The evaluation of proofs

Question

I searched this community upon proofs and found some interesting topics. Yet I would like to know how you guys achieved to write your own proofs. What I mean is, my course describes proof as something that comes with experience or learning by doing. I assume that this community has professional experience in mathematics and I would like to model your approach on how to become professional in writing proofs. This might be an unusual question, but I believe that many members of this community would appreciate your advice and experiences. For my part I really would appreciate your advice, since memorizing isn't enough for me to study math. But still when I start proofing theorems I just don't know where to start. Maybe you had the same experience once, yet somehow you managed to accomplish your goal. What did you do? Thank you for your time and support.

-Daniel

Answer 1

Try to prove theorems and results you already know by several different methods. I am not sure about your level of mathematics so the following examples may not be appropriate. If not pick problems that are. For example prove the Pythagorean theorem or find the roots of a second order polynomial. If you know the Peano axioms prove addition and multiplication are commutative. When you think you have a proof go over it carefully and see if the justification for each step is correct. You might want to wait a day or two for this step.

Answer 2

I would add to Jay's answer "try to prove theorems" by saying that you learn to write clear mathematics by writing and rewriting, preferably using Latex, for easy revision, so that the matter and the layout become as clear as you can make it. You also have to learn to read what you have written to find the errors, lack of clarity, and poor layout. My supervisor Henry Whitehead you should write your paper, then put it in drawer for six weeks, and then rewrite it without looking at your draft! I confess to have rarely followed this method.

I remember in my early years of teaching remarking to two students on their homework: "Please read what you have written, and see that it is nonsense, where I have marked in red!" One did precisely this, and went on to get better and better and got a very good degree. The other carried on exactly the same.

Many of us have benefited greatly from the detailed criticisms of supervisors, and referees, and their care in reading what we have written. To be of most benefit, students should be required to write and rewrite their work; this does not happen often on pure logistic grounds. But I have tried it on a small scale with first year students. For more details, see the course Ideas in mathematics.

Also you should realise that published work is usually the result of a series of approximations. A proof usually has an idea controlling it, but the first written version may have serious flaws, of various kinds. One may leave it for days, weeks, months, years, but on rereading there one sees something worthwhile, but needing a lot more work. This is where craftmanship and a professional approach are essential.

It was said of Grothendieck that he would work extremely hard to get the right concepts so that the proof became essentially tautologous! He also commented on "the difficulty of bringing concepts out of the dark!"