"In my own experience, mathematics in general and pure mathematics in particular has always seemed like secret gardens, special places where I could try to grow exotic and beautiful theories. You need a key to get in, a key that you earn by letting mathematical structures turn in your head until they are as real as the room you are sitting in.”
[Think about the mathematical object until you build a ‘intimate relation’ with it!]
“Mathematical thinking is logical and rational thinking. It’s not like writing poetry. In general, persons who can work effectively in mathematics must be mostly rational in thinking when doing such work. However, I can guess that one could have very specialized delusions, like some extreme and atypical religious cultic orientation and also be a good mathematician. This seems possible.”
[Mathematics is not all logical and rational: like creative art, you need to be ‘insane’ from time to time.]
“The surprise was that in higher dimensions, there may be many completely different smoothings. In particular, for the 7-dimensional surface of an 8-dimensional cube, there are twenty-eight essentially different smooth manifolds which can be obtained by carefully chosen smoothings — these are the so-called exotic 7-spheres. I wasn’t expecting such a result, or looking for it. Rather, I simply came upon an apparent contradiction when trying to describe possible manifolds in two different ways. The only way to resolve this contradiction was by positing the existence of such exotic spheres, a conclusion which has led to entirely new fields of research.”
[Do not have a preconception about what you will arrive at in a research. Wander around the garden and once in a while you can find a jewel, as suggested by Atiyah.]
““To understand” was the goal I gave for my high school yearbook , and this is still what drives me. I love to reach understanding : first, to see something (big or little) that doesn’t make sense or is simply discordant, then to reflect and ponder, to search and stare in my mind’s eye until sometimes, miraculously, vision is transformed and mist and muddle develop into form, order, and connection. Mathematics is not about numbers, equations, computations , or algorithms: it is about understanding.”
[To UNDERSTAND. Do more thinking and understanding rather than aimless calculations.And by doing that you build a personal relation with mathematics.]
“I am very slow in doing research. I don’t believe in boundaries between different areas of mathematics. I like to think about challenging problems that I am excited about, and follow wherever they lead me. This allows me to interact and learn from many smart colleagues. In a way, doing mathematics feels like writing a novel where your problem evolves like a live character . However, you have to be very precise in what you say: everything must fit together like the gears in a clock. …”
[Resonates with Milnor.]
“I grew up in a family of musicians…becoming a musician…And if I had done that, then in some ways my main activity in life would have been similar to what it actually turned out to be. Like a long mathematical proof, a substantial piece of music is a complex abstract entity that must satisfy strict constraints, and creating such an entity involves careful planning on many levels, from the global structure down to the little subproblems that arise when you try to make your higher-level ideas work. My father has always had a keen interest in mathematics, and it feels as though I have taken a path that in another life he might have liked to take himself.
…But the problems I first worked on…to solve them it was not enough to spot a clever trick.Instead, I had to use one of the most common methods of mathematical research, which was to take an existing argument that used a technique I would never have thought of for myself and modify it.
As my research progressed, I began to understand that there is more to mathematical skill than problem-solving power: also very important are how one goes about selecting problems to work on and how one persuades others that one’s research is interesting. In both cases, it helps greatly if there is a bigger project to which one’s work contributes.
…When one works on a problem that many other people have tried, a little voice in one’s ear is constantly saying, “If this approach worked, then the problem would have been solved long ago.” And the voice is right 99.9 percent of the time. But if one digs deep enough into a problem, one sometimes manages to identify and isolate a fundamental obstacle to solving it, and just occasionally one discovers that a technique has recently been developed that one can use to get round this obstacle.”
“In the second year, I took a mathematical physics course, and there I discovered a mysterious new phenomenon. Some of the students had something called “physical intuition,” which enabled them to give strange and wonderful answers to questions , which were greeted by the professor with delight but made no sense to me at all.
…I had learned an important lesson. It was the joy of getting a new idea, of finding something that no one had before thought of, of working out my own way to new results. I could never take mathematical advice; it did not stimulate me. Only my own peculiar point of view could get my blood rushing. I could read other peeple’s work, enjoy it, and be stimulated by it, but only in so far as I could get my own point of view could I make real progress.
…As I tell my students, point of view is everything, and, as in writing, one must find ones own voice to really contribute.”
[Think it through by yourself! This reminds me that Susskind once said, he could never read a textbook, writing one would be much easier than reading one for him.]
“I am trying to “understand.” I am not trying to discover something new, but rather see the “essential reasons” why some results are true. I return to the source, in an attempt to discover “the mother of all formulae.” Other mathematicians’ new ideas and results are irritating. I would desire very much to show that there is a simple reason why “all of that” is true (at least when I was young I had that arrogance). Sometimes I succeeded in finding “higher reasons” why a result was valid: an idea springs up from my past work and lands just there in front of me, ordering me to do something.”
[Resonate with Thurston. It could be difficult sometimes but that is the goal—to UNDERSTAND! If you do not try to understand, you are not doing mathematics, but some ‘math-ish’ things.]
“There are no formal classes at the Institute for Advanced Study, where I teach. We get young postdocs fresh from receiving their degree. It is important for them to expand their horizon and go beyond delving into their thesis topic. They have to learn other things as well and our role is to guide them and help them to become independent. Independence is very important. They must learn to judge by themselves what is interesting and worth doing, not just listen to the advice of others. When they come to me and ask what they should do next, that is the sign that they are not independent.
…Good science is always created. One needs to imagine how things may be and proceed from there. It is critical to be flexible and not to have preconceived ideas, not to force things to look the way you would like them to look. A danger in creative research is to get excited by certain ideas, overvalue their significance , and try to make them fit within what one knows.”
[Develop your own taste. And again have no preconceptions.]
“There is a saying that geometry is the art of thinking correctly with wrong figures. I agree, insisting on the plural. You have more than one picture for each mathematical object. Each of them is wrong but we know how each is wrong. That helps us determine what should be true. It enables you to jump from one setting to another. In mathematics, it is a pleasure when you realize that two things that appear to have nothing in common, in fact, are connected: creating a dictionary between two questions is a powerful tool. Often something will be obvious from one point of view but will give you surprising information if you look at it from another point of view.
…There are very different ways of thinking within mathematics . Some people are very algebraic, can think with formulae, and are very fast at computing. Some think only in terms of pictures. Some can be extremely precise. Others are very vague and can only give ideas. The diversity is useful because each way of thinking complements the others.
…In mathematics, there are not only theorems. There are, what we call, “philosophies” or “yogas,” which remain vague. Sometimes we can guess the flavor of what should be true but cannot make a precise statement. When I want to understand a problem, I first need to have a panorama of what is around it. A philosophy creates a panorama where you can put the things in place and understand that if you can do something here, you can make progress somewhere else. That is how things begin to fit together.”
“As an example, imagine you have a series of numbers such that if you add 1 to any number, you get the product of its left and right neighbors. Then this series will repeat itself at every fifth step! For instance, if you start with 3, 4, then the sequence continues : 3, 4, 5/3, 2/3, 1, 3, 4, 5/3, etc….what other things in mathematics it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, the Schrödinger equation of quantum mechanics , and certain models in quantum field theory. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful.Some mathematicians find formulas and special cases less interesting and care only about understanding the deep underlying reasons. Of course that is the final goal, but the examples let you see things for a particular problem differently.
…Mathematics is creative, not a mechanical procedure. It is very personal. Sometimes just from the statement of a result you can guess which mathematician did the work….you express your personality by the choices you made in discovering and proving it.”
[It is hard to believe that this innocuous looking series can have that many connection! Examples are incarnations and origins of ideas and we surly should not understate their importance. And, quote from Boltzmann’s Populäre Schriften: ‘ Wie der Musiker bei den ersten Takten Mozart, Beethoven, Schubert erkennt, so würde der Mathematiker nach wenigen Seiten, seinen Caüchy, Gauß, Jacobi, Helmholtz unterscheiden.’(一个音乐家在听到几个音节以后，就能辨认出来莫扎特、贝多芬或者舒伯特的音乐，同样一个数学家或物理学家，也能在念了几页文字以后，就辨认出来柯西、高斯、雅可比、亥姆霍兹或者克尔斯豪夫的工作。)]
“It is important to emphasize the incredible role that luck plays in this — having the right professor at the right time, being around when some new mathematical problem is coming into popularity, being exposed to interesting questions. I’m completely convinced that there are lots of mathematicians in who are in any measurable way much smarter than I am, higher IQs, the can learn more, they can learn faster, answer questions more quickly. I’m a much better mathematician in the sense that I’ve done better mathematics, ad this is very largely a question of luck, of being in the right place at the right time. You have to be pretty good at mathematics, but if you are pretty good at mathematics and around people who are interested in uninteresting mathematics, you’ll be pretty good at uninteresting mathematics.”
“I had felt it in work that I had done before , where many other mathematicians would have agreed that the work was one of “discovery.” Working on wavelets had felt exactly the same — yet, here most mathematicians viewed it as “construction” rather than discovery. This puzzle made me want to find the boundary between these two mathematical realms. I haven’t found it and I am now convinced that it doesn’t exist: all our mathematics is constructed. It is a construction we make in order to think about the world. I would even go so far as to say that mathematical thinking is the only way we have to think logically about things we observe. There are other ways in which we experience the world and in which we interact with it — ways that have to do more with emotions and sensual delight and that lead to other wonderful things, like love and art — but when we want to think logically, we basically are back to what is essentially mathematics.”
[There is no universal agreement on this question. I tend to believe that mathematics is the way we think but the form of it evolves in our interaction with the outside world, whatever that means.]