MacTutor
Walter Ledermann - Encounters of a Mathematician
The formative years: Berlin 1928-33
I was only seventeen years of age when I left school with a Certificate of Maturity (roughly equivalent to British A-levels). This document carried with it the right of admission to any university in Germany or Austria. German students had to rely on their own financial resources to a greater extent than, for example, would be the case in modern Britain. There were few scholarships, and they were awarded only after a rigorous means test. There were no halls of residence; students lived at home or in cheap lodgings. There was no check on the attendance of lectures and, generally, no examinations before the final examination, the timing of which was determined by the candidate. Also, it was permissible to visit lectures that were not part of one's own course.
One of the most agreeable privileges was the arrangement whereby a student could spend one or more semesters at different universities in Germany or Austria, and the courses he took there would be counted towards his degree.
To my mind one of the objectionable features of academic life were the arrogant and swaggering student fraternities whose members wore coloured caps and sashes. At their regular meetings they would drink inordinate amounts of beer between the renderings of noisy student songs. They were trained in sabre fencing and they were committed to fighting a duel "in defence of their honour." It was a matter of pride that they did not flinch when their faces were lacerated with deep cuts, which left permanent scars. Members maintained a life-long loyalty to their fraternity observing a strict hierarchy between the generations. This caused a good deal of nepotism; for example, it was said that a large proportion of the high judiciary in Germany was in the hands of the fraternities. One could see even a number of University professors with scarred faces.
The original fraternities were of some antiquity. They were fiercely nationalistic and closed to Jews. Later on, fraternities with a more liberal outlook were founded; ironically they copied the drinking habits and "honour" code.
Fortunately, students of mathematics generally kept aloof from the snobbish sub-culture. The atmosphere in the mathematics common room was friendly irrespective of race or religion. Such political radicalism, as it existed, had a bias towards the left, but it was always peaceful.
I was not tempted, as many students were, to spend my first semester at one of the more romantic places, like Heidelberg or Freiburg in the Black Forest. I did not seek distraction and was keen to start serious study as soon as possible. So I decided to stay in Berlin and I was duly accepted as a student at the Friedrich-Wilhelms-Universität (now Humboldt Universität) there. To become a university student was an exhilarating experience: in some ways a university -- as the name asserts -- was an all-embracing alma mater to whom one belongs as an alumnus, and I was looking forward to the freedom and independence which was part of student life.
Among other things it was now up to me to make my own study programme by selecting suitable courses for which I wished to register. The University Office provided each student with a Study Book in which he entered the titles of all the courses he had chosen for a particular semester. (See illustration.)
A small fee was paid at the Office for each course and a receipt was issued. After one of the first few lectures in the semester the student had to hand the receipt to the lecturer who in turn signed his name in the student's Study Book opposite the entry specifying the course. This procedure was known as signing-on. When the course involved written work that had to be marked or laboratory assignments, then the student had to present his book again at the end of the semester and ask for a second signature; this was known as signing-off, and it testified that the student's performance had been satisfactory. Since any check on attendance was precluded by academic freedom, it was possible for a student to be registered for a particular course by putting in a single appearance in order to sign-on and then remain absent for the remainder of the semester.
On entering the university two options were open to me: I could either seek a supervisor and work towards a doctorate, which in those days was not a postgraduate degree, or I could aim at the State Examination, which was the academic qualification for a career in the school service. I decided to adopt the second alternative, as it would hold out better prospects of employment. In any event, I was not sure whether I should succeed in doing original research work.
The requirements for the State Examination consisted of a high standard in two major subjects, which in my case were Mathematics and Physics, and of a more modest standard in a minor subject for which I chose Chemistry with some hesitation because I was ill-prepared for it from school. In addition, I was also obliged to pass an oral examination in philosophy; this was due to a quirk of organization: for, although the University of Berlin was founded in the early nineteenth century, it adopted the medieval structure whereby there are only four faculties, namely Divinity, Jurisprudence, Medicine and Philosophy. As a consequence, more than half of the students belonged to the faculty of Philosophy, including students of mathematics, physics or chemistry. Moreover, there was a regulation which stipulated that every student in the faculty of Philosophy had to pass an (oral) examination in philosophy at the end of the course.
Almost all undergraduates in Berlin who studied either mathematics or physics belonged to a student society called MAPHA, which, when translated into English, stood for 'mathematical and physical study group'.
The main purpose of this society was to help one another with our studies and to get advice about our choice of lectures. There were also social events like hikes, parties and talks by outside speakers. The atmosphere was very friendly and supportive.
My widely ranging interests and obligations resulted in a heavy programme of lectures and exercise classes, amounting to more than thirty contact hours per week. On the whole, I enjoyed attending lectures even when the speaker's expository skill did not always match his fame as a scholar. But the speakers took their tasks seriously; formally dressed, they began each lecture with "ladies and gentlemen." Whenever possible, they tried to reach an interesting goal when the lecture ended after forty-five minutes.
During my fourth semester I was tempted to indulge in my love of Greek literature, and I joined a seminar in which, under the guidance of a classics lecturer, we read Plato's Phaidros in the original language.
Although, in principle, we could choose any courses to make up the programme for a semester, there were some obvious limitations -- clearly, one would not register for Analysis II without having previously attended Analysis I. Also the programme was strongly influenced by the hierarchical order within the teaching staff. At the head of each branch of the subject was the 'Ordinarius', that is Full Professor. Only an Ordinarius was in receipt of a salary which presumably was quite generous; in fact, he was a Civil Servant with permanent tenure and entitlement to a pension. Undergraduates were not normally allowed to approach an Ordinarius unless they had previously spoken to one of his Assistants. All the important courses were given by the Ordinarius representing the subject. Next in the academic hierarchy came the Extra-Ordinarius (Associate Professor). He would occasionally obtain a contract to give certain lecture courses; but his position was inferior to that of an Ordinarius. At the bottom of the ladder was the Privat-Dozent (Private Lecturer). His remuneration consisted entirely of the modest fees paid by the students attending his class. Quite often the courses offered by a Private Dozent were on specialized topics and attracted only a small number of students. Thus the income was minimal; indeed, it would appear that under the prevailing system only a person of private means or outside income could embark on an academic career.
Most of the mathematicians whose lectures I attended made a lasting impression on me, and I still have vivid memories of them even after seventy years.
Erhard Schmidt was the Ordinarius for Analysis. Before coming to Berlin he had done brilliant work at Göttingen, and he enjoyed a world-wide reputation. He was a man of impressive appearance and charismatic personality. But he seemed utterly unapproachable to me. During the four semesters when I attended his lectures I never dared to address a word to him, and he took no notice of me. There was an air of improvisation in the way he delivered his lectures; indeed, one had the feeling that he was rediscovering the most intricate mathematical results on the spur of the moment. He despised learning by rote. His advice was: "Never try to memorize a formula. Either it represents an important fact; then you will meet it in your studies so often that you will not forget it, or it is not an important formula. Then there is no reason why you should remember it."
Surprisingly, Erhard Schmidt could be funny; in fact, he possessed considerable histrionic talents. He was of Baltic origin, and he spoke German with a heavy drawl, typical of his native dialect. I suppose, this feature made his jocular remarks all the more amusing. He began his first lecture with the words: "Ladies and Gentlemen, for the sake of brevity allow me to address you all henceforth as Gentlemen." Often, there were visiting students from other faculties who sat giggling at the back of the classroom. They had come to be entertained by this eccentric professor of mathematics, although they could not understand one word of what he said. It seemed that Erhard Schmidt had numerous evening engagements outside the university He therefore found it convenient to give his lectures from 6 p.m. till 8 p.m. because the venues of his appointments could be more readily reached from the university, which was situated in the centre of the city, than from his home on the outskirts. Sometimes he came to his lecture wearing a dress suit and the porter came to the class room at 8 p.m. and said: "Sir, your taxi is outside."
The difficulties which many students may have had with the intellectual challenge of Erhard Schmidt's lectures were mitigated by the invaluable help provided by Dr Georg Feigl. He was an Assistant to Erhard Schmidt. Feigl's course Introduction to Higher Mathematics was an outstanding achievement in academic teaching. It eased for us the transition from school to university and filled the gaps in our understanding of some of the difficult parts of Erhard Schmidt's lectures. Feigl also conducted exercise classes for Erhard Schmidt's course on Analysis. Evidently, the grand man eschewed the lowly task of setting weekly exercises, let alone marking them. Feigl was also a lecturer in his own right. I remember his course on Projective Geometry which was so popular that it had to be held in the Auditorium Maximum of the University. But there was such a heavy demand on this hall that the only slot that could be found for Feigl and his audience, to which I belonged, was from 7.15 a.m. till 9 a.m. on two days of the week. This situation, coupled with Erhard Schmidt's propensity to evening lectures made it necessary for me to spend 13 hours in the university on certain days, to which had to be added one hour's journey from and to my home.
The person who had by far the strongest influence on my career was Issai Schur, the Ordinarius for Algebra. He was a superb lecturer. His courses were extremely well structured and organized: they were divided into chapters and subsection, each with a separate number. He prepared his lectures so well that he delivered them without recourse to the notes, which, as I learned later, he carried in the pocket of his jacket. Schur's lectures were very popular. His introductory courses on algebra were attended by about 300 students. On some days I was able only to secure a seat in one of the back rows and had to use opera glasses in order to read his writing on the black board. I felt that Schur's lectures were perfect in form and content. I took rough notes of what he said during the lectures and later, during week-ends or vacations. I transcribed them into cloth-bound books. They comprise some 2000 pages of about 320 lectures given by Schur on various branches of algebra and number theory. I greatly treasured these hand-written books as a monument of Schur's supreme knowledge and wisdom and I frequently referred to them throughout my career.
Finally, in 1998 I donated the books to the Humboldt University, the name now given to the Friedrich-Wilhelms University where I was a student from 1928 till 1933.
Schur was born in Russia but he was educated at a German-speaking school and he spoke German so well that one could not detect that German was not his native language. He was Jewish. When the Nazis came to power in 1933 he was immediately dismissed and forbidden to enter the university. Just before the political catastrophe I had registered as a candidate for the Final Examination, and I had the good fortune that Schur agreed to supervise my dissertation. As I could no longer see him in his office at the university he kindly invited me to his home at the outskirts of Berlin in order to discuss my work. On these occasions, after we had finished our mathematical business, we shared our deep anxieties about the fate that would befall the Jews under the Nazi tyranny. Schur said: "I find it too distasteful to read German newspapers, as they have all been taken over by the Nazis. It is a blessing that the London Times is still obtainable here, which enables me to keep in touch with the world." For six years Schur and his wife endured harassment and persecution at the hands of the Nazis. Several of his former students, who had found refuge in Britain, tried to persuade Schur to join them; but he declined giving as one of his reasons that he was reluctant to change his language again. Finally, he was able to go to Switzerland, where his daughter lived. From there he reached Palestine. By now he was a broken man in spirit and body. He died in 1941 on his 66th birthday.
The third Ordinarius for pure mathematics was Ludwig Bieberbach. He was an eminent mathematician and the author of several useful text-books. But he was a poor lecturer. I remember him nervously flitting through the pages of a copy of his own book, which lay in front of him, in order to find some help in continuing the argument he tried to expound in the lecture.
Bieberbach disgraced himself by being one of the (happily few) German mathematicians who became active Nazis. Shortly after Hitler's accession to power Bieberbach strutted along the corridors of the University proudly sporting a brown uniform. He pushed racial doctrines to absurd limits by founding the journal Deutsche Mathematik (German Mathematics) in which only authors of Nordic blood were allowed to publish, and he asserted that their mathematical work was superior to that produced by authors of Latin or Jewish 'race'. I have not seen Bieberbach's journal. But I remember reading a review by G. H. Hardy which ended somewhat as follows: "In times of political upheaval we may occasionally make statements that we do not really mean and later regret. But in the case of Bieberbach I have come to the less charitable conclusion that he believes what he says."
The Ordinarius for Applied Mathematics was Richard von Mises, an Austrian nobleman of Jewish extraction. The field of his interests and creative contributions was very wide; it included most branches of mathematics, engineering, aviation, philosophy and literature, especially the work of the poet Rainer Maria Rilke. Mises was very sure of himself. He spoke and wrote with precision and authority and with an aristocratic demeanour. After the lecture an assistant stood behind the door, holding a brush. Ready to remove any speck of dust that might have settled on the master's well-cut suit. In those days, Applied Mathematics included not only Mechanics, but also Geometric Drawing, Numerical Analysis (before the existence of electronic computers!), Probability Theory and Statistics. I vividly remember his course on Probability, in which he expounded his original definition of randomness. I thought it was a brilliant idea, and I regretted that it was not more widely accepted. This was in 1931, the year in which he published his substantial treatise Wahrscheinlichkeitrechnung (Calculus of Probability). This was intended to be the first of a series of volumes on various topics in applied mathematics. Alas, it was not to be: for, being of Jewish origin, Mises was dismissed from his post by the Nazis. He accepted a position in Istambul (which, however, he soon left in order to go to America). At that time I was anxiously looking for a country to which I could escape from the Nazis. In my admiration of Mises I took the bold step of telephoning him at home and said: "Professor, please excuse my intrusion. I understand that you are going to Istanbul. Would you be willing to accept me as a student, and could I study for a doctorate at Istanbul?", to which he made the typically curt reply: "Of course, you can take a doctorate anywhere."
I also attended some courses given by junior members of the faculty, that is by a lecturer who was not an Ordinarius. Among these, the person who had the strongest influence on me was Dr. Heinz Hopf. Already before 1930 he had gained a high reputation through his fundamental contribution to algebraic topology, and it was not surprising that in 1931 he was offered a post of Ordinarius at the prestigious Swiss Federal Institute of Technology (ETH) in Zürich, where he remained for the rest of his life. It was fortunate for me that Hopf was still in Berlin during most of my student years. I was fascinated by his lectures on Topology 1930; it was then a new subject and rarely included in the undergraduate syllabus. Even more important for my mathematical education was the fact that in 1929 Hopf gave the main course on Functions of a Complex Variable. Normally, this course would have been given by the Ordinarius, that is by Erhard Schmidt. But in 1929 Schmidt had been elected to be the Rector (RECTOR MAGNIFICUS) of the University and he found it necessary to reduce his teaching commitments. So he asked Heinz Hopf to take over this course. Hopf carried out his task in a masterly fashion. His lectures were meticulously prepared and he delivered them in a clear voice and with excellent black board technique. The course was spread over two semesters. It began with very simple material, explaining even what a complex number was. But at the end of the second semester he introduced us to some deep results concerning the geometric aspect of function theory.
After Hopf left Berlin to take up his post a Zürich, I did not see him for fifteen years. In the meantime, I had settled in Great Britain and the world was beginning to recover from the horrors of Hitler's war. After the war I got married. My wife's parents lived in Zürich, where her father was a professor of the History of Art. Therefore we often went to Zürich to visit her parents. On these occasions I would call on Hopf, who kindly invited me to his house and sometimes went for a walk with me. He spoke to me about his years in Berlin, when I was his student. One of the outstanding events in the history of algebra was the publication of van der Waerden's book Moderne Algebra which brought about a strong tendency towards abstraction. Some of the younger members of the Faculty, including Hopf, thought it would be a good idea to hold a weekly seminar in which van der Waerden's book could be studied by all the members. But the academic etiquette in those days required that such a seminar could take place only with the permission of Issai Schur, who was the Professor of Algebra. There was a belief that Schur did not like abstract algebra and might therefore not approve of this seminar. So Hopf's colleagues said to him: "You are polite and well dressed (which some of the others were not). Go to Schur on our behalf and asked him for permission to run the seminar." Hopf had an interview with Schur, who readily gave his approval but added: "I shall not take part." Hopf was pleased that he had the opportunity to give the course on Complex Variables, and, in particular, that so many students attended it. "I used the fees for it to pay for my honeymoon," he told me.
Heinz Hopf visited the United Kingdom a few times. He was once the main speaker at a meeting of the British Mathematical Colloquium. After the meeting he was the guest at one of the Cambridge Colleges, I think it was Trinity. He dined at High Table the don who sat next to him was not a mathematician; but he tried to involve the distinguished foreign visitor in conversation. He asked: "Are you a Trinity man?" When Hopf replied: "No," he continued: "Were you at St Johns?" When the answer was again "no," his host went through another ten or so Colleges. Hopf got more and more puzzled but kept up his denials. Then there was a pause and Hopf hoped that the topic of the conversation would be changed. However, his neighbour now declared with conviction: "So you have been to Oxford."
The study of physics, which was the second subject for my State Diploma, was less enjoyable for me than my involvement with mathematics. This was partly due to the fact that I had had very little teaching of physics at school, but also because I felt that I had no talent for science. However, there was some incentive for pursuing my work in physics: at that time there were some famous physicists in Berlin, and I was curious to see them and to listen to their lectures. Most of the material contained in the remainder of this chapter was published in an article Physics in Berlin 1928-33, (Physics Bull. 38 (1987) (IOP Publishing Ltd.). I should like to thank the Editor for giving permission to reproduce the article here, with minor alterations.
In the first year, for two semesters, students of physics were expected to attend the course on experimental physics given by Walther Nernst. He was a colourful person originally a professor of Chemistry -- he had been awarded the Nobel Price for chemistry -- he became a professor of Physical Chemistry, a subject for which he had laid the foundation through his celebrated text-book. Finally, he was appointed to the professorship in Physics. He was fond of remarking that he was a successor both to Bunsen and to Helmholtz. To us undergraduates he seemed arrogant, pompous and unapproachable. Nevertheless, ironically, he addressed the students by the egalitarian term of "commilitones" (fellow soldiers).
His lectures were unsystematic and difficult to follow. The vast majority of students who had to register for his course stayed away after a single attendance when they handed over their vouchers for the fee paid and obtained Nernst's signature in their study book. But on the first few days of the semester the lecture theatre was overcrowded with many students sitting on the steps or standing. Nernst came into the class room and said amiably: "Dear commilitones, I am so sorry about the inconvenience so many are being put to because of the lack of seating accommodation. But I can assure you that there will be plenty of empty seats in a week from now." He was right. For the rest of the semester the lecture theatre was half empty. But I persisted with my attendance although his talent for communication was modest, I could sense that I was in the presence of a great scientist who had a deep insight in the laws of nature. His vanity could also lead to amusing incidents. One of these occurred when he wanted to demonstrate that a difference of electrical potential will arise when a disc of base metal touches a disc of gold. He told us that many years ago this experiment was shown to an audience of wealthy amateurs of science. The lecturer produced an iron disc and then said: "Gentlemen, I should be obliged if one of you could lend me a gold coin so that I can continue with the experiment." The request was immediately granted, several outstretched arms offering purses filled with gold sovereigns. Now Nernst turned to us: "Dear commilitones, I do not expect that any one of you could lend me a gold coin at this moment. So I have taken the precaution of bringing some of my own collection." He clapped his hands and the laboratory technician came in, carrying a tray covered with a velvet cloth, on which lay a number of gold medals. Nernst picked them up one by one making such comments as "This is the Nobel Prize. This is the Bunsen Medal, this is the Franklin Medal" and with audible aside: "as far as I know there is one other such medal in Europe; it belonged to my friend the late Herr Boltzmann in Vienna," and so on. It was a most effective performance despite its boastfulness and conceit.
When I started my course, the Ordinarius for Theoretical Physics was Max Planck. His personality was very different from that of Nernst. Planck was quiet-spoken and reticent about his achievements. He always wore a black frock-coat, old-fashioned even in the 1920's. He taught theoretical physics in a cycle of five one-semester courses, comprising general mechanics, mechanics of continua, electricity and magnetism, optics and thermodynamics. In the winter of 1929 it was once again the turn of thermodynamics, and there was a rumour that Planck would retire after completing this cycle. I was anxious not to miss the opportunity of having been a student of Max Planck. So I registered for the thermodynamics course although I was not really ready for it, because I had not attended the preceding four courses. There were several hundred students in his class. We queued up after one of the first few lectures to order to hand over our vouchers and to obtain Planck's signature in our Study Books. He dealt rapidly with the waiting students. But when my turn came, he paused and said to me: "If you have any difficulties, do not hesitate to come to my office. I shall be very pleased to answer your questions if I can." I was very touched by his encouragement; perhaps he felt that I needed it, because I was the youngest student in the class. However, I never accepted his invitation.
Planck's lectures were well prepared and, as one would expect, of high intellectual standard. To be sure, his mathematics was a little old-fashioned and did not always conform to the rigour demanded in our pure mathematics courses. Regrettably, the effectiveness of his lectures was diminished by an unfortunate blackboard technique: before each lecture the porter brought a sponge and a saucer filled with water. Planck carefully squeezed the sponge until it was thoroughly moist. He held the sponge in his left hand while rapidly writing on the board with his right hand. Then, after covering a few lines he deleted everything with a sudden swing of his left arm. Thus it was difficult to take notes unless you were sitting at the extreme right of the room.
Apparently, Planck was too senior or too old to undertake the labour of holding the exercise classes that supplemented his course on Thermodynamics. So this task was delegated to Max von Laue, himself a Nobel Laureate. Von Laue had a slight speech defect and did not enjoy lecturing to young students. His exercise class was offered as a collegium publicum; no fee was payable and we did not get his signature in our Study Books. Von Laue's problems were very hard, and I did not succeed in solving many of them, but neither did the other students. He gave us the solutions of those questions from the preceding week which nobody had been able to solve then he addressed us in his staccato voice: "Are there any questions? If not, then I will close for to-day," and the class was dismissed after about 25 minutes.
As had been anticipated, Planck retired on completion of his course on Thermodynamics. His successor was Erwin Schrödinger. He was very different from the professors I had encountered so far. When he came to Berlin, he was in his mid-forties, a good-looking man with a scintillating personality. For many of us his charm was enhanced by an attractive Austrian accent. He was lively and informal. On a summer's day he would come to his lecture in an open-neck shirt, an attire utterly alien to a Prussian professor. Once he was stopped by the security guard who thought he was a political agitator; one of his students had to arrange for his release, because Schrödinger carried no identity papers.
His lectures were well thought out but delivered in a somewhat improvised manner. He did not adopt Planck's five-semester cycle of clearly defined topics, but the divided the subject into field physics and corpuscular physics, devoting two semesters to each. Schrödinger had a complete mastery of mathematical techniques and a deep insight into mathematical ideas and principles. But his mathematics was always related to physical theories which it served to describe. He once said: "I am often asked for advice about the best way to study theoretical physics." My reply is: "In the first year you study only mathematics; in the second year you study only mathematics. In the third year you can come to me, and we shall talk about physics."
Despite his great intellectual power Schrödinger had retained an endearing boyish trait. One day, a Zeppelin was flying over Berlin and could be seen from the University. When he noticed this, he ran to the window of the class room and shouted: "Look at the Zeppelin."
When the Nazis came to power in March 1933 Schrödinger had been in his position only for a few years. Yet in the summer of 1933 he suddenly left and never returned to Berlin. He was not Jewish and his departure had nothing to do with the Nazi racial laws. I was given the following explanation of this extraordinary event: one of the first things Hitler tried to do was to coerce Austria by economic means and in particular to destroy Austria's tourist industry. So he decreed that German citizens who wished to travel to Austria would have to pay a fee of 1000 Mark for an exit visa. This was a very large sum of money and virtually stopped all travelling from Germany to Austria. As was his custom, Schrödinger wanted to spend the summer holiday in his native country. So he applied to the Nazi authorities for exemption from the prohibitive travel tax. But his request was refused on the grounds that he was now a German citizen after having accepted the appointment as a Full Professor at a German university. Schrödinger was very angry. Instead of going to Austria he went to Oxford, where, I am sure, he was well received. From there he sent a postcard to the head porter of the University of Berlin who had the (rather funny) name Pfannkuchen (Pancake) saying: "Dear Herr Pfannkuchen, please tell the Senate that I am not coming back." This was, of course, regarded as a deliberate insult by the Nazi authorities who swore that they would severely punish Schrödinger, whenever they could lay hands on him.
At that time the Chair of Physics in Edinburgh became vacant. Professor E. T. Whittaker who was the Professor of Mathematics in Edinburgh suggested that Schrödinger might be invited to accept the Chair of Physics. But he said: "It is not proper that a man of such distinction should be interviewed by a panel. So I will explain to him personally what the appointment would entail." Schrödinger was invited to come to Edinburgh and Whittaker met him at Waverley Station. I may mention that Whittaker was always dressed in a conventional manner: dark suit and coat, hat and, of course, carrying an umbrella. On the next day he gave us an account of his meeting with Schrödinger: "I suggested that we walk around Arthur's Seat while I would describe to him the duties on an Edinburgh professor. Then it started to rain, and, would you believe it, Professor Schrödinger pulled out of his pocket a close-fitting arrangement that he proceeded to put on his head." It was clear that for this and, no doubt, weightier reasons, Whittaker came to the conclusion that Schrödinger would not be happy with a position in Edinburgh. Shortly afterwards Schrödinger accepted an invitation to join the Institute for Advanced Study in Dublin, in which President de Valera took a personal interest (he had mathematical training). It is possible that Whittaker had something to do with this appointment because he had been on friendly terms with de Valera since the time when he was Astronomer Royal of Ireland before moving to Edinburgh in 1912.
Schrödinger's enmity towards the Nazis nearly had disastrous consequences for him. He was in Austria in 1938 when the Nazis invaded that country. They had not forgotten the insult he had inflicted on them in 1933. He was arrested and deprived of his freedom to travel abroad. But Schrödinger was too clever for them. He pretended that he had been converted to the Nazi ideology and he issued an open declaration expressing regret that he had misunderstood the great German revival and that he would now gladly join the ranks of his Aryan compatriots. His insult of 1933 was forgiven. Unhindered he slipped out of the country and went back to Ireland, where he spent the rest of his working life. In his old age he returned to Austria and he died there in 1961.
Albert Einstein's research institute was situated near Berlin. As he was also a member of the Prussian Academy he was automatically associated with the University. But as far as I know he gave no undergraduate courses in my time. However, it was our ambition to have Einstein as a speaker at one of our Mapha meetings. This proved to be difficult to arrange because Mrs Einstein always answered the door and prevented any caller from approaching her husband. However, some students had noticed that during the summer Einstein often worked in the roof garden of his villa near Berlin. So two representatives from Mapha climbed up the drain pipe and appeared on the wall of the garden. Somewhat taken aback by the intrusion Einstein asked: "What do you want," and one of the students replied: "Professor, please forgive the disturbance; but would you be willing to give a lecture for the Mapha?" Einstein consented and a date was fixed. The lecture was, in fact, open to all members of the University, who filled the Auditorium Maximum to capacity. He presented a popular and fascinating account of the General Theory of Relativity.
My Oral Examination for the State Examination took place in November 1933, when the Nazis had been in power for eight months. The political revolution casts its shadows even on purely academic pursuits. Candidates were allowed to express a wish as to who should be their examiner in any particular subject. I had a difficulty in choosing an examiner for the Philosophy. I was told that all the professors of philosophy at the university were Nazi sympathizers and would probably fail me because I was a Jew. Then somebody mentioned that Professor Metzner was a member of the examination board although he was on the staff of the Technical University. It was known that the Professor was a devout Roman Catholic. At that time, the Catholic Zentrum Party did not support Hitler. Some months before the examination was to be held I asked Professor Metzner for an interview. I told him that I was a mathematician and had also attended a course on mathematical logic. He accepted me as a candidate and gave me some reading to do. On the morning of the oral examination Professor Metzner was delayed. When he finally arrived he was flustered but apologetic. I was the first candidate. He said: "I know you are a mathematician. Would you say that mathematical logic has invalidated the classical work of Aristotle?" I knew what was in his mind. Aristotelian philosophy has played an important part in Catholic theology, and he was afraid that the mathematical approach had undermined Catholic teaching. I replied: "Mathematics offers a new approach to logic; but it has not diminished the importance of Aristotle's work." He was obviously pleased and said: "This is what I have always said. I give you Grade A. Next candidate, please."
It was very fortunate the examination in mathematics could proceed as planned. Schur, who had been dismissed in March of that year, was temporarily reinstated through the intervention of his colleague Erhard Schmidt and was therefore able to conduct the examination. The co-examiner, that is the person who acted as a secretary for the proceedings, was Ludwig Bieberbach. This was unpleasant for Schur and for me; for Bieberbach was an enthusiastic Nazi and appeared wearing his brown uniform. However, it should be said that he did not interfere in any hostile manner. I still remember Schur's first question:
"What, in your opinion, is the most important theorem of the differential calculus?" I said: "Taylor's Theorem," and he replied: "I disagree. Surely, once you have proved Rolle's theorem, the other main theorems can easily be deduced from it." He then went on to algebra. "How do you determine the rank of a matrix?" passing to complex analysis he asked "What is the addition theorem for the Weierstrass elliptic function?" I started to recall the rather complicated formula; but I got stuck and had to admit that I did not remember the last term in the denominator. After a few seconds Schur said: "I don't remember it either." It was then that Bieberbach spoke the only time during the examination; without hesitation he quoted the complete formula.
Schur was pleased that I could cite various details in the theory of conformal mappings. "How is it that you know all this?" he asked, and I replied: "I learned it in Dr Hopf's lectures." Schur had put me at my ease, and the Oral Examination, which I had feared would be an awful ordeal, turned out to be quite a pleasant experience.
This concluded my life as a mathematician in Berlin. I was soon to leave my native city. More than forty years elapsed before I saw Berlin again, when I accompanied my wife on her professional meeting that was held in that city. However, I gratefully remember the foundations for my career that I received from the University of Berlin and the enriching experiences I had when the cultural life of Berlin was at its peak.
One of the most agreeable privileges was the arrangement whereby a student could spend one or more semesters at different universities in Germany or Austria, and the courses he took there would be counted towards his degree.
To my mind one of the objectionable features of academic life were the arrogant and swaggering student fraternities whose members wore coloured caps and sashes. At their regular meetings they would drink inordinate amounts of beer between the renderings of noisy student songs. They were trained in sabre fencing and they were committed to fighting a duel "in defence of their honour." It was a matter of pride that they did not flinch when their faces were lacerated with deep cuts, which left permanent scars. Members maintained a life-long loyalty to their fraternity observing a strict hierarchy between the generations. This caused a good deal of nepotism; for example, it was said that a large proportion of the high judiciary in Germany was in the hands of the fraternities. One could see even a number of University professors with scarred faces.
The original fraternities were of some antiquity. They were fiercely nationalistic and closed to Jews. Later on, fraternities with a more liberal outlook were founded; ironically they copied the drinking habits and "honour" code.
Fortunately, students of mathematics generally kept aloof from the snobbish sub-culture. The atmosphere in the mathematics common room was friendly irrespective of race or religion. Such political radicalism, as it existed, had a bias towards the left, but it was always peaceful.
I was not tempted, as many students were, to spend my first semester at one of the more romantic places, like Heidelberg or Freiburg in the Black Forest. I did not seek distraction and was keen to start serious study as soon as possible. So I decided to stay in Berlin and I was duly accepted as a student at the Friedrich-Wilhelms-Universität (now Humboldt Universität) there. To become a university student was an exhilarating experience: in some ways a university -- as the name asserts -- was an all-embracing alma mater to whom one belongs as an alumnus, and I was looking forward to the freedom and independence which was part of student life.
Among other things it was now up to me to make my own study programme by selecting suitable courses for which I wished to register. The University Office provided each student with a Study Book in which he entered the titles of all the courses he had chosen for a particular semester. (See illustration.)
A small fee was paid at the Office for each course and a receipt was issued. After one of the first few lectures in the semester the student had to hand the receipt to the lecturer who in turn signed his name in the student's Study Book opposite the entry specifying the course. This procedure was known as signing-on. When the course involved written work that had to be marked or laboratory assignments, then the student had to present his book again at the end of the semester and ask for a second signature; this was known as signing-off, and it testified that the student's performance had been satisfactory. Since any check on attendance was precluded by academic freedom, it was possible for a student to be registered for a particular course by putting in a single appearance in order to sign-on and then remain absent for the remainder of the semester.
On entering the university two options were open to me: I could either seek a supervisor and work towards a doctorate, which in those days was not a postgraduate degree, or I could aim at the State Examination, which was the academic qualification for a career in the school service. I decided to adopt the second alternative, as it would hold out better prospects of employment. In any event, I was not sure whether I should succeed in doing original research work.
The requirements for the State Examination consisted of a high standard in two major subjects, which in my case were Mathematics and Physics, and of a more modest standard in a minor subject for which I chose Chemistry with some hesitation because I was ill-prepared for it from school. In addition, I was also obliged to pass an oral examination in philosophy; this was due to a quirk of organization: for, although the University of Berlin was founded in the early nineteenth century, it adopted the medieval structure whereby there are only four faculties, namely Divinity, Jurisprudence, Medicine and Philosophy. As a consequence, more than half of the students belonged to the faculty of Philosophy, including students of mathematics, physics or chemistry. Moreover, there was a regulation which stipulated that every student in the faculty of Philosophy had to pass an (oral) examination in philosophy at the end of the course.
Almost all undergraduates in Berlin who studied either mathematics or physics belonged to a student society called MAPHA, which, when translated into English, stood for 'mathematical and physical study group'.
The main purpose of this society was to help one another with our studies and to get advice about our choice of lectures. There were also social events like hikes, parties and talks by outside speakers. The atmosphere was very friendly and supportive.
My widely ranging interests and obligations resulted in a heavy programme of lectures and exercise classes, amounting to more than thirty contact hours per week. On the whole, I enjoyed attending lectures even when the speaker's expository skill did not always match his fame as a scholar. But the speakers took their tasks seriously; formally dressed, they began each lecture with "ladies and gentlemen." Whenever possible, they tried to reach an interesting goal when the lecture ended after forty-five minutes.
During my fourth semester I was tempted to indulge in my love of Greek literature, and I joined a seminar in which, under the guidance of a classics lecturer, we read Plato's Phaidros in the original language.
Although, in principle, we could choose any courses to make up the programme for a semester, there were some obvious limitations -- clearly, one would not register for Analysis II without having previously attended Analysis I. Also the programme was strongly influenced by the hierarchical order within the teaching staff. At the head of each branch of the subject was the 'Ordinarius', that is Full Professor. Only an Ordinarius was in receipt of a salary which presumably was quite generous; in fact, he was a Civil Servant with permanent tenure and entitlement to a pension. Undergraduates were not normally allowed to approach an Ordinarius unless they had previously spoken to one of his Assistants. All the important courses were given by the Ordinarius representing the subject. Next in the academic hierarchy came the Extra-Ordinarius (Associate Professor). He would occasionally obtain a contract to give certain lecture courses; but his position was inferior to that of an Ordinarius. At the bottom of the ladder was the Privat-Dozent (Private Lecturer). His remuneration consisted entirely of the modest fees paid by the students attending his class. Quite often the courses offered by a Private Dozent were on specialized topics and attracted only a small number of students. Thus the income was minimal; indeed, it would appear that under the prevailing system only a person of private means or outside income could embark on an academic career.
Most of the mathematicians whose lectures I attended made a lasting impression on me, and I still have vivid memories of them even after seventy years.
Erhard Schmidt was the Ordinarius for Analysis. Before coming to Berlin he had done brilliant work at Göttingen, and he enjoyed a world-wide reputation. He was a man of impressive appearance and charismatic personality. But he seemed utterly unapproachable to me. During the four semesters when I attended his lectures I never dared to address a word to him, and he took no notice of me. There was an air of improvisation in the way he delivered his lectures; indeed, one had the feeling that he was rediscovering the most intricate mathematical results on the spur of the moment. He despised learning by rote. His advice was: "Never try to memorize a formula. Either it represents an important fact; then you will meet it in your studies so often that you will not forget it, or it is not an important formula. Then there is no reason why you should remember it."
Surprisingly, Erhard Schmidt could be funny; in fact, he possessed considerable histrionic talents. He was of Baltic origin, and he spoke German with a heavy drawl, typical of his native dialect. I suppose, this feature made his jocular remarks all the more amusing. He began his first lecture with the words: "Ladies and Gentlemen, for the sake of brevity allow me to address you all henceforth as Gentlemen." Often, there were visiting students from other faculties who sat giggling at the back of the classroom. They had come to be entertained by this eccentric professor of mathematics, although they could not understand one word of what he said. It seemed that Erhard Schmidt had numerous evening engagements outside the university He therefore found it convenient to give his lectures from 6 p.m. till 8 p.m. because the venues of his appointments could be more readily reached from the university, which was situated in the centre of the city, than from his home on the outskirts. Sometimes he came to his lecture wearing a dress suit and the porter came to the class room at 8 p.m. and said: "Sir, your taxi is outside."
The difficulties which many students may have had with the intellectual challenge of Erhard Schmidt's lectures were mitigated by the invaluable help provided by Dr Georg Feigl. He was an Assistant to Erhard Schmidt. Feigl's course Introduction to Higher Mathematics was an outstanding achievement in academic teaching. It eased for us the transition from school to university and filled the gaps in our understanding of some of the difficult parts of Erhard Schmidt's lectures. Feigl also conducted exercise classes for Erhard Schmidt's course on Analysis. Evidently, the grand man eschewed the lowly task of setting weekly exercises, let alone marking them. Feigl was also a lecturer in his own right. I remember his course on Projective Geometry which was so popular that it had to be held in the Auditorium Maximum of the University. But there was such a heavy demand on this hall that the only slot that could be found for Feigl and his audience, to which I belonged, was from 7.15 a.m. till 9 a.m. on two days of the week. This situation, coupled with Erhard Schmidt's propensity to evening lectures made it necessary for me to spend 13 hours in the university on certain days, to which had to be added one hour's journey from and to my home.
The person who had by far the strongest influence on my career was Issai Schur, the Ordinarius for Algebra. He was a superb lecturer. His courses were extremely well structured and organized: they were divided into chapters and subsection, each with a separate number. He prepared his lectures so well that he delivered them without recourse to the notes, which, as I learned later, he carried in the pocket of his jacket. Schur's lectures were very popular. His introductory courses on algebra were attended by about 300 students. On some days I was able only to secure a seat in one of the back rows and had to use opera glasses in order to read his writing on the black board. I felt that Schur's lectures were perfect in form and content. I took rough notes of what he said during the lectures and later, during week-ends or vacations. I transcribed them into cloth-bound books. They comprise some 2000 pages of about 320 lectures given by Schur on various branches of algebra and number theory. I greatly treasured these hand-written books as a monument of Schur's supreme knowledge and wisdom and I frequently referred to them throughout my career.
Finally, in 1998 I donated the books to the Humboldt University, the name now given to the Friedrich-Wilhelms University where I was a student from 1928 till 1933.
Schur was born in Russia but he was educated at a German-speaking school and he spoke German so well that one could not detect that German was not his native language. He was Jewish. When the Nazis came to power in 1933 he was immediately dismissed and forbidden to enter the university. Just before the political catastrophe I had registered as a candidate for the Final Examination, and I had the good fortune that Schur agreed to supervise my dissertation. As I could no longer see him in his office at the university he kindly invited me to his home at the outskirts of Berlin in order to discuss my work. On these occasions, after we had finished our mathematical business, we shared our deep anxieties about the fate that would befall the Jews under the Nazi tyranny. Schur said: "I find it too distasteful to read German newspapers, as they have all been taken over by the Nazis. It is a blessing that the London Times is still obtainable here, which enables me to keep in touch with the world." For six years Schur and his wife endured harassment and persecution at the hands of the Nazis. Several of his former students, who had found refuge in Britain, tried to persuade Schur to join them; but he declined giving as one of his reasons that he was reluctant to change his language again. Finally, he was able to go to Switzerland, where his daughter lived. From there he reached Palestine. By now he was a broken man in spirit and body. He died in 1941 on his 66th birthday.
The third Ordinarius for pure mathematics was Ludwig Bieberbach. He was an eminent mathematician and the author of several useful text-books. But he was a poor lecturer. I remember him nervously flitting through the pages of a copy of his own book, which lay in front of him, in order to find some help in continuing the argument he tried to expound in the lecture.
Bieberbach disgraced himself by being one of the (happily few) German mathematicians who became active Nazis. Shortly after Hitler's accession to power Bieberbach strutted along the corridors of the University proudly sporting a brown uniform. He pushed racial doctrines to absurd limits by founding the journal Deutsche Mathematik (German Mathematics) in which only authors of Nordic blood were allowed to publish, and he asserted that their mathematical work was superior to that produced by authors of Latin or Jewish 'race'. I have not seen Bieberbach's journal. But I remember reading a review by G. H. Hardy which ended somewhat as follows: "In times of political upheaval we may occasionally make statements that we do not really mean and later regret. But in the case of Bieberbach I have come to the less charitable conclusion that he believes what he says."
The Ordinarius for Applied Mathematics was Richard von Mises, an Austrian nobleman of Jewish extraction. The field of his interests and creative contributions was very wide; it included most branches of mathematics, engineering, aviation, philosophy and literature, especially the work of the poet Rainer Maria Rilke. Mises was very sure of himself. He spoke and wrote with precision and authority and with an aristocratic demeanour. After the lecture an assistant stood behind the door, holding a brush. Ready to remove any speck of dust that might have settled on the master's well-cut suit. In those days, Applied Mathematics included not only Mechanics, but also Geometric Drawing, Numerical Analysis (before the existence of electronic computers!), Probability Theory and Statistics. I vividly remember his course on Probability, in which he expounded his original definition of randomness. I thought it was a brilliant idea, and I regretted that it was not more widely accepted. This was in 1931, the year in which he published his substantial treatise Wahrscheinlichkeitrechnung (Calculus of Probability). This was intended to be the first of a series of volumes on various topics in applied mathematics. Alas, it was not to be: for, being of Jewish origin, Mises was dismissed from his post by the Nazis. He accepted a position in Istambul (which, however, he soon left in order to go to America). At that time I was anxiously looking for a country to which I could escape from the Nazis. In my admiration of Mises I took the bold step of telephoning him at home and said: "Professor, please excuse my intrusion. I understand that you are going to Istanbul. Would you be willing to accept me as a student, and could I study for a doctorate at Istanbul?", to which he made the typically curt reply: "Of course, you can take a doctorate anywhere."
I also attended some courses given by junior members of the faculty, that is by a lecturer who was not an Ordinarius. Among these, the person who had the strongest influence on me was Dr. Heinz Hopf. Already before 1930 he had gained a high reputation through his fundamental contribution to algebraic topology, and it was not surprising that in 1931 he was offered a post of Ordinarius at the prestigious Swiss Federal Institute of Technology (ETH) in Zürich, where he remained for the rest of his life. It was fortunate for me that Hopf was still in Berlin during most of my student years. I was fascinated by his lectures on Topology 1930; it was then a new subject and rarely included in the undergraduate syllabus. Even more important for my mathematical education was the fact that in 1929 Hopf gave the main course on Functions of a Complex Variable. Normally, this course would have been given by the Ordinarius, that is by Erhard Schmidt. But in 1929 Schmidt had been elected to be the Rector (RECTOR MAGNIFICUS) of the University and he found it necessary to reduce his teaching commitments. So he asked Heinz Hopf to take over this course. Hopf carried out his task in a masterly fashion. His lectures were meticulously prepared and he delivered them in a clear voice and with excellent black board technique. The course was spread over two semesters. It began with very simple material, explaining even what a complex number was. But at the end of the second semester he introduced us to some deep results concerning the geometric aspect of function theory.
After Hopf left Berlin to take up his post a Zürich, I did not see him for fifteen years. In the meantime, I had settled in Great Britain and the world was beginning to recover from the horrors of Hitler's war. After the war I got married. My wife's parents lived in Zürich, where her father was a professor of the History of Art. Therefore we often went to Zürich to visit her parents. On these occasions I would call on Hopf, who kindly invited me to his house and sometimes went for a walk with me. He spoke to me about his years in Berlin, when I was his student. One of the outstanding events in the history of algebra was the publication of van der Waerden's book Moderne Algebra which brought about a strong tendency towards abstraction. Some of the younger members of the Faculty, including Hopf, thought it would be a good idea to hold a weekly seminar in which van der Waerden's book could be studied by all the members. But the academic etiquette in those days required that such a seminar could take place only with the permission of Issai Schur, who was the Professor of Algebra. There was a belief that Schur did not like abstract algebra and might therefore not approve of this seminar. So Hopf's colleagues said to him: "You are polite and well dressed (which some of the others were not). Go to Schur on our behalf and asked him for permission to run the seminar." Hopf had an interview with Schur, who readily gave his approval but added: "I shall not take part." Hopf was pleased that he had the opportunity to give the course on Complex Variables, and, in particular, that so many students attended it. "I used the fees for it to pay for my honeymoon," he told me.
Heinz Hopf visited the United Kingdom a few times. He was once the main speaker at a meeting of the British Mathematical Colloquium. After the meeting he was the guest at one of the Cambridge Colleges, I think it was Trinity. He dined at High Table the don who sat next to him was not a mathematician; but he tried to involve the distinguished foreign visitor in conversation. He asked: "Are you a Trinity man?" When Hopf replied: "No," he continued: "Were you at St Johns?" When the answer was again "no," his host went through another ten or so Colleges. Hopf got more and more puzzled but kept up his denials. Then there was a pause and Hopf hoped that the topic of the conversation would be changed. However, his neighbour now declared with conviction: "So you have been to Oxford."
The study of physics, which was the second subject for my State Diploma, was less enjoyable for me than my involvement with mathematics. This was partly due to the fact that I had had very little teaching of physics at school, but also because I felt that I had no talent for science. However, there was some incentive for pursuing my work in physics: at that time there were some famous physicists in Berlin, and I was curious to see them and to listen to their lectures. Most of the material contained in the remainder of this chapter was published in an article Physics in Berlin 1928-33, (Physics Bull. 38 (1987) (IOP Publishing Ltd.). I should like to thank the Editor for giving permission to reproduce the article here, with minor alterations.
In the first year, for two semesters, students of physics were expected to attend the course on experimental physics given by Walther Nernst. He was a colourful person originally a professor of Chemistry -- he had been awarded the Nobel Price for chemistry -- he became a professor of Physical Chemistry, a subject for which he had laid the foundation through his celebrated text-book. Finally, he was appointed to the professorship in Physics. He was fond of remarking that he was a successor both to Bunsen and to Helmholtz. To us undergraduates he seemed arrogant, pompous and unapproachable. Nevertheless, ironically, he addressed the students by the egalitarian term of "commilitones" (fellow soldiers).
His lectures were unsystematic and difficult to follow. The vast majority of students who had to register for his course stayed away after a single attendance when they handed over their vouchers for the fee paid and obtained Nernst's signature in their study book. But on the first few days of the semester the lecture theatre was overcrowded with many students sitting on the steps or standing. Nernst came into the class room and said amiably: "Dear commilitones, I am so sorry about the inconvenience so many are being put to because of the lack of seating accommodation. But I can assure you that there will be plenty of empty seats in a week from now." He was right. For the rest of the semester the lecture theatre was half empty. But I persisted with my attendance although his talent for communication was modest, I could sense that I was in the presence of a great scientist who had a deep insight in the laws of nature. His vanity could also lead to amusing incidents. One of these occurred when he wanted to demonstrate that a difference of electrical potential will arise when a disc of base metal touches a disc of gold. He told us that many years ago this experiment was shown to an audience of wealthy amateurs of science. The lecturer produced an iron disc and then said: "Gentlemen, I should be obliged if one of you could lend me a gold coin so that I can continue with the experiment." The request was immediately granted, several outstretched arms offering purses filled with gold sovereigns. Now Nernst turned to us: "Dear commilitones, I do not expect that any one of you could lend me a gold coin at this moment. So I have taken the precaution of bringing some of my own collection." He clapped his hands and the laboratory technician came in, carrying a tray covered with a velvet cloth, on which lay a number of gold medals. Nernst picked them up one by one making such comments as "This is the Nobel Prize. This is the Bunsen Medal, this is the Franklin Medal" and with audible aside: "as far as I know there is one other such medal in Europe; it belonged to my friend the late Herr Boltzmann in Vienna," and so on. It was a most effective performance despite its boastfulness and conceit.
When I started my course, the Ordinarius for Theoretical Physics was Max Planck. His personality was very different from that of Nernst. Planck was quiet-spoken and reticent about his achievements. He always wore a black frock-coat, old-fashioned even in the 1920's. He taught theoretical physics in a cycle of five one-semester courses, comprising general mechanics, mechanics of continua, electricity and magnetism, optics and thermodynamics. In the winter of 1929 it was once again the turn of thermodynamics, and there was a rumour that Planck would retire after completing this cycle. I was anxious not to miss the opportunity of having been a student of Max Planck. So I registered for the thermodynamics course although I was not really ready for it, because I had not attended the preceding four courses. There were several hundred students in his class. We queued up after one of the first few lectures to order to hand over our vouchers and to obtain Planck's signature in our Study Books. He dealt rapidly with the waiting students. But when my turn came, he paused and said to me: "If you have any difficulties, do not hesitate to come to my office. I shall be very pleased to answer your questions if I can." I was very touched by his encouragement; perhaps he felt that I needed it, because I was the youngest student in the class. However, I never accepted his invitation.
Planck's lectures were well prepared and, as one would expect, of high intellectual standard. To be sure, his mathematics was a little old-fashioned and did not always conform to the rigour demanded in our pure mathematics courses. Regrettably, the effectiveness of his lectures was diminished by an unfortunate blackboard technique: before each lecture the porter brought a sponge and a saucer filled with water. Planck carefully squeezed the sponge until it was thoroughly moist. He held the sponge in his left hand while rapidly writing on the board with his right hand. Then, after covering a few lines he deleted everything with a sudden swing of his left arm. Thus it was difficult to take notes unless you were sitting at the extreme right of the room.
Apparently, Planck was too senior or too old to undertake the labour of holding the exercise classes that supplemented his course on Thermodynamics. So this task was delegated to Max von Laue, himself a Nobel Laureate. Von Laue had a slight speech defect and did not enjoy lecturing to young students. His exercise class was offered as a collegium publicum; no fee was payable and we did not get his signature in our Study Books. Von Laue's problems were very hard, and I did not succeed in solving many of them, but neither did the other students. He gave us the solutions of those questions from the preceding week which nobody had been able to solve then he addressed us in his staccato voice: "Are there any questions? If not, then I will close for to-day," and the class was dismissed after about 25 minutes.
As had been anticipated, Planck retired on completion of his course on Thermodynamics. His successor was Erwin Schrödinger. He was very different from the professors I had encountered so far. When he came to Berlin, he was in his mid-forties, a good-looking man with a scintillating personality. For many of us his charm was enhanced by an attractive Austrian accent. He was lively and informal. On a summer's day he would come to his lecture in an open-neck shirt, an attire utterly alien to a Prussian professor. Once he was stopped by the security guard who thought he was a political agitator; one of his students had to arrange for his release, because Schrödinger carried no identity papers.
His lectures were well thought out but delivered in a somewhat improvised manner. He did not adopt Planck's five-semester cycle of clearly defined topics, but the divided the subject into field physics and corpuscular physics, devoting two semesters to each. Schrödinger had a complete mastery of mathematical techniques and a deep insight into mathematical ideas and principles. But his mathematics was always related to physical theories which it served to describe. He once said: "I am often asked for advice about the best way to study theoretical physics." My reply is: "In the first year you study only mathematics; in the second year you study only mathematics. In the third year you can come to me, and we shall talk about physics."
Despite his great intellectual power Schrödinger had retained an endearing boyish trait. One day, a Zeppelin was flying over Berlin and could be seen from the University. When he noticed this, he ran to the window of the class room and shouted: "Look at the Zeppelin."
When the Nazis came to power in March 1933 Schrödinger had been in his position only for a few years. Yet in the summer of 1933 he suddenly left and never returned to Berlin. He was not Jewish and his departure had nothing to do with the Nazi racial laws. I was given the following explanation of this extraordinary event: one of the first things Hitler tried to do was to coerce Austria by economic means and in particular to destroy Austria's tourist industry. So he decreed that German citizens who wished to travel to Austria would have to pay a fee of 1000 Mark for an exit visa. This was a very large sum of money and virtually stopped all travelling from Germany to Austria. As was his custom, Schrödinger wanted to spend the summer holiday in his native country. So he applied to the Nazi authorities for exemption from the prohibitive travel tax. But his request was refused on the grounds that he was now a German citizen after having accepted the appointment as a Full Professor at a German university. Schrödinger was very angry. Instead of going to Austria he went to Oxford, where, I am sure, he was well received. From there he sent a postcard to the head porter of the University of Berlin who had the (rather funny) name Pfannkuchen (Pancake) saying: "Dear Herr Pfannkuchen, please tell the Senate that I am not coming back." This was, of course, regarded as a deliberate insult by the Nazi authorities who swore that they would severely punish Schrödinger, whenever they could lay hands on him.
At that time the Chair of Physics in Edinburgh became vacant. Professor E. T. Whittaker who was the Professor of Mathematics in Edinburgh suggested that Schrödinger might be invited to accept the Chair of Physics. But he said: "It is not proper that a man of such distinction should be interviewed by a panel. So I will explain to him personally what the appointment would entail." Schrödinger was invited to come to Edinburgh and Whittaker met him at Waverley Station. I may mention that Whittaker was always dressed in a conventional manner: dark suit and coat, hat and, of course, carrying an umbrella. On the next day he gave us an account of his meeting with Schrödinger: "I suggested that we walk around Arthur's Seat while I would describe to him the duties on an Edinburgh professor. Then it started to rain, and, would you believe it, Professor Schrödinger pulled out of his pocket a close-fitting arrangement that he proceeded to put on his head." It was clear that for this and, no doubt, weightier reasons, Whittaker came to the conclusion that Schrödinger would not be happy with a position in Edinburgh. Shortly afterwards Schrödinger accepted an invitation to join the Institute for Advanced Study in Dublin, in which President de Valera took a personal interest (he had mathematical training). It is possible that Whittaker had something to do with this appointment because he had been on friendly terms with de Valera since the time when he was Astronomer Royal of Ireland before moving to Edinburgh in 1912.
Schrödinger's enmity towards the Nazis nearly had disastrous consequences for him. He was in Austria in 1938 when the Nazis invaded that country. They had not forgotten the insult he had inflicted on them in 1933. He was arrested and deprived of his freedom to travel abroad. But Schrödinger was too clever for them. He pretended that he had been converted to the Nazi ideology and he issued an open declaration expressing regret that he had misunderstood the great German revival and that he would now gladly join the ranks of his Aryan compatriots. His insult of 1933 was forgiven. Unhindered he slipped out of the country and went back to Ireland, where he spent the rest of his working life. In his old age he returned to Austria and he died there in 1961.
Albert Einstein's research institute was situated near Berlin. As he was also a member of the Prussian Academy he was automatically associated with the University. But as far as I know he gave no undergraduate courses in my time. However, it was our ambition to have Einstein as a speaker at one of our Mapha meetings. This proved to be difficult to arrange because Mrs Einstein always answered the door and prevented any caller from approaching her husband. However, some students had noticed that during the summer Einstein often worked in the roof garden of his villa near Berlin. So two representatives from Mapha climbed up the drain pipe and appeared on the wall of the garden. Somewhat taken aback by the intrusion Einstein asked: "What do you want," and one of the students replied: "Professor, please forgive the disturbance; but would you be willing to give a lecture for the Mapha?" Einstein consented and a date was fixed. The lecture was, in fact, open to all members of the University, who filled the Auditorium Maximum to capacity. He presented a popular and fascinating account of the General Theory of Relativity.
My Oral Examination for the State Examination took place in November 1933, when the Nazis had been in power for eight months. The political revolution casts its shadows even on purely academic pursuits. Candidates were allowed to express a wish as to who should be their examiner in any particular subject. I had a difficulty in choosing an examiner for the Philosophy. I was told that all the professors of philosophy at the university were Nazi sympathizers and would probably fail me because I was a Jew. Then somebody mentioned that Professor Metzner was a member of the examination board although he was on the staff of the Technical University. It was known that the Professor was a devout Roman Catholic. At that time, the Catholic Zentrum Party did not support Hitler. Some months before the examination was to be held I asked Professor Metzner for an interview. I told him that I was a mathematician and had also attended a course on mathematical logic. He accepted me as a candidate and gave me some reading to do. On the morning of the oral examination Professor Metzner was delayed. When he finally arrived he was flustered but apologetic. I was the first candidate. He said: "I know you are a mathematician. Would you say that mathematical logic has invalidated the classical work of Aristotle?" I knew what was in his mind. Aristotelian philosophy has played an important part in Catholic theology, and he was afraid that the mathematical approach had undermined Catholic teaching. I replied: "Mathematics offers a new approach to logic; but it has not diminished the importance of Aristotle's work." He was obviously pleased and said: "This is what I have always said. I give you Grade A. Next candidate, please."
It was very fortunate the examination in mathematics could proceed as planned. Schur, who had been dismissed in March of that year, was temporarily reinstated through the intervention of his colleague Erhard Schmidt and was therefore able to conduct the examination. The co-examiner, that is the person who acted as a secretary for the proceedings, was Ludwig Bieberbach. This was unpleasant for Schur and for me; for Bieberbach was an enthusiastic Nazi and appeared wearing his brown uniform. However, it should be said that he did not interfere in any hostile manner. I still remember Schur's first question:
"What, in your opinion, is the most important theorem of the differential calculus?" I said: "Taylor's Theorem," and he replied: "I disagree. Surely, once you have proved Rolle's theorem, the other main theorems can easily be deduced from it." He then went on to algebra. "How do you determine the rank of a matrix?" passing to complex analysis he asked "What is the addition theorem for the Weierstrass elliptic function?" I started to recall the rather complicated formula; but I got stuck and had to admit that I did not remember the last term in the denominator. After a few seconds Schur said: "I don't remember it either." It was then that Bieberbach spoke the only time during the examination; without hesitation he quoted the complete formula.
Schur was pleased that I could cite various details in the theory of conformal mappings. "How is it that you know all this?" he asked, and I replied: "I learned it in Dr Hopf's lectures." Schur had put me at my ease, and the Oral Examination, which I had feared would be an awful ordeal, turned out to be quite a pleasant experience.
This concluded my life as a mathematician in Berlin. I was soon to leave my native city. More than forty years elapsed before I saw Berlin again, when I accompanied my wife on her professional meeting that was held in that city. However, I gratefully remember the foundations for my career that I received from the University of Berlin and the enriching experiences I had when the cultural life of Berlin was at its peak.