How the “Mathematics” of Science Really Works.
Sometimes mathematicians and physicists have no idea what their equations really mean at the time they formulate them. That is what is truly freaky about mathematics when applied to physical phenomena. It works even when we do not know it.
One of the clearer expositions of how the “mathematics” of science actually works in practice was penned by the economist Brad DeLong.
“Max Planck comes up with an equation that works. In order to do so he has to make a “purely formal assumption.” And it is only half a decade later that Einstein realizes that the little h that appears in Max Planck’s equation is not a formal assumption or an “artifact” but instead tells us what is perhaps the most important thing about the guts of the universe.
For half a decade the first equation of quantum theory was there. But nobody knew how to read it.
It is this “what if we took this equation seriously?” factor that is, to my mind at least, the spookiest thing about the unreasonable effectiveness of mathematics in physics. Take the h in Max Planck’s equation seriously, and you have the quantum principle–something that was not in Planck’s brain when he wrote the equation down. Take seriously the symmetry in Maxwell’s equations between the force generated when you move a magnet near a wire and the force and the force generated when you move a wire near a magnet, and you have Special Relativity–something that was not in Maxwell’s brain when he wrote down the equation. Take Newton’s gravitational force law’s equivalence between inertial and gravitational mass seriously and you have General Relativity–something never in Newton’s mind. And take the mathematical pathology at r = 2M in the Schwarzchild metric for the space-time metric around a point mass seriously, and you have black holes and event horizons.”
Another example is that of Kepler when he proposed the three laws of motion among heavenly bodies that began modern mathematical physics. He believed he was “proving” God created harmonic relations among heavenly bodies. It was Newton years later who realized what Kepler actually proved was how and why things moved in nature. Go figure.