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Rigorousness of mathematics in physics classes

  1. Jan 28, 2007 #1
    Does anybody else here find that their physics classes - while providing an acceptable working understanding of the maths involved, don't treat the mathematics with enough rigorousness for their students to be able to understand other discussions on the subject?

    I ask because I'm quite often baffled by the mathematical discussions that go on here; and also struggle to understand many textbooks on areas that I nevertheless do quite well in. As an example, I am very happy with wave quantum mechanics and can get by with matrix mechanics, but I don't know what a Hilbert space is and don't seem to be able to find any sources to explain what one is in terms I understand.

    In general this phenomenon means I'm stuck with only my lecturer's interpretations of the subject and nothing else, because the amount of mathematical terms in most mathematical physics books I've found goes way over my head. Am I alone here?
     
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  3. Jan 28, 2007 #2

    ZapperZ

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    Wait... you're contradicting yourself.

    First you said that the mathematics do not have enough "rigorousness". But then, you said that most mathematical physics books goes way over your head. Have you ever considered that maybe those things that you don't understand ARE the "rigoriousness" that you were looking for. Just because you do not understand it, doesn't mean such a thing doesn't exist.

    I know that Arfken deals with this in his book. How "rigorous" you want it to be depends on you, because these books, and the physics classes, are not meant to teach you the mathematics, but rather how to use the mathematics as a means to a specific end, which is the physics. You can always pursue the mathematics itself, either on your own, or by taking extra classes from the math department. That is where they do everything rigorously, to death, if that is what you want.

    Zz.
     
  4. Jan 28, 2007 #3
    What I'm saying is that I wish my classes were more rigorous so that I could understand the level of mathematics involved in textbooks on the subject. As it stands, the style of teaching I receive makes it difficult to read around the subject because, unlike the class teaching, most authors treat the mathematics much more technically and thus beyond my understanding. To go back to dedicated mathematics textbooks below that level to get a grounding in the general principles first would take me far, far too long.
     
  5. Jan 28, 2007 #4
    What is stopping you from trying to learn the mathematics you do not understand? There are mathematics textbooks available in print and there are several online resources as well. Or you could just discuss it with someone in real life. Don't see problems, see solutions.
     
  6. Jan 28, 2007 #5
    In short, time. I could give up what free time I have to study mathematics on top of my course, but...well, I didn't come here to do mathematics. I care about it right now merely as a means to an end - understanding physics better.

    I get the impression that many british universities are actually similar in this respect. Physics lecturers seem to frame their material in a functional style, rather than delving into technicalities. This would be fine, if not for the different approach taken by authors. The two approaches don't mesh terribly well. Pragmatism, or laziness?
     
  7. Jan 28, 2007 #6
    You might try to find Arno Bohm's Quantum Mechanics book at the library. He takes more time with the mathematics of Hilbert space at the beginning.

    The reason physics books aren't as rigorous as mathematical physics books is that that level of detail is not needed to get ones work done, and there really isn't time to cover it. Take a look at some functional analysis books that cover Hilbert space: they are quite thick and dense.
     
  8. Jan 28, 2007 #7
    Trying to figure out how little effort one need to spend to barely reach ones goal is a very bad idea. There is nothing that is keeping you from using the time you do have more efficiently. You do not have to give up anything. Still, if you do, learn to sort out your priorities.

    Mathematics is a tool for problem solving and a vital tool for understanding physics. Mathematics is a part of physics, whether you like it or not.
     
  9. Jan 28, 2007 #8
    Should a physicist study math proofs? is found here:

    https://www.physicsforums.com/showthread.php?t=149605

    Generally, the physicists said that math is just a tool for them to get the physics job done. There are many, many mathematical tools that a theoretical physicist needs to know, and they basically said that they don't have the time nor the inclination to study the foundation behind these mathematical tools.

    I believe that a theoretical physicist probably knows more mathematical tools (relating to physics) than a mathematician knows, and that is because a physicist will just take up marginal time to learn the tools to know how to use them, whereas a mathematician will devote a great deal of time to work out the properties, theorems, abstractions, generalizations, etc... for each tool. A theoretical physicis probably knows about 5 times (rough number) as many mathematical tools (relating to physics) than a mathematician, but a mathematician probably understands each tool 10 times (again rough number) better than the physicist.
     
    Last edited: Jan 28, 2007
  10. Jan 28, 2007 #9
    Far from it. I'm more than happy with my current performance already - it's just that I get this sinking feeling over not having the deep understanding of theoretical physics that I'd like to. I'm just saying that I'd like to be able to feel more confident that I understand the formalisms of the theoretical stuff, but I'm not willing to devote half my life to doing so. All I want from lecturers is a bit more attention to detail, and "read this, this and this" occasionally. Not too much to ask.
     
  11. Jan 28, 2007 #10
    andytoh, this has nothing to do with studying proofs. At all. It is about understanding the mathematics behind it in order to understand the physics. See post #3 for a better definition of what the OP wants.

    Sojourner01, you cannot except that the world will change according to your needs, just because you want to learn the underlying mathematics in physics courses. If instructors started wasting time going over every single concept in mathematics that is in use in that particular course, they would waste hours and hours. Instead, you should adapt to the world.

    It is all about discipline. If there is something you do not understand to the extent that it is disturbing your physics studies, find information about it for yourself instead of blaming the instructor for your lack of understanding. That is not the way the world works. Just look up the underlying information about that particular section of mathematics. How hard can it be?
     
  12. Jan 28, 2007 #11

    mathwonk

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    I also struggled as a student to understand the mathematical parts of my physics classes. This was in my case because the mathematics clases had evolved to presenting the material in a sanitized way that ignored its physical beginnings.

    The physics use of math was actually linked to the rigorous math but no one took time to tell me how. for example in feymans lectures he gives a nice little approximation of the length of a planetary orbit, just by adding up polygonal terms. This blew my mind because i thought an integral was an antiderivative and had forgotten it was about an idealized way of adding things up.

    another time i was tempted in a problem to use integerals in a suggestive but imprecise way prompted by the leibniz notation. I spent an hour or so justifying that use to myself before using it in my homework, and was praised by the physics grader for being the first person in over a hundred homeworks to do so (this was at harvard).

    Again the rigor was achievable but it took time no one wanted to spend. I thionk this is aprtly because the correctness of the use of math in physics is not based on the rigor of the math but on the track record of its success in physics and on the intuition of physicists, which is actually more reliable than mathematical rigor.

    It is a challenge and an instructive one to bring the two into accord, but in some cases this is still an open challenge. In any case, when studying with physicists, one should try to acquire the physicists ability to know his use of math is correct without an epsilon and delta proof.

    However, this is a math guys point of view, and Zapper is more qualified on the use of math in physics than I.
     
  13. Jan 28, 2007 #12
    And mathwonk comes up with the most balanced and insightful response I've yet received to this problem, on the forum and elsewhere.
     
  14. Jan 28, 2007 #13
    One example is people just think that integration is just a way to calculate areas underneath curves...its scope is much broader than that, and if you don't understand the true meaning of integration and how it works, then you are screwed in most of calculus. Same goes for differentiation.

    Too much mathematical rigor can also be bad for you and your studies of physics. If you get too formal you lose sight of the intuitive nature of the mathematics. While a mathemeticians can take 5 to 6 lines of derivation to show a particular fact, you often can see that the result is quite obvious by thinking with less rigor and with a little more intuition and abstraction. Sure, physicists need to have a certain element of mathematical rigor in their work, but intuition is just as important.

    Another example that makes me cringe (and I hear it all the time) is when people refer to a differential as a 'rate of change'. They don't see the distinction between dy and dy/dx, for instance.

    Quit looking at derivations as just mathematical manipulations and whatnot, and start looking at derivations with more of an abstract mindset. Open your mind.
     
    Last edited: Jan 28, 2007
  15. Jan 28, 2007 #14

    mathwonk

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    crash course in hilbert space

    thank you for the kind response. i will try to remind you also that you probably do know what a hilbert space is.

    R^n has many important properties. the game of axiomatizing consists in taking some of those properties, assigning a name to them, and calling any other object sharing those properties by that name.

    For hilbert space we take quite a few of the properties of R^n, namely, ability to add vectors and multiply by scalars, ability to take dot products, and hence to define lengths using them, and finally the validity of the cauchy criterion for convergence. that is all.

    i.e. a hilbert space is any vector space having a positive definite dot product, such that all cauchy sequences converge in the distance |V| defined as the square root of V.V.

    Do not feel bad if you did not remember this definition of hilbert space, as one of david hilbert's most famous attributed quotes is something like "and what is this 'hilbert space'?".

    a consequence of the axioms choaen is that one can use infinite series, i.e. infinite linear combinations of vectors, which converge if and only if their absolute values do so.

    moreover one has a concept of "hilbert basis" which is an infinite orthonormal set, such that every vector in the space is either a finite or an infinite linear combination of those.

    this allows one to deal with a space of uncountable linear dimension, with a countable "hilbert basis" of vectors.

    then the usefulnes of this concept is that most of the basic theorems of fourier analysis go thropugh in this context. I.e. the canonical example of a hilbert space is the space of functions expresible as infinite series of sines and cosines, with the usual dot product, namely integral of the product over some convenient interval, like -pi to pi.

    abstractly, this is isomorphic, as are all hilbert spaces having a countable hilbert basis, to the space of "square summable sequences" of real or complex numbers.

    these are the smallest infinite dimensional hilbert spaces. they are called "separable". thus we have taken most properties of R^n except finite dimensionality.:tongue:
     
  16. Jan 28, 2007 #15

    mathwonk

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    in general try never to be intimidated by a big name attached to a concept, like hilbert or banach space, A HEISENBERG GROUP, OR A HAAR MEASURE, or galois cohomology, or a normal separable proper scheme of finite type over an artin ring. it is merely shorthand for some ist of proeperties which someone is too lazy to give you, and also a mask for frightening the uninitiated.

    grothendieck always tried to choose terminology for his ideas which was suggestive of their meaning (a separated scheme behaves like a topological space where one can separate distinct points by disjoint open sets), but too often concepts are merely named for someone a student or colleague wishes to honor, but who often had no role in their discovery.

    The "Kodaira dimension" of an algebraic variety e.g. is apparently a concept introduced by moishezon in the Steklov Institute seminar on surfaces and named later by others. This of course is not kodaira's fault.
     
    Last edited: Jan 28, 2007
  17. Jan 28, 2007 #16

    George Jones

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    Sojourner01, you might find this post of mine interesting.

    This type of stuff is important only to a small minority of phsyicists, but is becomng increasingly useful in some area of string theory, high energy physics, relativity, etc.

    As ZapperZ says, if you are interested in mathematics as mathematics, you can fill some of your options with pure math courses. This is what I did, both as an undergrad, and as a grad student.
     
  18. Jan 28, 2007 #17

    George Jones

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    This reminds me of story about John Wheeler (of MTW) and the popularization of Kruskal-Szekeres coordiantes that are so well known, and so important for the study of black holes.

    From

    http://www.princeton.edu/~paw/archive_old/PAW98-99/09-0210/0210irtx.html

    "In one amusing incident, Wheeler tells the story of an "elegant discovery" concerning spacetime and relativity made by Martin Kruskal, a mathematical physicist and colleague of his. At the time Kruskal made the discovery, he related it to Wheeler. When Wheeler found that, for whatever reason, Kruskal had not published the results, he wrote a paper up himself with Kruskal's name, and sent it to Physical Review. He forgot, however, to tell Kruskal he had done so. Kruskal only learned of "his" article upon receiving mysterious galley proofs from the journal; when he figured out what had happened, he suggested coauthorship, which Wheeler politely rejected."
     
  19. Jan 28, 2007 #18

    mathwonk

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    One of my collagues was also surprized to see a joint article with his own name on it in the bibliography of another department member who had added it out of homage to the contribution made by the other.

    I know of other cases but will not list them since the parties are still active and I could get it wrong.
     
  20. Jan 28, 2007 #19

    mathwonk

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    Sojourner what think you of the crash course in hilbert space?
     
  21. Jan 28, 2007 #20

    D H

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    Too much mathematical rigor can even be bad for the study of even applied mathematics. I took a graduate-level ODE class many years ago. The instructor was very rigorous, which I liked because it was appropriate to the subject at hand. I had the same instructor the following semester for optimal control theory, an applied math class.

    The first 50 or so pages of the text provided the context for the subject: The nature of the subject, necessary and sufficient conditions for the existence of an optimal controller, stability of the controller. The instructor got so hung up on the lack of rigor in this introductory section that we only made it through the first 60 pages of the book.:eek: We never got into the meat of the subject. I had to learn optimal control theory on my own.

    In reality, the book is one of the most rigorous applied math texts in my library. My advice: Beware a pure mathematician teaching an applied math course. You might not learn anything.
     
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