the power of genius, and make their exit. That scientific careers peak early is, of course, a commonplace—some might say a myth. Yet it would seem that those exits are, on the whole, earlier than those of poets and artists. Aging genius has been a topic of much discussion among scientists and their anatomists. Like any myth, that of the coupling of youth and scientific discovery is both exaggerated and compelling. Indeed, those to whom scientific genius is ascribed have done as much as any to propel the myth. G. H. Hardy, a numbers theorist who continued to produce well beyond middle age, nevertheless wrote in his memoir, A Mathematician's Apology, “No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game.” And we have Russell's own account of his dogged work on the Principia to remind us that scientific creativity requires immense energy:
So I persisted, and in the end the work was finished, but my intellect never quite recovered from the strain. I have been ever since definitely less capable of dealing with difficult abstractions than I was before.3
Mathematical theorizing seems especially suited to the young, nor is theoretical physics a country for old men. Thus, we have John Nash at twenty-two (equilibrium and game theory); John von Neumann at twenty (a definition of ordinal numbers); Carl Friedrich Gauss at twenty-one (the fundamen
tal theorem of algebra); Evariste Galois at twenty-one (recognized posthumously for his algebraic theories); Alan Turing at twenty-four (the Turing machine). Arguably, the average age of discovery has risen in recent years. Jordan Ellenberg reminds us that modern mathematics is itself a mature field, requiring years of study well beyond that of an eighteenth or nineteenth or even early twentieth century prodigy.
Still, Gödel was only twenty-five when in 1931 he proved the “incompleteness” of mathematics. And the early days of quantum physics saw a procession of theorists remarkable for their youth as well as their genius: Wolfgang Pauli and Werner Heisenberg were both twenty-five, and Paul Dirac was twenty-four, when they published their landmark contributions. Newton's insights into gravity and optics came when he was twenty-three, “in the prime of my age for invention” or his “annus mirabilis.” Einstein published his special theory of relativity at the age of twenty-nine in his own annus mirabilis of 1905. In the last four centuries, only Newton and Einstein among major theorists were able to surpass their earliest work—Newton with his universal law of gravitation in 1686 and Einstein with his general theory of relativity in 1916.
But even Einstein's gifts finally failed. In the late 1920s, while in his late forties—an almost Methuselah-like age for topflight creative work in theoretical physics—Einstein began his exit.
Why such a “running down” of energy happened to him, or happens to other scientific theorists, is a puzzle, as mysterious as the initial outburst of genius that occurs early in such careers. Explanations range from the physiological (a decrease in testosterone, according to the psychologist Satoshi Kanazawa) to the sociological (math and physics reward brash, revolutionary discoveries, and thus brash, youthful thinkers) to the biohistorical (age statistics are affected by life expectancy or the relative “age” of the field).4 Despite a mountain of studies, we know very little about why such supreme gifts appear or disappear.
Einstein's Genius Club Page 2