Is proof writing a learnable skill?

[xbomber88]

For some reason it seems like I am just incapable of doing proofs. I have trouble with even very simple proofs. Admittedly I probably haven't tried as heard as I could because writing proofs just annoys me. Last semester I was talking a course called Discrete Math at my college which basically turned out to be a course on proof writing. I absolutely hated the class and decided to drop it halfway through the semester. Maybe I should've tried to stick with it but at the same time I think it was going to destroy my very respectable GPA which I didn't want to happen. But it was so frustrating. Most of the class seemed to be able to do the work and I just simply couldn't do it. It made me feel like a complete idiot which is not something that I'm used to. I'm not sure why I have so much trouble with proofs. I'm not an idiot. I have a 3.9 GPA and I got an 800 on the quantitative section of the SAT. Also I was taking multivariable calculus at the same time which I didn't have near as much difficulty with. But for some reason it seems like I just can't do proofs. Is it a skill that I can learn or is it just something you either have or you don't? To me writing proofs just seems like a lot of hocus pocus. I can usually follow through a proof step by step and understand what's going on and follow it all the way to the result. But it almost seems like a magic trick or something to me. Logically, I can't figure out how anyone would ever come up with the steps they use to prove something. I would've gladly given up proofs forever but now I'm starting to realize how important higher mathematics is in theoretical physics which I find to be very interesting and I can't do any higher mathematics without the ability to do proofs. I really want to take some higher math courses at my school but at the same time I'm scared to death to do it because I don't want to be in the position of either taking a huge hit to my GPA or dropping the course again. Does anybody have any ideas as to why I have so much difficulty with proofs and how I can change that? Can I even change that?

 

[altcmdesc]

Proof writing is absolutely a learnable skill.

The best thing you can to do learn how to write proofs is read them. Some proofs do have tricks to them which seem to come out of nowhere, and those tricks are usually pretty applicable to other proofs in the same subject as well.

Also, get familiar with some basic logic and proof techniques such as proving the contrapositive (instead of proving "if P then Q" prove "if not Q then not P"), how to prove an "iff" statement, induction, etc. This helps a lot to find a starting point, which, for me, is the most difficult part of writing proofs.

Try looking at this website for some tips: http://zimmer.csufresno.edu/~larryc/proofs/proofs.html

 

[JasonJo]

Just keep practicing. Is there an intro to proofs course offered? I used to have some difficulty, and then you get used to all the proof techniques, which one is optimal, combining them, etc.

 

[rodigee]

Ya, the first encounter with proofs is difficult for some people. From my experiences most of the difficulties fall in one of three areas:

1) The mathematics involved. If you don't understand the math, you aren't going to be able to put together arguments about it.
2) Logic. You need to understand basic logic to put together a sound argument.
3) Style. This is an often unfamiliar form of writing, and that can bog people down.

xbomber88, do your troubles fall into any of those categories? But regardless of that... just keep practicing. You'll figure it out.

 

[Vanadium 50]

Well, xbomber, I have to say that a high GPA is less impressive if you've been avoiding classes that emphasize what you don't already know how to do. And the idea to keep avoiding classes that emphasize what you don't already know how to do - i.e. learning new things - to keep your GPA high kind of defeats the whole point of college, no?

First, you can learn how to do proofs. Nobody is born knowing this. Everybody has to learn how. However, like everything else, it takes practice. You won't get practice by dropping classes that require proofs.

Second, I often hear the complaint "I understand the math; I just can't do the proofs". In the vast majority of the cases, it's because they don't understand the math. It's possible to think you understand things when you don't, and being forced to do proofs can quickly remove this misperception.

Good luck!

 

[xbomber88]

Well, xbomber, I have to say that a high GPA is less impressive if you've been avoiding classes that emphasize what you don't already know how to do. And the idea to keep avoiding classes that emphasize what you don't already know how to do - i.e. learning new things - to keep your GPA high kind of defeats the whole point of college, no?

Admittedly that GPA might be a little bit misleading. I was a business major my freshman year and I'm a sophomore now. But I achieved that GPA with very little effort. I haven't gotten my grades back yet but this year as a physics major in my first semester I'm fairly certain that I'm going to have at least a 3.8 and probably a 3.9 and there's even a small chance that I might have a 4.0. So yeah, your point about my 3.9 GPA not being that impressive if I'm not taking hard classes might have some merit. But the idea that I'm not learning anything and defeating the point of college by avoiding classes that emphasize proofs is kind of ridiculous. I'm learning plenty of things. Just not how to do proofs. I'm a physics major not a math major so proofs aren't of primary concern to me. Which leads me to my next question. As a physics major should I even care that I can't do proofs? I think I'd rather do theoretical physics than experimental physics which I understand is a lot heavier on the math. But do theoretical physicists need to be able to understand where all of the math they use comes from and the proofs like a mathematician would or do they simply need to know how to apply it? If I really do need to learn how to do proofs I'm willing to spend next summer going through a book on proof writing and practicing but if not I'd be perfectly happy to never do another proof in my life.

 

[xbomber88]

Is there an intro to proofs course offered?

In response to you the course that I dropped last semester basically was the intro to proofs course offered at my school even if that's not what it was called. But I really didn't like the way that the professor taught it which was probably half of my problem. In fact he didn't really teach it at all. Basically all he would do is give us homework which all consisted of writing proofs and then he would have the students come in and write the proofs on the board. Occasionally he would tell us we were wrong if we were heading completely in the wrong direction. But even that he would only do occasionally. Also, the book we used was a tiny little paperback book that was only about 100 pages long and it pretty much took the same strategy as the professor did. It would give us a tiny bit of guidance but basically would just give us some proofs to do and then say go learn it on your own. I understand that the professor wasn't just being lazy and that the whole point of the class was that if we discovered something on our own rather than being told how to do it by the professor we'll understand it better. But is that really the best way to go about teaching a course on proof writing? It did seem to work for a lot of the people in the class but it certainly didn't work for me. I would just get so annoyed in class watching the students put up their proofs on the board when half the time it was obvious that they weren't doing the proof the right way or were taking a long roundabout way that could've been made much easier if the professor would've just given us a little help. I wanted to actually learn proofs from people who really know how to do them not from a bunch of people that are learning how to do them just like me. My other big problem was that I never had any idea where to start when I was writing a proof. I just felt completely lost and didn't have any clue what direction to go in. I just wish that he would've actually given us some insight and strategies on how to write proofs rather than just expecting us to do everything ourselves. Maybe he could've slowly transitioned into making us do everything ourselves as the semester progressed but that's how it was from day one. What do you guys think about the way that course was taught?

 

[Vanadium 50]

There certainly are proofs in theoretical physics:

1. Bell's Theorem.
2. Factorization Theorem
3. CPT Theorem
4. Coleman-Mandula Theorem

just to name a few more famous ones.

You should also recognize that if you continue your career in physics, you will run into more professors whose teaching styles do not mesh with your preferred mode of learning. You will also run into professors whose choice of material is not immediately obvious to be useful. The option of dropping classes taught by these professors will not always be present.

 

[will.c]

I think the way your teacher taught proof writing is just fine. There is no algorithm, no crank to turn. The only way to learn how to write proofs is to read and write a lot of proofs. Of course, there are a lot of topics that should be lectured on in an introductory class, like basic logic, contradiction, induction, etc. but after that it's practice practice practice.

And I must agree that proof is exceedingly important in physics. It's not just about "where the math comes from," it's that physicists must be able to explain physical phenomena entirely from a small set of physical postulates (Newton's laws, Maxwell's equations, the QM postulates etc.) How is this done? By proof.

 

[rodigee]

Second, I often hear the complaint "I understand the math; I just can't do the proofs". In the vast majority of the cases, it's because they don't understand the math. It's possible to think you understand things when you don't, and being forced to do proofs can quickly remove this misperception.
Good luck!

Words of wisdom.

 

[samspotting]

Vellemen's How to Prove it is a really good book on learning proofs.