Gödel was buried on 19 January in the Princeton Cemetery on Witherspoon Street; the funeral was small and private. But on 3 March there was a memorial service held at the Institute and presided over by André Weil. The speakers were Hao Wang, Hassler Whitney, and, as a last-minute fill-in for the prominent logician Robert Solovay, who had flown in from California but whose rented car had swerved into a snow ditch, Simon Kochen.
Kochen recalled in his tribute to Gödel that at his Ph.D. oral exam, his examiner, Stephen Kleene, had asked him to name five of Gödel’s theorems. The point of the question was that “Each of the theorems . . [was] the beginning of a whole branch of modern mathematical logic.” Proof theory, model theory, recursion theory, set theory, intuitionist logic; all had been transformed by, or, in certain cases, had gotten their inception from, Gödel’s work.
Kochen then compared Gödel’s work to Einstein’s, in terms of the way in which both grew out of their deep foundational thinking. “That was an obvious comparison to make,” Kochen told me.
He also, far more surprisingly, compared Gödel’s work to that of Kafka, not knowing that Gödel himself had been an admirer of the writer.14 Both men combined their strongly “legalistic” bent, in Kochen’s words, with an otherworldly, almost surreal, ability to create self-contained worlds, worlds that might seem at first blush to run counter to logic but which are compounded of the very stuff of logic. “There is an Alice-in-Wonderland quality to both men’s work,” Kochen said to me.
Kochen told me that had he had time to prepare his remarks, the analogy to Kafka would probably never have been made by him. It was because he had to come up with something quickly that he reached for the vague suggestion that Gödel’s work had always impressed on him, but that he had never bothered to think out. But everyone I spoke with who had been present at the memorial service remembers Kochen’s remark. The comparison with Kafka, another sui generis mind out of Central Europe, managed to capture something startling and true.
Incompleteness (All Over Again)
Einstein and Gödel shared, along with so much else, a preoccupation with the nature of time. Despite the popular distortions, to a certain extent encouraged by the vague suggestions of the word “relativity,” Einstein was, as we have already seen, as far from interpreting his famous theory in subjective terms as it is possible to be. On the contrary, on his interpretation, relativity theory offers a realist description of time that is startlingly distinct from our subjective experience of time. The great yawning chasm between the “out yonder” and the “in here” is stretched even wider, on the Einsteinian hypothesis, since objective time—the time that is described in the equations of relativity theory—is lacking the very feature that seems to provide the essential stab to our subjective experience of time: its inexorable flow, ultimately lighting all our yesterdays the way to dusty death.15 Is there anything we know more intimately than the fleetingness of time, the transience of each and every moment?
Einstein and Gödel walking on the suppressed path that went from Fuld Hall to Olden Farm.
Yet, strangely enough, it isn’t so . . . not if we take Einstein’s physics seriously. The nature of reality that spills forth from Einstein’s physics is so much more startling than the simplistic, undergraduate-beloved shibboleth: everything is relative to subjective points of view. In Einstein’s physics, there is no passage of time, no unidirectional flow away from the fixed past and toward the uncertain future. The temporal component of space-time is as static as the spatial components; physical time is as still as physical space. It is all laid out, the whole spread of events, in the tenseless four-dimensional space-time manifold. The distinctions we make between the past and the present and the future—distinctions that are so emotionally fraught and without which we can’t even begin to describe our inner worlds—only have relevance within those inner worlds. Objective time, as it is characterized in relativity, can’t support the distinction between the past and present and future. Or, as Einstein told Rudolf Carnap, “the experience of the now means something special for man, something essentially different from the past and the future, but this important difference does not and cannot occur within physics.”
Understanding relativity theory to imply that there is no absolute now flowing along on a relentless tide of temporality, Einstein, living “under the sword of Damocles,” seemed to take comfort in his vision of tenseless physical objectivity. In a condolence letter to the widow of Michele Besso, his longtime friend and fellow physicist, Einstein wrote: “In quitting this strange world he has once again preceded me by just a little. That doesn’t mean anything. For us convinced physicists the distinction between past, present, and future is only an illusion, albeit a persistent one.”
It is a vision of impersonal objectivity sufficient to extract the bitterness, at least for Einstein, from the thought of one’s own personal demise, than which there are few thoughts more unpalatable. Einstein’s imperturbability recalls, in its transcendence, the death of Socrates that had so inspired Plato and, through Plato, all of Western civilization. This is scientific realism carried to heroic heights. The physicist who discussed the meaning of time on his daily walks with the logician was dying, and he knew it.
Gödel, no less than Einstein, believed that time is nothing like what it seems to us to be. His personality may not have been of the sort to allow him to use his vision of time to transcend the fears—both real and imagined—that tormented his mortal existence; but nonetheless it did, perhaps, offer him some degree of comfort. His own work on relativity theory had provided him a model of time that seemed to have appealed to him on a deep level, to mesh with the very substance of the man, as his embrace of Platonism had done.
Gödel, of course, had a long-standing interest in physics. He had first entered the University of Vienna intending to study physics and did so for the first two or three years while a student there, before switching to math. His relationship with Einstein rekindled his earlier interest in physics, and at some point in their relationship Gödel began to ponder relativity theory for himself. He came up with an entirely unique model satisfying Einstein’s field equations in general relativity, a model as Alice-in-Wonderland-like as anything else he had ever done.
In Gödel’s model, time is cyclical. Not only are all events laid out in indifference to tensed distinctions between past, present, and future, but also endless repetitions of the patterns occur, and the parallelism between space and time, implicit in relativity theory, is extended further. “It turns out,” wrote Gödel, “that temporal conditions in these universes show . . . surprising features, strengthening the idealistic viewpoint (according to which all change is actually an illusion, nonobjective). Namely, by making a round trip on a rocket ship in a sufficiently wide curve, it is possible in these worlds to travel into any region of the past, present, and future, and back again, exactly as it is possible in other worlds to travel to distant parts of space.”
Gödel published his solution to Einstein’s equations in the Festschrift volume in honor of Einstein’s seventieth birthday.16 Einstein’s published remarks on the paper, also published in the Festschrift, acknowledge having been “disturbed” by the possibility of looping timelike lines, allowing one to return to the past, that Gödel gleefully expounded. Einstein’s response both pays tribute to the validity of Gödel’s deductions while also suggesting that Gödel’s solution might “be excluded on physical grounds.”
It is unclear how much of his cosmological work Gödel had shared with his daily walking partner before handing him over the results on his seventieth birthday; but Einstein’s reaction to the paper suggests that Gödel had shared little. Gödel’s closed loops of time, allowing one, at least theoretically, to return to the past, were accepted by Einstein as formally possible, in the sense that Gödel had shown that this model of time solves Einstein’s field equations. But as a physicist and a man of common sense, Einstein would have preferred that his field equations excluded such an Alice-in
-Wonderland possibility as looping time.
But the model of cyclical time seemed to have appealed to Gödel very much. Did Gödel actually like the idea of being able to go back and live his life all over again? Did he, too, like his friend Einstein, draw some sort of solace from his contemplation of the real nature of time, distinct from the unidirectional finality of our experience of it?
Who knows? The opacity of the logician prevails. However, there is one interesting fact that perhaps allows us a peek behind the opacity. Gödel took his quite extraordinary solution to Einstein’s equations so seriously that he descended, probably for the only time in his life, from the highest reaches of Reine Vernunft to try to acquire actual empirical (!) data to support his closed-looped model for time. John Archibald Wheeler and Kip Thorne, two of the most prominent physicists of their days, who had collaborated (with Charles Misner) on a marvelous book on gravitation, were closely questioned by Gödel in the early 1970s as to whether they had found any evidence for, or against, a preferred sense of rotation of the galaxies. Gödel was clearly disappointed in them, Wheeler reported, when they confessed that they just hadn’t looked into the question:
It turned out that he himself, as a preliminary step to get some evidence, had taken down the great Hubble atlas of the galaxies. Gödel, whom you think of as the mathematician among mathematicians, had taken a ruler and got the angle and made a statistics of these numbers and concluded that within the statistical error there was no preferred sense of rotation. . . .
About a year after our visit to Gödel I was down the hall here in the office of Jim Peebles [a prominent Princeton astrophysicist] talking to him about cosmology, and a student came in and threw down on the table a big thing. “Here it is, Professor Peebles!” So I said to him, “What is it?” He said, “It’s my thesis.” “What’s it about?” “It’s about whether there is any preferred sense of rotation in the galaxies.” “How marvelous,” I said, “Gödel will be so pleased.” “Who is Gödel?” “Well,” I said, “if you called him the greatest logician since Aristotle you’d be downgrading him.” “Are you kidding?” “No, no.” “What country does he live in?” “Right here in Princeton,” I answered. So I picked up the phone and dialed Gödel, reached him at home, and told him about this. Pretty soon his questions got to the point I couldn’t answer them. I turned it over to the student, and pretty soon it got to the point that the student couldn’t answer them. He gave the phone to Peebles, and when Peebles finally hung up he said, “My, I wish we talked to Gödel before we did the work.”
When Gödel presented his ideas on relativity at the Institute, which took place several years before the conversations with Wheeler, Thorne, and Peebles, the physicists present all expressed astonishment at how well the mathematical logician had grasped all the subtleties of the physical theory. But of course he had had the privilege of discussing the intricacies of the theory with the theoretician himself—even as the theoretician confessed that, at the end, he only went to his office to have the privilege of walking home each day with the logician, the two great minds of the twentieth century able to share, at least for a while, their intellectual exile with one another.
It is tempting to connect Gödel’s attraction to these closed time loops with a passing remark that Hao Wang made, indicating how Gödel had thought of his life as incomplete:
In philosophy Gödel has never arrived at what he looked for: to arrive at a new view of the world, its basic constituents, and the rules of their composition. Several philosophers, in particular Plato and Descartes, claim to have had at certain moments in their lives an intuitive view of this kind totally different from the everyday view of the world.
And, again, Wang made reference to a transcendental experience that Gödel had awaited all his life:
He also looked for (but failed to obtain) an epiphany (a revelation or sudden illumination) that would enable him to see the world in a different light. (In his conversations with me, he repeatedly said that Plato, Descartes, and Husserl all had such an experience.)
Philosophy had inspired Kurt Gödel’s formidable mathematical career from the beginning. It had been his focus ever since his first course at the University of Vienna in the history of philosophy, when Gödel, like so many lovers of abstraction, had found in Plato a vision of reality that answered to his intellectual love. As philosophy had been his end, so, too, it was by philosophy’s light that he judged his life, finally, incomplete. No longer believing that it was possible to change other people’s minds, not even by way of a priori proof, he awaited the epiphany that would change his own. With the sense of his own incompleteness—and perhaps, too, with the preserved death-terror of a child believing that his eight-year-old heart had been fatally damaged—he was drawn to his model of an eternal life of cyclical time, a model which undermines the reality of personal death.
If time loops back on itself, as Gödel in the tormented last years of his life sought empirically to corroborate, then a young Gödel will once again sit in a college classroom in Vienna, transfigured by the notion of the infinite eternal verities lying suspended beyond all the human imperfections, the confoundments and obfuscations and distortions that make him wonder how people can ever understand one another at all. And he will think about using the language of mathematics in a way that no one has thought to use it before, so that it can talk about itself—only precisely, because mathematically, so that everyone will understand. He will dream, silently and audaciously, of proving a mathematical theorem the likes of which has never before been seen, a mathematical theorem that will illuminate the nature of mathematics itself.
And then he will do it.
1 The earliest version of the ontological argument was St. Anselm’s, and it goes something like this: God is, by definition, that than which nothing greater can be conceived. God, therefore, cannot be conceived of as not existing, for otherwise we can conceive of Him as being greater, viz. by existing. It is therefore inconceivable that God not exist; ergo He exists.
2 This was in fact the last thing he published in his lifetime.
3 In the Nachlass there’s a sheet of paper on which Gödel had listed, probably in 1970 according to Hao Wang, all his unpublished work, from 1940 on. It reads something like:
1. About one thousand 6 x 8-inch stenographic pages of clearly written philosophical notes (= philosophical assertions).
2. Two philosophical papers almost ready for print [His paper on relativity and Kant’s philosophy and his paper on syntax and mathematics, originally intended for the Carnap Festschrift, but never published by him.]
3. Several thousand pages of philosophical excerpts and [notes on the] literature.
4. The clearly written proofs of my [his] cosmological results.
5. About six hundred clearly written pages of set theoretical and logical results, questions and conjectures (to some extent outstripped by recent developments).
6. Many notes on intuitionism and other foundational questions.
4 Time magazine, in commemoration of the end of the last millennium, devoted a few special issues to the 100 greatest minds of the last century. Kurt Gödel was cited as the century’s greatest mathematician. Interestingly, Ludwig Wittgenstein and Alan Turing also made the list, and Albert Einstein was chosen as the greatest mind of the century.
5 Wirtinger was a mathematician who had reportedly become embittered and withdrawn after his colleague, Professor Furtwängler, got a prize for an important result in algebraic number theory. What a way to earn a footnote in the story of the most important mathematical result of the twentieth century.
6 The axiom of choice is concerned with collections of sets, particularly infinite collections. There are various ways of stating the axiom. In fact there’s a whole book, by H. Rubin and J. Rubin, entitled Equivalents of the Axiom of Choice. A simple version of the axiom is: For any set of non-empty disjoint sets (sets that have no members in common), there exists a set consisting of exactly one member of each of the non-empty
sets. In other words, if you have a bunch of sets that don’t overlap with each other, then, roughly speaking, you can form a set by choosing one member of each set in the bunch. (You really need the axiom only when the bunch is infinite.) Another way of stating the axiom of choice is: For any set of non-empty sets, there exists a function that assigns to each one of these non-empty sets one of its members. Infinitely many choices (hence the name of the axiom) may be required, which is why the axiom has received so much attention. The axiom is saying that a certain set exists, even though the set is not really specified or constructed. The axiom of choice is probably the second most discussed axiom of mathematics, right after Euclid’s parallels postulate. Like the parallels postulate, the axiom of choice was proved to be independent of the other axioms, in this case of set theory. Gödel proved the first part of the independence by showing that the axiom is consistent with the other axioms of set theory; and then (as with the continuum hypothesis) Paul Cohen completed the proof (in 1963) by showing that the negation of the axiom of choice is consistent with the other axioms. Just as the proof of the logical independence of the parallels postulate gave rise to non-Euclidean geometry, so, too, there’s a non-Cantorian form of set theory that uses the negation of the axiom of choice. But though the very idea of the infinite number of choices involved in the axiom of choice might make mathematicians a bit queasy, most mathematicians don’t hesitate to avail themselves of the axiom in constructing their (nonconstructive) proofs, because it has so many important applications in practically all branches of mathematics that its rejection would seriously manacle mathematicians. It’s not clear when Gödel began to think about set theory, and it’s not clear when he proved that the axiom of choice is consistent with the other axioms of set theory. He didn’t tell anyone about his proof until the following year when he was back again in Princeton. Not surprisingly, it was von Neumann who received the confidence of the important new result, which Gödel published in 1938.
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