Category Archives: time management

Why writing by hand makes kids smarter

Time management — English proverb/ wisdom

Take care of the minutes and the hours will take care of themselves.

Einstein told Banesh Hoffmann, “I am slow…!”

Picked up from Reader’s Digest: Indian Edition: March 2020 : just for the joy of sharing and learning with my students. 

January 1968: The Unforgettable Albert Einstein: A professor remembers his encounters with Albert Einstein, and pays a glowing tribute to the man’s genius and his many accomplishments.

By Banesh Hoffmann:


It was one of the greatest scientists the world has ever known, yet if I had to convey the essence of Albert Einstein in a single word, I would choose ‘simplicity’. Perhaps, an anecdote will help. Once, caught in a downpour, he took off his hat and held it under his coat. Asked why, he explained, with admirable logic, that the rain would damage the hat, but his hair would be none the worse for its wetting. This knack for going instinctively to the heart of the matter was the secret of his major scientific discoveries — this and his extraordinary feeling for beauty.

I first met Albert Einstein in 1935, at the famous Institute for Advanced Study in Princeton, New Jersey. Einstein had been among the first to be invited to the Institute, and was offered carte blanche as to salary. To the director’s dismay, Einstein asked for an impossible sum. It was far too small !! The director had to plead with him to accept a larger salary.

I was awe in of Einstein, and hesitated before approaching him about some ideas I had been working on. My hesitation proved unwarranted. When I finally knocked on his door, a gentle voice said, “Come” — with a rising inflection that made the single word both a welcome and a question. I entered his office and found him seated on a table, calculating and smoking his pipe. Dressed in ill-fitting clothes, his hair characteristically awry, he smiled a warm welcome. His utter naturalness at once set me at ease.

As  I began to explain my ideas, he asked me to write the equations on blackboard so that he could see how they developed. Then, came the staggering — and altogether endearing —request: “Please go slowly. I do not understand things quickly.” This from Einstein ! He said it gently, and I laughed. From then on, all vestiges of fear were gone. 

BURST OF GENIUS Einstein was born in 1879 in the German city of Ulm. He had been no infant prodigy; indeed, he was so late in learning to speak that his parents feared he was a dullard. In school, though his teachers saw no special talent in him, the signs were already there. He taught himself calculus, for example, and he told me that his teachers seemed a little afraid of him because he asked questions that they could not answer. At the age of 16, he asked himself whether a light wave would seem stationary if one ran abreast of it. It seems an innocent question, but this shows Einstein going to the heart of a problem. From it there would arise, 10 years later, his theory of relativity.

Einstein failed his entrance examinations at the Swiss Federal Polytechnic School in Zurich, but was admitted a year later. There he went beyond his regular work to study the masterworks of physics on his own. Rejected when he applied for academic positions, he ultimately found work, in 1902, as a patent examiner in Berne, and there, in 1905, his genius burst into fabulous flower.

Among the extraordinary things he produced in that memorable year were his theory of relativity, with its famous offshoot E = mc^{2} (energy equals mass times the speed of light squared), and his quantum theory of light. These two theories were not only revolutionary but seemingly self-contradictory as well: the former was intimately linked to the theory that light consists of waves, while the latter said that it consists of somehow of particles. Yet this unknown young man boldy proposed both at once — and he was right in both cases, though how he could possibly have been is far too complex a story to tell here.


Collaborating with Einstein was an unforgettable experience. In 1937, the Polish physicist Leopold Infeld and I asked if we could work with him. He was pleased with the proposal, since he had an idea about gravitation waiting to be worked out in detail. Thus, we got to know not merely the man and the friend, but also the professional.

The intensity and depth of his concentration were fantastic. When battling a recalcitrant problem, he worried it as an animal worries its prey. Often, when we found ourselves up against a seemingly insuperable difficulty, he would stand up, put his pipe on the table, and say in his quaint English, “I will a little tink” (he could not pronounce “th”). Then, he would pace up and down, twirling a lock of his long greying hair around his forefinger.

A dreamy, faraway yet inward look would come over his face. There was no appearance of concentration, no furrowing of his brow — only a placid inner communion. The minutes would pass, and then suddenly Einstein would stop pacing as his face relaxed into a gentle smile. He has found the solution to the problem. Sometimes it was so simple that Infeld and I could have kicked ourselves for not having thought of it. But the magic had been performed invisibly in the depths of Einstein’s mind, by a process we could not fathom.

When his wife died, he was deeply shaken, but insisted that now more than ever was the time to be working hard. I vividly remember going to his house to work with him during that sad time. His face was haggard and grief-lined but he put forth a great effort to concentrate. Seeking to help him, I steered the discussion away from routine matters into more difficult theoretical problems, and Einstein gradually became absorbed in the discussion. We kept at it for some two hours, and at the end his eyes were no longer sad. As I left, he thanked me with moving sincerity, but the words he found sounded almost incongruous. “It was a fun,” he said. He had a moment of surcease from grief, and these groping words expressed a deep emotion.


Although Einstein felt no need for religious ritual and belonged to no formal religious group, he was the most deeply religious man I have known. He once said to me, “ideas come from God,” and one could hear the capital ‘G’ in the reverence with which he pronounced the word. On the marble fireplace in the mathematics building at Princeton University is carved, in the original German, what one might call his scientific credo: God is subtle, but He is not malicious.” By this Einstein meant that scientists could expect to find their task difficult, but not hopeless. The Universe was a Universe of law, and God was not confusing with deliberate paradoxes and contradictions.

Einstein was an accomplished amateur musician. We used at play duets; he at the violin, I at the piano. One day he surprised me by saying that Mozart was the greatest composer of all. Beethoven, he said, “created” his music but the music of Mozart was of such purity and beauty that one felt he merely “found” it — that it had always existed as part of the inner beauty of the Universe, waiting to be revealed.

It was this very Mozartian simplicity that most characterized Einstein’s methods. His 1905 theory of relativity, for example, was built on two simple assumptions. One is the so-called principle of relativity, which means, roughly speaking, that we cannot tell whether we are at rest or moving smoothly. The other assumption is that the speed of light is the same, no matter what the speed of the object that produces it. You can see how reasonable this is if you think of agitating a stick in a lake to create waves. Whether you wiggle the stick from a stationary pier. or from a rushing speedboat, the waves once generated are on their own, and their speed has nothing to do with that of the stick.

Each of these assumptions, by itself, was so plausible as to seem primitively obvious. But. together they were in such violent conflict that a lesser man would have dropped one or the other and fled in panic. Einstein daringly kept both — and by doing so he revolutionized physics. For he demonstrated that they could after all, exist peacefully side by side, provided we give up cherished beliefs about the nature of time.

Science is like a house of cards, with concepts like time and space at the lowest level. Tampering with time brought most of the house tumbling down, and it was this made Einstein’s work so important —- and so controversial. At a conference in Princeton in honour of his 70th birthday, one of the speakers, a Nobel prize winner, tried to convey the magical quality of Einstein’s achievement. Words failed him, and with a shrug of helplessness he pointed to his wrist-watch, and said in tones of awed amazement, “It all came from this.” His very ineloquence made this the most eloquent tribute I have heard to Einstein’s genius.


Although fame had little effect on Einstein as a person, he could not escape it; he was, of course, instantly recognizable. One autumn Saturday, I was walking with him in Princeton discussing some technical matters. Parents and almuni were streaming excitedly toward the stadium, their minds on the coming football game. As they approached us, they paused in sudden recognition, and a momentary air of solemnity came over them as if they had been reminded of a world far removed from the thrills of football. Yet Einstein seemed totally unaware of the effect he was having on them, and went on with the discussion as though they were not there.

We think of Einstein as one concerned only with the deepest aspects of science. But he saw scientific principles in every day things to which most of us would give barely a second thought.He once asked me if I had ever wondered why a man’s feet will sink into either dry or completely submerged sand, while sand that is merely damp provides a firm surface. When I could not answer, he offered a simple explanation. It depends, he pointed out, on surface tension, the elastic-skin effect of a liquid surface. This is what holds a drop together, or causes two small raindrops on a window pane to pull into one big drop the moment their surfaces touch.

When sand is damp, Einstein explained, there are tiny amounts of water between the grains. The surface tensions of these tiny amounts of water pull all the grains together, and friction then makes them hard to budge. When the sand is dry, there is obviously no water between grains. If the sand is fully immersed, there is water between grains, but there is no water surface between them to pull them together. This is not as important as relativity; yet as his youthful question, about running abreast of a light wave showed, there is no telling what seeming trifle will lead an Einstein to a major discovery. And, the puzzle of the sand gives us an inkling of the power and elegance of Einstein’s mind.


Einstein’s work, performed with pencil and paper, seemed remote from the turmoil of everyday life. But his ideas were so revolutionary that they caused violent controversy and irrational anger. Indeed, in order to be able to award him a belated Nobel Prize, the selection committee had to avoid mentioning relativity, and pretend that his prize was primarily due to his work on quantum theory. Political events upset the serenity of his life even more. When the Nazis came to power in Germany, his theories were officially declared false because they had been formulated by a Jew. His property was confiscated, and it is said that a price was put on his head.

When scientists in the United States fearful that the Nazis might develop an atomic bomb, sought to alert American authorities to that danger, they were scarcely heeded. In desperation, they drafted a letter, which Einstein signed and sent directly to President Roosevelt. It was this act that led to the fateful decision to go all-out on the production of an atomic bomb — an endeavour in which Einstein took no active part. When he heard of the agony and destruction that his E=mc^{2} had wrought, he was dismayed beyond measure and from then on there was a look of ineffable sadness in his eyes.

There was something elusively whimsical about Einstein. It is illustrated by my favourite anecdote about him. In his first year in Princeton, on Christmas Eve, so the story goes, some children sang carols outside his house. Having finished, they knocked on his door and explained that they were collecting money to buy Christmas presents. Einstein listened then said, “Wait a moment.” He put on his scarf and overcoat, and took his violin from its case. Then, joining the children, he accompanied their singing of “Silent Night” on his violin.

How shall I sum up what it meant to have known Einstein and his work? Like the Nobel prize winner who pointed helplessly at his watch, I can find no adequate words. It was akin to the revelation of the great art that lets one see what was formerly hidden. And, when for example, I walk on the sand of a lonely beach, I am reminded of his ceaseless search for cosmic simplicity —- and the scene takes on a deeper, sadder beauty.



Nalin Pithwa.

PS: Thinking takes time, practice, perseverance and solitude. The reward of an intellectual discovery, mathematical or other, is far richer and complete than instant gratification. :-))

You and your research ( You and your studies) : By Richard Hamming, AT and T, Bell Labs mathematician;

Although the title is grand (and quite aptly so)…the reality is that it can be applied to serious studies for IITJEE entrance, CMI entrance, highly competitive math olympiads, and also competitive coding contests…in fact, to various aspects of student life and various professional lifes…

Please read the whole article…apply it wholly or partially…modified or unmodified to your studies/research/profession…these are broad principles of success…


From Passive to Active Learning: India Today: Jamshed Bharucha: Aug 19 2019

(By Jamshed Bharucha; Vice Chancellor, SRM Amravati University)


On the value of time…essence of time management

Time is Life.

Money lost can come back, but time lost can never come back. Time is more valuable than money. (Bertrand Russell).

Safeguarding children from digital addiction — Kiran Bajaj, Principal, Greenlawns School.

Reproduced from The DNA newspaper, print edition, Mumbai, date April 17 2017, Monday; authored by Kiran Bajaj:

(From the Principal’s Desk) (The writer is Principal of Greenlawns High School, Mumbai.)

Generally, the word addiction brings to mind habits such as smoking, drinking or gambling, but in today’s digital age, addiction is more connected to technology. A child  could get addicted to iPads, television, or any kind of “screen”. Children develop an uncontrolled habit of indulging in certain activities even when they are warned that doing those things are not good for them.

Children are slowly becoming digital addicts. These days, some kids don’t just play with electronic toys but make it a part of their lives. Children carry their smartphones every where including the washroom as they feel disconnected from their social network without their phones. Today, young addicts include three year olds who scream when they can’t have their tablets to play on, or secondary school children who can’t quit “WhatsApping” or posting messages on Facebook, and others who compulsively play online games.

Toddlers are becoming couch potatoes almost as soon as they have the pram. No two children are alike and different children perceive the television medium differently. Studies have shown that watching TV at an early age does form a habit and that it has potentially damaging effects on their health. Some children watch TV while eating dinner, while doing homework, doing chores, etc. To prevent children from becoming couch potatoes, parents can encourage children to play board games, outdoor activities and socialize with friends.  Try to make the alternatives really fun at first, to help your child transition into watching less TV.

Families can help prevent addiction when there is a strong bond between children and parents, and a lot of parental involvement in the child’s life and discipline. Bringing in new games, books, and other activities will give children something better to do with their time while at home. For younger ones, simple sticker books and colouring books will keep them entertained for hours. Art and craft activities encourage children to use their imagination, as well as learning toys, puppet theaters, anything that gets your child thinking rather than just watching.



Nalin Pithwa.

PS: Thanks to DNA and Kiran Bajaj !

TV and video games or mobile games can spoil a kid’s concentration power

Vineeta Pandey

DNA India, Aug 5, 2010.

Study says attention problems may linger till child attains adolescence.

Children may find television viewing and playing video games more fun than playing with other children.

But this temporary, quick-fix solution comes with a whole set of long-lasting problems.

A study published in American Journal of Pediatrics said that viewing television and playing video games can cause serious attention problems among children. What’s worse is that the problems may linger till they attain adolescence and, in some cases, continue even in their youth.

The study says that this can be because most television programmes involve rapid changes in focus and frequent exposure to television has the potential to harm children’s abilities to sustain focus on tasks that are not inherently attention-grabbing.

Also, since most TV shows are exciting, children who frequently watch them have more difficulty paying attention to less exciting tasks like school work. Similar is the case with children playing video games.

Delhi-based psychologist Dr Aruna Broota said, “Frequent television viewing leads to attention problems among children. They lose interest in studies, books and reading newspapers. Even if they read, they tend to lose interest fast and often do not complete the full story or book. This is because on TV events jump from one theme to the next. Children’s emotions get blunted as a result of watching cartoons, etc, which are thrilling and exciting.”

“Similarly, video games that are often seen as gadgets to help gain concentration among children can, in fact, lead to concentration problems if played for more than half-an-hour,” Broota said.

Attention problems, often manifested in the form of attention-deficit/ hyperactivity disorder, are associated with negative outcomes among children and adolescents, which include poor performance in school and increased aggression.

The study says that exposure to television and video games was associated with greater attention problems among late adolescents and young adults. This indicated that a child’s attention span continued to remain affected irrespective of whatever age he or she was addicted to watching TV or playing video games.

Similar studies in TV viewing and playing video games have been linked with problems such as high blood pressure and disturbed sleep among children.

More tips on concentration and studies, later,

Nalin Pithwa

Tagore on Time Management

The butterfly counts not months, but moments, and has time enough. — Rabindranath Tagore, Poet Laureate.

Pursuit of Mathematics and Creativity in Mathematics

Reference: Adventures of a mathematician — Stanislaw Ulam.

While still a schoolboy in Lwów, then a city in Poland, Stanislaw Ulam signed his notebook “S. Ulam, astronomer, physicist and mathematician.”

Of these early interests perhaps it was natural that the talented young Ulam would eventually be attracted to mathematics; it is in this science that Poland has made its most distinguished intellectual contributions in this century. Ulam was fortunate to have been born into a wealthy Jewish family of lawyers, businessmen, and bankers who provided the necessary resources for him to follow his intellectual instincts and his early talent for mathematics. Eventually Ulam graduated with a doctorate in pure mathematics from the Polytechnic Institute at Lwów in 1933. As Ulam notes, the aesthetic appeal of pure mathematics lies not merely in the rigorous logic of the proofs and theorems, but also in the poetic elegance and economy in articulating each step in a mathematical presentation. This very fundamental and aristocratic form of mathematics was the concern of the school of Polish mathematicians in Lwów during Ulam’s early years.

The pure mathematicians at the Polytechnic Institute were not solitary academic recluses; they discussed and defended their theorems practically every day in the coffeehouses and tearooms of Lwów. This deeply committed community of mathematicians, in pursuing their work through collective discussion in public, allowed
talented young students like Ulam to observe the intellectual excitement and creativity of pure mathematics. Eventually young Ulam could participate on an equal footing with some of the most distinguished mathematicians of his day —- The long sessions at the cafes with Stefan Banach, Kazimir Kuratowski, Stanislaw Mazur, Hugo Steinhaus, and others set the tone of Ulam’s highly verbal and collaborative style early on. Ulam’s early mathematical work from this period was in set theory, topology, group theory, and measure. His experience with the lively school of mathematics in Lwów established Ulam’s lifelong, highly creative quest for new mathematical and scientific problems.


Becoming a mathematician in Poland

When I try to remember how I started to develop my interest in science I have to go back to certain pictures in a popular book on astronomy I had. It was a textbook called Astronomy of Fixed Stars, by Martin Ernst, a professor of astronomy at the University of Lwów. In it was a reproduction of a portrait of Sir Isaac Newton. I was nine or ten at the time, and at that age a child does not react consciously to the beauty of a face. Yet I remember distinctly that I considered this portrait— especially the eyes—as something marvelous. A mixture of physical attraction and a feeling of the mysterious emanated from his face. Later I learned it was the Geoffrey Kneller portrait of Newton as a young man, with hair to his shoulders and an open shirt. Other illustrations I distinctly remember were of the rings of Saturn and of the belts of Jupiter. These gave me a certain feeling of wonder, the flavor of which is hard to describe since it is sometimes associated with nonvisual impressions such as the feeling one gets from an exquisite example of scientific reasoning. But it reappears, from time to time, even in older age, just as a familiar scent will reappear. Occasionally an odor will come back, bringing coincident memories of childhood or youth.

Reading descriptions of astronomical phenomena today brings back to me these visual memories, and they reappear with a nostalgic (not melancholy but rather pleasant) feeling, when new thoughts come about or a new desire for mental work suddenly emerges.

The high point of my interest in astronomy and an unforgettable emotional experience came when my uncle Szymon Ulam gave me a little telescope. It was one of the copper or bronze tube variety and, I believe, a refractor with a twoinch objective. To this day, whenever I see an instrument of this kind in antique shops, nostalgia overcomes me, and after all these decades my thoughts still turn to visions of the celestial wonders and new astronomical problems.
At that time, I was intrigued by things which were not well understood—for example, the question of the shortening of the period of Encke’s comet. It was known that this comet irregularly and mysteriously shortens its threeyear
period of motion around the sun. Nineteenth century astronomers made several attempts to account for
this as being caused by friction or by the presence of some new invisible body in space. It excited me that nobody really knew the answer. I speculated whether the $latex 1/r^{2}$ law of attraction of Newton was not quite exact. I tried to imagine how it could affect the period of the comet if the exponent was slightly different from 2,
imagining what the result would be at various distances. It was an attempt to calculate, not by numbers and symbols, but by almost tactile feelings combined with reasoning, a very curious mental effort.

No star could be large enough for me. Betelgeuse and Antares were believed to be much larger than the sun (even though at the time no precise data were available) and their distances were given, as were parallaxes of many stars. I had memorized the names of constellations and the individual Arabic names of stars and their distances and luminosities. I also knew the double stars.

In addition to the exciting Ernst book another, entitled Planets and the Conditions of Life on Them, was strange. Soon I had some eight or ten astronomy books inmy library, including the marvelous NewcombEngelmann
Astronomie in German. The BodeTitius formula or “law” of planetary distances also fascinated me,
inspiring me to become an astronomer or physicist. This was about the time when, at the age of eleven or so, I inscribed my name in a notebook, “S. Ulam, astronomer, physicist, mathematician. My love for astronomy has never ceased; I believe it is one of the avenues that brought me to mathematics.” 

From today’s perspective Lwów may seem to have been a provincial city, but this is not so. Frequent lectures by scientists were held for the general public, in which such topics as new discoveries in astronomy, the new physics and the theory of relativity were covered. These appealed to lawyers, doctors, businessmen, and other laymen.

I had mathematical curiosity very early. My father had in his library a wonderful series of German paperback books—Reklam, they were called. One was Euler’Algebra. I looked at it when I was perhaps ten or eleven, and it gave me a mysterious feeling. The symbols looked like magic signs; I wondered whether one day I
could understand them. This probably contributed to the development of my mathematical curiosity. I discovered by myself how to solve quadratic equations. I remember that I did this by an incredible concentration and almost painful and not quiteconscious effort. What I did amounted to completing the square in my head without paper or pencil.

In high school, I was stimulated by the notion of the problem of the existence of odd perfect numbers. An integer is perfect if it is equal to the sum of all its divisors including one but not itself. For instance: 6 = 1 + 2 + 3 is perfect. So is 28 = 1 + 2 + 4 + 7 + 14. You may ask: does there exist a perfect number that is odd? The answer is unknown to this day.

In general, the mathematics classes did not satisfy me. They were dry, and I did not like to have to memorize certain formal procedures. I preferred reading on my

At about fifteen I came upon a treatise on the infinitesimal calculus in a book by Gerhardt Kowalevski. I did not have enough preparation in analytic geometry or even in trigonometry, but the idea of limits, the definitions of real numbers, the notion of derivatives and integration puzzled and excited me greatly. I decided to read a page
or two a day and attempt to learn the necessary facts about trigonometry and analytic geometry from other books.
I found two other books in a secondhand bookstore. These intrigued and fascinated me more than anything else for many years to come: Sierpinski’s Theory of Sets and a monograph on number theory. At the age of seventeen I knew as much or more elementary number theory than I do now.

I also read a book by the mathematician Hugo Steinhaus entitled What Is and What Is Not Mathematics and in Polish translation Poincaré’s wonderful La Science et l’Hypothèse, La Science et la Méthode, La Valeur de la Science, and his Dernières Pensées. Their literary quality, not to mention the science, was admirable. Poincaré molded portions of my scientific thinking. Reading one of’ his books today demonstrates how many wonderful truths have remained, although everything in mathematics has changed almost beyond recognition and in physics perhaps even more so. I admired Steinhaus’s book almost as much, for it gave many examples of actual mathematical problems.

The mathematics taught in school was limited to algebra, trigonometry, and the very beginning of analytic geometry. In the seventh and eighth classes, where the students were sixteen and seventeen, there was a course on elementary logic and a survey of history of philosophy. The teacher, Professor Zawirski, was a real scholar, a lecturer at the University and a very stimulating man. He gave us glimpses of recent developments in advanced modern logic. Having studied Sierpinski’s books on the side, I was able to engage him in discussions of set theory during recess and in his office. I was working on some problems on transfinite numbers and on the problem of the continuum hypothesis.

I also engaged in wild mathematical discussions, formulating vast and new projects, new problems, theories and methods bordering on the fantastic, with a boy named Metzger, some three or four years my senior. He had been directed toward me by friends of’ my father who knew that he too had a great interest in mathematics.

In the fall of 1927 I began attending lectures at the Polytechnic Institute in the Department of General Studies, because the quota of Electrical Engineering already was full. The level of the instruction was obviously higher than that at high school, but having read Poincaré and some special mathematical treatises, I naively expected
every lecture to be a masterpiece of style and exposition. Of course, I was disappointed.

As I knew many of the subjects in mathematics from my studies, I began to attend a second year course as an auditor. It was in set theory and given by a young professor fresh from Warsaw, Kazimir Kuratowski, a student of Sierpinski, Mazurkiewicz, and Janiszewski. He was a freshman professor, so to speak, and I a freshman student. From the very first lecture I was enchanted by the clarity, logic, and polish of his exposition and the material he presented. From the beginning I participated more actively than most of the older students in discussions with Kuratowski, since I knew something of the subject from having read Sierpinski’s book. I think he quickly noticed that I was one of the better students; after class he would give me individual attention. This is how I started on my career as a mathematician, stimulated by Kuratowski.

Soon I could answer some of the more difficult questions in the set theory course, and I began to pose other problems. Right from the start I appreciated Kuratowski’s patience and generosity in spending so much time with a novice. Several times a week I would accompany him to his apartment at lunch time, a walk of about twenty
minutes, during which I asked innumerable mathematical questions. Years later, Kuratowski told me that the questions were sometimes significant, often original, and
interesting to him.
My courses included mathematical analysis, calculus, classical mechanics, descriptive geometry, and physics. Between classes, I would sit in the offices of some of the mathematics instructors. At that time I was perhaps more eager than at any other time in my life to do mathematics to the exclusion of almost any other activity.

It was there that I first met Stanislaw Mazur, who was a young assistant at the University. He came to the Polytechnic Institute to work with Orlics, Nikliborc and Kaczmarz, who were a few years his senior.
In conversations with Mazur I began to learn about problems in analysis. I remember long hours of sitting at a desk and thinking about the questions which he broached to me and discussed with the other mathematicians. Mazur introduced me to advanced ideas of real variable function theory and the new functional analysis.

We discussed some of the more recent problems of Banach, who had developed a new approach to this theory.
Banach himself would appear occasionally, even though his main work was at the University. I met him during this first year, but our acquaintance began in a more meaningful, intimate, and intellectual sense a year or two later.

At the beginning of the second semester of my freshman year, Kuratowski told me about a problem in set theory that involved transformations of sets. It was connected with a well known theorem of Bernstein: if 2A = 2B, then A = B, in the arithmetic sense of infinite cardinals. This was the first problem on which I really spent arduous hours of thinking. I thought about it in a way which now seems mysterious to me, not consciously or explicitly knowing what I was aiming at. So immersed in some aspects was I, that I did not have a conscious overall view. Nevertheless, I managed to show by means of a construction how to solve the problem, devising a method of representing by graphs the decomposition of sets and the corresponding transformations. Unbelievably, at the time I thought I had invented the very idea of graphs.

I wrote my first paper on this in English, which I knew better than German or French. Kuratowski checked it and the short paper appeared in 1928 in Fundamenta Mathematicae, the leading Polish mathematical journal which he edited. This gave me self confidence.

I still was not certain what career or course of work I should pursue. The practical chances of becoming a professor of mathematics in Poland were almost nil—there were few vacancies at the University. My family wanted me to learn a profession, and so I intended to transfer to the Department of Electrical Engineering for my
second year. In this field the chance of making a living seemed much better.

Before the end of the year Kuratowski mentioned in a lecture another problem in set theory. It was on the existence of set functions which are “subtractive” but not completely countably additive. I remember pondering the question for weeks. I can still feel the strain of thinking and the number of attempts I had to make. I gave myself an ultimatum. If I could solve this problem, I would continue as a mathematician. If not, I would change to electrical engineering.
After a few weeks I found a way to achieve a solution. I ran excitedly to Kuratowski and told him about my solution, which involved transfinite induction. Transfinite induction had been used by mathematical workers many times in other connections; however, I believe that the way in which I used it was novel.

I think Kuratowski took pleasure in my success, encouraging me to continue in mathematics. Before the end of my first college year I had written my second paper, which Kuratowski presented to Fundamenta. Now, the die was cast. I began to concentrate on the “impractical” possibilities of an academic career. Most of what people
call decision making occurs for definite reasons. However, I feel that for most of us what is ultimately called a “decision” is a sort of vote taken in the subconscious, in which the majority of the reasons favoring the decision win out.

The mathematics offices of the Polytechnic Institute continued to be my hangout. I spent mornings there, every day of the week, including Saturdays. (Saturdays were not considered to be part of the weekend then; classes were held on Saturday mornings.)

Mazur appeared often, and we started our active collaboration on problems of function spaces. We found a solution to a problem involving infinitely dimensional vector spaces. The theorem we proved—that a transformation preserving distances is linear—is now part of the standard treatment of the geometry of function
spaces. We wrote a paper which was published in the Compte Rendus of the French Academy.

It was Mazur (along with Kuratowski and Banach) who introduced me to certain large phases of mathematical thinking and approaches. From him I learned much about the attitudes and psychology of research. Sometimes we would sit for hours in a coffee house. He would write just one symbol or a line like y = f(x) on a piece of paper, or on the marble table top. We would both stare at it as various thoughts were suggested and discussed. These symbols in front of us were like a crystal ball to help us focus our concentration. Years later in America, my friend Everett and I often had similar sessions, but instead of a coffee house they were held in an office with a blackboard. Mazur’s forte was making what he called “observations and remarks.” These stated—usually in a concise and precise form—some properties of notions. Once made, they were perhaps not so difficult to verify, for sometimes they were peripheral to the usual formulations and had gone unnoticed. They were often decisive in solving problems.


Banach enjoyed long mathematical discussions with friends and students. I recall a session with Mazur and Banach at the Scottish Café which lasted seventeen hours without interruption except for meals. What impressed me most was the way he could discuss mathematics, reason about mathematics, and find proofs in these conversations. Since many of these discussions took place in neighborhood coffee houses or little inns, some mathematicians also dined there frequently. It seems to me now the food must have been mediocre, but the drinks were plentiful. The tables had white marble tops on which one could write with a pencil, and, more important, from which notes could be easily erased. There would be brief spurts of conversation, a few lines would be written on the table, occasional laughter would come from some of the participants, followed by long periods of silence during which we just drank coffee and stared vacantly at each other. The café clients at neighboring tables must have been puzzled by these strange doings. It is such persistence and habit of concentration which somehow becomes the most important prerequisite for doing genuinely creative mathematical work.



Nalin Pithwa