Creating mathematics
People have a wide variety of feelings about math: one considers it horrifically painful whereas another considers it exquisitely beautiful. Some see it as the furthest thing from nature, a human construct of black-and-white thinking; others see it as the most natural thing: the enduring forms that underlie reality itself. But the issue I’m interested in discussing here—again one that gets very different opinions—is that of whether or not mathematics is creative.
People have a wide variety of feelings about math: one considers it horrifically painful whereas another considers it exquisitely beautiful. Some see it as the furthest thing from nature, a human construct of black-and-white thinking; others see it as the most natural thing: the enduring forms that underlie reality itself. But the issue I’m interested in discussing here—again one that gets very different opinions—is that of whether or not mathematics is creative.
It’s not often discussed, but all the math we have today was created, or “lifted out”, by a person like me or you. The person saw something in the world that gave them the impression of a clear pattern, and they watched and participated in that pattern until they could devise a way to keep track of some aspect of it unambiguously. For example, a person named Isaac created what we now call “calculus” in order to track how the location of a celestial body changes through time. The moon moves continuously, without skipping around, but how should you keep track of this phenomenon of continuity itself? We could imagine that long before Isaac’s work, a shepherd noticed that the very same sheep that leave an area in the morning should come back in the evening, though maybe in a different order, and they came up with numbers as a way to keep track of the phenomenon of preserved quantity.
I think of mathematical fields as accounting systems. You can account for sheep and coins using arithmetic: some aspect of what happens when you let two groups of sheep (or coins) come together in the same enclosed space is tracked by what happens when you add two numbers together. What’s tracked there is what addition means: 10 sheep and 12 more sheep come together to make 22 sheep, every time. Similarly, some aspect of how celestial bodies move without jumping around is tracked by what happens with a function like y=x^2 : if you change the input x-value only a little bit, then the output y-value changes only a little bit too. The math makes these notions precise, so that people can share their accounts and check each other’s work. We can be accountable to each other that way.
Category theory is a field of math, so according to the above ideas it should be an accounting system for something, but what? I would say that category theory is a system by which to account for structural patterns themselves, for the fact that analogies can work and be effective to varying degrees. How do we account for the way structures combine and interact? How do we account for why analogies—similarity of structure despite difference in content—help us think effectively? That’s definitely a pattern, right? Category theory as an accounting system says that what’s at play here—what should be tracked if we want to understand structure and analogy—are, roughly speaking, types of things, the ways they relate and combine, and the rules they follow. It makes those italicized words precise so that people can share their accounts and check each other’s work.
Returning to creativity, what I find sad (or better, “worth reconsidering”) is the fact that we’re taught in school to follow in the tracks of others—using arithmetic, using calculus, etc—but we aren’t taught that we can also track our own interests, the patterns we see in the world. Games can be fun, but if your only choice is to play someone else’s game, you might easily get bored; with games it’s nice to know you can stop playing or play a different game if you want to. I think it’s a loss that in most school math classes, all we do is play someone else’s game for years on end. No wonder some kids find math to be terrible: the math that’s being taught (e.g. “factor the polynomial x^2+3x+2”) just isn’t about—isn’t tracking—anything the kid can imagine ever caring about.
Kids, and people generally, are creative and they want to express it, but it’s not easily boxed: creativity comes in many forms. One person might find football creative: they make up new strategies and formations within football. Another finds fashion creative: they express something in their choices there, and they admire the creativity available in the medium of wearable fabric. Another finds chemistry creative: they make up new ways of producing reactions and materials with interesting properties. And there’s a whole other sort of creativity which is somewhat “meta”, namely making up one’s own game. Within their new game, they make room for the sort of creativity we find in football, fashion, or chemistry, but then there is additionally the creativity in making up the game itself. There one asks, “what do I want this game to be about? How can I make it fun for people to play with me?”
The point I want to make, which is probably clear to many but not clear enough to all, is that mathematics, mathematical disciplines big and small, are like games that can be created. If you see something you want to track—a phenomenon you want to give an account of—work with it until you can lift out a notion of what types of things are at play, what relationships and combinations are possible, and what rules you see them following. Football players arrange themselves dynamically to form protected pockets and channels by which a ball can be moved to a goal; what should be tracked here if we want to talk unambiguously about how it works?
I’d venture to say that the biggest game on the planet today is the financial system. When I tap my credit card on the reader, I’m playing a game that has very particular rules and involves billions of people. Do we like this game? I’ll speak for myself. The financial system has enabled the creation of the very device I’m using to type and the one I imagine you’re using to read, and as such I personally find it astonishingly powerful and creative. But it’s also enabled environmental devastation, famine, and other major crises, so I also find it deeply lacking and discordant.
If we are going to create a new game that transcends and includes whatever it is that the financial system is successful at, and yet do so in a way that works with nature—both the harmonizing and the creative and evolutionary aspects of nature—we need to find a way to account for the very issue of what “working with” nature means. Since the dawn of culture, people have seen patterns in what it means to “work with” nature, and they wrote them down in any form they could imagine: as art, theater, myth, science, etc.
Now here we are again today, able to notice all sorts of patterns in the world around us. If you can see some of them and articulate any of the types, relationships, and rules you see in these patterns, you too can create or lift out a new kind of mathematics to help us account for them and thus lend your insights to the formation of the living system.