Proof based math for physics student

[f25274]

Hello. I find myself struggling in my first proof based math class, number theory. I have taken math up to linear algebra and differential equations. It is elementary number theory so it really should not be that hard. It was probably the easiest class available that was proof based. However, the questions are all either too easy or too hard. Some questions I can immediately see the answer upon reading the question. Some questions, I go through all the definitions, theorems, all the proof techniques I know but I could still not prove it. When I later find the answer, I realize that I needed to use a clever trick or some identity that I would have never thought of using. Of course, the class doesn't teach those tricks or identities since they would be strategies specific to a problem i.e. unable to be generalised. Is math really just for those who already get it? It seems that you just have to be creative and know all those tricks to be good at proofs. Does it get easier? I originally planned to double major in math and physics, but I don't think I enjoy this part of math. Is proof based math even useful for a physics student? Of course number theory is not going to be useful but what about real analysis and abstract algebra and topology?

Also, I don't find it as fun as people told me it was going to be. In fact, I don't really like it at all.

 

[Seydlitz]

I agree with you that as a beginner also I find that sometimes proof use clever technique or idea that I can't just imagine how the author would be so creative as to finding that out. But the thing is I also find out that those tricks as subtle it might be can be used to prove similar question related to that particular area, albeit it can be challenging yes I agree to apply it so that you get what you want. I understand other and even senior mathematicians know this tricks, as I remember Ian Stewart identified this problem as finding the right "hinge".

I don't do many proofs yet but I can give an example in elementary limit theory. See this thread of mine:
https://www.physicsforums.com/showthread.php?t=702079

I think the trick here is to use triangle inequality as the way to go from one inequality to the next and finally to get the result we want. In many proofs though especially the formal one, I don't think the author explicitly mentions his trick or insight, hence you only see the finished, simple, and direct-to-the point result. I'm optimistic still that with enough experience anyone can tackle and do proofs. I even discovered a simple tiny probably obvious theorem that a symmetrical matrix is normal without the book saying so.

Also this thread should also be helpful for you:
https://www.physicsforums.com/showthread.php?t=699717

 

[homeomorphic]

About struggling with problems. I think you need to work on your problem-solving technique. Creativity is a skill that you can get better at, not just something that you are born with. First of all, you have to be persistent. You may have gotten used to be able to solve problems in one sitting. But when you do higher-level math, you can't expect to be able to solve it in one sitting. You have to stop working on it and come back to it fresh sometimes. A little twist on this idea of being persistent is actually one of the big things involved in being creative. To be creative, you can't get into the mind-set of just coming up with one or two ideas and thinking that is good enough. You have to come up with lots and lots of different ideas. Try everything you can think of. Another stumbling block for me, that I have noticed over the years is that sometimes I am making a mistake or wrong assumption somewhere that is invisible to me and is making the problem much harder than it should be. So, the technique there is to assume that you are making some mistake somewhere and try to find where it is--sometimes, it could be that you don't have the problem straight in your mind. Other than checking for possible errors, you should avoid repeating the same thoughts over and over again. You have to keep things moving. Try something new, rather than stubbornly insisting on pushing your first idea through. It is easy for your brain to get stuck in a rut, which is where these things like taking a break and thinking about something else and forcing yourself to try a different approach really help, in addition to providing additional ways in which you might arrive at a solution. Another thing to try is see if you can solve a simpler version of the problem. An easier special case, a slightly different problem, giving yourself stronger assumptions to work with. That often helps get a handle on it, and often, once you see how to do the simpler version, you can see how to generalize it. Another tip is that you really need to understand the material deeply. Often, that will make the problems easier. Also, think outside the box. And finally, do not be afraid to ask the professor for help if you are completely stuck. Typically, they will at least get you unstuck.

Does it get easier? For the most part, no. It's all uphill. Just gets harder and harder, although you can get used to doing proofs, so in some ways it could get easier, although that wasn't my experience at all, since I was a "natural" at proofs, so that I didn't really have to get used to it much.

As far as whether it's fun, there are two things I can say there. With regard to doing problems, some of the enjoyment is the challenge, and then the satisfaction of overcoming the challenge. Another thing that I can say is that what I like about math is not just solving problems. I like understanding deep ideas. Particularly, I like the experience of viewing something in just the right way so that it becomes obvious. Most often, for me, that means being able to picture it in my mind's eye. Another thing I enjoy about it is seeing how the subject all comes together. This idea leads to this one, and that leads to this other idea. You make such and such definition because it helps with so and so, etc. Unfortunately, mathematicians often tend to be very formal in their presentation, so that this sort of thing takes a back seat.

As for whether it's useful, yes, doing stuff that forces you to practice all the stuff I mentioned is extremely useful for anyone who has to solve hard problems, including physics students. I'm actually sort of wary of the whole, "we need to teach kids math because it improves their problem-solving skills" argument, as a reason for teaching kids basic algebra. It's not so clear that it really does transfer to anything else because there is no creative process going on. But the sort of process I was talking about above, I think, really does transfer to other contexts because creativity in different fields can be very similar. Now, if you're talking about content-specific stuff like real analysis, abstract algebra, and topology, that's very much dependent on what kind of physics you want to do. If you don't like that sort of stuff, you can easily get by without it, and if you do like it, then you could very well find a use for at least some of it, just by choosing the right path within physics.

 

[f25274]

Thanks for the good advice!