Introducing Students to the Unreasonable Effectiveness of Mathematics in Nature

[Ask a student to sing their favorite song.  Identify an musical interval: an octave or a fifth are the easiest to recognize. Using the picture below, talk about how our ears hear harmony when the length of two strings are in simple mathematical ratios.]

Now we can think of music as representing “the natural.” This is anything that occurs in nature–the motion of the planets, the reflection of light, how chemical elements bond, all of that.

But where can the number “one” be found in nature? Where is the number “seven?” “ten?” or “463?” Numbers are not there. They are artificial, man-made. Inventions of language.

You don’t at all need math to do music. You can make perfectly awesome music without knowing the first thing about simple arithmetic, let alone functions and exponents! And you don’t need music to do math. The two are completely separate.

And yet, the structure of musical harmony by some strange cosmic coincidence has the same simple structure as numbers.

Why isn’t nature just random? Why isn’t an octave a 1.8477539… ratio, instead of 2:1? [pause] Anyone have a guess? If these are really two completely separate activities, why does the same structure appear in both? There is no agreed upon answer to this question.

It’s like they are two parallel universes, which miraculously happen to have striking similarities.

I want to touch back on this theme throughout the semester. Music is the most striking, easiest-to-relate-to example of “the unreasonable effectiveness of mathematics” in nature, but there are plenty of others. The 16th century found it in the the movement of the planets; the 17th century, in the laws of gravity, which required the invention of calculus and the language of functions in order to adequately describe. We see it again in the 19th century, with the discovery of laws describing light waves, X-rays, and radio.

Math is also found in aspects of life other than science. Economics, statistics, or any activity that requires measurement and analysis. All these are interesting in their own right, but personally, science is the most interesting to me.