Reminiscences of an Octogenarian Mathematician
L J Mordell,
St John's College,
It is customary for the fellows of St John's College, Cambridge, to dine privately on December 27, the birthday of St John, the Evangelist. The Master proposes a toast to those fellows who have attained the age of eighty since the preceding December 27, and asks each of them to give a talk. As I became eighty on January 28, 1968, it was my turn to do so.
I started off by saying that this was a really great occasion in my life and that I was very grateful to our College for making it possible. I said that it was not an easy matter to make an appropriate speech on such an occasion. Fortunately it was not too difficult for me to do so, as I have recently been reading a book by the well-known and popular American author Dale Carnegie, entitled How to Slop Worrying and Start Living. In this, he makes the cogent remark that no man is so happy as when he is talking about himself. He says nothing about the feelings of his listeners.
There are two reasons why I propose to make myself thoroughly and unashamedly happy by talking about myself. The first is that on several occasions, both in England and America, I have been told that I am a legendary character. As it occurs to me that most legendary characters, for example King Arthur, are dead, I wish to show that I have actually existed and am very much alive, and so I shall give some account of the subject so that there will be no doubt about the matter.
The second reason is that there have been many stories, mostly apocryphal, as to how I, a natural born American, came to study at St John's College. The reason is a very simple and natural one. I do not mean to be boastful or vainglorious, and I wish to apologize if I seem so and to crave your indulgence. Though one has ups and downs, I have on the whole been very fortunate, and many people, far cleverer than I, have not been so lucky.
I was born in Philadelphia on January 28, 1888, and as a child went to what are called the public schools, that is, the state schools. From the age of 6 to 10, one went to the Primary School, from age 10 to 14 to the Grammar School as it was called, and from age 14 to 18 to the High School. In the Grammar School, I think I was good at arithmetic, but certainly was not in later life. One of my classmates there, now Professor F C Dietz, is Professor Emeritus of History at the University of Illinois at Urbana; I saw him in December last when I lectured there. I mentioned this to Professor Paul T Bateman, who is head of the Mathematical Department there, and he countered by saying that one of his early classmates had been electrocuted for murder.
But talking about crime reminds me that in a visit to Professor Jacques Hadamard in Paris while I was Professor at Manchester, I said I had once given a lecture at Strangeways Prison, Manchester. One reason for doing so was that one cannot leave a prison at a time one wishes to. He said to me that as editor of a mathematical journal, he received rather good papers from someone unknown to him, so he invited him to dinner. His correspondent wrote that owing to circumstances beyond his control, he could not accept the invitation, but he invited Hadamard to visit him. Hadamard did so and found to his great surprise that his author was confined to a criminal lunatic asylum. Apparently he was quite sane except for the murder of his aunts. His name was André Bloch, and he was a very good mathematician.
The Central High School, as it was called, was a well-known and comparatively old school. It is best described by saying that it corresponded to a good English grammar school except that it did not specialize in any subjects. The courses, however, in both arts and science were adequate for admission into any University. At the time I entered it, new buildings had been erected, and these were formally opened by President Theodore Roosevelt.
I don't think there was much mathematical background in our family. It is true that my father, who was a Hebrew scholar, had written a learned monograph, On the origin of letters and numerals according to the Sefer Yetzirah, but this had no influence on me.
Professor Winston Thomas, Professor Emeritus of Hebrew here, who dined in John's last week, reminded me that I was one of a committee at Manchester University that had interviewed him for the chair of Hebrew there. He said that I asked him if he knew the booklet, and he said no.
About the age of 14 before entering High School, I came across some old algebra books in the 5 and 10 cent counters of Leary's famous bookstore in Philadelphia, and for some strange reason the subject appealed to me.
One of these books was A Treatise on Algebra by Charles William Hackley, who was professor 1843-61 at what was then called Columbia College, New York. It is of interest that Sylvester was one of the candidates for the chair. My copy is the third edition, dated 1849 (the first was in 1846). It was really a good book, though not rigorous, and contained a great deal of material including the theory of equations, series, and a chapter on the theory of numbers. It, like the old algebras of those days, had a chapter on Diophantine analysis, a subject I found most attractive. It is not without interest that in later years much of my best research deals with this. In fact I have just written a book on the subject which appeared in 1969.
I began, however, to read more modern books, such as the algebras by Henry Sinclair Hall and Samuel Rratcliffe Knight, Charles Smith, and George Chrystal; the trigonometry by Ernest Hobson, and the coordinate geometry by Sidney Luxton Loney and Charles Smith. My father wished to get some idea of what progress I was making and got in touch with a Professor Isaac Joachim Schwatt at the University of Pennsylvania. He was the author of a book on college algebra and also one on operations with series. He very kindly agreed to give me a test paper on algebra which I took. I seemed to have impressed him, for he wrote on January 24, 1904 when I was 16 years old:
From the examination I have given your son in algebra, I am led to believe that he possesses more than ordinary ability for the study of the subject. I should not be surprised if with proper training he should turn out to be a man who will advance the science of mathematics.All I remember about the examination is that there was a question on Charles-François Sturm's theorem about equations, which I could not do then and cannot do now. Reading Schwatt's remarks now objectively, I do not think there was sufficient evidence for his statement; I think he was too flattering. Many British school boys would have been far cleverer than I. However, his prophecy seems to have been fulfilled, and also in another material way which no one could have foreseen.
The examples in the books I read were taken from scholarship examinations in Cambridge and the Mathematical Tripos. I naturally became rather curious about Cambridge and made inquiries. I got the Scholarship papers for some recent years and worked upon the questions. I finally conceived what I can only describe as a thoroughly mad and crazy idea of going to Cambridge and trying for a scholarship. I had no idea of the necessary standards, I was self-taught mathematically and had never participated in a competitive examination. A judge, whom my father knew and to whom he spoke about my plans, said: "The damn fool. Are not the American Universities good enough for him?" I think his appellation was justified.
From my point of view, the fact that one could specialize in mathematics and not do a great many subjects, as in American Universities, was a strong factor in favour of Cambridge.
Anyhow, my people had confidence in me and agreed to allow me to enter for a scholarship. There were two major groups, the Trinity group and the John's group. I selected the John's group because I thought there were more Colleges in its group, and so my chances of getting something would be better.
I went to Cambridge and sat for the scholarship examination in December 1906. I did not think I had done too badly, but of course I had no idea how I had done compared with other candidates. On Saturday, December 15, 1906, I wrote to my father as follows: "On the Friday, I went to see Mr Bushe-Fox [Loftus Henry Kendal Bushe-Fox], the college tutor. As soon as he saw me, he shook hands with me and congratulated me. I asked him what it was. He said I had done very well, that I had done better than any of the 100 or so competitors, and that I was way ahead of the second man. I certainly did feel fine and let out a yell."
So as I have said in the beginning of my talk, there was a very simple explanation of why I went to Cambridge. A Simon Muhr scholarship from my school helped me.
After I arrived in Cambridge, in October, 1907, I had to pass the Little Go. The subjects of the examination were Greek and Latin, an English set book, elementary algebra and geometry (for all of which my High School training was adequate), and also Paley's evidences of Christianity. However, one could substitute William Jevon's Logic for this, as I did.
Undergraduates lived like real gentlemen in my time. A gyp and a bedder were always on hand to lay a fire, set and clear the table for breakfast, lunch, and tea, prepare one's bath in a round tub, trim the lamp, etc. There was a boots to polish one's shoes and to deal with baggage. The College kitchen was open all day and would even bring tea to a punt in the backs.
Money really meant something solid in those days. A shave in a first class establishment cost 6d, but only 4d in the ordinary ones. Tobacco had gone up from 51/2d an ounce to 7d, whisky (not that I drank any) was 3/6 a bottle, a half day excursion to London cost 3s 9d. Dinner in hall was 2s 1d, the 1d being for the serviette.
I did not participate in any athletic activities; perhaps it would have done me good to have done so. In this connection, one does not know what the future has in store for one. Believe it or not, I became a rock climber in a mild sort of way. This was very strange, for the first hill that I went up in the Lake District was Helm Crag, and this seemed to me about 50 degrees beyond the perpendicular. But in 1925, I went for what I thought was a walking holiday in Scotland with the now eminent Sir Robert Robinson. We came to a mountain, Buchaille Etive in Glencoe, and after walking up a little distance he said, "We now put on the rope." We did the North Wall Chasm climb, and this I enjoyed very much.
So for diving: at the age of 28, I had the greatest difficulty in mustering up enough courage to dive from a 5 ft board. But when I was at Manchester, where they had a modern swimming pool, I was looked upon as a great man, not for so trivial a reason as being an F.R.S., but because I used to dive off a five-metre board.
I attribute my present good health to my outdoor activities.
In those days John's provided the mathematical courses for its students. There was Mr Webb [Robert Rumsey Webb], a very successful coach, who lectured in applied mathematics, He was a good teacher and an amusing personality. He used to say, "pp > jj" (plodding patience is greater than jumping genius). He could be very sarcastic, and would say to a student, "Write all you know on this piece of paper" - something about an inch square.
There was Dr Thomas Bromwich, who had just come back to Cambridge from Galway. He had just written his book on infinite series of which he was very proud. Finally, Dr Henry Baker was the director of studies.
Dr Baker was my supervisor. He was a geometer and tried to make me one, but did not succeed. He sent me in my first term to Herbert Richmond's lectures on higher plane curves, and to Andrew Forsyth's on differential geometry. I soon dropped them; I don't think he approved of me - I did not go to many lectures. I remember going to one of his, and he stood behind me and commented upon the fact that I wrote slantingly up the page. I thought, but did not say, that it was none of his business. I found him perhaps distant or unsympathetic. I should have been more fortunate if G H Hardy had been my director of studies. He would have known how to deal with a self-educated mathematician, who rather unwisely did not appreciate the advantages of lectures. He would have taught me worldly wisdom and the ways of the world as well as mathematics from a modern point of view. This would have been very important as the older books which I had first come across were lacking in rigour.
There was a great deal of discussion about the Mathematical Tripos in those days, and proposals were put forward for reforming it. The examination papers still contained a question based on Newton's methods of more than two centuries ago. Great emphasis was placed on being able to do all kinds of tricky questions, and no doubt this was useful for developing one's powers. It was perhaps not sufficiently realized that one of the most important things a mathematician could do was to advance the science of mathematics by producing new and important results. Hardy, my predecessor in the Sadleirian chair, took an active part in the reform. As a result of his efforts, the year 1909 was the last year for which there was an order of merit for the Mathematical Tripos. The first person was called the Senior Wrangler, and the speculation as to who he would be reminded one of the Derby. The tripos could be taken after two or three years, and I took my examination after two years. I was considered a strong candidate, but I blotted my copy book and was only third wrangler. I think I could have done better.
The Senior Wrangler was Percy John Daniell, who became Professor at Sheffield. The second one was Eric Harold Neville, who became Professor at Reading, and I was third and became Professor at Manchester and Cambridge. There were three bracketed fourth: Edward Hodgson Berwick, Professor at Bangor, Sir Charles Darwin, Professor at Edinburgh and Master of Christ's College, and George Henry Livens, Professor at Cardiff. I am the only survivor of this group, so according to the laws of statistical averages, I should live to a ripe old age.
After the Tripos, the real mathematicians took Part 3 of the Tripos. There was usually a fourth year spent on research. It was then that I took up the study of number theory in earnest. There was then no Ph.D. degree; it came in only after the First World War, so the present day mathematicians are much more learned than we were. There were two Smith Prizes in those days for which B.A.'s could compete. Neville got the first one, and I the second for an essay on The Diophantine equation y2 = x3 + k, a topic which has played a prominent part in my research even in the very latest years. I might mention that my Cambridge inaugural lecture, "A Chapter in the Theory of Numbers," dealt with this topic.
I continued my studies, staying on in Cambridge, and wrote another paper entitled, Indeterminate equations of the 3rd and 4th degrees. I was very unfortunate with this paper. It was rejected by the London Mathematical Society; I really don't know why. Perhaps they did not approve of my style, but it was a really important paper and has played a prominent part in the progress of number theory even in the present day.
I hope you will bear with me if I mention one of my results. I had proved that the integer solutions of the equation y2 = ax3 + bx2 + cx + d could be found from the representations of unity by binary quartics. Neither I nor the referees were aware that in 1909 Axel Thue had proved there could be only a finite number of representations. This meant that the cubic had only a finite number of solutions, a really important result. Some years later, Professor Carl Ludwig Siegel, one of the world's foremost mathematicians, generalized this result and communicated his results to me. I asked him whether he would not object to this being published by the London Mathematical Society. He did not reply and so I took it for granted that he did not object. Proof sheets were sent to him, and he was then very annoyed because the mathematicians of Frankfort had agreed not to publish anything for a few years. However, he agreed to let it appear anonymously as due to X. When I saw him a few years ago, he said it need no longer be anonymous.
Later in 1928 at the International Mathematical Congress held at Strasburg, Leonard Eugene Dickson, the eminent authority on number theory, included some of my results in one of three topics which he discussed. In very recent years, Dr Baker of Trinity College has used my results to find a bound for all the solutions, an outstanding accomplishment which one would have hardly thought possible.
Number theory was little studied in Britain in those days. There were only Professor George Ballard Mathews at Bangor, who wrote a book on number-theory, and John Hilton Grace of Peterhouse, who wrote a few papers on the subject. Anyhow, my work did not seem to have been appreciated.
I submitted this work and others for a fellowship, but was unsuccessful. My tutor, Mr Bushe-Fox, said I had not played my cards very well. A geometer, a protégé of Baker, was elected. I don't think he produced very much afterwards.
However, I was not disheartened. I said to myself, I would show the blighters!
I stayed on a few years doing research and some private tutoring. In 1912, an international mathematical congress was held in Cambridge, and I attended it. I am fully aware of the implications of the story I am going to tell. I went into the buffet room where all the distinguished mathematicians were gathered, and I thought to myself, "What an odd looking lot they are." I have no doubt that History repeats itself.
About 1912, I applied for a post at University College, Reading. The salary at such small places was 120 pounds per annum. At larger places, e.g., University College and Kings College, London, the salary was 150 pounds. Three of us were interviewed, and Professor Bowley [Arthur Lyon Bowley], a statistician who was head of the department, appointed a Scotch football player. It is the irony of fate that many years afterwards, I was on the selection Committee of the Royal Society when he was a Candidate. I did not hold his choice against him.
In 1913, I was appointed a lecturer at Birkbeck College in London. There I stayed some seven years, except for some 21/2 years during the war in the statistical department of the Ministry of Munitions. I had during all this time continued my studies and researches, and I began to gain recognition. As my stature increased, my thoughts turned to professorships, and around 1919 and 1920, I applied unsuccessfully for two. In 1920, I decided that a change of scene would be welcome, and I applied for a lectureship at the Manchester College of Technology. In the interview, they said that it seemed that London was running after me for a chair. Not quite, I said, I was doing the running. I was appointed, but did not expect to stay for more than two or three years.
I became a professor at Manchester University in 1923 and a Fellow of the Royal Society in 1924. I was very fortunate in having as my colleague Professor S Chapman, with whom I became very friendly and from whom I learned a great deal about worldly matters and how to run a department. Fortune was kind to me, and in later years I gathered around myself some brilliant young mathematicians as members of my staff or research students. There were Professor Harold Davenport, F.R.S., now at Cambridge (but who recently died), Professor Kurt Mahler, F.R.S., Professor at Manchester, Canberra, and Ohio State University, and Dr Paul Erdös, Professor at the Hungarian Academy of Science, all of whom have acquired world wide reputations.
Professor Erdős and I recently had a little competition about the number of universities at which we had lectured. I with 170 was slightly ahead of him, but of course he will soon overtake me.
It is not often that such a brilliant young trio could be found anywhere. We had also Professor Beniamino Segre, an Italian emigré, now President of the Lincei Academy at Rome, and Hans Heilbronn, F.R.S., Professor at Bristol and Toronto. It is not surprising that mathematics flourished and that the Manchester School became well known. As a result, I shone with a great deal of reflected glory.
In 1945, I was elected to the Sadleirian Chair of Pure Mathematics here in Cambridge, in succession to Professor Hardy. When I was asked if I was glad to leave Manchester, I said I was sorry to leave Manchester and glad to go to Cambridge. I could not help recalling that Berwick, who was at Leeds and appointed to a chair at Bangor, said at a meeting how glad he was to leave Leeds.
I was very fortunate again at Cambridge in having some very bright students. This was perhaps the beginning of the new number theory school here, now one of the best in the world under the leadership of (the late) Professor Davenport and Professor J W S Cassels, both of whom I am proud to say were my former students.
During my eight years as Sadleirian Professor, I did much lecturing abroad, though I took only one term off. It was after retiring in 1953 that I really did a lot of travelling. In fact I have been Visiting Professor for either a term or a year at some twelve Universities, namely, Chicago, Pennsylvania. Colorado, Arizona, Notre Dame, Illinois, Catholic University at Washington - all in the U.S.A., and at Toronto, Mt Allison, and Waterloo in Canada, and Ghana and Nigeria in West Africa. I have now lectured at some 190 Universities and institutions, including practically all the important Universities in the U.S.A. and in all the countries of Europe except Russia, Bulgaria, Portugal, and Greece, and at some seven Universities in India, Khartoum, Uganda, and West Africa.
It is difficult to escape hearing me lecture, and at many Universities I meet people who have heard me lecture elsewhere. In a lecture at the University of North Carolina, there were four people, one of whom heard me lecture in Calcutta, another in Berlin, a third in Toronto, and a fourth in Chicago. Yesterday, March 4, 1969, I was introduced to the American speaker, Professor Martin Davis, at the number theory seminar. I said I thought I had not met him before. Oh no, he said, he met me at a lecture I gave at New York University. I usually lecture upon my recent research. Number theory has the great advantage that it is not difficult to give some idea of the subject to a general mathematical audience.
It is customary in most after-dinner talks to introduce some irrelevant stories. I have a large stack of stories, but as these arose in the course of my travels, it will not be out of place here if I give a few.
In 1923, I attended a meeting of the American Mathematical Society held at Vassar College in New York State. Some one called George Yuri Rainich from the University of Michigan at Ann Arbor, gave a talk upon the class number of quadratic fields, a subject in which I was then very much interested. I noticed that he made no reference to a rather pretty paper written by one Rabinowitz from Odessa and published in Crelle's journal. I commented upon this. He blushed and stammered and said, "I am Rabinowitz." He had moved to the U.S.A. and changed his name. This story is known all over the U.S.A. Occasionally some one from Ann Arbor dines at John's and I ask them if they know Rainich. Yes, they say, there is a funny story about him. "Stop," I say, "let me tell you the story."
During the second world war, we had a country cottage at Chinley, about 20 miles from Manchester, where I was then Professor. My wife and I were coming home one week-end by train. We entered a compartment, and my wife sat diagonally opposite from me. In front of me was a youth and beside me a middle-aged man. Presently I noticed that the youth was reading a book entitled Teach Yourself Trigonometry. Hello, I thought, we are in the same profession. So I asked him whether it was an interesting subject. He did not reply, maintaining a stony silence. Obviously this was an important war secret, and Hitler was not going to get any information from him. Five minutes later I tried again and asked him whether it was a difficult subject. Again no reply, and so I tried no further. When we came to our local station, I got out, and my wife continued into town. She told me afterwards what took place. The other man turned to the youth and said, "You were very rude. Why did you not answer the gentleman?" The reply was, "What does he think he knows about mathematics?"
In 1953, when I was visiting Professor at Toronto, I went with my wife to buy a pair of socks. By the time I left, she and the salesman persuaded me to buy an overcoat. When I related this to a doctor friend, he said he knew a far better salesman. A woman, whose husband had died, went to buy a suit of clothes to bury him in. The salesman persuaded her to buy a suit with two pairs of trousers.
In 1958, I was Visiting Professor at the University of Colorado at Boulder. One day the phone rang, and a woman's voice said "I am Ann Lee and I want to give you a chance of winning 45 dollars." I said, "Oh." She then asked me, "What was the oldest dance in the world?" I said, "This is a difficult question and I don't know." She then asked me, "Where does the tango come from?" I said, "South America." "Good," she said, "you have answered the question, you are now entitled to 45 dollars of free dancing lessons." You may think for a moment what reply you would make to this, but I would get top marks. I asked her, "Who was the first President of the U.S.A.?" "George Washington," she said. "Good," I replied, "you have won your 45 dollars back again." This story is known all over the U.S.A.
In the course of one's travels, a great many unlikely coincidences arise. Perhaps the most surprising one occurred when I was going from the States to visit my son, who was dean of the Faculty of Engineering at McGill in Montreal. We had to change planes at Syracuse in New York State. Presently my wife said there was someone walking around and looking at me as if he knew me. The next time he passed by I said to him, "Excuse me, am I supposed to know you?" He said no, but that he had often seen me at Manchester University, that he was going to Montreal, and that my son, who had met him in Barbados, was going to meet him.
Some years ago, I attended a meeting of the Mathematical Association of America held at the U. S. Air Force Academy at Colorado Springs. In walking around, I noticed someone with a great deal of brass about him, and I said to him, "Are you a mathematician or only a general?" He said, "Only a general." I am only a mathematician, but I have taken seriously to heart Bacon's dictum that every man is a debtor to his profession. I have tried to be of help to mathematicians and in fact have learned much from them. In a modest way, I have tried to advance the science of mathematics in a manner Professor Schwatt never thought of when he said I would do so. As I have lectured in so many places, I have thought I should give others the opportunity of doing so, and so I have endowed an annual lecture in Pure Mathematics in this University. I had suggested that the first talk should be given by Professor Davenport on 'Number theory in Great Britain', but his death prevented this.
There is an old adage: Oats and beans and barley grow, but neither you nor I nor anybody else knows what makes oats and beans and barley grow. Neither you nor I nor anybody else knows what makes a mathematician tick. It is not a question of cleverness. As I have already said, I know many mathematicians who are far abler and cleverer than I am, but they have not been so lucky. An illustration may be given by considering two miners. One may be an expert geologist, but he does not find the golden nuggets that the ignorant miner does.
In some ways, a mathematician is not responsible for his activities. One sometimes feels there is an inner self occasionally communicating with the outer man. This view is supported by the statements made by Henri Poincaré and Jacques Hadamard about their researches. I remember once walking down St Andrews Street some three weeks after writing a paper. Though I had never given the matter any thought since then, it suddenly occurred to me that a point in my proof needed looking into.
I am very grateful to my inner self for his valuable help in the solution of some important and difficult problems that I could not have done otherwise.
I commenced this talk by saying a toast had been drunk to me by the Master and Fellows of St John's College. I might conclude by reciting one sent to me by Professor Leo Moser. Of him, it was said that he was writing a book and taking so long about it that his publishers became very much worried and went to see him. He said he was very sorry about the delay, but he was afraid that the book might have to be a posthumous one. Well, he was told, please hurry up with it.
Moser's toast was as follows:
Here's a toast to L J Mordell,
Young in spirit, most active as well,
He'll never grow weary,
Of his love, number theory,
The results he obtains are just swell.