How can I handle long proofs in mathematics?

[PhDorBust]

I'm reading Calculus on Manifolds by Munkres and I often encounter multiple page proofs that are very technical.

I can verify the argument in a reasonable amount of time, but to actually digest the proof (i.e. learn it such that I can reproduce it by memory weeks later) takes an inordinate amount of time.

How do you handle long proofs?

 

[micromass]

Put a lot of time into them. There's no other way.

You don't need to know all the proofs though. From a short look at the proof, I can usually judge whether it is important or not...

 

[MathematicalPhysicist]

It's Analysis on... ;-)

Anyway as everything that it's long (the same as learning a long score on guitar or whatever) break it down to small pieces of a proof which you can digest.

How much time it takes to digest the proof (score) depends on you, the more you read and practice the less time it takes.

 

[AlephZero]

I can verify the argument in a reasonable amount of time, but to actually digest the proof (i.e. learn it such that I can reproduce it by memory weeks later) takes an inordinate amount of time.

If "reproduce it by memory" means "reproduce it word-for-word the same as the book", that has little value.

Most long proofs are made from a few big ideas and a lot of fairly trivial detals. Remember the big ideas and learn how to re-invent the details for yourself. When the details start to become "obvious", then you are really learning the subject.

 

[RichardParker]

Sometimes I just skip the proofs. Would I perish in the future if I do this often? I'm in a hurry because I need to get math done before I can do research.

 

[klackity]

Most proofs aren't actually so complicated, once you get past the details and see the intuition.

When reading a proof, what you should be looking for most is the intuition behind it. After you read through all the details of a proof, you should try to summarize the major steps of the proof in natural language.

For example, suppose you had just looked at a formal proof of the extreme value theorem (a continuous function on a closed interval attains its supremal and infimal values). It would likely involve a lot of details that would make the proof hard to understand upon a first reading. But if you think about the meaning behind the details, it's much clearer. You might translate it first as follows:

"Well, if M is the supremum of f in [a,b], then we can find a sequence of points in the interval such that f(x) gets closer and closer to M. Because [a,b] is bounded, this sequence must have an accumulation point, c. But because [a,b] is closed, c must be in the interval. And because f is continuous, the limit of f(x) as we approach c must equal c."

 

[klackity]

RichardParker: You will absolutely perish in the future if you do that often. If you don't understand the proof, chances are you don't really understand the theorem that well either.

 

[wisvuze]

( *EDIT: oh I thought you were talking about the Spivak book, munkres is ANALYSIS on manifolds, get it straight :P ) I know how you feel, when I first saw the "inverse function theorem" proof in Calculus on Manifolds , I was just like holy schmoly I'm not reading that! But anyway, a good way to read long proofs is to do the following: before you fully digest the proof, you must fully digest the hypotheses and the theorem statement itself!
With that in mind, you can section off parts of the proof according to what parts of the theorem statement must be satisfied before pushing forward. ( for example, show that x is linear and invertible; look for the part that proves linearity, then look for the part that proves invertibility. ) This is especially easier if the theorem has multiple parts (i.e., if blahblahblahblahblah, then f is differentiable, continuous and ____ ). Also, you must look for the parts of the proof that uses each part of the hypothesis. Don't let any of the hypotheses go to waste! And if you end up realizing that certain parts of the hypothesis were not required at all, then that's cool too -- you'll probably understand the requirements of the theorem better, and you'll get a feel for what is redundant and what is not.

Skipping proofs is useful sometimes; whether or not you should read the proof depends on how much "structure" or concept of something the proof itself can reveal. If you care about x and y being so and so, there is a more conceptual quality behind those properties, and you could care less if you had to use the triangle inequality to prove such a thing. However, you *should* always be able to prove everything you know (this doesn't mean you aren't able to momentarily skip proofs )

 

[RichardParker]

-__- I guess a what I am doing is getting math ''undone''. Thanks.

 

[mal4mac]

Sometimes I just skip the proofs. Would I perish in the future if I do this often? I'm in a hurry because I need to get math done before I can do research.

Quite the opposite - you'll probably go far, as the 'must know the proof' neurotic perishes. For physicists, learning math proofs is like a mechanic making tools before he fixes the car. The tools are there! Fix the car already!

 

[MathematicalPhysicist]

Quite the opposite - you'll probably go far, as the 'must know the proof' neurotic perishes. For physicists, learning math proofs is like a mechanic making tools before he fixes the car. The tools are there! Fix the car already!

Sometimes physicists need to discover the maths by their own, so your analogy works well usually but every one and then comes some anamoly.

 

[Simfish]

Most long proofs are made from a few big ideas and a lot of fairly trivial detals. Remember the big ideas and learn how to re-invent the details for yourself. When the details start to become "obvious", then you are really learning the subject.

Yeahh - not realizing that was what made me quit math.