Hilbert

instructing Hurwitz in his own specialty, which became known as the

"Schubert calculus." He had also managed to convince Hurwitz's father,
a Jewish manufacturer as doubtful of the rewards of an academic life as
Judge Hilbert, that the gifted boy must be permitted to continue his study
of mathematics. Encouraged by Schubert, the father borrowed the necessary
money from a friend.
Hurwitz's first mathematical work was published in collaboration with
Schubert while he was still in the gymnasium. His later studies gave him
an exceptionally wide background in the mathematics of the day. He receiv-
ed his doctor's degree from Felix Klein, one of the most spectacular of the
younger mathematicians in Germany at that time. He attended the lectures
of the great men in Berlin and then moved on to Gottingen, where he did
impressive work in function theory.
He was a sweet-tempered young man who loved music almost as much
as mathematics and played the piano beautifully. But already, before he
came to Konigsberg, he had suffered a nearly fatal bout with typhoid fever.
He had frequent, very severe migraine headaches, which may have been at
least partially caused by the fact that he was a perfectionist in everything
he did.
Hilbert found the new teacher "unpretentious in his outward appearance"
but saw that "his wise and gay eyes testified as to his spirit." He and Min-
kowski soon established a close relationship with Hurwitz. Every afternoon,
"precisely at five," the three met for a walk "to the apple tree." It was at
this time that Hilbert found a way of learning infinitely preferable to poring
over dusty books in some dark classroom or library.
"On unending walks we engrossed ourselves in the actual problems of
the mathematics of the time; exchanged our newly acquired understandings,
our thoughts and scientific plans; and formed a friendship for life."
Learning "in the most easy and interesting way," the three young men
explored every kingdom of the mathematical world. Hurwitz with his vast
"well founded and well ordered" knowledge was always the leader. He
quite overwhelmed the other two.
"We did not believe," Hilbert recalled later, "that we would ever bring
ourselves so far."
But there was no need for them to feel like Alexander, who complained
to his schoolmates, "Father will conquer everything and there will be
nothing left for us to conquer."
The world of mathematics is inexhaustible.

14
 III

Doctor of Philosophy

Having completed the eight university semesters required for a doctor's

degree, Hilbert began to consider possible subjects for his dissertation. In
this work he would be expected to make some sort of original contribution
to mathematics. At first he thought that he might like to investigate a general-
ization of continued fractions; and he went to Lindemann, who was his
"Doctor-Father," with this proposal. Lindemann informed him that unfor-
tunately such a generalization had already been given by Jacobi. Why not,
Lindemann suggested, take instead a problem in the theory of algebraic
invariants.
Although the theory of algebraic invariants was considered a very modern
subject, its roots lay in the seventeenth century invention of analytic geom-
etry by Rene Descartes. On Descartes's map of the plane, horizontal coor-
dinates are real numbers which are designated by x; vertical coordinates,
real numbers designated byy. Since any point on the plane is then equivalent
to a pair of real numbers x, y, geometric figures can be formalized by alge-
braic equations and, conversely, algebraic equations can be graphed as
geometric figures. Concepts and relations in both fields are clarified -
geometric ideas becoming more abstract and easily handled; algebraic ideas,
more vivid and more intuitively comprehensible.
There is also a great gain in generality. Just as the size and the shape of
figures do not change when their position in relation to the axes is changed,
so certain properties of their related algebraic forms remain unchanged.
These "invariants" serve to characterize the given geometric figure. Thus,
quite naturally, the development of projective geometry, which concerns
itself with the often dramatic transformations effected by projection, led
eventually to a parallel development in algebra which concentrated on the
invariants of algebraic forms under various groups of transformations.

15
Hilbert

Because of its sheer sophisticated power, the algebraic approach soon out-
stripped the geometric one; and the theory of algebraic invariants became
a subject of consuming interest for a number of mathematicians.
The pioneers in the new theory had been Englishmen - Arthur Cayley
and his good friend, Joseph Sylvester, both men, as it happened, lawyers
turned mathematicians. But the Germans had been quick to take up the
theory; and now the great German mathematical journal, Mathematische
Annalen, was almost exclusively an international forum for papers on
algebraic invariants.
The problem which Lindemann suggested to Hilbert for his doctoral
dissertation was the question of the invariant properties for certain algebraic
forms. This was an appropriately difficult problem for a doctoral candidate,
but not so difficult that he could not be expected to solve it. Hilbert showed
his originality by following a different path from the one generally believed
to lead to a solution. It was a very nice piece of work, and Lindemann was
satisfied.
A copy of the dissertation was dispatched to Minkowski, who after his
father's recent death had gone to Wiesbaden with his mother.
"I studied your work with great interest," Minkowski wrote to Hilbert,
"and rejoiced over all the processes which the poor invariants had to pass
through before they managed to disappear. I would not have supposed that
such a good mathematical theorem could have been obtained in Konigs-
berg !"
On December 11, 1884, Hilbert passed the oral examination. The next
and final ordeal, on February 7, 1885, was the public promotion exercise
in the Aula, the great hall of the University. At this time he had to defend
two theses of his own choice against two fellow mathematics students
officially appointed to be his "opponents." (One of these was Emil Wiechert,
who later became a well-known seismologist.) The contest was generally
no more than a mock battle, its main function being to establish that the
candidate could perceive and frame important questions.
The two propositions which Hilbert chose to defend spanned the full
breadth of mathematics. The first concerned the method of determining
absolute electromagnetic resistance by experiment. The second pertained
to philosophy and conjured up the great ghost of Immanuel Kant.
It had been the position of Kant, who had lectured on mathematics as
well as philosophy when he taught at Konigsberg, that man possesses
certain notions which are not a posteriori (that is, obtained from experience)
but a priori. As examples of a priori knowledge he had cited the fundamental

16
 1884-1886

concepts of logic, arithmetic and geometry - among these the axioms of

Euclid.
The discovery of non-euclidean geometry in the first part of the nine-
teenth century had cast very serious doubt on this contention of Kant's; for
it had shown that even with one of Euclid's axioms negated, it is still possible
to derive a geometry as consistent as euclidean geometry. It thus became
clear that the knowledge contained in Euclid's axioms was a posteriori -
from experience - not a priori.
Could this also possibly be true of the fundamental concepts of arithmetic?
Gauss, who was apparently the first mathematician to be aware of the
existence of non-euclidean geometries, wrote at one time:
"I am profoundly convinced that the theory of space occupies an entirely
different position with regard to our knowledge a priori from that of
[arithmetic]; that perfect conviction of the necessity and therefore the
absolute truth which is characteristic of the latter is totally wanting in our
knowledge of the former. We must confess, in all humility, that number is
solely a product of our mind. Space, on the other hand, possesses also a
reality outside our mind, the laws of which we cannot fully prescribe a
priori."
Hilbert appears to have felt this way too; for in his second proposition he
maintained:
That the oijections to Kant's theory of the a priori nature of arithmetical judg-
ments are unfounded.
There is no record of his defense of this proposition. Apparently, his
arguments were convincing; for at the conclusion of the disputation he was
awarded the degree of Doctor of Philosophy.
The Dean administered the oath:
"I ask you solemnly whether by the given oath you undertake to promise
and confirm most conscientiously that you will defend in a manly way true
science, extend and embellish it, not for gain's sake or for attaining a vain
shine of glory, but in order that the light of God's truth shine bright and
expand."
That night the new Doctor of Philosophy and his celebrating friends
wired the news to Minkowski.
Hilbert was now set upon the first step of an academic career. If he were
fortunate and qualified, he would arrive ultimately at the goal, the full
professorship - a position of such eminence in the Germany of the day
that professors were often buried with their title and the subject of their
specialty on their gravestones. As a mere doctor of philosophy, however,

17
Hilbert

he was not yet eligible even to lecture to students. First he had to turn out
still another piece of original mathematical research for what was known as
"Habilitation." If this was acceptable to the faculty, he would then be
awarded the venia legendi, which carried with it the title of Privatdozent and
the privilege of delivering lectures without pay under the sponsorship of
the university. As such a docent he would have to live on fees paid by stu-
dents who chose to come to hear his lectures. Since the courses which all
the students took, such as calculus, were always taught by a member of the
official faculty, he would be fortunate to draw a class of five or six. It was
bound to be a meager time. Eventually, however, if he attracted attention
by his work and abilities (or better yet, it was rumored, if he married a
professor's daughter), he would become an Extraordinarius, or associate
professor, and receive a salary from the university. The next step would
be an offer of an Ordinariat, or full professorship. But this final step was by
no means automatic, since the system provided an almost unlimited supply
of docents from which to draw a very limited number of professors. Even
in Berlin, there were only three mathematics professors; at most Prussian
universities, two; at Konigsberg, only one.
As a hedge against the vicissitudes of such a career, a young doctor could
take the state examination and qualify himself for teaching at the gymnasium
level. This was not a prize to be scorned. Although many, their eyes on the
prestige-ladened professorship, didn't consider the alternative, one needed
only to match the number of docents with the number of professorial
chairs which might reasonably become vacant in the next decade to see its
advantages. Hilbert now began to prepare himself for the state examination,
which he passed in May 1885.
That same summer lvfinkowski returned to Konigsberg, received the
degree of doctor of philosophy, and then left almost immediately for his
year in the army. (Hilbert was one of his official opponents for the promo-
tion exercises.)
Hilbert had not been called up for military service. He considered a study
trip, and Hurwitz urged Leipzig ~ and Felix Klein.
Although Klein was only 36 years old, he was already a legendary figure
in mathematical circles. When he had been 23 (Hilbert's present age) he
had been a full professor at Erlangen. His inaugural lecture there had made
mathematical history as the Erlangen Program ~ a bold proposal to use the
group concept to classify and unify the many diverse and seemingly unre-
lated geometries which had developed since the beginning of the century.
Early in his career he had shown an unusual combination of creative and

18
 1884-1886

organizational abilties and a strong drive to break down barriers between

pure and applied science. His mathematical interest was all-inclusive.
Geometry, number theory, group theory, invariant theory, algebra - all
had been combined for his master work, the development and completion
of the great Riemannian ideas on geometric function theory. The crown of
this work had been his theory of automorphic functions.
But the Klein whom Hilbert now met in Leipzig in 1885 was not this
same dazzling prodigy. Two years before, in the midst of the work on
automorphic functions, a young mathematician from a provincial French
university had begun to publish papers which showed that he was concen-
trating his efforts -in the same direction. Klein immediately recognized the
caliber of his competitor and began a nervous correspondence with him.
With almost super-human effort he drove himself to reach the goal before
Henri Poincare. The final result was essentially a draw. But Klein collapsed.
When Hilbert came to him, he had only recently recovered from the long
year of deep mental depression and physical lassitude which had followed
his breakdown. He had passed the time writing a little book on the icosahe-
dron which was to become a classic; but the future direction of his career
was still not determined.
Hilbert attended Klein's lectures and took part in a seminar. He could
not have avoided being impressed. Klein was a tall, handsome, dark haired
and dark bearded man with shining eyes, whose mathematical lectures were
universally admired and circulated even as far as America. As for Klein's
reaction to the young doctor from Konigsberg - he carefully preserved the
Vortrag, or lecture, which Hilbert presented to the seminar and he later
said: "When I heard his V ortrag, I knew immediately that he was the com-
ing man in mathematics."
Since his breakdown, Klein had received two offers of positions - the
first, to Johns Hopkins University in America, he had refused; the second,
to Gbttingen, he had just recently accepted. Apparently Hilbert had already
absorbed a feeling for Gbttingen, the university of Gauss, Dirichlet and
Riemann. Inspired by Klein's appointment, he now scribbled one of his
little verses on the inside cover of a small notebook purchased in Leipzig.
The writing is so illegible that the German cannot be made out exactly,
but the sense of the verse is essentially "Over this gloomy November day!
Lies a glow, all shimmery! Which Gbttingen casts over us! Like a youthful
memory."
In Leipzig, Hilbert soon became acquainted with several other young
mathematicians. One of these was Georg Pick, whose knowledge of pollina-

19
Hilbert

tion and breeding as well as his admiration for Hurwitz's work recommend-
ed him to Hilbert. Another was Eduard Study, whose main interest, like
Hilbert's own, was invariant theory. The two should have had a lot in
common, but this was not to be the case. Study was a "strange person,"
Hilbert wrote to Hurwitz, "and almost completely at opposite poles from
my nature ttnd, as I think I can judge, from yours too. Dr. Study approves,
or rather he knows, only one field of mathematics and that's the theory of
invariants, very exclusively the symbolic theory of invariants. Everything
else is unmethodical 'fooling around' .... He condemns for this reason all
other mathematicians; even in his own field he considers himself to be the
only authority, at times attacking all the other mathematicians of the sym-
bolic theory of invariants in the most aggressive fashion. He is one who
condemns everything he doesn't know whereas, for example, my nature
is such I am most impressed by just that which I don't yet know." (Hurwitz
wrote back, "This personality is more repugnant to me than I can tell you;
still, in the interest of the young man, I hope that you see it a little too
darkly.")
There were a considerable number of people at Leipzig who were inter-
ested in invariant theory; but Klein went out of his way to urge both Study
and Hilbert to go south to Erlangen to pay a visit to his friend Paul Gordan,
who was universally known at that time as "the king of the invariants."
For some reason the expedition was not made. Perhaps because Hilbert did
not care to make it with Study.
Hilbert was soon a member of the inner mathematical circle in Leipzig.
At the beginning of December 1885, a paper of his on invariants was
presented by Klein to the scientific society. On New Year's Eve he was
invited to a "small but very select" party at Klein's - "Professor Klein,
his honored spouse, Dr. Pick and myself." That same evening Minkowski,
stranded and cold at Fort Friedrichsburg in the middle of the Pregel, was
sending off New Year's greetings to his friend with the plaintive question,
"Oh, where are the times when this poor soldier was wont to busy himself
over the beloved mathematics?" But at the Kleins' the conversation was
lively - "on all possible and impossible things." Klein tried to convince
Hilbert that he should go to Paris for a semester of study before he returned
to Konigsberg. "He said," Hilbert wrote to Hurwitz, "that Paris is at this
time a beehive of scientific activity, particularly among the young mathe-
maticians, and a period of study there would be most stimulating and
profitable for me, especially if I could manage to get on ~he good side of
Poincare. "

20
 1884-1886

Klein himself in his youth had made the trip to Paris in the company of
his friend Sophus Lie, and both he and Lie had brought away their knowl-
edge of group theory, which had played an important part in their careers.
Now, according to Hurwitz, Klein always tried to send promising young
German mathematicians to Paris.
Hurwitz himself seconded Klein's recommendation: "I fear the young
talents of the French are more intensive than ours, so we must master all
their results in order to go beyond them."
By the end of March 1886, Hilbert was on his way.

21
 IV

Paris

On the train to Paris, Hilbert had the good luck to be in the same com-
partment with a student from the Ecole Poly technique who knew all the
French mathematicians "at least from having looked at them." But in Paris,
of necessity, he had to join forces with the disagreeable Study, who was
already established there, also on Klein's recommendation.
Together, Hilbert and Study paid the mathematical visits which Klein
had recommended. When they wrote to Klein, they read their letters aloud
to one another so that they would not repeat information.
As soon as Hilbert was settled, he wrote to Klein. The letter shows how
important he considered the professor. It was carefully drafted out with
great attention to the proper, elegant wording, then copied over in a large,
careful Roman script rather than the Gothic which he continued to use in
his letters to Hurwitz.
"The fact that I haven't allowed myself at an earlier point in time to
entrust the international post with a letter to you is due to the various
impediments and the unforeseen cares which are always necessary on
the first stay in a strange country. Fortunately, I have now adjusted to the
climate and accommodated myself to the new environment well enough
that I can start to spend my time in the way that I wish .... "
He tried hard to follow Klein's instructions about becoming friendly
with Poincare. The Frenchman was eight years older than he. Already he
had published more than a hundred papers and would shortly be proposed
for the Academy with the simple statement that his work was "above
ordinary praise." In his first letter, Hilbert reported to Klein that Poincare
had not yet returned the visit which he and Study had paid on him; however,
he added, he had heard him lecture at the Sorbonne on potential theory and
the mechanics of fluids and had later been introduced to him.

22
 1886

"He lectures very clearly and to my way of thinking very understandably

although, as a French student here remarks, a little too fast. He gives the
impression of being very youthful and a bit nervous. Even after our intro-
duction, he does not seem to be very friendly; but I am inclined to attribute
this to his apparent shyness, which we have not yet been in a position to
overcome because of our lack of linguistic ability."
By the time that Hilbert wrote to Klein again, Poincare had returned the
visit of the young Germans. "But about Poincare I can only say the same -
that he seems reserved because of shyness, but that with skillful treatment
he would open up."
In replying to the letters from Paris, Klein (who was now established in
Gottingen) played no favorites between his two young mathematicians.
"It is thoroughly necessary that you and Hilbert have personal contact
with Gordan and Noether," he wrote to Study. "Next time," he concluded
his letter, "I shall write to Dr. Hilbert." But Hilbert seems to have placed
a higher value on Klein's letters, for he preserved those written to Study
as well as those written to himself.
The French mathematicians - Hilbert wrote to Klein - welcomed him
and Study with great warmth. Jordan was most kind "and he is the one
who presents the most devoted greetings to you." He gave a dinner for
Hilbert and Study "to which only Halphen, Mannheim and Darboux were
invited." Since, however, everyone spoke German in deference to the
visitors, the conversation on mathematics was "very superficial."
Hilbert was not impressed with the mathematical lectures he heard.
"French students do not have much that would interest us." Picard's lectures
seemed "least elementary." Although Hilbert found Picard's pronunciation
hard to understand, he attended his lectures regularly. "He gives the im-
pression of being very energetic and positive in his conversation as well
as in his teaching."
Some of the well-known mathematicians were a disappointment. "Con-
cerning Bonnet - the trouble we went through to find him - a cruel fate
sent us to three different houses first - was scarcely in proportion to the
advantage we expected from such an old mathematician. He is obviously
no longer responsive to mathematical things."
Hilbert and Study attended the meeting of the Societe Mathematique in
the hope of becoming acquainted with some young - or, at least, younger-
mathematicians: "one reason is to be not always towered over by men
who are so much greater than we are." Among those they met, Hilbert
found Maurice d'Ocagne especially outstanding "because of his pleasant

23
Hilbert

manners and approachability." In the course of the Societe meeting he

saw how he could sketch out a more direct proof than the one given by
d'Ocagne of a theorem in his communication. "So I took courage, sup-
ported by Halphen, to point out this way of proving it." D'Ocagne asked
Hilbert to write down his proof and offered to correct the French in case
he wanted to publish it in the Comptes Rendus. "But I do not want to go
into this thing because I think neither the theorem itself nor the proof is
important enough to be put in the Comptes Rendus."
"As to the publications of Poincare," Klein commented in this connection,
"I have always the impression that there is the intention to publish some-
thing even if none or few new results are present. Do you approve of this?
Have you happened to hear in Paris that people there have the same
opinion?"
Among the French mathematicians, it was Hermite who seemed the most
attractive to Hilbert;
"He not only showed us all his well-known politeness by returning our
visit promptly, but also showed himself very kind ... by offering to spend
the morning with me when he doesn't have a lecture."
The young Germans went back for a second visit. Hermite seemed very
old to them - he was 64 - "but extraordinarily friendly and hospitable."
He talked of his law of reciprocity in binary forms and encouraged them
to extend it to ternary forms. In fact, much of the conversation was on
invariants, since that was the subject in which Hermite knew his young
visitors were most interested. He directed their attention to the most famous
unsolved problem in the theory - what was known as "Gordan's Problem"
after Klein's friend Gordan at Erlangen - and told them at length about
his correspondence with Sylvester concerning the latter's efforts to solve it.
"The way Hermite talked about other, non-scientific topics proves that
he has kept his youthful attitude into old age," Hilbert wrote admiringly to
Klein.
While Hilbert was having these stimulating corttacts in Paris, Minkowski
was still soldiering in Konigsberg. "I have stood guard duty at 20°, and
been forgotten to be relieved on Christmas Eve .... " But he was hoping
soon "to renew an old acquaintance with Frau Mathematika," and begged
for news in the smallest detail of all that had happened to his friend "in
enemy territory."
"And if one of the great gentlemen, Jordan or Hermite, still remembers
me, please give him my best regards and make it clear that I less through
nature than through circumstance am such a lazybones."

24
 1886

In Paris, Hilbert was concentrating singlemindedly on mathematics. The

letters to Klein record no sightseeing expeditions but mention only his
desire to visit the observatory. In addition to the mathematical calls and
lectures, he was attempting to edit and copy "in pretty writing" the paper
for his habilitation. The work was progressing well.
At the end of April 1886, Study went home to Germany and reported
to Klein in person on the activities in Paris.
"Not as much about mathematics as I expected," Klein commented
disapprovingly to Hiibert. He then proceeded to fire off half a dozen ques-
tions and comments which had occurred to him while glancing through
the most recent number of the Comptes Rendus: "Who is Sparre? The so-
called Theorem of M. Sparre is already in a Munich dissertation (1878,
I think). Who is Stieltjes? I have an interest in this man. I have come across
an earlier paper by Humbert - it would be very interesting if you could
check on the originality of his work (perhaps via Halphen?) and find out
for me a little more about his personality. It is strange that the geometry
in the style of Veronese-Segre happens to be coming back in fashion
again ... " The tone was more intimate than when Klein had been writing
to the two young men together. "Hold it always before your eyes," he
admonished Hilbert, "that the opportunity you have now will never come
again."
This letter of Klein's found poor Hilbert spending a miserable month.
The doctor diagnosed an illness of acclimatization "while I think it is a
terrible poisoning of the stomach from H 2 S04 , which one has to drink here
in a thin and pallid form under the name of wine." There were no more
calls and the copying of the habilitation paper had to be postponed. He
managed only to drag himself to lectures and meetings. "Everything stops
when the inadequacy of the human organism shows itself .... "
He may also have been just a little homesick.
By the end of June, on his way back to Konigsberg, he was happy and
full of enthusiasm. He stopped in Gottingen and reported to Klein on the
Paris experiences. It was his first visit to the University, and he found
himself charmed by the little town and the pretty, hilly countryside, so
different from the bustling city of Konigsberg and the flat meadows beyond
it. He also stopped in Berlin, where he "paid a visit to everything that has
anything to do with mathematics." This included even the formidable
Leopold Kronecker.
Kronecker was a tiny man, scarcely five feet tall, who had so successfully
managed his family's business and agricultural affairs that he had been able

25
Hilbert

to retire at the age of 30 and devote the rest of his life to his hobby, which
was mathematics. As a member of the Berlin Academy, he had regularly
taken advantage of his prerogative to deliver lectures at the University.
He was now 63 and only recently, since the retirement of Kummer, had he
become an official professor.
Kronecker had made very important contributions, especially to the
higher algebra; but he once remarked that he had spent more time thinking
about philosophy than about mathematics. He was now disturbing his
fellow mathematicians, particularly in Germany, by his loudly voiced doubts
about the soundness of the foundations of much of the contemporary
mathematics. His principal concern was the concept of the arithmetic
continuum, which lies at the foundation of analysis. The continuum is the
totality of real numbers - positive and negative - integers, fractions or
rationals, and irrationals - which provides mathematicians with a unique
number for every point on a line. Although the real numbers had been used
for a long time in mathematics, it was only during the current century that
their nature had been clarified in a precise and rigorous manner in the work
of Cauchy and Bolzano and, more recently, of Cantor and Dedekind.
The new formulation did not satisfy Kronecker. It was his contention
that nothing could be said to have mathematical existence unless it could
actually be constructed with a finite number of positive integers. In his
view, therefore, common fractions exist, since they can be represented as
a ratio of two positive integers, but irrational numbers like 7t do not exist -
since they can be represented only by an infinite series of fractions. Once,
discussing with Lindemann the proof that 7t is transcendental, Kronecker
objected: "Of what use is your beautiful investigation regarding 7t? Why
study such problems when irrational numbers do not exist?" He had not
yet made his remark that "God made the natural numbers, all else is the
work of man," but already he was talking confidently of a new program
which would "arithmetize" mathematics and eliminate from it all "non-
constructive" concepts. "And if I can't do this," he said, "it will be done
by those who come after me!"
Although a man of many admirable qualities, Kronecker had been viru-
lent and very personal in his attacks on the men whose mathematics he
disapproved. ("In fact," recalled Minkowski in a letter to Hilbert, "I did
not hear much good about Kronecker even when I was in Berlin.") The
distinguished old Weierstrass had been reduced almost to tears by Kron-
ecker's remarks about "the incorrectness of all those conclusions with
which so-called analysis works at present." The high-strung, sensitive

26
 1886

Cantor, as a result of Kronecker's attacks on the theory of sets, had broken

down completely and had had to seek asylum in a mental institution.
Hilbert had been warned not to expect a welcome from Kronecker, but
surprisingly he was received - he wrote to Klein - "in a very friendly way."
Back home in Konigsberg, he settled down to the serious business of
habilitation. The work he had prepared was a much more ambitious paper
than the doctoral dissertation had been, but still on the subject of invariant
theory. To a later mathematician, who studied "every line" of Hilbert's
work during his own student days, the habilitation paper was to seem a
curiously false start: "He begins with the claim that it is a most important
point of view, then it just goes out like a burnt match. Nothing came of
it .... I was always surprised that for several years Hilbert went around in
a direction that didn't lead anywhere, perhaps because of the too formal
point of view which he took; and this may have been partly due to his
contact with Study."
In addition to his paper, the candidate for habilitation also had to deliver
a lecture on a topic selected by the faculty from a choice which he offered.
Hilbert proposed "The Most General Periodic Functions" and "The
Concept of the Group." The faculty selected the first topic, which was also
the one which he himself preferred. The lecture was presented to the satis-
faction of everyone concerned; the colloquium examination, passed suc-
cessfully. On July 8, 1886, Hilbert was able to write to Klein: "The title
with which you undeservedly addressed me in your last letter is now in
actuality mine."
Earlier, there had been some discussion between Hilbert and Klein over
the advisability of Hilbert's habilitating at Konigsberg. The East Prussian
capital was very much on the outskirts of mathematical activity. Few mathe-
matics students were willing to come that far, so few, in fact, that Linde-
mann had to refuse Minkowski's request to habilitate at Konigsberg when
he got out of the army.
"But, after all, I am content and full of joy to have decided myself for
Konigsberg," Hilbert wrote to Klein. "The constant association with
Professor Lindemann and, above all, with Hurwitz is not less interesting
than it is advantageous to myself and stimulating. The bad part about
Konigsberg being so far away from things I hope I will be able to overcome
by making some trips again next year, and perhaps then I will get to meet
Herr Gordan .... "
Almost half of the great creative years between twenty and thirty were
gone.

27
 v
Gordan's Problem

Hilbert was resolved that as a docent he would educate himself as well

as his students through his choice of subjects and that he would not repeat
lectures, as many docents did. At the same time, on the daily walk to the
apple tree, he and Hurwitz set for themselves the goal of "a systematic
exploration" of mathematics.
The first semester he prepared lectures on invariant theory, determinants
and hydrodynamics, the last at the suggestion of Minkowski, who was
habilitating at Bonn and showing an interest in mathematical physics. There
were not many who took advantage of this earliest opportunity to hear
David Hilbert. Only in the lectulies on invariant theory was he able to draw
the number of students required by the University for the holding of a
class. "Eleven docents depending on about the same number of students,"
he complained disgustedly to Minkowski. In honor of his new status he
had a formal picture taken. It showed a young man with glasses, a some-
what straggly moustache and already thinning hair, who looked as if he
might be expected to go after what he wanted.
In Bonn, Minkowski was having his troubles. He did not find the other
docents congenial, and the mathematics professor had been taken ill. "I feel
his absence especially. He was the only one here to whom I could put a
mathematical question, or with whom I could speak at all on a mathematical
subject." Whenever he had the opportunity, he returned to Konigsberg
and joined Hilbert and Hurwitz on their daily walks.
During these years the friendship between Hilbert and Minkowski
deepened. Minkowski was a frequent vacation guest at Rauschen. Receiving
the photograph of Hilbert after one of the Rauschen visits, Minkowski
wrote, "If I had not seen you in it so stately and dignified, I would other-
wise have had to think of the outlandish impression which you made on

28
 1886-1892

me in your Rauschen outfit and hairstyle at our brief meeting this summer."
He added, musingly: "That we, although so close, could not at all open up
to one another was for me more than a little surprising."
In their correspondence they continued to address each other by the
formal pronoun "Sie"; but Hilbert, sending Minkowski a reprint of his
first published work - the paper which Klein had presented to the Leipzig
Academy the previous year - inscribed it: "To his friend and colleague in
the closest sense ... from the author."
That first year as a docent, Hilbert made none of the trips which he had
so optimistically planned in order to compensate himself for the isolation
of Konigsberg. Later he was to recall the years in the "security" of his
native city as a time of "slow ripening." The second semester he gave the
lectures on determinants and hydrodynamics which he had originally hoped
to give the first semester. He began to plan lectures on spherical harmonics
and numerical equations. In spite of the variety of his lectures, his own
published work continued to be entirely in the field of algebraic invariants;
but he also interested himself in questions in other fields .
Finally, at the beginning of 1888, he felt that he was at last ready to take
the trip which he had so long promised himself. He drew up an itinerary
which would allow him to call on 21 prominent mathematicians, and in
March he set out. In his letters to Minkowski he jokingly referred to himself
as "an expert invariant-theory man." Now he went first to Erlangen, where
the "king of the invariants" held his court.
Paul Gordan was an impressive personality among the mathematicians
of the day. Twenty-five years older than Hilbert, he had come to science
rather late. His merchant father, while recognizing the son's unusual com-
putational ability, had refused for a long time to concede his mathematical
ability. A one-sided, impulsive man, Gordan was to leave a curiously
negative mark upon the history of mathematics; but he had a sharp wit, a
deep capacity for friendship, and a kinship with youth. Walks were a neces-
sity of life to him. When he walked by himself, he did long computations
in his head, muttering aloud. In company he talked all the time. He liked
to "turn in" frequently. Then, sitting in some cafe in front of a foaming
stein of the famous Erlangen beer, surrounded by young people, a cigar
always in his hand, he talked on, loudly, with violent gestures, completely
oblivious of his surroundings. Almost all of the time he talked about the
theory of algebraic invariants:
It had been Gordan's good fortune to enter this theory just as it moved
onto a new level. The first years of development had been devoted to deter-

29
Hilbert

mining the laws which govern the structure of invariants; the next concern
had been the orderly production and enumeration of the invariants, and
this was Gordan's meat. Sometimes a piece of his work would contain
nothing but formulas for 20 pages. "Formulas were the indispensable
supports for the formation of his thoughts, his conclusions and his mode
of expression," a friend later wrote of him. Gordan's strength, however,
in the invention and execution of the formal algebraic processes was con-
siderable. At the beginning of his career, he had made the first break-through
in a famous invariant problem. For this he had been awarded his title as
king of the invariants. The general problem, which was still unsolved and
now the most famous problem in the theory, was called in his honor
"Gordan's Problem." This was the problem which Hermite had discussed
with Hilbert and Study in Paris.
"Gordan's Problem" was far removed from the "solving for x" with
which algebra had begun so many centuries before. It was a sophisticated
"pure mathematical" question posed, not by the physical world, but by
mathematics itself. The internal structure of all invariant forms was by this
time known. Although there would be certain ambiguities and repetitions,
different invariant forms of specified order and degree could be written
down and counted, at least in principle. The next question was of a quite
different nature, for it concerned the totality of invariants. Was there a
basis, a finite system of invariants in terms of which all other invariants,
although infinite in number, could be expressed rationally and integrally?
Gordan's great achievement, exactly 20 years before the meeting with
Hilbert, had been to prove the existence of a finite basis for the binary
forms, the simplest of all algebraic forms. Characteristically, his proof had
been a computational one, based on the nature of certain elementary opera-
tions which generate invariants. Today it is dismissed as "crude computa-
tion"; but that it was, in its day, a high point in the history of invariant
theory is apparent from the fact that in 20 years of effort by English, German,
French and Italian mathematicians, no one had been able to extend Gordan's
proof beyond binary forms, although in certain specific cases the theorem
was known to be true. The title won in 1868 remained unchallenged. Just
before Hilbert's arrival in Erlangen, Gordan had published the second part
of his "Lectures on Invariant Theory," the plan of this work being primarily
"to expound and exemplify worthily" (as a writer of the day explained) the
theorem which he had proved at that time.
Hilbert had been familiar with Gordan's Problem for some time; but
now, listening to Gordan himself, he seems to have experienced a phenom-

30
 1886-1892

enon which he had not experienced before. The problem captured his
imagination with a completeness that was almost supernatural.
Here was a problem which had everyone of the characteristics of a great
fruitful mathematical problem as he himself was later to list them:
Clear and easy to comprehend ("for what is clear and easily comprehended
attracts, the complicated repels").
Difficult ("in order to entice us") yet not completely inaccessible ("lest it
mock our efforts").
Significant ("a guidepost on the tortuous paths to hidden truths").
The problem would not let him go. He left Gordan, but Gordan's Prob-
lem accompanied him on the train up to Gottingen, where he went to visit
Klein and H. A. Schwarz. Before he left Gottingen, he had produced a
shorter, more simple, more direct version of Gordan's famous proof of
the theorem for binary forms. It was, according to an American mathema-
tician of the period, "an agreeable surprise to learn that the elaborate proofs
of Gordan's theorem formerly current could be replaced by one occupying
not more than four quarto pages."
From Gottingen, Hilbert went on to Berlin and visited Lazarus Fuchs,
who was now a professor at the university there; also Helmholtz; and
Weierstrass, who had recently retired. He then paid another call on Kron-
ecker. He had a great deal of admiration for Kronecker's mathematical
work, but still he found the older man's authoritarian attitude toward the
nature of mathematical existence extremely distasteful. Now he discussed
with Kronecker some plans for future investigations in invariant theory.
Kronecker does not seem to have been much impressed. He cited a work
of his own and said, Hilbert noted, "that my investigation on the subject
is contained therein." They had a long talk, however, about Kronecker's
ideas on what constitutes mathematical existence and his objections to
Weierstrass's use of irrational numbers. "Equal is only 2 = 2 .. " Only the
discreet and singular have significance," Hilbert wrote in the little booklet
in which he kept notes on the conversations with the mathematicians he
visited. The importance the conversation with Kronecker had in Hilbert's
mind at this time is indicated by the fact that he devoted four pages of his
notebook to it - the other mathematicians visited, including Gordan,
never received more than a page.
He left Kronecker, still thinking about Gordan's Problem.
Back home in Konigsberg, the problem was with him in the midst of
pleasure and work, even at dances, which he loved to attend. In August he
went up to Rauschen, as was still his custom; and from Rauschen, on Sep-

31
Hilbert

tember 6, 1888, he sent a short note to the Nachrichten of the Gottingen

Scientific Society. In this note he showed in a totally unexpected and original
way how Gordan's Theorem could be established, by a uniform method,
for forms in any desired number of variables.
No one was prepared for the announcement of the solution of the famous
old problem, and the first reaction was almost sheer disbelief.
Since Gordan's own solution of the simplest case, the solution of the
general problem had been sought in essentially the same manner, by means
of the same kind of elaborate algorithmic apparatus which had been used
so successfully by Gordan. With many variables and a complicated trans-
formation group, this approach became fantastically difficult. It was not
unusual for a single formula to run from page to page in the Annalen.
"Comparable only to the formulas which describe the motion of the moon I"
a later mathematician complained. In this atmosphere of absolute formalism
it had occurred to Hilbert that the only way to achieve the desired proof
would be to approach it from a path entirely different from the formalistic
one which all investigators to date had taken and found impenetrable. He
had set aside the whole elaborate apparatus and rephrased the question
essentially as follows:
"If an infinite system of forms be given, containing a finite number of
variables, under what conditions does a finite set of forms exist, in terms
of which all the others are expressible as linear combinations with rational
integral functions of the same variables for coefficients?"
The answer he came to was that such a set of forms always exists.
The foundation on which this sensational proof of the existence of a
finite basis of the invariant system rested was a lemma, or auxiliary theorem,
about the existence of a finite basis of a module, a mathematical idea he had
obtained from the study of Kronecker's work. The lemma was so simple
that it seemed almost trivial. Yet the proof of Gordan's general theorem
followed directly from it. The work was the first example of the charac-
teristic quality of Hilbert's mind - what one of his pupils was to describe
as "a natural naivete of thought, not coming from authority or past experi-
ence."
When the proof of Gordan's Theorem appeared in print in December,
Hilbert promptly fired off a copy to Arthur Cayley, who half a century
before had laid the foundation of the theory. ("The theory of algebraic
invariants," a later mathematician once wrote, "came into existence some-
what like Minerva: a grown-up virgin, mailed in the shining armor of
algebra, she sprang forth from Cayley's jovian head. Her Athens, over which

32
 1886-1892

she ruled and which she served as a tutelary and beneficent goddess, was
projective geometry. From the beginning she was dedicated to the proposi-
tion that all projective coordinate systems are created equal .... ")
"Dear Sir," Cayley replied politely from Cambridge on January 15, 1889,
"I have to thank you very much for the copy of your note .... It [seems] to
me that the idea is a most important valuable one, and that it ought to lead
to a demonstration of the theorem as to invariants, but I am unable to
satisfy myself as yet that you have obtained such a demonstration."
By January 30, however, having received two explanatory letters from
Hilbert in the intervening time, Cayley was congratulating the young
German: "My difficulty was an a priori one, I thought that the like process
should be applicable to semi-invariants, which it seems it is not; and now
I quite see .... I think you have found the solution of a great problem."
Hilbert had solved Gordan's Problem very much as Alexander had
untied the Gordian Knot.

At Gordium [Plutarch tells us] he saw the famous chariot fastened with cords made of
the rind of the cornel-tree, which whosoever should untie, the inhabitants had a tradition,
that for him was reserved the empire of the world. Most authors tell the story that
Alexander, finding himself unable to untie the knot, the ends of which were secretly
twisted round and folded up within it, cut it asunder with his sword. But Aristobulus
tells us it was easy for him to undo it, by only pulling the pin out of the pole, to which
the yoke was tied, and afterwards drawing of the yoke itself from below.

To prove the finiteness of the basis of the invariant system, one did not
actually have to construct it, as Gordan and all the others had been trying
to do. One did not even have to show how it could be constructed. All one
had to do was to prove that a finite basis, of logical necessity, must exist,
because any other conclusion would result in a contradiction - and this
was what Hilbert had done.
The reaction of some mathematicians was similar to what must have been
the reaction of the Phrygians to Alexander's "untying" of the knot. They
were not at all sure that he had untied it. Hilbert had not produced the basis
itself, nor had he given a method of producing it. His proof of Gordan's
Theorem could not be utilized to produce in actuality a finite basis of the
invariant system of even a single algebraic form.
Lindemann found his young colleague's methods "unheimlich" - uncom-
fortable, sinister, Iveird. Only Klein seemed to recognize the power of the
work - "wholly simple and, therefore, logically compelling" - and it was
at this time that he decided he must get Hilbert to Gottingen at the first

33
Hilbert

opportunity. Gordan himself announced in a loud voice that has echoed in

mathematics long after his own mathematical work has fallen silent:
"Das ist nicht Mathematik. Das ist Theologie."
Hilbert had now publicly taken a position in the current controversy
provoked by Kronecker over the nature of mathematical existence. Kron-
ecker insisted that there could be no existence without construction. For
him, as for Gordan, Hilbert's proof of the finiteness of the basis of the
invariant system was simply not mathematics. Hilbert, on the other hand,
throughout his life was to insist that if one can prove that the attributes
assigned to a concept will never lead to a contradiction, the mathematical
existence of the concept is thereby established.
In spite of the philosophical difference, Hilbert was at this time greatly
under the influence of the mathematical ideas of Kronecker - in fact, the
fundamental significance of his work in invariants was later to be seen as
the application of arithmetical methods to algebraic problems. He sent a
copy of every paper he published to Kronecker. Nevertheless, Kronecker
remarked petulantly to Minkowski that he was going to stop sending
papers to Hilbert if Hilbert did not send papers to him. Hilbert promptly
composed a letter which managed to be formal and respectful but firm:
"I remember exactly, and my list of mailed papers also shows it clearly,
that I have taken the liberty of sending you a copy of each paper without
exception immediately after its publication; and you have had the kindness
to send your thanks on postcards for some of the last mailings. On the other
hand, most honorable professor, it has never happened that a reprint of one
of your papers has arrived as a gift from you to me. When I had the honor
of calling on you about a year ago, however, you mentioned that you
would choose something from your papers and send it to me. Under the cir-
cumstances I believe that there must be some misunderstanding, and I
write these lines to remove it as fast and as surely as possible."
Then, with many crossings-out, he struggled to express the idea that
what he had written should not be construed as expressing any other
meaning than the stated one: not reproaches, but just explanations.
He finally gave up, and simply signed himself, "Most respectfully, David
Hilbert."
During the next two years, Hilbert, still a docent, sent two more notes to
the Nachrichten and then in 1890 brought all his papers on algebraic forms
together into a unified whole for the Annalen. By this time the revolutionary
effect of Hilbert's work was being generally recognized and accepted.
Gordan, offering another proof of one of Hilbert's theorems, was deferen-

34
 1886-1892

tial to the young man - Herr Hilbert's proof was "completely correct,"
he wrote, and his own proof would not even have been possible "if Herr
Hilbert had not utilized in invariant theory concepts which had been devel-
oped by Dedekind, Kronecker and Weber in another part of mathematics."
While Hilbert was thus involved in the purest of pure mathematics, Min-
kowski was moving increasingly away from it. Heinrich Hertz, two years
after his discovery of the electromagnetic waves predicted by Maxwell,
and still only 31 years old, had recently become professor of physics at
Bonn. Minkowski, complaining of "a complete lack of half-way normal
mathematicians" among his colleagues, found himself attracted more and
more by Hertz and by physics. At Christmas he wrote that, contrary to his
custom, he would not be spending the vacation at Konigsberg:
"I do not know if I need console you though, since this time you would
have found me thoroughly infected with physics. Perhaps I even would
have had to pass through a 10-day quarantine period before you and Hur-
witz would have admitted me again, mathematically pure and unapplied,
to your joint walks."
At another time he wrote:
"The reason that I am now almost completely swimming in physical
waters is because here at the moment as a pure mathematician I am the only
feeling heart among wraiths. So for now," he explained, "in order to have
points in common with other mortals, I have surrendered myself to magic -
that is to say, physics. I have my laboratory periods at the Physics Institute;
at home I study Thomson, Helmholtz and consorts. And from the end of
next week on, I will even work several days a week in a blue smock in an
institute for the production of physical instruments, a technician, therefore,
and as practical as you can imagine!"
But the diverging of scientific interests did not affect the friendship; and,
in fact, it was at this time that the two young men made the significant
transition in their correspondence from the formal pronoun "Sie" to the
intimate "du."
The Privatdozent years seemed to stretch out interminably. The letters
were much concerned with the possibility of promotion. In 1891 Minkowski
wrote that he had been told that he might be proposed for a position in
Darmstadt. "But this ray of hope could easily shine so long that it shines
upon mostly grey hair." That same year - apparently with special permission
from the University - Hilbert was delivering his lectures on analytic func-
tions to only one student - an American from Baltimore - a man somewhat
older than the young lecturer but, in his opinion, "very sharp and extra-

35
Hilbert

ordinarily interested." This was Fabian Franklin, an important man in

invariant theory and the successor of Sylvester at Johns Hopkins.
Because there were few mathematics students at Konigsberg, Hilbert
attended the meetings of the natural scientists as well as those of the math-
ematicians. But Konigsberg was surprisingly full of congenial young people.
Wiechert was a docent too; and he had recently been joined by a student
named Arnold Sommerfeld, with whom he was devising a harmonic
analyzer. Both Wiechert and Sommerfeld were to become masters of elec-
trodynamic theory, but when "Little Sommerfeld" heard Hilbert lecture
on ideal theory, he became convinced that his interest lay entirely with the
most pure and abstract mathematics. "Already," he later commented, "it
was clear that a spirit of a special sort was at work."
There was lots of happy social life. Hilbert was a gay young man with a
reputation as a "snappy dancer" and a "charmeur," according to a relative.
He flirted, outrageously, with a great number of girls. His favorite partner
for all activities, however, was Kathe Jerosch, the daughter of a Konigsberg
merchant, an outspoken young lady with an independence of mind that
almost matched his own.
Even after the work of 1890, Gordan's Problem still would not let Hilbert
go. As a mathematician he preferred an actual construction to a proof of
existence. "There is," as one mathematician has said, "an essential difference
between proving the existence of an object of a certain type by constructing
a tangible example of such an object, and showing that if none existed one
could deduce contrary results. In the first case one has a tangible object, while
in the second one has only the contradiction." He would very much have
liked to produce for old Kronecker, Gordan and the rest a constructive
proof of the finiteness of the basis of the invariant system. At the moment
there was simply no method at hand.
In the course of the next two years, however, the nature of his work
began to change. It became infused with the ideas of algebraic number
fields. Again, Kronecker's ideas were important. And it was here that Hil-
bert found at last the powerful new tools he had been seeking. In a key
work, in 1892, he took up the question of exactly what was needed to
produce in actuality a full system of invariants in terms of which all the
other invariants could be represented. Using as a foundation the theorem
which he had earlier proved, he was able to produce what was in essence a
finite means of executing the long sought construction.
Although Hilbert was not the first to make use of indirect, non-construc-
tive proofs, he was the first to recognize their deep significance and value

36
 1886-1892

and to utilize them in dramatic and extremely beautiful ways. Kronecker

had recently died; but to those who like Kronecker still declared that
existence statements are meaningless unless they actually specify the object
asserted to exist, Hilbert was always to reply:
"The value of pure existence proofs consists precisely in that the individ-
ual construction is eliminated by them, and that many different constructions
are subsumed under one fundamental idea so that only what is essential to
the proof stands out clearly; brevity and economy of thought are the raison
d'hre of existence proofs .... To prohibit existence statements ... is tanta-
mount to relinquishing the science of mathematics altogether."
Now, through a proof of existence, Hilbert had been able to obtain a
construction. The impetus which his achievement gave to the use of exis-
tential methods can hardly be overestimated.
Minkowski was utterly delighted:
"For a long while it has been clear to me that it could be only a question
of time until the old invariant question was settled by you - only the dot
was lacking on the 'i'; but that it all turned out to be so surprisingly simple
has made me very happy, and I congratulate you."
He was inspirtd to literary flight and an assortment of metaphors. The
first existence proof might have got smoke in Gordan's eyes, but now
Hilbert had found a smokeless gunpowder. The castle of the robber barons
- Gordan and the rest - had been razed to the ground with the danger
that it might never rise again. Hilbert would be doing a service to his fellow
mathematicians if he would bring together the materials in this area on
which one could rebuild. But he probably would not want to spend his
time doing that. There were still too many other things that he was capable
of doing!
Gordan himself conceded gracefully.
"I have convinced myself that theology also has its merits."
When Klein went to Chicago for what was billed as an "International
Congress of Mathematicians" to celebrate the founding of the University
of Chicago, he took with him a paper by Hilbert in which that young man
matter-of-factly summarized the history of invariant theory and his own
part in it:
"In the history of a mathematical theory the developmental stages are
easily distinguished: the naive, the formal, and the critical. As for the theory
of algebraic invariants, the first founders of it, Cayley and Sylvester, are
together to be regarded as the representatives of the naive period: in the
drawing up of the simplest invariant concepts and in the elegant applica-

37
Hilbert

tions to the solution of equations of the first degrees, they experienced the
immediate joy of first discovery. The inventors and perfecters of the sym-
bolic calculation, Clebsch and Gordan, are the champions of the second
period. The critical period finds its expressions in the theorems I have listed
above .... "
The theorems he referred to were his own.
It was a rather brash statement for a young mathematician who was still
not even an Extraordinarius, but it had considerable truth in it. Cayley
and Sylvester were both alive, one at Cambridge and the other at Oxford.
Clebsch was dead, but Gordan was one of the most prominent mathemati-
cians of the day. Now suddenly, in 1892, as a result of Hilbert's work,
invariant theory, as it had been treated since the time of Cayley, was fin-
ished. "From the whole theory," a later mathematician wrote, "the breath
went out."
With the solution of Gordan's Problem, Hilbert had found himself and
his method - an attack on a great individual problem, the solution of which
would turn out to extend in significance far beyond the problem itself. Now
something totally unexpected occurred. The problem which had originally
aroused his interest had been solved. The solution released him.
At the conclusion of his latest paper on invariants he had written: "Thus
I believe the most important goals of the theory of function fields gener-
ated by invariants have been obtained." In a letter to Minkowski, he an-
nounced with even more finality: "I shall definitely quit the field of invar-
iants."

38
 VI

Changes

During the next three years Hilbert rose in the academic ranks, did all
the things that most young men do at this time of their lives, married,
fathered a child, received an important assignment, and made a decision
which changed the course of his life.
This sudden series of events was set into motion by the death of Kron-
ecker and the game of "mathematical chairs" which ensued in the German
universities. Suddenly it seemed that the meager docent years might be
coming to an end. Minkowski calling in Berlin on Friedrich Althoff, who
was in charge of all matters pertaining to the universities, heralded the
news:
"A. says ... the following are supposed to receive paid Extraordinariats:
you, I, Eberhard, and Study. I have not neglected to represent you to A. as
the coming man in mathematics .... As to Study, in conscience I could only
praise his good intentions and his diligence. A. is very devoted to you and
Eberhard."
At almost the same time Hurwitz, who had been an associate professor
(Extraordinarius) at Konigsberg for eight years, received an offer of a full
professorship from the Swiss Federal Institute of Technology in Zurich.
This meant an end to the daily mathematical walks, but opened up the
prospect of Hilbert's being appointed to Hurwitz's place.
"Through this circumstance," Minkowski wrote affectionately, "your
frightful pessimism will have been allayed so that one dares again to venture
a friendly word to you. In some weeks now, hopefully, the Privatdozent-
sickness will be definitely over. You see - at last com(:s a spring and a
summer."
In June, Hurwitz married Ida Samuels, the daughter of the professor of
medicine. Hilbert had recently become engaged to Kathe Jerosch, and

39
Hilbert

after Hurwitz's wedding he was increasingly impatient with the slow pace
of promotion. At last, in August, the faculty unanimously voted him to
succeed to Hurwitz's place. He announced the setting of the date of his
wedding at the same time he communicated the news of the promotion to
Minkowski.
Minkowski replied happily with his congratulations: "You will now
have finally been converted to the idea that those in the decisive positions
are sincerely well disposed toward you. Your prospects for the future,
therefore, are excellent."
The Hilbert and Jerosch families had long been friends. From the outset
it was generally agreed that Hilbert had found the perfect mate for himself.
"She was a full human being in her own right, strong and clear," one of
Hilbert's earliest pupils wrote of Kathe, "and always stood on the same
footing with her husband, kindly and forthright, always original."
A photograph, taken about this time, shows the young couple. He is 30;
she is 28. Already they look rather like one another. They are almost the
same height, mouths wide and firm, strong noses, a level clear-eyed look.
Hilbert's head seems relatively small. He has grown a beard. Already his
hair has receded until the high scholar's forehead stands out impressively.
Neither pretty nor homely, Kathe has good features, but she seems more
interested in things other than her own appearence. Her dark hair is parted
in the middle, drawn back rather severely, and coiled on the top of her head
toward the back.
On October 12, 1892, Hilbert and Kathe Jerosch were married.
("The pleasant frame of mind in which you find yourself cannot help but
have repercussions in your scientific work," Minkowski wrote. "I expect
another great discovery. ")
At almost the same time that Hilbert succeeded Hurwitz in Konigsberg,
Minkowski received his promised associate professorship in Bonn. He had
hoped to go somewhere else, but "it will be better for you to remain in
Bonn," Althoff told him. By now Heinrich Hertz had been struck down
with the illness which was soon to take his life at the age of 37; Minkowski's
interest in physics had abated; and he had returned to his first love, the
theory of numbers. But later he once said to Hilbert that if "Papa" Hertz
had lived he might have become a physicist instead of a mathematician.
Minkowski's approach to number theory was geometrical, it being his
aim to express algebraic conjectures about the rational numbers in terms of
geometric figures, an approach which frequently made the proofs more
obvious. He was deeply absorbed in a book on this new subject, and his

40
 1892-1895

letters to Hilbert were filled with his concern about the presentation of his
material. All must be "klipp und klar" before it went to the publisher.
Although he called Poincare "the greatest mathematician in the world,"
he told Hilbert, "I could not bring myself to publish things in the form in
which Poincare publishes them."
The book frequently kept Minkowski from Konigsberg at vacation time.
Hilbert complained about a lack of mathematical conversation now that
Hurwitz was gone. "I am in a much more unhappy situation than you,"
Minkowski reminded him. "Just as closed off as Konigsberg is from the
rest of the world, just so closed off is Bonn from all other mathematicians.
One is here a pure mathematics Eskimo!"
By the beginning of the new year (1893) Minkowski was happier. The
book was half finished, accompanied by praise from Hermite which Hilbert
found very touching.
"Y ou are so kind as to call myoId research works a point of departure
for your magnificent contribution," the old Frenchman wrote to Minkowski,
"but you have left them so far behind that they cannot claim now any other
merit than to have suggested to you the direction in which you have chosen
to proceed."
Hilbert began the year with a new proof of the transcendence of e (first
proved by Hermite) and of n (proved by Lindemann). His proof was a
considerable improvement over these earlier ones, astonishingly simple
and direct. Here was the great work which Minkowski had been anticipating
since the previous fall. Receiving it, he sat down and wrote immediately.
"An hour ago I received your note on e and n ... and I cannot do other
than to express to you right away my sincere heartfelt astonishment ....
I can picture the exhilaration of Hermite upon reading your paper and, as I
know the old gentleman, it won't surprise me if he should shortly inform
you of his joy that he is still permitted to experience this."
Along with the professional and personal changes in his life, Hilbert was
beginning to show a new mathematical interest. "I shall devote myself to
number theory from now on," he had told Minkowski after the completion
of the last work on invariants. Now he turned to this new subject.
Gauss, as is well known, placed the theory of numbers at the pinnacle of
science. He described it as "an inexhaustible storehouse of interesting
truths." Hilbert saw it as "a building of rare beauty and harmony." He was
as charmed as Gauss had been by "the simplicity of its fundamental laws,
the economy of its concepts, and the purity of its truth"; and both men were
equally fascinated by the contrast between the obviousness of the many

41
Hilbert

numerical relationships involved and the "monstrous" difficulty of dem-

onstrating them. Yet, in spite of the similarity of their comments, they were
talking about two different versions of number theory.
Gauss was praising the classical theory of numbers, which goes back to
the Greeks and deals with the relationships which exist among the ordinary
whole, or natural, numbers. Most important are those between the prime
numbers, called the "building blocks" of the number system, and the other
numbers which, unlike the primes, can be divided by some number other
than themselves and 1. By Gauss's time the concept of number had been
extended far beyond the natural numbers. But Gauss himself had become
the first mathematician to extend the notions of number theory itself beyond
the rational "field" in which every sum, difference, product and (unlike
among the natural numbers) quotient of two numbers is another number
in the field. He did this for those numbers of the form a + b V- 1 where a
and b are rational numbers. These numbers also form a field, an algebraic
number field, as do the numbers of the form a + b fl", and so on; and
they are among the fields which are the subject of what is called algebraic
number theory. It was this development, the number theory creation 0 f
Gauss, which Hilbert praised.
The greatest obstacle to the extension of number theory to algebraic
number fields had been the fact that in most algebraic number fields the
fundamental theorem of arithmetic, which states that the representation of
any number as the product of primes is unique, does not hold. This obstacle
had been eventually overcome by Kummer with the invention of "ideal
numbers." Since Kummer, two mathematicians with very different mathe-
matical approaches had been at work in algebraic number fields. Even
before Hurwitz had left for Zi.irich, he and Hilbert had been devoting their
daily walks to discussions of the modern number theory works of these
two. "One of us took the Kronecker demonstration for the complete
factorization in prime ideals and the other took Dedekind's," Hilbert
later recalled, "and we found them both abominable." Now he began his
work in algebraic number fields in much the same way that he had opened
his attack on Gordan's Problem. He went back and thought through the
basic idea. His first paper in the new subject was another proof for the unique
decomposition of the integers of a field into prime ideals.
Hilbert had scarcely settled down into his new position as an assistant
professor with a salary and a wife when there was welcome news. Linde-
mann had received an offer from Munich and would be leaving Konigs-
berg.

42
 1892-1895

"I take it for granted - and with any sense of justice Lindemann cannot
think otherwise - that you should be his successor," Minkowski wrote to
Hilbert. "If he succeeds in putting it through, he will at least leave with
honor the place which he has occupied for 10 years."
Hilbert of course agreed. The final decision in the matter was not Linde-
mann's, however, but Althoff's. The faculty nominated Hilbert and three
other more established mathematicians for the vacant professorship and
sent the list to Berlin.
Althoff was no bureaucrat, but an administrator who had been academ-
ically trained. His great goal was to build up mathematics in Germany. He
was a good friend of Klein's - the two had served in the army together
during the Franco-Prussian War - and he thought very highly of Klein's
opinion. Now, from the faculty's impressive list of names, he selected that
of the 31-year-old Hilbert. He then proceeded to consult him about the
appointment of a successor to his post as Extraordinarius - something
almost unheard of.
Here was an opportunity to bring Minkowski back to Konigsberg. In
spite of the difficult situation which existed at Bonn because of the long
illness of the professor of mathematics, Hilbert embarked enthusiastically
upon the unfamiliar course of academic diplomacy. He wrote to Minkowski
of the possibility that they might soon be together again.
"I would consider it special luck to step into your place at Konigsberg,"
Minkowski replied. "The association with my mathematical colleagues
here is really deplorable. One complains of migraine; as for the other,
his wife trots in every five minutes in order to give another, non-mathe-
matical direction to the conversation. If I could exchange this association
with yours, it would be the difference between day and night for my scien-
tific development."
But the sick professor at Bonn wanted to hang on to Minkowski, to
whom he had become accustomed. Althoff liked to keep his professors
happy. The negotiations dragged on.
In the meantime, in the new household, things were going along well
and according to form. On August 11, 1893, at the seaside resort of Cranz,
a first child, a son, was born to the Hilberts and named Franz.
A few weeks after Franz's birth, Hilbert went south to Munich for the
annual meeting of the German Mathematical Society, which had recently
been organized by a group of mathematicians - Hilbert among them - for
the purpose of providing more contact among the different branches of
mathematics. At the meeting Hilbert presented two new proofs of the

43
Hilbert

decomposition of the numbers of a field into prime ideals. Although he

had only begun to publish in the area of algebraic number theory, his
competence apparently impressed the other members. One of the
Society's projects was the yearly publication of comprehensive surveys of
different fields of mathematics (the first had been on the theory of invar-
iants); and now it was voted that Hilbert and Minkowski, who was of
course already well known as a number theorist, be asked to prepare such a
report on the current state of affairs in the theory of numbers "in two
years." The note of urgency in the assignment was occasioned by the fact
that the revolutionary work of Kummer, Kronecker and Dedekind was so
extremely complicated or so far in advance of its time that it was still incom-
prehensible to most mathematicians. That Hilbert and Minkowski could
be expected to rectify this situation was a tribute, not only to their math-
ematical ability, but also to the simplicity and clarity of their mathematical
presentation.
That fall the letters that went between Konigsberg and Bonn were devot-
ed almost equally to three topics: the organization of the report for the
Mathematical Society, the progress of the negotiations to bring Minkowski
to Konigsberg again, and the fact that baby Franz could already "out-
shriek" all other babies in his father's opinion.
The situation in Bonn did not improve; by New Year's Day 1894, Min-
kowski wrote that he had given up almost all hope of obtaining the appoint-
ment in Konigsberg. Then three days later, following an interview with
Althoff, he sent a joyful letter to Hilbert.
"End good, all good. . . . Hearty thanks for all your kind efforts which
have led to this happy result; and may we have a pleasant and profitable
collaboration which will make the prime numbers and the reciprocity laws
wiggeln und waggeln."
On his way up to Konigsberg in March, Minkowski stopped in Gottin-
gen. H. A. Schwarz had by this time moved on to Berlin - his place being
taken by Heinrich Weber - and Klein had a free hand to put into practice
his ideas. Minkowski seems to have been tremendously impressed by the
stimulating situation which Klein had already created at the University.
"Who knows when I shall have another opportunity to inspire the mathema-
tical workshop which is now of the highest repute?"
With Minkowski's arrival in the spring of 1894, the daily walks to the
apple tree and the number theory discussions were happily resumed. It was
Hilbert's feeling that he could not have had a better collaborator on the
Zahlbericht, as the number theory report was called. In spite of Minkowski's

44
 1892-1895

mild disposition, he was fundamentally critical, insisted on literary as well

as intellectural clarity, "and even to the work of others applied a strict
standard."
The Zahlbericht now began to take shape in Hilbert's mind. Such an
assignment as the one made by the Mathematical Society might be expected
to be an unwelcome chore to a young mathematician, but this was not to
be the case with Hilbert. Already his own work showed that his particular
interest was the extension of the reciprocity laws to algebraic number
fields. Now he willingly set aside these plans, seeing in the assigned report
an opportunity to lay the foundation needed for deeper investigations.
Although he still had no fondness for learning from books, he read every-
thing that had been published on number theory since the time of Gauss.
The proofs of all known theorems would have to be weighed carefully.
Then he would have to decide in favor of those "the principles of which
are capable of generalization and most useful for further research." But
before such a selection could be made, the "further research" itself would
have to be carried out. The difficulties of thought and style which had
barred the way to general appreciation and understanding of his prede-
cessors' work would have to be eliminated. It had been decided that the
report should be divided into two parts. Minkowski would treat rational
number theory; Hilbert, algebraic number theory. During the year 1894
Hilbert laid the foundations of his share of the Zahlbericht.
But, again, the two friends were not to be together for long. Early in
December a letter labeled "Very Confidential" arrived from Gottingen.
"Probably you do not yet know," Klein wrote to Hilbert, "that Weber is
going to Strassburg. This very evening we will have a meeting of the faculty
to choose a committee to set up a list; and as little as I can predict the
results, I want to inform you that I shall make every effort to see that no
one other than you is called here.
"Y ou are the man whom I need as my scientific complement because of
the direction of your work and the power of your mathematical thinking and
the fact that you are still in the middle of your productive years. I am count-
ing on it that you will give a new inner strength to the mathematical school
here, which has grown continuously and, as it seems, will grow even more -
and that perhaps you will even exercise a rejuvenating effect upon me ....
"I can't know whether I will prevail in the faculty. I know even less
whether the offer will follow from Berlin as we propose it. But this one
thing you must promise me, even today: that you will not decline the call
if you receive it!"

45
Hilbert

There is no record that Hilbert ever considered declining. In fact, he wrote

to Klein, "Without any doubt I would accept a call to Gottingen with
great joy and without hesitation." But he may have had some doubts. Klein
was the acknowledged leader of mathematics in Germany. He was a regal
man, the word "kingly" being now used most frequently to describe him.
Sometimes even "kingly" wasn't strong enough, and one former student
referred to him as "the divine Felix." A man who knew him well and was
proud of the fact that Klein once took his advice in a personal matter,
later confessed that he felt even to the end a distance between himself and
Klein "as between a mortal and a god."
As for Klein's feelings - already it was clear that Hilbert questioned any
authority, personal or mathematical, and went his own way. Klein was not
unaware of the reasons against his choice. When in the faculty meeting his
colleagues accused him of wanting merely a comfortable younger man, he
replied, "I have asked the most difficult person of all."
Hilbert worked very hard on his reply to Klein's letter, crossing out and
rewriting extensively to get exactly the effect he wanted. When he was
satisfied, he had Kathe copy his letter in her best handwriting. It was a
custom he was to follow often throughout his career.
"Your letter has surprised me in the happiest way," he began. "It has
opened up a possibility for the realization of which I might have hoped
at best in the distant future and as the final goal of all my efforts ....
"Decisive for me above all would be the scientific stimulation which
would come from you and the greater sphere of influence and the glory
of your university. Besides, it would be the fulfillment of mine and my
wife's dearest wish to live in a smaller university town, particularly one
which is so beautifully situated as Gottingen."
Upon receiving this letter from Hilbert, Klein proceeded to plan out a
campaign.
"I have already told Hurwitz that we will not propose him this time so
that we will be more successful in proposing you. We will call Minkowski
in second place. I have discussed this with Althoff and he thinks that will
make it easier then for Minkowski to get your place in Konigsberg."
Within a week he was writing triumphantly to Hilbert:
"This has been just marvellous, much faster than I ever dared to hope it
could be. Please accept my heartiest welcome!"

46
 VII

Only Number Fields

The red-tiled roofs of Gottingen are ringed by gentle hills which are
broken here and there by the rugged silhouette of an ancient watch tower.
Much of the old wall still surrounds the inner town, and on Sunday after-
noons the townspeople "walk the wall" - it is an hour's walk. Outside
the wall lie the yellow-brick buildings of the Georg August Universitiit,
founded by the Elector of Hannover who was also George II of England.
Inside, handsome half-timbered houses line crooked, narrow streets. Two
thoroughfares, Prinzenstrasse and Weender Strasse, intersect at a point
which the mathematicians call the origin of the coordinates in Gottingen.
The center of the town, however, is the Rathaus, or town hall. On the wall
of its Ratskeller there is a motto which states unequivocally: Away from
Gifttingen there is no life.
The great scientific tradition of Gottingen goes back to Carl Friedrich
Gauss, the son of a man who was at different times a gardener, a canal-
tender and a brick-layer. Gauss enrolled at the University in the autumn of
1795 as the protege of the Duke of Brunswick. During the next three years
he had so many great mathematical ideas that he could often do no more
than record them in his journal. Before he left the University, at the age
of 21, he had virtually completed one of the masterpieces of number theory
and of mathematics, the Disquisitiones Arithmeticae. Later he returned to
Gottingen as director of the observatory with incidental duties of instruc-
tion. He spent the rest of his l~fe there, leaving his mark on every part of
pure and applied mathematics. But when he was an old man and had won a
place with Archimedes and Newton in the pantheon of his science, he always
spoke of the first years he had spent at Gottingen as "the fortunate years."
Hilbert arrived in Gottingen in March 1895, almost exactly one hundred
years after Gauss. It was not immediately apparent to the students that

47
Hilbert

another great mathematician had joined the tradition. Hilbert was too
different from the bent, dignified Heinrich Weber whom he replaced and
the tall, commanding Klein. "I still remember vividly," wrote Otto Blu-
menthal, then a student in his second semester, "the strange impression I
received of the medium-sized, quick, unpretentiously dressed man with a
reddish beard, who did not look at all like a professor."
Klein's reputation drew students to Gottingen from all over the world,
but particularly from the United States. The Bulletin of the newly founded
American Mathematical Society regularly listed the courses of lectures to
be given in Gottingen, and at one time the Americans at the University
were sufficient in number and wealth to have their own letterhead: The
American Colony of Cdttingen. "There are about a dozen ... in our lectures,"
a young Englishwoman named Grace Chisholm (later Mrs. W. H. Young)
wrote to her former classmates at Cambridge. "We are a motley crew: five
are Americans, one a Swiss-French, one a Hungarian, and one an Italian.
This leaves a very small residuum of German blood."
The center of mathematical life was the third floor of the Auditorienhaus.
Here Klein had established a reading room, the Lesezimmer, which was
entirely different from any other mathematical library in existence at that
time. Books were on open shelves and the students could go directly to
them. Klein had also established on the third floor what was to become
almost his signature: a tremendous collection of mathematical models
housed in a corridor where the students gathered before lectures. Although
not in actuality a room, it was always referred to as the Room of the Mathe-
matical Models.
Klein's lectures were deservedly recognized as classics. It was his custom
often to arrive as much as an hour before the students in order to check
the encyclopedic list of references which he had had his assistant prepare.
At the same time he smoothed out any roughness of expression or thought
which might still remain in his manuscript. Before he began his lecture,
he had mapped out in his mind an arrangement of formulas, diagrams and
citations. Nothing put on the blackboard during the lecture ever had to be
erased. At the conclusion the board contained a perfect summary of the
presentation, every square inch being appropriately filled and logically
ordered.
It was Klein's theory that students should work out proofs for them-
selves. He gave them only a general sketch of the method. The result was
that a student had to spend at least four hours outside class for every hour
spent in class if he wished to master the material. Klein's forte was the

48
 1895-1898

comprehensive view. "He possessed the ability to see the unifying idea in
far apart problems and knew the art of explaining this insight by amassing
the necessary details," a student has said. In the selection of his lecture
subjects, Klein pursued a characteristically noble plan: "to gain in the course
of time a complete view of the whole field of modern mathematics."
In contrast, Hilbert delivered his lectures slowly and "without frills,"
according to Blumenthal, and with many repetitions "to make sure that
everyone understood him." It was his custom to review the material which
he had covered in the previous lecture, a gymnasium-like technique dis-
dained by the other professors. Yet his lectures, so different from Klein's,
were shortly to seem to many of the students more impressive because they
were so full of "the most beautiful insights."
In a well-prepared lecture by Hilbert the sentences followed one another
"simply, naturally, logically." But it was his custom to prepare a lecture in
general, and often he was tripped up by details. Sometimes, without espe-
cially mentioning the fact, he would develop one of his own ideas spontan-
eously in front of the class. Then his lectures would be even farther from
the perfection of Klein's and exhibit the rough edges, the false starts, the
sometimes misdirected intensity of discovery itself.
In the eight and a half years of teaching at Konigsberg, Hilbert had not
repeated a single subject "with the one small exception" of a one-hour
course on determinants. In Gottingen now he was easily able to choose his
subjects to adjust to Klein's wishes. The first semester he lectured on deter-
minants and elliptic functions and conducted a seminar with Klein every
Wednesday morning on real functions.
Although Hilbert had accepted the professorship in Gottingen with
alacrity, there were two aspects of the new situation that bothered him.
Kathe was not happy. The society in Gottingen, while more scientifically
stimulating for him, lacked the warmth to which she had been accustomed
in Konigsberg. Carefully observed distinctions of rank cut the professors
off from the docents and advanced students. In spite of Klein's kindness,
he maintained with the Hilberts, as he did with everyone else, a certain
distance. Mrs. Klein (granddaughter of the philosopher Hegel) was a very
quiet woman, not the kind who likes to gather peopll~ around her. The
Klein house at 3 Wilhelm Weber Strasse, big, square and impressive with
a bust of Jupiter on the stairs that led to Klein's study, looked already like
the institute it was eventually to become. For Hilbert "comradeship" and
"human solidarity" were essential to scientific production. Like Kathe,
he found the atmosphere at Gottingen distinctly cool.

49
Hilbert

Hilbert was also concerned, in the beginning, that he might not prove
worthy of the confidence which Klein had shown in him. He recognized
that he had been taken on faith. Before he had left Konigsberg, he had
written to Klein, "My positive achievements - which I indeed know best
myself - are still very modest." In the draft of a later letter he had returned
to this same subject, adding hopefully, "As to my scientific program, I think
that I will eventually succeed in shaping the theory of ideals info a general
and usable tool (applicable also to analytic functions and differential equa-
tions) which will complement the great and promising concept of the
group." Then he had carefully crossed out this sentence and noted in the
margin: I have not written this.
Now, in Gottingen, Hilbert concentrated all his powers on his share of
the number theory report for the German Mathematical Society, which he
saw as the necessary foundation for his future hopes.
In Konigsberg, Minkowski almost immediately received the appointment
as his friend's successor. "The whole thing has taken place so quickly that
I still have not come to complete consciousness of my astounding luck. In
any case, I know I have you alone to thank for everything. I shall see I
break out of my cocoon so that no one will hold it against you for proposing
me." Minkowski was happy in his new position - professors now went
out of their way to describe to him the virtues of their daughters - but
since Hilbert's departure, he wrote, he had walked "not once" to the apple
tree.
With encouragement from Hilbert, Minkowski now took advantage of
the fact that he was a full professor to deliver a course of lectures on Cantor's
theory of the infinite. It was at a time when, according to Hilbert, the work
of Cantor was still actually "taboo" in German mathematical circles, partly
because of the strangeness of his ideas and partly because of the earlier
attacks by Kronecker. Although Minkowski admired Kronecker's mathe-
matical work, he deplored as much as Hilbert the way in which the older
man had tried to impose his restrictive personal prejudices upon mathematics
as a whole.
"Later histories will call Cantor one of the deepest mathematicians of
this time," Minkowski said. "It is most regrettable that opposition based
not alone on technical grounds and coming from one of the most highly
regarded mathematicians could cast a gloom over his joy in his scientific
work."
As the year 1895 progressed, the letters between Gottingen and Konigs-
berg became less frequent.

50
 1895-1898

"We both try in silence to crack the difficult and not really very tasty
nut of our common report," Minkowski wrote, taking up the correspond-
ence again, "you perhaps with sharper teeth and more exertion of energy."
The idea of the joint report did not really appeal to Minkowski. "I
started somewhat too late with my share," he wrote unhappily. "Now I
find many little problems it would have been nice to dispose of." He was
more interested in his book on the geometry of numbers. "The complete
presentation of my investigations on continued fractions has reached almost
a hundred printed pages but the all-satisfying conclusion is still missing:
the vaguely conceived characteristic criterion for cubic irrational numbers
.... But I haven't been able to work on this problem because I have really
been working on our report."
Hilbert, on the other hand, was devoting himself wholeheartedly to the
report. He was fascinated by the deep connections which had recently been
revealed between the theory of numbers and other branches of mathematics.
Number theory seemed to him to have taken over the leading role in algebra
and function theory. The fact that this had not occurred earlier and more
extensively was, in his opinion, due to the disconnected way in which
number theory had developed and the fact that its treatment had always
been chronological rather than conceptual. Now, he believed, a certain
and continuous development could be effected by the systematic building
up of the theory of algebraic number fields.
After the Wednesday morning seminars he walked with the students up
to a popular restaurant on the Hainberg for lunch and more mathematics.
On these excursions he talked freely to them "as equals," according to
Blumenthal, but always the subject of conversation at this time was "only
algebraic number fields."
By the beginning of 1896, Hilbert's share of the Zablbericht was almost
finished, but Minkowski's was not. In February Hilbert proposed that either
Minkowski's share should be published with his as it stood, or else it should
be published separately the following year.
"I accept your second plan," Minkowski wrote gratefully. "The decision
... is hard on me only insofar as I'll have the guilty feeling for a whole year
that I didn't meet the expectations of the Society and, in some degree, your
expectations. You, it is true, haven't made any remark of this kind, but ....
The reproaches may lose some of their force if now the biggest part of my
book is appearing and the rest is following soon. Finally, I can imagine
that I am doing what I think is in the interest of the project. I beg you not to
think I left you in the lurch."

51
Hilbert

Within a month after receiving this letter, Hilbert had completed his
report on algebraic number fields. It was exactly a year since his arrival in
Gottingen. The manuscript, which was to run to almost 400 pages in print,
was carefully copied out by Kathe Hilbert in her clear round hand and sent
to the printer. The proof-sheets were mailed to Minkowski in Konigsberg
as they arrived. Minkowski's letters during this period show the affectionate
and yet sharp and unrelenting care with which he read them.
"One more remark seems to be necessary on page 204." "I have read
till where the long calculations start. They still seem pretty tangled." "This
thought is not so simple that it can be silently omitted."
Minkowski had recently received an offer of a position in Zurich. Such
an offer, known as "a call," was customarily the subject of complicated
ritual and negotiation, since it was the only means by which a man who had
become a full professor could further improve his situation. Minkowski
had no gift for such parrying. Althoff, he wrote to Hilbert, did not seem
eager to keep him at Konigsberg. Rather regretfully, he finally accepted
the position in Zurich for the fall of 1896.
In Zurich, however, he was again in the company of Hurwitz ("just the
same except for a few white hairs"), and the two friends read the remaining
proof-sheets of Hilbert's report together. Corrections and suggestions kept
coming to Gottingen.
Hilbert began to grown impatient.
Minkowski soothed him: "I understand that you want to be through
with the report as soon as possible ... but as long as there are so many
remarks to be made, I can't promise you any great speed .... " "A certain
care is advisable .... " "Comfort yourself with the thought that the report
will be finished soon and will gain high approval."
The careful proofreading continued.
By this time Hilbert was beginning to feel more at home in Gottingen.
He had found a congenial colleague in Walther Nernst, a professor of
physics and chemistry who, like himself, was the son of a Prussian judge.
But Hilbert also liked to be with younger people, and now he cheerfully
ignored convention in choosing his friends. These included Sommerfeld,
who had come to Gottingen to continue his studies and had become
Klein's first assistant. He also selected the brightest, most interesting
students in his seminar for longer walks. His "Wunderkinder," he called
them.
Although even advanced students and docents stood in awe of Klein,
they easily fell into a comradely relationship with Hilbert. His Konigsberg

52
 1895-1898

accent with its distinctive rhythm and inflection seemed to them to give a
unique flavor to everything he said. They delighted in mimicking his manner
and opinions, were quick to pick up the" Aber nein!" - But no! -with which
he announced his fundamental disagreement with an idea, whether in mathe-
matics, economics, philosophy, human relations, or simply the management
of the University. ("It was very characteristic the way he said it, but very
difficult to catch in English, even in twenty words.")
In the seminar they found him surprisingly attentive to what they had
to say. As a rule he corrected them mildly and praised good efforts. But if
something seemed too obvious to him he cut it short with "Aber das ist
doch ganz einfach!" - But that is completely simple! - and when a student
made an inadequate presentation he would chastize him or her in a manner
that soon became legendary. "Ja, Fraulein S-----, you have given us a very
interesting report on a beautiful piece of work, but when I ask myself what
have you really said, it is chalk, chalk, nothing but chalk!" And he could
also be brutal. "You had better think twice before you uttered a lie or an
empty phrase to him," a later student recalled. "His directness could be
something to be afraid of."
After a year in Gottingen, the Hilberts decided to build a house on Wil-
helm Weber Strasse, the broad linden-lined avenue favored by professors.
("Very likely," wrote Minkowski, "Fate will feel challenged now and try
to seduce you from Gottingen with many spectacular offers.") The house
was a forthright yellow-brick structure with none of the "new style" or-
nateness favored by its neighbors. It was large enough that the activities
of 4-year-old Franz would not disturb his father as they had in the apartment.
The yard in back was large too. They got a dog, the first of a long line of
terriers, all to be named Peter. Hilbert, who worked best "under the
free sky," hung an i8-foot blackboard from his neighbor's wall and
built a covered walk-way so that he could be outdoors even in bad
weather.
The house was almost finished when Hilbert wrote the introduction to
the Zahlbericht. To a later student with a love of language not characteristic
of most mathematicians, the introduction was to seem one of the most
beautiful parts of German prose, "the style in the literary sense being the
accurate image of the way of thinking." In it Hilbert emphasized the esteem
in which number theory had always been held by the greatest mathemati-
cians. Even Kronecker was quoted approvingly as "giving expression to
the sentiment of his mathematical heart" when he made his famous pro-
nouncement that God made the natural numbers ....

53
Hilbert

"I still find many things to criticize," Minkowski wrote patiently. " ... Will
you not in your foreword perhaps mention the fact that I read the last
three sections in manuscript?"
Thus instructed, Hilbert wrote an acknowledgment of what he owed to
his friend. Minkowski was still not satisfied.
"That you omitted the thanks to Mrs. Hilbert both Hurwitz and I find
scandalous and this simply can't be allowed to remain so."
This last addition was made in the study of the new house at 29 Wilhelm
Weber Strasse. The final date on the introduction to the Zahlbericht was
April 10, 1897.
"I wish you luck that finally after the long years of work the time has
arrived when your report will become the common property of all mathe-
maticians," Minkowski wrote upon receiving his specially bound copy,
"and I do not doubt in the near future you yourself will be counted among
the great classicists of number theory. . .. Also I congratulate your wife
on the good example which she has set for all mathematicians' wives,
which now for all time will remain preserved in memory."
The report on algebraic number fields exceeded in every way the expecta-
tions of the members of the Mathematical Society. They had asked for a
summary of the current state of affairs in the theory. They received a master-
piece, which simply and clearly fitted all the difficult developments of recent
times into an elegantly integrated theory. A contemporary reviewer found
the Zahlbericht an inspired work of art; a later writer called it a veritable
jewel of mathematical literature.
The quality of Hilbert's creative contribution in the report is exempli-
fied by that theorem which is still known today simply as "Satz 90." The
development of the ideas contained in it were to lead to homological algebra,
which plays an important role in algebraic geometry and topology. As
another mathematician has remarked, "Hilbert was not only very thorough,
but also very fertile for other mathematicians."
For Hilbert, the spring of 1897 was a memorable one - the new house
completed, the Zahlbericht at last in print. Then came sad news. His only
sister, Elise Frenzel, wife of an East Prussian judge, had died in childbirth.
According to a cousin, the relationship between brother and sister was
reputed in the family to have been "cool." But for Minkowski, writing to
Hilbert at the time, it seemed impossible to find comforting words:
"Whoever knew your sister must have admired her for her always cheerful
and pleasant disposition and must have been carried along by her happy
approach to life. I still remember... how gay she was in Munich, and

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how she was in Rauschen. It is really unbelievable that she should have left
you so young. How close she must have been to your heart, since you have
no other brother or sister and you grew up together for so many years!
It seems sometimes that through a preoccupation with science, we acquire
a firmer hold over the vicissitudes of life and meet them with greater calm,
but in reality we have done no more than to find a way to escape from our
sorrows."
Minkowski's next letter, however, contained happy personal news. He
had become engaged to Auguste Adler, the daughter of the owner of a
leather factory near Strassburg. "My choice is, I am convinced, a happy one
and I certainly hope ... it will be good for my scientific work." In a post-
script he added a little information about his fiancee for the Hilberts. "She
is 21 years old, she looks very sympathisch, not only in my judgment, but
also in the judgment of all those who know her. She has grown up with
six brothers and sisters, is very domestic, and possesses an unusual degree of
intelligence. "
Minkowski planned to be married in September, but first there was an
important event. An International Congress of Mathematicians was going
to take place in August in Zurich, which being Swiss was considered
appropriately neutral soil. Klein was asked to head the German delegation.
"Which will have the consequence," Minkowski noted, "that nobody will
come from Berlin."
Although for some reason Hilbert did not attend this first congress, he
read the papers which were presented and was most impressed by two of the
featured addresses. One of these was a lecture on the modern history of the
general theory of functions by Hurwitz. The other was an informal talk by
Poincare on the way in which pure analysis and mathematical physics serve
each other.
Shortly after the Congress, Minkowski was married in Strassburg.
He did not write to Hilbert again until the end of November:
"After my long silence, you must think that my marriage has changed me
completely. But I stay the same for my friends and for my science. Only I
could not show any interest for some time in the usual manner."
With the Zahlbericht completed, Hilbert was now involved in investiga-
tions of his own which he had long wished to pursue. The focal point of
his interest was the generalization of the Law of Reciprocity to algebraic
number fields. In classical number theory, the Law of Quadratic Reciprocity,
known to Euler, had been rediscovered by Gauss at the age of 18 and
given its first complete proof. Gauss always regarded it as the "gem" of

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Hilbert

number theory and returned to it five more times during his life to prove
it in a different way each time. It describes a beautiful relationship which
exists between pairs of primes and the remainders of squares when divided
by these.
For treating the Law of Reciprocity in the generality which he had in
mind, Hilbert needed a broad foundation; and this he had achieved in the
Zahlbericht. In its introduction he had noted that "the most richly equipped
part of the theory of algebraic number fields appears to me the theory of
abelian and relative abelian fields which has been opened up by Kummer
for us through his work on the higher reciprocity law and by Kronecker
through his investigation of the complex multiplication of elliptic functions.
The deep insights into this theory which the works of these two mathemati-
cians give us show at the same time that ... an abundance of the most
precious treasures still lies concealed, beckoning as rich reward to the
investigator who knows their value and lovingly practices the art to win
them."
Hilbert now proceeded to go after these treasures. As a result of his work
on the Zahlbericht he had a knowledge of the terrain that was both "intimate
and comprehensive." He moved cautiously but with confidence.
"It is a great pleasure," a later mathematician noted, "to watch how, step
by step, in a succession of papers ascending from the particular to the
general, the adequate concepts and methods are evolved and the essential
connections come to light."
By studying the classical Law of Quadratic Reciprocity of Gauss, Hilbert
was able to restate it in a simple, elegant way which also applied to algebraic
number fields. From this he was then able to guess with brilliant clarity
what the reciprocity law must be for degrees higher than 2, although he did
not prove his conjectures in all cases. The crown of his work was the paper
published the year after the Zahlbericht and entitled "On the theory of rela-
tive abelian fields." In this paper, which was basically programmatic in
character, he sketched out a vast theory of what were to become known as
"class-fields," and developed the methods and concepts needed to carry
out the necessary investigations. To later mathematicians it was to seem
that he had "conceived by divination" - nowhere else in his work is the
accuracy of his mathematical intuition so apparent. Unlike the work in
invariants, which had marked the end of a development, the work in alge-
braic number fields was destined to be a beginning. But for other mathe-
maticians.
Hilbert himself now turned abruptly away.

56
 VIII

Tables, Chairs, and Beer Mugs

The announcement that Hilbert would lecture during the winter of

1898-99 on the elements of geometry astonished the students to whom he
had talked "only number fields" since his arrival in Gbttingen three years
before. Yet the new interest was not entirely without antecedent.
In his docent days Hilbert had attended a lecture in Halle by Hermann
Wiener on the foundations and structure of geometry. In the station in
Berlin on his way back to Konigsberg, under the influence of Wiener's
abstract point of view in dealing with geometric entities, he had remarked
thoughtfully to his companions: "One must be able to say at all times -
instead of points, straight lines, and planes - tables, chairs, and beer mugs."
In this homely statement lay the essence of the course of lectures which he
now planned to present.
To understand Hilbert's approach to geometry, we must remember that
in the beginning mathematics was a more or less orderless collection of
statements which either seemed self-evident or were obtained in a clear,
logical manner from other seemingly self-evident statements. This criterion
of evidence was applied without reservation in extending mathematical
knowledge. Then, in the third century B. c., a teacher named Euclid or-
ganized some of the knowledge of his day in a form that was commonly
followed. First he defined the terms he would use - points, lines, planes, and
so on. Then he reduced the application of the criterion of evidence to a
dozen or so statements the truth of which seemed in general so clear that
one could accept them as true without proof. Using only these definitions
and axioms (as the statements were later called), he proceeded to derive
almost five hundred geometric statements, or theorems. The truth of these
was in many cases not at all self-evident, but it was guaranteed by the fact
that all the theorems had been derived strictly according to the accepted

57
Hilbert

laws of logic from the definitions and the axioms already accepted as
true.
Although Euclid was not the most imaginative of the Greek geometers
and the axiomatic method was not original with him, his treatment of geom-
etry was greatly admired. Soon, however, mathematicians began to recog-
nize that in spite of its beauty and perfection there were certain flaws in
Euclid's work; particularly, that the axioms were not really sufficient for
the derivation of all the theorems. Sometimes other, unstated assumptions
crept in - especially assumptions based on visual recognition that in a
particular construction certain lines were bound to intersect. It also seemed
that one of Euclid's axioms - the Parallel Postulate - went so far beyond
the immediate evidence of the senses that it could not really be accepted as
true without proof. In its various forms the Parallel Postulate makes a
statement essentially equivalent to the statement that through any point
not on a given line in a plane, at most one line can be drawn which will not
intersect the given line. Generally, however, this flaw and the others in
Euclid were dismissed as things which could be easily removed, first by
enlarging the original number of axioms to include the unstated assumptions
and then by proving the particularly questionable axiom as a theorem, or
by replacing it with another more intuitively evident axiom, or - finally -
by demonstrating that its negation led to a contradiction. This last and most
sophisticated method of dealing with the problem of the Parallel Postulate
represents the first appearance in mathematics of the concept of consistency,
or freedom from contradiction.
Gauss was apparently the first mathematician to whom it occurred,
perhaps as early as 1800, that the negation of Euclid's parallel postulate
might not lead to a contradiction and that geometries other than Euclid's
might be possible. But this idea smacked so of metaphysical speculation
that he never published his investigations on the subject and only com-
municated his thoughts to his closest friends under pledges of secrecy.
During the 1830's, however, two highly individualistic mathematicians
tried independently but almost simultaneously to derive from a changed
parallel axiom and the other, unchanged traditional axioms of euclidean
geometry what theorems they could. Their new axiom stated in essence
that through any point not on a given line, infinitely many lines can be
drawn which will never meet the given line. Since this was contrary to
what they thought they knew as true, the Russian Lobatchewsky and the
Hungarian]. Bolyai expected that the application of the axiomatic method
would lead to contradictory theorems. Instead, they found that although

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the theorems established from the new set of axioms were at odds with the
results of everyday experience (the angles of a triangle, for instance, did
not add up to two right angles as in Euclid's geometry), none of the expected
contradictions appeared in the new geometry thus established. It was
possible, they had discovered, to build a consistent geometry upon axioms
which (unlike Euclid's) did not seem self-evidently true, or which even ap-
peared false.
Surprisingly enough, this discovery of non-euclidean geometries did not
arouse the "clamors of the Boeotians" which, according to Gauss (in a letter
to Bessel on January 27, 1829), had deterred him from publishing his own
investigations on the subject. In fact, there was not very much interest in
the discovery among mathematicians. For the majority it seems to have
been too abstract.
It was not unti11870 that the idea was generally accepted. At that time
the 21-year-old Felix Klein discovered a "model" in the work of Cayley by
means of which he was able to identify the primitive objects and relations
of non-euclidean geometry with certain objects and relations of euclidean
geometry. In this way he established that non-euclidean geometry is every
bit as consistent as euclidean geometry itself; for a contradiction existing
in the one will have of necessity to appear in the other.
Thus the impossibility of demonstrating the Parallel Postulate was at
last shown to be "as absolutely certain as any mathematical truth what-
soever." But, again, the full impact of the discovery was not immediately
and generally felt. The majority of mathematicians, although they now
recognized the several non-euclidean geometries resulting from various
changes in the Parallel Postulate, held back from recognizing the fact, which
automatically followed, that Euclid's other axioms were equally arbitrary
hypotheses for which other hypotheses could be substituted and that still
other non-euclidean geometries were possible.
A few mathematicians did try to achieve treatments of geometry which
would throw into relief the full implication of the discovery of the non-
euclidean geometries, and would at the same time eliminate all the hidden
assumptions which had marred the logical beauty of Euclid's work. Such
a treatment had been first achieved by Moritz Pasch, who had avoided
inadvertently depending on assumptions based on visual evidence by reduc-
ing geometry to a pure exercise in logical syntax. Giuseppe Peano had gone
even farther. In essence he had translated Pasch's work into the notation
of a symbolic logic which he himself had invented. Peano's version of geom-
etry was completely abstract - a calculus of relations between variables.

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Hilbert

It was difficult to see how Hilbert could hope to go beyond what had
already been done in this area of mathematical thought. But now in his
lectures he proceeded to reverse the trend toward absolutely abstract
symbolization of geometry in order to reveal its essential nature. He returned
to Euclid's points, straight lines and planes and to the old relations of inci-
dence, order and congruence of segments and angles, the familiar figures.
But his return did not signify a return to the old deception of euclidean
geometry as a statement of truths about the physical universe. Instead -
within the classical framework - he attempted to present the modern point
of view with even greater clarity than either Pasch or Peano.
With the sure economy of the straight line on the plane, he followed to
its logical conclusion the remark which he had made half a dozen years
before in the Berlin station. He began by explaining to his audience that
Euclid's definitions of point, straight line and plane were really mathemat-
ically insignificant. They would come into focus only by their connection
with whatever axioms were chosen. In other words, whether they were
called points, straight lines, planes or were called tables, chairs, beer mugs,
they would be those objects for which the relationships expressed by the
axioms were true. In a way this was rather like saying that the meaning of an
unknown word becomes increasingly clear as it appears in various contexts.
Each additional statement in which it is used eliminates certain of the mean-
ings which would have been true, or meaningful, for the previous statements.
In his lectures Hilbert simply chose to use the traditional language of
Euclid:
"Let us conceive three distinct systems of things," he said. "The things
composing the first system we will call points and designate them by the
letters A, B, C, .... "
The "things" of the other two systems he called straight lines and planes.
These "things" could have among themselves certain mutual relations
which, again, he chose to indicate by such familiar terms as are situated,
between, parallel, congruent, continuous, and so on. But, as with the "things"
of the three systems, the meaning of these expressions was not to be deter-
mined by one's ordinary experience of them. For example, the primitive
terms could denote any objects whatsoever provided that to every pair of
objects called points there would correspond one and only one of the objects
called straight lines, and similarly for the other axioms.
The result of this kind of treatment is that the theorems hold true for
any interpretation of the primitive notions and fundamental relationships
for which the axioms are satisfied. (Many years later Hilbert was absolutely

60
 1898-1899

delighted to discover that from the application of a certain set of axioms

the laws governing the inheritance of characteristics in the fruit fly can be
derived: "So simple and precise and at the same time so miraculous that no
daring fantasy could have imagined it !")
In his lectures Hilbert now proposed to set up on this foundation a simple
and complete set of independent axioms by means of which it would be
possible to prove all the long familiar theorems of Euclid's traditional
geometry. His approach - the original combination of the abstract point of
view and the concrete traditional language - was peculiarly effective. "It
was as if over a landscape wherein but a few men with a superb sense of
orientation had found their way in murky twilight, the sun had risen all at
once," one of his later students wrote. By developing a set of axioms for
euclidean geometry which did not depart too greatly from the spirit of
Euclid's own axioms, and by employing a minimum of symbolism, Hilbert
was able to present more clearly and more convincingly than either Pasch
or Peano the new conception of the nature of the axiomatic method. His
approach could be followed by the students in his class who knew only the
original Elements of Euclid. For established mathematicians, whose first
introduction to real mathematics had invariably been the Elements, it was
particularly attractive, "as if one looked into a face thoroughly familiar and
yet sublimely transfigured."
At the time of these lectures on geometry arrangements were being made
in Gbttingen for the dedication of a monument to Gauss and Wilhelm
Weber, the two men - one in mathematics, the other in physics - from
whom the University's twofold scientific tradition stemmed. To Klein the
dedication ceremony seemed to offer an opportunity to emphasize once
again the organic unity of mathematics and the physical sciences. Gauss's
observatory had been no ivory tower. In addition to his mathematical
discoveries, he had made almost equally important contributions to physics,
astronomy, geodesy, electromagnetism, and mechanics. The broadness of
his interest had been reinforced by a collaboration with Wilhelm Weber.
The two men had invented an electromagnetic telegraph which transmitted
over a distance of more than 9,000 feet; the monument was to show the
two of them examining this invention. Carrying on and extending the tradi-
tion of mathematical abstraction combined with deep interest in physical
problems was central to Klein's dream for Gbttingen. So now he asked
Emil Wiechert to edit his recent lectures on the foundations of electro-
dynamics for a celebratory volume, and asked Hilbert to do the same for
his lectures on the foundations of geometry. (This was the same Wiechert

61
Hilbert

who had been Hilbert's official opponent for his promotion exercises at
Konigsberg, now also a professor at Gottingen.)
For the published work, as a graceful tribute to Kant, whose a priori view
of the nature of the geometrical axioms had been discredited by the new
view of the axiomatic method, Hilbert chose as his epigraph a quotation
from his fellow townsman:
"All human knowledge begins with intuitions, then passes to concepts,
and ends with ideas."
Time was short, but he took time to send the proof-sheets of the work to
Zurich so that Minkowski could go over them. As always, Minkowski was
appreciative and prophetic. The work was, in his opinion, a classic and
would have much influence on the thinking of present and future mathe-
maticians.
"It is really not noticeable that you had to work so fast at the end," he
assured Hilbert. "Perhaps if you had had more time, it would have lost
the quality of freshness."
Minkowski was not too happy in Switzerland. "An open word - take
the surprise easy - I would love to go back to Germany." His style of
thinking and lecturing was not popular in Zurich "where the students,
even the most capable among them, ... are accustomed to get everything
spoon-fed." But he hesitated to let his availability be known in Germany.
"I feel that even if I had some hope of getting a position, I would still make
myself ridiculous in the eyes of many."
Hilbert tried to cheer him up by inviting him to Gottingen for the dedica-
tion ceremonies of the Gauss-Weber monument. The days spent there
seemed "like a dream" to Minkowski when at the end of a week he had to
return to the "hard reality" of Zurich. "But their existence cannot be denied
any more than your 18 = 17 + 1 axiom of arithmetic .... No one who
has been in Gottingen recently can fail to be impressed by the stimulating
society there."
As soon as Hilbert's lectures, entitled in English The Foundations of Geom-
etry, appeared in print, they attracted attention all over the mathematical
world.
A German reviewer found the book so beautifully simple that he rashly
predicted it would soon be used as a text in elementary instruction.
Poincare gave his opinion that the work was a classic: "[The contemporary
geometers who feel that they have gone to the extreme limit of possible
concessions with the non-euclidean geometries based on the negation of the
Parallel Postulate] will lose this illusion if they read the work of Professor

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 1898-1899

Hilbert. In it they will find the barriers behind which they have wished to
confine us broken down at every point."
In Poincare's opinion, the work had but one flaw.
"The logical point of view alone appears to interest Professor Hilbert,"
he observed. "Being given a sequence of propositions, he finds that all
follow logically from the first. With the foundation of this first propo-
sition, with its psychological origin, he does not concern himself ....
The axioms are postulated; we do not know from whence they come; it is
then as easy to postulate A as C .... His work is thus incomplete, but this
is not a criticism I make against him. Incomplete one must indeed resign
oneself to be. It is enough that he has made the philosophy of mathematics
take a long step forward .... "
The American reviewer wrote prophetically, "A widely diffused knowl-
edge of the principles involved will do much for the logical treatment of
all science and for clear thinking and writing in general."
The decisive factor in the impact of Hilbert's work, according to Max
Dehn, who as a student attended the original lectures, was "the character-
istic Hilbertian spirit... combining logical power with intense vitality,
disdaining convention and tradition, shaping that which is essential into
antitheses with almost Kantian pleasure, taking advantage to the fullest of
the freedom of mathematical thought!"
To a large extent, Hilbert, like Euclid himself, had achieved success
because of the style and logical perfection of his presentation rather than its
originality. But in addition to formulating the modern viewpoint in a way
that was attractive and easily grasped, he had done something else which
was to be of considerable importance. Having set up in a thoroughly
rigorous modern manner the traditional ladder of thought - primitive
notions, axioms, theorems - he had proceeded to move on to an entirely
new level. In after years, when the approach would have become common,
it would be known as metamathematics -literally, "beyond mathematics."
For, unlike Euclid, Hilbert required that his axioms satisfy certain logical
demands:
That they were complete, so that all the theorems could be derived from
them.
That they were independent, so that the removal of anyone axiom from
the set would make it impossible to prove at least some of the theorems.
That they were consistent, so that no contradictory theorems could be
established by reasoning with them.
The most significant aspect of this part of Hilbert's work was the attempt-

63