Walter Ledermann: Encounters of a Mathematician


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(Childhood in Berlin: 1911 - 1920 )
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(The formative years: Berlin 1928-33 )

Mathematical awakening and musical abundance: Berlin 1920 - 28

Soon after the war The Köllnische Gymnasium was closed down by the new socialist government and turned into an educational establishment for the less privileged members of the community. I became a pupil of the Leibniz Gymnasium. This, too, was a conventional grammar school, in which boys from the ages of nine to eighteen received a secondary education with strong emphasis on the classics. In a bizarre manner the nine classes were numbered in Latin from six to one, in the reverse order, the most junior form being the sixth form, and the three top forms being divided into lower and upper classes, each running for two years. So the names of the nine classes from the bottom upwards were: sexta, quinta, quarta, lower tertia, upper tertia, lower secunda, upper secunda, lower prima, upper prima. Latin was studied for nine years and Greek for six years, each with at least six lessons a week. It is therefore not surprising that little time was left for the more modern aspects of education. However, French was taught from the third year onwards, though mostly for the purpose of reading rather than speaking the language. Despite the strong political antagonism the view was still widely held in Germany that France was the centre of European civilization and that every cultured person should have knowledge of French. Unbelievably, English was almost totally ignored. During the last three years pupils were offered either English or Hebrew, at two lessons per week. The Hebrew option was not intended for the Jewish pupils, of whom there were only a small number, but for those who wanted to become ministers in the Protestant Church; for they were required to know enough Hebrew and Greek to read the Old and the New Testament in their original languages.

In schools of this kind mathematics was usually regarded as a topic of secondary importance. However, the Leibniz Gymnasium was exceptional in this respect: it had taken its name from one of the greatest German mathematicians, and it was considered a matter of honour and pride that a high standard in mathematics should be attained. Geometry was introduced in the third year (the quarta).

When I reached this stage in 1922, there was still a shortage of teachers with academic qualifications because training had been disrupted by the Great War. So, my first mathematics lesson was given by the Arts teacher, a benevolent elderly gentleman called Herr Wüster. In his teaching he closely followed the prescribed text-book which according to custom in those days, was a simplified version of Euclid. After producing the usual definitions of a straight line, parallels and angles, Mr Wüster reached the first substantial theorem namely that in every triangle the sum of the angles equals two right-angles. Although this is not a world-shaking fact, it had the quality of universal and unassailable truth, which impressed me greatly. The statement refers to all triangles whether drawn in the past or in the future. I realized that mathematical results possess an absolute certainty that make them superior to the dogmas of philosophy and religion, which we are expected to accept "without proof." It was there and then that I decided to become a mathematician and I am grateful to Herr Wüster for having kindled the flame of enthusiasm.

Later on two academically trained mathematicians were added to the teaching staff of the Leibniz Gymnasium. One of them was Herr Anders. He had been a naval officer in the War and conducted his lessons with military precision and efficiency. He demanded strict discipline. When the bell rang at the end of the lesson he called out in a clipped voice "stand up -- class dismissed." Anders was an ardent nationalist and later supported the Nazis.

But he was not unfair to me and even, at my father's request, supplied a six-line testimonial stating that my performance in mathematics had been "very good." Shortly after my emigration to Britain a former class mate spoke to Anders and, incidentally, expressed regret that I had left the country, to which Anders replied curtly "when you plane down wood, there will be shavings."

The other mathematician was Herr Satow. He was a good mathematician of flamboyant temperament. He boasted that he had studied very advanced mathematics at the University, including "elliptic functions" -- but he never explained to us what they were. I enjoyed Satow's lessons, they were both instructive and entertaining. His real passions were mountaineering and the music of Wagner and Richard Strauss. He was fond of using various quotations from mountaineering and the operas of Richard Wagner and Richard Strauss (he never mentioned Mozart or Verdi). His lessons were interspersed with accounts of his latest exploits in the Austrian Alps or with lengthy quotations from Wagner's Ring der Nibelungen. "Boys, what was the most important event of the year 1911?" Answer: "The first performance of Der Rosenkavalier." On the whole, I benefited from Satow's lessons and I am grateful that I received a good mathematical education at the Leibniz Gymnasium, true to the spirit of its patron.

Although mathematics was my main interest, I also derived some enjoyment from studying the classics, especially Greek when the teacher guided us through some of the original works of Homer, Sophocles and Plato. At one stage we were told to select our own project, and I presented Euclid's proof of Pythagoras's theorem in the original language.

There were only a few Jewish pupils at the Leibniz Gymnasium, it is curious that there were some Jewish teachers at the school, two of whom had a strong influence on my education; Herr Moritz Baum taught me Latin for several years and at one time was our 'Form Master' (Klassen Lehrer); he was a very efficient teacher. Herr Bruno Strauss, who emigrated to the United States, taught German literature and impressed me with his original approach and insight.

On the whole I received a good general education at the Leibniz Gymnasium. Since I received good reports both in the classics and in mathematics, I was allowed to skip six months in the top form. I was therefore awarded the leaving certificate at an earlier age than would normally be the case. Consequently, I was able to complete my university course just before the Nazis would have forced me to abandon my studies.

I encountered no anti-Semitic aggression at school. On the contrary, I was on good terms with all my class mates. They appointed me to be their representative ("Speaker") when they wanted to make some requests to the teacher. Some of the hefty Nordic lads, quite voluntarily and without being asked by me, undertook to be a body-guard for me. I remember one occasion, when a boy from another class tried to bully me in the play ground. He was immediately set upon by my protectors and beaten up.

My school years 1920-1928 were a period of much political unrest and tribulation in Germany. Soon after the end of the war there were clashes between the army and revolutionary groups who tried to establish a soviet-style regime. The Leibniz Gymnasium was within walking distance from my home. I remember some days when on my way to school I heard shouts "clear the streets" and I ran for shelter in a near-by house, before the army sprayed the street with machine gun fire in order to drive back its opponents. For some time I was haunted by the death of Rosa Luxemburg, the communist revolutionary. She was murdered by army officers and her body was thrown into the canal that traverses Berlin. The swimming bath, to which my father took us every day in the summer, was situated at the same canal, and I was afraid that the body of the dead woman would come up just when I was doing my exercises in the water. Atrocities committed by right-wing fanatics were not confined to communists. In June 1922 Walther Rathenau, the German foreign minister, was assassinated. "This is the beginning of the end," as my father prophetically remarked when he heard the news. It was no coincidence that both Luxenburg and Rathenau were Jews. The people who committed these crimes would soon form the nucleus of the Nazi party. In fact, it was in the following year that Hitler was involved in the unsuccessful violent attempt to overthrow the Bavarian government. The political situation rapidly deteriorated. The French occupied the Ruhr district, where much of Germany's industry was situated. This drastic action was taken ostensibly because Germany had failed to fulfil its obligation for restitution but it greatly increased the anger and the hatred towards the French caused by what the Germans regarded as deliberate humiliation. I remember one morning when our modern language teacher, who had lost his right arm in the war, was asked by the headmaster to take our class into the school hall for an assembly; I hard him mutter: "I would much rather lead you into battle against the French." (We were then 14 years old.) About that time R. Weizenböck published his important treatise on The Theory of Invariants. If one underlines the first letter of each sentence in the Preface one would compose the words "Nieder mit den Franzosen." [Down with the French!]

Of all the political calamities that befell Germany in those years it is the run-away inflation of 1922/3 that left me with the most vivid and poignant memories. The value of the currency declined so rapidly that the exchange rate of the American dollar was officially quoted twice a day and displayed in shop windows. Wages had to be paid each day as a weekly wage packet would have lost most of its value at the end of the week. The Reichsbank [State Bank] printed bank notes day and night, sometimes with quite unsuitable designs; for example in December 1922 it was intended to put a 1,000 Mark note into circulation, with the picture of a historical person engraved upon it. By the time the note was ready to go into production it had lost virtually all its value. So the Bank had printed across its front and back in large red letters: Eine Milliarde Mark [A thousand million Mark]. When finally a new currency was introduced, the exchange rate was 1 New Mark = 1,000,000,000,000 Old Mark. My father, as a patriotic German, had put all his savings into Government stocks and shares. After the new currency was introduced, he received a letter from his bank saying: "Dear doctor, we have for safe keeping all your bonds and shares. If you will kindly come to this office, we shall be pleased to hand them to you. We regret that we cannot send them to you by post, since the cost of the postage stamps would exceed the value of your portfolio." So, after having worked as a doctor for twenty-five years, he had become penniless and had to start again when he was fifty-two years old and a father of four children.

The financial chaos caused hardship to many professional people who were forced to look for ways of supplementing their income. I was then eleven years old and I had recently started playing the violin. My parents wanted to find a good teacher for me. By chance my mother had heard that one of the senior professors at the Academy of Music, who would normally only train professional musicians, was willing to give private lessons. My mother and I were received for an interview. I was rather frightened at this grumpy old man with a white beard. However he agreed to accept me as a pupil. When my mother asked him what his fee was, he replied: "There is no point in mentioning a figure. Pay me the equivalent of a loaf of bread for each lesson." Before I went for my first lesson, my mother sent me to the baker's shop in order to ask what the price of a loaf was on that day. I have forgotten the exact figure, perhaps it was 35,000 Mark. My mother put the money into an envelope and sent me off. I did not enjoy the lesson. The professor was very harsh and shouted at me: "You play like a street urchin." (To this day I have not discovered how a street urchin plays the violin.) After the lesson I handed him the envelope containing all the bank notes which would buy a loaf of bread on this day. He counted the money and after a few moments said: "Tell your mother that I am an old man and my stomach is not very good. I can only digest white bread." (This was considered to be a luxury by most people.) Fortunately, the lessons with the professor came to an end after the inflation was terminated by the introduction of a new and stable currency.

I was transferred to another teacher, Herr Gehwald. He was very kind and helpful and I enjoyed his lessons. For more than thirty years Herr Gehwald was sub-leader of the orchestra of the State Opera (formerly Royal Opera) in Berlin. He possessed a beautiful Amati violin which was too precious to be taken on public transport. Therefore he had another violin which was quite nice but much less valuable, and this violin was left permanently in the orchestra pit and he used it on all rehearsals and performances. When Herr Gehwald retired, he sold this violin to my father. I played on it for many years: it now belongs to my granddaughter Tamara.

There was a great deal of political unrest. By the 1930's the political situation was deteriorating rapidly. A large part of the population had lost trust in the liberal Weimar Republic and yearned for a drastic change. Many people, especially among the younger generation, and also some intellectuals embraced communism in the belief that this system would eliminate unemployment and bring about a more just distribution of wealth. Others and, I think, it was perhaps the majority of the German people, still bore deep resentment against the "disgraceful" peace treaty of Versailles which ended the First World War. It was indeed a fateful error on the part of some Western politicians to place humiliating conditions many years after the war. For example, as late as 1924 Germans were excluded from international meetings like the International Congress of Mathematicians. It is not surprising that many Germans gave up hope that the "spirit of Versailles" could be changed by peaceful means. They joined the Nazi Party because Hitler had vowed that he would restore Germany's pride. Both the communists and the Nazis wished to destroy the Weimar Republic and to that extent, ironically, they had a common aim. I remember during a strike of the Berlin Underground Railway, two pickets guarded the entry to the Underground Station from which I normally travelled to the University. One man wore a swastika on his uniform and the other carried a red flag embossed with hammer and sickle. A sinister precursor of the Molitov-Ribbentrop Pact which, about seven years later gave the starting signal to the Second World War.

Despite the calamities in politics and economics, life in Berlin in the 1920's and early 1930's was not totally blighted by gloom and despondency. The flowering of the arts during that period was perhaps unsurpassed in the history of the city. Numerous theatres offered excellent performances both of the classics and of modern plays, some of which had a somewhat revolutionary and daring tinge. Berlin has always been well supplied with museums and art galleries. I remember attending some highly illuminating lectures on classical and on modern art. But my most profound and lasting experiences were provided by the abundance and variety of first-rate musical performances which I was privileged to attend. There were three first class opera houses. Their musical directors were Erich Kleiber, Otto Klemperer and Bruno Walter respectively. Moreover, there was the superb Berlin Philharmonic Orchestra conducted in turn by Wilhelm Furtwängler and Bruno Walter. Some of the most memorable recitals which I attended were Arthur Schnabel's cycle of all the Beethoven Sonatas.

Initially my love of music was almost exclusively directed towards opera. My mother's father had a beautiful gramophone cabinet; its upper part contained the turntable, which had to be wound up with a handle; the lower part was reserved for the collection of records, almost all them operatic, predominately by Wagner. My grandfather must have observed that I was fascinated when he played a record for me during one of our visits. I cannot have been more than seven years old then because he died before I had reached the age of eight. He stipulated in his will that I should inherit the gramophone cabinet and all the records. Shortly after his death I went down with scarlet fever and had to be kept in strict isolation from the other children. But his gramophone cabinet stood by my bedside and was a great consolation for me. I had asked my parents to give me the librettos of Wagner's operas. By the end of my illness I knew large parts of Der Ring der Nibelungen by heart.

I was about twelve years old when I started to play piano trios with two school friends and I also developed a keen interest in orchestral and choral music. It was very fortunate for me that we had a personal connection with the Berlin Philharmonic: the porter who served this celebrated group of musicians was Franz Jastrau. As a boy he had been a patient of my father and he had retained a deep love and respect for his "uncle doctor." Jastrau had no interest in music; but he was an indispensable factotum for the members of the orchestra and he was usually allowed to claim two complimentary tickets for the "public rehearsal" which was held on a Sunday morning prior to the official concert by the Berlin Philharmonic Orchestra which took place on Monday evening. Invariably Jastrau gave these two tickets to my father. My mother did not like to go out on a Sunday morning; so I had the great privilege and pleasure to attend these wonderful concerts with my father. Although they were called rehearsals, they were in fact polished performances, conducted in turn by Bruno Walter and Furtwängler. They played an invaluable part in my musical education. I remember only one occasion which caused me disappointment and frustration. A concert had been arranged at which Rachmaninov was going to play the solo part of his latest piano concerto accompanied by the Berlin Philharmonic Orchestra conducted by Furtwängler. The tickets had been sold out for months; but Jastrau was confident that he would obtain admission for us. However, my father was unable to come, as he had to attend to a patient. So I went alone to the Philharmonic Hall and patiently waited at the agreed place near the artists door in the hope that Furtwängler's secretary would come with the ticket. The second bell had gone and there was no sight of her. Finally she approached holding a ticket. But before I could take it from her, a tall man behind me snatched it out of her hand. I looked around to see who this evildoer was and I recognized Richard Strauss, who had come all the way from Bavaria to attend this concert. The concert began almost immediately. Crestfallen I made my way home where I arrived while the rest of the family were having their Sunday lunch. My mother said: "Why are you not at the concert?" to which I replied meekly: "Richard Strauss has taken my ticket."

For many years since my adolescence my chief interest in music remained opera. I hardly ever went to see plays at a theatre since a drama was for me simply an "opera without music." The State Opera is a historical building in the centre of Berlin. It used to be called the Royal Opera House. The kings of Prussia and, later, the Emperor of Germany enjoyed operatic performances. A large space in the dress-circle was reserved for the royal visitors. In my time, the musical director of the State Opera was the eminent conductor Erich Kleiber. He was supported by world famous conductors including Leo Blech and Georg Szell. My father had a season ticked for the State Opera which provided him with two tickets for very good seats in the second circle. Sometimes, when my mother did not feel inclined to go out in the evening, my father would take me with him and we enjoyed the opera together. It was easy to get cheap tickets for the gallery. The booking office opened on Sunday morning and tickets were sold only for performances during the following week. Quite often I would take an early underground train on a Sunday morning and join the queue of opera enthusiasts and connoisseurs who were discussing the merits of the forthcoming performances before the box office opened. I had a great passion for Wagner's music which I shared with many people in Germany. When I was still at school there was a year during which I had heard Die Meistersinger fourteen times. On a certain Sunday morning I was in the queue at the box office in order to buy a ticket for Tristan und Isolde. Dr Hempel, who taught German literature at the Leibniz Gymnasium, also stood in the queue. He asked me what opera I wanted to hear. When I told him he asked: "How often have you heard Tristan und Isolde?" I said: "Four times." He replied: "I shall also buy a ticket for this opera. I have heard it one hundred and five times so far." In the West of Berlin was the Deutsche Oper (German Opera) which was maintained by the city of Berlin. Its musical director was Bruno Walter. I remember some beautiful performances directed by him, especially of operas by Mozart. Somewhat later a third opera house was established, again supported by the State of Prussia. It was situated in the centre of Berlin. It was called the Kroll Oper. Its director was Otto Klemperer. Thus three operatic performances of the highest standard were offered every evening except during the summer months and on the few occasions when the excellent orchestra of the opera house performed a symphony concert on the stage. A music lover was indeed fortunate to live in Berlin in the 1920's and 1930's. It is odd that despite their insistence on musical perfection, all operas were sung in German. As long as I lived in Germany I never heard an opera in Italian or in French. It was absurd that some of the leading singers at the State Opera like Violetta de Strozzi and Tino Patiera had to sing Puccini's operas in German. I can recall only one exception: the great Russian bass Shaliapin gave a small number of guest performances at the State Opera. I attended Boris Gudenov, in which he sang the title part magnificently in Russian while the rest of the cast harangued him in German.

My predilection for Wagner's music persisted into my student years. In 1931, after I had earned a little money by coaching school pupils, I made a "pilgrimage" to Bayreuth. It was only two years before Hitler came to power. There was an exaggerated veneration of Wagner throughout the little town of Bayreuth, some of it in bad taste: when souvenir shops sold vessels made of red glass to represent the Holy Grail which was believed to contain the blood of Christ. Wagner's house "Wahnfried" was open to visitors. Cosima Wagneer, who had vigorously pursued the anti-Semitic tradition of Richard Wagner, had died, and Jewish visitors were no longer denied entrance. I went into the house and saw the grand piano and the open score on it, from which Wagner requested the (Jewish) resident pianist to play to him. I also went into the garden and paid my respect at Wagner's grave. But the atmosphere was tense and rather unpleasant. My ticket was for this evening's performance of Tannhäuser, to be conducted by Toscanini. However, there was a rumour going around to the effect that the performance may have to be cancelled because at a rehearsal at that morning Toscanini had quarrelled with the orchestra and declared that he would not conduct any more. But when I arrived at the Festival Theatre in the afternoon, I was told that all would be well, because Winifred Wagner (Richard Wagner's English daughter-in-law) had begged the maestro literally on her knees (so it was said) not to carry out his threat. Even, if he had done so, it might not have been disastrous, because Furtwängler sat behind me in the audience and would perhaps have taken over the direction. In the event it was a wonderful experience to hear Toscanini conducting the splendid orchestra, which in Bayreuth is placed underneath the stage and produces a magical sound. Unfortunately, the soloists were not all of the first class. I had heard better singing at the Berlin State Opera. On the next day I heard Parsifal, also beautifully conducted by Toscanini.

I did not stay on at Bayreuth but I travelled around Bavaria for a few more days. I felt that I was rather a stranger there, perhaps on account of Berlin accent. I was amused by the beer drinking habits of the Bavarian people. When sitting down for lunch at a restaurant I was asked by the waiter not "Do you wish to have a glass of beer with your lunch, Sir?" but "What kind of beer would you like to have with your meal?"

When I arrived at Munich I discovered that the German air line Lufthansa offered reduced fares to students at the prices of a third class rail ticket provided that they produced a registration document from a German university. As I did not carry such a document with me I telephoned the registry office of the university of Berlin and asked them to send by express a copy of the registration document to be collected at the main post office in Munich, which was kept open all night. The paper arrived at midnight, and I was able to claim my student ticket at the airport the next day on my return to Berlin. I was told that concession tickets would be issued only as far as the next stop on the way to Berlin, which was Leipzig, and that I would have to apply again for a ticket to the next stop. I readily agreed to this condition and was excited to become airborne, the first member of my family to fly. Unhappily, it was nearly also my last flight, because a passenger tried to commit suicide during the flight by locking himself in the lavatory and attempting to blow up the aeroplane by setting light to the fuel tank. Fortunately, the action was discovered in time by the cabin staff. The man was arrested and the plane landed safely at Leipzig and I continued my journey to Berlin.

Although, generally, the political atmosphere in Germany in the 1920's was gloomy, there were a few bright spots. One of these was an attempt on the part of the League of Nations to foster the reconciliation of the French and German nation by arranging an exchange of school children who would live in each other's family during the summer vacation. I had always been interested in foreign travel and I offered to take part in this exchange. The organizers of this project accepted me as a candidate; they evidently regarded me as a typical German school boy. They could not find the family of a French doctor to match my father's profession. Instead they proposed that I should accept the invitation of Monsieur Ploussard, a pharmacist at Châlons-sur-Marne, a town in the Eastern part of France. He had a son, Pierre, who was fifteen years old, as I was. So one day in the summer of 1926 I made my way into France. I interrupted the journey at Saarbrücken, where I stayed for one night with a distant relative; Saarbrücken was then a German town. On the next day I crossed into France and stopped for a few hours at Metz, a historical town which was then French but had previously been German. After some sight-seeing I wanted to go back to the railway station. I asked a man for direction to the station in what I thought was acceptable French. However, to my shame, he answered me in German. Finally, I arrived at Châlons-sur-Marne. Monsieur Ploussard met me at the train. He was very friendly and took me to his home, which was a large and comfortable apartment situated above the pharmacy. I was introduced to the family: Madame Plousard and their children, Pierre was their eldest child; he had a sister and a brother. Pierre had been studying German at school for some time and was keen to improve his knowledge of the language. Of course, at home only French was spoken. Madame Ploussard wanted to make sure that I should benefit as much as possible from the conversation at meal time. When she suspected that I had not understood the trend of what had been said, she called out: "Pierre, explique!"

Since in France the schools break up later for their summer holiday than in Germany, I accompanied Pierre to school during the first week of my visit. It was interesting for me to attend lessons in mathematics and Latin in French; but I scored a notable success in the German lesson when the teacher asked me to read aloud a poem, which I was able to do with a better accent than either he or his pupils could achieve.

During the school holidays Pierre and I made many excursions on bicycles. We explored the beautiful countryside and visited some places of historical interest. But eight years after the end of the Great War the grim reminders of the horrendous events were still visible. The battle of the Marne was one of the turning points of the war, where the German army was halted and Paris was saved. But the loss of lives was enormous. We passed military cemeteries which contained around twenty thousand graves; those of allied soldiers were marked by white crosses, and behind a hedge there were black crosses over the graves of German soldiers. I felt that it would take a long time for France to recover from the physical and spiritual damage inflicted upon it. Seven years later, when I was desperately looking for a country where I could find refuge from Nazi Germany, I decided that it would not be safe to go to France as it would not be strong enough to withstand Hitler's onslaught.

My stay in France was a valuable experience. Although I was there for only five weeks, I picked up the language quite well. Pierre was a good friend. He came to live with my family for a similar period later that summer. We got on very well and his knowledge of German made good progress.

After the war I got into touch with Pierre again. He had been a prisoner of war for a time, but was exchanged fairly soon with a German doctor who had been captured by the allies. Pierre's father had died and Pierre had taken over the pharmacy at Châlons-sur-Marne. Many years later when travelling through France during our summer holiday we stopped for a night at Châlons-sur Marne. I was pleased to introduce Rushi and Jonathan to Pierre, who entertained us to a luxurious French dinner, sadly, this was the last time I saw him. When we stopped again at a subsequent visit to Châlons-sur -Marne we were told that Pierre had been killed in a road accident a few weeks before.


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JOC/EFR May 2009