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The clarified version

As far as I understand, Wigner considers a "miracle" the fact that it is even possible to find a mathematical equation that describes a natural phenomenon.

It is not exactly what I was wondering about though. Lets say such an equation has been found. What exactly does it describe?

  1. Do we treat the phenomenon itself as just a black box that happens to "output" numbers that fit into the equation?
    This idea is supported by the fact not every intermediate step in solving the system of equations has an obvious physical interpretation.

  2. The system of equations mirror the internal "structure/working" of the phenomenon?
    On the other hand, this is supported by the following example. Kirchhoff's rule "the algebraic sum of currents in a network of conductors meeting at a point is zero" clearly follows from the fact no additional charges enter or leave the circuit.

  3. Is it a mix of the both options above?

  4. Maybe throughout the history it has been discovered empirically that coming up with equations and then solving them works for physics, but no one really knows why and how it works?
    An answer along these lines is perfectly fine with me too. I just have not seen the way/method math is used in physics discussed anywhere -- and so wonder if I'm missing something obvious to everyone else.


The original question

My question is a general one. But to explain what it is asking let's first a look at "solving" of an electrical circuit using Kirchhoff's laws as an example.

Solving an electrical circuit

So to find out the directions and amounts of the currents we have written down the equations based on the Kirchhoff's laws.
And up to this point we were staying in the physics' "land" -- because Kirchhoff's laws intuitive/physical interpretation is not hard to see.

Once we had the system of equations we used the usual/general math techniques to solve the equations.
I guess, the math techniques used to solve equations were discovered much earlier than the concept of the electric circuit (and the task of solving it) was invented/discovered. Also it does not seem possible to "interpret/map" each step taken to solve the equations in terms of the physical phenomena actually happening in the circuit. But still solving the equations let us find the amounts and directions of currents.
In other words we went outside of the physics' "land" and into the mathematics' "land" but in the end still came up with the physically correct answers.

To sum it up, my question is: Mathematical techniques used to describe physical phenomena are not necessarily specifically invented for physics and do not necessarily have any meaningful physical interpretation. How come these techniques are able to produce correct (can be verified by experiment) results?

And on the same note, who came up with the idea of using math for describing things in physics, how did this person come up with the idea?

Hopefully, it is possible to understand what I'm asking about. I've tried as hard as possible to make the question clear and concise. But, honestly, I find it challenging to express this question clearly. Anyway, I will be glad to clarify it further as much as needed.

Thank you in advance!

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closed as too broad by user36790, Cosmas Zachos, David Z Jul 11 '16 at 16:06

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Math originates in counting things in nature. Physics boils down to observing how nature is now and predicting how nature will be in the future. They both originate in observing nature, but math focused more on manipulating the counts of things (sometimes in such an advanced way that counting isn't even involved anymore) and physics more on the prediction side. It is natural for physics to use math. It observes the count of something at one point and predicts the count of it at a different point. Gotta use math to communicate that $\endgroup$ – Jim Jul 11 '16 at 12:57
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    $\begingroup$ Eugene Wigner wrote an essay, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". It might be of interest. $\endgroup$ – Paul T. Jul 11 '16 at 13:11
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    $\begingroup$ Think for a moment about what happens to proposed descriptions of reality whose math doesn't work for describing the system they pertain to. Kirchhoff's laws didn't end up in the texts because the man's name is fun to say. $\endgroup$ – dmckee Jul 11 '16 at 13:15
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    $\begingroup$ I actually think this is off topic, as it's really an epistemiology question (on understanding how and why science works). $\endgroup$ – David Z Jul 11 '16 at 13:43
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    $\begingroup$ You should try to think about why math works for describing and solving any sort of problems. In short, it is a problem solving language. $\endgroup$ – skullpetrol Jul 11 '16 at 15:28
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One very popular view (as espoused by Max Tegmark) is that (quoting count_to_10) :

math works because the universe is based on math

http://www.scientificamerican.com/article/is-the-universe-made-of-math-excerpt/
https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis

Such a view was common from the time of Pythagoras, through to Kepler and Newton, with attempts to find mystical mathematical patterns in nature, and the description of God as a Geometer. Galileo wrote in 1623 : "The book of nature is written in the language of mathematics."


An alternative view which is more "down to Earth" is that mathematics developed from the attempt to describe the world using numbers - not simply counting but also measuring (distance, angle, area, volume, weight, etc). This is obvious in the case of Geometry (literally, 'land measurement'). Trigonometry also developed for use in surveying, navigation and astronomy (in the latter case for predicting floods or auspicious astrological events). Probability was developed to answer questions about gambling. Calculus developed from trying to account for the shape of celestial orbits. More recently, the mathematics of chaos arose from weather prediction, and fractal geometry from the practical question of measuring the length of a coastline.

Throughout most of its history mathematics developed as a tool of science and technology, from the time of Archimedes to the era of Euler, Lagrange, Gauss and Legendre. So it should not be surprising that it "works" in physics. It was not until about 1850 that Pure Mathematics became recognised as a separate subject.

As Paul T points out, the issue was addressed by Eugene Wigner in a famous essay, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" ( http://www.maths.ed.ac.uk/~aar/papers/wigner.pdf.) However, I think this description of "unreasonable effectiveness" clashes with the reality of mathematical physics.

Take a look inside Landau & Lifschitz or any other graduate text in mathematical physics. Seeing the horrendous mathematics required to solve many differential equations (Fourier Transforms, Bessel Functions, etc), most of which have no analytical solution anyway, you might then question whether the description of "unreasonable effectiveness" is really appropriate. Even more so when you realise that these complex solutions are still only an approximation to reality since the differential equations have themselves arisen only after making several simplifying assumptions.

In Quantum Mechanics only the most simple problems can be solved analytically. Some are resolvable only into transcendental equations (eg finite potential barrier). Others are tractable only as "perturbations" of known solutions, or in QED require the summing of infinite series of terms. In some fields special tricks like Renormalization and Regularization are needed to deal with infinities.

That linear algebra applies quite well in numerous macroscopic situations of interest is due to the facts that (1) many phenomena are approximately linear over the narrow region of interest, and (2) they are only weakly coupled to each other. Then empirical laws like Hooke's Law and Ohm's Law give sufficiently accurate results without making the calculations too difficult.

The Law of Large Numbers, which is the basis of statistical mechanics, is also a great help in getting round the difficulties of solving non-linear equations at the molecular level.

Most notably in the case of turbulence, although we can write the Navier-Stokes Equation - which again rests on simplifying assumptions - nobody has yet worked out how to solve it. But even with a system as simple as the Double Pendulum, we can write its equation of motion but we cannot always predict its behaviour.

As dmckee says :

Think for a moment about what happens to proposed descriptions of reality whose math doesn't work for describing the system they pertain to. Kirchhoff's laws didn't end up in the texts because the man's name is fun to say.

When mathematics doesn't provide a solution to a physics problem, it is left out of the textbooks. Or we simplify until the problem is solvable. We concentrate on the problems we can solve, and avoid those we can't. That leaves the impression that mathematics can solve every physics problem.


So in summary my answer is that :

  1. mathematics works in physics because it was developed (in part) for the purpose of describing the world, and
  2. it doesn't actually work anywhere near as well as some people make out.

Response to The clarified Version

  1. We only treat the phenomenon as a black box when we are totally clueless about what is going on. Then we develop empirical equations - we select parameters and vary them to match experimental results. This rarely happens in physics, more so in engineering.

  2. Usually we aim to make the equations model the inter-relations of relevant variables : ie mirror the internal structure of the phenomenon. However, in solving those equations we are not restricted to mimicking the phenomenon - unless we're running a simulation. We can use any mathematical short-cuts (eg integration, analogy, symmetry) to predict the end result.

  3. Yes, we sometimes use a mixture of these two approaches : eg the Semi-Empirical Mass Formula in nuclear physics, and the various Equations of State for Real Gases. Dimensional Analysis might also come under this category : we choose which variables are relevant, and look for consistent relationships between them.

  4. I don't agree with Wigner that there is such a big mystery about the process and its success, that it is a "miracle" and that "nobody knows how it works." I am, as Geremia says, a disciple of Aristotle as Wigner is of Plato. Is it a miracle that we just happen to live on the only habitable planet within sight? Or is that a tautology, since we cannot do otherwise? Likewise I think it is no more a miracle that we've had amazing success applying mathematics to physics than that we've had amazing success applying our minds to developing aerospace, computer and communications technologies.

The success of applying mathematics has spurred us to using it almost exclusively, perhaps at the expense of other approaches. As I said above, we tend to focus on problems to which maths can be applied, and neglect those to which it can't. And we're not satisfied that we understand something until we can write down and solve the governing equation(s).

When existing mathematics fails to apply to a problem, we try or invent new tools, concepts or branches of mathematics to deal with it - such as topology, non-Euclidean geometries, catastrophe theory, fractal geometry, chaos, self-organizing systems and emergence. We forget the many failures which PhD students have had in trying to apply inappropriate mathematics to a stubborn problem.

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    $\begingroup$ To add, math is essentially a language of logic that applies to nature only when nature obeys the logic. The fundamental logic that nature obeys in the situation described is that of kirchhoff's laws. Once this observation is made you may adapt a mathematical approach to this logic in order to obtain information hidden to the observer of similar situations that follow the same laws. If the expression $1 = 1$ is logical in nature, then we know that $x + 1 = 1$ $x$ must be $0$. $\endgroup$ – obliv Jul 11 '16 at 18:09
  • $\begingroup$ sammy, you've edited this answer far too often. Making trivial or excessive edits to a post is inappropriate; each time you edit, you should go through and fix everything that needs to be fixed. As a general rule of thumb, if you find yourself editing the same post more than 3 or 4 times, you're probably being too loose with your edits. Please don't keep editing this unless you have something that really needs to be fixed. $\endgroup$ – David Z Jul 11 '16 at 18:37
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    $\begingroup$ @DavidZ : Thank you for your advice, David, but why should it matter? Is it causing a problem? $\endgroup$ – sammy gerbil Jul 12 '16 at 2:41
  • $\begingroup$ Yes, it does cause a problem. It displaces other posts from the front page of the site and takes up contributors' time checking the content of the latest edits. $\endgroup$ – David Z Jul 12 '16 at 10:06
  • $\begingroup$ Thank you, @sammygerbil! I read your answer and Eugene Wigner's essay. And also tried to follow the links posted in the comments. It seems in the process I realized how my question can be clarified. Likely it will sound somewhat different from the way it was expressed originally. Please take a look when you have a chance. $\endgroup$ – Nik Jul 12 '16 at 17:38

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