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.
There is a continuum in the way one understands a theorem.
At one end of the spectrum 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 on 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 black 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.
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⊗kˉk≅¯kn .
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 C⊗RC is not a field are “wrong” since I know, by the definition of separable I have interiorized, that C⊗RC=C2.
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.