Gödel, however, declined to take any courses in the language of the republic in which he was living (the native language of most of the students was German). Klepetař recalled to Dawson that Gödel was the only one of his fellow students he never heard speak a word of Czech and that, especially after October 1918 when the Czechoslovak Republic declared its independence, “Gödel considered himself always Austrian and an exile in Czechoslovakia.” So the sense of exile began early in his life and, in its various senses, some more pernicious than others, it is doubtful that it ever left him.
Gödel entered the University of Vienna in 1924, intending to study physics, but, as he later told Hao Wang, “his interest in precision led him from physics to mathematics and to mathematical logic.” Gödel’s interest in physics had begun at the age of 15, when he read Goethe’s theory of colors, which was embedded in a general attack on Newtonian physics. His transition to mathematics was also encouraged, he told Hao Wang, by the excellent teaching of some of his professors at the university. The course on number theory, given by Professor Phillip Furtwängler, attracted such huge numbers of students (up to 400) that it was necessary to issue alternate-day seating passes. Gödel was one of these rapt students, and he later said that they were the most wonderful lectures he had ever heard.
Intending to concentrate on number theory, he switched his major to mathematics in 1926, but in 1928 he began to work in mathematical logic. He was already a committed Platonist in 1926 when he turned from physics to mathematics. His metaphysical commitment had been forged the year before, when he took a course in the history of philosophy with Professor Heinrich Gomperz, whose father, Theodore, was a distinguished professor of ancient philosophy.
It is no easy task to penetrate the inner life of Kurt Gödel. One knows enough to recognize that it is markedly different from those of others, so reasoning by analogy will get us only so far. Then, too, he was the most reticent of men, assertively nondemonstrative in all things other than mathematics—where “demonstration” means, of course, something quite unique. He was a man of deep passions, as his life will bear out; but these passions were kept scrupulously hidden and they were rigorously intellectual.
I think it is fair to say, however, that like so many of us Gödel fell in love while an undergraduate. He underwent love’s ecstatic transfiguration, its radical reordering of priorities, giving life a new focus and meaning. One is not quite the same person as before.
Kurt Gödel fell in love with Platonism, and he was not quite the same person as he was before.
What is the evidence for so transformative a passion roiling within the opaquely self-contained logician? That is, what is the evidence, in addition to the incompleteness theorems themselves?
Some of the evidence lies in the Nachlass, Gödel’s literary remains, which are housed in Princeton’s Firestone Library. The Nachlass had been left to molder in the Institute’s basement, until John Dawson undertook the formidable task of becoming Gödel’s archivist. (Gödel used a sort of shorthand script, Gabelsberger, that he had learned in high school, so the job involved, on top of everything else, translation.) Gödel had apparently kept almost every scrap of paper that had ever intercepted his life. There are journal articles, clothing bills, manuscripts, family pictures, student exercises, library slips for books he had borrowed in Vienna and Princeton. I found (in the Institute’s collection of Gödeliana) those little Bible studies published by the Jeho-vah’s Witnesses, the kind that their itinerants will urge on you if you happen to be home in the middle of the day and answer the door. These contained careful underlinings and marginalia in the logician’s hand.
More telling are the many drafts of letters that were never posted, the manuscripts for articles that he had promised to deliver, that he had labored over with revision upon painstaking revision, and then never released for publication. One receives the impression of a cautiousness of hysterical proportions. Not a cautiousness of cerebration, for Gödel’s intellectual ambitions were audacious, his intuitions fierce, his willingness to carry them to their logical conclusion undeterable. Rather there was a hysteria of discretion in presenting his thoughts to the external world.
Among the documents over which Gödel labored, and then never delivered, are the responses he made to a questionnaire that had been prepared for him by a sociologist. Burke D. Grandjean had made repeated attempts to interview Gödel and finally devised a questionnaire for him in 1974. (This is two years before the logician’s death.) There are two slightly different versions of the completed questionnaire in the Nachlass, as well as a typed, unsigned, and unsent letter addressed to Mr. Grandjean, dated 19 August 1975, and which begins in a rather bristling manner: “Dear Mr. Grandjean: Replying to your inquiries I would like to say first that I don’t consider my work ‘a facet of the intellectual atmosphere of the early 20th century’ but rather the opposite.” One can imagine the sociologist’s reverential letter that had prompted this testy retort. In the context of so reticent a life, this letter, together with the two sets of replies to the questionnaire, is revealing. Gödel’s bristling tone in his unsent response to Grandjean strengthens the sense that his life, especially after the death of Einstein, was characterized by a profound sense of intellectual isolation—a sense of isolation deepened by misinterpretations of his famous result.
Grandjean had listed various thinkers and asked Gödel to indicate which ones had influenced him, and Gödel made clear how far from the mark Grandjean’s assumptions were. There seems to be a lifetime of exasperation behind the responses. Leibniz is not even listed.
To Grandjean’s question: “Are there any influences to which you attribute special significance in the development of your philosophy?” Gödel’s entire answer consisted of: “Heinrich Gomp. [erz] Prof[essor] of Ph[ilosophy] of Vienna.” A strange answer, but, then again, not. It was in Professor Gomperz’s class that Gödel’s transfigurative intellectual love had been engendered. Though Gödel may have sat rapt in Professor Furtwängler’s class on number theory, it was in Professor Gomperz’s Introduction to the History of Philosophy that the true rapture transpired.
Plato has always had a strong appeal to the mathematically inclined. Plato himself was mathematically inclined. Written over the entrance to the Academy, the Athenian school of higher education he founded (essentially, the first European university), were the words: “Let no one enter herein who has not first studied geometry.”
The ancient Greek philosopher’s disdain for the Sophists, particularly for such men as Protagoras, gave the negative connotation to the word for these itinerant teachers. (The root of the word Sophist is the ancient Greek word for knowledge. Philosophy, literally “love of knowledge,” shares the root.) Protagoras had meant his assertion that “man is the measure of all things” to apply most directly to the moral sphere; he had been arguing for what we now call “moral relativism,” the claim that there is no objective difference between right and wrong, only different opinions, relativized either to individuals or to conglomerates of individuals who roughly share the same values (i.e., to societies). “True,” when attached to moral opinions is an abbreviation for “true for x,” where x is an individual or a society of ethically like-minded individuals.
Plato took on the relativists. It was his lifelong occupation. He not only argued for the objectivity of moral truth but he also founded his claim of objective truth—in the moral as well as other spheres—on his assertion of the objectivity of an abstract reality, graspable not through the senses but through reason.
The one area in which Platonism has proved most stubbornly resilient is mathematics, or rather metamathematics. A mathematician’s sense that he is discovering objective truths, rather than simply constructing systems, is a commitment to Platonism. The conviction that such things as numbers and sets serve as models for our systems, which systems are true only insofar as they describe the nature of such things as numbers and sets, is likewise a commitment to Platonism.
First expos
ure to Plato can be an extremely heady experience for those with a passion for abstraction. (I remember my own.) It can amount to a sort of ecstasy. Plato himself argued that the beauty of the abstract realm, which immeasurably exceeds that of any single particular, can and ought to kindle a passion far larger than any prompted by individual beautiful persons (fickle, imperfect creatures who cannot even be counted on to love us in return and whose beauty not only cannot compete with the transcendent sort but also is subject to the corrosive actions of time). To have used the expression “fell in love” in relation to Gödel’s undergraduate experience is to echo Plato himself, who used the most erotically charged language to describe the mind’s approach toward and possession of the beauties of abstract objectivity.
“Here is the life, Socrates, my friend,” said the Manitean visitor, “that a human being should live—studying the beautiful itself. Should you ever see it, it will not seem to you to be on the level of gold, clothing, and beautiful boys and youths, who so astound you now when you look at them that you and many others are eager to gaze upon your darlings, and be together with them all the time. You would cease eating and drinking, if that were possible, and instead just look at them and be with them. What do we think it would be like,” she said, “if someone should happen to see the beautiful itself, pure, clear, unmixed, and not contaminated with human flesh and color and a lot of other mortal silliness, but rather if he were able to look upon the divine, uniform, beautiful itself. Do you think,” she continued, “it would be a worthless life for a human being to look at that, to study it in the required way, and be together with it?”
A “symposium” actually meant a drinking party in Plato’s Athens, and in the dialogue to which Plato wryly gave that name he urges us to leave off lesser intoxications, including those associated with the sensual love of beautiful young things, and to become drunk on the beauty of truth—the sort of necessary and immutable truth acquired through pure reason, for which mathematics serves as the model. An aspect of the Platonic vision is a rejection of the easy bifurcation between passion, on the one side, and reason, on the other. Plato is urging us toward impassioned reason, the higher intoxication. Of course, susceptibility to the higher intoxication is predicated on the ability to grasp the intellectual love object, the beauties of pure abstraction, “to look at that, to study it in the required way, and be together with it.” The young Kurt Gödel was singularly susceptible.
Gödel’s reaction to the headiness of Plato’s rapturous vision of truth was, it seems, the resolve to devote himself only to mathematics of (to recall Einstein’s phrase) “genuine importance.” It would have to be mathematics that had metasignificance, that was philosophically porous so that the objective source of all abstract truth could be seen to shine.
At first, Gödel had been drawn toward number theory because he believed that it would provide the strongest evidence for, and the clearest application of, conceptual realism. It was in 1928, when he was 22, that his mathematical interests began to shift toward mathematical logic. The fact that he had been attracted to number theory precisely because of his Platonist commitment, as he told Hao Wang, and that he then veered toward mathematical logic is tantalizing. Hao Wang did not ask the follow-up question. Exactly when did Gödel glimpse that logic might yield the metamathematical conclusions that he was seeking?
It is tempting to speculate about this, and informed speculation is the most that we have. We have no good idea of the path that led him to his theorems, by way of an ingenious form of argument the likes of which had never before been seen.
In contrast, we know a great deal about the preoccupations that had led Einstein to his special theory of relativity. It is all part of the public record of the scientist who performed the role of the professional genius in the collective imagination of the world. We know how, beginning at the age of 16, he used to perform Gedanken-experiments, thought-experiments, imagining himself hitching a ride on a light beam, or running along beside it, trying to deduce how the laws of physics would look from the point of view of an observer moving at the speed of light.
But Gödel’s genius was never put on public display the way Einstein’s was. The sources of his inspiration, the play of mind, revealing how ancient paradox could be transformed into a proof for conclusions shot through with meta-overtones, are unknown. He must somehow have glimpsed the metamathematical potential of logic, even when logic was, as it was then, far less mathematically respectable than his own work would render it. We do not know exactly when he proved his first incompleteness result. Not even his dissertation advisor (he had by then advanced to doing graduate work) knew what he was up to. But we do know that by 7 October 1930 he had the proof for the first incompleteness theorem.
The logician Jaakko Hintikka wrote:
It is a measure of Gödel’s status that the most important moment of his career is the most important moment in the history of twentieth-century logic, maybe in the history of logic in general. This Sternstunde was October 7, 1930. The setting was a conference on the foundations of mathematics in Königsberg on October 5–7, 1930.
What happened at Königsberg on 7 October 1930 was that Kurt Gödel, a relatively unknown graduate student attending a conference on metamathematics dominated by the leaders in the field, dropped a parsimonious few words indicating that he had a proof for the incompleteness of arithmetic. He was basically ignored by everyone present, with the exception of one mathematician, who happened to be there to represent a metamathematical position deeply at odds with Gödel’s Platonism but who was astute enough to draw for himself the implications of Gödel’s wildly muted “announcement.”
Yet there is something wrong with what Hintikka says. The most important moment in Gödel’s career did not come in the public revelation of the first incompleteness theorem. That moment just seems like the most important because that is when Gödel gave some slight public indication of what he had been up to. The most important moments of his career were, in fact, those about which we know nothing: the moments of intuitions or thought-experiments or God-knows-what that brought him to the proof itself.
His Platonist conviction must have convinced him, sans proof, that mathematical reality must exceed all formal attempts to contain it; but how did he lay hands on the strategy by which to prove incompleteness? How did it occur to him, in particular, to transform the structural features of self-referential paradoxes into a proof? How did the inspired idea of what we now call “Gödel numbering” come to him, the technique by means of which statements of mathematics would acquire double-entendres, making metamathematical statements as well? The overall strategy of the proof is astoundingly simple, the details that had to be worked out are astoundingly complicated, and both astounding features make us wish we knew more about how he came up with it all. But all we have is the result: the proof that forever changed our understanding of mathematics, and, in doing so, perhaps helped to change our understanding of ourselves.
So it is not Königsberg that is the scene of the real drama but rather Vienna—the Vienna of the late twenties and early thirties, a city utterly unique in its cultural and intellectual aspects. No thinker reflects in an utter vacuum—not even the purest of pure mathematicians up there on the topmost turret of Reine Vernunft. Not even a thinker so unwaveringly loyal to the integrity of his own intuitions as Kurt Gödel is utterly indifferent, if even in the spirit of opposition, to the prevailing opinions of his day, to the sorts of questions floating like spores in the intellectual atmosphere.
The city of Vienna in that period between the two world wars, the strange intensity of the thinking and the creating that were pursued there, plays its role in the story of Gödel’s theorems. Vienna was then a city with a hugely disproportionate number of seminal thinkers and artists—scientists, musicians, poets, visual artists, philosophers, architects—who collectively seemed drawn into one sustained and intense conversation, pursued across every discipline and art form. Gödel, as reticent as he was, also en
ded up participating in this conversation.
Here, in this highly dramatic city, in which even intellectual life achieved a theatricality, even Gödel, the last person in the world to seek outward drama, attained a certain degree of it.
Out from the Muddle of the Old: A City in Search of New Foundations
If Princeton is a high-energy intellectual vortex disguising itself as a pleasantly bland spot on the suburban New Jersey landscape, the Vienna of the 1920s, when Kurt Gödel arrived there as a student, was “the research laboratory for world destruction” in the famous words of one of its contemporary chroniclers, the journalist and satirist Karl Kraus. The novelist Herman Kesten saw it as a city of “brilliant creation in a nonetheless decaying culture.”
The teeming intellectual life of the city was carried out in broad sight, not only in university lecture halls and professors’ offices but also in the numerous cafés that seem still to display the essence of Viennese life. Much had changed in the city, and the country as a whole, after the First World War and the collapse of the Hapsburg Empire in 1916. But Vienna remained, in its feel, a small large city—the undisputed cultural center of its country. The sense that it was almost an entirely self-enclosed entity within the greater country, sharing little in the way of outlook with the rest of the population, has perhaps a counterpart in contemporary New York City’s relationship with the United States, though the discontinuity between city and general culture seems to have been far greater in the case of Vienna and Austria.
In Gödel’s day, Vienna still had the only real university in Austria, and this was contained almost entirely in one building. This physical concentration of academic life was indicative of the intellectual life in general. It was a city whose thinkers all seemed to be at least marginally acquainted with one another, influencing each other’s thinking across disciplines, so that mathematicians, physicists, historians, philosophers, novelists, poets, musicians, architects, and artists were engaged, in a sense, in the same conversation. The overall topic was the moral and intellectual death and decay of all that had come before, and the need to construct entirely new methodologies, forms, and foundations. It was this sustained theme in so many diverse fields that provoked the emergence of what we have come to call modernity, and even postmodernity: in literature, music, architecture, art, philosophy, psychology, and even, to some extent, science.
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