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Frustated by the lack of mathematical rigour in physics topics

  1. May 11, 2012 #1
    I have become so annoyed by the lack of mathematical detail in the physics textbooks. For example, to study Foundations of Quantum Mechanics, I had to spend hours and hours doing background reading from books, the net and Wikipedia until I got my way aroung vectors spaces and its nooks and crannies. I think I will have to do the same next year when I take the General Relativity course. This is becoming so annoying and actually I am beginning to think I should have studied maths instead of physics, so I could have known more mathematics and felt satisfied and then I could have specialised in theoretical physics.

    This is actually making me wonder if I should self-study mathematics from the freshman upto senior level just so the mathematics is not so demotivating?
     
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  3. May 12, 2012 #2

    Vanadium 50

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    Are you complaining that the mathematics used in physics is not rigorous? Or are you complaining that the mathematics used in physics is not taught in physics textbooks?
     
  4. May 12, 2012 #3

    micromass

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    Why not minor or double major in mathematics??
     
  5. May 12, 2012 #4
    For me that was part of the frustration. Switching between rigorous mathematics and sloppy physics from one class to the next. Even so called mathematical physics courses are sloppy compared to courses taught in the maths department.
     
  6. May 13, 2012 #5
    Oh! I mean that the maths is not rigorous. But even if physicists only managed to use correct notation in every case (for example, f(x) is not a function; f is the function and f(x) is the output) and managed to show the representation of any mathemcatical expression in all its generality, then I guess the subject could have been more satisfying, because even though we would be still be using approximations in our calculations, at least we would know where the approximations have been made and why the approximations do not cause a huge difference between predictions and experimental observations.

    But, there's this other point that vector spaces are an abstract concept and so it's must better to introduce the idea of vector spaces and all its properties by appealing to examples of functions, matrices, Euclidean vectors and other exotic objects and to show how their properties are all similar whenever a new property of a vector space is introduced. The end result is that we have to spend hours trying to convince ourselves of the validity of the statements so that we avoid taking the mathematical statements for granted.
     
  7. May 13, 2012 #6
    Exactly! I can be satisfied with not knowing as much abt vector space as a maths undergrad, but at least i wish whatever little i knew i knew rigorously.
     
  8. May 13, 2012 #7

    micromass

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    The thing is that physicists use such advanced math at times that it becomes impossible to do everything rigorously. Ideally, you could study the math first and then do the physics, but then it would likely take a huge amount of time before you get to see anything interesting.

    For example, in calculus you have seen that in some cases

    [tex]\int_a^b\int_c^d f(x,y)dxdy=\int_c^d\int_a^b f(x,y)dydx[/tex]

    The proof for this result is quite annoying. So what is the benifit, for a physics major, to learn that proof?? It's so much easier to accept it as a given. Instead of getting into the details of the proof, one could spend his time better with the details of physics problems. After all, physics majors pay to learn physics, not annoying math proofs.

    I mean, if the choice is "knowing much physics, but not knowing the rigorous math" and "knowing the rigorous math, but not much physics", it makes sense that physics educations go for the first option and that math educations go for the second option.

    I guess that the only thing you can do about it, is self-study the math or take math courses in the math department.
     
  9. May 13, 2012 #8
    Mathematicians will detest this, but some things, with a solid intuitive understanding, just make sense. The thing you brought just makes sense if you use an intuitive understanding of integration as an infinite sum.

    I believe the OP is not complaining about the absence of rigorous proofs. He is complaining about the absence of intuitive introduction to the topics. Or perhaps he's complaining about the book using mathematical concepts without explicitly defining their use. (So maybe his book had a vector space, used properties of a vector space, and never once said "Hey, this is a vector space." So it treated that fact like it was completely regular and not even worth stating.) I could be wrong, though.
     
    Last edited: May 13, 2012
  10. May 13, 2012 #9

    micromass

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    Yes, it makes sense. Except that it's wrong some times. To discover when the formula is true, one needs the rigorous proof.
     
  11. May 13, 2012 #10
    Well, I think this goes back to what the OP MIGHT be saying (I mean, he really needs to clarify). In this particular example, I don't think he would want a rigorous proof of this attribute. I think he would just want an intuitive description of how it makes sense alongside warning about when it might fail (combined with more intuitive understanding of why it might). I suspect in the vector spaces example, the book barely defined they were even using one, and only through a large amount of Googling and research (as he stated) did he even find out he needed to know this math.
     
  12. May 13, 2012 #11
    If you want to study physics and be a physics major do not expect your classes to be taught like math classes. Physicists and mathematicians do not think in the same way, even on the theoretical side. If physics classes spent time mathematically proving and being fastidious about mathematical notation they wouldn't get enough physics done! Be a double major and learn to jump between the styles. Its definitely worth it and I certainly enjoy it.

    For example in Griffiths he gives some overview of linear algebra in the back of his QM book. He mentions that although it isn't hard to show that the e.vectors of a finite dimensional Hermitian matrix span the vector space, the proof used does not carry over to infinite dimensional cases. It's still true (I think...) but infinite dimensional matrices was beyond the scope of my proof based linear algebra class so is obviously beyond any quantum class.

    My advice is to simply suck it up and deal with it for now. Both kinds of classes have their benefits and although physics classes aren't rigorous, they teach you to hone and work with your intuition. Don't underestimate its usefulness. The Standard Model, which is amazingly accurate, is a QFT with a gauge symmetry and guess what QFT is not mathematically rigorous! I mean progress has been made but the functional integral is notoriously illdefined. And developments in physics have spurred research in mathematics (especially in topology and knot theory) so its no longer this one way relationship where we physicists just borrow and abuse notation :)
     
  13. May 13, 2012 #12
    Also, if you stay just as a physics major, don't worry about trying to the mathematics in full. Like I said, the subtleties are not immediately relevant. You'll learn to pick up what you need to and when you get more advanced in your studies the mathematics will become more advanced and rigorous.
     
  14. May 13, 2012 #13

    atyy

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    Rigour is the last thing you want in physics - first get the correct theory, then try to construct it rigourously. Even most of mathematics is that way, ideas first, rigour second - take numbers and the calculus, those came before ZFC. Logic is perhaps the one place where it's rigour first.

    Apparently it's still physicists who abuse notation. Turaev says in the Introduction of his book that the path integral approach to the Jones polynomial has still not been justified.
     
    Last edited: May 13, 2012
  15. May 13, 2012 #14
    Most of the math is rigorous. However, the purpose of an introductory textbook is not to present the technical details of the theory with full rigor. The purpose is to teach you how to do example calculations and understand the conceptual implications of the theory.
     
  16. May 13, 2012 #15

    AlephZero

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    I'm still not quite sure what the OP is complaining about. If he/she tried to do a QM course without being told what the prerequiste math was, that's one problem. If he/she skipped the prerequisite math courses "because they were not rigorous enough", that's a different problem!

    I'm with Micromass and Roshan on this one. I have a math degree but spent most of my working life in engineering companies. I once invented a neat numerical method but was concerned there was possibiltiy it would loop forever. The condition for that could be expressed in terms of the ranks of a sequence of submatrices of a matrix, and the conditions for selecting the next member of the sequence. It could also be described by a thought experiment about how to assemble a certain type of "rube goldberg" structure, using temporary supports in a particular way to stop the partly assembled object falling apart. There wasn't any point wasting time doing research into an obscure corner of linear algebra, when a straightforward physical argument answered the question!
     
  17. May 13, 2012 #16
    That's because physicists aren't mathematicians. Physics textbooks focus on the physics and not the math.

    That won't work. The point of theoretical physics is to describe nature. Sometimes you have to abuse math in order to do that. In most situations, having a rigourous proof turns out to be unnecessary of you have something that "just works."
     
  18. May 13, 2012 #17
    In physics and engineering there is a tendency to be sloppy, unorganized, and unrigorous (ie unmethodological) with the mathematics. I've had many courses where formulas and equations are shot-gunned onto the white board with no justification and no derivation. If you care about being organized with your mathematics, this gets very frustrating and I can see why some people switch from physics/engineering to pure math.
     
  19. May 15, 2012 #18
    When I was studying electrical engineering in college, I stared at those CMOS equations and thought, "There is no rigor in physics."
     
  20. May 15, 2012 #19
    If you're fluent in analysis, you can translate physics into mathematical terms. Alternatively, you can use textbooks that utilize the theorem-proof style. Check out Classical Mechanics by Gregory.
     
  21. May 16, 2012 #20
    I kinda understand the OP point but theres usually a large amount of material that needs to be covered and mathematical rigour is usually pretty low on the priority list. Most of the time it serves no purpose in a physical sense.

    I'm an EE/Math dual major but I'm much farther ahead in my EE degree. Its nice when I learn something in a math class after I've seen it in engineering and the two sorta click together. Really helps in my understanding when I understand the point and the reason.
     
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