Good Book for Learning Mathematical Proofs

[NathanaelNolk]

Hello guys,
I didn't see any recommendation for a good introductory textbook about mathematical proofs. (Did I completely miss it?) Do you know which one I could take ? It would really be helpful.

 

[verty]

Learning to ride a bicycle takes more than reading about bicycles. In the same way, learning to write proofs takes more than reading about proofs. Then also, there are two types of theory, the descriptive type (these are the brakes, these are the pedals) and the prescriptive type (if you want to slow down, squeeze the brake lever).

So with proofs there is a question whether to learn logic (descriptive) or how to prove (prescriptive), and there needs to be a practice phase. Some books will tend to be prescriptive and recipe-oriented. Some will tend to be descriptive and mathematical. And some will have both, some factual stuff and some recipes. It seems obvious to want both but that may not be the best way to learn: hand-picked exercises that fit recipes perfectly may not give one a full appreciation.

There is also the following problem. Someone like yourself who wants to learn how to write proofs does not yet know what knowledge is needed. So at the beginning, there is a lot of background knowledge needed and not all of it seems relevant. But one can only know what is relevant when one comes to use it. So there needs to be a committal, one needs to say, I am going to learn this even if it seems not to be terribly relevant, because someone more knowledgeable believes it is relevant. And that can be difficult.

The way to overcome it is to use the right books. These are what I recommend.

https://www.amazon.com/dp/0486477673/?tag=pfamazon01-20

Study chapters 1-3, know it very well.

https://www.amazon.com/dp/0486616304/?tag=pfamazon01-20

Cover chapters 1-6. This is also for the practice phase: go through it a second time, proving every theorem. You won't need to do it all, by about chapter 5 you will have had all the practice you should need.

PS. Obviously this is not the only way to learn this, it is one way among many. I think it is a good way.

 

[NathanaelNolk]

Thanks for your answer verty, I'll take a look at those books you listed.

 

[verty]

Thanks for your answer verty, I'll take a look at those books you listed.

You're most welcome. I truly believe this is a case where there isn't an easy route to learning, one has to just commit to someone's recipe and follow it, if not mine then someone else's. At the end, it'll make sense.

One thing though, set theory although still very relevant is not quite the unifying foundation it used to be. For that reason, only the parts that relate to other areas are important. So you certainly don't need to bother with the transfinite stuff, and even some of that cardinal/ordinal stuff is not such a big deal.

Anyway, if you do choose to follow my method, try to enjoy it, you are learning something cool, how to prove mathematical claims.

 

[NathanaelNolk]

One thing though, set theory although still very relevant is not quite the unifying foundation it used to be. For that reason, only the parts that relate to other areas are important. So you certainly don't need to bother with the transfinite stuff, and even some of that cardinal/ordinal stuff is not such a big deal.

Okay, then I'll be sure to skip those parts. Thanks!

Anyway, if you do choose to follow my method, try to enjoy it, you are learning something cool, how to prove mathematical claims.

I don't learn this just because I find it useful, but rather because I'm truly interested on how these proofs are constructed. I'll sure enjoy this!