Mathematics is useful. Eugene Wigner refers to this as “mysterious” and a “miracle” in a well-known article, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Just why is it that math is so useful? I dunno. Therefore, this is a deep, permanently unresolvable mystery etched into the stone of the laws of physics that can never be explained under any circumstances.

Actually, I believe Wigner has mistaken not knowing an answer with there not being one. Most of us probably do not immediately know why math is useful, but that does not imply there is no reason. (See the Wikipedia references for more arguments for and against Wigner.)

Let's solve the mystery. I suggest that mathematics is well-defined language. In a given mathematical language, all words have single, unambiguous meanings. For instance, in the language of the real numbers, a real number is well-defined and unambiguous. If you don’t know what a real number is, you can look it up, and you will get essentially the same answer from any source.

Contrast this definition with Wigner’s, that “mathematics is the science of skillful operations with concepts and rules invented just for this purpose.” What does that mean? His definition is so vague that it begs the question: Has he simply defined a mystery into mathematics rather than finding one? I believe so. His “mathematics” is so poorly defined that it’s not surprising he found a mystery. Perhaps if he had taken a lesson from mathematics and defined mathematics sufficiently well, he wouldn’t have found a mystery at all.

Well-defined language is useful because we have to communicate with each other. If Newton hadn’t defined “force,” and only presented his results, no one would have known what he was talking about. There are many more examples of the power of definition. Look up terms like “synchrotron radiation,” “diffusion coefficient,” and “Hilbert space.” In physics, these terms have unambiguous meanings. And good thing, because we can’t do without them. Conversely, to see the destructive power of not using well-defined language, look up “consciousness,” “qualia,” and even “information.” Enormous confusion inundates these concepts, because no one knows what the hell everybody else is talking about.

We have almost solved the mystery. But there is another question lurking beneath the surface: Mathematical theories can be used to make predictions that actually turn out to be correct. How can it be that we are able to derive new facts about the world using well-defined language? This seems like another magical, miraculous, unresolvable, mystery problem. But actually I think the answer is straightforward. When we construct mathematical models, we construct systems (in our heads, computers, notebooks, networks and so on), that are analogous to the systems we are studying. In the same way that a toy car and a car both have wheels, a model and its corresponding physical system both share properties.

Why are some mathematical models better analogies than others? For the same reason that some physical systems are more common than others. Suppose we tried to explain things in the universe by analogy to our sun. A lot of things in the universe can be explained this way—many other stars are analogous to our sun. But suppose we tried to explain things in the universe by analogy to American Idol. Well, there are some things that are like American Idol. But while there are hundreds of billions of stars in our galaxy alone that can be explained via analogy to our sun, there are only a few shows on television that can be explained via analogy to American Idol. So American Idol not a very useful model. Likewise, some mathematical models are analogous to far more physical systems than others. Those are, almost by definition, the ones which are most useful. This also suggests the possibility that there exist mathematical models that are not analogous to anything but themselves, which refutes the suggestion that all mathematics is physically real, except in the trivial sense that the models themselves are real.

That’s it. Math is useful because it is well-defined. And some mathematical models are actually good analogies to physical systems.