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Frustrated by how physics is generally explained/taught

  1. May 29, 2016 #1
    I'm neither a physicist, nor working in a physics-related domain but I've tried for years to get interested in physics because of a fascination for questions like "what is the world" and "how does it works". I suppose this is a feeling shared by other people here. The thing is I have often became stuck in my attempt to understand, even sometimes about things that are said to be "simple". Maybe it takes a lifetime to get those things. Certainly I'm only clever enough to reach a given level of understanding.

    However over the years I became aware that there is a recurrent obstacle in books, classes, explanations, articles that have always kept me either frustrated or confused, most of the time both. I would call it an excessive concern for computational aspect over conceptual ones.

    As an example, the idea of a physical system I was taught younger was to think of a "real" piece of matter on which we could make experiments and then attach numbers. In this approach every property is a sort of "number". The piece of matter is for the intuitive physical aspect: something you can touch and imagine. Numbers are ... for calculation only. Actually those numbers are most of the times numbers-that-depends-on-other-numbers but this is only a technical detail you know. Learn the formula, learn multi-variable calculus and that's it. At the contrary, after my studies it was a real eye-opener to understand (as a self-taught) a physical system could be defined itself as a configuration/phase space and related properties as fields (or even better bundle sections) on that space. Suddenly a lot of things made sense because the physical semantics of the system could be described inside a mathematical structure.

    First I must say I'm definitely a math-inclined guy. Second I deeply believe that maths can (and should) encode the semantics of theories. But as illustrated above I've always noticed maths are more often considered as a mere pragmatical tool that help us creating supposedly abstract nonrealistic "models" used for computation. On the contrary I see maths as the best language to speak about physical objects and concepts themselves. When for example I consider spacetime as a manifold, I don't think I'm creating an ideal object in my mind or in my notebook that "looks like" reality. I consider the idea that spacetime can be thought as a particular manifold itself: that somehow it makes sense to speak about the "real" spacetime using the vocabulary and the concepts of differential geometry (of course that may be partially inaccurate: I know the hole argument against manifold substantialism in this precise example, but I do not think it invalidates the general approach).

    In my view, looking for a structure for a theory in physics should imply that :
    • central concepts should be present even if with a mathematical form
    • obviously distinct concepts should be distinct kinds of mathematical objects
    • non physical information (like any computational devices) should never be present as a necessary core component of the structure
    But here is the kind of things we generally encounter:
    • arbitrary coordinates everywhere : space coordinates that refers to arbitrary directions and arbitrary center, the so-called canonical coordinates on the phase space that are absolutely not canonical, tensors-as-arrays, vectors-as-tuples, matrices, Christoffel symbols, Lie groups described as matrices groups, etc
    • other relative objects : frames of reference that inject privileged viewpoints called "observers", all quantities that are dependent on the frames of reference (speed, kinetic energy, work, …) position vector/angular momentum/torque/moment of inertia which all depend on an entirely non-physical geometrical point, wavefunctions that depends on both coordinate systems and frames, etc
    • wrongly-categorised objects : vectors that should be bivectors or points, "values" that are in fact fields, fields, that should be measures, gradients that should be forms, states as Hilbert space vectors, groups instead of homogeneous spaces, etc
    • completely inconsistent (but supposedly more intuitive) objects: differentials as infinitesimal quantities, eigenvectors for unbounded operators, etc
    What deeply irritates me (at best) or drives me to dispair in this kind of approach is that:
    • it results in a mix between physical information and representational artifacts. In the end what is physical and what is not becomes awefully blurry
    • many distinctions are flattened to fit primitive mathematical entities so that you end up with a gigantic list of long equations between a bunch of numerical values. ##\mathbb{R}^n## is suddenly everywhere. I even came up with a name for this process: the ##\mathbb{R}^n##-ization of physics.
    I don't deny that there exists approaches that priviledge a more conceptual mindset but:
    • They are generally few except if you consider higher level domains of physics. If you're just trying to have a very clear and precise understanding of more basic stuff you're stuck with the usual material (show me a sound introduction to classical thermodynamics)
    • They rarely covers all aspects (AFAIK analytical mechanics cannot handle non conservative forces)
    • They generally still contain a certain degree of arbitrariness (again analytical mechanics can be coordinate-free but is still frame-dependent)
    • It's not always shown how they are related to more widely used approaches so you have to invest energy and time to create the links (because often you have no choice)
    My question is basically: are there other people here with the same mindset and how do they cope with it without giving up entirely on physics ? I tried to adapt my manner of thinking. I did tried. But it feels so much unnatural.
    Last edited: May 29, 2016
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  3. May 29, 2016 #2
    It seems to me that you have the same central complaint that led Michael Spivak to start a textbook series "https://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322", currently only Mechanics has been done. But if you haven't read it, it sounds like you will be quite happy with it.
    Last edited by a moderator: May 7, 2017
  4. May 31, 2016 #3
    Actually I had already given a quick look over it. That may more be rigorous than most of the other books on the topic but that certainly does not avoid a lot of the pitfalls I've mentioned before: coordinates, coordinates, coordinates everywhere. And structures nowhere
    Last edited by a moderator: May 7, 2017
  5. May 31, 2016 #4
    It sounds like you want to do for physics what the Bourbaki group did for math. I honestly don't know if this has been done before, but the person who probably would know would be Spivak.
    Last edited by a moderator: Jun 1, 2016
  6. Jun 1, 2016 #5
    I think you can already find on certain topics more math-oriented conceptual approaches. The thing is again that those "Bourbaki-like" approaches seems to be more frequent when you're considering more complex theories. I don't deny that apparently plenty of people are more comfortable with the kind of approach I reject (because the explanations rely more on intuition and a so-called "common sense" and the computational aspects are only requiring simple mathematical knowledge) I'm just fed up it is so pervasive for a huge part of physics. It just looks like everyone should think the same manner. Adapt or die.

    Note that I don't necessarily want something as terse and austere as Bourbaki. I just consider that on the continuous line between austere-and-too-abstract VS informal-and-overconcerned-by-computation there is a huge bias in favor of the latter.
  7. Jun 16, 2016 #6
    Apparently nobody seems to be in the same situation as me :frown: My post was moved in the educational forum but I didn't intend to focus on the teaching aspect but rather on a more epistemological "what does it mean to understand physics" question.

    Nevertheless, as I suppose there are many teachers around, if I can ask an additional related question: Do you try to mix several approaches when presenting physics to your audience? Have you already met students which were obviously not on the same page as you (and with who communication and teaching was problematic) because they were using different perspective, mindset, metaphors, representations than you to understand and explain the same physical notions?
  8. Jun 21, 2016 #7

    Andy Resnick

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    Yes- that's good teaching practice.

    Yes, because introductory physics is required for many non-physics majors.
  9. Jun 22, 2016 #8
    Sure but my question was more about students of physics themselves.
  10. Jun 22, 2016 #9

    Andy Resnick

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    What is a 'student of physics'?
  11. Jun 22, 2016 #10
    I meant someone majoring in physics.
  12. Jun 22, 2016 #11


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    If I understand your frustration correctly, your main objection is the introduction of specific frames of reference and coordinates. But aren't these constructs necessary in order to do any useful calculations? The vast majority of students learning physics will not go on to become theoretical physicists who plumb the depths of our description of nature. Virtually all of them, if they use physics at all, will use it to do specific calculations in applied fields like engineering. So the techniques necessary to do calculations in real setting are an important part of learning physics. For those few who are interested in the foundational aspects of theoretical physics, the material is out there for them to study. Misner, Thorne, and Wheeler do an excellent job of discussing GR in a coordinate-free manner. I personally like the approach taken by Doran and Lasenby in "Geometric Algebra for Physicists".
  13. Jun 22, 2016 #12
    No my main objection is what I consider as a real obsession for them and for calculation at the detriment of conceptual understanding. I don't pretend that calculation tools are useless or should not be taught, I say they could come second after a coordinate-free formal description (you can include in "coordinates" anything that can be considered as non-physical).

    Maybe I don't belong to the (supposed) "vast majority of students learning physics". I don't plan to become a theoretical physicist but still this approach has slowed me down or stuck me in confusion. What's the use of learning how to use a computation tool when you don't understand what the computation semantically represents?

    Again I don't deny they exists but again look: you're already talking about GR. What about this kind of approach for classical thermodynamics, newtownian mechanics (including rigid body kinematics, fluid mechanics, continuum mechanics), optics, classical electromagnetic theory ... ? Additionally your remark implies that "those few who are interested in the foundational aspects of theoretical physics" are "those whose want go deeper", presupposing they have already understood the usual coordinate-based approach. What if some of them got stuck because of it?
  14. Jun 22, 2016 #13


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    After all this, I STILL don't have an understanding what the problem here is. Can someone give me a simple, coordinate-free explanation?

    @burakumin : Have you applied the approach that you want and tried to teach someone else using your preferred methodology? Has it worked consistently over the various topics that you described? Have you also considered if you are the "select few" or the only one who had trouble with the "traditional" methods?

  15. Jun 22, 2016 #14
    They'll get stuck in a coordinate-free approach as well.
    It is much harder to understand when you can't picture something.

    As far as I can tell the only way to describe classical (i.e. Newtonian) mechanics in a coordinate-free way is to look at manifolds and other difficult stuff.
    And without previous knowledge in the coordinate approach I wouldn't even understand what we were trying to do.
    Coordinate freedom comes at the cost that you need more knowledge of the deep mathematical nature of the theory.
    Look e.g. at Arnolds book on classical mechanics, there's a reason it is part of the "graduate texts in mathematics" series
  16. Jun 22, 2016 #15
    Sorry I must be too stupid to make myself clear.

    Clarifying this was exactly the purpose of my questions:

    I don't. I must be a counter-example.

    I don't understand this remark. Coordinates are precisely what kills all geometric insights with me, they are precisely what prevent me to picture anything.
    Last edited: Jun 22, 2016
  17. Jun 22, 2016 #16
    Can you give an example? Because I don't see how this works.
    I'm interested though.
  18. Jun 22, 2016 #17

    George Jones

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    I personally find that coordinate-free notation can sometimes be useful for calculation, and that coordinate-free notation often is useful conceptually.

    Even math grad students, however, have trouble doing calculations in a coordinate-free manner. From Zee's book "Einstein Gravity in a Nutshell":
    This was a physics student enrolled in a pure math grad course. :biggrin:
    Last edited: Jun 22, 2016
  19. Jun 22, 2016 #18

    Andy Resnick

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    Oh, ok. Then my first answer remains the same (use several pedagogical approaches) but my second answer is now 'no, not really', because by the time I get my hands on physics majors, which is second semester junior year and beyond, they are largely assimilated. To be sure, different students learn differently and are comfortable with different approaches, but as a whole they are familiar with the same subset of metaphors, representations, etc.
  20. Jun 22, 2016 #19


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    As a follow-up to what Andy said, I'm going to be tacky and refer to something I wrote elsewhere quite a while ago, and which has some relevancy here.

    I think the OP needs to be aware that we ALL have our own ways of learning things. Part of being a student, and part of going through an academic program, is for one to figure out for oneself how one learns, or what method is the most effective for one to acquire a knowledge or an understanding efficiently. We all had to go through this on our own terms. And this is why I claim that learning how to learn is the most important thing that I learned about being a physicist.

  21. Jun 25, 2016 #20


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    Hey burakumin.

    What you are trying to ask is whether a different language exists (with certain properties/constraints) to describe physics.

    I guess what you should ask yourself is based on your understanding of physics (which is something you will have to elaborate on), what are the differences between how you would represent information to define physics and how the existing "body" of knowledge represents information.

    You mention things like co-ordinate systems and mechanics and your "dis-satisfaction" of how things are organized (which is the purpose of language) but perhaps you could outline what sort of language you think would be appropriate to use in place of what already exists and why you think it would be more appropriate.

    I'm sure if you did that you'd get a lot more specific/directed "responses" for your question - particularly by those with significant education/training in the "field".
  22. Jul 19, 2016 #21
    Sorry to have been so long to answer.

    So in the end it seems we totally agree. It's just that precisely the traditional manner of explaining physics seldom "makes something click in my head". And the fact that many teachers apparently follow Resnik's path does not help me :
    Now about these questions:
    When for example I'm asked to think about the point at the top of the pyramid of Cheops I can perfectly imagine it without any notion of coordinate. Now if I start to refer to this location only by ##(x, y, z)## I'm loosing precious information on the actual structure I'm actually considering. And ##\mathbb{R}^3## can be many many things: a commutative group, a vector space, an homogeneous space, an affine space, a (commutative or Lie) algebra, a topological space, a metric space, a hilbert space, a (topological, differential, riemmanian or pseudo-riemmanian) manifold, an ordered set, a measured space, a probability space, etc etc. Nobody can tell me that taking the euclidean norm of ##(x, y, z) ## is mathematically unsound. But does that norm tell me anything about the pyramid? I can compare ##(x, y, z) ## to any other triple using the lexicographic order. Does this comparison have the least physical importance ? If I'm talking about a circle everybody knows what I'm talking about and everybody must understand that in this context the notion of distance must make sense otherwise the concept of circle is meaningless. Even if the argument may seem contrieved technically speaking one cannot be sure that the numerical equation ##x^2 + y^2 = 1 ## represents a circle in absense of any additional information. I can give you a coordinate system where this describes a square. Mathematics can encompass this kind of differences because in maths not everything has to be a number.

    Example of the kind of approaches I like:
    By opposition to these ones:

    Now like said before there are many other fields where it's very difficult to find references with the approach I expect.
    Last edited: Jul 19, 2016
  23. Jul 19, 2016 #22


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    If you are talking about information not being a number then you have to describe it's structure and algebra.

    The most basic structures in mathematics are numbers and sets and if you have another structure you should mention it.

    Co-ordinate systems are just bases in mathematics meaning that they organize information in a very particular constrained way.

    Numbers order information in a particular way consistent with arithmetic and vector/geometric algebra. If you don't want numbers then you need to specify the states, how they are organized, the algebras used on them and why you would prefer an alternative structure as opposed to what exists within science/engineering.

    If you can do that then what you're trying to say would make a lot more sense.
  24. Jul 20, 2016 #23
    Sorry Chiro but I don't understand the purpose of your answer. I already know what are structures and coordinates.

    Haven't I already given examples of that? Have you seen the references I added?
  25. Jul 20, 2016 #24

    Andy Resnick

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    Ah! Now I think I understand what you mean. I'm not a mathematician, so my language may be wrong, but I think you are asking why functional analysis/algebraic topology is not used by most pedagogical materials. Yes?
  26. Jul 20, 2016 #25
    General topology, differential geometry, general algebraic structures (monoids, groups, rings, fields, algebras, modules, vector spaces, ...), order theory, measure theory, set theory, graph theory, etc, etc. Plenty of already existing concepts could be used in physics. Another concrete example for @chiro is this thread.

    To avoid any misunderstanding, I'm pretty aware that some may answer "because we don't want to wait for students to learn too much maths before starting to teach physics". But again I don't pretend that this kind of approach should always be used and for everybody. I would just like that there also exists this kind of alternative approach for many subfields of physics. And often it doesn't.
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