Mathematical proofs?

Question

I'm great at computational math (Calculus sequence, Algebra, etc) but now I'm making the transition to advanced mathematics.

I'm learning it myself over the summer so I can be more at ease come the real thing.

Have any good tips on studying mathematical proofs? Please post them. I find the whole process of making an hypothesis, justifying it and then forming the conclusion strange.

Answer 1

The transition to doing proofs is usually a difficult one for math students. In fact, I know of nobody for whom this step was a smooth one. There is a huge difference between reading a proof and creating one of your own. When reading a proof, there are two competing mistakes that students have a tendency to do. The first is to skim the proof for the main ideas and think that is enough to understand it. Quite often, the devil is in the details. The other main mistake is to get so caught up in the small stufff that the overall flow of the proof is missed. It is crucial to try to figure out why the proof works and how you can motivate it.

As for places to study proofs, I would suggest a beginning abstract algebra book or an introductory number theory book. For the first, there should be a treatment of the integers and divisibility at the beginning with a transition to group theory towards the middle of the book. Herstein's Abstract Algebra book is a good one, although it is challenging for your level.

One place that many students have difficulty is with the 'small words' of mathematics: 'there exists, for every, if...then, or, and, is equivalent to'. A good foundation in symbolic logic (as done in many philosophy departments) can be an immense help. For example, knowing the difference between

There exists x such that for every y...

and

For every y, there exists x such that...

and having *no* confusion between the two is crucial for understanding many proofs. Also, knowing techniques to prove an implication, an existence statement, an 'or' statement, an 'and' statement, etc will serve you well.

Finally, don't expect to read a mathematical proof at the same rate that you read a calculus book. Spend time with each proof until you understand it both at the level of each step *and* at the level of overall motivation. Why was that technique used to prove the result? If you can't answer that, you won't be able to do your own proofs.

Good luck!

Answer 2

Another little hint to add is this:

When doing many if and only if (iff, if you don't know what that is, you will soon enough, so come back and read this again then) remember that you don't have to prove both directions at the same time. In high school algebra and introductory calculus you are given many problems where you are asked to prove an iff (for example: prove A is true iff B is true). In many of these cases you can say "A is true iff C is true, and C is true iff D is true . . and then finally you finish with iff B is true."

When dealing with complex mathematical proofs, you need to understand it will rarely work out that nicely. You can do a proof in one direction using induction, and then come back and use a proof by contradiction in the other directions. Remember A iff B is really two problems:
A implies B
and B implies A

and will usually be treated as two problems.

Answer 3

math really has given a great answer to this question. i have only a few small points to add:

-- When studying a published proof, when the proof takes a step or makes an assertion that you do not *KNOW* to be true, **S*T*O*P**. Do not gloss over this line the way that you might gloss over a word you half-understand in a history text. Pull out a pencil and a piece of paper and prove to yourself that the step is valid or that the assertion is true. You will understand the overall proof much better when each statement of the proof is bedrock truth to you.

-- I LOVE CHICAGO NOTATION!! The upside-down A for "for all," the backwards E for "there exists," the whole works. Not every mathematician agrees with me, but I do love Chicago notation. It gets more of the semantic garbage of the english language out of the way. My proofs look (to me) more like what I was thinking while I was proving them.

-- Mark your trail. And proof that you make from which you will derive any real learning will be composed of many sub-proofs. Find some way to delimit those sub-proofs for later reference. My favorite way of marking my proofs is by margin spacing. Much as computer programmers mark conditional and looping structures in their programs by indentation, I indent special cases, intermediate lemma proofs, inductions, reductio ad absurdem, etc. so that I can see the structure and flow of the logic more clearly. The result is sort of poetic in an "Ogden Nash"/"ee cummings" sort of way.