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Physical reason vs. Mathematical reason

  1. Sep 4, 2014 #1


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    Sometimes people ask a question and when I answer them in terms of equations, they say they want the physical reason, the real reason..

    1- Do you think there is a distinction between physical and mathematical reasons?
    2- If your answer to the last question was yes, explain the distinction! if it was no, explain how would you clarify the issue!

    I think there is no distinction and mathematics is simply an extension of logic to places where human language and imagination can't reach.

    What is your idea?
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  3. Sep 4, 2014 #2


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    Yes of course.

    Just because you tell me "here's the equations relevant to the question and here's how to manipulate the equations to get the desired result(s)" doesn't mean you've given me any insight into what's going on physically. You've just shown me how a physical phenomenon drops out of manipulations of some equations. But I want to know the deeper physical reasoning for the phenomenon so that I can build an intuition for the subject. It's easy to just give equations and show the end result. It's much harder, in my opinion, to provide the correct insight. It's basically the difference between a book like Thornton and Marion and the Feynman lectures.

    Actually a relevant example just took place in my class yesterday. In Peskin and Schroeder the authors provide the mathematical distinction between the retarded Green's function and the Feynman propagator as being evaluations of the Green's function along different contours. That's straightforward enough and is just a bunch of simple calculations but what's the physical distinction between these Green's functions in the context of QFT and when do we even care about the retarded Green's function? Thankfully my TA was kind enough to provide this (very helpful) insight.
  4. Sep 4, 2014 #3


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    I know what you mean. But I somehow feel two sided about your answer. One side is that you're right and there is this distinction. The other side is you're talking about the physical interpretation of equations given as answer rather than a completely physical math-free answer to the question.
    Honestly, I can't exactly say which one is my feeling about your answer. Looks like I didn't have a clear picture of the question when I was asking it! I think that's because I don't quite understand those people who ask for "real reasons"!
    I agree with you that of course equations themselves are only a set of symbols and unless you give them meaning and interpretation, they can tell you nothing. That's something no one can question.

    Let me give you an example of one of my experiences:

    Layman: Why time stops at the speed of light?

    Me: Its in fact only a limiting behaviour BTW I can explain it using the Lorentz transformation for time between two events happening at the same place [itex] \Delta t'=\frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}} [/itex]. As you can see, as the speed approaches the speed of light, [itex] \Delta t' [/itex] approaches infinity and that means time is not passing.

    Layman: I know about that formula. I wanna know its physical reason!

    How would you answer such a question?

    Maybe I should have explained the things that lead to Lorentz transformations but that's not going to give a direct "physical reason"! Just think what would be the reaction if you tell them this frame can consider itself as being at rest and now the other one's time is getting slow!!!
    Last edited: Sep 4, 2014
  5. Sep 4, 2014 #4


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    Totally disagree. Because math is "just a language", it is at worst as valid as any other for offering descriptions and because it is quantitative, it is typically more precise.

    For example, what's "really" happening with the inverse square law? Geometry! That's it, there is nothing more.
  6. Sep 4, 2014 #5
    I would show them the postulates and then derive the Lorentz transformations from them. I think they he/she will be satisfied if you showed them how fixing the speed of light leads to the transformation. I could be wrong though.
  7. Sep 4, 2014 #6


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    I disagree with that. The inverse square law is geometry, plus the fact that we are talking about some quantity which is conserved in some way. The geometry is math. The conservation law is physics.
  8. Sep 4, 2014 #7
    The issue here is that the questioner believes there is a distinction. If you want to continue the discussion with them you have to find out what they think the distinction is. I would suppose they could give you examples of what they think are "real, physical" reasons and what are "mathematical reasons," and from those examples you could figure out what they mean.
  9. Sep 5, 2014 #8


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    Geometry is the correct answer. It is kind of a vague answer that doesn't really convey much information, though.

    You're taking a certain amount of stuff that propagates spherically, which means its being spread out over a larger and larger surface area.

    With the formula for surface area of a sphere being 4*pi*r^2

    Geometry may be a correct answer, but I think the answer needs a few more details to really help the person asking the question.

    And that's almost always the case when explaining something via mathematical formula. Almost always, since sometimes the formula simply models something with no implied reasons for its truth.

    A mathematical model of the Earth's gravity, for example. The Earth's oblateness and other perturbations explain why the mathematical model is needed. There is no explanation for the value of the constants except those are the values needed for the mathematical model to accurately reflect what we observe.
  10. Sep 6, 2014 #9
    Philosophy v. Math

    I would view material reasoning to be Philosophical in nature and math to be just the agreed upon symbolic language to describe observed natural phenomena from a sentient point of view.

    The Pythagorean concept of all things are number and ‘knowledge of knowledge’ or the Greek phrase ‘nóesis noéseos’ and René Descartes Latin saying, “Cogito, ergo sum.” or “I think, therefore I am.” Can sum up the whole debate of natural reason or material observation (philosphy) v. math.
  11. Sep 6, 2014 #10
    "Before I studied the art, a punch to me was just like a punch, a kick just like a kick. After I learned the art, a punch was no longer a punch, a kick no longer a kick. Now that I've understood the art, a punch is just like a punch, a kick just like a kick. The height of cultivation is really nothing special. It is merely simplicity; the ability to express the utmost with the minimum." - Bruce Lee
  12. Sep 6, 2014 #11
    Totally agree!
  13. Sep 6, 2014 #12
    Seems like good advice to me.

    We all, including laymen, come equipped with some 'naive physics', e.g. a rough and ready 'feel' for Newton's third law that is totally in line with the obvious equation (R = -F). But we don't have any feel for Einstein's equation! If they aren't happy with accepting the equation as a reason then I'd just shrug my shoulders and say, "I am".

    Anyway, what's the reason for Newton's third law? ... ask them that. There isn't any 'reason' - it just 'is'. It's really the same for your Einstein time equation.
  14. Sep 6, 2014 #13


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    Consider a bouncing ball. Each bounce is some proportion of the previous bounce (determined by coefficient of restitution).

    The mathematical reason the ball stops bouncing in a given time is related to the sum of an infinite geometric series.

    But, a ball can't bounce an infinite numbers of times. Physically, eventually, the ball has too little energy even to leave the ground at all.

    That, to me, is a clear distinction between a mathematical model and the physical reality.

    (And, yes, you could develop a more sophisticated mathematical model, but that's not the point.)
  15. Sep 6, 2014 #14


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    Considering bounces that can't be seen by naked eyes( because of very tiny energies and so heights) and quantum fluctuations, the ball actually does bounce an infinite number of times. For me, this suggests even a more relevance of mathematics to physics that tells us something we didn't put in!
  16. Sep 6, 2014 #15


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    Okay, then, I have a physical challenge for you! It's called the gambler's hell. You can find it here:


    The challenge is to get rid of all your coins. It can be done mathematically. But, you'll struggle to find a physical process that could even produce an infinite number of coins in the first place.

    If there were no distinction between maths and physics then the gambler's hell problem would represent a realisable physical process.

    But, it is purely a mathematical model that has no realisable physical equivalent.
  17. Sep 6, 2014 #16
    Both maths and physics are objective, though, physics relies on experiments while mathematics dwells somewhere in the abstract.
  18. Sep 6, 2014 #17


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    Hey, not so fast bro!
    I didn't say there is no distinction between math and physics. How can I say that???!!!
    I just said math and physics are more relevant to each other than you think.
    But even now, you can find weird things happening in physics that were someday only pure math.
    For example, mathematically, in a 2D plane, there are infinite pairs of basis vectors and you can choose each of them as a basis. And a vector in a basis, can be decomposed to a linear combination of two vectors belonging to another basis.
    Now consider this, you have a linearly polarized light which is palarized in the x direction and you send it to a polarizer which makes an angle of 45 degrees with the polarization direction of the light. Then you send the outgoing light, to another polarizer making an angle of 45 with the polarization direction of the first polarizer and this time you get a light polarized in the y direction.
    Mathematically its simple. You just had a vector in the x direction. It means it has two contributions in the y direction that are cancelling each other. So you just remove one of them so that the other can show itself. Then you remove the x component and only the y component remains.
    But what happens physically? Can you say something apart from the last paragraph?
    The electric field was oscillating in the x direction, where the hell that y component came from?
    In fact if you look careful enough, there will be a physical explanation for it. You just need to analyze very carefully the behaviour of light, those polarizers and their interaction. But my point is that, before doing such complicated calculations, vector analysis is giving you the answer, without putting in it something that makes it give the right answer. People who invented vector analysis, didn't know, or weren't considering this experiment while working on vector analysis.
    To me, that means there is a deep connection between mathematics and physics that don't understand yet.
  19. Sep 6, 2014 #18
    Recall that the entire field of topology was ignored for many decades of early 20th century because of a perceived lack of physical application.
  20. Sep 6, 2014 #19


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    Neither of these statements agree with your original post, which said:

    "I think there is no distinction and mathematics is simply an extension of logic to places where human language and imagination can't reach."

    The mathematical reasons for something (the Polar Decomposition Theorem, for example) do not depend on and are not directly related to the physical reasons relating to the deformation of solid bodies.

    The mathematical reasons (e.g. for Polar Decomposition of matrices) are independent of the physical application.
  21. Sep 6, 2014 #20
    Try telling your layman friends that! Take a ball, let it bounce - it obviously doesn't bounce an infinite number of times! That's the reality. You are using a model of quantum fluctuations to add unnecessary complexity to a simple Newtonian bouncing ball model.
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