Skip to comments.Gödel and Einstein: Friendship and Relativity
Posted on 12/21/2004 7:47:50 PM PST by snarks_when_bored
In the summer of 1942, while German U-boats roamed in wolf packs off the coast of Maine, residents in the small coastal town of Blue Hill were alarmed by the sight of a solitary figure, hands clasped behind his back, hunched over like a comma with his eyes fixed on the ground, making his way along the shore in a seemingly endless midnight stroll. Those who encountered the man were struck by his deep scowl and thick German accent. Speculation mounted that he was a German spy giving secret signals to enemy warships. The dark stranger, however, was no German spy. He was Kurt Gödel, the greatest logician of all time, a beacon in the intellectual landscape of the last thousand years, and the prey he sought was not American ships bound for Britain but rather the so-called continuum hypothesis, a conjecture made by the mathematician Georg Cantor about the number of points on a line.
Gödel was spending the summer vacationing at the Blue Hill Inn with his wife, Adele, although fellow visitors at the inn rarely saw either of them. They materialized for dinner but were never observed actually eating. To the locals Gödel's scowl betrayed a dark disposition, but the inn-keeper saw things differently. For her it was the expression of a man lost in thought. His last word to Blue Hill would not decide the issue. He sent a letter accusing the innkeeper of stealing the key to his trunk.
The place Gödel would return to in the fall was a long way from Blue Hill -- the prestigious Institute for Advanced Study, in Princeton, N.J. There he would no longer have to walk alone, arousing the suspicions of neighbors. He had a walking companion, a colleague at the institute and his best friend. There was no danger that Gödel's reputation would intimidate his companion. For his friend, another German-speaking refugee with a mathematical bent, was the most famous scientist of all time, Albert Einstein, whose own meditative strolls already irritated the residents of Princeton.
"From a distance," a biographer wrote, "the [residents of Princeton] chuckled discreetly over [Einstein's] habit of licking an ice cream on Nassau Street on his way home from Fine Hall and were astonished by his utterly un-American long walks through the streets of Princeton." Indeed, toward the end of his career, when he was more or less retired, Einstein commented that his own work no longer meant much to him, and that he now went to his office "just to have the privilege of walking home with Kurt Gödel." Ironically, it was not the scowling Gödel but his smiling companion who had once given indirect aid to the German U-boats, when, during World War I, although a courageous and committed pacifist, Einstein had helped improve the gyroscopes used by the German navy. Gödel's research would also, in the end, relate to gyroscopes, but these spun in the center of the universe, not in the dank bowels of submarines.
Washed up onto America's shores by the storm of Nazism that raged in Europe in the 1930s, the two men awakened to find themselves stranded in the same hushed academic retreat, the Institute for Advanced Study, the most exclusive intellectual club in the world, whose members had only one assigned duty: to think. But Gödel and Einstein already belonged to an even more exclusive club. Together with another German-speaking theorist, Werner Heisenberg, they were the authors of the three most fundamental scientific results of the century.
Each man's discovery, moreover, established a profound and disturbing limitation. Einstein's theory of relativity set a limit -- the speed of light -- to the flow of any information-bearing signal. And by defining time in terms of its measurement with clocks, he set a limit to time itself. It was no longer absolute but henceforth limited or relative to a frame of measurement. Heisenberg's uncertainty principle in quantum mechanics set a limit on our simultaneous knowledge of the position and momentum of the fundamental particles of matter. This was not just a restriction on what we can know: For Heisenberg it signified a limit to reality. Finally, Gödel's incompleteness theorem -- "the most significant mathematical truth of the century," as it would soon be described in a ceremony at Harvard University -- set a permanent limit on our knowledge of the basic truths of mathematics: The complete set of mathematical truths will never be captured by any finite or recursive list of axioms that is fully formal. Thus, no mechanical device, no computer, will ever be able to exhaust the truths of mathematics. It follows immediately, as Gödel was quick to point out, that if we are able somehow to grasp the complete truth in this domain, then we, or our minds, are not machines or computers. (Enthusiasts of artificial intelligence were not amused.)
Einstein, Gödel, Heisenberg: three men whose fundamental scientific results opened up new horizons, paradoxically, by setting limits to thought or reality. Together they embodied the zeitgeist, the spirit of the age. Mysteriously, each had reached an ontological conclusion about reality through the employment of an epistemic principle concerning knowledge. The dance or dialectic of knowledge and reality -- of limit and limitlessness -- would become a dominant theme of the 20th century. Yet Gödel's and Einstein's relation to their century was more uneasy than Heisenberg's.
The zeitgeist took root most famously in quantum mechanics. Here Gödel and Einstein would find themselves in lonely opposition to Heisenberg, who, on the wrong side in the war of nations, chose the winning team in the wars of physics. Heisenberg was champion of the school of positivism, known in quantum physics as the Copenhagen interpretation, in deference to Heisenberg's mentor, Niels Bohr. What had been a mere heuristic principle in Einstein's special relativity -- deducing the nature of reality from limitations on what can be known -- became for Heisenberg a kind of religion, a religion Gödel and Einstein had no wish to join. Some, however, claimed to see in Gödel's theorem itself an echo of Heisenberg's uncertainty principle. The group did not include Gödel.
Einstein, himself one of the great pioneers of quantum mechanics, had known and inspired Heisenberg in Germany. In 1911 in Prague, years before Heisenberg came on the scene, Einstein once pointed out to his colleague Philipp Frank the insane asylum in the park below his study and remarked, "Here you see that portion of lunatics who do not concern themselves with quantum theory." By Einstein's lights, a bad situation became even worse after Heisenberg. In an early encounter, Heisenberg, on the defensive against Einstein's harangue against quantum mechanics, fought back: "When I objected that in [my approach] I had merely been applying the type of philosophy that he, too, had made the basis of his special theory of relativity, [Einstein] answered simply: 'Perhaps I did use such philosophy earlier, and also wrote it, but it is nonsense all the same.'"
The two parted before the war, Einstein emigrating to the United States, Heisenberg remaining in Germany, to which he would remain loyal to the end. In Princeton, Einstein -- pacifist, bohemian, socialist, and Jew -- was a man apart. To be sure, he found Gödel, but together they remained isolated and alone, not least because of their opposition to Heisenberg's positivist worldview, which ruled the intellectual scene even as Heisenberg's fatherland was attempting to dominate the world. Gödel and Einstein were not merely intellectual engineers, as so many of their brethren, inspired by positivism, had become, but philosopher-scientists. Ironically, while their stars had begun to wane, the sheer size of their reputations made them unapproachable. Not to each other, however. "Gödel," wrote their colleague Freeman Dyson, "was the only one of our colleagues who walked and talked on equal terms with Einstein."
Their tastes, however, remained distinct. Einstein, a violinist, could never bring his friend to subject himself to the likes of Beethoven and Mozart. Gödel, in turn, had no more success in dragging Einstein to Snow White and the Seven Dwarfs, his favorite movie. History, sadly, does not record which of the seven dwarfs was Gödel's favorite, but we do know why he favored fairy tales: "Only fables," he said, "present the world as it should be and as if it had meaning." (That meaning, of course, may be dark. It is not known whether Alan Turing acquired an affection for Snow White from Gödel when visiting the institute in the 1930s, but some have heard an echo of the dark side of Snow White in Turing's decision to end his life by eating a poisoned apple when, as a reward for his having broken the "Enigma" code of the German navy, the British government ordered him to receive hormone injections as a "cure" for his homosexuality.)
Einstein, before fleeing Germany, had already become a refugee from mathematics. He later said that he could not find, in that garden of many paths, the one to what was fundamental. He turned to the more earthly domain of physics, where the way to the essential was, he thought, clearer. His disdain for mathematics earned him the nickname "lazy dog" from his teacher, Hermann Minkowski, who would soon recast the "lazy dog's" special relativity into its characteristic four-dimensional form. "You know, once you start calculating," Einstein would quip, "you shit yourself up before you know it."
Gödel's journey, by contrast, was in the opposite direction. Having befriended Gödel, Einstein commented that he knew now, at last, that in mathematics, too, one could find a path to the fundamental. In befriending Einstein, Gödel was reawakened to his early interest in physics. On their long walks home from the office, Einstein, forever cheerful, would attempt to raise the spirits of the gloomy and pessimistic Gödel by recounting his latest insights into general relativity. Sadly, however, pessimism blossomed into paranoia. The economist Oskar Morgenstern, calling one day on his good friend, was shocked to find the great Gödel hiding in the cellar behind the furnace.
From their long walks together, from their endless discussions, something beautiful would soon be born. The scene was pregnant with possibility. Time, which has taunted thinkers from Plato to St. Augustine to Kant, had finally met its match in Einstein. While the U-boats of his former fatherland were stalking the Allied fleet, this most un-German of Germans was hunting a more elusive prey. He had amazed the world decades earlier when he alone succeeded in capturing and taming time itself in the equations of relativity. "Every boy in the streets of Göttingen," his countryman David Hilbert wrote, "understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians." Relativity had rendered time, the most elusive of beings, manageable and docile by transforming it into a fourth dimension of space, or rather, of relativistic space-time. Sharing with Gödel his latest thoughts on the four-dimensional universe of space-time that he himself had conjured into being, Einstein was sowing the seeds of relativity in the mind of a thinker who would later be described as a combination of Einstein and Kafka.
If Einstein succeeded in transforming time into space, Gödel would perform a trick yet more magical: He would make time disappear. Having already rocked the mathematical world to its foundations with his incompleteness theorem, Gödel now took aim at Einstein and relativity. Wasting no time, he announced in short order his discovery of new and unsuspected cosmological solutions to the field equations of general relativity, solutions in which time would undergo a shocking transformation. The mathematics, the physics, and the philosophy of Gödel's results were all new. In the possible worlds governed by these new cosmological solutions, the so-called "rotating" or "Gödel universes," it turned out that the space-time structure is so greatly warped or curved by the distribution of matter that there exist timelike, future-directed paths by which a spaceship, if it travels fast enough -- and Gödel worked out the precise speed and fuel requirements, omitting only the lunch menu -- can penetrate into any region of the past, present, or future.
Gödel, the union of Einstein and Kafka, had for the first time in human history proved, from the equations of relativity, that time travel was not a philosopher's fantasy but a scientific possibility. Yet again he had somehow contrived, from within the very heart of mathematics, to drop a bomb into the laps of the philosophers. The fallout, however, from this mathematical bomb was even more perilous than that from the incompleteness theorem. Gödel was quick to point out that if we can revisit the past, then it never really "passed." But a time that fails to "pass" is no time at all.
Einstein saw at once that if Gödel was right, he had not merely domesticated time: He had killed it. Time, "that mysterious and seemingly self-contradictory being," as Gödel put it, "which, on the other hand, seems to form the basis of the world's and our own existence," turned out in the end to be the world's greatest illusion. In a word, if Einstein's relativity theory was real, time itself was merely ideal. The father of relativity was shocked. Though he praised Gödel for his great contribution to the theory of relativity, he was fully aware that time, that elusive prey, had once again slipped his net.
But now something truly amazing took place: nothing. Although in the immediate aftermath of Gödel's discoveries a few physicists bestirred themselves to refute him and, when this failed, tried to generalize and explore his results, this brief flurry of interest soon died down. Within a few years the deep footprints in intellectual history traced by Gödel and Einstein in their long walks home had disappeared, dispersed by the harsh winds of fashion and philosophical prejudice. A conspiracy of silence descended on the Einstein-Gödel friendship and its scientific consequences.
An association no less remarkable than the friendship of Michelangelo and Leonardo -- if such had occurred -- has simply vanished from sight. To this day, not only is the man on the street unaware of the intimate relationship between the two giants of the 20th century, even the most exhaustive intellectual biographies of Einstein either omit all mention of this friendship or at best begrudge a sentence or two. Whereas a whole industry has grown up in search of Lieserl, the "love child" of Einstein's first marriage, the child of the imagination that was born of the friendship of Einstein and Gödel has been abandoned.
Only in the last few years has this child, the Gödel universe, received any glimmer of recognition. This comes from the redoubtable Stephen Hawking. Revisiting the rotating Gödel universe, Hawking was moved to deliver the highest of compliments. So threatening did he find Gödel's results to the worldview of sober physicists that he put forward what amounts to an anti-Gödel postulate. If accepted, Hawking's famous chronology-protection conjecture would precisely negate Gödel's contribution to relativity. So physically unacceptable did Hawking find conclusions like Gödel's that he felt compelled to propose what looks like an ad hoc modification of the laws of nature that would have the effect of ruling out the Gödel universe as a genuine physical possibility.
Hawking's attempt to neutralize the Gödel universe shows how dangerous it is to break the conspiracy of silence that has shrouded the Gödel-Einstein connection. Not only does this mysterious silence hide from the world one of the most moving and consequential friendships in the history of science, it also keeps the world from realizing the true implications of the Einstein revolution. It is one thing to overturn, as Einstein did, Newton's centuries-old conception of the absoluteness and independence of space and time. It is quite another to demonstrate that time is not just relative but ideal. Unlike Einstein, a classicist who forever sought continuity with the past, Gödel was at heart an ironist, a truly subversive thinker. With his incompleteness theorem he had shaken the foundations of mathematics, prompting the great mathematician Hilbert to propose a new law of logic just to refute Gödel's results. The Gödel universe, correctly understood, shares with his incompleteness theorem an underlying methodology and purpose. It is a bomb, built from cosmology's most cherished materials, lobbed into the foundations of physics.
In the footsteps of Gödel and Einstein, then, can be heard an echo of the zeitgeist, a clue to the secret of the great and terrible 20th century, a century that, like the 17th, may well go down in history as one of genius. The residents of Blue Hill, preoccupied with war and the enemy out at sea, had failed to take the full measure of their man.
Palle Yourgrau is a professor of philosophy at Brandeis University. This essay is from his new book, A World Without Time: The Forgotten Legacy of Gödel and Einstein, to be published next month by Basic Books. Copyright © 2004 by Palle Yourgrau.
The joke is that Gödel proved that mathematicians will always have work, because any logical system within which they work will contain statements that require new, stronger logical systems to prove, and of course somebody's got to come up with those new, stronger logical systems and the proofs they allow...but then the process starts over again within the new systems, and so on ad infinitum (or at least until the cosmos disappears)!
Thanks for the ping. :-)
Glad to do it, bud. The headline screw-up is aggravating, but maybe the Mod can fix it.
Was that during their Vaudeville days?
Looks like it, huh? (laugh)
Gödel and Einstein together again for the first time.
I saw that movie. The Blue Angel, starring Marlene Dietrich. (Sorry, couldn't resist.)
Probably true to life at that time.
When I was a kid I looked at graphics of atoms in textbooks, and wondered about whether atoms were solar systems, or whether protons or neutrons were Universes, was the Earth a proton, etc. Maybe this whole Universe is just boring down the tunnel of an atom smasher somewhere....
Nice photo, snarks!
(...but I can't help but be reminded of the Bartles & James wine-cooler geezers on the porch.)
We could call it Waiting for Gödel.
(Ducks away quickly to avoid the hail of well-aimed vegetables.)
Thanks for the ping!
We had a long-ago thread on this, with a copy of his essay on the subject:
Why Socialism? (Albert Einstein).
I hate it when people pronounce his name "Girdle".
I own a copy of one of Goedels books. I will be prepared to read it in about 200 years if my brain doesn't overheat and melt and run out my ears before then.
This leads me to make the proposal that all philosophy professors should have their fingers broken so they cannot afflict the rest of the world their horrific writing styles.
Good article, but it's lost in today's flood of crevo threads. I've got to abandon watching this one. If something comes up, I hope I'll get pinged.
"Incompleteness Theorem, shows (roughly) that any logical system strong enough to contain arithmetic will also contain statements the truth of which can be known, but not proved using only the axioms of the given logical system. "
This theorem shows what has become the primary hindrance to widespread acceptance of String Theory. That is that we currently have no way of proving it.
Your lack of knowledge about Einstein's worth outside of being a scientist might also be characterized as 'dangerous or stupid'. I give you but one example:
Albert Einstein's letter to Franklin D. Roosevelt, August 2, 1939:
Old Grove Road
Peconic, Long Island
August 2, 1939
F. D. Roosevelt,
President of the United States,
Washington, D. C.
Some recent work by E.Fermi and L. Szilard, which has been communicated to me in a manuscript, leads me to expect that the element uranium may be turned into a new and important source of energy in the immediate future. Certain aspects of this situation which has arisen seem to call for watchfulness and, if necessary, quick action on the part of the Administration. I believe therefore that it is my duty to bring to your attention the following facts and recommendations:
In the course of the last four months it has been made probable - through the work of Joliot in France as well as Fermi and Szilard in America - that it may become possible to set up a nuclear chain reaction in a large mass of uranium,by which vast amount s of power and large quantities of new radium-like elements would be generated. Now it appears almost certain that this could be achieved in the immediate future.
This new phenomena would also lead to the construction of bombs, and it is conceivable - though much less certain - that extremely powerful bombs of a new type may thus be constructed. A single bomb of this type, carried by boat and exploded in a port, m ight very well destroy the whole port together with some of the surrounding territory. However, such bombs might very well prove to be too heavy for transportation by air.
The United States has only very poor ores of uranium in moderate quantities. There is some good ore in Canada and the former Czechoslovakia, while the most important source of uranium is Belgian Congo.
In view of this situation you may think it desirable to have some permanent contact maintained between the administration and the group of physicists working on chain reactions in America. One possible way of achieving this might be for you to entrust with this task a person who has your confidence and who could perhaps serve in an inofficial capacity. His task might comprise the following:
a) to approach Government Departments, keep them informed of the further development, and put forward recommendations for Government action, giving particular attention to the problem of securing a supply of uranium or for the United States;
b) to speed up the experimental work,which is at present being carried on within the limits of the budgets of University Laboratories, by providing funds, if such funds be required, through his contacts with private persons who are willing to make contrib utions for this cause, and perhaps also by obtaining the co-operation of industrial aboratories which have the necessary equipment.
I understand that Germany has actually stopped the sale of uranium from the Czechoslovakian mines which she has taken over. That she should have taken such an early action might perhaps be understood on the ground that the son of the German Under-Secretary of State, von Weizsacker, is attached to the Kaiser-Wilhelm-Institute in Berlin where some of the American work on uranium is now being repeated.
Yours very truly,
For the record (as if anybody's keeping records!), even if I don't respond to each comment posted to me on a thread I start, you can be sure that I've read them all (and all of the others, too). Often I decide not to respond just in order to keep the noise level down; but I always laugh at the jokes (intentional and otherwise).
And with that last, mildly snarky remark (got to keep up appearances), I'll say,
Merry Christmas and best regards to all...
And a very merry Christmas to you and yours.
I wonder if Einstein would have written the letter on Sept 2 1939? The nazis and the communists were allies then and Einstein probably followed the partyline.
Thanks for the ping.
If Einstein succeeded in transforming time into space, Gödel would perform a trick yet more magical: He would make time disappear. ..it turned out that the space-time structure is so greatly warped or curved by the distribution of matter that there exist timelike, future-directed paths...
He was forced to leave Europe in 1933, and, after arriving in Princeton, never left there. He became an American citizen in 1940.
He was a gentle man who sought peace and well-being for all, but who also could see clearly enough the nature of the threat posed to the world by Hitler and the Nazis. That is why he agreed to sign the letter posted above by IonImplantGuru.
Didn't Hawking change his mind about time travel a few years back?
Are you referring to Hawking's recent re-canting of his view that material sucked into a black hole could re-emerge into another universe (thus violating the principle of the conservation of information)? Here's a link for that story:
The link is to Preskill's webpage, Preskill being the guy who won the bet with Hawking.
Also, doesn't the following sound like a weapon in waiting?
If Einstein succeeded in transforming time into space, Gödel would perform a trick yet more magical: He would make time disappear. ..it turned out that the space-time structure is so greatly warped or curved by the distribution of matter that there exist timelike, future-directed paths...
Unless we can figure out a way to spin the entire cosmos like a top, we won't be able to use Gödel's solutions of Einstein's field equations to produce a weapon. I found the following using Google (it's by Stephen Speicher and may be found here):
Kurt Goedel was the first to demonstrate the existence of closed timelike curves (CTCs) in an exact solution to the Einstein field equations of general relativity ("An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation," _Reviews of Modern Physics_, 21: pp. 447-450, 1949). Goedel himself interpreted the CTCs as demonstrating that any objective lapse of time is an illusion: "...that for _every_ possible definition of a world time one could travel into regions of the universe _which are passed_ according to that definition. This again shows that to assume an objective lapse of time would lose every justification in these worlds."
In Goedel's universe one can travel to the past, and, as Goedel said in one of his later manuscripts, "in whatever way one may [try to] introduce an absolute 'before,' there always exist either temporally incomparable events or cyclically ordered events." It was Goedel's contention, and echoed later by physicists such as Roger Penrose, that our ordinary perceptions of time as past, present, and future is obviated by the very existence of closed timelike curves. The idea being that physics treats time as it does the spatial dimensions, and the "flow of time" is just an illusion. The closed timelike curve in the Goedel universe permits one to travel towards one's causal future, but eventually wind up at one's local past.
Einstein acknowledged Goedel's work here as being
"an important contribution to the general theory of relatvity, especially to the analysis of the concept of time. The problem here involved disturbed me already at the time of the building up of the general theory of relativity, without my having succeeded in clarifying it." 
And Einstein explicated the issue:
"...and if the series is closed in itself. In that case the distinction 'earlier-later' is abandoned for world-points which lie far apart in the cosmological sense, and those paradoxes, regarding the _direction_ of the causal connection, arise, of which Mr. Goedel has spoken." 
Like Stephen Hawking, Einstein also questioned Goedel's solution on physical grounds: "It will be interesting to weigh whether these are not to be excluded on physical grounds." And, Hawking says: "This suggests that the solution is not very physical." And, indeed, Goedel's rotating universe was later understood not to be the universe in which we live. However, the Goedel solution is certainly of interest, and one will find it discussed in many advanced texts on relativity (for instance, in Hawking & Ellis, "The Large Scale Structure of Space-time," and Ciufolini & Wheeler, "Gravitation and Inertia").
There are other perspectives from other theorists, but, I think it fair to say that the modern perspective in general relativity is to treat the spacetime manifold as an abstraction within which we analyze in order to make real-world predictions of actual events.
 "Albert Einstein: Philosopher-Scientist," Edited by Paul Arthur Schilpp, _MJF Books_, p. 687, 1949/1970.
 Ibid. p. 688.
You don't get it, do you? How do you think we "defeated" the Soviet Union? In the 1970s and '80s, when the commies were on a roll, it was often asked aloud if the US would be able to survive. And here we are, just a few years later, having waged no war against the foe, yet they're gone and we're Number One.
Even a casual observer must agree that we live in the most improbable of times. Do you think it happened by accident?
I agree with you that deep insight and high achievement in one area does not translate into the same in other areas. Clearly, Einstein was a much better scientist than he was a political observer.
Substitute 'do not translate' for 'does not translate' in the first sentence of the second paragraph of my previous post.
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.