Mathematics

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TL;DR

Most people probably think of math as being about numbers and equations and, I don’t know, triangles. But where do numbers and triangles come from? Like, why are these things that anyone ever needs to talk about? You know, every math student ever has probably at some point though, math class is boring and stupid and when am I ever going to use this? Besides which, how do we know that math is true? Why do four plus three turn out to be seven instead of twelve, and how does that kind of truth relate to scientific truth, which is based on evidence rather than I guess, pure Vulcan logic?

I think most mathematicians are probably with Plato on this. They’d tell you the mathematical truth is more pure and essential and fully known than anything in science. you’ve probably heard that evolution is just a theory, even gravity is just a theory because science can’t ever fully prove anything. But maths has proofs and we know for sure they’re right. But why is science stuck defending theories about the imperfect real world while maths has these perfect truths? Plato’s idea was that there are these ideal forms, like perfect circles, which exist in a kind of maths heaven somewhere. And the things in maths heaven work as maths does and we keep realizing they exist when we see the regular world struggling to imitate them with its imperfect triangles and circles. I’m not being very fair to this idea, because to me it just sounds crazy. I mean if ideal forms don’t exist somewhere in the physical universe then isn’t the notion of their reality a metaphor? And if so, what’s it a metaphor for? Are ideal forms of the property of nature, or do they exist in our minds, or what? There are a bunch of other ideas. The intuition viewpoint says that the source of mathematical truth is human intuition; that we verify the truth of mathematical propositions subjectively by thinking through them and directly experiencing their truth.

Well, I don’t know about you, but I have an annoying tendency to directly experience the truth of things that turn out to be false! So, how do we deal with that? And then there’s the social constructionist view that mathematics is just a kind of language game that mathematicians play with one another, making up the rules as they go along like some super nerdy version of Calvin ball. Like, the only reason four plus three is seven is that everybody says so. As a kid, did you ever just keep asking why, why, why, and eventually your parents or whoever would break down and say “because I said so”? maybe that was the answer! I love to mock social constructionists but there’s a valid, important point here that the consensus of mathematicians, like any human agreement, is to some extent vulnerable to mass allusions and groupthink and all the usual in-group/out-group drama that pervades human social life.

Still, if it’s strictly true that there’s little or no connection to any underlying reality, then it’s really weird that the rules mathematicians have happened to agree on are so great for describing the world around us. And I think that’s pretty much the average scientist’s view of mathematics: It some kind of language of nature. You need it to describe how the universe acts and who cares why it works? We’ve found that it does. At least here we’re on familiar territory for scientists. Mathematical truth if nothing else is backed up by empirical observations of the world just like any scientific theory. But does that mean that mathematical proof isn’t proof? When we want to know whether a mathematical idea is true, do we have to go out and find a relevant example triangle, or an unaccountably infinite set, and confirm the theory describes something in nature? Okay, so obviously I don’t buy any of these arguments, so here’s what I think: I got this idea from Haskell Curry, an all-around awesome mathematician who outlined it in a little paper in 1951. I’m not sure Curry did a great job defending his idea from hostile philosophers, but he characterized math in a way that, at least for me, finally clarifies this vague hunch humanity has had for so long that mathematical reasoning is somehow valid. Besides observing perfect circles, there’s another very physical and empirical way that we interact with math. We perform calculations and we observe their results. What Curry thought, and what I think, is that maths is a science on the same footing as physics or chemistry. Mathematical rules themselves are phenomenon that exists in the real world and can be observed and studied. There may not be any perfect circles, but they're certainly are real definitions of them in the rules of geometry. And following the rules of geometry to reach conclusions about their properties is a real phenomenon with observable results.

Maths, in other words, is the science that studies the implications of systems of rules. And any rules count: maths is not just the stuff taught in grade school, like the straight edge and compass rules of geometry, or the equation-balancing rules of algebra, but also non-euclidean geometries and exotic algebras with arbitrary numbers of operations that are commutative or not or distributive whichever way, and the rules of games like tic-tac-toe and chess, even the abstracted rules of grocery checkout lines or stock options or gravity or quantum mechanics. Even straightforward rules can have surprising implications, so it’s not like there’s nothing to discover just because you know the rules. The Pythagoreans, for example, were scandalized to learn that the rules of geometry led undeniably to irrational numbers like the square root of two. But there it is, and we have proof. A proof that is, if you look at it another way, a reproducible experiment. Maybe we can’t prove things in maths anymore than in science, but by following the rules of geometry, we can show ourselves again and again that the hypotenuse of a right triangle with two sides of equal length can’t be the ratio between any two whole numbers. So in the science of mathematics, the hypotheses are ideas we have about the implications of rules, like “you can guarantee a draw against any opponent in tic-tac-toe just by playing well”, and then experiments are individual applications of the rules, like checking a proof, to verify that they play out in a certain way.

The experiments in maths, in other words, are computations Like the intuition idea that maths is founded in the activity of human brains computing an answer, Curry’s view focuses on the empirical observation of computations. But the idea isn’t to directly experience their intuitive truth, so much is to see that the faithful application of rules does indeed lead to a particular conclusion. Putting computation at the center of mathematics like this makes me wonder about the connection between maths and computer science. I think maths is the science of computation. After all, the deep connection between mathematical proof and commutable, which was discovered as part of the investigation into the foundations of maths in the early twentieth century, is what kicked off the field of theoretical computer science in the first place. Somebody in computer science, maybe Edsger W. Dijkstra, is supposed to have said that computer sciences are no more about computers than astronomy is about telescopes. If throughout history mathematicians used their brains to conduct mathematical experiments in the same way that astronomers observed the night sky with their naked eyes, then computers are like telescopes through which we can better investigate the computational nature of reality. But what about that kid asking “when am I ever gonna use this?” If nature is all computational and mathematical, shouldn’t math be useful? Well, of course, it is.

Maths is everywhere. When kids ask “when am I ever going to use this,” what they mean is “you’re explaining this so badly that I don’t even know what it is. You’ve taught me how to swim without ever showing me a body of water big enough to swim in.” We give them little buckets, like contrived “a train leaves Chennai Central” problems, and then drill them endlessly on techniques for solving these non-problems. Traditional math classes are so eager to cover every clever technique in a subject that we end up covering them all using boring little problems that nobody needs to solve, and we leave people confused about why they learned to solve them in the first place, which they promptly forget how to do anyway. If we covered fewer techniques, we’d have more time to tackle real problems, and kids would have a chance to get an idea of what math is. Or, at least, what it’s good for.