Do you view a theorem's proof as an exercise?

[andytoh]

Whenever I read a corollary, even a lemma, I try to prove it myself first before reading the textbook's proof. This is because I usually find the level of difficulty of the proof comparable to the exercises in the textbook, and thus the textbook's proof is its (correct) solution (which is a real bonus since textbooks almost never show its solution to its exercises). With theorems, sometimes they can be proved by me too, sometimes they are too hard to prove, but I always try to think of a proof first before reading its proof. Do you guys do the same thing? Or do you read theorems and their proofs back to back?

 

[andytoh]

And when I do read a theorem's proof (after giving up trying to prove it myself), I will only read just a little bit of it until I've received enough hints, and then try to finish the rest of the proof myself. By the way, I rarely write out the proof formerly, just a good sketch of it. But the point is that after doing the "exercise", not only do I improve in my solving abilities, but I remember the theorem more and understand it better as well.

 

[pivoxa15]

I can only dream of doing something like what you do. I use to not even do the exercises in maths books. I however did do it once and thoroughly and found I really enjoyed the book. It seemed like reading the book was a 'distraction' and couldn't wait until the exercises came. However once I started not able to do some of the exercises, I got depressed and interest wained. I have to overcome this weakness someday.

 

[Math Jeans]

Well, for me, it depends on the course. Does the course that you are taking involve proofs in the exams?

I remember I once took a multivariable calculus course that not only involved the material of a regular multivariable calculus course, but in the exams (that were the entire grade), they made the problems overly difficult (basically much more difficult than anything we practiced), and made us do proofs in them.

(They even made us do multivariable epsilon-delta proofs )

But generally I will try to figure out a proof on my own in a non-proof oriented course as those don't usually take a lot of time. For a proof oriented course, I will just look at the proof and note the method because I need all of the time I can get.

 

[andytoh]

For me, reading a theorem and then reading the proof immediately after is like reading the solution of a problem immediately after reading the problem.

 

[futurebird]

Whenever I read a corollary, even a lemma, I try to prove it myself first before reading the textbook's proof. This is because I usually find the level of difficulty of the proof comparable to the exercises in the textbook, and thus the textbook's proof is its (correct) solution (which is a real bonus since textbooks almost never show its solution to its exercises). With theorems, sometimes they can be proved by me too, sometimes they are too hard to prove, but I always try to think of a proof first before reading its proof. Do you guys do the same thing? Or do you read theorems and their proofs back to back?

It depends on if I'm having a hard time with the class on not. For my east Linear Algebra class I do this all the time, some of the time I can even guess what the next theorem will be. But for complex I'm happy to just understand each proof and be able to produce it step by step on my own. I think it's the best thing to do... when you can.

 

[pivoxa15]

For me, reading a theorem and then reading the proof immediately after is like reading the solution of a problem immediately after reading the problem.

If they prove the theorem in the book then it usually is because it is too difficult to be left as an exercise.

It also depends on one's time. However getting into the habit of doing as many excercises yourself definitely is a great benefit.

 

[andytoh]

If they prove the theorem in the book then it usually is because it is too difficult to be left as an exercise.

It also depends on one's time. However getting into the habit of doing as many excercises yourself definitely is a great benefit.

A theorem's proof is sometimes easier than some of the problems in a book (at least from the books I've read). I found this to be the case for about 80% of all the theorems' proofs in Munkres' Topology.

 

[pivoxa15]

(They even made us do multivariable epsilon-delta proofs )

I've found these proofs are interesting if you actually understand them concretely.

 

[Math Jeans]

I've found these proofs are interesting if you actually understand them concretely.

I know, but they are your worst nightmare when you just start them. They get cool.

 

[NeoInTheMatrix]

I can only dream of doing something like what you do. I use to not even do the exercises in maths books.

Not doing the exercises in a math book is like not even bothering to read the book.

 

[pivoxa15]

Not doing the exercises in a math book is like not even bothering to read the book.

True. People say it is not possible to define maths but maybe one can define maths by saying it is about problem solving. Doing proofs is problem solving as well.

So if one does maths without solving any problems oneself but merely reads it then one is not doing maths at all.

 

[andytoh]

By the way, proving theorems (that you've never seen the proof of before) is like being a mathematician. Pretend that no one has ever proven the theorem before (and so the theorem is currently a conjecture). Prove the theorem, and your work will be published and you will be showered with glory. It's so much fun!

 

[radou]

What you're doing is good, andytoh, because you are actually thinking about what you're learning. Of course, this requires a certain amount of experience with proofs, i.e. the structure of proofs. Corolaries are usually not hard to prove, since they are conclusions which follow easily when combining certain theorems, propositions, etc. Of course, it all depends on the course you're dealing with and your personal motivation. In general, books which contain only sketches and outlines of some proofs, or which leave some simple proofs as exercises to the reader, are more useful, since they make the reader think about the material he's digesting, which is probably the most important thing when learning mathematics.

 

[pivoxa15]

By the way, proving theorems (that you've never seen the proof of before) is like being a mathematician. Pretend that no one has ever proven the theorem before (and so the theorem is currently a conjecture). Prove the theorem, and your work will be published and you will be showered with glory. It's so much fun!

So you think you can prove theorem like Steift Van Kampen theorem and the Van something sequence in homology the first time learning these subject mattes?

 

[wildman]

So you think you can prove theorem like Steift Van Kampen theorem and the Van something sequence in homology the first time learning these subject mattes?

Of course not, but one ought to at least try. The harder you try, the more you will learn.

 

[andytoh]

For the impossible theorems, just read the beginning of the first paragraph, and figure out what the first subtask is. Try to complete the subtask yourself. Then try to figure out what the next subtask is. If you cannot read a little bit of the next phase, and then try to complete the next subtask yourself. Continue this process. Meanwhile, try to figure out how these subtasks fit together to complete the entire task. If you cannot do that, then read ahead of time the beginning of each phase in the entire proof to figure out what all the subtasks are and try to arrive at the conclusion assuming these subtasks have been carried out. Doing this is like doing several exercises.

For example, trying to prove that the axiom of choice implies the well-ordering theorem is hopeless for a student. But Munkres actually breaks down the proof into several exercises in section 11 of his textbook, whose results can then be used to arrive at the desired conclusion. This breaking down of long difficult proofs into subroutines can be with any proof. You type out the subexercise questions yourself and carry out the subsolutions. Indicate using brace brackets the solutions of the subexercises found in the theorem's entire proof in the textbook.