I need a better "method" for working out proofs and problems

[dkotschessaa]

I will even take book recommendations, though I have read Polya's "how to solve it," and Vellemans similarly titled "How to Prove it."

I think I am looking more for how to organize my thoughts, and much of this overlaps with "how to study," which, I am still trying to learn how to do.

My method of working is, well, not working:

Look at the problem in a book, write it down clearly (this is the one to hand in).

Another piece of paper - write down the problem.

I'm showing P implies Q...

Write down the bits of information I know that might be relevant - theorems, definitions.

Look at P... what does it imply? Which of the theorems might be relevant to P in particular?

Start stringing things together.

Out of paper

Grab another... start scribbling implications, stringing things together.

Ok, if Q was true, what would that imply?

more scribbling... more paper...

Then the "staring" phase begins.. it looks like I am not working, but of course I am "thinking." Except I am not thinking... "thinking" implies some sort of linear process. I'm perhaps contemplating or waiting for an idea to strike.

I very rarely get struck with ideas like some people do. My experience is more along the lines of "You don't understand things in math, you just get used to them." (Said some mathematician)

Revelations often come to me like when somebody reveals the secret of a magic trick. Not a delighted "OH so THAT'S how it's done!" But more of a "Oh..so THAT'S how it's done?"

I don't get inspirations on walks or during showers while doing the dishes. I don't wake up at 2:00 in the morning with an idea of how to solve something, though I do tend to have math floating through my head all night, which I suppose is my mind processing something.

I very often end up googling problems or parts of problems, and I feel very bad about this. I do not copy verbatim, but try to convey the idea of the proof in language that I understand, even at the risk of it being wrong. (I'd rather hand in something incorrect and original than something correct and unoriginal, even though it has cost me a grade to do so).

I know there is no "algorithm" to writing proofs, but there must be something better than what I am doing...

-Dave K

 

[jedishrfu]

How about "Proofs from The Book' based on Erdos notion of God keeping the best proofs in The Book.

https://www.amazon.com/dp/3540404600/?tag=pfamazon01-20

http://en.wikipedia.org/wiki/Paul_Erdős

 

[dkotschessaa]

How about "Proofs from The Book' based on Erdos notion of God keeping the best proofs in The Book.

https://www.amazon.com/dp/3540404600/?tag=pfamazon01-20

http://en.wikipedia.org/wiki/Paul_Erdős

Yes, thank you for mentioning it. I had thought about that one before. One thing I have trouble with is seeing how multiple proofs for the same theorem are equivalent. I often fail to see the equivalence. This book seems it could help with that.

Speaking of "book" proofs, that is actually another weakness of mine. I'm obsessed with the idea of an "elegant" proof to perhaps an unhealthy extreme. When I start to realize a proof requires a lot of detail I tend to assume there's got to be a better one. But I can't be sure I'm being elegant or just lazy...

 

[jedishrfu]

We have a similar situation in programming and what we focus on:
- first make it work
- second make it work well
- third make it work fast

In your case, time is limited:
- first prove it
- second rework your proof to prove it well
- third rework it again for elegance

After step one, you have something to hand in and you can stop and go on to the next proof and revisit it later.

For me, I always liked how geometric proofs were constructed via the two column method: statement then reason for all steps. I'm not sure that works well for algebraic proofs though.

One interesting point, my son discovered in law school is that writing legal briefs are very similar to writing math proofs and that often mathematicians can do a better job than lawyers in writing these briefs as evidenced by some mathematicians becoming lawyers (probably for the pay...)

 

[dkotschessaa]

We have a similar situation in programming and what we focus on:
- first make it work
- second make it work well
- third make it work fast

In your case, time is limited:
- first prove it
- second rework your proof to prove it well
- third rework it again for elegance

After step one, you have something to hand in and you can stop and go on to the next proof and revisit it later.

That's good to keep in mind. If I am choosing to skip over details it is probably because I am being lazy or avoiding something I do not understand. So if I have to write out the gory bloody details, I probably should, and if I find an elegant re-work of the idea I can still do that later. Thanks. (I'm not sure if you realized that's what you said just now, but you did...)

For me, I always liked how geometric proofs were constructed via the two column method: statement then reason for all steps. I'm not sure that works well for algebraic proofs though.

I think it's not a bad idea to adopt such a system, even if that's not the proof I ultimately hand in. For the proofs I am handing in (as homework) there does seem to be an artform to when you are supposed to quote a result or theorem or when you assume that it's common knowledge at this point in the program. (I found that my algebra teacher graded less harshly when I gave less information, which was interesting.)

One interesting point, my son discovered in law school is that writing legal briefs are very similar to writing math proofs and that often mathematicians can do a better job than lawyers in writing these briefs as evidenced by some mathematicians becoming lawyers (probably for the pay...)

Yes, I have always suspected this was the case. We were meeting with some lawyers awhile ago and I think we were mutually pleasantly surprised at how fluently we were able to converse with one another, despite me having no legal background. (Of course I am also a skeptic nerd, so I am fluent in rhetorical logical fallacies and such).

When I mentioned to them that I was a math major, they both cringed. It was pretty funny. "But you're lawyers!" I said. "Yes, because we couldn't do math." One replied.

-Dave K