“Most of us ordinary mortals don’t have enough psychic energy to split between two passions; we cannot be good at both mathematics and cello, we cannot be satisfyingly creative in both mathematics and cabinet making, we cannot be effective both mathematics and politics. To have but one passion, to channel all one’s energies into one major activity, is especially
important for a professional at the apprenticce stage of his development. A graduate student in mathematics doesn’t -shouldn’t- have time to do anything except be a graduate student in mathematics. ” p.53
“Even after I finally mastered epsilons, I didn’t immediately begin to like analysis. I didn’t like it at all; Doob’s influence was insidiously slow-acting. In retrospect I can seethe same pattern as I encountered in linear algebra: I didn’t understand it, and therefore (a) I didn’t like it, and (b) I had to work hard to learn it; having worked hard I came to understand it better than the subjects that seemed easy, and therefore ultimately I came to like it more. ” p.56
“By now, having been a student for over 60 years, registered in schools of various sorts (from grade school through Ph.D.) for 16 of them, I think I know how to study. After prelims I didn’t, not yet, but I began to grope toward finding a method that would be right for me. If I had to describe my conclusion in one word, I’d say examples. They are, to me, of paramount importance. Every time I learn a new concept (free group, or pseudodifferential operator, or paracompact space), I look for examples -and, of course, non-examples.” p.62
“Here you sit, an undergraduate with a calculus book open before you, or a pre-thesis graduate student with one of those books whose first ten pages, at least, you would like to master, or a research mathematician (established or would-be) with an article fresh off the press—what do you do now? How do you study, how do you penetrate the darkness, how do you learn something? All I can tell you for sure is what I do, but I do suspect that the same sort of thing works for everyone. It’s been said before and often, but it cannot be overemphasized: study actively. Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? Another way I keep active as I read is by changing the notation; if there is nothing else I can do, I can at least change (improve?) the choice of letters. Some of my friends think that’s silly, but it works for me.” p.69
“Changing the notation is an attention-focusing device, like taking notes during lectures, but it’s something else too. ”
…
“I believe that changing the notation of everything I read, to make it harmonious with my own, saves me time in the long run. If I can do it well, I don’t have to waste time fitting each new paper on the subject into the notational scheme of things”
…
“you want to learn group theory, it is not a good idea to open a book on page 1 and read it, working all the problems in order, till you corne to the last page. It’s a bad idea. The material is arranged in the book so that its linear reading is logically defensible, to be sure, but we readers are human, all different from one another and from the author, and each of us is likely to find something
difficult that is easy for someone else. My advice is to read till you come to a definition new to you, and then stop and try to think of examples and non-examples, or till you come to a theorem new to you, and then stop and try to understand it and prove it for yourself—and, most important, when you come to an obstacle, a mysterious passage, an unsolvable problem, just skip it. Jump ahead, try the next problem, turn the page, go to the next chapter, or even abandon the book and start another one. Books may be linearly ordered, but our minds are not.” p.70
Hakan ERGÜL 