One day in fifth grade, I was playing with numbers, scribbling down products and quotients—you know, typical cool-kid stuff—when I noticed a pattern. Take any pair of numbers that are two apart (like 6 and 8, or 9 and 11). Multiply them together. Then add one.
You’ll get the square of the number in between them!
This blew my mind. The numbers were hiding secret alliances, passing coded messages amongst themselves, and I’d somehow broken inside. I was a number spy.
Tapping into what little I knew of algebraic notation, I rewrote my discovery in abstract terms:
I carried that formula with me for the next three years. I wore it like a locket and recited itlike an oath. It was pure and crystalline and true. Never mind that I hadn’t proved it. It was right, and it was mine.
When I hit Algebra 1, and learned not merely to write algebraic symbols but to manipulate them, I found that my formula wasn’t just known. It was a trivial consequence of the distributive property.
I suffered ten seconds of stunned disappointment that my treasured conjecture was nothing but old hat. Then I smiled. When you’re standing on a mountain, you don’t care that others have stood on that same spot before. You’re just enjoying the view.
From time to time, the formula and my “discovery” drifted back into my thoughts, like an old phone number or a favorite lyric. Then, just recently, I was toying with the idea of multiplication as arrays. It’s an old, simple idea, one that my sister teaches to 2nd graders. I thought about square numbers:
Then I pictured subtracting one:
Then I pictured taking the last column, turning it on its side, and sticking it below the bottom row:
Suddenly the pattern threw off its disguise, and there it was, shining up at me, a diamond in a field of dots. It was a new proof of my formula. I’d never imagined the elegance and simple insight that the geometric perspective might bring.
There is nothing novel or historic or remotely significant about my observation. It is a fact that has been discovered and re-discovered thousands of times, across all the continents. It is old news. It is—to use the mathematician’s favorite word of sneering derision—trivial.
But it is also a pattern grander and deeper than the minds that stumble upon it, so this fact belongs to no one and everyone at once. It is mine, just as it is the intellectual property of some Babylonian who derived it millennia ago and murmured, “Trivial… but interesting.”
It’s like literature. I don’t believe that the great writers invent new themes. The human condition is only so complicated, and the messages which strike deepest at our hearts (“Power corrupts”; “Love wins out”; “We are all alone at death”) have been circulating in our plays and poetry for centuries. The job of the writer is not necessarily to pour new ideas into the reservoir of human thought. It’s to draw water from that reservoir, and pour it into the vessels of character and story and phrase. The job of a writer isn’t to invent or discover truth. It’s to bring truth to our lips, so we can drink.
So it is sometimes with math. As the X-Files promised, the truth is out there. And to discover it makes us mathematicians, no matter whether others have already walked the same path.