One very popular view (as espoused by Max Tegmark) is that (quoting count_to_10) :
math works because the universe is based on math
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 :
- mathematics works in physics because it was developed (in part) for the purpose of describing the world, and
- it doesn't actually work anywhere near as well as some people make out.
Response to The clarified Version
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.
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.
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.
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.