remarlzable mathematicians

Remarlzable Mathematicians

From Euler to von Neumann

loan James Mathematical Institute, Oxford

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From Cantor to Hilbert

DAVID HILBERT (1862-1943) We now return to Germany. An inspiring teacher as well as a great researcher, Hilbert was arguably the leading figure in the mathematical world in the years following the death of Poincare. Many of the mathematicians who came to Gottingen, as students, as faculty members or as visitors, fell under his spell. Through the reminiscences of some of them we can obtain a faint idea of Hilbert's exceptional charisma. He revolutionized several branches of mathematics and opened up new fields of research, while his stimulating writings have exerted a powerful influence throughout the mathematical world. The most important phase of Hilbert's career lies in the first quarter of the twentieth century.

David Hilbert was born on January 23, 1862 in Wehlau, near Konigsberg, then the principal city of East Prussia. He was descended from a Protestant middle-class family which had settled near Freiberg in Saxony in the seventeenth century. On his father's side his more immediate forbears were professional men who practised in the capital and coronation city of the Prussian kings, which had developed a special cultural tradition. His

David Hilbert (1862- 1943)

father Otto was a judgei his mother, Maria Theresa (nee Erdtmann), was interested in mathematics, and Hilbert is said to have inherited his gifts from her side of the family. The future mathematician was their only son, but he had a younger sister who died in childbirth at the age of twenty-eight.

When Otto Hilbert became a more senior judge the family moved into Konigsberg itself, where his son was enrolled at the Royal Friedrichskolleg at the age of eight. This was a traditional grammar school where the curriculum emphasized the classicsi no science was taught, apart from a little mathematics. It was not the most suitable school in Konigsberg

for someone like David Hilbert, and those who would later become close friends of his attended the more progressive Wilhelms-Gymnasium. He only began to display his true abilities when he completed his school career with a year at the Gymnasium, followed by three years as a student at the University of Konigsberg, apart from the customary two semesters elsewhere which he spent in Heidelberg.

Another Konigsberg student at that time was the precocious Hermann Minkowski, already mentioned in the proB.le of Henry Smith. Minkowski was two years younger than Hilbert but being more advanced academically was in a position to act as his mentor. They formed a lasting friendship. Another early influence was Adolf Hurwitz from Gottingen, three years senior to Hilbert and already an associate professor at the university. In Hilbert's student years the leading mathematician was the able and versatile Heinrich Weber, Dedekind's collaborator on the memoir 'Algebraic functions of a single variable'. In 1883 Weber left and was replaced by Lindemann, celebrated for having proved the transcendence of n:. Lindemann was hardly

in the first rank of mathematicians, but he had a wide range of interests, one of which was the theory of invariants. This was the subject on which Hilbert wrote his dissertation.

Hilbert passed the doctoral examination at the end of 1884 and embarked upon the customary tour of other centres. He went first to visit Klein in Leipzig, thus beginning a long-lasting friendship, and then went on

to Paris, where he met Hermite, Jordan, and Poincare amongst others. On the way back to Konigsberg he stopped in Berlin and saw Kronecker, who had just succeeded Kummer at the university. By 1886 Hilbert had qualified as a privatdozent at Konigsberg, and remained in this position for the next six years. In 1892 he was appointed associate professor to replace Hurwitz who had moved to Zurich. In the same year he married Kathe Jerosch, the independent-minded daughter of a local merchanti their only child Franz was born the following year. He suffered from mental illnessi from the age

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of twenty-one he was often a patient in a psychiatric hospital, and never lived a normal life.

Hilbert's primary research interest during this period was in invariant theory, and in 1893 he created a sensation with the paper he contributed to the Chicago Mathematics Congress describing the outcome of his work.

First Cayley and Sylvester in England, and then Clebsch and Gordan in Germany, had been laboriously constructing tables of invariants. Their investigations led them to conjecture that the set of invariants always had a finite basis, but the algorithmic methods they used had only very limited success in proving this. By a tour de force of abstract algebra, but using ideas which went back to Dedekind, Hilbert proved the conjecture in a few pages. When he heard of this Gordan exclaimed, 'That is not mathematics. That is theology. '

Much later Courant described Hilbert's method of dealing with a problem as follows: 'He was a most concrete, intuitive mathematician who invented, and very consciously used, a principle: namely, if you want to

solve a problem first strip the problem of everything that is not essential. Simplify it, specialize it as much as you can without sacrificing its core. Thus it becomes simple, as simple as it can be made, without losing any of its punch, and then you solve it. The generalization is a triviality which

you do not need to pay too much attention to. This principle of Hilbert's proved extremely useful for him and also for others who learned it from him; unfortunately it has been forgotten.'

However in 1893 von Lindemann moved from Konigsberg to Munich and Hilbert, who was in his first year as associate professor, was chosen to

succeed him, against strong competition. Although only just turned thirty, he was now a full professor at a first-rate university. After some wrangling Minkowski, who had moved to Bonn the previous year, returned to take the post Hilbert had just vacated, although it was not long before he followed Hurwitz to Zurich. During the ten years he was at Konigsberg, Hilbert lectured on a wide range of subjects, from number theory, function theory, and projective and differential geometry to Galois theory, hydrodynamics,

and differential equations. Moreover, with the exception of a one-hour course on determinants, he never lectured on the same subject twice. He spoke slowly and without frills. When giving a course it was his practice to

devote the first part of each lecture to a summary of the previous one, as would be done at school.

Although Hilbert was deeply attached to his hometown he had ambitions to move to a place where there were better mathematics students


than at Konigsberg. The Hilberts used to sit each morning reading the

newspaper at breakfast just for news about the state of health of profes

sors of mathematics all over Europe. It was a very healthy period but in 1895 the opportunity he had been waiting for came up. When a certain

mathematician died this led to a cascade of professorial appointments at

German universities. At the end of this process Klein had secured Hilbert

for the Georgia Augusta, the university with which he was to be so closely

identified and where he remained for the rest of his professional career. At the same time Hilbert succeeded Klein as principal editor of the Annalen.

According to Courant, 'He took this very seriously and the violence with

which he rejected papers was completely without sympathy. As long as he

and Caratheodory and such people were editors it was a very high ranking,

maybe the highest ranking, mathematical journal in the world, and it really meant something to have a paper printed there.' In these years Hilbert was

in his prime while Klein, although immensely powerful, had long since

ceased research activity. Klein's emphasis on the importance of unifying

pure and applied mathematics was in contrast to Hilbert's view in which a secure axiomatic foundation was essential and intuitive arguments were

not worth much.

They also differed in other ways. Unlike Hilbert, Klein believed in

maintaining the traditional distant relationship between professors and

students. According to Courant:

In the Gottingen society, if you read old chronicles, a Gottingen

professor was a demi-god and very rank conscious - the professor, and

particularly the wife of the professor. Hilbert came to Gottingen and it

was very, very upsetting. Some of the older professors' wives met and

said 'Have you heard about this new mathematician who has come? He is upsetting the whole situation here. I learned that the other night

he was seen in some restaurant, playing billiards in the backroom

with privatdozents.' It was considered completely impossible for a full

professor to lower himself to be personally friendly with younger

people. But Hilbert broke this tradition completely, and this was an

enormous step towards creating scientific life; young students came to

his house and had tea or dinner with him. Frau Hilbert gave big, lavish

dinner-parties for assistants, students, etc. Hilbert went with his

students and also everybody else who wanted to come, for hour-long

hikes in the woods during which mathematics, politics and economics were discussed.

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Courant also gave a picture of Hilbert at work:

He spent his whole time gardening and in between gardening and little

chores, he went to a long blackboard, maybe twenty feet long, covered

so that also in the rain he could walk up and down, doing his

mathematics in between digging some flower beds. All day one could

observe him. I happened to have a student room on the fifth floor from which I could look out of the window and see Hilbert in his garden. He had a bicycle and practised little stunts on it. It was a very

harmless and pleasant life, alone with colleagues and students, and

very inspiring for everybody who had contact with him.

After invariant theory Hilbert turned to algebraic number theory. He spent the next four years preparing his famous Bericht tiber die Theorie der algebraischen Zahlkorper (Report on the theory of algebraic number

fields). As explained in the profile of Henry Smith, this was intended to

form part of a joint work with Minkowski, but the latter's contribution

never materialized. While many number theorists swore by the Zahlbericht, others thought it set back the development of algebraic number theory by

several decades. The preface of the report concludes with the words:

The theory of number fields is an edifice of rare beauty and harmony.

The most richly executed part of the building as it appears to me is the

theory of Abelian fields which Kummer by his work on the higher laws of reciprocity and Kronecker by his investigations of the complex

multiplication of elliptic functions have opened up to us. The deep

glimpses into the theory which the work of these two mathematicians

affords, reveal at the same time that there still lies an abundance of

priceless treasures hidden in this domain, beckoning as a rich reward to the explorer who knows the value of such treasures and with love

pursues the art to win them.

The Chicago Congress of 1893 was followed by the first proper

International Congress of Mathematicians at Zurich in 1897. Subsequently

Congresses have been held regularly every four years at various places, starting in 1900, apart from interruptions in wartime. Hilbert did not attend

the first of the series but at the second, held in Paris in 1900, he delivered an

address, at the end of which he described ten mathematical problems which

he thought particularly important, starting with the proof of the continuum

hypothesis. This famous list of problems, expanded to twenty-three when it was published, has received much attention, perhaps more than it deserves


and certainly more than Hilbert intended. It seems to have been prepared

at quite short notice from the questions he was thinking about at the time.

One or two of the problems were solved almost at once. Today, however, apart from some which were not of a nature for the word solution to be

appropriate, about the only problem on which little progress has been made

is the Riemann hypothesis, concerning the distribution of the zeroes of the

zeta function.

Hilbert's mathematical interests were remarkably broad; to describe

them properly would occupy disproportionate space. As Courant put it: 'the most impressive thing was the great variety, the wide spectrum, of

his interests. Enormously important to Hilbert throughout his life was the

variety in all aspects of mathematics. ' Characteristically he would make a decisive impact on some branch of mathematics and then move on to

something quite different. Thus algebraic number theory, which followed

invariant theory, was followed in turn by the foundations of geometry. He

attacked Waring's problem in number theory, and found the solution on a

visit to Brouwer in 1908. At another time he was thinking about Dirichlet's

principle, which Weierstrass had called into question, and succeeded in

rehabilitating this useful method in the calculus of variations. Later on

Hilbert became particularly interested in physics, with the special theory of

relativity and other sensational ideas indicating that a major revolution was

in progress. However it must be said that his contributions to mathematical

physics are not in the same class as his contributions to mathematics

itself. One of Hilbert's most important books is the Uber die Grundlagen

der Geometrie (On the foundations of geometry) of 1899, not so much for

the ideas it contained, which were not particularly novel, as the force with

which they were presented. Its conclusion is often quoted:

The present treatise is a critical enquiry into the principles of

geometry; we have been guided by the maxim so as to discuss every

problem in such a way as to examine whether it could not be solved in

some prescribed manner and by some restricted aids. In my opinion

this maxim contains a general and natural prescription; indeed,

whenever in our mathematical considerations we meet a problem or

guess a theorem, our desire for knowledge would not be satisfied as

long as we have not secured the complete solution and the exact proof

or clearly understood the reason for the impossibility and the necessity of our failure. Indeed, the present geometrical enquiry tries


to answer the question which axioms, suppositions or aids are

necessary for the proof of an elementary geometric truth; afterwards it

will depend on the standpoint which method of proof one prefers.

This statement of Hilbert's philosophy may be compared with the

one he gave at the Paris Congress, before producing his list of problems:

The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics

which builds the linking bridges and gives the ever more reliable

forms. From this it has come about that our entire contemporary

culture, inasmuch as it is based on the intellectual penetration and the

exploitation of nature, has its foundations in mathematics. Already Galileo said: one can understand nature only when one has learned the

language and the signs in which it speaks to us; but this language is

mathematics and these signs are mathematical figures . Kant made the

pronouncement: I assert that, in any particular natural science, one

encounters genuine scientific substance only to the extent that

mathematics is present. Indeed we do not master a scientific theory until we have shelled and completely prised free its mathematical

kernel. Without mathematics, the astronomy and physics of today

would be impossible; these sciences, in their theoretical branches,

virtually dissolve into mathematics. They, along with the many other

applications, are responsible for whatever esteem mathematics may enjoy in the eyes of the general public.

After quoting Gauss and Kronecker, Hilbert ended by saying: 'We

ought not to believe those who today, adopting a philosophical air and with

a tone of superiority, prophesy the decline of culture and are content with

the "unknowable" in a self-satisfied way. For us there is no unknowable, and in my opinion there is also none whatsoever for the natural sciences.

In place of this foolish "unknowable", let our watchword on the contrary

be: we must know- we shall know.'

Although Hilbert respected Klein, he needed a colleague whose in

terests and personality were closer to his own. The ideal would be his old friend Minkowski, who was looking for an opportunity to return to

Germany after almost ten years in Zurich. At Hilbert's insistence, the

Georgia Augusta created a new chair to which Minkowski was appointed.

Like Hilbert himself, Minkowski had broad interests. At Zurich he created

the beautiful geometry of numbers, at Gottingen he initiated the study of

space as a four-dimensional, non-euclidean, space-time continuum. Hilbert

David Hilbert (1862-1943)

and Minkowski were attempting to improve the mathematical foundations

of the special theory of relativity, which was then new. Einstein, who

was a former student of Minkowski's, was not impressed - he described

Hilbert's ideas as child-like- but much later he adopted Minkowski's ideas

about space-time in the general theory of relativity. A promising partner

ship between Hilbert and Minkowski was brought to a sudden end when

Minkowski died on January 12, 1909 as the result of a ruptured appendix. Minkowski's collected works, which were published the following year,

were edited by Hilbert, who was greatly distressed by his friend's untimely

death.

Hilbert's wide range of interests had its drawbacks. According to

Weyl, who attended his undergraduate lectures around 1910, Hilbert tended

to move from one theory to another, and from one discipline to the next, without providing motivations, without explanations of the historical back

ground, without giving explicit references to his sources, without stopping

to work out any particular ideas, without proving any assertion in detail,

but claiming all the while to possess a unified view of such matters. Weyl

was deeply impressed by Hilbert but he commented that the understanding

he imparted to students did not run very deep. This may be contrasted with

Courant's description of Hilbert's lecturing style:

His lectures were not perfect in a formal way, and it happened quite

often that he had not prepared quite enough, so that at the end of the

hour he would run out of material and had to improvise, which made

him stumble and fumble. His friends and students made fun of him

and gave him all kinds of ironical gifts for his birthday, to help him to

stretch the content. He also made mistakes and got stuck in proofs,

and so you had the chance to observe him struggling with sometimes

very simple questions of mathematics, and finding his way out. This was more inspiring than a wonderfully perfect performance lecturing.

The Franco-Prussian War had begun when Hilbert was a boy; the

siege of Paris took place when he was at school. Although nationalism was intense in both France and Germany after the war, and there was

certainly rivalry in the mathematical sphere, it was possible for Hilbert to

say, when introducing Poincare at Gottingen: 'You know, highly honoured

colleague, as do we all, how steady and close the mathematical interests of

rran.ce. an.d Ge.rman.'J have been. an.d con.t'mue to be ... 'Ibe mathematical threads tying France and Germany are, like no two other nations, diverse

and strong, so that from a mathematical perspective we may view Germany

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254 From Cantor to Hilbert

and France as a single land.' A few years later, Germany invaded France and the two countries were at war. Klein, as we have seen, was a signatory to the

Declaration to the World of Culture, Hilbert was not. Klein was expelled from the Paris Academy, Hilbert was not. He thought the war was stupid and did not hesitate to say so. As a result some of the students boycotted

his lectures, but nothing more. Poincare, 'the supreme genius', was six years older than Hilbert.

When he died, in 1912, Hilbert took his place in the general estimation as the greatest living mathematician. When Hurwitz, the companion of his Konigsberg years, died in 1919, there was an opening in Zurich, and Hilbert considered the idea of moving there. However, although conditions

in post-war Germany were appalling, he decided to remain where he was. Klein, as we have seen, was in poor health, and although his presence was still felt in Gottingen he played no real part in its post-war reconstruction. Hilbert had never had much to do with the 'mathematical arrangements',

as he called them. It was under the leadership of his former student Richard Courant that mathematics at Gottingen began to flourish again (how this came about will be told in the penultimate chapter).

In spite of the glow which surrounded Gottingen in the twenties there was a certain amount of hostility towards the faculty both from within Germany and from further afield. Partly this was jealousy, but it

must be admitted that there was a tendency at the Georgia Augusta not to worry too much about what was happening elsewhere and as a result a certain carelessness about attributions. The students even had a word for it: 'nostrification', whether conscious or unconscious. Hilbert was rather

notorious for unconscious or careless nostrification, but so was Courant.

The latter had this to say on the subject:

[Hilbert] was completely open, open to criticism and open to different points of view and every student. Everybody who had contact with

him felt that although he was such a mental giant and such a really great force in science one could talk to him on an equal footing- if one had something to talk about. Hilbert was not a scholar in the sense that he knew everything that happened in the world. He did not read every paper nor have a little catalogue in which he could find out everything that existed. On the contrary, it was one of his strengths but also one of his shortcomings, that he listened very carefully and caught inspiration, but then frequently forgot whence the inspiration came.

David Hilbert (1862-1943)

Hilbert had a lifelong interest in foundational questions; the Grundlagen of 1899 was just one example of his preference for axiomatics. For him

consistency was all important, if not for mathematics as a whole at least for certain parts of it, such as geometry. His opponent in what became a bitter

dispute was Brouwer, who held that it was truth rather than consistency

that mattered, and who rejected reductio ad absurdum arguments. The

formalism of Hilbert suffered a devastating blow in 1931 when Kurt Godel published the work on which the modern approach to mathematical logic is

based. The dispute with Brouwer over foundations became unduly personal, and did not show either of these formidable men at their best. However, as

Pavel Aleksandrov relates in his autobiography:

Plans were made to bring about a reconciliation between Brouwer and

Hilbert. An evening meal at Emmy Noetl1er's place was selected. In

the cozy attic which served as the study, the living room and the

dining room in Emmy Noether's apartment, Brouwer, Hilbert,

Courant and Landau were seated, along with some younger

mathematicians, including Hop£ and myself. It was my task to direct the conversation towards the reconciliation. Now it is a well-known

fact that the best way to bring two people back together is to centre

the focus on some third party whom the first two parties are eager to

criticize. Following this principle I brought the conversation round to

the function theoretician Luckenwald - the one who had achieved fame for his work on the theory of uniformity. The success of this

undertaking surely exceeded our boldest expectations, for in a short time Hilbert and Brouwer were pressing each other in a high-spirited

exchange of opinion, and drawing closer and closer to a common point

of view regarding the third party. All the while they nodded to each other in a friendlier and friendlier fashion. At last they united in a

mutual toast to each other.

Unfortunately the reconciliation between the two men did not last.

It was followed by an entirely separate quarrel concerning the question

of whether or not Germany should participate in the 1928 International Congress. Under pressure from certain influential French mathematicians,

particularly Picard, mathematicians from the former Central Powers had

been excluded from the 1920 International Congress in Strasbourg, the

first after the war; as a result the Strasbourg congress is not universally

recognized as one of the series. The issue divided the mathematical community; within France, for example, Painleve disagreed strongly with Picard's

255


position. At the 1924 Congress in Toronto such mathematicians were again

excluded, but after heated argument it was agreed that they should be

invited to the next Congress in Bologna. The German mathematicians were themselves deeply divided as to whether to accept. The nationalists, to

whose views Brouwer adhered, thought the way the invitation had been

given was insulting, while the internationalists, such as Hilbert, wanted to put the war behind them.

In the end Geheimrat Hilbert, although in poor health, led the German delegation of seventy-six mathematicians to Bologna. When they

entered the auditorium there was silence for a few moments, then everyone

rose to their feet and applauded. However Brouwer and certain others were

unable to see how Germans could attend the Congress 'without mocking

the memory of Gauss and Riemann, the humanistic character of mathemat

ical science, and the independence of the human spirit'. Hilbert responded

by getting Brouwer dropped from the editorial board of the Annalen, and, as we shall see, Brouwer, deeply hurt, responded by founding his own journal,

Compositio Mathematicae.

By this time Hilbert's most productive years were over. His illness was

diagnosed as pernicious anaemia, at that time a life-threatening condition: the symptoms are both physical and psychological, as we saw in the case of

Sophus Lie. Although Hilbert was able to receive a new treatment developed

at Harvard University, through the good offices of George Birkhoff, he never

fully recovered. Undoubtedly this affected his behaviour during the last part of his career. He retired in 1929, although he continued to lecture

occasionally. A street in Gottingen was named after him, while Konigsberg,

his birthplace, made him an honorary citizen. In 1932 Gottingen celebrated

his seventieth birthday with a torchlight procession. After 1934 he never again set foot in the Mathematical Institute. On one occasion he was seated

next to the new Nazi Minister of Education at a banquet when his neighbour

asked him whether it was true, as rumoured, that the Mathematical Insti

tute had suffered after the removal of the Jews and their friends . 'Suffered?',

Hilbert replied bitterly, 'It hasn't suffered, Herr Minister. It just doesn't exist any n1ore.'

Hilbert began to experience loss of memory and appeared to believe he was still living in Konigsberg. After a period of second childhood he

died in Gottingen on February 13, 1943, ten years after the Nazis came to

power. It was the middle of the war and barely a dozen people attended

his funeral. As for his physical appearance in his prime, in 1955 someone

who had known him well wrote: 'If I were a painter, I could draw Hilbert's


portrait, so strongly have his features engraved themselves into my mind,

forty years ago when he stood on the summit of his life. I still see the

high forehead, the shining eyes looking firmly through the spectacles, the

strong chin accentuated by the short beard, even the bold panama hat, and

his sharp East Prussian voice still sounds in my ears.' That is just how he

appears in the portrait shown here. Let us conclude with the words of Weyl,

the most gifted of Hilbert's students:

The impact of a scientist on his epoch is not directly proportional to

the scientific weight of his research. To be sure, Hilbert's mathematical work is of great depth and universality, and yet his

tremendous influence is not accounted for by that alone. Gauss and

Riemann are certainly of no lesser stature than Hilbert but they made little stir among their contemporaries and no school of devoted

followers formed around them. But Hilbert's was of a nature filled

with a zest for living, seeking intercourse with other people, above all

with younger scientists, and delighting in the exchange of ideas. His

optimism, his spiritual passion, his unshakeable faith in the supreme value of science, and his firm confidence in the power of reason to find

simple and clear answers to simple and clear questions were

irresistibly contagious.

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remarlzable mathematicians

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