The “Eerie” Feeling Of Math

I studied math before I studied philosophy and in my youth I was always that person who liked math. I’ve been re-engaging with math after a long hiatus and it has so great — fun, interesting, awe-inspiring. One thing I had forgotten is how math engenders so many different emotions in people: anxiety and fear about not being “good at math,” comfort and calm about being in a place with a “right answer,” curiosity and indignation about whether math is perceived to be discovered or invented.

One of the emotions of math that gets talked about less is what the physicist Eugene Wigner calls an “eerie” feeling. In the 1950s, Wigner gave a lecture on “The Unreasonable Effectiveness of Mathematics in Science.” It’s a miracle, he says, that math works. Math concepts are developed by mathematicians according to aesthetic criteria, including “look at me, I’m so ingenious!”-based considerations. Yet these same concepts are massively useful in physics. It’s strange; it’s surprising; It’s a miracle we neither understand nor deserve.

Wigner’s paper is either famous or infamous, depending on how you feel about his topic. In philosophy, there are whole cottage industries devoted to “is it really surprising, though?” and “maybe you only notice the successful ones?” and “no, it actually is a miracle.”

I don’t want to talk philosophy, though, I want to talk feelings. I love the way Wigner opens his paper with the simple story of a statistician and his old classmate from school days. The statistician shows a paper on population trends to his friend, a paper full of complex and sophisticated mathematics. The friend looks at the complex symbolism and is like “wait, what”? … Then the friend points to π and says “and what is this symbol here?” “Oh,” says the statistician, “that is π” —“the ratio of the circumference of the circle to its diameter.” “Now you are pushing your joke too far!” says the classmate, “surely the population has nothing to do with the circumference of the circle.”

Wigner says the story gives him an “eerie” feeling. Surely, the reaction of the classmate betrays “only plain common sense.” Like: yeah, what does the ratio of the circumference of the circle to its diameter have to do with population statistics?

I have this eerie feeling about math all the time. I get it about the application of math in science, and I also get it about the application of math to other math. You’ll be going along learning some thing, and suddenly a concept from some completely different concept not only pops up, but turns out to be exactly the thing for situation.

Just look at the Wikipedia page for the constant e and the bewildering array of seemingly unrelated applications: compound interest, probability theory, optimization using calculus, number theory, etc. etc.

In connection with the eeriness of math, I only recently learned more about the emergence and significance of the complex numbers — numbers like a+bi where i is the “imaginary” square root of minus one. They appear in Cardano’s work in the sixteenth century in connection with finding the solutions to polynomial equations — equations like x^2+1=0. If you try to find two numbers that add to 10 and multiply to 40 — that is, solutions to x^2-10x+40=0 — you find that there are no such real numbers, but that 5+√–15 and 5-√–15 work just fine.

While the square roots of negative numbers are not ordinary numbers, Cardano wasn’t uncomfortable with them: “√–9,” he wrote, “is neither +3 or –3 but is some recondite third sort of thing.” 

Complex numbers were essentially thought up to solve mathematical problems, not practical or physics problems. And as Wigner himself says, if you ask a mathematician to justify their interest in complex numbers, they will point (“with some indignation”) to their many uses in “beautiful” theorems in the theory of equations and other branches of math. So it is a bit weird when you find out later on that complex analysis is one of the most directly applied parts of math there is and that imaginary numbers are everywhere in physics and other applications.

A thing I wondered about for many years was why complex numbers were ubiquitous and other analogous ways of extending the real number structure less so. If you’re thinking abstractly, the addition of “i” according to certain principles is just an extension of real numbers with a new symbol according to calculation rules regarding how that symbol works with the existing numbers and relations. So — I never understood: why are we always studying that one extension of the real numbers and not some other one? Why not add multiple new symbols instead of just one? 


Then a few months ago I read Numbers: A Very Short Introduction, and I learned that “it is not possible to construct an augmented number system that contains [the complex numbers] and also retains all the normal laws of algebra.” Aha! That is a clear explanation of the specialness of the complex numbers: you can’t go bigger and still keep the rules you want to keep.

Of course, it’s a clear mathematical explanation. It explains how the complex numbers are mathematically special. But then the complex numbers are also special in applications of math — so much so that while you can take a “pure math” course on “Complex Analysis,” there are also courses on “Applied Complex Analysis.” Does the mathematical specialness of the complex numbers somehow carry over to the scientific context?  

It is strange! For me, the deeper I go, the eerier it gets.