The Life of Kurt Otto FrierdrichsBy Constance ReidWritten in 1983 for a mathematical journal and later published as an introduction to the two volume selected works of K.O. Friedrichs |

I knew K. O. Friedrichs only during the last dozen years of his life when, for a long period, we worked very closely on a book about Richard Courant. Since his life

and Courant’s were intimately connected, he told me a great deal about himself in the course of telling me about Courant. I have drawn also on the recollections of his

family and a few colleagues — especially Hans Lewy — and on the account of his scientific work which he himself wrote for McGraw-Hill’s Modern Scientists and

Engineers.

Friedrichs was born on September 28, 1901, in Kiel, a Baltic port and the capital of the German state of Schleswig-Holstein. He was christened Kurt Otto, but in

later years he objected to the sharp, abrupt sound of “Kurt,” which he felt didn’t suit him, and preferred to be called “Frieder”.

He was the second of three children. (An older sister became a teacher of English and a brother, a distinguished English translator and interpreter.) The mother,

Elisabeth Entel Friedrichs, was from Silesia. She was an exceptionally lovely woman whom the boy Friedrichs adored and for whom he was to name his only

daughter. The father, Karl Friedrichs, was a lawyer and legal writer, an expert on laws affecting the Civil Service. He could be maddeningly fussy in small matters —

“quite impossible” — but superb and very wise in the big things, so Friedrichs described him. It was a description which his own sons would later feel could be even

more appropriately applied to him.

The Friedrichs family had long been established and politically active in Schleswig and Holstein — the great-grandfather having fought for their independence from

Denmark — but the family left Kiel shortly after Friedrichs’s birth and by the time he was ready for school had settled in Dusseldorf. There the elder Friedrichs

became acquainted with another lawyer, the brother of the mathematician Felix Klein, the powerful director of mathematics at the University of Gottingen —

commonly known, even beyond the borders of Germany, as “the mecca of mathematics”.

Throughout his youth, Friedrichs was plagued by asthma and rarely able to participate in the activities of the other young people. He remembered himself as very

timid; however, after he had made the Abitur at the Realgymnasium in Dusseldorf, he moved — following the custom of the day — from university to university: to

Greifswald for a term, to Freiburg for a year, and to the Austrian university of Graz for a term.

“Then I said to myself, ‘Now I must go to the mecca of mathematics!”’

In the fall of 1922 he arrived in G6ttingen with a letter of introduction from Felix Klein’s lawyer brother. Richard Courant had just recently succeeded Klein as

director of mathematics and was now attempting in his own way to recreate the scientifically stimulating environment which he had known at the university during his

student days.

The times had become much harder since the war of 1914—1918. Inflation had reached staggering proportions. A university mess hail had replaced glamorous dining

clubs; and, as unappetizing soups and stews were ladled from a long trough, students flirted with serving girls in the most often unfulfilled hope of getting larger

portions. Constantly thinking about food, they found it impossible to concentrate on their books. Friedrichs, like many others, lived in a former prisoner-of-war camp.

“[Yet] in spite of a high degree of enforced frugality,” Courant later recalled, “the general intellectual life in the university was intense, idealistic, permeated by profound

desires to find a firm position in the struggle with the problems of the times, political, philosophical, religious, humanistic, artistic, and literary, not to speak of the

challenges which came from the new emerging scientific fields of relativity, quantum mechanics, genetics, aerodynamics, etc.”

Friedrichs did not come unprepared for such an intellectual life. In Freiburg he had been “infected,” as he put it, by the philosophy of Husseri and Heidegger. His

original scientific interest had been mathematical physics; and his mathematical hero was Hermann Weyl, who was making contributions to the foundations of

mathematics as well as to relativity theory in the early 1920’s.

Another student who arrived in Gottingen the same year as Friedrichs was Hans Lewy, with whom he later collaborated on several important papers. Lewy, two

years younger, was impressed by Friedrichs’s “vast and well organized” knowledge; but Friedrichs himself was overwhelmed by the superior knowledge of the young

lecturers and assistants in Gdttingen. These included such future luminaries as Carl Ludwig Siegel and Emil Artin — “such a bunch of people who knew everything

about everything!”

He was also captivated by the informal and scientifically exciting atmosphere which existed around Courant and by Courant’s approach to mathematics (although it

would always be quite different from his own). Reading the Hurwitz-Courant book on function theory, which appeared in 1922, he found the two sections by Hurwitz

“very neat, very clear, to the point, you could learn from them,” but the third section by Courant on geometric function theory— “I started reading one morning, I read

morning and night without stopping. It was the most breathtaking book I have ever read in mathematics.” Inaccuracies which would bother other mathematicians did

not faze him — “because, in spite of them, the essence came through so marvelously!”

On his side, Courant quickly recognized Friedriclis’s gifts in spite of the first impression which the young man made (“as if he could not ‘his drei zahien”’). Only a few

years later, in 1929, he noted that even as a student Friedrichs had stood out because of the independence and originality of his observations and the seriousness of

his scientific aspirations.

It is the opinion of Lewy that a less keen observer of human abilities and human nature than Courant would not have put much stock in Friedrichs at that time.

“He was a very retiring person, not much at ease with himself or the rest of the world. I guess on most people he would have made a negative impression as to his

abilities. Oh yes, I was impressed by him; but I was, you see, a student and on the same level with him. But thinking how he would have appeared to a teacher. I think

it took keen observation as well as a really intense human interest on Courant’s part to see what was there.”

(However, when Courant was asked a year before his death who of all hismany students had most surprised him by his scientific achievement, his reply was

“Friedrichs.”)

While still a student, Friedrichs solved a problem of relativity theory. His dissertation (1925) treated the theory of elastic plates. Courant, his “doctor-father,” found it

an “admirably powerful” work and, looking back, later, was to see it as “the point of departure for [Friedrichs’s] further development as one of the most fruitful and

successful representatives of the field of partial differential equations of mathematical physics.”

During the next two years, while Otto Neugebauer served as assistant in administrative matters, Friedrichs was Courant’s assistant in mathematics and was beginning

to work with him on the second volume of Courant-Hilbert. It was this period that saw the production of the now classic Courant-Friedrichs-Lewy paper. The

primary goal of the work was purely theoretical. At Courant’s suggestion, Friedrichs and Lewy used finite differences to develop an existence theory of hyperbolic

equations of the first order. The result which was to be significant for numerical analysis — but only after the development of the modern high-speed computer — was

the observation that a hyperbolic differential equation cannot be replaced by a difference equation in an arbitrary manner without a severe restriction on the ratio of the

time difference to the space difference. Friedrichs was always very modest about his share in this work, crediting Courant with great intuition in suggesting the

direction of the research and Lewy with great originality in making the observation mentioned above. According to Lewy, what Friedrichs himself brought to the

collaboration was his orderly mind and his gift for seeing formal relationships: “He was a thoroughly independent and original thinker with a way of making penetrating

remarks which seemed at first contradictory.”

In 1927, after Friedrichs had been an assistant in Gottingen for two years, Courant arranged for him to go to Aachen as assistant to Theodor von Karmin, the director

of the aerodynamics institute there and a former “G6ttinger.” Although Friedrichs was not happy at the thought of leaving the congenial “mathematical family” he had

found in Gottingen, he recognized the force of Courant’s argument that it would be good for him to get experience in applied mathematics, since otherwise — as they

both believed at the time — he would not be able to compete against the many gifted pure mathematicians then in Germany.

“Courant forced him out into what at the time were unpleasant situations,” Lewy says. “I remember saying goodbye to Friedrichs at the station when he left for

Aachen, and he was in a daze and very unhappy. He was a man, you see, who wouldn’t want to change, who would, so you would say, gradually have reduced to

smaller and smaller the radius of the circle in which he was operating. I think if it had not been for Courant he would have become a gymnasium teacher.”

Friedrichs spent two years in Aachen with von Karmin. By the time he returned to G6ttingen in 1929, he had published several works and had several more awaiting

publication. Courant, recommending his Habilitation, pointed out to the faculty that Friedrichs was the only one among the young mathematicians in Germany who was

qualified “to build the bridge from modern analysis to mechanics and to hammer out its applications.” It was true that he still made a somewhat poor first impression

because of his shyness, “but in spite of that he pleases on account of his excellent character and his helpfulness to colleagues and students and enjoys a special

popularity.” He was sure Gottingen would not be able to keep him for long; and, in fact, within a year Friedrichs was “called” to a full professorship at the Technische

Hochschule in Braunschweig.

The Hochschule was a comparatively small institution, but Friedrichs (the youngest full professor on the faculty) had to teach several hundred first- and second-year

students.

“A tremendous teaching load, many hours, and I had never taught the courses before so I had to work from morning to night to prepare them. Yet I have never done

scientific work more intensely than during that year. It’s a general observation of mine that if you have all the time you want for your scientific research, you just end up

consulting a psychiatrist. If you have to fight against obstacles, if you have to fight for your time, you do much better. Unless, of course, it goes too far.”

Also contributing decisively to Friedrichs’s creativity at the time was his belated discovery of the great paper in which John von Neumann introduced the modern

theory of operators in Hilbert space.

“I had known von Neumann in Gottingen in 1926. His main interest then had been logic, and we in the group around Courant were somewhat doubtful about him. He

offered a very abstract approach. Brilliant, but is it really substantial? We wondered. And I remember having discussions with Courant about von Neumann at that

time. Oh well, this abstractness for its own sake — we don’t believe in that! But we didn’t know it. And when I got hold of von Neumann’s paper, it opened my

eyes. It was a revelation. And I immediately translated a paper I had written earlier into von Neumann’s language of general Hilbert space.”

Years later he remarked to Peter Lax that all he needed mathematically he got from Hermann Weyl — “except for Hilbert space, and that I got from von Neumann.”

In 1931 he became the first to apply the von Neumann theory to partial differential operators, showing that the initial-value problem for linear hyperbolic equations

could be solved by using energy integrals and, later, showing that the same method could be applied to elliptic equations.

“In this work Lebesgue theory played no role,” he noted in the account of his scientific work for Modern Scientists and Engineers, “the underlying Hilbert spaces were

defined by completion of spaces of smooth functions. Operators were defined by completion (strong extension) or by adjointness (weak extension); the identity of

these two extensions is one of the fundamental results, proved with the aid of so-called mollifiers. These methods are rather simple and direct; they preceded the

development of the theory of distributions of Laurent Schwartz as applied to partial differential equations.”

Somewhat later, after Franz Rellich’s discovery that the spectrum is moved smoothly by specified smooth perturbations for an operator with only a discrete spectrum,

Friedrichs treated the problem for an operator with a continuous spectrum. He found that, again under appropriate conditions, the character of the spectrum is

unchanged; however, in general drastic changes are possible. In 1948 he showed that the theory of these perturbations is fundamental to the theory of scattering.

Later, as he pointed out, the pair of operators he had introduced turned out to be identical with the forward- and backward-wave operators which formed the basis

for subsequent development.

Although living in Braunschweig, Friedrichs continued to visit Gottingen, giving talks in the seminar on occasion and working with Courant on the second volume of

Courant-Hilbert. He was disturbed, however, by a change he sensed in the atmosphere. “The easy carefree attitude of earlier days was no longer so common.” The

students in Gottingen — as well as his students in Braunschweig —were different.” Some even protested against relativity theory: “For us Germans, there is something

absolute!” Then — on January 31, 1933 — Adolf Hitler was named chancellor of the Reich.

‘It was for many a great surprise. Many couldn’t believe that it would mean too much.”

But Lewy left Germany immediately; and Friedrichs, who had carefully read Mein Kampf, could not share the optimistic hope of many that governmental responsibility

would now temper the rowdy radicalism of Hitler’s brown shirts, especially in regard to the Jews.

‘Still also I myself didn’t realize how deep all this would go.”

It was only four days after Hitler became chancellor that Friedrichs attended a benefit theatrical ball — the social event of the year in Braunschweig — in the hope of

seeing there a young Jewish girl whom he had noticed each day on his way home from his morning lecture. He immediately spotted her dark straight hair, bobbed in

the style of the American movie actress Colleen Moore; and, after the entertainment of the evening, never having spoken a word to her before, he marched

determinedly across the ballroom and invited Nellie Bruell to dance. Thus began the great romantic adventure of his life — a romance and adventure story which is

told in charming detail by Nellie Bruell Friedrichs in her Erinnerungen aus meinem Leben in Braunschweig, 1912—193 7 (No. 4 in the Kleine Shrissten of the

Stadtarchiv und Stadtbibliothek of Braunschweig).

Although the new regime took no interest at the time in the budding romance of the Aryan professor and the young Jewish woman, Courant and a number of other

Jewish professors were very soon dismissed from their positions. Friedrichs and Neugebauer circulated a petition in support of Courant and submitted it to the

ministry but without success. After a year’s sojourn in England as a visiting professor, Courant emigrated to the United States and a modest position at New York

University. By that time Nellie Bruell had had to abandon her position as an assistant at the Hochschule and was giving English lessons to Jews who were desirous of

emigrating. She had no problem keeping busy. It was clear to Friedrichs that for personal and political reasons he was going to have to leave Germany soon.

In the summer of 1935 he visited Courant in New York, ostensibly to continue work on the second volume of Courant-Hilbert but in actuality “to see ifl could

emigrate.” He concluded that he “had very definite feelings that this is a country where I could live.” Courant promised to see what he could do about finding an

academic position for him in the United States, no easy task at a time when most young American Ph.D.’s were without jobs.

While Friedrichs was on his way back to Braunschweig, the Nurnberg laws, which forbade marriage between Aryans and non-Aryans, were passed. From that time

on, he was able to meet with Nellie Bruell only infrequently and then secretly at the apartment of a friend of hers who was not Jewish.

In the spring of 1936 Courant wired an invitation to give a few lectures at NYU during the coming summer. When word of the invitation reached the Nazi minister of

education, he wrote angrily to the Braunschweig rector: “I can find no explanation why Professor Friedrichs has not already declined the invitation from Professor

Courant, one of the Jewish emigrees who, although he knows German, writes in English.”

Friedrichs dutifully refused the invitation — in German; but he added in a note to the rector that he saw no reason why an invitation should not be written in the

language of the institution extending it, even when a Jewish emigrant from Germany was a member of the administration of the institution. That summer, with great

secrecy, he materialized at the Czechoslovakian spa where Courant was vacationing and announced that he was now willing to take any American job he could get,

except possibly construction work, which he didn’t think he could do very well.

Back in Germany, in the fall of 1936, he began surreptitiously to arrange to emigrate during the following spring vacation. He told even his parents nothing of his plans

so that, if they were questioned later, they could truthfully say that they had not known he was going to go. For their sake, he was determined that everything he did

would be done according to the law and whatever he said or wrote would be factually correct. To get permission to leave the country (necessary because he was of

military age), he explained that he wanted to visit his sister, who was at the time in Paris. To obtain a visa from the American consul in Hamburg, he produced the

required documents showing that he had permanent residence and employment in Germany. These had been signed by landlady and a clerk at the Hochschule. He

was certain that both of them suspected that he was not planning to return.

As soon as she heard that Friedrichs was safely across the German border, Nellie Bruell, who had a French passport and could still travel freely, left for her father’s

home in Lyon to wait until Friedrichs had obtained a job in the United States and could send for her.

On March 4, 1937, Friedrichs arrived in New York, essentially penniless, since he had been permitted to take only 10 marks out of Germany. Courant found him a

place to live.

“So I was his assistant again. I lived in a room near him in New Rochelle. In fact, I didn’t even know the precise details of that. I was not too concerned. I thought

that somehow it would work out.”

A few days after his arrival he sent a letter to Germany formally resigning his position as a German professor. Still concerned about the possible effect of his emigration

upon his parents, he took the letter to Princeton to mail so that it would not bear a New Rochelle postmark, which would connect him with Courant.

During the spring and summer of 1937, after a dozen years of more or less unproductive effort, Courant with Friedrichs’s help completed the second volume of

Courant-Hilbert. In the course of the writing, however, Friedrichs had recognized that they had grown so far apart mathematically that they would never be able to

work together on the projected third volume.

Courant wrote letters about Friedrichs’s presence in the United States to everyone he knew who was interested in the development of applied mathematics. In fact,

he campaigned so energetically on Friedrichs’s behalf that Lewy, who was by then an assistant professor at Berkeley, warned him that as a recent arrival from Europe

he might jeopardize his own position at NYU. Undeterred, he continued to “spread the news” about Friedrichs.

“That was something Courant always did for Friedrichs,” Lewy says. “I have known few people among gifted mathematicians who were as adverse as Friedrichs was

to putting themselves forward.”

In trying to place Friedrichs in America, Courant emphasized also to him the importance of his work in applied mathematics.

“He told me, look, the situation is not easy in America to get positions for mathematicians, but there is a great need in America to develop applied mathematics; and

since I had a background in applied mathematics, that was my best chance. And, in fact, Courant wanted — independent of me — to develop applied mathematics

here. So in a way my coming fitted his plans.”

That summer, with some help from people interested in aeronautics —“including, I always understood, Lindbergh” — Friedrichs was appointed to a temporary

professorship at NYU. Nellie Bruell was immediately sent for. In August, following the complicated procedure recommended at the time, she and Friedrichs drove to

Canada so that he, officially emigrating, could truthfully say (since he had not yet married a non-Aryan) that there existed no reason why he could not return to

Germany if he so wished. As soon as they could find a justice of the peace on the American side of the border, they were married.

It was a well made and fortunate marriage for both of them, her outgoing and supportive nature and her exceptional talent for friendship perfectly complementing his

needs and gifts.

“Due to Courant and to Nellie, his wife — and also due to Hitler probably —Friedrichs really blossomed,” Lewy says. “He became a different person from what he

was as a student.”

The same year that Friedrichs came to NYU, James J. Stoker, a young applied mathematician from Pittsburgh, a former mining engineer, also joined the faculty. For

the next quarter of a century, until Courant’s retirement in 1961, these three very different men and very different mathematicians were to be the core of the

development which became the Courant Institute of Mathematical Sciences.

In spite of their different national and scientific backgrounds, Friedrichs and Stoker were able to collaborate very effectively. Friedrichs always said that, as in the case

of Lewy, his attitude toward mathematics was strongly influenced by working with Stoker. He spoke with special pleasure of their collaboration on the boundary

problem of the buckled plate with large deflections.

“I think that piece of work I did with Stoker is a piece of work I like myself more than Stoker does. I had more mathematical background, he had more engineering,

and we worked very well together. In any case the main point is that there was a piece of work where mathematical methods were used, and still it clarified an issue of

significance in mechanical engineering.”

Stoker — who did in fact recall this same work with special pleasure at the memorial service held for Friedrichs at NYU — emphasized on that occasion the quality

of daring in Friedrichs’s life and in his mathematics:

“[He] always picked out the dangerous problems to work on, the really tough ones, in which there was no certainty of getting through, and then he would push them

through.”

Friedrichs later published several other papers having to do with plate theory. In fact, throughout his career in America, he alternated periods of working on more

applied problems with periods of working on problems which were of a purely mathematical character.

He considered, so he wrote, that probably the most significant work he did after 1937 was that on symmetric hyperbolic linear differential equations. The work, in

which he utilized powerful tools which he had developed earlier, is considered a breakthrough in many ways. Later he extended the theory to accretive equations.

During World War II Friedrichs returned to fluid dynamics. In spite of a number of contributions to theory made while working on problems of flow past airfoils,

shock waves, and combustion, he considered that his most valuable wartime contribution was the manual Supersonic Flow and Shock Waves, which he and Courant

wrote. After the war, when it had been declassified, the two authors decided to publish it as a book. Cathleen Morawetz, then a graduate mathematics student at

NYU, was selected to edit the volume. She has described the process of collaboration in a way that highlights the difference in the approaches of the two men:

“One or the other of them would take a section from the manual and rewrite it. If Courant did it, then it went to Friedrichs. And Friedrichs would look at it and

grumble that it wasn’t sufficiently exact. He would rewrite it, and it would become all “ifs” and “buts”. Then Courant would take it and he would mumble and groan

that it was much too complicated. Then he would rewrite it. Then Friedrichs would take it back and say it wasn’t precise enough. The process went on many, many

times for each section. When it came back to Friedrichs, he would put in again some of what he had had before, but not so much. Then the next time Courant wouldn’

t take out so much. They were both pretty determined about what the end product should be, and they were both quite willing to do an awful lot of work. But I never

remember a single session where they both sat down together over the manuscript.”

During the war and after, Friedrichs also analyzed the mathematical aspects of deflagrations and detonations. He had always been interested in what goes on in the

boundary layer. “He put it under the microscope, so to speak,” his long-time colleague Fritz John explains. Now he showed that a boundary-layer type of argument,

which he had introduced earlier in airfoil theory, would explain why certain types of detonations (weak) and deflagrations (strong) would not occur. Later he showed

that the Lundquist equation which governs magneto-fluid dynamics could be written as a symmetric hyperbolic system and, still later, that the same is true in the

relativistic case.

In a series of five papers on the quantum theory of fields, he attempted to put that theory on a sound mathematical foundation. These papers appear in his book,

Mathematical Aspects of the Quantum Theory of Fields. A somewhat different approach to the theory is contained in his later book, Perturbations of Spectra in

Hi/bert Space.

Between 1940 and 1950 Friedrichs and his wife became the parents of four sons and one daughter; and these members of the household in New Rochelle provided

the obstacles to uninterrupted research which he had recognized since his Braunschweig days as being conducive rather than detrimental to his mathematical

productivity. Organization and self-discipline, as well as an exceptionally understanding and cooperative wife, enabled him to protect himself from any interruption

during the time which he set aside for writing and research. Even Courant, who liked to drop in on neighboring colleagues in New Rochelle, never got beyond the

living room and a cup of tea with Nellie Friedrichs unless he had a previous appointment.

Friedrichs’s five children, in short talks which each gave at the memorial service in New Rochelle two days after his death, recalled a father whose earliest individual

contacts with them had been “lessons” on his lap in his study. Christopher, the second son, who spoke last, expanded upon these recollections.

“To a large extent I think my father’s very idea of what it meant to be a parent was bound up with the idea of educating his children,” he said. “. . . In our youngest

years, of course, there was not much ‘educational content’ in these lessons, but I don’t think that mattered to Dad at all — the important thing was that a ‘lesson’

seemed to him to be a very natural form of interaction between a parent and a child. And, of course, as we grew a bit older some more real content was slipped into

these lessons. Eventually the ‘lessons’ were dropped, but teaching and learning continued.”

Very much in the same way Friedrichs’s idea of being a mathematician was bound up with the idea of being a teacher. He considered the expression “mathematician

and educator” redundant.

Thoroughly prepared, he would start his lectures at one side of the blackboard with a series of little boxes which by the end of the hour would stretch to the other

side. Although he was not a showman, he would sometimes leave a box empty until he could fill it at exactly the moment when the information he gave would have the

maximum effect.

Former students who spoke at the NYU service recalled that his lectures were not polished performances. He was often difficult to follow because of the originality of

his thought and his willingness “to make a mess.”

“He believed that the first part of a course should be confusion,” Jerome Berkowitz explained. The student should get the feel of the problem and learn what would

not work. “[The second part] should be deconfusion, clarification, and illumination. ‘Of course,’ he said, ‘I sometimes don’t have time for the second part.’’’

(Peter Lax recalls that Friedrichs sometimes described himself as a great teacher “in the weak sense.”)

Friedrichs’s teaching was not limited to his courses. Especially after his retirement, he enjoyed giving expository lectures. These would range from a single talk to a

mathematical “mini series.” In them he would try always to reach a wider public than that of the specialist.

(Fritz John thinks that he would have agreed with the statement that no mathematical theory is completely realized until you can explain it to the first man you meet on

the street.)

In 1965 he published a little book entitled From Pythagoras to Einstein. Typically, he wrote in the preface that the book was not addressed to a “well defined”

audience. The first chapter, in fact, was based on a talk he had given to a special sixth grade math class, the middle chapters were hopefully intended for high school

students, and the last chapter “might just as well be read by college seniors.”

“Professionally, of course, it was hard to imagine Dad as anything other than a mathematician..., Christopher Friedrichs recalled. “Yet at the very same time, he had

such a wide range of knowledge and talent that he could, I think, have followed many other paths. I think, for example, that he could also have been a great physicist.

He could have been a good philologist and certainly ... he could have been a very good historian. He would have been a better than average naturalist. He had a solid

knowledge of philosophy and a sophisticated interest in things like archeology and modern German literature. And he had also a deep and fundamental concern with

the problems of society and political ethics.”

Since his youth Friedrichs had had a general philosophical bent. This had made him concentrate in his work, as he said, “as much on points of view as on proofs.”

When Lewy asked what he intended to do when he retired, he replied promptly, “Go back to philosophy.” He found it personally significant that, unlike most

scientists, he was not attracted to positivism or pragmatism: “the effect was, so to say, not to be doctrinaire, as so many scientists are.” He often remarked, “A

question which has an answer does not belong to philosophy.” Lewy also remembers his “smiling” paradox: “Philosophy is the growing recognition of the impossibility

of reducing known facts to unknown ones. The science which does that is called theoretical physics.”

According to Fritz John, it was Friedrichs’s philosophical propensity which drove him to search for the deeper reasons for the occurrence of symmetric hyperbolic

systems in physics in connection with the conservation laws. In a note in 1970 with Lax and in a later paper he was able to establish that general systems of

conservation equations which possess a convex extension can be reduced to symmetric hyperbolic systems. He also became very interested in the “observable

quantities” of physicists. He thought that these were misused or wrongly interpreted. Trying to clarify the subject in a 1979 paper, “On the Notion of State in Quantum

Mechanics,” he showed that the future values of unobserved observables are determined in terms of their unobserved initial values, “thus in some sense validating

causality.”

From 1971 to 1976 he devoted a great deal of time and effort to working with me on the book Courant in Gottingen and New York, the writing of which he had

suggested. He thought it was important to keep alive for future generations of young mathematicians the unique scientific spirit of Richard Courant, which had played

such an important role in his own life and career. Appropriately, the last talk which Courant himself ever gave was at the celebration of Friedrichs’s seventieth

birthday. On that occasion he spoke with great feeling of the man whom he had known for almost half a century as student, colleague, and friend:

“He is one of the rare scientists whose intellectual and scientific development has never slackened but has gone forward continually. ... One of the very wonderful

aspects of all this is the fact that even now, at the age of seventy years, Friedrichs has not stopped or slackened in his endeavors and the radiant inspiration that

emanates from him. All his friends and many others in many countries in the world are profoundly aware of the fact that he has been and has become steadily more

truly a great man of science. What a great human being he is everybody near to him knows very well.”

There were, of course, many honors bestowed on Friedrichs during the course of his career. Writing the account of himself for Modern Scientists and Engineers, he

mentioned particularly Director and Distinguished Professor of the Courant Institute of New York University, Josiah Willard Gibbs Lecturer of the American

Mathematical Society, election to the National Academy of Sciences and to the American Academy of Arts and Sciences, the Applied Mathematics and Numerical

Analysis Award of the National Academy, and (in 1977) the National Medal of Science, the nation’s highest scientific award.

There were also honorary doctorates from Aachen, Uppsala, Braunschweig, Columbia, and NYU.

During the last decades he received invitations to lecture in many different parts of the world. He had always been interested in seeing new places; and he and his wife

were now able to travel even more extensively than in the past — to Israel, Japan, India, Africa — but also, many times, back to Germany. Because of the asthma

with which he was still afflicted, although not nearly so severely as in his youth, it was his custom to leave New Rochelle during the ragweed season. In 1982 he spent

most of that time in Braunschweig, where he had begun his professional career.

When he returned in October, he was mortally ill. Increasing deafness had already cut him off from many social contacts, and in the next few months he gradually and

quietly withdrew from life itself He died at his home in New Rochelle in the final moments of 1982, with his wife and two of his sons at his side.

and Courant’s were intimately connected, he told me a great deal about himself in the course of telling me about Courant. I have drawn also on the recollections of his

family and a few colleagues — especially Hans Lewy — and on the account of his scientific work which he himself wrote for McGraw-Hill’s Modern Scientists and

Engineers.

Friedrichs was born on September 28, 1901, in Kiel, a Baltic port and the capital of the German state of Schleswig-Holstein. He was christened Kurt Otto, but in

later years he objected to the sharp, abrupt sound of “Kurt,” which he felt didn’t suit him, and preferred to be called “Frieder”.

He was the second of three children. (An older sister became a teacher of English and a brother, a distinguished English translator and interpreter.) The mother,

Elisabeth Entel Friedrichs, was from Silesia. She was an exceptionally lovely woman whom the boy Friedrichs adored and for whom he was to name his only

daughter. The father, Karl Friedrichs, was a lawyer and legal writer, an expert on laws affecting the Civil Service. He could be maddeningly fussy in small matters —

“quite impossible” — but superb and very wise in the big things, so Friedrichs described him. It was a description which his own sons would later feel could be even

more appropriately applied to him.

The Friedrichs family had long been established and politically active in Schleswig and Holstein — the great-grandfather having fought for their independence from

Denmark — but the family left Kiel shortly after Friedrichs’s birth and by the time he was ready for school had settled in Dusseldorf. There the elder Friedrichs

became acquainted with another lawyer, the brother of the mathematician Felix Klein, the powerful director of mathematics at the University of Gottingen —

commonly known, even beyond the borders of Germany, as “the mecca of mathematics”.

Throughout his youth, Friedrichs was plagued by asthma and rarely able to participate in the activities of the other young people. He remembered himself as very

timid; however, after he had made the Abitur at the Realgymnasium in Dusseldorf, he moved — following the custom of the day — from university to university: to

Greifswald for a term, to Freiburg for a year, and to the Austrian university of Graz for a term.

“Then I said to myself, ‘Now I must go to the mecca of mathematics!”’

In the fall of 1922 he arrived in G6ttingen with a letter of introduction from Felix Klein’s lawyer brother. Richard Courant had just recently succeeded Klein as

director of mathematics and was now attempting in his own way to recreate the scientifically stimulating environment which he had known at the university during his

student days.

The times had become much harder since the war of 1914—1918. Inflation had reached staggering proportions. A university mess hail had replaced glamorous dining

clubs; and, as unappetizing soups and stews were ladled from a long trough, students flirted with serving girls in the most often unfulfilled hope of getting larger

portions. Constantly thinking about food, they found it impossible to concentrate on their books. Friedrichs, like many others, lived in a former prisoner-of-war camp.

“[Yet] in spite of a high degree of enforced frugality,” Courant later recalled, “the general intellectual life in the university was intense, idealistic, permeated by profound

desires to find a firm position in the struggle with the problems of the times, political, philosophical, religious, humanistic, artistic, and literary, not to speak of the

challenges which came from the new emerging scientific fields of relativity, quantum mechanics, genetics, aerodynamics, etc.”

Friedrichs did not come unprepared for such an intellectual life. In Freiburg he had been “infected,” as he put it, by the philosophy of Husseri and Heidegger. His

original scientific interest had been mathematical physics; and his mathematical hero was Hermann Weyl, who was making contributions to the foundations of

mathematics as well as to relativity theory in the early 1920’s.

Another student who arrived in Gottingen the same year as Friedrichs was Hans Lewy, with whom he later collaborated on several important papers. Lewy, two

years younger, was impressed by Friedrichs’s “vast and well organized” knowledge; but Friedrichs himself was overwhelmed by the superior knowledge of the young

lecturers and assistants in Gdttingen. These included such future luminaries as Carl Ludwig Siegel and Emil Artin — “such a bunch of people who knew everything

about everything!”

He was also captivated by the informal and scientifically exciting atmosphere which existed around Courant and by Courant’s approach to mathematics (although it

would always be quite different from his own). Reading the Hurwitz-Courant book on function theory, which appeared in 1922, he found the two sections by Hurwitz

“very neat, very clear, to the point, you could learn from them,” but the third section by Courant on geometric function theory— “I started reading one morning, I read

morning and night without stopping. It was the most breathtaking book I have ever read in mathematics.” Inaccuracies which would bother other mathematicians did

not faze him — “because, in spite of them, the essence came through so marvelously!”

On his side, Courant quickly recognized Friedriclis’s gifts in spite of the first impression which the young man made (“as if he could not ‘his drei zahien”’). Only a few

years later, in 1929, he noted that even as a student Friedrichs had stood out because of the independence and originality of his observations and the seriousness of

his scientific aspirations.

It is the opinion of Lewy that a less keen observer of human abilities and human nature than Courant would not have put much stock in Friedrichs at that time.

“He was a very retiring person, not much at ease with himself or the rest of the world. I guess on most people he would have made a negative impression as to his

abilities. Oh yes, I was impressed by him; but I was, you see, a student and on the same level with him. But thinking how he would have appeared to a teacher. I think

it took keen observation as well as a really intense human interest on Courant’s part to see what was there.”

(However, when Courant was asked a year before his death who of all hismany students had most surprised him by his scientific achievement, his reply was

“Friedrichs.”)

While still a student, Friedrichs solved a problem of relativity theory. His dissertation (1925) treated the theory of elastic plates. Courant, his “doctor-father,” found it

an “admirably powerful” work and, looking back, later, was to see it as “the point of departure for [Friedrichs’s] further development as one of the most fruitful and

successful representatives of the field of partial differential equations of mathematical physics.”

During the next two years, while Otto Neugebauer served as assistant in administrative matters, Friedrichs was Courant’s assistant in mathematics and was beginning

to work with him on the second volume of Courant-Hilbert. It was this period that saw the production of the now classic Courant-Friedrichs-Lewy paper. The

primary goal of the work was purely theoretical. At Courant’s suggestion, Friedrichs and Lewy used finite differences to develop an existence theory of hyperbolic

equations of the first order. The result which was to be significant for numerical analysis — but only after the development of the modern high-speed computer — was

the observation that a hyperbolic differential equation cannot be replaced by a difference equation in an arbitrary manner without a severe restriction on the ratio of the

time difference to the space difference. Friedrichs was always very modest about his share in this work, crediting Courant with great intuition in suggesting the

direction of the research and Lewy with great originality in making the observation mentioned above. According to Lewy, what Friedrichs himself brought to the

collaboration was his orderly mind and his gift for seeing formal relationships: “He was a thoroughly independent and original thinker with a way of making penetrating

remarks which seemed at first contradictory.”

In 1927, after Friedrichs had been an assistant in Gottingen for two years, Courant arranged for him to go to Aachen as assistant to Theodor von Karmin, the director

of the aerodynamics institute there and a former “G6ttinger.” Although Friedrichs was not happy at the thought of leaving the congenial “mathematical family” he had

found in Gottingen, he recognized the force of Courant’s argument that it would be good for him to get experience in applied mathematics, since otherwise — as they

both believed at the time — he would not be able to compete against the many gifted pure mathematicians then in Germany.

“Courant forced him out into what at the time were unpleasant situations,” Lewy says. “I remember saying goodbye to Friedrichs at the station when he left for

Aachen, and he was in a daze and very unhappy. He was a man, you see, who wouldn’t want to change, who would, so you would say, gradually have reduced to

smaller and smaller the radius of the circle in which he was operating. I think if it had not been for Courant he would have become a gymnasium teacher.”

Friedrichs spent two years in Aachen with von Karmin. By the time he returned to G6ttingen in 1929, he had published several works and had several more awaiting

publication. Courant, recommending his Habilitation, pointed out to the faculty that Friedrichs was the only one among the young mathematicians in Germany who was

qualified “to build the bridge from modern analysis to mechanics and to hammer out its applications.” It was true that he still made a somewhat poor first impression

because of his shyness, “but in spite of that he pleases on account of his excellent character and his helpfulness to colleagues and students and enjoys a special

popularity.” He was sure Gottingen would not be able to keep him for long; and, in fact, within a year Friedrichs was “called” to a full professorship at the Technische

Hochschule in Braunschweig.

The Hochschule was a comparatively small institution, but Friedrichs (the youngest full professor on the faculty) had to teach several hundred first- and second-year

students.

“A tremendous teaching load, many hours, and I had never taught the courses before so I had to work from morning to night to prepare them. Yet I have never done

scientific work more intensely than during that year. It’s a general observation of mine that if you have all the time you want for your scientific research, you just end up

consulting a psychiatrist. If you have to fight against obstacles, if you have to fight for your time, you do much better. Unless, of course, it goes too far.”

Also contributing decisively to Friedrichs’s creativity at the time was his belated discovery of the great paper in which John von Neumann introduced the modern

theory of operators in Hilbert space.

“I had known von Neumann in Gottingen in 1926. His main interest then had been logic, and we in the group around Courant were somewhat doubtful about him. He

offered a very abstract approach. Brilliant, but is it really substantial? We wondered. And I remember having discussions with Courant about von Neumann at that

time. Oh well, this abstractness for its own sake — we don’t believe in that! But we didn’t know it. And when I got hold of von Neumann’s paper, it opened my

eyes. It was a revelation. And I immediately translated a paper I had written earlier into von Neumann’s language of general Hilbert space.”

Years later he remarked to Peter Lax that all he needed mathematically he got from Hermann Weyl — “except for Hilbert space, and that I got from von Neumann.”

In 1931 he became the first to apply the von Neumann theory to partial differential operators, showing that the initial-value problem for linear hyperbolic equations

could be solved by using energy integrals and, later, showing that the same method could be applied to elliptic equations.

“In this work Lebesgue theory played no role,” he noted in the account of his scientific work for Modern Scientists and Engineers, “the underlying Hilbert spaces were

defined by completion of spaces of smooth functions. Operators were defined by completion (strong extension) or by adjointness (weak extension); the identity of

these two extensions is one of the fundamental results, proved with the aid of so-called mollifiers. These methods are rather simple and direct; they preceded the

development of the theory of distributions of Laurent Schwartz as applied to partial differential equations.”

Somewhat later, after Franz Rellich’s discovery that the spectrum is moved smoothly by specified smooth perturbations for an operator with only a discrete spectrum,

Friedrichs treated the problem for an operator with a continuous spectrum. He found that, again under appropriate conditions, the character of the spectrum is

unchanged; however, in general drastic changes are possible. In 1948 he showed that the theory of these perturbations is fundamental to the theory of scattering.

Later, as he pointed out, the pair of operators he had introduced turned out to be identical with the forward- and backward-wave operators which formed the basis

for subsequent development.

Although living in Braunschweig, Friedrichs continued to visit Gottingen, giving talks in the seminar on occasion and working with Courant on the second volume of

Courant-Hilbert. He was disturbed, however, by a change he sensed in the atmosphere. “The easy carefree attitude of earlier days was no longer so common.” The

students in Gottingen — as well as his students in Braunschweig —were different.” Some even protested against relativity theory: “For us Germans, there is something

absolute!” Then — on January 31, 1933 — Adolf Hitler was named chancellor of the Reich.

‘It was for many a great surprise. Many couldn’t believe that it would mean too much.”

But Lewy left Germany immediately; and Friedrichs, who had carefully read Mein Kampf, could not share the optimistic hope of many that governmental responsibility

would now temper the rowdy radicalism of Hitler’s brown shirts, especially in regard to the Jews.

‘Still also I myself didn’t realize how deep all this would go.”

It was only four days after Hitler became chancellor that Friedrichs attended a benefit theatrical ball — the social event of the year in Braunschweig — in the hope of

seeing there a young Jewish girl whom he had noticed each day on his way home from his morning lecture. He immediately spotted her dark straight hair, bobbed in

the style of the American movie actress Colleen Moore; and, after the entertainment of the evening, never having spoken a word to her before, he marched

determinedly across the ballroom and invited Nellie Bruell to dance. Thus began the great romantic adventure of his life — a romance and adventure story which is

told in charming detail by Nellie Bruell Friedrichs in her Erinnerungen aus meinem Leben in Braunschweig, 1912—193 7 (No. 4 in the Kleine Shrissten of the

Stadtarchiv und Stadtbibliothek of Braunschweig).

Although the new regime took no interest at the time in the budding romance of the Aryan professor and the young Jewish woman, Courant and a number of other

Jewish professors were very soon dismissed from their positions. Friedrichs and Neugebauer circulated a petition in support of Courant and submitted it to the

ministry but without success. After a year’s sojourn in England as a visiting professor, Courant emigrated to the United States and a modest position at New York

University. By that time Nellie Bruell had had to abandon her position as an assistant at the Hochschule and was giving English lessons to Jews who were desirous of

emigrating. She had no problem keeping busy. It was clear to Friedrichs that for personal and political reasons he was going to have to leave Germany soon.

In the summer of 1935 he visited Courant in New York, ostensibly to continue work on the second volume of Courant-Hilbert but in actuality “to see ifl could

emigrate.” He concluded that he “had very definite feelings that this is a country where I could live.” Courant promised to see what he could do about finding an

academic position for him in the United States, no easy task at a time when most young American Ph.D.’s were without jobs.

While Friedrichs was on his way back to Braunschweig, the Nurnberg laws, which forbade marriage between Aryans and non-Aryans, were passed. From that time

on, he was able to meet with Nellie Bruell only infrequently and then secretly at the apartment of a friend of hers who was not Jewish.

In the spring of 1936 Courant wired an invitation to give a few lectures at NYU during the coming summer. When word of the invitation reached the Nazi minister of

education, he wrote angrily to the Braunschweig rector: “I can find no explanation why Professor Friedrichs has not already declined the invitation from Professor

Courant, one of the Jewish emigrees who, although he knows German, writes in English.”

Friedrichs dutifully refused the invitation — in German; but he added in a note to the rector that he saw no reason why an invitation should not be written in the

language of the institution extending it, even when a Jewish emigrant from Germany was a member of the administration of the institution. That summer, with great

secrecy, he materialized at the Czechoslovakian spa where Courant was vacationing and announced that he was now willing to take any American job he could get,

except possibly construction work, which he didn’t think he could do very well.

Back in Germany, in the fall of 1936, he began surreptitiously to arrange to emigrate during the following spring vacation. He told even his parents nothing of his plans

so that, if they were questioned later, they could truthfully say that they had not known he was going to go. For their sake, he was determined that everything he did

would be done according to the law and whatever he said or wrote would be factually correct. To get permission to leave the country (necessary because he was of

military age), he explained that he wanted to visit his sister, who was at the time in Paris. To obtain a visa from the American consul in Hamburg, he produced the

required documents showing that he had permanent residence and employment in Germany. These had been signed by landlady and a clerk at the Hochschule. He

was certain that both of them suspected that he was not planning to return.

As soon as she heard that Friedrichs was safely across the German border, Nellie Bruell, who had a French passport and could still travel freely, left for her father’s

home in Lyon to wait until Friedrichs had obtained a job in the United States and could send for her.

On March 4, 1937, Friedrichs arrived in New York, essentially penniless, since he had been permitted to take only 10 marks out of Germany. Courant found him a

place to live.

“So I was his assistant again. I lived in a room near him in New Rochelle. In fact, I didn’t even know the precise details of that. I was not too concerned. I thought

that somehow it would work out.”

A few days after his arrival he sent a letter to Germany formally resigning his position as a German professor. Still concerned about the possible effect of his emigration

upon his parents, he took the letter to Princeton to mail so that it would not bear a New Rochelle postmark, which would connect him with Courant.

During the spring and summer of 1937, after a dozen years of more or less unproductive effort, Courant with Friedrichs’s help completed the second volume of

Courant-Hilbert. In the course of the writing, however, Friedrichs had recognized that they had grown so far apart mathematically that they would never be able to

work together on the projected third volume.

Courant wrote letters about Friedrichs’s presence in the United States to everyone he knew who was interested in the development of applied mathematics. In fact,

he campaigned so energetically on Friedrichs’s behalf that Lewy, who was by then an assistant professor at Berkeley, warned him that as a recent arrival from Europe

he might jeopardize his own position at NYU. Undeterred, he continued to “spread the news” about Friedrichs.

“That was something Courant always did for Friedrichs,” Lewy says. “I have known few people among gifted mathematicians who were as adverse as Friedrichs was

to putting themselves forward.”

In trying to place Friedrichs in America, Courant emphasized also to him the importance of his work in applied mathematics.

“He told me, look, the situation is not easy in America to get positions for mathematicians, but there is a great need in America to develop applied mathematics; and

since I had a background in applied mathematics, that was my best chance. And, in fact, Courant wanted — independent of me — to develop applied mathematics

here. So in a way my coming fitted his plans.”

That summer, with some help from people interested in aeronautics —“including, I always understood, Lindbergh” — Friedrichs was appointed to a temporary

professorship at NYU. Nellie Bruell was immediately sent for. In August, following the complicated procedure recommended at the time, she and Friedrichs drove to

Canada so that he, officially emigrating, could truthfully say (since he had not yet married a non-Aryan) that there existed no reason why he could not return to

Germany if he so wished. As soon as they could find a justice of the peace on the American side of the border, they were married.

It was a well made and fortunate marriage for both of them, her outgoing and supportive nature and her exceptional talent for friendship perfectly complementing his

needs and gifts.

“Due to Courant and to Nellie, his wife — and also due to Hitler probably —Friedrichs really blossomed,” Lewy says. “He became a different person from what he

was as a student.”

The same year that Friedrichs came to NYU, James J. Stoker, a young applied mathematician from Pittsburgh, a former mining engineer, also joined the faculty. For

the next quarter of a century, until Courant’s retirement in 1961, these three very different men and very different mathematicians were to be the core of the

development which became the Courant Institute of Mathematical Sciences.

In spite of their different national and scientific backgrounds, Friedrichs and Stoker were able to collaborate very effectively. Friedrichs always said that, as in the case

of Lewy, his attitude toward mathematics was strongly influenced by working with Stoker. He spoke with special pleasure of their collaboration on the boundary

problem of the buckled plate with large deflections.

“I think that piece of work I did with Stoker is a piece of work I like myself more than Stoker does. I had more mathematical background, he had more engineering,

and we worked very well together. In any case the main point is that there was a piece of work where mathematical methods were used, and still it clarified an issue of

significance in mechanical engineering.”

Stoker — who did in fact recall this same work with special pleasure at the memorial service held for Friedrichs at NYU — emphasized on that occasion the quality

of daring in Friedrichs’s life and in his mathematics:

“[He] always picked out the dangerous problems to work on, the really tough ones, in which there was no certainty of getting through, and then he would push them

through.”

Friedrichs later published several other papers having to do with plate theory. In fact, throughout his career in America, he alternated periods of working on more

applied problems with periods of working on problems which were of a purely mathematical character.

He considered, so he wrote, that probably the most significant work he did after 1937 was that on symmetric hyperbolic linear differential equations. The work, in

which he utilized powerful tools which he had developed earlier, is considered a breakthrough in many ways. Later he extended the theory to accretive equations.

During World War II Friedrichs returned to fluid dynamics. In spite of a number of contributions to theory made while working on problems of flow past airfoils,

shock waves, and combustion, he considered that his most valuable wartime contribution was the manual Supersonic Flow and Shock Waves, which he and Courant

wrote. After the war, when it had been declassified, the two authors decided to publish it as a book. Cathleen Morawetz, then a graduate mathematics student at

NYU, was selected to edit the volume. She has described the process of collaboration in a way that highlights the difference in the approaches of the two men:

“One or the other of them would take a section from the manual and rewrite it. If Courant did it, then it went to Friedrichs. And Friedrichs would look at it and

grumble that it wasn’t sufficiently exact. He would rewrite it, and it would become all “ifs” and “buts”. Then Courant would take it and he would mumble and groan

that it was much too complicated. Then he would rewrite it. Then Friedrichs would take it back and say it wasn’t precise enough. The process went on many, many

times for each section. When it came back to Friedrichs, he would put in again some of what he had had before, but not so much. Then the next time Courant wouldn’

t take out so much. They were both pretty determined about what the end product should be, and they were both quite willing to do an awful lot of work. But I never

remember a single session where they both sat down together over the manuscript.”

During the war and after, Friedrichs also analyzed the mathematical aspects of deflagrations and detonations. He had always been interested in what goes on in the

boundary layer. “He put it under the microscope, so to speak,” his long-time colleague Fritz John explains. Now he showed that a boundary-layer type of argument,

which he had introduced earlier in airfoil theory, would explain why certain types of detonations (weak) and deflagrations (strong) would not occur. Later he showed

that the Lundquist equation which governs magneto-fluid dynamics could be written as a symmetric hyperbolic system and, still later, that the same is true in the

relativistic case.

In a series of five papers on the quantum theory of fields, he attempted to put that theory on a sound mathematical foundation. These papers appear in his book,

Mathematical Aspects of the Quantum Theory of Fields. A somewhat different approach to the theory is contained in his later book, Perturbations of Spectra in

Hi/bert Space.

Between 1940 and 1950 Friedrichs and his wife became the parents of four sons and one daughter; and these members of the household in New Rochelle provided

the obstacles to uninterrupted research which he had recognized since his Braunschweig days as being conducive rather than detrimental to his mathematical

productivity. Organization and self-discipline, as well as an exceptionally understanding and cooperative wife, enabled him to protect himself from any interruption

during the time which he set aside for writing and research. Even Courant, who liked to drop in on neighboring colleagues in New Rochelle, never got beyond the

living room and a cup of tea with Nellie Friedrichs unless he had a previous appointment.

Friedrichs’s five children, in short talks which each gave at the memorial service in New Rochelle two days after his death, recalled a father whose earliest individual

contacts with them had been “lessons” on his lap in his study. Christopher, the second son, who spoke last, expanded upon these recollections.

“To a large extent I think my father’s very idea of what it meant to be a parent was bound up with the idea of educating his children,” he said. “. . . In our youngest

years, of course, there was not much ‘educational content’ in these lessons, but I don’t think that mattered to Dad at all — the important thing was that a ‘lesson’

seemed to him to be a very natural form of interaction between a parent and a child. And, of course, as we grew a bit older some more real content was slipped into

these lessons. Eventually the ‘lessons’ were dropped, but teaching and learning continued.”

Very much in the same way Friedrichs’s idea of being a mathematician was bound up with the idea of being a teacher. He considered the expression “mathematician

and educator” redundant.

Thoroughly prepared, he would start his lectures at one side of the blackboard with a series of little boxes which by the end of the hour would stretch to the other

side. Although he was not a showman, he would sometimes leave a box empty until he could fill it at exactly the moment when the information he gave would have the

maximum effect.

Former students who spoke at the NYU service recalled that his lectures were not polished performances. He was often difficult to follow because of the originality of

his thought and his willingness “to make a mess.”

“He believed that the first part of a course should be confusion,” Jerome Berkowitz explained. The student should get the feel of the problem and learn what would

not work. “[The second part] should be deconfusion, clarification, and illumination. ‘Of course,’ he said, ‘I sometimes don’t have time for the second part.’’’

(Peter Lax recalls that Friedrichs sometimes described himself as a great teacher “in the weak sense.”)

Friedrichs’s teaching was not limited to his courses. Especially after his retirement, he enjoyed giving expository lectures. These would range from a single talk to a

mathematical “mini series.” In them he would try always to reach a wider public than that of the specialist.

(Fritz John thinks that he would have agreed with the statement that no mathematical theory is completely realized until you can explain it to the first man you meet on

the street.)

In 1965 he published a little book entitled From Pythagoras to Einstein. Typically, he wrote in the preface that the book was not addressed to a “well defined”

audience. The first chapter, in fact, was based on a talk he had given to a special sixth grade math class, the middle chapters were hopefully intended for high school

students, and the last chapter “might just as well be read by college seniors.”

“Professionally, of course, it was hard to imagine Dad as anything other than a mathematician..., Christopher Friedrichs recalled. “Yet at the very same time, he had

such a wide range of knowledge and talent that he could, I think, have followed many other paths. I think, for example, that he could also have been a great physicist.

He could have been a good philologist and certainly ... he could have been a very good historian. He would have been a better than average naturalist. He had a solid

knowledge of philosophy and a sophisticated interest in things like archeology and modern German literature. And he had also a deep and fundamental concern with

the problems of society and political ethics.”

Since his youth Friedrichs had had a general philosophical bent. This had made him concentrate in his work, as he said, “as much on points of view as on proofs.”

When Lewy asked what he intended to do when he retired, he replied promptly, “Go back to philosophy.” He found it personally significant that, unlike most

scientists, he was not attracted to positivism or pragmatism: “the effect was, so to say, not to be doctrinaire, as so many scientists are.” He often remarked, “A

question which has an answer does not belong to philosophy.” Lewy also remembers his “smiling” paradox: “Philosophy is the growing recognition of the impossibility

of reducing known facts to unknown ones. The science which does that is called theoretical physics.”

According to Fritz John, it was Friedrichs’s philosophical propensity which drove him to search for the deeper reasons for the occurrence of symmetric hyperbolic

systems in physics in connection with the conservation laws. In a note in 1970 with Lax and in a later paper he was able to establish that general systems of

conservation equations which possess a convex extension can be reduced to symmetric hyperbolic systems. He also became very interested in the “observable

quantities” of physicists. He thought that these were misused or wrongly interpreted. Trying to clarify the subject in a 1979 paper, “On the Notion of State in Quantum

Mechanics,” he showed that the future values of unobserved observables are determined in terms of their unobserved initial values, “thus in some sense validating

causality.”

From 1971 to 1976 he devoted a great deal of time and effort to working with me on the book Courant in Gottingen and New York, the writing of which he had

suggested. He thought it was important to keep alive for future generations of young mathematicians the unique scientific spirit of Richard Courant, which had played

such an important role in his own life and career. Appropriately, the last talk which Courant himself ever gave was at the celebration of Friedrichs’s seventieth

birthday. On that occasion he spoke with great feeling of the man whom he had known for almost half a century as student, colleague, and friend:

“He is one of the rare scientists whose intellectual and scientific development has never slackened but has gone forward continually. ... One of the very wonderful

aspects of all this is the fact that even now, at the age of seventy years, Friedrichs has not stopped or slackened in his endeavors and the radiant inspiration that

emanates from him. All his friends and many others in many countries in the world are profoundly aware of the fact that he has been and has become steadily more

truly a great man of science. What a great human being he is everybody near to him knows very well.”

There were, of course, many honors bestowed on Friedrichs during the course of his career. Writing the account of himself for Modern Scientists and Engineers, he

mentioned particularly Director and Distinguished Professor of the Courant Institute of New York University, Josiah Willard Gibbs Lecturer of the American

Mathematical Society, election to the National Academy of Sciences and to the American Academy of Arts and Sciences, the Applied Mathematics and Numerical

Analysis Award of the National Academy, and (in 1977) the National Medal of Science, the nation’s highest scientific award.

There were also honorary doctorates from Aachen, Uppsala, Braunschweig, Columbia, and NYU.

During the last decades he received invitations to lecture in many different parts of the world. He had always been interested in seeing new places; and he and his wife

were now able to travel even more extensively than in the past — to Israel, Japan, India, Africa — but also, many times, back to Germany. Because of the asthma

with which he was still afflicted, although not nearly so severely as in his youth, it was his custom to leave New Rochelle during the ragweed season. In 1982 he spent

most of that time in Braunschweig, where he had begun his professional career.

When he returned in October, he was mortally ill. Increasing deafness had already cut him off from many social contacts, and in the next few months he gradually and

quietly withdrew from life itself He died at his home in New Rochelle in the final moments of 1982, with his wife and two of his sons at his side.

THE FRIEDRICHS FAMILY WEB SITE Contact: Marty .Friedrichs@Gmail.com |

with Richard Courant 1965