Intuition

It appears some of the best scientists have an intuitive grasp of the world around them. Take Richard Feynman.

Julian Schwinger closed Feynman's obituary in this way:

An honest man, the outstanding intuitionist of our age, and a prime example of what may lie in store for anyone who dares to follow the beat of a different drum.

Murray Gell-Mann described (I believe sardonically) the Feynman Problem-Solving Algorithm:

1. Write down the problem.
   
2. Think very hard.
   
3. Write down the answer.

Hans Bethe went as far as to say:

In science, as well as in other fields of human endeavor, there are two kinds of geniuses: the “ordinary” and the “magicians.”

An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it.

It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark. They seldom, if ever, have students because they cannot be emulated and it must be terribly frustrating for a brilliant young mind to cope with the mysterious ways in which the magician’s mind works.

Richard Feynman is a magician of the highest caliber.

Freeman Dyson explained Feynman's science in this manner:

The reason Dick's physics was so hard for ordinary people to grasp was that he did not use equations . . . Dick just wrote down the solutions of out of his head without ever writing down the equations. He had a physical picture of the way things happen, and the picture gave him the solutions directly with a minimum of calculation. It was no wonder that people who had spent their lives solving equations were baffled by him. Their minds were analytical; his was pictorial.

Robert Oppenheimer wrote a letter to Raymond Birge, the chairman of the physics department at the University of California-Berkeley, in an attempt to secure Feynman for a professorship:

As you know, we have quite a number of physicists here, and I have run into a few who are young and whose qualities I had not known before. Of these there is one who is in every way so outstanding and so clearly recognized as such, that I think it appropriate to call his name to your attention, with the urgent request that you consider him for a position in the department at the earliest time that that is possible. You may remember the name because he once applied for a fellowship in Berkeley: it is Richard Feynman. He is by all odds the most brilliant young physicist here, and everyone knows this. He is a man of thoroughly engaging character and personality, extremely clear, extremely normal in all respects, and an excellent teacher with a warm feeling for physics in all its aspects. He has the best possible relations both with the theoretical people of whom he is one, and with the experimental people with whom he works in very close harmony.

The reason for telling you about him now is that his excellence is so well known, both at Princeton where he worked before he came here, and to a not inconsiderable number of "big shots" on this project, that he has already been offered a position for the post war period, and will most certainly be offered others. I feel that he would be a great strength for our department, tending to tie together its teaching, its research and its experimental and theoretical aspects. I may give you two quotations from men with whom he has worked. Bethe has said that he would rather lose any two other men than Feyman from this present job, and Wigner said, "He is a second Dirac, only this time human."

Don P. Mitchell, one of Feynman's former students, has written:

The best lecture I recall started out with Feynmann suggesting that he stop the course, because it wasn't really getting anyplace. Then he decided to talk about what he was doing right then, as an example of real research. He was interested in quantum chromodynamics, and the big frustration at the time was that people had a theory, but it was too difficult to evaluate it and predict numerical results of experiments. He explained that in cases like this, it was hard to know where to start. He wanted to "understand" aspects of the theory, and develop intuition. For example, asymptotic confinement (quarks seem to bind together more tightly as you pull them apart).

He told us that he approaches these issues by rehearsal. 'Think of a simpler problem that seems similar. If you can't solve that, think of a simpler one. If you can solve it, then SOLVE IT. Don't just say you know how to solve it. After that, you might think of a way to attack the harder problem. You might realize something.' Basically, keep poking around and give your intuition a chance to develop and wait for ideas to pop into your head.

He suggested looking at an hydrogen atom in 2D. He noted that in that case, there was an infinite ionization energy, all the states were bound. He did some of the work on the board with class participation (and people occasionally yelling out minor corrections to his math). Everyone was very excited and eager. He computed the energy levels. Then he computed the width of the energy bands. They overlapped! What did that mean? Was the energy really quantized, or did it behave like a continuum then? It was amazing that such a trivial problem would quickly lead to a mystery.

He ended the lecture by charging everyone with the task of "learning something new about two dimensions" that they could report to the class. "Anything new. I don't care how trivial it is." And he meant "anything". He wasn't the least bit afraid to do something trivial, but maybe with a different viewpoint. Then show it to people with delight. He wasn't the least bit afraid to ask a dumb question at a talk--often a question that lots of other people wanted to ask.

Paul Davies writes in the preface to Six Easy Pieces:

To place QED on a sound basis it was necessary to make the theory consistent not only with the principles of quantum mechanics but with those of the special theory of relativity too. These two theories come with their own distinctive mathematical machinery, complicated systems of equations that can indeed be combined and reconciled to yield a satisfactory description of QED. Doing this was a tough undertaking, requiring a high degree of mathematical skill, and was the approach followed by Feynman’s contemporaries. Feynman himself, however, took a radically different route—so radical, in fact, that he was more or less able to write down the answers straightaway without using any mathematics!

To aid this extraordinary feat of intuition, Feynman invented a simple system of eponymous diagrams. Feynman diagrams are a symbolic but powerfully heuristic way of picturing what is going on when electrons, photons, and other particles interact with each other. These days Feynman diagrams are a routine aid to calculation, but in the early 1950s they marked a startling departure from the traditional way of doing theoretical physics.

[. . .]

Physics is an exact science, and the existing body of knowledge, while incomplete, can’t simply be shrugged aside. Feynman acquired a formidable grasp of the accepted principles of physics at a very young age, and he chose to work almost entirely on conventional problems. He was not the sort of genius to beaver away in isolation in a backwater of the discipline and to stumble across the profoundly new. His special talent was to approach essentially mainstream topics in an idiosyncratic way. This meant eschewing existing formalisms and developing his own highly intuitive approach. Whereas most theoretical physicists rely on careful mathematical calculation to provide a guide and a crutch to take them into unfamiliar territory, Feynman’s attitude was almost cavalier. You get the impression that he could read nature like a book and simply report on what he found, without the tedium of complex analysis.

Indeed, in pursuing his interests in this manner Feynman displayed a healthy contempt for rigorous formalisms. It is hard to convey the depth of genius that is necessary to work like this. Theoretical physics is one of the toughest intellectual exercises, combining abstract concepts that defy visualization with extreme mathematical complexity. Only by adopting the highest standards of mental discipline can most physicists make progress. Yet Feynman appeared to ride roughshod over this strict code of practice and pluck new results like ready-made fruit from the Tree of Knowledge.

[. . .]

Although quantum mechanics had made the textbooks by the early 1930s, it is typical of Feynman that, as a young man, he preferred to refashion the theory for himself in an entirely new guise. The Feynman method has the virtue that it provides us with a vivid picture of nature’s quantum trickery at work. The idea is that the path of a particle through space is not generally well defined in quantum mechanics. We can imagine a freely moving electron, say, not merely traveling in a straight line between A and B as common sense would suggest, but taking a variety of wiggly routes. Feynman invites us to imagine that somehow the electron explores all possible routes, and in the absence of an observation about which path is taken we must suppose that all these alternative paths somehow contribute to the reality. So when an electron arrives at a point in space—say a target screen—many different histories must be integrated together to create this one event.

Feynman’s so-called path-integral, or sum-over-histories approach to quantum mechanics, set this remarkable concept out as a mathematical procedure. It remained more or less a curiosity for many years, but as physicists pushed quantum mechanics to its limits—applying it to gravitation and even cosmology—so the Feynman approach turned out to offer the best calculational tool for describing a quantum universe. History may well judge that, among his many outstanding contributions to physics, the path-integral formulation of quantum mechanics is the most significant.

Here's an excerpt from an interview titled "The Smartest Man in the World" with Omni Magazine:

Omni: Maybe it’s the way the textbooks are written, but few people outside science appear to know just how quickly real, complicated physical problems get out of hand as far as theory is concerned.

Feynman: That’s very bad education. The lesson you learn as you grow older in physics is that what we can do is a very small fraction of what there is. Our theories are really very limited.

Omni: Do physicists vary greatly in their ability to see the qualitative consequences of an equation?

Feynman: Oh, yes — but nobody is very good at it. Dirac said that to understand a physical problem means to be able to see the answer without solving equations. Maybe he exaggerated; maybe solving equations is experience you need to gain understanding — but until you do understand, you’re just solving equations.

[. . .]

Omni: To someone looking at high-energy physics from the outside, its goal seems to be to find the ultimate constituents of matter. . . . But with the big accelerators, you get fragments that are more massive than the particles you started with, and maybe quarks that can never be separated. What does that do to the quest?

Feynman: I don’t think that ever was the quest. Physicists are trying to find out how nature behaves; they may talk carelessly about some “ultimate particle” because that’s the way nature looks at a given moment, but ..Suppose people are exploring a new continent, OK? They see water coming along the ground, they’ve seen that before, and they call it “rivers.” So they say they’re exploring to find the headwaters, they go upriver and sure enough, there they are, it’s all going very well. But lo and behold, when they get up far enough they find the whole system’s different. . . . As long as it looks like the way things are built is wheels within wheels, then you’re looking for the innermost wheel — but it might not be that way, in which case you’re looking for whatever the hell it is that you find!

Omni: But surely you must have some guess about what you’ll find; there are bound to be ridges and valleys and so on . . . ?

Feynman: Yeah, but what if when you get there it’s all clouds? You can expect certain things, you can work out theorems about the topology of watersheds, but what if you find a kind of mist, maybe, with things coagulating out of it with no way to distinguish the land from the air? The whole idea you started with is gone! That’s the kind of exciting thing that happens from time to time. One is presumptuous if one says, “We’re going to find the ultimate particle, or the unified field laws,” or “the” anything. If it turns out surprising, the scientist is even more delighted. You think he’s going to say, “Oh, it’s not like I expected, there’s no ultimate particle, I don’t want to explore it”? No, he’s going to say, “What the hell is it, then?”

Omni: You’d rather see that happen?

Feynman: Rather doesn’t make any difference: I get what I get.

[. . .]

Omni: Do you have any guesses on [the history of cosmology]?

Feynman: No.

Omni: None at all? No leaning either?

Feynman: No, really. That’s the way I am about almost everything. Earlier, you didn’t ask whether I thought that there’s a fundamental particle, or whether it’s all mist; I would have told you that I haven’t the slightest idea. Now, in order to work hard on something, you have to get yourself believing that the answer’s over there, so you’ll dig hard there right? So you temporarily prejudice or predispose yourself — but all the time, in the back of your mind, you’re laughing. Forget what you hear about science without prejudice. Here, in an interview, talking about the Big Bang, I have no prejudices — but when I’m working, I have a lot of them.

Omni: Prejudices in favor of . . . what? Symmetry, simplicity . . . ?

Feynman: In favor of my mood of the day. One day I’ll be convinced there’s a certain type of symmetry that everybody believes in, the next day I’ll try to figure out the consequences if it’s not, and everybody’s crazy but me. But the thing that’s unusual about good scientists is that while they’re doing whatever they’re doing, they’re not so sure of themselves as others usually are. They can live with steady doubt, think “maybe it’s so” and act on that, all the time knowing it’s only “maybe.” Many people find that difficult; they think it means detachment or coldness. It’s not coldness! It’s a much deeper and warmer understanding, and it means you can be digging somewhere where you’re temporarily convinced you’ll find the answer, and somebody comes up and says, “Have you seen what they’re coming up with over there?”, and you look up and say, “Jeez! I’m in the wrong place!” It happens all the time.

Feynman relates the following story in "The Dignified Professor":

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.

However, Feynman's mother Lucille Feynman said after Omni Magazine named Feynman the world's smartest man:

If that's the world's smartest man, God help us!