1) Read “A Beautiful Mind: A Biography of John Forbes Nash, Jr.” by Sylvia Nasar.
“On the afternoon of the Nobel announcement, Nash said that he had won for game theory and that he felt that game theory was like string theory, a subject of great intrinsic intellectual interest that the world wishes to imagine can be of some utility. He said it with enough skepticism in his voice to make it funny.”
–“A Beautiful Mind: A Biography of John Forbes Nash, Jr.” (1998)
2) Sometimes one can get considerable inspiration from famous scholars:
Writer and Statesman Johann von Goeth:
“What advantage does he derive from the system of bookkeeping by double-entry! It is among the finest inventions of the human mind.”
—Wilhelm Meister’s Apprenticeship
Physicist and scholar Richard Feynman:
Books and Audio & Video Tapes of Richard Feynman
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On Academics
Now you may ask, “What is mathematics doing in a physics lecture?” We have several possible excuses: first, of course, mathematics is an important tool, but that would only excuse us for giving the formula in two minutes. On the other hand, in theoretical physics we discover that all our laws can be written in mathematical form; and that this has a certain simplicity and beauty about it. So, ultimately, in order to understand nature it may be necessary to have a deeper understanding of mathematical relationships. But the real reason is that the subject is enjoyable, and although we humans cut nature up in different ways, and we have different courses in different departments, such compartmentalization is really artificial, and we should take our intellectual pleasures where we find them. — The Feynman Lectures on Physics – Vol. 1 (Algebra)
On teaching and research
In any thinking process there are moments when everything is going good and you’ve got wonderful ideas. Teaching is an interruption, and so it’s the greatest pain in the neck in the world. And then there are the longer periods of time when not much is coming to you. You’re not getting any ideas, and if you’re doing nothing at all, it drives you nuts! You can’t even say, “I’m teaching my class.”
If you’re teaching a class, you can think about the elementary things that you know so very well. These things are kind of fun and delightful. It doesn’t do any harm to think of them over again. Is there a better way to present them? The elementary things are easy to think about; if you can’t think of a new thought, no harm done; what you thought about it before is good enough for the class. If you do think of something new, you’re rather pleased that you have a new way of looking at it.
The questions of the students are often the source of new research. They often ask profound questions that I’ve thought about at times and then give up on, so to speak, for a while. It wouldn’t do any harm to think about them again and see if I can go any further now. The students may not be able to see the things I want to answer, or the subtleties I want to think about, but they remind me of a problem by asking questions in the neighborhood of that problem. It’s not so easy to remind yourself of these things.
So I find that teaching and the students keep life going, and I would never accept any position in which somebody has invented a happy situation for me where I don’t have to teach. Never. — Surely You’re Joking, Mr. Feynman (The Dignified Professor)
Then I had another thought: Physics disgusts me a little bit now, but I used to enjoy physics. Why did I enjoy it? I used to play with it. I used to do whatever I felt like doing – it didn’t have to do with whether it was important for the development of nuclear physics, but whether it was interesting and amusing for me to play with. When I was in high school, I’d see water running out of a faucet growing narrower, and wonder if I could figure what determines that curve. I found it was rather easy to do. I didn’t have to do it; it wasn’t important for the future of science; somebody else had already done it. This didn’t make any difference: I’d invent things and play with things.
So I got this new attitude. Now that I am burned out and I’ll never accomplish anything, I’ve got this nice position at the university teaching classes which I rather enjoy, and just like I read the Arabian Nights for pleasure, I’m going to play with physics, whenever I want to, without worrying about importance whatsoever.
Within a week I was in the cafeteria and some guy, fooling around, throws a plate in the air. As the plate went up in the air I saw it wobble, and I noticed the red medallion of Cornell on the plate going around. It was pretty obvious to me that the medallion went around faster than the wobbling.
I had nothing to do, so I start to figure out the motion of the rotating plate. I discovered that when the angle is very slight, the medallion rotates twice as fast as the wobble rate – two to one. It came out of a complicated equation! Then I thought, “Is there some way I can see in a more fundamental way, by looking at the forces or the dynamics, why it’s two to one?
I don’t remember how I did it, but I ultimately worked out what the motion of the mass particles is, and how all the accelerations balance to make it two to one.
I still remember going to Hans Bethe and saying, “Hey, Hans! I noticed something interesting. Here the plate goes around so, and the reason it’s two to one is … ” and I showed him the accelerations.
He says, “Feynman, that’s pretty interesting, but what’s the importance of it? Why are you doing it?”
“Hah!” I say. “There’s no importance whatsoever. I’m just doing it for the fun of it.” His reaction didn’t discourage me; I had made up my mind I was going to enjoy physics and do whatever I liked.
I went on to work out equations of wobbles. Then I thought about how electron orbits start to move in relativity. Then there’s the Dirac Equation in electrodynamics. And then quantum electrodynamics. And before I knew it (it was a very short time) I was “playing” – working, really – with the same old problem that I loved so much, that I had stopped working on when I went to Los Alamos: my thesis-type problems; all those old-fashioned, wonderful things.
It was effortless. It was easy to play with these things. It was like uncorking a bottle: Everything flowed out effortlessly. I almost tried to resist it! There was no importance to what I was doing, but ultimately there was. The diagrams and the whole business that I got the Nobel Prize for came from that piddling around with the wobbling plate. — Surely You’re Joking, Mr. Feynman (The Dignified Professor)
Now that we had got the classical theory right, Wheeler said, “Feynman, you’re a young fella – you should give a seminar on this. You need experience in giving talks. Meanwhile, I’ll work out the quantum theory part and give a seminar on that later.
So it was my first technical talk, and Wheeler made arrangements with Eugene Wigner to put it on the regular seminar schedule.
A day or two before the talk I saw Wigner in the hall. “Feynman,” he said, “I think that work you’re doing with Wheeler is very interesting, so I’ve invited Russell to the seminar.” Henry Norris Russell, the famous, great astronomer of the day, was coming to the lecture!
Wigner went on. “I think Professor von Neumann would also be interested. “Johnny von Neumann was the greatest mathematician around. “And Professor Pauli is visiting from Switzerland, so it happens, so I’ve invited Professor Pauli to come” – Pauli was a very famous physicist – and by this time I’m turning yellow. Finally, Wigner said, “Professor Einstein only rarely comes to our weekly seminars, but your work is so interesting that I’ve invited him specially, so he’s coming too.”
So I prepared the talk, and when the day came, I went in and did something that young men who have had no experience in giving talks often do – I put too many equations up on the blackboard. You see, a young fella doesn’t know how to say, “Of course, that varies inversely, and this goes this way . . . ” because everybody listening already knows; they can see it. But he doesn’t know. He can only make it come out by actually doing the algebra – and therefore the reams of equations.
As I was writing these equations all over the blackboard ahead of time, Einstein came in and said pleasantly, “Hello, I’m coming to your seminar. But first, where is the tea?” I told him, and continued writing the equations.
Then the time came to give the talk, and here are these monster minds in front of me, waiting! I remember very clearly seeing my hands shaking as they were pulling out my notes from a brown envelope.
But then a miracle occurred, as it has occurred again and again in my life, and it’s very lucky for me: the moment I start to think about the physics, and have to concentrate on what I’m explaining, nothing else occupies by mind – I’m completely immune to being nervous. So after I started to go, I just didn’t know who was in the room. I was only explaining the idea, that’s all. — Surely You’re Joking, Mr. Feynman (Monster Minds)
U.S. Economist George Stigler:
On Graduate School
The better students, moreover, are at the better schools. Most important scholars I have know received their training at major graduate schools, no matter how ordinary their undergraduate education. On the other hand, many extremely promising undergraduates must go to the lesser graduate centers, because of personal circumstances, ignorance of their quality, better financial aid, or whatever. Why have so few of these latter students succeeded in research? After all, the material that is taught in the major graduate centers is in print and available everywhere. My explanation is that in the leading graduate centers the students learn primarily from one another. They learn to impose higher standards upon themselves, both in the selection of problems to work on and in the adequacy of the solutions they provide to these problems. Bull sessions are a more effective method of teaching and learning than classroom lectures or discussion. One colleague has said that he considered his role in the classroom to be that of providing topics for bull sessions. I don’t think that the successes achieved by the graduates of the major schools are due simply to an old-boy network. Anyone who does outstanding work is in strong demand even if, like the hypothetical mathematician I referred to, he has few other redeeming traits. — Memoirs of an Unregulated Economist
Author Robert Kanigel:
On (The Genius) Ramanujan
The popular English magazine Strand had long carried a page, entitled “Perplexities,” devoted to intriguing puzzles, numbered and charmingly titled, like “The Fly and the Honey,” or “The Tessellated Tiles,” the answers being furnished the following month. Each Christmas, though, “Perplexities” expanded, the author fitting his puzzles into a short story. Now, in December 1914, “Puzzles at a Village Inn” took readers to the imaginary town of Little Wurzelfold, where the main topic of interest was what had just happened in Louvain.
In late August, pursuing an explicit policy of brutalization against civilian populations, German troops began burning the medieval Belgian city of Louvain, on the road between Liege and Brussels. House by house and street by street they set Louvain to the torch, destroying its great library, with its quarter million books and medieval manuscripts, and killing many civilians. The burning of Louvain horrified the world, galvanized public opinion against Germany, and united France, Russia, and England more irrevocably yet. “The March of the Hun,” English newspapers declared. “Treason to Civilization.” It was an early turning point of the war, doing much to set its tone. Louvain came to symbolize the breakdown of civilization. And now it reached even the “Perplexities” page of Strand.One Sunday morning soon after the December issue appeared, P. C. Mahalanobis sat with it at a table in Ramanujan’s rooms in Whewell’s Court. Mahalanobis was the King’s College student, just then preparing for the natural sciences Tripos, who had found Ramanujan shivering by the fireplace and schooled him in the nuances of the English blanket. Now, with Ramanujan in the little back room stirring vegetables over the gas fire, Mahalanobis grew intrigued by the problem and figured he’d try it out on his friend.
“Now here’s a problem for you,” he yelled into the next room. “What problem? Tell me,” said Ramanujan, still stirring. And Mahalanobis read it to him.
“I was talking the other day,” said William Rogers to the other villagers gathered around the inn fire, “to a gentleman about the place called Louvain, what the Germans have burnt down. He said he knowed it well…used to visit a Belgian friend there. He said the house of his friend was in a long street, numbered on this side one, two, three, and so on, and that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him. Funny thing that! He said he knew there was more than fifty houses on that side of the street, but not so many as five hundred. I made mention of the matter to our person, and he took a pencil and worked out the number of the house where the Belgian lived. I don’t know how he done it.” Perhaps the reader may like to discover the number of that house.
Through trial and error, Mahalanobis (who would go on to found the Indian Statistical Institute and become a Fellow of the Royal Society) had figured it out in a few minutes. Ramanujan figured it out, too, but with a twist. “Please take down the solution,” he said – and proceeded to dictate a continued fraction, a fraction whose denominator consists of a number plus a fraction, that fraction’s denominator consisting of a number plus a fraction, ad infinitum. This wasn’t just the solution to the problem, it was the solution to the whole class of problems implicit in the puzzle. As stated, the problem had but one solution – house no. 204 in a street of 288 houses; I + 2 + . . . 203 = 205 + 206 + . . . 288. But without the 50-to-500 house constraint, there were other solutions. For example, on an eight-house street, no. 6 would be the answer: I + 2 + 3 + 4 + 5 on its left equaled 7 + 8 on its right. Ramanujan’s continued fraction comprised within a single expression all the correct answers.
Mahalonobis was astounded. How, he asked Ramanujan, had he done it?
“Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, which continued fraction? And the answer came to my mind.”
The answer came to my mind. That was the glory of Ramanujan – that so much came to him so readily, whether through the divine offices of the goddess Namagiri, as he sometimes said, or through what Westerners might ascribe, with equal imprecision, to “intuition.” And yet, it was the very power of his intuition that, in one sense, undermined his mathematical development. For it blinded him to intuition’s limits, gave him less reason to learn modern mathematical tools, shielded him from his own ignorance.
“The limitations of his knowledge were as startling as its profundity,” Hardy would write. — The Man Who Knew Infinity: A Life of the Genius Ramanujan