Hilbert

be an outstanding number theorist - in this area of mathematics, the Min-

kowski of the new generation. He had refused to serve in the army and had
been confined in a mental institution which was located next to the clinic
owned by Landau's father. Thus young Carl Ludwig Siegel, extremely
gifted but without money, had become acquainted with the Gottingen
professor. To Siegel, Landau presented a quite different picture from the
spoiled cherub whom Norbert Wiener saw at about this same time.
"If it had not been for Landau," Siegel says simply, "I would have died."
When, however, Siegel came to Gottingen as a student in 1919, he worked
almost entirely alone. "I was very eager to show what 1 could do by myself."
He had no direct personal contact with Hilbert, but he was always to re-
member a lecture on number theory which he heard from Hilbert at this
time. Hilbert wanted to give his listeners examples of the characteristic
problems of the theory of numbers which seem at first glance so very simple
but turn out to be incredibly difficult to solve. He mentioned Riemann's
hypothesis, Fermat's theorem, and the transcendence of 2V2" (which he had
listed as his seventh problem at Paris) as examples of this type of problem.
Then he went on to say that there had recently been much progress on
Riemann's hypothesis and he was very hopeful that he himself would live
to see it proved. Fermat's problem had been around for a long time and
apparently demanded entirely new methods for its solution - perhaps
the youngest members of his audience would live to see it solved. But as
for establishing the transcendence of 2V2, no one present in the lecture hall
would live to see that!
The first two problems which Hilbert mentioned are still unsolved. But
less than ten years later a young Russian named Gelfond established the
transcendence of 2v=2. Utilizing this work, Siegel himself was shortly able
to establish the desired transcendence of 2V2.
Siegel wrote to Hilbert about the proof. He reminded him of what he
had said in the lecture of 1920 and emphasized that the important work was
that of Gelfond. Hilbert was frequently criticized for "acting as if every-
thing had been done in Gottingen." Now he responded with enthusiastic
delight to Siegel's letter, but he made no mention of the young Russian's
contribution. He wanted only to publish Siegel's solution. Siegel refused,
certain that Gelfond himself would eventually solve this problem too.
Hilbert immediately lost all interest in the matter.
After a semester in Hamburg with Heeke, who was now a professor
there, Siegel returned to Gottingen as assistant to Courant and later became
a Privatdozent. The money he earned was so little that Courant, who wanted

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a cycling companion, had to arrange an extra stipend so that Siegel could

afford to buy a bicycle.
Courant liked to keep Klein and Hilbert in touch with the gifted young
people. It was through him that Siegel had his first personal contacts with
the famous mathematicians of Gottingen. Because of the post-war housing
shortage he lived for a while at Klein's house. But even living under the
same roof he felt the distance which people had always felt between them-
selves and Klein. He worried constantly that he "would say the wrong
thing." Later he was taken by Courant to swim in that part of the Leine
river which was roped off for the faculty. He met Hilbert in the little shed
where the professors changed into their bathing suits. Young Siegel,
Courant explained to Hilbert, had recently found another proof of a theorem
of Heeke's connected with the Riemann hypothesis. Hilbert was very
enthusiastic. "He always liked to make young people feel hopeful." In the
bathhouse there with Hilbert, Siegel felt none of the constraint he had felt
in Klein's house.
Soon after this meeting with Hilbert, Siegel was asked by Courant to
referee a paper for the Annalen, of which Hilbert was still one of the principal
editors. The young man found the paper inaccurate in many places and,
even where accurate, unnecessarily roundabout in its methods. He reported
to Hilbert that in his opinion the paper was not publishable.
"No, no, I must publish it!" Hilbert insisted. "In 1910 this man was a
member of the committee that gave me the Bolyai Prize, and now I simply
cannot refuse to publish his paper! Take it and change whatever should be
changed. But I must publish it!"
The paper appeared in an improved form in the Annalen. Several months
later, when Siegel was sure that Hilbert had forgotten all about the matter,
a package was delivered to his rooms. It contained the two volumes of
Minkowski's collected works, inscribed "With friendly thoughts from the
editor."
One of the most fertile circles of research in post-war Gottingen revolved
around Emmy Noether. The desired position of Privatdozent had at last
been obtained for her in 1919. This was still the lowest possible rank on the
university scale, not a job but a privilege. But Emmy Noether was delighted
with the appointment. In the thirteen years which had passed since she had
had to defend her doctoral dissertation before Gordan, she had come a
long way. Already she had achieved important results in differential invar-
iants, which the Soviet mathematician Paul Alexandroff was to consider
sufficient to secure her a reputation as a first-rate mathematician, "hardly

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less a contribution to mathematical science than the notable researches of

Kowalewski." She herself was always to dismiss these works as standing
to the side of her main scientific path, on which at last she was now, at the
age of 39, taking her first step - the building up on an axiomatic basis of a
completely general theory of ideals. This work would have its source in the
early algebraic work of Hilbert, but in her hands the axiomatic method
would become no longer "merely a method for logical clarification and
deepening of the foundations [as it was with Hilbert] but a powerful weapon
of concrete mathematical research." Gordan's picture still hung over her
desk in Gottingen; but although she had been so thoroughly under his
influence in her youth that her dissertation had concluded with a table of
the complete system of covariant forms for a given ternary quartic and had
contained more than three hundred forms in symbolic representation - a
maiden work which she in later years dismissed as "Formelgestriipp!" -
a jungle offormulas - she was destined in the next decade to make Hilbert's
"theology" look like mathematics.
In 1922 she became a "nicht beamteter ausserordentlicher Professor"-
an unofficial extraordinary, or associate, professor. There were no obliga-
tions connected with this new title - and no salary, such an extraordinary
professor being considered more than usually inferior to an ordinary
professor. The title could be explained only by a Gottingen saying to the
effect that "an extraordinary professor knows nothing ordinary and an
ordinary professor knows nothing extraordinary." By this time, however,
inflation had so reduced the students' ability to pay fees that if the Privat-
dozents were not to starve away they had to be given some small sums by
the University for delivering lectures in their specialties. Such a "Lehrauf-
trag" for algebra was now awarded to Emmy Noether, the first and only
salary she was ever to be paid in Gottingen.
She and her work were not on the whole much admired in her native
land. She was never even elected to the Gottingen Scientific Society. "It
is time that we begin to elect some people of real stature to this society,"
Hilbert once remarked at a meeting. "Ja, now, how many people of stature
have we indeed elected in the past few years?" He looked thoughtfully
around at the members. "Only - zero," he said at last. "Only zero!"
A Dutchman, attending one of Emmy Noether's lectures for the first
time, remembers that she greeted him, "Ah, another foreigner! I get only
foreigners!" But among the foreigners who came to her were van der
Waerden from Holland, Artin from Austria, Alexandroff from Russia.
It was Alexandroff who christened her ceder Noether", der being the

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definite article which precedes all masculine nouns in German. But he later
said: "Her femininity appeared in that gentle and subtle lyricism which
lay at the heart of the far-flung but never superficial concerns which she
maintained for people, for her profession, and for the interests of all man-
kind."
She was not a good lecturer and her classes usually numbered no more
than five or ten. Once though, she arrived at the appointed hour to find more
than a hundred students waiting for her. "You must have the wrong class,"
she told them. But they began the traditional noisy shuffling of the feet
which, in lieu of clapping, preceded and ended each university class. So
she went ahead and delivered her lecture to this unusually large number of
students. When she finished, a note was passed up to her by one of her
regular students who was in the group. "The visitors," it read, "have
understood the lecture just as well as any of the regular students."
It was true, she had no pedagogical talents. Her mind was open only to
those who were in sympathy with it. Her teaching approach, like her think-
ing, was wholly conceptual. The German letters which she chalked up on
the blackboard were representatives of concepts. It seemed to van der
Waerden that "her touching efforts to clarify these, even before she had
quite verbalized them ... had the opposite effect." But of all the new genera-
tion in Gottingen, Emmy Noether was to have the greatest influence on
the course of mathematics.
While these widening circles of varying mathematical activity were form-
ing themselves around Courant, Landau and Emmy Noether, a group of
exceptionally gifted young physicists were gathering around Max Born,
who (like Courant, still in his thirties) had become professor of theoretical
physics after the war. From the beginning, it was Born's goal to establish
at Gottingen a physics institute comparable to Sommerfeld's institute at
Munich. When the opportunity arose, he arranged that his best friend,
James Franck, join him in Gottingen as professor of experimental physics.
It was a stroke reminiscent of Hilbert's bringing Minkowski to the Uni-
versity in 1902. But even before Franck's arrival in 1922, the spectacular
series of students who would make their way to Gottingen during the 1920's
had begun; and Born's first assistants were Wolfgang Pauli and Werner
Heisenberg.
Since the war, the Germans had been barred from most international
scientific gatherings; but now it seemed once again that in Gottingen an
international congress was permanently in session.

167
 xx
The Infinite!

The highpoint of the mathematical week at Gbttingen during the 1920's

was the regular session of the Mathematics Club.
The club was a very informal kind of organization without officers,
members or dues. Anyone with a doctor's degree could come to meetings,
and because of the quality of mathematics at Gbttingen it was always "a
very high class affair." Sometimes the speaker was a distinguished visitor
who reported on his own recent work or that of his students. More often
he was a member of the Gbttingen circle - professor, docent or student.
The bright young newcomers who saw the famous Hilbert in action for
the first time at these events were struck by his slowness in comprehending
ideas which they themselves "got" immediately. Often he did not under-
stand the speaker's meaning. The speaker would try to explain. Others
would join in. Finally it would seem that everyone present was involved in
trying to help Hilbert to understand.
"That I have been able to accomplish anything in mathematics," Hilbert
once said to Harald Bohr, "is really due to the fact that I have always found
it so difficult. When I read, or when I am told about something, it nearly
always seems so difficult, and practically impossible to understand, and then
I cannot help wondering if it might not be simpler. And," he added, with
his still childlike smile, "on several occasions it has turned out that it really
was more simple!"
Some of the young people were irritated by the precious time consumed
by Hilbert's questions; others found it fascinating to watch Hilbert's mind
in action.
"Scientifically, he did not grasp complicated things at a flash and absorb
them. This kind of talent he did not have," Courant explains. "He had to
go to the bottom of things."

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 1924-1925

Hilbert still set high standards of simplicity and clarity for the talks to the
Mathematics Club. His guiding rule for the speaker was "only the raisins
out of the cake." If computations were complicated, he would interrupt
with, "We are not here to check that the sign is right." If an explanation
seemed too obvious to him, he would reprove the speaker, "We are not in
tertia" - tertia being the level of the gymnasium in which the student is
12 to 14 years old.
The brutality with which he could dispose of someone who did not meet
his standards was well known. There were important mathematicians in
Europe and America who dreaded a speech before the Mathematics Club
in Gottingen. It seemed now sometimes to Ostrowski that Hilbert was un-
necessarily rough on speakers-as if he no longer attended so carefully as in the
past to the problem of the superior individual living among lesser individuals.
One young Scandinavian, today highly esteemed, came to Gottingen and
spoke about his work - "really important and beautiful and very difficult"
in Ostrowski's opinion. Hilbert listened and, when the visitor was through,
demanded only, "What is it good for?"
On another occasion he interrupted the speaker with, "My dear colleague,
I am very much afraid that you do not know what a differential equation is."
Stunned and humiliated, the man turned instantly and left the meeting,
going into the next room, which was the Lesezimmer. "You really shouldn't
have done that," everyone scolded Hilbert. "But he doesn't know what a
differential equation is," Hilbert insisted. "Now, you see, he has gone to
the Lesezimmer to look it up!"
Still another time the speaker was the young Norbert Wiener. The
importance of his talk in Gottingen can be gauged by the fact that many
years later in his autobiography he devoted more than a dozen pages to it.
After Wiener's talk to the Mathematics Club, everyone hiked up to Der
Rohns, as was the custom, and had supper together. Hilbert began in a
rambling way during supper to talk about speeches which he had heard
during the years he had been at Gottingen.
"The speeches that are given nowadays are so much worse than they
used to be. In my time there was an art to giving speeches. People thought
a lot about what they wanted to say and their talks were good. But now the
young people cannot give good talks any more. It is indeed exceptionally
bad here in Gottingen. I guess the worst talks in the whole world are given
in Gottingen. This year especially they have been very bad. There have
been - no, I have heard no good talks at all. Recently it has been especially
bad. But now, this afternoon, there was an exception -"

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Hilbert

The young "ex-prodigy" from America prepared himself to accept the

compliment.
"This afternoon's talk," Hilbert concluded, "was the worst there ever
has been!"
In spite of this remark (which was not reported in the autobiography),
Wiener continued to see Hilbert as "the sort of mathematician I [would like]
to become, combining tremendous abstract power with a down-to-earth
sense of physical reality."
The presence of Klein was still felt in Gottingen during the early twenties,
but now like the sunset rather than the high noon-day sun. The editing
of the collected works was completed, each paper accompanied by detailed
notes on the historical context in which it had originated - a history of the
mathematics of his time as well as of his own career. It often seemed to
Courant that Klein felt his own life was also completed. He continued to
take on projects, such as the editing of his war-time lectures on the history
of nineteenth century mathematics, "but with the knowledge that these
would have to be finished by others."
When a young mathematician did not immediately follow up a suggestion,
Klein dismissed him with, "I am an old man, I can't wait."
Young Norbert Wiener went to pay a call on Klein in the spring of 1925.
"The great man sat in an armchair behind a table, with a rug about his
knees. He ... carried about him an aura of the wisdom of the ages ... and
as he spoke the great names of the past ceased to be the mere shadowy authors
of papers and became real human beings. There was a timelessness about
him which became a man to whom time no longer had a meaning."
The 1920's were "the beautiful years" when modern physics was develop-
ing at an almost magical rate within a triangle which had as its vertices
Cambridge, Copenhagen and Gottingen. The 20-year-old Werner Heisen-
berg, still wearing the khaki shorts of the Youth Movement, came from
Munich to Gottingen in 1921. He recalls himself as "much impressed" by
the number of young physicists who were interested in the particular
problem that was currently interesting Hilbert - "a problem which at that
time exceeded by far my own mathematical and physical knowledge."
Hilbert had recently returned to his war-time ideas on relativity; and for a
while, according to Weyl, hopes ran high in the Hilbert circle of a unified
field theory. But, on the whole, it was Hilbert's spirit rather than his person
which was felt in physics at this time.
From 1922 on, Hilbert was no longer a physicist. The seminar on the
Structure of Matter, which he had instituted with Debye during the war,

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 1924-1925

was now carried on by Born and Franck. The members included at various
times during the twenties Heisenberg, Wolfgang Pauli, Robert Oppen-
heimer, K. T. Compton, Pascual Jordan, Paul Dirac, Linus Pauling, Fritz
Houtermans, P. M. S. Blackett among others. Hilbert rarely attended.
His own personal achievement in physics had been a disappointment,
"in no way comparable," Weyllater said in summary, "to the mathematical
achievement of any single period of his career." The axiomatization of
physics, which had been his goal when he first began the joint study with
Minkowski, always eluded him.
To Weyl, who himself made important contributions to mathematical
physics, it seemed that "the maze of experimental facts which the physicist
has to take into account is too manifold, their expansion too fast, and their
aspect and relative weight too changeable for the axiomatic method to
find a firm enough foothold, except in the thoroughly consolidated parts of
our physical knowledge. Men like Einstein or Niels Bohr grope their way in
the dark toward their conceptions of general relativity or atomic structure
by another type of experience and imagination than those of the mathemati-
cian, altough no doubt mathematics is an essential ingredient."
Hilbert's real contribution to physics was to lie in the mathematical
methods which he had created in his work on integral equations and in the
unification which this work had brought about. When, at the end of 1924,
Courant published the first volume of his Methods oj Mathematical Physics,
he placed the name of Hilbert on the title page with his own. This act
seemed justified, Courant wrote in the preface, by the fact that much material
from Hilbert's papers and lectures had been used as well as by the hope that
the book expressed some of Hilbert's spirit, "which had such decisive
influence on mathematical research and education."
"Actually it is more than a mere act of dedication that Hilbert's name
stands next to that of Courant on the title page," Ewald pointed out in a
review of the book for Naturwissenschaften. "Hilbert's spirit radiates from
the entire book - that elemental spirit, passionately seeking to grasp
completely the clear and simple truths, pushing trivialities aside and with
masterful clarity establishing connections between the high points of
recognition - a spirit that filled generations of searchers with enthusiasm
for science."
"Courant-Hilbert," as the book immediately became known, represented
a tremendous advance over previous classics of applied mathematics. There
had, in fact, really been nothing like it. In the past theoretical physicists
had for the most part had to obtain their mathematics from the work

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Hilbert

of Rayleigh and other physicists. Now they welcomed "Courant-

Hilbert."
Hilbert continued to have an assistant to keep him informed on the latest
developments in physics. Beginning in 1922, this position was held by
Lothar Nordheim, who like all of the other assistants was chosen for Hil-
bert by Sommerfeld.
Hilbert, in Nordheim's opinion, still had hopes at this time for the achieve-
ment of his goal of the axiomatization of physics. To his assistant, however,
he was no longer the legendary "great thinker." He was not well. He seemed
to live much in the past, had difficulty accepting changes, was prejudiced
in many things, his egoism having become more marked. "He could not
imagine any greater privilege for a young man than to be his assistant."
Nordheim would have preferred a position in Born's institute. Working
with Hilbert now in his home, he felt very much out of the mainstream of
physics.
But in spite of these signs of apparently early aging, Hilbert continued to
maintain his close contacts with youth.
At the same time that Nordheim was coming regularly to Hilbert's
house, another young man was also a frequent visitor. John von Neumann
had studied in Berlin with Erhard Schmidt, Hilbert's former student who,
at the beginning of the century, had so significantly forwarded the Hilbert
work on integral equations. He was a young man who was in one respect
at least the exact opposite of Hilbert. Whereas Hilbert was "slow to under-
stand," von Neumann was equipped with "the fastest mind I ever met,"
according to Nordheim. He frequently expressed his opinion that the mathe-
matical powers decline after the age of 26, but that a certain prosaic shrewd-
ness developing from experience manages to compensate for this gradual
loss. (During his own life, he slowly raised the limiting age.)
Von Neumann was 21 in 1924, deeply interested in Hilbert's approach
to physics and also in his ideas on proof theory. The two mathematicians,
more than forty years apart in age, spent long hours together in Hilbert's
garden or in his study.
Hilbert's real collaborator during these days, however, was Bernays. To
some people it seemed that he was even exploiting his logic assistant.
Bernays was no young student but a man in his middle thirties, a mature
mathematician. As Hilbert's assistant, he received a salary and, having
habilitated shortly after his arrival in Gottingen, also received fees from the
students who attended his lectures. He could live on what he received, but
certainly he could not marry.

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 1924-1925

Hilbert was very opposed to marriage for young scientists anyway. He

felt that it kept them from fulfilling their obligations to science. Later,
when Wilhelm Ackermann, with whom he had worked and collaborated
on a book, married, Hilbert was very angry. He refused to do anything
more to further Ackermann's career; and as a result, not obtaining a uni-
versity position, the gifted young logician had to take a job teaching in a
high school. When, sometime later, Hilbert heard that the Ackermanns
were expecting a child, he was delighted.
"Oh, that is wonderful!" he said. "That is wonderful news for me.
Because if this man is so crazy that he gets married and then even has a
child, it completely relieves me from having to do anything for such a
crazy man!"
In addition to preparing his own lectures, Bernays helped Hilbert prepare
his lectures, accompanied him to class and often took over the teaching for
part of the hour, supervised Hilbert's students who were working for the
doctoral degree, studied and digested the literature necessary for their work,
and did a great deal of writing on their joint book, which was to be entitled
Grundlagen der Mathematik. In Bernays, Hilbert had found someone as
interested in the foundations of mathematics as he was. He had no compunc-
tion about working his assistant as hard as he worked himself. "Genius is
industry," he liked to tell his students and his assistants, quoting Lichten-
berg. He himself was, as Weyllater recalled, "enormously industrious."
The two men sometimes, however, got into rather violent arguments
over the subject of foundations. Bernays attributes the emotional quality of
these arguments to a fundamental "opposition" in Hilbert's feelings about
mathematics.
"For Hilbert's program," he explains, "experiences out of the early part
of his scientific career (in fact, even out of his student days) had considerable
significance; namely, his resistance to Kronecker's tendency to restrict
mathematical methods and, particularly, set theory. Under the influence of
the discovery of the antinomies in set theory, Hilbert temporarily thought
that Kronecker had probably been right there. But soon he changed his
mind. Now it became his goal, one might say, to do battle with Kronecker
with his own weapons of finiteness by means of a modified conception of
mathematics ....
"In addition, two other motives were in opposition to each other - both
strong tendencies in Hilbert's way of thinking. On one side, he was con-
vinced of the soundness of existing mathematics; on the other side, he had
- philosophically - a strong scepticism."

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Hilbert

An example was Hilbert's attitude toward the question of the solvability

of every definite mathematical problem. At Paris he had spoken in ringing
tones of the axiom of the solvability of every problem, "the conviction
which every mathematician ~hares, although it has not yet been supported
by proof." He was convinced that in mathematics at least "there is no
ignorabitnus." Yet, at Zurich, he listed among the epistemological questions
which he felt should be investigated the question of the solvability in principle
of every mathematical question.
"The problem for Hilbert," Bernays explains, "was to bring together
these opposing tendencies, and he thought that he could do this through the
method of formalizing mathematics."
Bernays did not always agree with Hilbert about their program, but he
appreciated the fact that, passionate though Hilbert was in his disputation,
he never held it against his assistant personally when he took the opposite
side.
After their work was finished, Hilbert and Bernays often argued about
politics. Hilbert enjoyed expressing his views on the subject in extreme and
paradoxical ways.
Although he was generally considered conservative, he surprised every-
body by proposing Kathe Kollwitz, who was known to be very strongly
oriented to the left, for the Star of the Order of Merit, Peace Class. She had
become one of the great women artists of all time. ("I have never seen such
a drawing by the hand of a woman," the sculptor Constantin Meunier
said.) Her subject matter reflected her feeling for the sufferings of humanity.
"Of course what she draws is horrible to look at," Hilbert told his fellow
wearers of the Star. "But when we were young in Konigsberg and used to
dance, she was one of the first girls to dance without her corset!"
In spite of his conservative background, Hilbert was always a liberal in
the sense that he never considered himself bound to any certain political
view. In his arguments with his assistant, he often criticized "liberals"
for seeing things as they wished them to be and not as they were.
"Sometimes," he said, "it happens that a man's circle of horizon becomes
smaller and smaller, and as the radius approaches zero it concentrates on
one point. And then that becomes his point of view."
He liked to remind his younger assistant: "Mankind is always the con-
stant. "
Music often brought peace after the arguments, logical or political.
Bernays loved music and had played "four hands" with Hurwitz when he
was in Zurich. He was impressed by how much Hilbert's musical knowledge

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 1924-1925

and appreciation had developed over the years as a result of his love for
his phonograph, a new model of which was still being supplied regularly
to him by the manufacturer. Now he was a member of a group of professors
and their wives who attended concerts together in Gottingen and traveled
to Leipzig or Hannover for special musical events.
Hilbert sometimes seemed to have very little appreciation of the arts
other than music. Yet he was drawn to literature and, as Courant says,
"wanted to be aware." He appreciated Goethe and Homer, but he insisted
on action in novels. One of his "flames" once set out to educate him in
literature. She began by giving him a historical novel about the Swiss
civil wars, quite a bloody tale. Hilbert shortly returned it. "If I am given a
book to read," he said, "it should be one in which something really happens.
Describing the soul and the variations of mood - that I can do for
myself!"
There is one story about his attitude toward literature which also reveals
a great deal about his feeling for mathematics. It seems that there was a
mathematician who had become a novelist. "Why did he do that?" people
in Gottingen marvelled. "How can a man who was a mathematician write
novels?" "But that is completely simple," Hilbert said. "He did not have
enough imagination for mathematics, but he had enough for novels."
The current creation of Hilbert's own mathematical imagination was
his proof theory. At Zurich, in 1917, he had announced the general idea
and aims of the theory - but not the means of investigation. "For indeed,"
Bernays was later to comment, "the theory was not to rely on the current
mathematical methods." In the first communication on the new theory
(the attack on Brouwer and Weyl in Hamburg in 1922), Hilbert had argued
that mathematicians could regain elementary objectivity by formalizing the
statements and proofs of mathematics in the language of symbolic logic
and then taking the represented formulas and proofs directly as objects for
study. That same year, at Leipzig, he had added further refinements which
reduced the problem of proving the consistency of a formalized domain of
arithmetic - the task which he had set for the new century in Paris in 1900.
"Thus it seemed," Bernays later wrote, "that carrying out proof theory
was only a matter of mathematical technique."
On the occasion of a celebration at Munster in honor of Weierstrass,
Hilbert chose to talk "On the Infinite." He felt that the occasion was
appropriate for the fullest exposition of his program of Formalism up to
that time. The analysis of Weierstrass and the concept of the infinite as it
appeared in the work of Cantor had been prime targets for Kronecker. In

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Hilbert

the current program of Brouwer, many of the achievements of Weierstrass

and Cantor would be among the sacrifices required.
Hilbert was not at all well at the time of his talk in Munster. Recently it
had become clear that the deterioration noted by Nordheim was not that
of age alone, but the nature of the illness was still undetermined. In spite
of his poor health, however, Hilbert spoke as enthusiastically and opti-
mistically as ever.
He began his talk by pointing out that the present "happy state of affairs"
in analysis was entirely due to Weierstrass and his penetrating critique of
its methods. And yet - disputes about the foundations of analysis did
continue up to the present day. This was, in his opinion, because the mean-
ing of the infinite, as that concept was used in mathematics, had not yet been
completely clarified.
The infinite was nowhere to be found in reality; yet it existed in a very
real sense, in his opinion, as an "over-all negation." From time immemorial
the idea of the infinite had stirred men's emotions as no other subject.
Therefore, he felt, the definitive clarification of its nature went far beyond
the sphere of specialized scientific interest: it was needed for the dignity
of the human intellect itself!
The deepest insight into the nature of the infinite to date had been obtained
by a theory which came closer to a general philosophical way of thinking
than to mathematics. This theory, created by Georg Cantor, was set theory.
"It is, I think, the finest product of mathematical genius," Hilbert said,
"and one of the supreme achievements of purely intellectual human
activity."
But it was in Cantor's set theory, simply as a result of employing defini-
tions and deductive methods which had become customary in mathematics,
that the catastrophic antinomies had begun to appear.
" ... the present state of affairs ... is intolerable. Just think, the defini-
tions and deductive methods which everyone learns, teaches, and uses in
mathematics, the paragon of truth and certitude, lead to absurdities! If
mathematical thinking is defective, where are we to find truth and certi-
tude?"
There was, however, "a completely satisfactory way of avoiding the
paradoxes of set theory without betraying our science." Mathematicians
must establish throughout mathematics the same certitude for their deduc-
tions as exists in the ordinary arithmetic of whole numbers, "which no one
doubts and where contradictions and paradoxes arise only through our own
carelessness. "

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But if men were to remain within the domain of such purely intuitive
and finitary statements - as they must - they would have to have, as a
rule, more complicated logical laws. The logical laws which Aristotle had
taught and which men had used since they began to think, would not hold.
"We could, of course, develop logical laws which do hold for the domain
of finitary statements. But ... we do not want to give up the use of the
simple laws of aristotelian logic .... What then are we to do?
"Let us remember that we are mathematicians and that as mathematicians
we have often been in precarious situations from which we have been res-
cued by the ingenious method of ideal elements .... Similarly, to preserve
the simple formal rules of aristotelian logic, we must supplement the finitary
statements with ideal statements."
Mathematics, under this view, would become a stock of two kinds of
formulas: first, those to which the meaningful communications correspond
and, secondly, other formulas which signify nothing but which are the ideal
structures of the theory.
"But in our general joy over this achievement, and in our particular joy
over finding that indispensable tool, the logical calculus, already developed
without any effort on our part, we must not forget the essential condition
of the method of ideal elements - a proof of consistency."
For the extension of a domain by the addition of ideal elements is legiti-
mate only if the extension does not cause contradictions to appear.
This problem of consistency could be "easily handled." It was possible,
in his opinion, to obtain in a purely intuitive and finitary way - the way
in which truths are obtained in elementary number theory - the insights
which would guarantee the validity of the mathematical apparatus. Then
the test of the theory would be its ability to solve old problems, for the
solution of which it had not been expressly designed. He cited as an example
of such a problem the continuum hypothesis of Cantor, which he had listed
as the first of the Paris Problems. He now devoted the last part of his talk
to sketching an attack on this famous problem.
"No one," he promised his fellow mathematicians, "will drive us out of
this paradise that Cantor has created for us!"

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 XXI

Borrowed Time

On a soft warm evening in June 1925, Felix Klein died.

Everyone at Gottingen had long been prepared for Klein's death.
"But the event after it happened touched us all deeply and affected us
painfully," Hilbert said in a little speech to his colleagues the next morning.
"Up until yesterday Felix Klein was still with us, we could pay him a visit,
we could get his advice, we could see how highly interested he was in us.
But that is now all over."
Everything they saw around them in Gottingen was the work of Klein,
the collection of mathematical models in the adjoining corridor, the Lese-
zimmer with all the books on open shelves, the numerous technical institutes
that had grown up around the University, the easy relation they had with
the education ministry, the many important people from business and
industry who were interested in them .... They had lost "a great spirit,
a strong will, and a noble character."
An era had come to an end.
A few months later at a memorial meeting of the Gottingen Scientific
Society, Courant recalled the dramatic story of the great Felix: the meager
beginnings, the spectacular successes ("If today we are able to build on the
work of Riemann, it is thanks to Klein."), the tragic breakdown, and then-
"the wonderful turning point" - the seemingly broken man who had
lived another 43 years and displayed the most varied sides as researcher,
teacher, organizer and administrator.
And yet Klein's life had not been without its inner tragedy. The power
of synthesis had been granted to him to an extraordinary degree. The other
great mathematical power of analysis had been to a certain extent withheld.
His ability to bring together the most distant, abstract parts of mathematics
had been remarkable, but the sense for the formulation of an individual

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problem and the absorption in it had been lacking. "He was like a flier who,
soaring high over the world, discovers and looks over new fields ... but
cannot land his plane in order to take actual possession, to plow and to
harvest." Perhaps Klein had himself been unaware of this deep schism but,
in Courant's opinion, it had been one of the causes of the decisive break-
down during his competition with Poincare. Certainly he had perceived
"that his most splendid scientific creations were fundamentally gigantic
sketches, the completion of which he had to leave to other hands."
Sometimes he had failed to preserve purely human relationships. "Many
who knew him only as an organizer ... found him too harsh and violent,
so he produced much opposition to his ideas ... which a gentler hand would
easily have overcome. " Yet his nearest relatives and colleagues and the
great majority of his students had known always that behind the relentlessly
naive drive, a good human being stood.
They placed on his grave a simple inscription: "Felix Klein, A Friend,
Sincere and Constant."
The same year that Klein died Runge retired; his place was taken by
Gustav Herglotz.
The ailing Hilbert's condition steadily worsened. In the fall of 1925 it
was at last recognized that he was suffering from pernicious anemia. The
disease, which was generally considered fatal, had gone so long undetected
because the first symptoms, occurring in someone of his age, seemed merely
an early failing of powers. Now the doctors gave him at best a few months,
maybe even weeks.
In spite of what the doctors said, Hilbert remained optimistic about his
condition. He insisted that he actually did not have pernicious anemia -
it was some other, less serious disease which merely had all the same symp-
toms.
It was Hilbert's sheer luck that earlier in the year 1925 G. H. Whipple
and F. S. Robscheit-Robbins had discovered the beneficial effects of raw
liver on blood regeneration, and by 1926 their work was being applied
to the treatment of pernicious anemia by G. R. Minot in America. A pharma-
cologist friend in Gottingen now chanced to read about Minot's work in
the Journal of the American Medical Association and showed the article to
Hilbert. In addition to describing the new treatment - still, it stressed, in a
highly experimental state - the article also described vividly the mortal
seriousness of "P. A." But Hilbert reading it completely ignored all the
distressing details. He concentrated only upon the hopes raised by Minot's
work.

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Hilbert

Mrs. Landau was the daughter of Paul Ehrlich, who had discovered
salvarsan, the "magic bullet" treatment for syphilis; and she had many
contacts in the medical world. With the help of Courant, she drafted a
long telegram to Minot, who was at Harvard. "It was the longest telegram
I ever sent," Courant says. At the same time another telegram went to
Oliver Kellogg who, in 1902, had been the first Hilbert student to write
his doctoral dissertation on integral equations. Now a mathematics professor
at Harvard, Kellogg rallied support among the mathematicians for the
request from Gottingen.
At first Minot and his associates were not too receptive. They had very
little of the treatment substance, which would have to be administered to the
patient for the rest of his life. People. were dying of pernicious anemia
within a few miles of Harvard University ....
George Birkhoff, the leading American mathematician and also a Harvard
mathematics professor, had recently seen a play in which a doctor was able
to save but 10 men. How should he choose the 10? "On the basis of their
value to mankind" was the playwright's answer. In conversation with Mi-
not, Birkhoff quoted The Doctor's Dilemma of George Bernard Shaw.
Mathematics makes mathematicians persistent men. Minot gave in.
Instructions were wired to the pharmacologist in Gottingen for concocting
large amounts of raw liver which would serve for treatment until the more
concentrated experimental substance arrived from the United States. E. U.
Condon, visiting Gottingen in the summer of 1926, heard Hilbert complain-
ing that he would rather die than eat that much raw liver.
Eventually, though, Minot's preparation arrived.
At this late stage it was probably not possible to reverse completely the
progress of the disease; however, everyone in Gottingen noticed that Hil-
bert's condition began to improve almost immediately. All during his illness
he had continued to work to the best of his ability - even turning his dining
room into a lecture hall when he was not well enough to go to the University.
Now when a former student inquired after his health, he replied firmly,
"That illness - well, it no longer exists."
The two years, beginning in 1925, were the "Wunderjahre" of what was
known in Gottingen as "boy physics" because so many of the great discov-
eries were being made by physicists still in their twenties. Early in 1925
Heisenberg came to Born with the seemingly weird mathematics that had
developed in a new theory of quantum mechanics which he had created.
Heisenberg thought that this was the one thing that still had to be corrected
in his theory. In actuality, it was his great discovery. Born promptly iden-

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tified the weird mathematics as matrix algebra, the germ of which had
existed in the quaternions developed by William Rowan Hamilton more
than three-quarters of a century before.
In matrix algebra multiplication is not commutative: a X b may not
equal b X a but something entirely different. Prior to Heisenberg's work,
matrices had rarely been used by physicists, although one exception had
been Born's earlier work on the lattice theory of crystals. But now even
Born had to consult his old friend Otto Toeplitz about certain properties
of matrices and considered himself fortunate to obtain as his assistant
Pascual Jordan, whom he just happened to meet when Jordan overheard
him talking about matrices to a companion in a train compartment and pro-
ceeded to introduce himself. Jordan had been one of Courant's assistants
in the preparation of Courant-Hilbert and was therefore very familiar with
matrix algebra.
The Heisenberg paper was followed just 60 days later by the great Born-
Jordan paper, which provided the necessarily rigorous mathematical founda-
tion for the new matrix mechanics. The next year saw the publication of
Born's famous statistical interpretation, for which he later received the
Nobel Prize.
Hilbert never went as deeply into quantum mechanics as he had gone
into relativity, but he still demanded that his physics assistant teach him
the new theory.
"Generally he tried to give a course on what he was learning," says
Nordheim. "He was a person for whom it was difficult to understand others.
He always had to work things through for himself. That seemed to be his
only way of really understanding. So when there was a new development,
he tried to give a course on it. Usually this also contained some old material,
for nothing grows up entirely of itself. For the new parts we had to
make drafts. After that he would try to put the ideas into his own
words."
In the spring of 1926 Hilbert announced his first lectures on quantum
mechanics. Nordheim recalls how he had to extract "rather laboriously"
the essence from the papers of Born and his collaborators for Hilbert, who
was still at this time not well.
"Of course, he knew a lot about matrix algebra and differential equations
and so on, and all of these things are the mathematical tools of quantum
mechanics. In this respect my job was made easier. I went to his home two
or three times a week, as required, and we discussed the general situation.
Then he would ask for writeups on specific points or the development of

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formulas on particular applications. The next time we would talk these

over - whether everything was correct and understood."
The matrix mechanics of Heisenberg was followed in short order by the
wave mechanics of Erwin Schrodinger. The two papers, although they
were on the same subject and led to the same results, astonished physicists;
for, as one of them marvelled, "they started from entirely different physical
assumptions, used entirely different mathematical methods, and seemed to
have nothing to do with each other."
The equivalence of Heisenberg's and Schrodinger's theories, however,
was soon established.
The whole development gave Hilbert "a great laugh," according to
Condon:
" ... when [Born and Heisenberg and the Gottingen theoretical physi-
cists] first discovered matrix mechanics they were having, of course, the same
kind of trouble that everybody else had in trying to solve problems and to
manipulate and to really do things with matrices. So they had gone to Hil-
bert for help and Hilbert said the only times he had ever had anything to do
with matrices was when they came up as a sort of by-product of the eigen-
values of the boundary-value problem of a differential equation. So if you
look for the differential equation which has these matrices you can probably
do more with that. They had thought it was a goofy idea and that Hilbert
didn't know what he was talking about. So he was having a lot of fun
pointing out to them that they could have discovered Schrodinger's wave
mechanics six months earlier if they had paid a little more attention to
him."
As a result of his almost miraculous recovery Hilbert lived to see what
has been called "one of the most dramatic anticipations in the history of
mathematical physics."
The Courant-Hilbert book on mathematical methods of physics, which
had appeared at the end of 1924, before both Heisenberg's and Schrodinger's
work, instead of being outdated by the new discoveries, seemed to have been
written expressly for the physicists who now had to deal with them. Hilbert's
own work at the beginning of the century on integral equations, the theory
of eigenfunctions and eigenvalues of 1903-04 and the theory of infinitely
many variables of 1905-06, turned out to be the appropriate mathematics
for quantum mechanics (as was first established by Born in a joint paper
with Heisenberg and Jordan).
"Indirectly Hilbert exerted the strongest influence on the development
of quantum mechanics in Gottingen," Heisenberg was later to write. "This

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influence can be fully recognized only by one who studied in Gottingen

during the twenties. Hilbert and his colleagues had created there an atmos-
phere of mathematics, and all the younger mathematicians were so trained
in the thought processes of the Hilbert theory of integral equations and
linear algebra that each project which belonged in this field could develop
better in Gottingen than in any other place. It was an especially fortunate
coincidence that the mathematical methods of quantum mechanics turned
out to be a direct application of Hilbert's theory of integral equations .... "
To Hilbert himself this was yet another example of that pre-established
harmony which seemed to him almost the embodiment and realization of
mathematical thought.
"I developed my theory of infinitely many variables from purely mathe-
matical interests," he marvelled, "and even called it 'spectral analysis'
without any presentiment that it would later find an application to the
actual spectrum of physics I"
What happened next was also impressive to Hilbert, for it underlined
the continuity of mathematical effort. Hilbert's theory of infinitely many
variables - which had become known as "Hilbert Space" theory - now
turned out to be in several respects not quite equal to the task of handling
quantum mechanics. At this point young John von Neumann, inspired by
Erhard Schmidt, formulated Hilbert's concept of a quadratic form more
abstractly so that the extended Hilbert theory was able to meet completely
the needs of the physicists.
Hilbert's last publication in physics was a collaboration with Nordheim
and von Neumann on the axiomatic foundations of quantum mechanics,
in which, although he did almost none of the work, the spirit was quite
definitely that of Hilbert. While the effort turned out to be not mathemat-
ically rigorous, it served to introduce von Neumann to quantum mechanics
and inspired him to create his later famous analysis of the foundations of
the subject.
In 1927 Nordheim left Gottingen and Eugene Wigner became Hilbert's
special assistant for physics. He recalls that he saw Hilbert "only about five
times." When Wigner left in 1928, his place was taken by a mathematics
student named Arnold Schmidt and the position became a second assist-
antship in logic. Hearing a lecture by Schrodinger on the new physics in
1928 or 1929, Hilbert grumbled to his former student Paul Funk: "I don't
see how anybody understands what is happening in physics today. Even I
don't understand much which I would like to learn from physics books.
But with me, if I don't understand something, then I go to the telephone

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Hilbert

and call up Debye or Born, and they come and explain it to me. And then I
understand it - but what do other people do?"
He himself was still, after his illness, deep in the work on the foundations
of mathematics.
The enthusiasm for Brouwer's Intuitionism had definitely begun to
wane. Brouwer came to Gottingen to deliver a talk on his ideas to the
Mathematics Club.
"You say that we can't know whether in the decimal representation of
7t ten 9's occur in succession," someone objected after Brouwer finished.

"Maybe we can't know - but God knows!"

To this Brouwer replied dryly, "I do not have a pipeline to God."
After a lively discussion Hilbert finally stood up.
"With your methods," he said to Brouwer, "most of the results of modern
mathematics would have to be abandoned, and to me the important thing
is not to get fewer results but to get more results."
He sat down to enthusiastic applause.
The feeling of most mathematicians has been informally expressed by
Hans Lewy, who as a Privatdozent was present at Brouwer's talk in Got-
tingen:
"It seems that there are some mathematicians who lack a sense of humor
or have an over-swollen conscience. What Hilbert expressed there seems
exactly right to me. If we have to go through so much trouble as Brouwer
says, then nobody will want to be a mathematician any more. After all,
it is a human activity. Until Brouwer can produce a contradiction in clas-
sical mathematics, nobody is going to listen to him.
"That is the way, in my opinion, that logic has developed. One has
accepted principles until such time as one notices that they may lead to
contradiction and then he has modified them. I think this is the way it will
always be. There may be lots of contradictions hidden somewhere; and as
soon as they appear, all mathematicians will wish to have them eliminated.
But until then we will continue to accept those principles that advance us
most speedily."
Hilbert's program, however, also received its share of criticism. Some
mathematicians objected that in his formalism he had reduced their science
to "a meaningless game played with meaningless marks on paper." But to
those familiar with Hilbert's work this criticism did not seem valid.
" ... is it really credible that this is a fair account of Hilbert's view,"
Hardy demanded, "the view of the man who has probably added to the
structure of significant mathematics a richer and more beautiful aggregate

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of theorems than any other mathematician of his time? I can believe that
Hilbert's philosophy is as inadequate as you please, but not that an ambitious
mathematical theory which he has elaborated is trivial or ridiculous. It is
impossible to suppose that Hilbert denies the significance and reality of
mathematical concepts, and we have the best of reasons for refusing to
believe it: 'The axioms and demonstrable theorems,' he says himself, 'which
arise in our formalistic game, are the images of the ideas which form the
subject-matter of ordinary mathematics.'''
By 1927 Hilbert was well enough to go again to Hamburg "to round
out and develop my thoughts on the foundations of mathematics, which I
expounded here one day five years ago and which since then have kept me
most actively occupied." His goal was still to remove "once and for all"
any question as to the soundness of the foundations of mathematics. "I
believe," he said, "I can attain this goal completely with my proof theory,
even though a great deal of work must still be done before it is fully devel-
oped."
In the course of his talk, he took up various criticisms of his program,
"all of which I consider just as unfair as can be." He went back even as far
as Poincare's remarks on the Heidelberg talk. "Regrettably, Poincare, the
mathematician who in his generation was the richest in ideas and the most
fertile, had a decided prejudice against Cantor's theory that kept him from
forming a just opinion of Cantor's magnificent conceptions." As for the
most recent investigations, of which the program advanced by Brouwer
formed the greater part, "the fact that research on foundations has again
come to attract such lively appreciation and interest certainly gives me the
greatest pleasure. When I reflect on the content and the results of these
investigations, however, I cannot for the most part agree with their ten-
dency; I feel, rather, that they are to a large extent behind the times, as if
they came from a period when Cantor's majestic world of ideas had not
yet been discovered."
The whole talk had a strongly polemical quality: "Not even the sketch
of my proof of Cantor's continuum hypothesis has remained uncriticized \"
Hilbert complained, and took up this proof again at length.
The formula game "which Brouwer so deprecates," he pointed out,
enabled mathematicians to express the entire thought-content of the science
of mathematics in a uniform manner and develop it in such a way that, at
the same time, the interconnections between the individual propositions
and the facts become clear. It had, besides its mathematical value, an impor-
tant general philosophical significance.

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Hilbert

"For this formula game is carried out according to certain definite rules,
in which the technique oj ollr thinking is expressed. These rules form a closed
system that can be discovered and definitively stated. The fundamental idea
of my proof theory is none other than to describe the activity of our under-
standing, to make a protocol of the rules according to which our thinking
actually proceeds .... If any totality of observations and phenomena
deserves to be made the object of a serious and thorough investigation, it
is this one - since, after all, it is a part of the task of science to liberate us
from arbitrariness, sentiment and habit and to protect us from the subjec-
tivism that already made itself felt in Kronecker's views and, it seems to me,
finds its culmination in Intuitionism .... "
It was true, Hilbert conceded, that the consistency proof of the formalized
arithmetic which would "determine the effective scope of proof theory and
in general constitute its core" was not yet at hand. But, as he concluded his
address, he was thoroughly optimistic: such a proof would soon be pro-
duced.
"Already at this time I should like to assert what the final outcome will
be: mathematics is a presuppositionless science. To found it I do not need
God, as does Kronecker, or the assumption of a special faculty of our
understanding attuned to the principle of mathematical induction, as does
Poincare, or the primal intuition of Brouwer, or, finally, as do Russell and
Whitehead, axioms of infinity, reducibility, or completeness .... "
When Hilbert finished, Hermann Weyl rose to make a few remarks.
Weyl's love for his old teacher had not been affected by the five years of
controversy. Also his enthusiasm for Brouwer's ideas had abated. He
nevertheless felt that at this point he should defend Brouwer:
"Brouwer was first to see exactly and in full measure how [mathematics]
had in fact everywhere far exceeded the limits of contentual thought. I
believe that we are all indebted to him for this recognition of the limits of
contentual thought. In the contentual considerations that are intended to
establish the consistency of formalized mathematics, Hilbert fully respects
these limits, and he does so as a matter of course; we are really not dealing
with artificial prohibitions by any means. Accordingly, it does not seem
strange to me that Brouwer's ideas have found a following; his position
resulted of necessity from a thesis shared by all mathematicians before
Hilbert proposed his formal approach and forms a new, indubitable funda-
mental logical insight that even Hilbert acknowledges.
"That from this point of view only a part, perhaps only a wretched part
of classical mathematics is tenable is a bitter and inevitable fact. Hilbert

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could not bear this mutilation. And it is again a different matter that he
succeeded in saving classical mathematics by a radical reinterpretation of its
meaning without reducing its inventory, namely by formalizing it, thus
transforming it in principle from a system of intuitive results into a game
with formulas that proceeds according to fixed rules.
"Let me now by all means acknowledge the immense significance and
scope of this step of Hilbert's, which evidently was made necessary by the
pressure of circumstances. All of us who witnessed this development are full
of admiration for the genius and steadfastness with which Hilbert, through
his proof theory of formalized mathematics, crowned his axiomatic life work.
And, I am very glad to confirm, that there is nothing that separates me from
Hilbert in the epistemological appraisal of the new situation thus created."
In contrast to Weyl, Brouwer had become, like Kronecker, a fanatic in
the service of his cause. He looked upon Hilbert as "my enemy," and once
left a house in Amsterdam where he was a guest when van der Waerden,
who was also a guest, referred to Hilbert and Courant as his friends.
The ill-feeling was undoubtedly intensified by the fact that circumstances
placed Hilbert constantly in opposition to Brouwer.
Both men were on the editorial staff of the Annalen. Hilbert was one of
the three principal editors, a position which he had held since 1902; Brou-
wer, a member of the seven-man editorial board. At about this time Brou-
wer began to insist that all papers by Dutch mathematicians and all papers
on topology be submitted directly to him. Everyone objected, especially
Dutch topologists, since it was well known that when a paper got into
Brouwer's hands, it did not get out for several years. Although personally
unaffected, Hilbert was repelled by Brouwer's dictatorial demands. When
he had been in good health, he had been confident of his ability to protect
the integrity of the Annalen. Since his illness, however, he had begun to
fear that if anything happened to him, Brouwer would take over the journal
to the detriment of mathematics. So he now called together his friends to
devise a way of removing Brouwer from the editorial board.
Caratheodory, who was himself a member of the board, came up with
a solution. Since Brouwer alone could not be asked to resign, the entire
seven-man board should be dismissed. Hilbert promptly acted. The change
is reflected in the covers of Vol. 100 of the Annalen and Vol. 101, on which
only the names of Hilbert, Heeke and Blumenthal remain.
(It should be mentioned that Einstein, disturbed by the controversy,
resigned from his position as one of the three principal editors. "What is
this frog and mouse battle among the mathematicians?" he asked a friend.)

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Hilbert

Hilbert and Brouwer were then placed in opposition by another circum-

stance.
Since the war, the German mathematicians had not been invited to any
international meetings. In 1928, however, the Italians, planning the first
official International Congress since 1912, determined to make it truly
international. Once again, invitations were sent to all German schools and
mathematical organizations. Many Germans did not want to accept. The
leader of this group was a professor at the University of Berlin named
Ludwig Bieberbach. In his opposition to accepting the invitation of the
Italians, he was seconded by Brouwer, who although Dutch was an ardent
German nationalist. In the spring of 1928 Bieberbach sent a letter to all
German secondary schools and universities urging them to boycott the
congress at Bologna. Hilbert responded by sending out a letter of his
own:
"We are convinced that pursuing Herr Bieberbach's way will bring
misfortune to German science and will expose us all to justifiable criticism
from well disposed sides .... The Italian colleagues have troubled them-
selves with the greatest idealism and expense in time and effort . . . . It
appears under the present circumstances a command of rectitude and the
most elementary courtesy to take a friendly attitude toward the Congress."
In August, although suffering from a recurrence of his illness, Hilbert
personally led a delegation of 67 mathematicians to the Congress. At the
opening session, as the Germans came into an international meeting for the
first time since the war, the delegates saw a familiar figure, more frail than
they remembered, marching at their head. For a few minutes there was not
a sound in the hall. Then, spontaneously, every person present rose and
applauded.
"It makes me very happy," Hilbert told them in the familiar accents,
"that after a long, hard time all the mathematicians of the world are repre-
sented here. This is as it should be and as it must be for the prosperity of
our beloved science.
"Let us consider that we as mathematicians stand on the highest pinnacle
of the cultivation of the exact sciences. We have no other choice than to
assume this highest place, because all limits, especially national ones, are
contrary to the nature of mathematics. It is a complete misunderstanding
of our science to construct differences according to peoples and races, and
the reasons for which this has been done are very shabby ones.
"Mathematics knows no races .... For mathematics, the whole cultural
world is a single country."

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 1925-1929

Hilbert's scientific paper for presentation at the Congress concerned

again the fundamental problems of mathematics. Recently there had been
signs that his expectation that carrying out proof theory was only a matter
of mathematical technique was perhaps overly optimistic. The first attempt
at a substantial consistency proof (in Ackermann's dissertation) had required
an essential restriction of the formal system not in the original plan. Simi-
larly, in a paper of von Neumann's, the consistency proof, following Hil-
bert's line of reasoning, had not applied to the full system. Now, though,
Ackermann's proof had been revised and simplified; and it seemed at the
moment that the consistency of formalized number theory, at least, had
been proved.
Hilbert now added to the problem of consistency another problem, that
of the completeness of the formal system.
When Hilbert went to pay his hotel bill, he was informed that it had
already been paid for him by the committee in charge of planning the Con-
gress.
"Ah, if I had only known that," he said, "I would have eaten a great
deal more."
The Hilbert career was almost over.
The year after the Bologna Congress, he was permitted to see what Felix
Klein had not lived to see - the dedication of a handsome building to house
the Mathematical Institute of Gottingen.
The new institute had been made possible by Courant's friendship with
the Bohr brothers and the entree which they had provided to the Rocke-
feller Foundation. The funds given by the Foundation were then matched
by the German government. Thus the Institute was a joint project of
German and American money and effort.
"There will never be another institute like this!" Hilbert exulted. "For
to have another such institute, there would have to be another Courant -
and there can never be another Courant!"

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 XXII

Logic and the Understanding of Nature

The mandatory age of retirement for a professor was 68, an age Hilbert
would attain on January 23, 1930; in Gottingen a bittersweet feeling of
anticipation and regret was in the air.
During the winter semester 1929-30, Hilbert delivered his "Farewell to
Teaching." For his subject he went back to the foundations of his fame
and lectured for almost the first time in 40 years on the invariants. Pro-
fessors crowded into the lecture hall with the students. A street was named
Hilbert Strasse. "A street named after you!" Mrs. Hilbert exclaimed. "Isn't
that a nice idea, David?" Hilbert shrugged. "The idea, no, but the execution,
ah - that is nice. Klein had to wait until he was dead to have a street named
after him!"
He saw another student through the doctoral. Appropriately this was
Haskell Curry, an American. Curry had little contact with Hilbert though.
He remembers he came to class on a warm spring day wearing a fur-lined
coat. He was always accompanied by Bernays, who sometimes had to step
in and lecture for a bit. Curry had most of his conferences with Bernays;
but since Bernays was not a full professor, he had to take the final exami-
nation with Hilbert.
"I rather enjoyed my final examination with him .... He did not ask me
any questions having to do with logic, but only to do with general mathe-
matics. One question was about the uniformization of algebraic functions.
It happened that I had just had a course on that subject with Professor
Osgood at Harvard. Although it was a way off from my special field, I
gave as precise an answer as anyone would expect in a field so far removed
from the candidate's specialty; he was quite impressed with me and turned
to me and said, 'Where did you learn that?' Although he seemed rather frail,
he was razor-sharp and alert."

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As the time of the retirement approached, the choice of a successor was

discussed. There was, it was generally agreed, but one possible choice. If
Courant had shown himself to be the Klein of the new generation, Weyl
was the Hilbert.
A decade before, Weyl had refused an offer from Gottingen because of
the uncertainties of life in Germany after the war. Hilbert said, "It is easy
to call Weyl but hard to get him." This time Weyl again had difficulty in
making up his mind. He had recently returned from England; and the
pessimism which he found expressed in the newspapers and letters which
had piled up on his desk during his absence filled him with apprehension
about returning to Germany. He was also concerned that he might not be
the right choice for Gottingen at this time. By now he was 45 years old.
He knew he was close to the end of his creative period. Perhaps the Institute
should get someone like young Emil Artin, "from whom great results can
still be expected." Yet he was tempted. He loved and revered Hilbert -
the Pied Piper who had led all the young rats down into the deep river of
mathematics. He knew that he was more completely in the mathematical-
physical tradition of Gottingen than Artin. He would enjoy the opportunity
of working with Courant, Born and Franck. The situation in Germany
seemed to be improving. The Dawes Plan had helped to relieve the economic
problems. The lunatic fringe which muttered about the "Jewish physics"
of his friend Einstein still seemed on the fringe. In the end, this time, Weyl
wired his acceptance.
"I don't have to tell you with how much joy and how much pride I was
filled to be called as your successor," he wrote to Hilbert. " ... I am looking
forward most eagerly to working with the colleagues you have gathered
around yourself, you to whom the mathematical-scientific faculty owes its
strength and harmony." The dark clouds that hung over Germany might
not disappear quickly. "But I hope it will be granted to me to live many happy
years near you .... Please do not be angry about my tardiness in accepting."
The Gottingen which greeted Weyl in the spring of 1930 was at the height
of its new glory. More than ever before, it could be said that an international
congress of mathematicians was perpetually in session in the quiet little
town with its linden-lined streets and the solid respectable, now old "Ju-
gendstil" houses. On the outskirts a series of scientific industries and
laboratories surrounded the city like another wall. The Mathematical Insti-
tute was housed in its handsome new building, the Lesezimmer a long well-
lighted library. Extra Gottingen non est vita. The Latin motto was still
blazoned on the wall of the Ratskeller. In the sunshine outside, students

191
Hilbert

and professors sat at small tables and argued about politics, love and science.
The little goose girl gazed tranquilly down into her fountain. Weyl, return-
ing to the beloved town of his college years, must have agreed. Away from
Gottingen there was no life.
Of all the honors being showered on Hilbert during the retirement year,
the one that seemed to please him most came from his native city. The
Konigsberg town council voted to present its famous son with "honorary
citizenship." The presentation was scheduled to be made in the fall at the
meeting of the Society of German Scientists and Physicians, which was
being held that year in Konigsberg.
Hilbert gave considerable thought to the selection of a topic for his
acceptance address. It must be something of wide and general interest. In
Konigsberg, the birthplace of Kant, it must be philosophical in tone. It
must also be a fitting conclusion to the career that had begun long ago at
the university in Konigsberg. When he thought of the university, he
remembered the statue of Kant on the grounds and the laconic inscription
"Kant" - so expressive in its brevity. He also remembered Jacobi, from
whom the mathematical tradition of Konigsberg derived as in Gottingen
it derived from Gauss. He wanted a topic which would weave together
these great names and all the separate strands of his career - Konigsberg
and Gottingen, Jacobi, Gauss, Kant, mathematics and science, science and
experience, the great developments in knowledge and in thought through
which he had lived.
Naturerkennen - the understanding of nature - und Logik. This would
be his subject.
During the past decade he had become increasingly interested in reaching
a greater audience with mathematical ideas. He had frequently accepted the
opportunity of giving popular lectures in the Saturday morning series "for
all the Faculties of the University." He took subjects like "Relativity Theory"
or "The Infinite" or "The Principles of Mathematics" and tried, by finding
examples from familiar fields outside mathematics, to make the fundamental
concepts comprehensible to laymen.
"An enormous amount oflabor was devoted to this task,"Nordheim recalls.
"We had to prepare preliminary outlines either of new material or from old
lectures. These were then worked and re-worked practically every morning
and in this process flavored with Hilbert's own inimitable brand of logic
and humor."
During this period Hilbert and some of the other mathematics professors
regularly attended the lectures of a zoologist. Hilbert had developed a

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great interest in genetics. He delighted in the laws determining the heredity

of Drosophila, which could be obtained by the application of certain of the
geometric axioms. He was fascinated by the Pferdespulwurm - "that
creature with the most modest number of chromosomes, which corre-
sponds therefore to the hydrogen atom with only one electron." But he was
also impressed by the ability of the biologists to make their subject interest-
ing and understandable to laymen.
"The biologists understand popular presentation especially well," he
told Paul Funk one time. "In order to prevent the fatigue which straining
thoughts bring forth in laymen, one must occasionally insert a little dessin
(a French word meaning pattern, or design), and at that the biologists are
pre-eminent." Pronouncing the French word sharply in his Konigsberg
dialect, he went on to say: "For us mathematicians, popular presentation
is much harder, but still it must happen - if we go about it right - that
we find a beautiful dessin."
Now, in the summer of 1930, among his mature fruit trees, he sought
such a beautiful dessin as he began to prepare his speech for the Konigsberg
meeting, stripping away all vague generalities from his subject, putting
his ideas into simple non-technical language for a general audience (that
"man in the street" he had mentioned in Paris, to whom one should always
be able to explain any fully realized mathematical theory).
He had returned many times over the years to his native city, but there
was a special quality to this return. Kurt Reidemeister and Gabor Szego,
the mathematics professors now at the University, noted how pleased he
was at the social gathering arranged in connection with his speech, so
exuberant "that his wife must always again call him back." But Konigsberg
seemed colder than in the old days, and Hilbert had to borrow a fur coat
from Szego to keep warm.
The honorary citizenship was presented at the opening session. Then
Hilbert took his place at the rostrum. His head was now almost entirely
bald, the broad scholar's forehead contrasting more sharply than ever with
the delicate chin; the white moustache and small beard were neatly trimmed.
(Ostrowski was reminded of the head of Lenin.) He looked out at his
audience through the familiar rimless glasses, the blue eyes still sharp and
searching, and still so innocent. He placed his hands firmly on the manuscript
in front of him and began to speak very slowly and carefully:
"The understanding of nature and life is our noblest task."
In recent times richer and deeper knowledge had been obtained in
decades than had previously been obtained in the same number of centuries.

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Hilbert

The science of logic had also progressed until there was now, in the axio-
matic method, a general technique for the theoretical treatment of all scien-
tific questions. Because of these developments - he told his audience - the
men of today were better equipped than the philosophers of old to answer an
ancient philosophical question: "the part which is played in our under-
standing by Thinking on the one side and Experience on the other."
It was a worthy question with which to conclude a career; for, funda-
mentally, to answer it would be to ascertain by what means general under-
standing is achieved and in what sense "all the knowledge which we collect
in our scientific activities is truth."
Certain parallels between nature and thought had always been recognized.
The most striking of these was a pre-established harmony which seemed to
be almost the embodiment and realization of mathematical thought, the
most magnificent and wonderful example of which was Einstein's theory
of relativity.
But it seemed to him that the long recognized accord between nature
and thought, experiment and theory, could only be understood when one
took into account the formal element and the mechanism linked with it
which exists on both sides, in nature and in thought. The extension of the
methods of modern science should lead to a system of natural laws which
corresponded with reality in every respect. Then we should need only
pure thought - abstract deduction - in order to gain all physical knowl-
edge. But this was not, in his opinion, the complete answer: "For what is
the origin of these laws? How do we obtain them? How do we know that
they correspond with reality? The answer is that we can obtain these laws
only through our own experience .... Whoever wants nevertheless to deny
that universal laws are derived from experience must contend that there is
still a third source of understanding .... "
Konigsberg's great son, Immanuel Kant, had been the classical exponent
of this point of view - the point of view which Hilbert 4S years ago had
defended at his public promotion for the degree of doctor of philosophy.
Now, before his talk, he had smilingly commented to a young relative that
a lot of what Kant had said was "pure nonsense" - but that, of course, he
could not say to the citizens of Konigsberg.
Kant had stated that man possesses beyond logic and experience certain
a priori knowledge of reality.
"I admit," Hilbert told his audience, "that even for the construction of
special theoretical subjects certain a priori insights are necessary .... I even
believe that mathematical knowledge depends ultimately on some kind of

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such intuitive insight .... Thus the most general basic thought of Kant's
theory of knowledge retains its importance .... The a priori is nothing more
or less than ... the expression for certain indispensable preliminary condi-
tions of thinking and experiencing. But the line between that which we
possess a priori and that for which e~perience is necessary must be drawn
differently by us than by Kant - Kant has gre<l.tly overestimated the role and
the extent of the a priori."
Men now knew that many facts previously considered as holding good
a priori were not true, the most striking being the notion of an absolute
present. But it had also been shown, through the work of Helmholtz and
Gauss, that geometry was "nothing more than a branch of the total con-
ceptual framework of physics." We had forgotten that the geometrical
theorems were once experiences!
"We see now: Kant's a priori theory contains anthropomorphic dross
from which it must be freed. After we remove that, only that a priori will
remain which also is the foundation of pure mathematical knowledge."
In essence, this was the attitude which he had characterized in his recent
work on the foundations of mathematics.
"The instrument which brings about the adjustment of differences be-
tween theory and practice, between thought and experiment, is mathematics.
It builds the connecting bridge and continually strengthens it. Thus it
happens that our entire present culture, insofar as it is concerned with the
intellectual understanding and conquest of nature, rests upon mathematics!"
The effect of Hilbert's speech on the audience has been recalled by Oystein
Ore, who, as a young man on his honeymoon, was there in Konigs-
berg:
"I remember that there was a feeling of excitement and interest both in
Hilbert's lecture and in the lecture of von Neumann on the foundations of
set theory - a feeling that one now finally was coming to grips with both
the axiomatic foundation of mathematics and with the reasons for the appli-
cations of mathematics in the natural sciences."
In the final part of his speech, Hilbert carefully made the point that in
spite of the importance of the applications of mathematics, these must
never be made the measure of its value. He concluded with that defense of
pure mathematics which he had wanted so long ago to make in answer to
the speech given by Poincare at the first International Congress of Mathe-
maticians.
"Pure number theory is that part of mathematics for which up to now no
application has ever been found. But it is number theory which was consid-

195
Hilbert

ered by Gauss [who himself made untold contributiorls to applied mathe-

matics] as the queen of mathematics .... "
Kronecker had compared the mathematicians who concerned themselves
with number theory to Lotus-eaters who "once having consumed this food
can never give it up."
"Even our great Konigsberg mathematician Jacobi felt this way ....
When the famous Fourier maintained that the purpose of mathematics lies
in the explanation of natural phenomena, Jacobi objected, 'A philosopher
like Fourier should know that the glory of the human spirit is the sole aim
of all science!' .... Whoever perceives the truth of the generous thinking
and philosophy which shines forth from Jacobi's words will not fall into
regressive and barren scepticism."
Reidemeister and Szego had made arrangements for Hilbert to repeat the
conclusion of his speech over the local radio station; and when the session
adjourned, they accompanied him to the broadcasting studio.
There, as Hilbert spoke into the unfamiliar instrument, it seemed that his
voice rang out again with the enthusiasm and optimism of the vigorous man
who in the prime of his life had sent his listeners out to seek the solution to 23
problems which, he was certain, would lead to progress in mathematics.
"In an effort to give an example of an unsolvable problem, the philosopher
Comte once said that science would never succeed in ascertaining the secret
of the chemical composition of the bodies of the universe. A few years later
this problem was solved ....
"The true reason, according to my thinking, why Comte could not find
an unsolvable problem lies in the fact that there is no such thing as an un-
solvable problem."
He denied again, at the end of his career, the "foolish ignorabimus" of du
Bois-Reymond and his followers. His last words into the microphone were
firm and strong:
"Wir miissen wissen. Wir werden wissen."
We must know. We shall know.
As he raised his eyes from his paper and the technician snapped off the
recording machine, he laughed.
The record which he made of this last part of his speech at Konigsberg
is still in existence. At the end, if one listens very carefully, he can hear
Hilbert laugh.
"Wir miissen wissen. Wir werden wissen."
We must know. ff7e shall know.
It was in every respect a great last line.

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 1929-1930

However, lives do not always end on great last lines.

At almost the same time that Hilbert was making his speech at Konigs-
berg, a piece of work was being brought to a conclusion which was to deal
a death blow to the specific epistemological objective of the final program
of Hilbert's career. On November 17, 1930, the Monalshefle fiir Mathematik
lind Physik received for publication a paper by a 25-year-old mathematical
logician named Kurt Godel.

197
 XXIII

Exodus

When Hilbert first learned about Godel's work from Bernays, he was
"somewhat angry."
The young man had taken up both of the problems of completeness
which Hilbert had proposed at Bologna. He had established completeness
for the case of the predicate calculus. But then he had proceeded to prove -
with all the finality of which mathematics is uniquely capable - the inCOlll-
pleteness of the formalized number theory. He had also proved a theorem
from which it follows that a finitist proof of consistency for a formal system
strong enough to formalize all finitist reasonings is impossible.
In the highly ingenious work of Godel, Hilbert saw, intellectually, that
the goal toward which he had directed much effort since the beginning of
the century - the final unanswerable answer to Kronecker and Brouwer
and the others who would restrict the methods of mathematics - could not
be achieved. Classical mathematics might be consistent and, in fact, probably
was; but its consistency could never be established by mathematical proof,
as he had hoped and believed it could be.
The boundless confidence in the power of human thought which had led
him inexorably to this last great work of his career now made it almost
impossible for him to accept Godel's result emotionally. There was also
perhaps the quite human rejection of the fact that Godel's discovery was a
verification of certain indications, the significance of which he himself had
up to now refused to recognize, that the framework of formalism was not
strong enough for the burden he wanted it to carry.
At first he was only angry and frustrated, but then he began to try to deal
constructively with the problem. Bernays found himself impressed that
even now, at the very end of his career, Hilbert was able to make great
changes in his program. It was not yet clear just what influence Godel's

198
 1930-1933

work would ultimately have. Gbdel himself felt - and expressed the thought
in his paper - that his work did not contradict Hilbert's formalistic point
of view; and it soon became apparent that proof theory could still be
fruitfully developed without keeping to the original program. Broadened
methods would permit the loosening of the requirements of formalizing.
Hilbert himself now took a step in this direction. This was the replacing of
the schema of complete induction by a looser rule called "unendliche
Induktion." In 1931 two papers in the new direction appeared.
Although he had retired, he continued to lecture regularly at the Uni-
versity. He still prepared in only the most general way, still frequently got
stuck. When he found himself unable to work through a proof on the
blackboard, he would dismiss it with a wave of his hand as "completely
elementary." He sometimes stumbled over details, rambled impossibly,
repeated himself. "But still, one out of three lectures was superb!"
Hilbert's career having come to its official end with his retirement,
plans were made to begin the collecting and editing of his mathematical
works. Blumenthal, who had observed and studied the personality and
achievements of his teacher since 1895, was asked to compose a biography
for the final volume. Although Blumenthal had been a professor at Aachen
now for many years, he had never lost his strong feeling for Gbttingen,
returning time after time to be (as he said) "refreshed." Wherever he was,
even at the front during the first world war, he always organized the former
inhabitants of Gbttingen into a social club. He took on the assignment of
a life history with pleasure and painstaking care.
Volume I of the collected works was to be devoted to the Zahlbericht
and the other number theory papers. For Hilbert, as for Gauss, the first
years at Gbttingen had been "the fortunate years." The papers on algebraic
number fields were now recognized as the deepest and most beautiful of all
his mathematical works. Helmut Hasse, who with Emmy Noether, van der
Waerden, Artin, Takagi and others had taken part in carrying out the
program for class-fields which Hilbert had outlined in the last number
theory paper, was asked to write an evaluation of Hilbert's contribution in
this area.
It seemed to Hasse, in retrospect, that Hilbert's work on algebraic number
theory, like so much of his work, had stood in time and content at the turn
of two centuries. On one side, treating problems in great generality with
new methods which far surpassed the earlier methods in elegance and
simplicity, he had thrown into relief the works of the number theorists of the
old century. On the other side, "with wonderful farsightedness," he had

199
Hilbert

sketched out paths to the positive final treatment of the whole complex of
problems and had indicated the direction for the new century.
Three young mathematicians were brought to Gottingen to assist with the
editing. One of these was a young woman named Olga Taussky. She had been
trained in number theory by Philipp Furtwangler, who, although he had never
studied with Hilbert, had proved a number of theorems which Hilbert had
conjectured. Hilbert still enjoyed talking to young women. Mostly he talked
to Fraulein Taussky about his health and about his wish to return some day
and live out his life in Rauschen, the little Baltic fishing village where he had
spent the vacations of his youth. But one day, looking back over his career
and the many fields of mathematics in which he had worked, he remarked to
her that, much as he admired all branches of mathematics, he considered
number theory the most beautiful.
(That same year at the International Congress of Mathematicians at
Zurich, in connection with a talk by his former student, Rudolf Fueter,
Hilbert stated that the theory of complex multiplication of elliptic modular
functions, which brings together number theory and analysis, was not only
the most beautiful part of mathematics but also of all science.)
In the course of her work on Hilbert's papers, Fraulein Taussky was
astonished to discover many technical errors of varying degree. Although she
recognized that, because of Hilbert's powerful mathematical intuition, the
errors had not affected the ultimate results, she felt that they should be
corrected in the collected works.
Fraulein Taussky never consulted Hilbert about the correction of errors,
nor was she expected to. He had long ago put out of his mind his own work in
the field and had not followed the work of later mathematicians who had
carried out his program. At one point, with some trepidation, she pointed out
in a footnote (p. 506) that two conjectures had been incorrect.
The number theory volume was to be presented to Hilbert on his sev-
entieth birthday. A celebration was scheduled, a whole day of festivities.
It was all very bothersome, Hilbert complained to Bernays, but it would
be "good for mathematics."
Hermann Weyl wrote the birthday greeting that appeared in Naturlllis-
senschaften. Throughout his scientific career, as he wrote to his old friend
Robert Konig, he had kept before him a simple motto: "True to the spirit
of Hilbert." The birthday of Hilbert, Weyl now noted in his greeting, was
the high feast day for German mathematicians, celebrated year after year
in warm personal veneration for the master but also in personal affirmation
of their own beliefs and unity.

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 1930-1933

"Without doubt on the whole globe today Hilbert's name represents

most concretely what mathematics means in the framework of objective
spirits and how making mathematics, as a fundamental creative activity of
mankind, is alive among us."
And yet, Weyl had to concede, Hilbert's own brand of optimism, his
supreme confidence in the power of reason to come up with simple and
clear answers to simple and clear questions, was "not popular nowadays"
among the younger generation.
"Admittedly, one sentence or another of Hilbert's lecture [on logic and
the understanding of nature, which he had given in Konigsberg in 1930]
comes dangerously close to the opening words of Gottfried Keller's novel
Das Sinngedicht, in which he mocks his scientist Reinhardt - 'About twenty-
five years ago, when once again the natural sciences were at their highest
peak ... .'
"However, we do Hilbert an injustice if we confuse his rationalism with
that of Haeckel .... He would rightly be called presumptious if, Faust-like,
he had striven after the kind of magical knowledge which unlocks the very
core of being to the intellect .... Such knowledge is different from the
knowledge of reality that must prove itself by accurate prediction [and] can
be advanced only by the mathematical method ....
"Hilbert seems to me to be an outstanding example of a man through
whom the immensely creative power of naked scientific genius manifests
itself .... I remember how enthralled I was by the first mathematics class I
ever attended [at the University] .... It was Hilbert's famous course on
the transcendence of e and n ....
"W oe to the youth that fails to be touched to the core by such a man as
Hilbert !"
On the birthday itself, Ferdinand Springer, who was the publisher of the
collected works, came to Gottingen to present personally to Hilbert the
special white and gold leather-bound copy of the first volume. The beautiful
cover contained, however, not the printed pages, but only the proofs of the
pages; for Fraulein Taussky was still not satisfied. Hilbert made no comment
on the unfinished nature of the volume. But later, in his presence, Fraulein
Taussky declined a certain brand of cigarette as being too strong for her.
Somebody said that one really couldn't tell one brand from another.
"Aber nein!" Hilbert said. "Fraulein Taussky can tell the difference. She
is capable of making the finest, the very finest distinctions." She knew he was
needling her for taking so seriously errors which he considered unimportant,
and suspected that he had seen the footnote on p. 506.

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Hilbert

On the birthday evening there was a party in the magnificent new building
of the Mathematical Institute. Former colleagues and students came from
all over Germany and many from abroad. Although it was during the
Depression, everyone managed to look very elegant in shabby formal
dress. Olga Taussky remembers that she purchased a beautiful evening gown
for about two dollars, and it was much admired. There was a banquet with
many loving speeches and toasts. Arnold Sommerfeld read to Hilbert a little
verse which he had written: "Seiner Freunde treuester Freund I Hohler
Phrase argster Feind." (To his friends, truest friend I To the hollow phrase, bitter-
est enemy.)
Then Hilbert made a short speech. He recalled the great good luck with
which he had been blessed: the friendships with Minkowski and Hurwitz,
the study time in Leipzig with Felix Klein, the Easter trip of 1888 when he
had visited Gordan and Kronecker and many other mathematicians, his
appointment by Althoff as Lindemann's successor at an unusually early age.
And in his native city, he reminded the guests, he had had the good fortune
to find his wife "who since then in faithful comradeship has taken a decisive
part in my whole activity and especially in my concerns for the younger
generation." Minkowski's name was mentioned frequently. His sudden
death, Hilbert recalled, had left a "deep emptiness, both human and scien-
tific," but life had had to go on. Edmund Landau had come to take Min-
kowski's place. Now Felix Klein's great goal had at last been achieved, and
he himself was celebrating his seventieth birthday "in this beautiful Institute."
There was dancing after the banquet, and the guest of honor danced
almost every dance. A procession of students carrying torches marched
through the snow to the entrance of the brightly lighted building on Bunsen
Strasse, and shouted for Hilbert. He came out and stood on the steps,
bundled in his big coat with the fur collar, and somebody took a picture.
From every window of the Institute famous faces looked out.
Here at the end was the highest honor which the students could give to a
professor.
"For mathematics," Hilbert exhorted the shouting students, "hoch -
hoch - hoch!" In English, it would have been "Hip hip hooray!"
A few days after the birthday celebration, Hasse expressed to Mrs. Hilbert
"my ardent desire to talk once in my life personally to the great man."
Mrs. Hilbert invited him to come to tea and afterwards left him alone in
the garden with Hilbert.
"I began talking to him about what interested me most in those days -
the theory of algebraic numbers and in particular class-field theory. On

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 1930-1933

this theory I had written a report, in continuation of Hilbert's celebrated

Zahlbericht; and I began telling him what I had done in this theory, based
on his own famous results of the late nineties. But he interrupted me
repeatedly and insisted that I explain to him the basic conceptions and
results of that theory before he could listen to what I wanted to tell him.
So I explained to him the very foundations of class-field theory. About this
he got very enthusiastic and said, 'But that is extremely beautiful, who
has created it?' And I had to tell him that it was he himself who had laid
that foundation and envisaged that beautiful theory. After that he listened
to what I had to tell him about my own results. He listened attentively,
but more politely than intelligently."
In the Reichstag elections in the year of Hilbert's seventieth birthday,
the National Socialist Party made great gains. The following January,
President von Hindenburg appointed Adolf Hitler the chancellor of Ger-
many. Almost immediately came the first measure designed to break that
"satanical power" which had "grasped in its hands all key positions of
scientific and intellectual as well as political and economic life." The uni-
versities were ordered to remove from their employment almost every
full-blooded Jew who held any sort of teaching position.
The Hilbert school was perhaps the hardest hit. Hilbert's devotion to
his science had aiways been complete. No prejudice - national, sexual
or racial - had ever been allowed to enter into it. In 1917 an appro-
priate memorial had had to be written for Darboux even though his country
was at war with Germany. A position had had to be obtained for Emmy
Noether even though a woman had never been a Privatdozent at Gottingen.
Since the earliest friendships with Minkowski and Hurwitz, scientists had
never been classified by Hilbert as Aryan and non-Aryan. There were only
two kinds of scientists: those who solved problems of recognized worth,
and those who did not.
Now, in the Mathematical Institute itself, to whom did the ultimatum
apply? To Courant, who had replaced Klein and brought to reality Klein's
great dream. To Landau, who had come to Gottingen after the death of
Minkowski and made the University the center of research in the theory
of numbers. To Emmy Noether, who - in spi~e of the fact that she still
received no more than a trifling stipend - was the center of the most fertile
circle of research at that time in Gottingen. To Bernays, who had been
Hilbert's assistant and collaborator for almost sixteen years. In the Physics
Institute, both Born and Franck were Jews. A distinction, however, was
made between them by the new government. Franck, who had already

203
Hilbert

received his Nobel Prize, was exempted from the order. Born, who would
not receive his Nobel for some years, had to go. The ultimatum applied
to many others; sometimes it seemed to everyone.
Hilbert was extremely upset when he heard that many of his friends were
being put on "forced leave," as the current euphemism had it.
"Why don't you sue the government?" he demanded of Courant. "Go
to the state court? It is illegal for such a thing to happen!"
It seemed to Courant that Hilbert was completely unable to understand
that lawlessness had taken over. Since his birthday, it had been hard to get
him to listen and to accept innovations at the Institute. But chiefly his
difficulty seemed to be that he still believed the old system of justice pre-
vailed. He continued to retain the deep Prussian faith in the law which had
been inculcated in him by Judge Hilbert. It is exemplified by the story of
how when Frederick the Great was disturbed by the sound of a peasant's
mill and threatened to confiscate it, the peasant replied to the king with
complete confidence, "No - in Prussia there are still judges!" Frederick,
ashamed, had had the peasant's words inscribed across the portico of his
summer palace, where they still stood in 1933.
There was at first no general agreement among those affected about what
was to be done. How far would it all go? "If you knew the German people,
you knew it would go all the way." Young Hans Lewy decided to leave
Germany when Hitler was appointed chancellor. By the first of April he
was already in Paris. Some people who did not have to go left in protest.
Franck aligned himself with his fellow Jews. Others thought that something
of the greatness of Gbttingen could still be salvaged. Landau was allowed
to stay on because he had been a professor under the Empire. Further
exceptions would be made. Courant had been gassed and wounded in the
stomach fighting for Germany; surely that made him a German. Letters
were sent to the Minister about the case of Fraulein Noether. She held such
a minor job, received so little for her services. "I don't think there was
ever such a distinguished list of recommendations," Weyl later said.
Hilbert's name was at the top. But all the distinguished names had no
effect.
"The so-called Jews are so attached to Germany," Hilbert said plaintively,
"but the rest of us would like to leave."
Otto Neugebauer, now an associate professor, was placed at the head
of the Mathematical Institute. He held the famous chair for exactly one day,
refusing in a stormy session in the Rector's office to sign the required
loyalty declaration. The position of the head of the Mathematical Institute

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 1930-1933

passed to Weyl. Although his wife was part Jewish, he was one of those
who thought that something might yet be salvaged. All during the bitter
uncertain spring and summer of 1933 he worked, wrote letters, interviewed
officials of the government. But nothing could be changed.
By late summer nearly everyone was gone. Weyl, vacationing with his
family in Switzerland, still considered returning to Gottingen in the hope
that somehow he could keep alive the great scientific tradition. In America,
his many friends worried about him and wrote long letters, advising, urging,
begging that he leave Germany before it was too late. Abraham Flexner
offered him a position at the Institute for Advanced Study. Finally Einstein,
who had already been at the newly created Institute for several years,
prevailed upon the younger man to come and join him there.
In Gottingen, Hilbert was left almost alone. He kept Bernays on as his
assistant at his own expense. The Foundations of Mathematics, which he and
Bernays had written in collaboration, was almost ready for publication. He
put away his general mathematical books and became progressively more
distant. With Bernays's help, he saw Arnold Schmidt and Kurt Schutte
through the doctorate. Schutte was the last of 69 mathematicians (40 of
them during the years from 1900 to 1914) to receive their degrees from
Hilbert. In actuality, however, all of Schutte's contacts were through Ber-
nays. He saw Hilbert only once.
"When I was young," Hilbert said to young Franz Rellich, one of the
few remaining members of the old circle, "I resolved never to repeat what
I heard the old people say - how beautiful the old days were, how ugly the
present. I would never say that when I was old. But, now, I must."
Sitting next to the Nazis' newly appointed minister of education at a
banquet, he was asked, "And how is mathematics in Gottingen now that it
has been freed of the Jewish influence?"
"Mathematics in Gottingen?" Hilbert replied. "There is really none any
more."

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 XXIV

Age

In the center of the town the swastika flew above the Rathaus and cast
its shadow on the little goose girl. The university bulletin and publications
appeared again in the traditional German script, the first page of each one
bearing the statement that it appeared under Herr Goebbe1s's sponsorship.
A Nazi functionary became the head of the Mathematical Institute.
During the winter semester 1933-34 Hilbert lectured for one hour a week
on the foundations of geometry. After the end of the semester, he never
again came to the Institute.
Landau continued to lecture; but when he announced a course in cal-
culus, an unruly mob prevented his entering the lecture hall. "It is all
right for you to teach advanced courses," he was told, "but these are
beginners and we don't want them taught by a Jew." Siegel, now a
professor at Frankfurt, attempted to get support for his old teacher from
a group of professors who were safe in their positions. He was not
successful.
After a while Landau too was gone from Gottingen. Unlike the others,
he did not leave the country, being tied to his native land by the fact of his
wealth and possessions. Hardy arranged for him to deliver a series of lec-
tures in England: "It was quite pathetic to see his delight when he found
himself again in front of a blackboard and his sorrow when his opportunity
came to an end."
By the spring of 1934 the situation had become so bad for the Jews that
Bernays felt he must leave Germany, and he returned to Zurich. The
Mathematical Institute continued to pay the salary of Hilbert's remaining
assistant, Arnold Schmidt, who worked with him in his home on problems
of logic and foundations.

206
 1933-1945

"There were brief failings of memory which might make strangers think
he was not so sharp," Schmidt says, "but those who worked with him in
this area knew differently."
Now Helmut Hasse was made the head of the Institute. This was a great
improvement; for although Hasse had long been a convinced nationalist,
he was a first-rate mathematician.
That summer Emmy Noether, for whom a place had been found in
America at Bryn Mawr, returned to Gottingen. "Her heart knew no malice,"
Weyllater explained. "She did not believe in evil- indeed it never entered
her mind that it could playa role among men." Things were not so clear
then as they were later to seem, and she wished Hasse only success in his
efforts to rebuild the great tradition of Gottingen after the exodus of the
previous year. At the end of the summer she returned to Bryn Mawr. She
was at the height of her powers, her imagination and her technique having
reached the maximum point of perfect balance. In her hands "the axiomatic
method, no longer merely a method for logical clarification and deepening
of the foundations, [had become] a powerful weapon of concrete mathe-
matical research." Already, with it, she and van der Waerden and others
had laid the foundations of modern algebra.
At first both the Hilberts had spoken out in such a forthright way against
the new regime that their friends remaining in Gottingen were frightened
for their safety. But they did not trust many of the people who were left,
nor the new people who came, and after a while they too fell silent.
"Well, Herr Geheimrat, how do you fare?" one of the now infrequent
visitors inquired of Hilbert.
"I - well, I don't fare too well. It fares well only with the Jews," he re-
plied in the old unexpected way. "The Jews know where to stand."
Von Hindenburg died in the summer of 1934, purportedly leaving a will
which bequeathed the presidency of the Reich to Hitler, who would then
be both president and chancellor. An election was scheduled for August
with the alternatives, yes or no. The day before the election, the newspapers
carried a proclamation announcing that Hitler had the support of German
science. The list of signatures included the name of Hilbert. Whether Hilbert
actually signed the proclamation is not known. Arnold Schmidt, who was
at that time seeing him almost every day, was not aware of the existence of
such a proclamation until he was shown a copy of the newspaper report of
it more than thirty years later. Signing would have been contrary to every-
thing Schmidt knew from personal experience that Hilbert believed. He
had to concede, however, that "at that time it is possible that Hilbert

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Hilbert

would have signed anything to get rid of someone who was bothering
him."
In 1935 the final volume of the collected works, which contained the
life history written by Blumenthal, was published. Hilbert wrote a little
note to his oldest student, commenting on this last great piece of luck -
that he should have such a splendid interpreter of his life and work. Blu-
menthal placed the note in his own copy of Hilbert's collected works.
For his biographical article, Blumenthal had called up his memories of
his teacher since that day when the "medium-sized, quick, unpretentiously
dressed man, who did not look at all like a professor" had come to Gottin-
gen in the spring of 1895 as the successor of Heinrich Weber. But in spite
of the warmth and affection, the life history remained objective.
"For the analysis of a great mathematical talent," Blumenthal concluded,
"one has to differentiate between the ability to create new concepts and the
gift for sensing the depth of connections and simplifying fundamentals.
Hilbert's greatness consists in his overpowering, deep-penetrating insight.
All of his works contain examples from far-flung fields, the inner relatedness
of which and the connection with the problem at hand only he had been
able to discern; from all these the synthesis - and his work of art - was
ultimately created. As far as the creation of new things is concerned, I
would place Minkowski higher, and from the classical great ones, for in-
stance, Gauss, Galois, Riemann. But in his sense for discovering the syn-
thesis only a very few of the great have equaled Hilbert."
In the spring of 1935 Emmy Noether died in the United States following
an operation.
In his office at the Institute for Advanced Study, Einstein wrote a letter to
the editor of The New York Times, which had reported her death only
briefly: "In the judgment of most competent living mathematicians, Fraulein
Noether was the most significant creative mathematical genius [of the
female sex] thus far produced ....
"Beneath the effort directed toward the accumulation of worldly goods
lies all too frequently the illusion that this is the most substantial and desir-
able end to be achieved; but there is, fortunately, a minority composed of
those who recognize early in their lives that the most beautiful and satisfying
experiences open to human kind are not derived from the outside but are
bound up with the individual's own feeling, thinking and acting ....
However inconspicuously the lives of these individuals run their course,
nonetheless, the fruits of their endeavors are the most valuable contri-
butions which one generation can make to its successors."

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 1933-1945

The editors of the Annalen decided to risk publishing a memorial article

by van der Waerden. After the journal appeared in print, they waited for
the blow to fall; but nothing happened. Taking courage, they published a
paper by Blumenthal, who was still listed on the cover of the Annalen as
one of the editors, although as a consequence of the Nuremberg Laws he
had recently been removed from his professorship in Aachen. Again, noth-
ing happened.
But the general scientific situation in Germany was progressively dete-
riorating. A strong supporter of the Third Reich was Ludwig Bieberbach,
who had so passionately opposed the German mathematicians' attending
the International Congress at Bologna. He and others analyzed the dif-
ferences in the creative styles of German mathematicians and Jewish
mathematicians. Death was no protection. When Klein was listed in a
Jewish encyclopedia, his antecedents were carefully examined and it was
determined finally that he was "a great German mathematician." Hilbert's
antecedents were also examined. There was a joke that there was only one
Aryan mathematician in Gottingen and in his veins Jewish blood flowed.
The joke depended upon the fact that during Hilbert's illness, he had
received a blood transfusion from Courant. Now the question was seriously
raised if it was not suspicious for an Aryan mathematician to have the name
David. It finally became necessary for Hilbert to produce the autobiography
of Christian David Hilbert to show that David was a family name and that
the other family names indicated that the Hilberts had at one time been
Pietists.
In the late summer of 1936 the mathematicians of the world met again
for another International Congress, this time at Oslo. Although Hilbert
did not attend, he was remembered with a telegram of greeting from the
delegates. Courant, who came from the United States, where he was now
teaching at New York University, telephoned from Oslo; but Hilbert did
not know what to say to his former pupil and colleague, and the conversa-
tion consisted of his fumbling over and over, "Well, what should I say?
What should I ask now? Let me think for a moment."
In 1937 Hilbert was 75. A newspaper reporter came to interview him and
ask him about places in Gottingen connected with the history of mathemat-
ics. "I actually know none," he said without (to the reporter's surprise) a
trace of embarrassment at his ignorance. "Memory only confuses thought -
I have completely abolished it for a long time. I really don't need to know
anything, for there are others, my wife and our maid - they will know."
As the reporter then began to express "a courteous doubt" whether one

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Hilbert

could so eliminate memory and history, Hilbert put back his head and gave
a little laugh.
"Ja, probably I have even been known to be especially gifted for forget-
ting. For that reason indeed did 1 study mathematics."
Then he closed his eyes.
The reporter refrained from disturbing any further the old man, "the
honorary doctor of five universities, who with easy serenity could completely
forget everything - house, streets, city, names, occurrences and facts -
because he had the power in each remaining moment to derive and develop
again a whole world."
That night there was a birthday party at the Hilberts', a comparatively
large affair for the new days. While the congratulatory speeches were being
made, Hilbert sat in another room with his arms around the two young
nurses who came regularly to the house to give him some physical treat-
ments. When Heeke, who had come from Hamburg for the party, reminded
him that he really should listen to the glowing speeches being made about
him and his work, he laughed, "This is much better!"
Elizabeth Reidemeister took a birthday picture. She reminded him of
what seemed to her some important event in which they had both partici-
pated, and was surprised that he remembered nothing of it. "I am interested
only in the stars," he explained.
During this period Franz was home again. With age he had come.to look
disconcertingly like his father. He patterned himself after him, loudly spoke
out his opinion on all subjects - a tragic parody - "the sound without the
substance," as the people in Gottingen observed. He never held any real
job.
But he also studied various subjects very thoroughly - Goethe, theology.
He was a real "Kenner," according to Arnold Schmidt - an expert on the
fields of his interests. He spoke often of learning mathematics so that he
could appreciate his father's work.
The nextyear-1938 - saw the last birthday party in the house on Wilhelm
Weber Strasse. There were only a few old friends for lunch. Heeke
came from Hamburg, Caratheodory from Munich. Siegel, who was
now at the Institute in Gottingen, was there. Also present was Blu-
menthal.
"What subjects are you lecturing on this semester?" Hilbert asked.
"I do not lecture any more," Blumenthal gently reminded him.
"What do you mean, you do not lecture?"
"I am not allowed to lecture any more."

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 1933-1945

"But that is completely impossible! That cannot be done. Nobody has a

right to dismiss a professor unless he has committed a crime. Why do you
not apply for justice?"
The others tried to explain Blumenthal's situation, but Hilbert became
increasingly angry with them.
"I felt," Siegel says, "that he had the impression we were trying to play
a bad joke on him."
Shortly afterwards, Blumenthal's name had to be removed from the
cover of the Annalen. With the help of his friends he left Germany for
Holland.
Across the ocean in America, George P6lya, who was now at Stanford
University, reminded Weyl that it was 1938 and that by its terms their
wager on the future of Intuitionism was now up. Weyl conceded that he
had indeed lost, but he asked P6lya please not to make him concede pub-
licly.
That same year Edmund Landau died.
Life went on in Gottingen. Mrs. Hilbert, who was gradually losing her
sight, mourned that the people who used to come to visit no longer came.
Sometimes, though, there were still small social gatherings around the
Hilberts.
On one of these occasions there was a discussion about which German
town was the most beautiful. Some of the guests said Dresden, others said
Munich. But Hilbert insisted, "No, no - the most beautiful town in Ger-
many is still Konigsberg!" When his wife protested, "But, David, you can't
really say that - Konigsberg is not all that beautiful," he replied: "But,
Kathe, after all I must know, for I have spent my whole life there." Even
when she reminded him that in fact they had come to Gottingen more than
forty years before, he shook his head: "Ah, a few little years - I have spent
my whole life in Konigsberg!"
"So," Hasse, who was present, thought sadly, "his mind has condensed
all the forty fruitful years of his wonderful achievement in so many branches
of mathematics into a few little years."
In 1939 an agreement signed in Munich among Germany, France, Eng-
land and Italy seemed for the moment to guarantee "peace in our time."
The Swedish Academy of Science announced the first Mittag-Leffler Prize,
which was to go to David Hilbert and Emile Picard. The old Frenchman
received his prize from the emissary of the Academy, Torsten Carleman,
at a large banquet in Paris. After a glowing tribute, Carleman presented
Picard with a complete, beautifully bound set of Mittag-Leffler's Acta

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Hilbert

Mathematica, the journal which had been one of the first to publish Cantor's
works. From Paris, Carleman went to Gottingen. He expected that there he
would enjoy a repetition of banquet, speech and presentation. Disappointed
when such was not forthcoming, he still insisted that he wanted to award
the prize to Hilbert in person. Hasse and Siegel finally located Hilbert, who
was in the nearby Harz mountains with his wife, and drove Carleman to
the inn where they were staying. Hilbert listened silently to Carleman's
tribute. Shortly afterwards, the 72 red leather volumes of Acta Mathematica
appeared on the shelves of the library of another mathematician, to whom
Hilbert had almost immediately sold them.
It was August again. On the first day of September Germany invaded
Poland. Within a week France and England had declared war on Germany.
Hilbert's assistant now was the gifted young logician, Gerhard Gentzen,
who, following the new and less restrictive methods of "transfinite induc-
tion," had been able to achieve the long-sought proof of the consistency of
arithmetic. The proof had been managed, however, only by substantially
lowering the standards Hilbert had originally set up. Gentzen came regularly
to Hilbert's house and read aloud - at Hilbert's request - the poems of
Schiller. After a while Gentzen too was gone. He died in 1945 following
arrest and imprisonment in Prague.
Holland was invaded. In England efforts were made by Ewald and others
to get Blumenthal to safety. But it was too late.
Siegel had vowed at the end of the first war that he would not remain in
Germany during another war. In March 1940, he received an invitation to
deliver a lecture in Oslo. He knew that he would not see the Hilberts again,
and so he went to say goodbye. They were not at the house on Wilhelm
Weber Strasse, the furnace having broken down; but he found them in a
shabby hotel where, Mrs. Hilbert told him, Hermann Amandus Schwarz
always used to stay when he was in Gottingen. Schwarz had been responsible
for bringing Klein to the University and had been dead now some twenty
years. The Hilberts were having breakfast in their room, Hilbert sitting on
the bed and eating from a jar of caviar that Niels Bohr had sent him from
Copenhagen. Siegel said goodbye. In Oslo he found that the Bohr brothers
and Oswald Veblen had already arranged passage to the United States,
where a place at the Institute for Advanced Study would be waiting for him.
Two days after he left Oslo, the Germans invaded Norway.
In December 1941, a month before Hilbert's eightieth birthday, the
United States entered the war. Although there was no party on the eightieth
birthday, a tribute to Hilbert appeared as usual. It was prepared by Wal-

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ther Lietzmann, who in 1902 had headed the delegation from the Math-
ematics Club who had pleaded with Hilbert to refuse the tempting offer
from Berlin and remain in Gottingen. The story of Hilbert's life and
his achievements, everything was there in Lietzmann's tribute except the
names of the many Jews (other than Minkowski and Hurwitz) who had
played such an important part in his career. Blumenthal was circumspectly
quoted merely as "the author of the life history in the collected works."
The picture that accompanied the tribute was a recent one, and the eyes which
had looked so firmly and innocently out at the world seemed now distrustful.
In Holland, Blumenthal dedicated a paper to Hilbert in honor of the
eightieth birthday.
The Berlin Academy voted to commemorate the birthday with a special
citation for that work which of all the influential Hilbert works had had the
most pervasive influence on the progress of mathematics - the little 92-page
book on the foundations of geometry.
The day that this award was voted by the Academy, Hilbert fell on the
street in Gottingen and broke his arm. He died, a little more than a year
later, on February 14, 1943, of complications arising from the physical
inactivity that resulted from the accident.
Not more than a dozen people attended the morning funeral service in
the living room of the house on Wilhelm Weber Strasse. From Munich
came one of his oldest friends. Standing beside the coffin, Arnold Sommer-
feld spoke of Hilbert's work.
What had been his greatest mathematical achievement?
"The invariants? The number theory, which was so loved by him? The
axiomatics of geometry, which was the first great achievement in this
field since Euclid and the non-euclidean geometries? What Riemann and
Dirichlet surmised, Hilbert established by proof at the foundation of func-
tion theory and the calculus of variations. Or were integral equations the
high point .... Soon in the new physics ... they bore most beautiful fruits.
His gas theory had a fundamental effect on the new experimental knowledge,
which has not yet been played out. Also his contributions to general rela-
tivity theory are of permanent value. Of his final endeavors in connection
with mathematical knowledge, the last word has still not been spoken. But
when in this field a further development is possible, it will not by-pass
Hilbert but go through him."
Caratheodory had also planned to come from Munich for the funeral,
but he had fallen ill. The tribute which he had written was read by Gustav
Herglotz with tears streaming down his face.

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