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How much math should a physicist know?

  1. May 5, 2015 #1
    Math in the sense of more than just calculation skills or techniques. Should a physicist be able to proof math theorems?

    For example,
    1. prove the chain rule in single-variable and multi-variable calculus
    2. prove that any partition of a set is associated with exactly one equivalence relation and vice versa
    3. prove that the well-ordering principle, the principle of finite induction and the principle of complete induction are equivalent
    4. prove that ##\sqrt2## is an irrational number

    (Roughly speaking, I've arranged the examples according to how closely related I think they are to physics.)
     
    Last edited: May 5, 2015
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  3. May 5, 2015 #2

    micromass

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    No definitely not. But I guess you should make an (artificial) distinction between "derivations" and "proofs". A physicists should definitely be able to do mathy derivations (a stupid example: trigonometric identities). But actual proofs like the multivariable chain rule are not really all that necessary (the singe-variable chain rule is definitely something you should be able to prove by yourself though).

    Of course, a lot depends on you. If you are not able to accept the mathematics without a good (formal) reason, then you should do proofs. But it is definitely possible to be a good physicists without being good in proofs.
     
  4. May 5, 2015 #3
    There was one time when I wanted to do a line integral along the diagonal line ##y=x## and I wrote

    ##r=\sqrt{x^2+y^2}##
    ##dr=\frac{\partial r}{\partial x}dx +\frac{\partial r}{\partial y}dy##

    which is wrong (since it should be ##dr=dx+dy##).

    And I was so confused and I couldn't understand why what I wrote was wrong.

    Do you think that if we do not study the proofs, we may not actually know what we are doing with all the math? We may just know how to calculate and have a vague understanding, but a slight twist may throw us into confusion.
     
  5. May 5, 2015 #4

    micromass

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    That might be true. I've sometimes seen physicists get nonsense results which they couldn't explain why they were false. Not having a decent understanding of the foundations of math might get you that. But is it really a problem? Sure, if you don't like these situations, then yes. But you can definitely be a physicists without this bothering you:

    1) Even the mathematician Euler did many "wrong" things in math, but did get good results, and very very few nonsense results. The thing is that he intuitively knew when something would get you nonsense and when something would be a meaningful result. So the intuition of when something is allowed is very important, and I think most physicists do have this (or learn this).

    2) In the end, it is the experimental results which are important for a physicist. If your math is horrible, but you get a meaningful result which agrees with the experiments, then as a physicist, you did a good job.

    3) It happens way more often you reach good results with horrible math, then you get nonsense results anyway.

    So is it worth learning proofs? That's up to you. If you feel uncomfortable having a vague understand of the math, then sure. But you can be a physicsts without this.
     
  6. May 5, 2015 #5
    Such as?

    What do you mean by this?
     
  7. May 5, 2015 #6

    ZapperZ

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    I will quote the Preface in Mary Boas's classic text "Mathematical Methods in the Physical Science", a book that I've touted since forever:

    So there!

    Zz.
     
  8. May 5, 2015 #7

    radium

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    I would say what is important is to be able to use math to develop/complement physical and intuition. For example, linear algebra and group theory are incredibly important in understanding quantum mechanics. The basic postulate of quantum mechanics is that the state of a system can be represented by a state vector living in a Hilbert space. There are certain details about infinite dimensional Hilbert spaces that are swept under the rug in quantum mechanics which can create problems. However, you can still do most things without needing to worry about these issues. The properties of Hilbert space are very important in understanding quantum mechanics, like the fact that it is a complex vector space, not a real vector space. This has to do with time evolution and relates to how the phase of a wave function is defined (we see that it is the relative phase that matters). We can also see that the uncertainty principle is really just an instance of the Cauchy-Schwartz inequality. Group theory is very important since it can tell you things about how states are related. Group representations can be used to characterize degeneracies in eigenvalue spectrums. They can also be used to classify particles. The states of a particle can labelled by mass and four momentum and transform under the given irreducible representation of the Lorentz group.
     
  9. May 6, 2015 #8
    From this, would it be safe to assume that, after beginning graduate courses, Physicists would have to learn more math beyond the topics covered in Boas?

    I'm currently an undergrad taking upper div Physics classes, and there are a lot of Differential Equation techniques I learned in my lower div DE class that I have yet to see being used in Physics. Examples include the integrating factor, exact differential equations, homogeneous differential equations, Bernoulli differential equations, Cauchy-Euler differential equations (which are a specific form of linear, variable coefficient, ODE's). Whenever we come across a L-VC-ODE, we use Series Solutions. Now that we have computers, we can always numerically solve differential equations that get too complicated.

    I also wonder if it is common to solve PDE's other than by Separation of Variables (which of course, limits the the solution to products of explicit functions of independent variables). So far, in my QM class, the wavefunction is basically a Fourier Series or Fourier Integral made up of linear independent solutions to Schrodinger's Equation found through Separation of Variables. What if we can solve SE using methods that can get other forms of solutions?
     
    Last edited: May 6, 2015
  10. May 6, 2015 #9

    atyy

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    If you take thermodynamics, you will encounter a famous integrating factor that results in the state variable known as the "entropy" :)
     
  11. May 6, 2015 #10

    aleazk

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    Take any QFT book written for physicists and by physicists: most of the "proofs" you will find in those books are, to put it provocatively, pure mathematical phantasy, i.e., they are only formal, non rigorous, proofs. So, to answer your question in this particular interpretation: no, you don't need to know your math rigorously (at the mathematician's level) to work in physics. All you need is to have a good intuition with the objects you are working with and a lot of practice. Physicists often learn the correct way to do something through a lot practice, trial and error, etc. In general, they do have a rigorous intuition and basis about the definitions and theorems of, e.g., basic calculus and algebra, etc. But when things get more sophisticated, they rely more and more on formal arguments. This because of many reasons: sometimes, we simply don't know how to make things rigorous; sometines, to learn the actual math is to embark in a journey of pure math that will distract you from physics, your actual career, for which your boss pays you!

    After you cultivate that basic set of skills, indispensable for any physicist (usually, you maturate it through undergrad and first graduate years), how much you deepen your math skills will depend on your taste, personality, research interests, etc.

    All of the above is based on my experience with other fellow physicists and physics students. They really get results working in that way.

    If you ask me in my personal case: I love math and it's very difficult for me to think properly without laying down first all of the math (at least the definitions and basic setup) rigorously. In fact, in my personal case, I always feel that whenever I improve my understanding of the mathematical structure behind a physical theory, I understand its general foundations (the mathematical, obviously, but also the physical) better. But that's just me, I have no intention of making a general claim. I feel I'm on firm ground, not in some kind of cloud in the sky. Of course, this evidently shows that my tastes are more in the foundations, mathematical foundations of the theories. In general, mathematical physicists work in this area. People working in this area often know some advanced math topics at more or less the mathematician's level, or, directly, they are actually mathematicians.

    But, even with all that, I don't have the time nor the stamina to study absolutely all the math that I would like to know in all of its details. It's already very difficult to do that with the math I actually need! I have a nice book on spectral theory on my desk. I'm constantly reading it. But I also need to go and solve actual physics problems too.

    One needs to find some sort of equilibrium. Usually, I prefer to have a rigorous knowledge of the basic setups (basic definitions and theorems) and a good intuition. In this sense, I can work in peace because I know what's licit and what's not. Also, sometimes I just read the proofs of the, perhaps, more tangential theorems and try to understand them at that moment. After a while, you forget them, but at least I know that at some moment, a previous version of myself understood the proof. And those proofs actually go to your subconscious and improve your intuition.
     
  12. May 6, 2015 #11
    Hello Happiness, I am a physics major. I shall attempt to tackle your question for each numbered item on the list, and then in general.

    For #1:

    As a physics major, I tutor both the introductory Physics I class for engineers at my university (through my school's SI program), and the theoretical-based Calculus 1 courses. During one of my sessions in the calculus review sessions, I made a specific point to prove the Chain Rule because I found that much of the difficulty students displayed with not understanding what was going on, such as during differentiation rules and when performing maneuvers like "u-substitutions," stemmed from a poor grasp of the nested structure created by the chain rule. During these calculus review sessions, I occasionally make "plugs" into how the chain rule relates to known formulas regarding kinematic equations or harmonic motion, in order to prime them for their physics sequence (which the engineers typically take the semester after Calc 1 if they are on the standard track).

    During my SI session with the physics classes, I find that a great deal of struggle with them understanding the "Why?" of what we are doing (and thereby their struggle with solving problems) corresponds to a lack of perception for how the physical ideas are being communicated through the mathematical substrate upon which these ideas are being communicated. The disconnect between what they do understand and what they are expected to understand seems to lie in how well we explain the derivations (in class, as well as in the textbook), and a great many of the derivations in physics involve the chain rule somewhere/somehow.

    Since so many of the various formulas used in physics to solve introductory mechanics problems "pop-out" of Newton's Second Law, in order to have students understand how all of this makes sense/is even possible, they need both a good sense of algebraic shuffling (which they should already have) and a good sense for how the chain rule links together the hierarchy of interesting quantities obtained through summations (integrations) and "rates of accumulation/change" (differentiations). For example, in order for a student to more fully understand precisely why a string acquires the amount of tension it does when you stretch it, or why a block slows down as much as it does when friction is present, showing them how to calculate the integral that breaks-down the situation, rather than just showing them the memorizable-algebraic expression, goes a long way to produce and cement such understanding. Likewise, in order to understand why a string's tension changes at a certain rate in proportional to its stretching, and why the block moving across friction decelerates at the rate that it does, it is vastly helpful to have a great mental picture of the exact definition of a derivative. Then, when we start to add new levels of complexity to our physical problems, various parameters may be changing all at once (the spring force, the force due to air resistance, the force due to friction, etc. may all be changing at the same time!), and without a proper mental picture of the simultaneous action of partial derivatives, and how we describe those partial derivatives mathematically, it is difficult to really know what you are looking at. The total derivative of a system, which takes into account the action of all of the partial derivatives, is obtained using the chain rule.

    And there is also the ever-important propagation of error, which requires an understanding of the chain rule to understand where the "error" is coming from.

    #2:

    When we study wave motion, we describe a set of frequencies built off of a "fundamental" frequency, which all together describe the possible standing waves characteristic of each of the normal modes. This is a denumerable set which can be partitioned into subsets, based upon qualities of interest. If we consider the musical chromatic scale, then each note represents an equivalence class, where the elements of the class are the doublings/halves of the "root note." When you describe constructive interference, you can obtain maxima by specifying integer multiples of a phase difference of 2 Pi, which can then be further broken down into the set of even integer multiples and odd integer multiples. Likewise, for destructive interference you can obtain minima by specifying odd integer multiples of a phase difference of Pi. In quantum mechanics, we similarly use integer multiples to describe the class of energy levels. These are all examples of how sets are obtained in physics, and how from this follows that such sets can be partitioned. Proving theorems regarding equivalence relations and partitions of sets (or anything else about sets) helps prime you to be able to better understand both where these analyses came from, and how to elaborate upon them in the future for novel physical problems which might benefit from similar methods.

    #3:

    We want to know that our formula for constructive interference applies for both the cases where n=1, and for each n=k, where arbitrary k is an element of the integers such that k is greater than or equal to 2, as k approaches infinity. So we prove the case for arbitrary k+1 to show this. However, we are lucky that our recursion relationship is simple for these scenarios of interference. With more complicated recursion relationships, you may find that you need the strong principle of induction, which allows you to assume more information, and still acquire a rigorous proof. Understanding which type of induction you need, based on how complicated your recursion is, can help you know the best method to tackle your physical problem, and show that your result holds for all cases. The Well-Ordering Principle can be also used in "reverse" to perform a proof by minimum counter-example, which can sometimes be easier than regular induction.

    #4

    If you can understand how to compute one irrational number, you are better able to compute another irrational number. If you know how to prove why an irrational number is irrational (as opposed to rational), you are better able to understand why we would care to compute irrational numbers in the first place. Once you both know how to compute irrational numbers, and also care about their significance, you are better able to understand the fundamental constants of nature.

    The fundamental constants appear everywhere in your calculations in physical problems. Best to appreciate what they stand for, and how we got them put there.

    -=-=-=-=-=-=-=-=-=
    In General:

    In my university, the best "hard math" students graduating each year with mathematics degrees also tend to be physics double-majors.

    As a physicist, it is never a bad thing to be able to speak the language of science (math) as fluently as possible. Like a good poet with the English language, we can only reorganize Math to tell a new story once we know that Math very very very well.

    Math is our sandbox.
     
    Last edited: May 6, 2015
  13. May 6, 2015 #12
    Much of the math is covered in the specialized physics courses. Then as others have pointed out it is stylistic. I'm interested in computational chemistry/biology. Talking to professors at various universities I got into for graduate school, I encountered some who felt that taking pure math courses was essential. Others thought it was a waste of time, either because it was not useful or because they believed I should be capable of picking it up as needed. Oddly I didn't encounter anybody who seemed truly indifferent.

    I've always suspected that the gulf in rigor is really due to different goals. The mathematician wants the mathematics to be standalone from the applications and therefore must deal with numerous questions the physicist has obvious answers to since s/he has an intuitive understanding of the behavior of the system in question. The notion that these proofs are fantasy seems dubious in this context, although you are obviously being somewhat tongue in cheek.

    Again a lot of the language mathematicians build is designed to generalize the mathematics away from a specific application as far as I can tell, which would account for the fact that, in many cases, papers written by physicists do not incorporate much of this language. The question I suppose is whether or not this is a shortcoming.
     
  14. May 6, 2015 #13

    atyy

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    That's not provocative. I remember my physics lecturers saying things like "I don't know whether anything I'm telling you is actually correct, but it matches experiment" or "There's the ergodic theorem, but it is irrelevant to real time scales, so statistical mechanics is based on a prayer".

    Physicists are well known for studying non-existent mathematical objects, eg. the KPZ equation, which had been studied by physicists for maybe 20 years although it was only shown to be mathematically meaningful very recently by Hairer, an achievement for which he won the Fields medal.
     
  15. May 6, 2015 #14
    Did physicists obtain incorrect results when they used the KPZ equation prior to Hairer's rigorous study of its properties? Or were physicists simply unable to use it consistently?
     
  16. May 6, 2015 #15

    atyy

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    Honestly, I don't know - I'm a biologist who needs to know some stat mech, and can read non-rigourous physics stuff and if it makes handwavy sense I'm happy. I tried reading Hairer's paper since KPZ is stat mech, but I could not even understand a single symbol. So just going by the press release, it seems that in Hairer's sense, everything the physicists did is correct, but they missed that their "nonsense" equation makes perfect mathematical sense in ways they didn't even dream of.
     
  17. May 6, 2015 #16
    Which has been the case for the entire history of mathematics, with notable exceptions like integral calculus, invented with physics directly in mind. The mathematicians generalize, but there is nothing stopping you or I from taking their generalizations and pulling them around to suit our needs; that is, "calling upon them" to serve a specific purpose in the process of solving a physical problem, based upon a unique vision for how the math fits into the problem.

    The issue is not one of lack of applicability of what the mathematicians are building. That is not the reason you see papers written by physicists which do not incorporate these new, lofty mathematical generalizations. The issue is sheer volume of published works, the sheer volume of avenues of mathematical inquiry that have opened up, and the time constraints placed upon average human beings, who have many more responsibilities in their careers and lives besides sitting in front of a book everyday, all day.

    We are reaching a point where our academic systems and working environments have created a perception of some kind of boundary line between math and physics, one which people seem to interpret, incorrectly, as a boundary produced by a sort of difference of essence revealed between the subjects rather than a boundary imposed by the social and biological constraints that exist upon human beings. But there is no substantial difference in essence between how math and physics grow, other than the fact that math plays the naming-game with archetypes, and physics plays the naming-game with actors-who-have-an-identity. The logical processes, mental models, and visualizations follow the same basic cognitive structure. That is, they are, and always will be, highly related. They blend together as much as our imaginations allow them to.

    So the responsibility is on us the seek the connections that will always be there. The only shortcoming would be the mistake of assuming that the connections are not there to be found, thereby choosing not to go seeking them, and giving up on the chance to find them before any attempt was ever made.
     
    Last edited: May 6, 2015
  18. May 6, 2015 #17

    aleazk

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    And we would agree on that :smile: As I said: "This because of many reasons: (...) sometimes, to learn the actual math is to embark in a journey of pure math that will distract you from physics, your actual career, for which your boss pays you!"

    I said "fantasy" because, from the point of view of rigorous math, those arguments would not be considered valid (in the cases in which the same result can be proved rigorously; usually, the physicists' proofs tend to be incomplete and sloppy rather than "wrong", that's all). In some other cases, the word fantasy can be taken even more literally since the mathematical objects used lack rigorous definitions. I actually took the phrase from Folland's QFT book for mathematicians since I found it funny, as well as his analogy that, for someone mathematically minded, to study QFT is like the story in Goethe's Faust (Folland says "the devil offers you very effective tools and results, but you have to compromise your hardcore mathematical soul"; it's also a tongue in cheek analogy, directed at the specific audience of that book, mathematicians interested in the rudiments of QFT).

    And we would agree too :smile: As I said: "This because of many reasons: sometimes, we simply don't know how to make things rigorous (at least at the time in which the objects are introduced for the first time)". We could add the Dirac delta and so many other things.

    The provocation was to the idea that one needs to do the math 100% right in order to do physics, not an attack to physicists.
     
    Last edited: May 6, 2015
  19. May 6, 2015 #18

    atyy

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    Hmmm, mathematicians seem to be obsessed with the devil. :devil:

    "Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." - Michael Atiyah http://divisbyzero.com/2010/07/26/algebra-the-faustian-bargain/

    "In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." - Hermann Weyl http://www-history.mcs.st-andrews.ac.uk/Quotations/Weyl.htm

    Fortunately, physics is of God, since STRINGS is Solid Theoretical Research in Natural Geometric Structures. :angel: http://physics.princeton.edu/strings2014/slides/Maldacena.pdf
     
  20. May 6, 2015 #19

    aleazk

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    Well, the devil states his offers very clearly, that's why mathematicians like him, :woot:. With god, you never know, he works in mysterious ways, they say.
     
  21. May 6, 2015 #20
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