Somewhere in my college days, we read Aristotle. I recall that he droned on about truth and beauty, but for the life of me, I have no clue about what he said. Or maybe I’m just recovering snatches from Zen and the Art of Motorcycle Maintenance.
Regardless, as a defrocked mathematician, I have an ambivalence towards these two notions. This may strike you as strange, “beauty” not often associated with mathematics, so let me explain. In mathematics, we work from guesses called “conjectures,” which we then try to establish by a finite series of logical steps, reasoning from assumptions (axioms) and from results already proven to be true. Once established, they get promoted from “conjecture” to “theorem.” It all sounds so dull when you put it this way.
To make it fun, the conjectures have to be interesting, like Fermat’s Last Theorem first stated in 1637 or the Four-Color Map Theorem from 1852. You can easily explain these to freshman algebra students (the Map Theorem to any reasonably bright 8-year-old), but their proofs turned out to be fiendishly difficult. In other words, they were fun problems to work on, like a crossword puzzle from The New York Times. In fact, they were both still conjectures when I got my degree in 1971.
And that’s what math is, thinking up interesting problems and then solving them. If you want to make your name in the field, you have to think up a really interesting problem, preferably simple to state, at least to practitioners in your field, and then present an astoundingly elegant solution. Your colleagues will go “Wow! That is so cool! How did you ever think of that?” Actually, that’s what they might tell you; what they really think can’t be included in a family blog. And you don’t show the rubes how the magic was done, the drafts, the sketches, the false starts and blind alleys. Ever. Just the final simple, elegant, beautiful result.
“Elegance” is the mathematical equivalent of beauty, and it virtually always implies “short,” brevity being the soul of wit, after all. You can have a long proof if that’s all you can think up, but most of it will be preparatory stuff, like the lead-up in a magic act. Then, bang! the rabbit pops out of the hat. The “bang!” is where the beauty lies. Without the bang!, you may have truth–any proof, even an ugly one, will move a guess from conjecture to theorem–but you won’t have beauty.
I want to throw out a conjecture of my own on the relation of mathematics and physics. It’s not necessary that they have any relationship: Why should thinking up and solving logical puzzles have any relationship to what’s happening in the observable universe? But as Richard Feynman concluded, mathematics is the language of physics. “Physicists cannot make a conversion to any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language she speaks in.” (The Character of Physical Law, p. 52) This seems reasonable, but that’s not where I’m going.
Here’s my conjecture: The mathematics that nature speaks is the beautiful mathematics, the elegant mathematics. In fact, I’ll add a corollary conjecture: The more beautiful the mathematics, as mathematicians would appreciate, the more likely it is to correspond to something in nature, as physicists will determine.
I got this idea from a recent article at Phys.org, “Einstein’s ‘spooky action’ common in large quantum systems, mathematicians find.” I was reading merrily along, minding my own business, when this jumped out:
The researchers used mathematics where analysis, algebra and geometry meet, Szarek said. The math applies to hundreds, thousands or millions of dimensions. “We put together several things from different parts of mathematics, like a puzzle, and adapted them,” he said. “These are mathematical tools developed largely for aesthetical reasons, like music.” (emphasis added; note the part about “like a puzzle”)
They can’t be implying that the mathematics was developed in order to analyze music–that doesn’t take much in the way of real mathematics. What they’re saying is that the mathematicians developed these tools because they were beautiful, elegant, fun to play with. It’s quite likely that the mathematicians who came up with these things wouldn’t know a strange quark from a truth quark (or a beauty quark … sorry). Nor did they care. They, like all artists–and mathematicians are artists in the same way that poets and, yes, composers are–were just creating beauty. Just having fun.
Perhaps Feynman, who seemed to dance all around my conjecture without actually stating it, did put his finger on the fundamental principle:
You can recognize truth by its beauty and simplicity. It is always easy when you have made a guess and done two or three little calculations to make sure that it is not obviously wrong, to know that it is right. When you get it right, it is obvious that it is right–at least if you have any experience–because what usually happens is that more comes out than goes in. … The truth always turns out to be simpler than you thought. (Feynman, op. cit., 165)
PS — yes, I know that “truth” and “beauty” are no longer the universally accepted names for these particles, but they fit too well to pass up.