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Having Trouble With Mathematical Proofs - How Important for Physics?

  1. Jan 11, 2013 #1
    I'm having a lot of difficulty with proofs. I can understand them when I read them, but when asked to prove things myself, I usually fall flat on my face.

    How important are proofs for physics? Do I really need an intimate understand of the mathematics, or would that be irrelevant?

    I was thinking of buying "How to Prove It", because I become frustrated with myself. If I can't prove it, maybe I don't understand it well enough.
     
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  3. Jan 11, 2013 #2

    MarneMath

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    I'm not a physicist. Most that I do know, can give me an idea behind a proof, even if they can't figure out all the details that makes a proof solid. My personal opinion is the ability to prove many mathematical statements may not be an important tool for Physicist as much as the ability to translate real world problems into mathematical statements. However, I do think if you truly have no idea why a mathematical idea is true and when the idea fails, you might find yourself abusing techniques when you shouldn't be using them, Other than the intutitve, I don't think a deep understanding is truly needed.
     
  4. Jan 11, 2013 #3
    I can give an idea behind it to, but that's all it is, just an idea. If you told me to make a go at proofing it, I'd fail. I can't help but feel that it's a problem.
     
  5. Jan 11, 2013 #4
    In my experience, if you can prove it, you have mastered the background of the concept in your head and can move to learning something else. That's how I teach myself stuff.

    It may not work for everyone, but if you get better at proofs, you will probably be better at physics.
     
  6. Jan 11, 2013 #5
    I wouldn't know about physics importance, but I do have a tip that should be able to help with proving things. At least, proving that one thing's equal to another.

    Backsolve. Look at the final expression, manipulate it if necessary, and see what you can do to the initial expression to make it look more like the final expression.

    Taking a look at a proof for the product rule for derivatives should give you a good example of what I'm talking about.
     
  7. Jan 11, 2013 #6

    Redbelly98

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    Mathematical proofs, of the type often encountered in math courses, are not that important for doing physics. But you will occasionally have to show that some described physical quantity is equivalent to some given algebraic expression.

    p.s. to OP: since this is posted in the mathematics part of the forum, the responses may be skewed more toward the mathematicians' point of view.
     
  8. Jan 11, 2013 #7
    Perhaps you could move this to the General Physics forum?
     
  9. Jan 12, 2013 #8

    MarneMath

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    I think that mostly has to do with a lack of mathematical maturity, which is really just a term for having a good deal of mathematical knowledge at your finger tips and the ability to know when to use it. It comes with practice and more focus on mathematics. Believe me, a good deal of mathematician have the opposite problem with it comes to physics. A good number of them look at your mathematics and just go "huh?" , even if in the end it makes sense. For the most part, both are skills that come with a lot of time and practice.
     
  10. Jan 12, 2013 #9
    Despite still being learning (and pretty early in this, I've stuck mainly to the Art of Problem Solving so far,) I can somewhat confirm this. Physics just seems too ... messy. Though I do like a good line integral! (e.g. magnetic field generated by a wire)
     
  11. Jan 12, 2013 #10
    I had the same feeling when I started, especially in the first year of studies when I had to know all the usual Analysis and Linear Algebra theorems and proofs.
    In the physics courses I did after that, until now, I never had to do any mathematical proof. Knowing the theorems is much more important than knowing how to prove them, in physics.

    It doesn't mean that knowing how to do such proofs is useless. They help physicists to get introduced to the way of thinking that is absolutely necessary in physics. Even if this introduction is hard to get in place, after some work you will see that things start to fit in your mind naturally.
    When dealing with physics problems, the maximum that I had to know in that aspect is the existence of some theorem or lemma, and then even if I don't remember it precisely the most important thing is to know what name to look up in a mathematics book, i.e. you have to have some intuition of what to use and when, and what fits or not the conditions and hypothesis of the problem you're working on.
     
  12. Jan 12, 2013 #11
    It helped me a lot. Proof writing made me a lot more organized in my thoughts and presentation. And, then, there are those times where a physics question is more or less a (not-completely) proof.
     
  13. Jan 12, 2013 #12
    That's quite different from, and much easier than, a proof in the way mathematicians use the work IMO.
     
  14. Jan 12, 2013 #13
    Depends on what kind of physics.

    But, really, you don't have to go far in math before being able to handle proofs becomes useful. Linear algebra, in particular--very important in physics. It's in poor taste to do computations without knowing why they work, and that involves something that at least approximates what you have to do to prove things in a math class.
     
  15. Jan 12, 2013 #14

    symbolipoint

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    This is how proofs for Math and proofs for Physics differ:

    Proofs for Math are usually for the purpose of exploring and exposing how Math works, and for further developing the science of Mathematics. Proofs for Physics are for the purpose of finding some relationship among physical quantities, from best of my memory, they are derivations. In physical things, one wants to find a formula for a physical situation.

    Since I did not go too far in Physics, I must wonder: Are there mathematical proofs in Physics? Are there just derivations of varying level of difficulty?
     
  16. Jan 13, 2013 #15

    Redbelly98

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    Agreed.

    I don't recall ever having to prove a mathematical theorem in going through my physics education (includes both undergrad and Ph.D. programs), but I was more an experimentalist than a theorist. Lots of derivations for sure, as you said above, but no proofs. There are lots of approximations or assumptions justified on physical grounds -- stuff I doubt you would get away with in a mathematical proof.

    A Mentor has moved the thread to Academic Guidance. :smile:
     
  17. Jan 13, 2013 #16
    I'm not sure that "derivations" are really any different from proofs. I think that's an artificial distinction. It's the same thing, except in math, you are just more finicky about the details and getting things right logically. If you justify things on physical grounds, that's the same things mathematicians do. Mathematicians justify things, initially based on intuitive, heuristic reasoning all the time, which is completely analogous to making physical assumptions. It's just that they subsequently have to prove that their guesses were rigorously correct. It's just adding that extra step--the process is really pretty similar.

    There's a reason mathematicians do it that way, though. They aren't just being finicky for the sake of being finicky. If you want to use the mathematical tools with complete safety, you have to be very careful--and you really get a feel for that once you have done a little math research. Until then, even after years of math classes, it can be hard to fully appreciate how precarious things can be. My thesis was on the verge of falling apart many times because I had not been careful enough.

    The other difference is that physics often just deals with stuff that has a lot of formulas, whereas, in math, you get a lot of stuff that just involves lots of words and concepts. You get formulas in math, too, of course, like if you want to calculate stuff in differential geometry. So, I think, the difference is really one of subject matter more than anything.
     
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