I studied and learnt Math the hard way. If a young, aspiring student of Mathematics is properly guided on how to study Mathematics, he or she will gradually become more competent, skilled, experienced, knowledgeable and experience a spurt in self-confidence. It is all summarized in the sayings, “start at a young age” and “slow and steady wins the race”.
Some views, albeit classical, are the best. Like “As a young child, Einstein was known as the devil of Math. He would pick up a Math book and solve it cover to cover”. Also, Srinivasa Ramanujan became a mathematician by solving 6000 problems on his own. Somebody had asked the immortal Norwegian mathematician, Niels Henrik Abel, “How did you become a mathematician so fast?”. He said, “By studying from the masters, not the pupils”.
also, I find the views of Paul Halmos, one of the finest expositors of Math, in his autobiography,”I want to be a Mathematician — An Automathography” most easily accessible, most detailed for someone who wants to be a real mathematics student. I am presenting some jewels of his wisdom below (verbatim):
I like words more than numbers, and I always did.
Then why, you might well ask, am I a mathematician? I don’t know.
The sentence I began with explains the way I feel about a lot of things, and how I got that way. It implies, for instance, or in any event I mean for it to imply, that in mathematics, I like the conceptual more than the computational. To me the definition of a group is far clearer and more important and more beautiful than the Cauchy integral formula. Is it unfair to compare a concept with a fact? Very well, to me the infinite differentiability of a once differentiable complex function is far superior in beauty and depth to the celebrated Campbell-Baker-Hausdorff formula about non-commutative exponentiation.
The beginning sentence includes also the statement that I like to understand mathematics, and to clarify it for myself and for the world, more even than to discover it. The joy suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me — both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension. At the same time, discovering a new truth, similar in subjective pleasure to understanding an old one, is in one way quite different. The difference is the pride, the feeling of victory, the almost malicious satisfaction that comes from being first. “First” implies that someone is second; to want to be first is asking to be “graded on the curve”. I seem to be saying almost, that clarifying old mathematics is more moral than finding new, and that’s obviously silly — but, let me say instead that insight is better without an accompanying gloat than with. Or am I just saying that I am better at polishing than at hunting and I like more what I can do better?
I advocate the use of words more than numbers in exposition also, and in the exposition of mathematics in particular. The invention of subtle symbolism (for products, for exponents, for series, for integrals — for every computational concept) is often a great step forward, but its use can obscure almost as much as it can abbreviate.
Consider an example of the point I am trying to make, a well-known elementary theorem with a well-known elementary proof. The theorem is Bessel’s inequality in inner-product spaces; it says that if is an orthonormal set, x is a vector and , then .
A standard way of presenting the proof is to write
and then compute
Rigorous and crisp, but not illuminating. To my mind, the ideal proof is the one sentence: “Form the inner product of with itself and multiply out.” That should be enough to get the active reader to pick up his pencil and reproduce the chain of equations displayed above, or, if he prefers, to lean back in his chair, close his eyes, and look at those equations inside his head. If he is too lazy for that and wants more of his work done for him, the sentence can be padded,”The result is positive and it consists of four terms: the first is , orthonormality implies that the last is , and both of the cross product terms are equal to . Cancel one of them against the positive sum, what remains is the asserted inequality.”
As far as length is concerned, the words use about the same amount of paper as the symbols. The words are clearer, and they have, moreover, the advantage that they can be communicated during the walk back from class to the office, with no blackboard or chalk handy. They are also more likely to deepen insight by leading to the discovery of the appropriate general concepts and context (such as the pertinence of projections).
As for writing — I write all the time, and I have done so as far back as I can remember. I write letters, from time to time I have kept diaries, and I write notes to myself, explicit notes with sentences, not just “try power series expansion” or “see Dunford-Schwartz for ”. I think by writing. In college I wrote notes — that is, I transcribed abbreviations scribbled in class to legible and grammatical sentences. Later, when I started trying to prove theorems (the acceptably low-key phrase for the more stuffy-sounding “doing research”). I would keep writing, as if I were conducting a conversation between me and myself. “What happens if I restrict to the ergodic case? Well, let me see, I have already looked at what happens when S is ergodic, but the useful case is when both S and T are…”I am tempted to preach, and to say: that is the right way, do it this way, do it any way or else you will fail — but all I can be sure of is that this is the right way for me, there is no other way that I can do things.
Study or worry
In my memory (nostalgia?) I see my teachers of those days, the ones I was afraid of as well as the ones I liked, as serious people who really knew and cared about their subjects. Even in my sub-teens I began to get the idea that learning was good — better than just studying and getting good grades.
Here you sit, an undergraduate student 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 case? 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. When I reported on Chapter VII of Stone’s book (the chapter on multiplicity theory, a complicated subject) to a small seminar containing Ambrose and Doob, my listeners poked fun at me for having changed the letters, but I felt it helped me to my eye on the ball as I was trying to change, organize and systematize the material. I feel that subtleties are less likely to escape me if I must concentrate on the bricks and mortar as well as gape admiringly at the architecture, I choose letters (and other symbols) that I prefer to the ones the author chose, and more importantly, I choose the same ones through out the subject, unifying the notations of the part of the literature that I am studying.
Changing the notation is an attention focusing device, like taking notes during lectures, but it’s something else too. It tends to show up the differences in the approaches of different authors, and it can therefore serve to point to something of mathematical depth that the more complacent reader would just nod at — yes, yes, this must be the same theorem I read in another book yesterday. 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 do not have to waste time fitting each new paper on the subject into the notational scheme of things. I have already thought that through and I can now go on to more important matters. Finally, a small point, but with some psychological validity, as I keep changing the notation to my own, I get a feeling of being creative, tiny but non-zero — even before I understand what is going on, and long before I can generalize it, improve it, or apply it, I am already active, I am doing something.
Learning a language is different from learning a mathematical subject, in one the problem is to acquire a habit, in the other to understand the structure. The difference has some important implications. In learning a language from a book, you might as well go through the book as it stands and work all the exercises in it, what matters is to keep practising the use of the language. If, however, 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 come to the last page. It’s a bad idea. The material is arranged in the book so that it’s linear reading is logically defensible, to be sure, but we readers are human, all different from one another and from the author (Paul Halmos), and each of us likely to find something difficult that is easy for someone else. My advice is to read to 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 a 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.
More later from the maestro’s automathography,
till then aufwiedersehen,