Studying mathematics: Is proving things yourself worth the time?

Question

When studying mathematics, is proving things yourself (before reading the proof given in the text) worth the time? This approach takes significantly longer than simply trying to follow along, but you understand everything better and learn to come up with mathematical tricks when they're needed.

Compare to the following alternative: first read the proof, and then reconstruct it later. You will still understand the structure of the proof, why the conditions in the hypothesis are needed, etc., but you have the benefit of knowing any tricks that are needed beforehand. Is coming up with these tricks yourself worth the time?

Answer 1

Absolutely, it's worth it.

Arguably, all of mathematical knowledge is the product of mathematicians figuring out the tricks to proving something. Many would say that's the fun part.

My number theory course in college was like this. At the beginning of the semester, the professor said, "Yes, I know you can look up the proofs. Please don't." It was the process of working the proofs out that was the important part.

Answer 2

It depends on what type of math appeals to you.

If you want to apply math, you want to be able to apply propositions to real world problems. In that case knowledge of the propositions is more important than the proofs behind them.

If you want to understand math and possibly come up with new propositions, you will need to learn how to prove propositions.

Answer 3

I'd say it very much depends on the context; if its a field you are comfortable in and you have a good grasp of the basic concepts then it definitely is.

If, however on the other hand, you're relatively new to an area of maths then maybe you're best spent trying to figure a proof out on your own or seeking hints from a friend / expert rather than simply reading it from a book!

Answer 4

Personally, I am not of the school of thought which says that every reader needs to independently verify every theorem. If you are just beginning to work with proofs, you should read all of them, but you do not necessarily need to work out every proof in all of its detail yourself.

However, I think it's a good idea to pause between reading the statement of a theorem and its proof to consider yourself what a proof may involve. Ideally you would prove it yourself, but this may be difficult and it's not always worth your time. At the very least, you should ask yourself the questions:

1) Is the theorem plausible? Does it match my intuitions? If not, what is going wrong? Try to construct a counterexample.

2) Is the theorem interesting? Why did the author choose to state it? Is it a "technical" lemma, uninteresting in itself but perhaps useful later? If so, it might be wise to postpone reading the proof until you understand why the statement is useful.

3) Is there a clear connection between the hypothesis and the conclusion - that is, do they involve similar-sounding definitions? If so, it's probably an indication that the proof is straightforward. On the other hand, if there seems to be a yawning gulf between the hypothesis and the conclusion, this might be an indication that some "high-powered" theorems are necessary to proceed. Can you identify which ones are necessary?

4) What is the central difficulty of the problem? Often when you first read a proof there is some niggling doubt in the back of your mind - some confusion that you can't put your finger on. Try to pinpoint this confusion by putting in the form of a question that is as specific as possible. "How on earth could I get $X$ only knowing $Y$?" "Doesn't this violate Thoerem 8.7?" "Isn't this trivial because of $Z$?" This sort of frustration is a good sign. It's much, much preferable to a general haze and confusion - having no idea what the author is trying to do and why it's relevant. Try to hold onto these questions - don't let them pass until you have your answer.

The point of all this thinking in-between the reading of the statement of a theorem and its proof is to get the largest possible "ah-hah!" moment when you read the proof. Then when you think about the problem later, you will remember the "ah-hah!" moment even when you've forgotten the other details. If you can remember the clever part of the proof, you can reconstruct the other details later when you want to.