When Einstein Walked with Gödel: Excursions to the Edge of Thought
By Jim Holt
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About this ebook
From Jim Holt, the New York Times bestselling author of Why Does the World Exist?, comes an entertaining and accessible guide to the most profound scientific and mathematical ideas of recent centuries in When Einstein Walked with Gödel: Excursions to the Edge of Thought.
Does time exist? What is infinity? Why do mirrors reverse left and right but not up and down? In this scintillating collection, Holt explores the human mind, the cosmos, and the thinkers who’ve tried to encompass the latter with the former. With his trademark clarity and humor, Holt probes the mysteries of quantum mechanics, the quest for the foundations of mathematics, and the nature of logic and truth.
Along the way, he offers intimate biographical sketches of celebrated and neglected thinkers, from the physicist Emmy Noether to the computing pioneer Alan Turing and the discoverer of fractals, Benoit Mandelbrot. Holt offers a painless and playful introduction to many of our most beautiful but least understood ideas, from Einsteinian relativity to string theory, and also invites us to consider why the greatest logician of the twentieth century believed the U.S. Constitution contained a terrible contradiction—and whether the universe truly has a future.
Jim Holt
Jim Holt is a prominent essayist and critic on philosophy, mathematics, and science. He is a frequent contributor to the New York Times Book Review, New York Review of Books and Prospect magazine. He lives in New York City. He is the author of Stop Me If You've Heard This [9781846681097] also published by Profile.
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Reviews for When Einstein Walked with Gödel
40 ratings4 reviews
- Rating: 4 out of 5 stars4/5Thoughtful and provocative essays about math and science and philosophy- and also very much about mathematicians and scientists and philosophers. A bit of redundancy because the essays were written for a variety of publications and a few bits were recycled. Very well written.
- Rating: 3 out of 5 stars3/5The format necessarily creates a lot of repetitions and doesn't allow for any in-depth explanations of any but the most superficial elements of the science discussed but it contains a lot of interesting observations and historical notes. It also questions some common assumptions showing alternate views and is probably the first book I ever read that doesn't propagate the now tiresome myth of Ada Lovelace.
- Rating: 4 out of 5 stars4/5Since I first read it a couple of years ago, I've returned again and again to Jim Holt's excellent "Why Does the World Exist?" More than I have than any other book. His off-the-cuff conversations with a gaggle of famous philosophers, cosmologists, and astrophysicists let the lay reader grapple with enormously complex ideas related to a single world-class conundrum, and the commentaries that followed each chapter really highlighted how good the author is at making awesome, fantastically complex philosophical ideas accessible to non-PhD candidates.
"When Einstein Walked with Gödel isn't quite that good, but it's still a worthwhile read. Holt's ability to make mind-stretching ideas accessible is still evident on every page, but this one is basically a collections of essays, mostly short, that Holt's written over the years. There's a lot of interesting stuff here, of course. We hear about different views of infinity, whether Einstein thought time actually existed, string theory, and people who equate numbers to platonic forms. Holt takes the troubled, alluring Ada Lovelace down a peg and takes a potshot at Richard Dawkins. He considers the ultimate fate of the universe. Amazingly enough, I think that I sort of understand most of what's going on here, and I've taken a college-level science course. Holt seems to have an almost preternatural ability to make big ideas accessible to the layperson. And that's a rare and valuable thing.
At the end of the day, though, I missed the aforementioned book's laser focus and found myself wishing that most of these chapters were longer. But who knows? I plan on reading more Holt in the future, of course. And "Why Does the World Exist?" seems to have gotten much more traction than you'd figure a book with that title would, so I'm hoping he'll have something new out soon. This book doesn't quite live up to it's snazzy title, and even its best moments are spoonful-sized, but you can still consider this review a recommendation. - Rating: 3 out of 5 stars3/5My contribution to Holt’s Edge of Thoughts in the form of an article too:
An unauthorised and short version of physics.
How did scientists first deduce that the universe had hidden dimensions, dimensions that are curled up so tight we can't see them? Until recently SOCK THEORY was the ruling paradigm. It was thought that Theodor Kaluza and Oscar Klein deduced the existence of at least one additional dimension from well known tendency of socks to disappear and then re-appear in unlikely places. How else to explain the mysterious behaviour of hosiery? Latterly a new paradigm, STRING THEORY, has superseded sock theory. Leave a length of string or anything long, thin and flexible lying undisturbed for even a day and you will find it has somehow got itself tied into knots. This can only be explained if we assume at least one additional dimension. String theory also gave birth to QUANTUM FIELD THEORY. Richard Feynman found that if you stored several discreet pieces of string in a cupboard for an hour or so they would become inextricably entangled. Feynman realised that given half a chance everything would get ENTANGLED with everything else. String theory also gave rise to superstring theory which in turn morphed into the theory of branes. If you've read this far you're probably a P-BRANE.
Quantum theory is flawed and quantum proponents are in denial. String theory is in crisis (it has recently been described as dangerous nonsense). The Copenhagen interpretation is under attack (by recent experiments - even though this not allowed by Copenhagen). Neutrinos don't actually exist (did your Neutrino lose it's flavour on the bedpost overnight?).
LOOP QUANTUM GRAVITY RULES! Determinism rules OK!
NB (not part of the article above): The references to one of Gödel's Incompleteness theorems in Holt’s first article suggest a slight misunderstanding of the meaning of Godel's work. What Gödel showed was not that the axioms of mathematics must be taken on faith (this insight is much older and relatively harmless) but something more subtle. Gödel showed that in any reasonably powerful mathematics, there must be perfectly legal statements which cannot be proven within the framework of that system, but will require additional axioms to plug the gap. And that this more powerful system will in turn necessarily include legal statements that cannot be demonstrated to be true or false without resort to still more new axioms, and so on... In other words, that no systems of mathematics along the lines of the Principia of Whitehead and Russell can ever be self contained. Yet another way of saying this is that the mathematical backbone of thought is a convention or a construct, not a pure, freestanding Platonic ideal. This was a startling insight, because prior to this discovery it had been assumed by all who are equipped to assume such things (e.g., Hilbert, Russell, etc) that a proof of the completeness of mathematics would be positive, not negative. Mathematicians are Platonists in their souls - it's profoundly disturbing to find out that the universe is not Platonic. Turing's and Church's related insight (the discovery of well-formed problems which no computer program can solve) was even more unsettling and of far greater practical significance. (The strategy of reducing a new problem to the halting problem, and thereby demonstrating it to be unsolvable is routine even for undergraduates, and applies to a universe of problems that come up frequently in practical applications, which undecidability does not.) Godel's theorem is simply the formal-logic manifestation of the same drubbing that Einstein, Plank, Heisenberg, Turing, Darwin, Freud, Wittgenstein, Lyell, et al, gave in other fields to our formerly rather poetical understanding of the nature of knowing.
Gödel's incompleteness proof shows that axioms, formulated in the artificial language of Peano arithmetic (the five Peano's axioms that is), could not be reducible to logic. They required supplementing with other branches of mathematics such as set theory. Effectively that requirements for completeness and consistences in any logical system were violated - hence the need for supplementing logic with other constructs.
Book preview
When Einstein Walked with Gödel - Jim Holt
Preface
These essays were written over the last two decades. I selected them with three considerations in mind.
First, the depth, power, and sheer beauty of the ideas they convey. Einstein’s theory of relativity (both special and general), quantum mechanics, group theory, infinity and the infinitesimal, Turing’s theory of computability and the decision problem,
Gödel’s incompleteness theorems, prime numbers and the Riemann zeta conjecture, category theory, topology, higher dimensions, fractals, statistical regression and the bell curve,
the theory of truth—these are among the most thrilling (and humbling) intellectual achievements I’ve encountered in my life. All are explained in the course of these essays. My ideal is the cocktail-party chat: getting across a profound idea in a brisk and amusing way to an interested friend by stripping it down to its essence (perhaps with a few swift pencil strokes on a napkin). The goal is to enlighten the newcomer while providing a novel twist that will please the expert. And never to bore.
My second consideration is the human factor. All these ideas come with flesh-and-blood progenitors who led highly dramatic lives. Often these lives contain an element of absurdity. The creator of modern statistics (and originator of the phrase nature versus nurture
), Sir Francis Galton, was a Victorian prig who had comical misadventures in the African bush. A central figure in the history of the four-color theorem
was a flamboyantly eccentric mathematician/classicist called Percy Heawood—or Pussy
Heawood by his friends, because of his feline whiskers.
More often the life has a tragic arc. The originator of group theory, Évariste Galois, was killed in a duel before he reached his twenty-first birthday. The most revolutionary mathematician of the last half century, Alexander Grothendieck, ended his turbulent days as a delusional hermit in the Pyrenees. The creator of the theory of infinity, Georg Cantor, was a kabbalistic mystic who died in an insane asylum. Ada Lovelace, the cult goddess of cyber feminism (and namesake of the programming language used by the U.S. Department of Defense), was plagued by nervous crises brought on by her obsession with atoning for the incestuous excesses of her father, Lord Byron. The great Russian masters of infinity, Dmitri Egorov and Pavel Florensky, were denounced for their antimaterialist spiritualism and murdered in Stalin’s Gulag. Kurt Gödel, the greatest of all modern logicians, starved himself to death out of the paranoiac belief that there was a universal conspiracy to poison him. David Foster Wallace (whose attempt to grapple with the subject of infinity I examine) hanged himself. And Alan Turing—who conceived of the computer, solved the greatest logic problem of his time, and saved countless lives by cracking the Nazi Enigma
code—took his own life, for reasons that remain mysterious, by biting into a cyanide-laced apple.
My third consideration in bringing these essays together is a philosophical one. The ideas they present all bear crucially on our most general conception of the world (metaphysics), on how we come to attain and justify our knowledge (epistemology), and even on how we conduct our lives (ethics).
Start with metaphysics. The idea of the infinitely small—the infinitesimal—raises the question of whether reality is more like a barrel of molasses (continuous) or a heap of sand (discrete). Einstein’s relativity theory either challenges our notion of time or—if Gödel’s ingenious reasoning is to be credited—abolishes it altogether. Quantum entanglement calls the reality of space into question, raising the possibility that we live in a holistic
universe. Turing’s theory of computability forces us to rethink how mind and consciousness arise from matter.
Then there’s epistemology. Most great mathematicians claim insight into an eternal realm of abstract forms transcending the ordinary world we live in. How do they interact with this supposed Platonic
world to obtain mathematical knowledge? Or could it be that they are radically mistaken—that mathematics, for all its power and utility, ultimately amounts to a mere tautology, like the proposition A brown cow is a cow
? To make this issue vivid, I approach it in a novel way, by considering what is universally acknowledged to be the greatest unsolved problem in mathematics: the Riemann zeta conjecture.
Physicists, too, are prone to a romantic image of how they arrive at knowledge. When they don’t have hard experimental/observational evidence to go on, they rely on their aesthetic intuition—on what the Nobel laureate Steven Weinberg unblushingly calls their sense of beauty.
The beauty = truth
equation has served physicists well for much of the last century. But—as I ask in my essay The String Theory Wars
—has it recently been leading them astray?
Finally, ethics. These essays touch on the conduct of life in many ways. The eugenic programs in Europe and the United States ushered in by the theoretical speculation of Sir Francis Galton cruelly illustrate how science can pervert ethics. The ongoing transformation of our habits of life by the computer should move us to think hard about the nature of happiness and creative fulfillment (as I do in Smarter, Happier, More Productive
). And the omnipresence of suffering in the world should make us wonder what limits there are, if any, to the demands that morality imposes upon us (as I do in On Moral Sainthood
).
The last essay in the volume, Say Anything,
begins by examining Harry Frankfurt’s famous characterization of the bullshitter as one who is not hostile to the truth but indifferent to it. It then enlarges the picture by considering how philosophers have talked about truth—erroneously?—as a correspondence
between language and the world. In a slightly ludic way, this essay bridges the fields of metaphysics, epistemology, and ethics, lending the volume a unity that I hope is not wholly specious.
And lest I be accused of inconsistency, let me (overconfidently?) express the conviction that the Copernican principle,
Gödel’s incompleteness theorems,
Heisenberg’s uncertainty principle,
Newcomb’s problem,
and the Monty Hall problem
are all exceptions to Stigler’s law of eponymy (vide p. 292).
J.H.
New York City, 2017
PART I
The Moving Image of Eternity
1
When Einstein Walked with Gödel
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 112 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 might 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 naive. 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 D. 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.
Decades before, 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. 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, that 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 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 these papers had a touch of magic about them and upset some 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. He began 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. Because 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 their frame of reference. Still, it was bold of Einstein to embrace the light principle, for its consequences seemed downright absurd.
Suppose—to make things vivid—that the speed of light is a hundred miles an hour. Now suppose I am standing by the side of the road and I see a light beam pass by at this speed. Then I see you chasing after it in a car at sixty miles an hour. To me, it appears that the light beam is outpacing you by forty miles an hour. But you, from inside your car, must see the beam escaping you at a hundred miles an hour, just as you would if you were standing still: that is what the light principle demands. What if you gun your engine and speed up to ninety-nine miles an hour? Now I see the beam of light outpacing you by just one mile an hour. Yet to you, inside the car, the beam is still racing ahead at a hundred miles an hour, despite your increased speed. How can this be? Speed, of course, equals distance divided by time. Evidently, the faster you go in your car, the shorter your ruler must become and the slower your clock must tick relative to mine; that is the only way we can continue to agree on the speed of light. (If I were to pull out a pair of binoculars and look at your speeding car, I would actually see its length contracted and you moving in slow motion inside.) So Einstein set about recasting the laws of physics accordingly. To make these laws absolute, he made distance and time relative.
It was the sacrifice of absolute time that was most stunning. Isaac Newton believed that time was objective, universal, and transcendent of all natural phenomena; the flowing of absolute time is not liable to any change,
he declared at the beginning of his Principia. Einstein, however, realized that our idea of time is something we abstract from our experience with rhythmic phenomena: heartbeats, planetary rotations and revolutions, the ticking of clocks. Time judgments always come down to judgments of simultaneity. If, for instance, I say, ‘That train arrives here at 7 o’clock,’ I mean something like this: ‘The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events,’
Einstein wrote in the June paper. If the events in question are at some distance from each other, judgments of simultaneity can be made only by sending light signals back and forth. Working from his two basic principles, Einstein proved that whether an observer deems two events to be happening at the same time
depends on his state of motion. In other words, there is no universal now. With different observers slicing up the timescape into past,
present,
and future
in different ways, it seems to follow that all moments coexist with equal reality.
Einstein’s conclusions were the product of pure thought, proceeding from the most austere assumptions about nature. In the more than a century since he derived them, they have been precisely confirmed by experiment after experiment. Yet his June 1905 paper on relativity was rejected when he submitted it as a dissertation. (He then submitted his April paper, on the size of atoms, which he thought would be less likely to startle the examiners; they accepted it only after he added one sentence to meet the length threshold.) When Einstein was awarded the 1921 Nobel Prize in Physics, it was for his work on the photoelectric effect. The Swedish Academy forbade him to make any mention of relativity in his acceptance speech. As it happened, Einstein was unable to attend the ceremony in Stockholm. He gave his Nobel lecture in Gothenburg, with King Gustav V seated in the front row. The king wanted to learn about relativity, and Einstein obliged him.
* * *
In 1906, the year after Einstein’s annus mirabilis, Kurt Gödel was born in the city of Brno (now in the Czech Republic). Kurt was both an inquisitive child—his parents and brother gave him the nickname der Herr Warum, Mr. Why?
—and a nervous one. At the age of five, he seems to have suffered a mild anxiety neurosis. At eight, he had a terrifying bout of rheumatic fever, which left him with the lifelong conviction that his heart had been fatally damaged.
Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians. In the philosophical world of 1920s Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like 2 + 2 = 4
true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.
Gödel was introduced into the Vienna Circle by one of his professors, but he kept quiet about his Platonist views. Being both rigorous and averse to controversy, he did not like to argue his convictions unless he had an airtight way of demonstrating that they were valid. But how could one demonstrate that mathematics could not be reduced to the artifices of logic? Gödel’s strategy—one of preternatural cleverness and, in the words of the philosopher Rebecca Goldstein, heart-stopping beauty
—was to use logic against itself. Beginning with a logical system for mathematics, a system presumed to be free from contradictions, he invented an ingenious scheme that allowed the formulas in it to engage in a sort of doublespeak. A formula that said something about numbers could also, in this scheme, be interpreted as saying something about other formulas and how they were logically related to one another. In fact, as Gödel showed, a numerical formula could even be made to say something about itself. Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, I am not provable.
At first, this looks like a paradox, recalling as it does the proverbial Cretan who announces, All Cretans are liars.
But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying when it asserts, I am not provable
? No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete, because there is at least one true proposition about numbers (the one that says I am not provable
) that cannot be proved within it. The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem.
Wittgenstein once averred that there can never be surprises in logic.
But Gödel’s incompleteness theorems did come as a surprise. In fact, when the fledgling logician presented them at a conference in the German city of Königsberg in 1930, almost no one was able to make any sense of them. What could it mean to say that a mathematical proposition was true if there was no possibility of proving it? The very idea seemed absurd. Even the once great logician Bertrand Russell was baffled; he seems to have been under the misapprehension that Gödel had detected an inconsistency in mathematics. Are we to think that 2 + 2 is not 4, but 4.001?
Russell asked decades later in dismay, adding that he was glad [he] was no longer working at mathematical logic.
As the significance of Gödel’s theorems began to sink in, words like debacle,
catastrophe,
and nightmare
were bandied about. It had been an article of faith that armed with logic, mathematicians could in principle resolve any conundrum at all—that in mathematics, as it had been famously declared, there was no ignorabimus. Gödel’s theorems seemed to have shattered this ideal of complete knowledge.
That was not the way Gödel saw it. He believed he had shown that mathematics has a robust reality that transcends any system of logic. But logic, he was convinced, is not the only route to knowledge of this reality; we also have something like an extrasensory perception of it, which he called mathematical intuition.
It is this faculty of intuition that allows us to see, for example, that the formula saying I am not provable
must be true, even though it defies proof within the system where it lives. Some thinkers (like the physicist Roger Penrose) have taken this theme further, maintaining that Gödel’s incompleteness theorems have profound implications for the nature of the human mind. Our mental powers, it is argued, must outstrip those of any computer, because a computer is just a logical system running on hardware and our minds can arrive at truths that are beyond the reach of a logical system.
Gödel was twenty-four when he proved his incompleteness theorems (a bit younger than Einstein was when he created relativity theory). At the time, much to the disapproval of his strict Lutheran parents, he was courting an older Catholic divorcée by the name of Adele, who, to top things off, was employed as a dancer in a Viennese nightclub called Der Nachtfalter (the Moth). The political situation in Austria was becoming ever more chaotic with Hitler’s rise to power in Germany, although Gödel seems scarcely to have noticed. In 1936, the Vienna Circle dissolved after its founder was assassinated by a deranged student. Two years later came the Anschluss. The perilousness of the times was finally borne in upon Gödel when a band of Nazi youths roughed him up and knocked off his glasses, before retreating under the umbrella blows of Adele. He resolved to leave for Princeton, where he had been offered a position by the Institute for Advanced Study. But, the war having broken out, he judged it too risky to cross the Atlantic. So the now married couple took the long way around, traversing Russia, the Pacific, and the United States and finally arriving in Princeton in early 1940. At the institute, Gödel was given an office almost directly above Einstein’s. For the rest of his life, he rarely left Princeton, which he came to find ten times more congenial
than his once beloved Vienna.
Although Gödel was still little known in the world at large, he had a godlike status among the cognoscenti. "There it was, inconceivably, K. Goedel, listed just like any other name in the bright orange Princeton community phonebook," writes Rebecca Goldstein, who came to Princeton University as a graduate student of philosophy in the early 1970s, in her intellectual biography Incompleteness: The Proof and Paradox of Kurt Gödel (2005). "It was like opening up the local phonebook and finding B. Spinoza or I. Newton. She goes on to recount how she
once found the philosopher Richard Rorty standing in a bit of a daze in Davidson’s food market. He told me in hushed tones that he’d just seen Gödel in the frozen food aisle."
So naive and otherworldly was the great logician that Einstein felt obliged to help look after the practical aspects of his life. One much-retailed story concerns Gödel’s decision after the war to become an American citizen. Gödel took the matter of citizenship with great solemnity, preparing for the exam by making a close study of the U.S. Constitution. On the appointed day, Einstein accompanied him to the courthouse in Trenton and had to intervene to quiet Gödel down when the agitated logician began explaining to the judge how the U.S. Constitution contained a loophole that would allow a dictatorship to come into existence.¹
Around the same time that Gödel was studying the Constitution, he was also taking a close look at Einstein’s relativity theory. The key principle of relativity is that the laws of physics should be the same for all observers. When Einstein first formulated the principle in his revolutionary 1905 paper, he restricted all observers
to those who were moving uniformly relative to one another—that is, in a straight line and at a constant speed. But he soon realized that this restriction was arbitrary. If the laws of physics were to provide a truly objective description of nature, they ought to be valid for observers moving in any way relative to one another—spinning, accelerating, spiraling, whatever. It was thus that Einstein made the transition from his special
theory of relativity of 1905 to his general
theory, whose equations he worked out over the next decade and published in 1916. What made those equations so powerful was that they explained gravity, the force that governs the overall shape of the cosmos.
Decades later, Gödel, walking with Einstein, had the privilege of picking up the subtleties of relativity theory from the master himself. Einstein had shown that the flow of time depended on motion and gravity and that the division of events into past
and future
was relative. Gödel took a more radical view: he believed that time, as it was intuitively understood, did not exist at all. As usual, he was not content with a mere verbal argument. Philosophers ranging from Parmenides, in ancient times, to Immanuel Kant, in the eighteenth century, and on to J.M.E. McTaggart, at the beginning of the twentieth century, had produced such arguments, inconclusively. Gödel wanted a proof that had the rigor and certainty of mathematics. And he saw just what he wanted lurking within relativity theory. He presented his argument to Einstein for his seventieth birthday, in 1949, along with an etching. (Gödel’s wife had knit Einstein a sweater, but she decided not to send it.)
What Gödel found was the possibility of a hitherto unimaginable kind of universe. The equations of general relativity can be solved in a variety of ways. Each solution is, in effect, a model of how the universe might be. Einstein, who believed on philosophical grounds that the universe was eternal and unchanging, had tinkered with his equations so that they would yield such a model—a move he later called my greatest blunder.
Another physicist (a Jesuit priest, as it happens) found a solution corresponding to an expanding universe born at some moment in the finite past. Because this solution, which has come to be known as the big bang model, was consistent with what astronomers observed, it seemed to be the one that described the actual cosmos.
But Gödel came up with a third kind of solution to Einstein’s equations, one in which the universe was not expanding but rotating. (The centrifugal force arising from the rotation was what kept everything from collapsing under the force of gravity.) An observer in this universe would see all the galaxies slowly spinning around him; he would know it was the universe doing the spinning and not himself, because he would feel no dizziness. What makes this rotating universe truly weird, Gödel showed, is the way its geometry mixes up space and time. By completing a sufficiently long round trip in a rocket ship, a resident of Gödel’s universe could travel back to any point in his own past.
Einstein was not entirely pleased with the news that his equations permitted something as Alice in Wonderland–like as spatial paths that looped backward in time; in fact, he confessed to being disturbed
by Gödel’s universe. Other physicists marveled that time travel, previously the stuff of science fiction, was apparently consistent with the laws of physics. (Then they started worrying about what would happen if you went back to a time before you were born and killed your own grandfather.) Gödel himself drew a different moral. If time travel is possible, he submitted, then time itself is impossible. A past that can be revisited has not really passed. And the fact that the actual universe is expanding, rather than rotating, is irrelevant. Time, like God, is either necessary or nothing; if it disappears in one possible universe, it is undermined in every possible universe, including our own.
Gödel’s strange cosmological gift was received by Einstein at a bleak time in his life. Einstein’s quest for a unified theory of physics was proving fruitless, and his opposition to quantum theory alienated him from the mainstream of physics. Family life provided little consolation. His two marriages had been failures; a daughter born out of wedlock seems to have disappeared from history; of his two sons, one was schizophrenic, the other estranged. Einstein’s circle of friends had shrunk to Gödel and a few others. One of them was Queen Elisabeth of Belgium, to whom he confided, in March 1955, that the exaggerated esteem in which my lifework is held makes me very ill at ease. I feel compelled to think of myself as an involuntary swindler.
He died a month later, at the age of seventy-six. When Gödel and another colleague went to Einstein’s office at the institute to deal with his papers, they found the blackboard covered with dead-end equations.
After Einstein’s death, Gödel became ever more withdrawn. He preferred to conduct all conversations by telephone, even if his interlocutor was a few feet distant. When he especially wanted to avoid someone, he would schedule a rendezvous at a precise time and place and then make sure he was somewhere far away. The honors the world wished to bestow upon him made him chary. He had shown up to collect an honorary doctorate in 1953 from Harvard, where his incompleteness theorems were hailed as the most important mathematical discovery of the previous hundred years, but he later complained of being thrust quite undeservedly into the most highly bellicose company
of John Foster Dulles, a co-honoree. When he was awarded the National Medal of Science in 1974, he refused to go to Washington to meet Gerald Ford at the White House, despite the offer of a chauffeur for him and his wife. He had hallucinatory episodes and talked darkly of certain forces at work in the world directly submerging the good.
Fearing that there was a plot to poison him, he persistently refused to eat. Finally, looking like (in the words of a friend) a living corpse,
he was taken to the Princeton Hospital. There, two weeks later, on January 14, 1978, he succumbed to self-starvation. According to his death certificate, the cause of death was malnutrition and inanition
brought on by personality disturbance.
A certain futility marked the last years of both Gödel and Einstein. What might have been most futile, however, was their willed belief in the unreality of time. The temptation was understandable. If time is merely in our minds, perhaps we can hope to escape it into a timeless eternity. Then we could say, like William Blake, I see the Past, Present, and Future existing all at once / Before me.
In Gödel’s case, it might have been his childhood terror of a fatally damaged heart that attracted him to the idea of a timeless universe. Toward the end of his life, he told one confidant that he had long awaited an epiphany that would enable him to see the world in a new light but that it never came.
Einstein, too, was unable to make a clean break with time. To those of us who believe in physics,
he wrote to the widow of a friend who had recently died, this separation between past, present, and future is only an illusion, if a stubborn one.
When his own turn came, a couple of weeks later, he said, It is time to go.
2
Time—the Grand Illusion?
Isaac Newton had a peculiar notion of time. He saw it as a sort of cosmic grandfather clock, one that hovered over the rest of nature in blithe autonomy. And he believed that time advanced at a smooth and constant rate from past to future. Absolute, true, mathematical time, of itself, and from its own nature, flows equably without relation to anything external,
Newton declared at the beginning of his Principia. To those caught up in the temporal flux of daily life, this seems like arrant nonsense. Time does not strike us as transcendent and mathematical; rather, it is something intimate and subjective. Nor does it proceed at a stately and unvarying pace. We know that time has different tempos. In the run-up to New Year’s Eve, for instance, time positively flies. Then, in January and February, it slows to a miserable crawl. Moreover, time moves faster for some of us than for others. Old people are being rushed forward into the future at a cruelly rapid clip. When you’re an adult, as Fran Lebowitz once observed, Christmas seems to come every five minutes. For little children, however, time goes quite slowly. Owing to the endless novelty of a child’s experience, a single summer can stretch out into an eternity. It has been estimated that by the age of eight, one has subjectively lived two-thirds of one’s life.
Researchers have tried to measure the subjective flow of time by asking people of different ages to estimate when a certain amount of time has gone by. People in their early twenties tend to be quite accurate in judging when three minutes had elapsed, typically being off by no more than three seconds. Those in their sixties, by contrast, overshot the mark by forty seconds; in other words, what was actually three minutes and forty seconds seemed like only three minutes to them. Seniors are internally slow tickers, so for them actual clocks seem to tick too fast. This can have its advantages: at a John Cage concert, it is the old people who are relieved that the composition 4´33˝ is over so soon.
The river of time may have its rapids and its calmer stretches, but one thing would seem to be certain: it carries all of us, willy-nilly, in its flow. Irresistibly, irreversibly, we are being borne toward our deaths at the stark rate of one second per second. As the past slips out of existence behind us, the future, once unknown and mysterious, assumes its banal reality before us as it yields to the ever-hurrying now.
But this sense of flow is a monstrous illusion—so says contemporary physics. And Newton was as much a victim of this illusion as the rest of us are.
It was Albert Einstein who initiated the revolution in our understanding of time. In 1905, Einstein showed that time, as it had been understood by physicist and plain man alike, was a fiction. Einstein proved that whether an observer deems two events at different locations to be happening at the same time
depends on his state of motion. Suppose, for example, that Jones is walking uptown on Fifth Avenue and Smith is walking downtown. Their relative motion results in a discrepancy of several days in what they would judge to be happening now
in the Andromeda galaxy at the moment they pass each other on the sidewalk. For Smith, the space fleet launched to destroy life on earth is already on its way; for Jones, the Andromedan council of tyrants has not even decided whether to send the