After Minkowski's arrival in Gottingen in the fall of 1902, Hilbert was no longer lonely: "A telephone call, or a few steps down the street, a pebble tossed up against the little corner window of his study, and there he was, always ready for any mathematical or non-mathematical undertaking."
Instead of conducting seminars with Klein, Hilbert now conducted them with Minkowski.
On Sunday mornings the two friends regularly set out with their wives on a picnic excursion.
The Hilberts had by this time left the Reformed Protestant Church in which they had been baptized and married. It was told in Gottingen that when Franz had started to school he could not answer the question, "What religion are you?" "If you do not know what you are," he was informed by the son of the philosopher Edmund Husserl, a Jew recently converted to Christianity, "then you are certainly a Jew."
The Sunday excursions were later enlarged to include the children of both families. Most frequently the destination was a resort called Mariaspring, where there was dancing outside under the trees. Here Hilbert would seek out his current "flame" - the pretty young wife of some colleague - and whirl her around the dance floor, much to the embarrassment of the little Minkowski girls, who found his energetic dancing very old fashioned. "It was a sport to him!" They were further ,embarrassed when, after the music stopped, he enveloped his partner in his great loden cape and made a show of hugging and kissing her.
The frequent parties at the Hilbert house were now more enjoyable for Hilbert because of the quiet presence of Minkowski. There was dancing at these affairs too - the rug rolled away, music furnished by the gramophone
a manufacturer had presented to the famous mathematics professor, the commands given in French by Hilbert. The table was always ladened with a variety of food, but the staple was talk. A subject would come up, somebody would ask Hilbert what he thought about it. Astrology, for instance. What did he think about that? Without an instant's hesitation he would answer firmly in the still uncorrupted East Prussian accent which made everything he said sound amusing and memorable: "When you collect the 10 wisest men of the world and ask them to find the most stupid thing in existence, they will not be able to find anything stupider than astrology!" Perhaps the guests would be discussing Galileo's trial and someone would blame Galileo for failing to stand up for his convictions. "But he was not an idiot," Hilbert would object. "Only an idiot could believe that scientific truth needs martyrdom - that may be necessary in religion, but scientific results prove themselves in time." Minkowski did not offer his opinion so frequently as Hilbert. When he did speak, his observation - often in the form of an appropriate quotation from Faust - went to the heart of the matter, and Hilbert listened. But Hilbert was always the more intrepid in expressing opinions. What technological achievement would be the most important? "To catch a fly on the moon." Why? "Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind." What mathematical problem was the most important? "The problem of the zeros of the zeta function, not only in mathematics, but absolutely most important!"
Sometimes the little Hilbert boy Franz, who was not shown at parties, would stop at the door and listen.
In front of a group, Minkowski suffered from "Lampenfieber" - in English, stagefright. He was still embarrassed by attention, even of much younger people; and in Zurich his shy, stammering delivery had completely put off a student named Albert Einstein. But in G6ttingen ("the shrine of pure thought," as it was called) the students recognized immediately that in Minkowski they had the privilege of hearing "a true mathematical poet." It seemed to them that every sentence he spoke came into being as he spoke it.
This was, at least once, quite literally true. Lecturing on topology, Minkowski brought up the Four Color Theorem - a famous unsolved problem in that field of mathematics. (The theorem states that four colors are always sufficient to color any map in such a way that no two adjoining regions will have the same color.)
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"This theorem has not yet been proved, but that is because only mathematicians of the third rank have occupied themselves with it," Minkowski announced to the class in a rare burst of arrogance. "I believe I can prove it."
He began to work out his demonstration on the spot. By the end of the hour he had not finished. The project was carried over to the next meeting of the class. Several weeks passed in this way. Finally, one rainy morning, Minkowski entered the lecture hall, followed by a crash of thunder. At the rostrum, he turned toward the class, a deeply serious expression on his gentle round face.
"Heaven is angered by my arrogance," he announced. "My proof of the Four Color Theorem is also defective."
He then took up the lecture on topology at the point where he had dropped it several weeks before. (The Four Color Theorem remained unproved until 1976.)
Hilbert was now beginning to devote himself to integral equations with the same exclusiveness which he had earlier lavished on invariants and number fields. He had begun his investigations in a way reminiscent of his approach in the earlier subjects. In his first paper, sent as a communication to the Gottingen Scientific Society, he had presented a simple and original derivation of the Fredholm theory which exposed the fundamental idea more clearly than Fredholm's own work. There were also already glimpses of fresh, fruitful ideas to come. With an intuitive grasp of the underlying relationships existing among the different parts of mathematics and between mathematics and physics, Hilbert had recognized that Fredholm equations could open up a whole series of previously inaccessible questions in analysis and mathematical physics. It was now his goal to encompass within a uniform theoretical arrangement of equations the greatest possible domain of the linear problems of analysis.
Minkowski was occupying himself again with his beloved theory of numbers. It concerned him, according to Hilbert, that many mathematicians hardly get a breath of what he called the "special air" of number theory; and during the winter of 1903-04 he delivered a series of relatively untechnical lectures, later published as a book, in which he presented the methods he had created and some of his own most significant results in a way in which they could be easily grasped. Hilbert was just as interested as Minkowski in emphasizing "the insinuating melodies of this powerful music" - this metaphor was also Minkowski's - and when Legh Reid, one of his former American students, wrote a book on the subject, Hilbert
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endorsed it with enthusiasm. Number theory was "the pattern for the other sciences, ... the inexhaustible source of all mathematical knowledge, prodigal of incitement to investigations in all other domains .... " A number theory problem was never dated, it was "as timeless as a true work of art." Thanks to Minkowski, Germany had recently become, once again, the number theory center of the world. "But every devotee of the theory of numbers will desire that it shall be equally a possession of all nations and be cultivated and spread abroad, especially among the younger generation, to whom the future belongs."
During 1903, Hermann Weyl arrived in Gottingen. He was an 18 year old country boy, seemingly inarticulate, but with lively eyes and a great deal of confidence in his own abilities. He had chosen the University because the director of his gymnasium was the cousin of one of the mathematics professors "by the name of David Hilbert."
"In the fullness of my innocence and ignorance," Weyl wrote many years later from the Institute for Advanced Study in Princeton, New Jersey, "I made bold to take the course Hilbert had announced for that term, on the notion of number and the quadrature of the circle. Most of it went straight over my head. But the doors of a new world swung op~n for me, and I had not sat long at Hilbert's feet before the resolution formed itself in my young heart that I must by all means read and study whatever this man had written."
Hilbert's "optimism, his spiritual passion, his unshakable faith in the supreme value of science, and his firm confidence in the power of reason to find simple and clear answers to simple and clear questions" were irresistible. Weyl heard "the sweet flute of the Pied Piper ... , seducing so many rats to follow him into the deep river of mathematics." That summer he went home with a copy of the Zahlbericht under his arm and, although he did not have any previous knowledge of the mathematics involved, worked his way through it during the vacation.
He was a young man with a taste tor words as well as for mathematics, and he found the peculiarly Hilbertian brand of thinking admirably reflected in the great lucidity of Hilbert's literary style:
"It is as if you are on a swift walk through a sunny open landscape; you look freely around, demarcation lines and connecting roads are pointed out to you before you must brace yourself to climb the hill; then the path goes straight up, no ambling around, no detours."
The summer months spent studying the Zahlbericht were, Weyl was always to say, the happiest months of his life.
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It was also at this time, during Minkowski's first years, that Max Born, the son of a well-known medical researcher in Breslau, came to Gottingen on the advice of his friends, Ernst Hellinger and Otto Toeplitz. They had informed him that Gottingen was now "the mecca of German mathematics."
Born's stepmother had known Minkowski in Konigsberg; and not long after his arrival, the new student was invited to lunch by the professor and presented to Guste Minkowski and the two little daughters. After lunch, Hilbert and Kathe came over and the whole party hiked to Die Plesse, a ruined castle overlooking the valley of the Leine and the red-tiled roofs of Gottingen.
Born was never to forget the afternoon. "The conversation of the two friends was an intellectual fireworks
display. Full of wit and humor and still also of deep seriousness. I myself had grown up in an atmosphere to which spirited discussion and criticism of traditional values was in no way foreign; my father's friends, most of them medical researchers like himself, loved lively free conversation; but doctors are closer to everyday life and as human beings simpler than mathematicians, whose brains work in the sphere of highest abstraction. In any case, I still had not heard such frank, independent, free-ranging criticism of all possibile proceedings of science, art, politics."
To Weyl and Born and the other students it seemed that Hilbert and Minkowski were "heroes," performing great deeds, while Klein, ruling above the clouds, was "a distant god." The older man now devoted more and more of his time and energy to the realization of his dream of Gottingen as the center of the scientific world. Before the turn of the century he had brought together economic leaders and scientific specialists in an organization called the Gottingen Association for Advancement of Applied Mathematics and Mechanics. As a result of the activities of this group (familiarly known as the Gottingen Association) the University was gradually being ringed by a series of scientific and technical institutes - the model for scientific-technological complexes which were later to grow up around various universities in America.
Sometimes Klein was now a little amusing for the seriousness with which he took himself and his many projects. It was said that he had but two jokes, one for the spring semester and one for the fall. He did not allow himself the pleasures of ordinary men. Every moment was budgeted. Even his daughter had to make an appointment to talk with her father.
Without ever making an issue of the matter, Hilbert and Minkowski saw to it that they themselves were never organized. Once, after Klein had
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completely filled a very large blackboard with figures on the German middle schools (the scientific education of which he was also trying to reform), he asked his colleagues if they had any comments. "Doesn't it seem to you, Herr Geheimrat," Minkowcki asked softly, "that there is an unusually high proportion of primes among those figures?" On another occasion, when Klein, written agenda in hand, tried to turn the informal weekly walks of the mathematics professors into department meetings, Hilbert simply failed to turn up for the next walk. But for the most part the three men, so different in their personalities, worked together in rare harmony.
In 1904, when the Extraordinariat for applied mathematics became vacant, Klein proposed to Althoff that a full professorship in that subject be established, the first in Germany specifically designated for applied mathematics. For the new position he had in mind Carl Runge, then at Hannover. Runge was not only a distinguished experimental physicist, famous for his measurement of spectral lines, but also a first-rate mathematician, whose name is attached to the approximation of analytic functions by means of polynomials.
Runge had known and admired Klein for almost ten years and had recently become acquainted with Hilbert. "Hilbert is a charming human being," he had written his wife. "His idealism and his friendly disposition and unassuming honesty cause a person to like him very much." The possibility of associating with these two gifted mathematicians was so exciting to Runge, who had felt very much alone in Hannover, that he went to Berlin to talk to Althoff about the new position with the feeling that it was too good to be true. "However," as his daughter later wrote, "he had not reckoned with the fact that for Klein's most broad and comprehensive plans there was always more sympathy from Althoff than for the individual plan of another." The new position was his if he wanted it. The salary would be somewhat less than what he had been receiving in Hannover, however.
"But you must not let yourself be influenced by financial considerations," his wife wrote emphatically when she heard the news. "We will come through, even with a thousand Marks less, and it will not hurt either me or the children."
At the beginning of the winter semester 1904-05, Runge joined the faculty. The mathematics professors, a quartet now, took up the practice of a weekly walk every Thursday afternoon punctually at three o'clock. Klein gave up preparing agendas. The walks became pleasant informal rambles during the course of which anything, including department busi-
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ness, could be discussed; and, as Hilbert happily remarked, "science did not come out too short."
Runge had a gift for computation which impressed even his new colleagues. Once, when they were trying to schedule a conference several years in the future, it became necessary to know the date of Easter. Since the determination of Easter is no simple matter, involving as it does such things as the phases of the moon, the mathematicians began to search for a calendar. But Runge merely stood silent for a moment and then announced that Easter that year would fall on such and such a date.
It was equally amazing to the mathematicians how Runge was able to handle mechanics. When the Wright brothers made their first flight, he constructed a model of their plane from paper scraps which he weighted down with needles and then allowed to glide to the ground. In this way he estimated "rather correctly" the capacity of the motor, the details of which were still a secret.
When Runge arrived in G6ttingen, the scientific faculty most closely connected with the mathematicians was also impressive. Physicists were Eduard Riecke and Woldemar Voigt. H. T. Simon was the head of the Institute for Applied Electricity; Ludwig Prandtl, of the Institute for Applied Mechanics; and Emil Wiechert, the Institute for Geophysics. Karl Schwarzschild was professor of astronomy.
But the stimulating society was not limited to these high-ranking men. Otto Blumenthal, who was to be distinguished for the rest of his life
as "Hilbert's oldest student," was very close to the professors although he was still merely a Privatdozent. He was a gentle, fun-loving, sociable young man who spoke and read a number of languages and was interested in literature, history and theology as well as mathematics and physics. Although born a Jew, he eventually became a Christian and frequently spoke of "we Protestants."
The unusually close relationship existing between the docents and the professors is evidenced by the fact that when Blumenthal and Ernst Zerme-10, another docent, wished to give some experimental lectures on elementary arithmetic, Hilbert and Minkowski regularly attended to give more authority to their project.
Zermelo was somewhat older than Blumenthal, a nervous, solitary man who preferred whisky to company. He liked to prove at this time, which was before Peary's expedition, the impossibility of reaching the North Pole. The amount of whisky needed to reach a latitude, he maintained, is proportional to the tangent of the latitude, i. e., approaches infinity at the
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Pole itself. When newcomers to Gottingen asked him about his curious name, he told them, "It used to be Walzermelodie, but then it became necessary to discard the first syllable and the last."
It was Zermelo who had recently pointed out to Hilbert a disturbing antinomy in set theory - the same antinomy which the young English logician Bertrand Russell pointed out to Gottlob Frege as Frege was ready to send to the printer his definitive work on the foundations of arithmetic. This antinomy - a contradiction reached by using methods of reasoning which had been accepted by mathematicians and everyone else since the time of Aristotle - had to do with the commonly recognized fact that some sets are members of themselves and others are not. For instance, the set of all sets having more than three members is a member of itself because it has more than three members. On the other hand, the set of all numbers is not a member of itself, since it is not a number. But now Zermelo and Russell, independently, had brought up the question of the set of all sets that are not members of themselves. Since the members of this set are the sets which are not members of themselves, the set is a member of itself if, and only if, it is not a member of itself.
By 1904, after its publication by Russell, the antinomy was having - in Hilbert's opinion - a "downright catastrophic effect" in mathematics. One after another, the great gifted workers in set theory - Frege himself as well as Dedekind - had all withdrawn from the field, conceding defeat. The simplest and most important deductive methods, the most ordinary and fruitful concepts seemed to be threatened; for this antinomy and others had appeared simply as a result of employing definitions and deductive methods which had been customary in mathematics. Even Hilbert had now to admit that perhaps Kronecker had been right - the ideas and methods of classical logic were in fact not equal to the strong demands of set theory.
In the past Hilbert had believed that Kronecker's doubts about the soundness of set theory and certain parts of analysis could be removed by substituting consistency, or freedom from contradiction, for construction by means of integers as the criterion of mathematical existence and then obtaining the necessary absolute proof of the consistency of the arithmetic of real numbers. Up until the discovery of the antinomies it had been his opinion that the desired consistency proof could be achieved relatively easily by a suitable modification of known methods of reasoning in the theory of irrational numbers. But with the discovery of the antinomies in set theory, upon which much of this reasoning was based, he saw that he was going to have to alter his view. In the late summer of 1904, when the third Inter-
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national Congress of Mathematicians met at Heidelberg, he departed from integral equations for the moment to take up the subject of the foundations of mathematics.
It had been Kronecker's contention that the integer was the foundation of arithmetic and that construction by means of a finite number of integers was therefore the only possible criterion of mathematical existence. Hilbert still, as always, passionately opposed such a restriction of mathematics and mathematical methods. Like Cantor, he firmly believed that the essence of mathematics is its freedom; and he saw in arbitrary restriction a real danger to the science. He was sure that there was a way to eliminate the antinomies without making the sacrifices which Kronecker's views demanded. His solution, however, involved going even farther than Kronecker had.
Hilbert now insisted that the integer itself "can and must" have a foundation.
"Arithmetic is often considered to be a part of logic, and the traditional fundamental logical notions are usually presupposed when it is a question of establishing a foundation for arithmetic," he told the mathematicians gathered at Heidelberg. "If we observe attentively, however, we realize that in the traditional exposition of the laws of logic certain fundamental arithmetical notions are already used; for example, the notion of set and, to some extent, also that of number. Thus we find ourselves turning in a circle, and that is why a partly simultaneous development of the laws of logic and of arithmetic is required if the antinomies are to be avoided."
He was convinced, he told them, that in this way "a rigorous and completely satisfying foundation" could be provided for the notion of number -"number" which would include, not only Kronecker's natural numbers and their ratios (the common fractions), but also the irrational numbers, to which Kronecker had always so violently objected, but without which "the whole of analysis," in Hilbert's opinion, "would be condemned to sterility."
It was Hilbert's proposal at Heidelberg that for the first time in the history of mathematics, proof itself should be made an object of mathematical investigation.
Poincare commented several times, unfavorably, upon the idea. The Frenchman was convinced that the principle of complete, or mathematical, induction was a characteristic of the intellect ("in Kronecker's language," as Hilbert once explained in discussing Poincare's position, "created by God") and that, therefore, the principle could not be established except by complete induction itself.
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Hilbert did not follow up the Heidelberg proposal. Instead, he continued to work on his theory of integral equations and on the side, in the company of Minkowski and at Minkowski's suggestion, began a study of classical physics.
Minkowski already had considerable technical knowledge in the field of physics; Hilbert had almost none and was familiar only with the broad outlines of the subject. Nevertheless he took it up with enthusiasm. For the second time since leaving school- the first time had been in connection with the Zahlbericht - he embarked on a course of "book study." Blumenthal, who had already begun what was to be a life-long study of his teacher's character and personality, was most impressed. He remembered an occasion during his own student days when in the course of his reading he had discovered to his dismay that the most beautiful development in his dissertation had already appeared in another paper. Hilbert, he recalled, had merely shrugged and said, "Why do you also know so much literature?"
Klein followed the joint physics study with interest. When he was 17, he had been assistant to Julius Plucker in Bonn. At that time he had been determined that "after the attainment of the necessary mathematical knowledge" he would devote his life to physics. Then, two years later, Plucker had died. A transfer to Gottingen - where the mathematicians were a far livelier group than the physicists - had made Klein a mathematician instead of a physicist.
As the physics study progressed, Minkowski became increasingly fascinated by the riddles of electrodynamics as recently formulated in the works of H. A. Lorentz. But Hilbert did not waver in his personal concentration on integral equations. In 1904 he sent a second communication to the scientific society in which he developed a significant extension of Fredholm's idea. In his classic work, Fredholm had recognized the analogy between integral equations and the linear equations of algebra. Hilbert now went on to set up the analogue of the transformation of a quadratic form of n variables onto principal axes. Out of the resulting combination of analysis, algebra and geometry, he developed his theory of eigenfunctions and eigenvalues, a theory which, as it turned out, stood in direct relation to the physic:.l theory of characteristic oscillations.
The spirit and significance of this work can best be glimpsed by a layman in the evaluation of it by a later student of Hilbert's:
"The importance of scientific achievement is often not alone in the new material which is added to material already on hand," Richard Courant
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has written. "Not less important for the progress of science can be an insight which brings order, simplicity and clarity into an existing but hard to reach area and thus facilitates or first makes possible the survey, comprehension and mastery of the science as a unified whole. We should not forget this point of view in connection with Hilbert's works in the field of analysis, ... for [all of these] exemplify his characteristic striving to find in the solution of new problems the methods which make the old difficulties easy, to establish new connections in existing materials and to bring the many branching streams of individual investigations back into a single bed."
It was during this happy, productive time that Hilbert received yet another offer to leave Gottingen. Leo Koenigsberger would give up his own chair in Heidelberg if Hilbert would accept it.
Although Kiithe favored the change, Hilbert refused. He did not neglect, however, to use the offer to negotiate for further
advantages for mathematics as the price of his remaining in Gottingen. To one of his proposals Althoff objected: "But we do not have that even in Berlin!"
"Ja," Hilbert replied happily, "but Berlin is also not Gottingen!"
