I recently finished the novel Shantaram, by Gregory David Roberts. As I’m not a professional book reviewer, let me instead quote from the Amazon review:

Crime and punishment, passion and loyalty, betrayal and redemption are only a few of the ingredients in Shantaram, a massive, over-the-top, mostly autobiographical novel. Shantaram is the name given Mr. Lindsay, or Linbaba, the larger-than-life hero. It means “man of God’s peace,” which is what the Indian people know of Lin. What they do not know is that prior to his arrival in Bombay he escaped from an Australian prison where he had begun serving a 19-year sentence. He served two years and leaped over the wall. He was imprisoned for a string of armed robberies performed to support his heroin addiction, which started when his marriage fell apart and he lost custody of his daughter. All of that is enough for several lifetimes, but for Greg Roberts, that’s only the beginning.

He arrives in Bombay with little money, an assumed name, false papers, an untellable past, and no plans for the future. Fortunately, he meets Prabaker right away, a sweet, smiling man who is a street guide. He takes to Lin immediately, eventually introducing him to his home village, where they end up living for six months. When they return to Bombay, they take up residence in a sprawling illegal slum of 25,000 people and Linbaba becomes the resident “doctor.” With a prison knowledge of first aid and whatever medicines he can cadge from doing trades with the local Mafia, he sets up a practice and is regarded as heaven-sent by these poor people who have nothing but illness, rat bites, dysentery, and anemia. He also meets Karla, an enigmatic Swiss-American woman, with whom he falls in love. Theirs is a complicated relationship, and Karla’s connections are murky from the outset.

While it was a cracking good read, what struck me particularly were the surprising mathematical allusions that the author used throughout the novel. In this mini-series, I’d like to explore the ones that I found.

In this third installment, the narrator a sudden realization that he had.

I put all of my focus on the beating of my heart, trying by force of will to slow its too-rapid pace. It worked, after a time. I closed around a single, still thought. That thought was of [a mafia don], and the formula he’d made me repeat so often: “The wrong thing, for the right reasons.” And I knew, as I repeated the words in the fearing dark, that the fight with [another mafia don], the war, the struggle for power, was always the same, everywhere, and it was always wrong.

[My mafia don], no less than [other mafia dons] and all the rest of them, were pretending that their little kingdoms made them kings; that their power struggles made them powerful. And they didn’t. They couldn’t. I saw that then so clearly that it was like understanding a mathematical theorem for the first time. The only kingdom that makes any man a king is the kingdom of his own soul. The only power that has any real meaning is the power to better the world. And only men like [my noble friends, not in the mafia] were such kings and had such power.

Shantaram, Chapter 41

The author’s choice of language is music to my ears: “I saw that then so clearly that it was like understanding a mathematical theorem for the first time.” There have been many, many times throughout my education and career that I struggled to understand some theorem. But the moment that I figured it out, I couldn’t believe what had taken me so long to finally get it. That’s the type of epiphany that the author seems to be describing.

I again quote at length from Richard P. Feynman, who did a far better job of explaining the emotions of such a sudden realization after being stuck in a rut than I ever could:

Then I had another thought: Physics disgusts me a little bit now, but I used to enjoy doing 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 out 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. That didn’t make any difference: I’d invent things and play with things for my own entertainment.

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 any 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 discover 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 come out 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.

Richard P. Feynman, “The Dignified Professor,” from Surely You’re Joking, Mr. Feynman!

I recently finished the novel Shantaram, by Gregory David Roberts. As I’m not a professional book reviewer, let me instead quote from the Amazon review:

Crime and punishment, passion and loyalty, betrayal and redemption are only a few of the ingredients in Shantaram, a massive, over-the-top, mostly autobiographical novel. Shantaram is the name given Mr. Lindsay, or Linbaba, the larger-than-life hero. It means “man of God’s peace,” which is what the Indian people know of Lin. What they do not know is that prior to his arrival in Bombay he escaped from an Australian prison where he had begun serving a 19-year sentence. He served two years and leaped over the wall. He was imprisoned for a string of armed robberies performed to support his heroin addiction, which started when his marriage fell apart and he lost custody of his daughter. All of that is enough for several lifetimes, but for Greg Roberts, that’s only the beginning.

He arrives in Bombay with little money, an assumed name, false papers, an untellable past, and no plans for the future. Fortunately, he meets Prabaker right away, a sweet, smiling man who is a street guide. He takes to Lin immediately, eventually introducing him to his home village, where they end up living for six months. When they return to Bombay, they take up residence in a sprawling illegal slum of 25,000 people and Linbaba becomes the resident “doctor.” With a prison knowledge of first aid and whatever medicines he can cadge from doing trades with the local Mafia, he sets up a practice and is regarded as heaven-sent by these poor people who have nothing but illness, rat bites, dysentery, and anemia. He also meets Karla, an enigmatic Swiss-American woman, with whom he falls in love. Theirs is a complicated relationship, and Karla’s connections are murky from the outset.

While it was a cracking good read, what struck me particularly were the surprising mathematical allusions that the author used throughout the novel. In this mini-series, I’d like to explore the ones that I found.

In this second installment, the narrator describes a conversation with a new acquaintance.

“My father was a teacher of chemistry and mathematics… My father was a stubborn man — it is a kind of stubbornness that permits one to become a mathematician, isn’t it? Perhaps mathematics is itself a kind of stubbornness, do you think?”

“Maybe,” I replied, smiling. “I never thought about it that way, but maybe you’re right.”

Shantaram, Chapter 26

When I think of stubbornness, I think of the determination of a marathon runner to push through fatigue to keep running hour after hour to complete all 26.2 miles of the course. I don’t usually think of a mathematician.

Nevertheless, the author certainly hit on something with this allusion. Mathematicians certainly need a healthy dose of stubbornness when staring a conjecture and trying to figure out its proof; it’s normal for that frustration to last for weeks, months, or even years. That said, I wouldn’t say that this is unique to mathematicians — researchers in just about any field of study need to be persistent to discover something that nobody else has figure out before.

Nobel Prize laureate Richard P. Feynman had a couple of vivid descriptions about the frustration of getting stuck on a research project and the stubbornness that was necessary to break out of that rut.

I don’t believe I can really do without teaching. The reason is, I have to have something so that when I don’t have any ideas and I’m not getting anywhere I can say to myself, “At least I’m living; at least I’m doing something; I’m making some contribution”–it’s just psychological.

When I was at Princeton in the 1940s I could see what happened to those great minds at the Institute for Advanced Study, who had been specially selected for their tremendous brains and were now given this opportunity to sit in this lovely house by the woods there, with no classes to teach, with no obligations whatsoever. These poor bastards could now sit and think clearly all by themselves, OK? So they don’t get any ideas for a while: They have every opportunity to do something, and they’re not getting any ideas. I believe that in a situation like this a kind of guilt or depression worms inside of you, and you begin to worry about not getting any ideas. And nothing happens. Still no ideas come.

Nothing happens because there’s not enough real activity and challenge: You’re not in contact with the experimental guys. You don’t have to think how to answer questions from the students. Nothing!

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.”

Richard P. Feynman, “The Dignified Professor,” from Surely You’re Joking, Mr. Feynman!

Also from Feynman:

The problem was to find the right laws of beta decay. There appeared to be two particles, which were called a tan and a theta. They seemed to have almost exactly the same mass, but one disintegrated into two pions, and the other into three pions. Not only did they seem to have the same mass, but they also had the same lifetime, which is a funny coincidence. So everybody was concerned about this…

At that particular time I was not really quite up to things: I was always a little behind. Everybody seemed to be smart, and I didn’t feel I was keeping up…

Anyway, the discovery of parity law violation was made, experimentally, by Wu, and this opened up a whole bunch of new possibilities for beta decay theory, It also unleashed a whole host of experiments immediately after that. Some showed electrons coming out of the nuclei spun to the left, and some to the right, and there were all kinds of experiments, all kinds of interesting discoveries about parity. But the data were so confusing that nobody could put things together.

At one point there was a meeting in Rochester–the yearly Rochester Conference. I was still always behind, and Lee was giving his paper on the violation of parity. He and Yang had come to the conclusion that parity was violated, and flow he was giving the theory for it.

During the conference I was staying with my sister in Syracuse. I brought the paper home and said to her, “I can’t understand these things that Lee and Yang are saying. It’s all so complicated.”

“No,” she said, “what you mean is not that you can’t understand it, but that you didn’t invent it. You didn’t figure it out your own way, from hearing the clue. What you should do is imagine you’re a student again, and take this paper upstairs, read every line of it, and check the equations. Then you’ll understand it very easily.”

I took her advice, and checked through the whole thing, and found it to be very obvious and simple. I had been afraid to read it, thinking it was too difficult.

Richard P. Feynman, “The 7 Percent Solution,” from Surely You’re Joking, Mr. Feynman!