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When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my studies -- sequence two of Calculus -- but what I'm trying to understand right now, though, is how one actually goes about studying so that when finished with a good text, there's more of an intuitive understanding than superficial.

After reading "The Art Of Problem Solving" from the Final Perspectives section of part eight in 'The Princeton Companion to Mathematics', it seems to hint at approaching studying in that very way. A quote in particular, from Eisenstein, that caught my attention was the following -- I'm not going to paraphrase much:

The method used by the director was as follows: each student had to prove the theorems consecutively. No lecture took place at all. No one was allowed to tell his solutions to anybody else and each student received the next theorem to prove, independent of the other students, as soon as he had proved the preceding one correctly, and as long as he had understood the reasoning. This was a completely new activity for me, and one which I grasped with incredible enthusiasm and an eagerness for knowledge. Already, with the first theorem, I was far ahead of the others, and while my peers were still struggling with the eleventh or twelfth, I had already proved the hundredth. There was only one young fellow, now a medicine student, who could come close to me. While this method is very good, strengthening, as it does, the powers of deduction and encouraging autonomous thinking and competition among students, generally speaking, it can probably not be adapted. For as much as I can see its advantages, one must admit that it isolates a certain strength, and one does not obtain an overview of the whole subject, which can only be achieved by a good lecture. Once one has acquired a great variety of material through [...] For students, this method is practicable only if it deals with small fields of easily, understandable knowledge, especially geometric theorems, which do not require new insights and ideas.

I feel that this type of environment is something you don't often see, especially in the US -- perhaps that's why so many of our greats are foreign born. As I understand it, he does go on to say that he wouldn't particularly recommend that method of study for higher mathematics, though.

A similar question was posed to mathoverflow where Tim Gowers (Fields Medal) went on to say that he recommended similar methods to study: link

I'm not quite certain that I understood the context of it all, though. Upon asking a few people whose opinion mattered to me, I was told that it if time were precious to me, it would be a waste going about studying mathematics in that way, so I'd like to get some perspective from you math.stackexchange. How do you go about studying your texts?

Edit: Broken link added fixed.

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There are two kinds of mathematicians: those who do not try to prove all theorems before reading the proofs and liars. –  Georges Elencwajg Mar 9 '12 at 9:16
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I tried to. But I don't have that kind time. –  Jeff Mar 9 '12 at 10:49
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@Georges: +1!! I was going to write something along the lines of, "If we're being honest, none of us try to prove all the theorems we encounter before reading the proofs." But you said it much better. –  Pete L. Clark Mar 9 '12 at 17:19
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As far as studying, I don't think you should ever try to learn mathematics linearly. As a prof of mine once remarked, it's written linearly to be easily checked for correctness, not to be understood. –  Neal Mar 10 '12 at 2:48
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I got the Princeton Companion lying in front of me. Your quote is from Eisenstein, not Einstein. –  begeistzwerst Apr 6 '12 at 10:02
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3 Answers

up vote 54 down vote accepted

There is a continuum in the way one understands a theorem.
At one end mathematicians just try to understand the statement and use it as a black box .
At the other end they understand the theorem so well that they improve it: this is called research.

An important thing to keep in mind is that your attitude toward a result is not fixed for ever: you may first consider it as a block box and solve exercises by blindly using it, then see how it is quoted in proving corollaries or other theorems and finally come back to it and realize that it is actually quite natural.

Professors have the advantage that they really have to understand a theorem if they want to teach it well and answer the students' questions.
One of the great aspects of this site is that everybody can be a teacher: I strongly advise you to try and answer questions here. They are at all possible levels and I am sure you can find some that you will answer very competently.

A paradoxical way of expressing what it means to have understood a theorem is to say that ideally you have to reach the stage where you consider that all its proofs in the literature are "wrong": it is a patently absurd statement but it conveys the idea that the theorem is now yours because you have integrated it into your own mathematical world.

Edit
Since Neal asks about this in his comment, let me emphasize that when I say that proofs in the literature are "wrong" I mean that , although they are technically 100% correct, they don' t correspond to the subjective way one has organized one's understanding of the subject.

For example, the definition I like for a finite field extension $K/k$ to be separable is that it is étale i.e. that the tensor product with an algebraic closure of $k$ is split: $K\otimes_k\bar k\cong \overline {k} ^n$ .
I know this is rather idiosyncratic and of course I know the equivalence with the usual definition, but then I feel that long proofs that $\mathbb C\otimes _\mathbb R\mathbb C$ is not a field are "wrong" since I know, by the definition of separable I have interiorized, that $\mathbb C\otimes _\mathbb R\mathbb C=\mathbb C^2$.
Let me emphasize that all this a completely personal and secret [till today :-)] attitude within myself and that I absolutely don't advocate that other mathematicians should change their definition of separable.

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Another big +1. As a student of mathematics, it is important to realize (i) there are many different levels of understanding. If you try to ignore this you will either get weighed down on very basic stuff and never move out of that small domain or (worse) will not have a deep understanding of anything; and (ii) the level of one's understanding of any given part of mathematics is a function of time...which is not even guaranteed to be non-decreasing, I'm sorry to say. –  Pete L. Clark Mar 9 '12 at 17:23
    
Thanks, @Pete: it feels great for me to to know that we hold similar views on the subject. –  Georges Elencwajg Mar 9 '12 at 18:02
    
@Georges, would you mind elaborating on what you mean by considering proofs in the literature "wrong"? –  Neal Mar 10 '12 at 2:40
    
I see. Thank you! –  Neal Mar 10 '12 at 12:40
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See my question on mathoverflow for books that are designed for this sort of study:

http://mathoverflow.net/questions/12709/are-there-any-books-that-take-a-theorems-as-problems-approach

Also, from a short bio of S.Ramanujan:

At sixteen, Ramanujan borrowed the English text "Synopsis of Pure Mathematics". This work was to prove a deep influence on Ramanujan's development as a mathematician, for it offered mathematical theorems without accompanying proofs, thereby prompting Ramanujan to prove the material by his own mathematical cunning.

Full text here: http://myhero.com/go/hero.asp?hero=s_ramanujan

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Regarding Ramanujan and Carr's Synopsis, while it is an inspiring story, one should bear in mind how exceptional Ramanujan was as a mathematician. (I think that most would agree that this was not caused by his learning from Carr's Synopsis, but rather that his ability to extract what he did from Carr's Synopsis was a sign of his remarkable nature). Regards, –  Matt E Mar 10 '12 at 4:53
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When I was a kid, they did not give the proof. So I had to proof all of them my self. When my teacher first taught me about similar triangles, I noticed that I can proof Pythagoras theorems out of it. Of course that's a bad proof because there are many simpler ways to show pythagoras theorems.

After a while, I realized that learning in graduate or undergraduate school is actually easier because they tell you the proof.

I then don't bother trying to proof things my self first. I just read it, then do it my self.

I consider my self "understand" the proof if I can remake the proof without reading the proof. If I can proof things my self as if I am solving theorems as problems.

In fact, I do not think doing homework is useful in learning math. Doing proof is all.

Most people do not proof the theorems and I think that's sad. It's sad because it's not difficult. It's sad, because seeing theorems as black boxes are often far more complicated. Theorems are very complicated black box.

I mean, if you launch a cannon at angle B relative to a slope with angle A, the distance is 2V^2 sin (B) * cos(A+B) / (g cos(A))

In Asia, people are expected to memorize that. Are you crazy? It can very easily derived. Just compute the time required for the cannon to land and then notice that the movement on true x direction is just constant speed.

Let's see area of circle as Pi * r^2. It's easily proofed by looking at the base of the circle as a "curvy" triangle with height r. So the area of the circle is just 1/2 *r * the circumference. Tada, you got 1/2 * r * 2* pi * r = pi *r^2.

What about the volume of a ball? Just see the ball as a pyramid with curvy base and height of r. Use 1/3 *r * area of ball, tada you got 4/3 * pi * r^3 very quickly.

I like fast derivation I made up my self like that. I often do not like complicated proofs and come up with simpler and simpler proofs.

Sometimes proof in theorems often contain too many logic. I simplify that with straight forward A = ... = ... =... = result

I think I should wrote a book about it one day.

For those who have a hard time accepting that a circle is a triangle, just see the circle as sums of triangle where the circumferences are the total bases and you'll get the same thing.

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