Applying

“... nature seems very conversant with the rules of pure mathematics, as our mathematicians have formulated them in their studies, out of their own inner consciousness and without drawing to any appreciable extent on their experience of the outer world.”

- James Jeans, The Mysterious Universe

Mystery is at its best when it gets more and more confusing, deeper and darker, before even the faintest clues show up. In high school, learning how to apply mathematics gave me not only mysteries, but subtle and confusing ones – to this day I have trouble describing them.

I first caught a hint of these mysteries as a junior while watching a short film in Mick Freeman’s introductory (algebra-based) physics class. The details of the film long ago slipped from memory, but what struck me, and stuck with me, was the ending – a “perfect” sine curve, graphed by some physical apparatus (which was probably a simple harmonic oscillator) as part of an experiment. Freeman made a remark afterwards suggesting the outcome was entirely artificial, that the makers of the film specifically constructed the apparatus and experiment to result in the tidy graph at the end.

And that’s just it: the sine wave was contrived. It did not come about without human intervention in so-called natural processes, those processes, I suppose, that occur in the physical world independently of what we humans do. If our physical theories result in part from constructing, describing, and explaining such artificial processes and their outcomes, as they apparently do in the experimental sciences, then how do they describe the natural world, that world supposedly out there, independent of what we humans see or do? And, more to the point, what do mathematical objects like “perfect” sine curves invoked in these theories have to do with that world?

Mathematics lies along that mysterious boundary between mind and world. I sometimes read that the universe has a mathematical structure, and until my mid-twenties or so, statements like this enamored me. In high school, I had only a shadowy notion of what it might mean to claim the physical universe has a mathematical structure, but in my naivete I believed something like this was the case, and that the structure was somehow revealed by our most advanced and therefore most accurate physical theories.

I wanted to learn more physical theory to see what that structure is. I looked at that “perfect” sine curve and wondered what its relationship was to the uncontrived world out there. Were sine curves, even seemingly perfect ones, embedded somehow in the physical world around us (ocean waves, electromagnetic waves)? Were they part of an underlying mathematical structure, out there? Is the physical world thereby intelligible? And if it is, does that help us answer broader questions, about its origin, or even creation, and about our own minds and how we learn and understand?

In short, I wanted to understand how mathematics is applied, and why it is possible to apply it so successfully, as we do in our physical theories. That understanding did eventually come to me, but I had a lot to learn first, and the answers I found were much different from what I expected them to be.