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“Every other subject is about something. Poetry is about something. Even most modern art is about something. Math is about nothing. Math describes much of the world but is all about itself, and it has the most fantastic conundrums. But it is not about the world.”
Yes, mathematics is an entirely self-referential language. That’s the very reason why it’s so useful. Complaining that math isn’t about some thing is like complaining that paint isn’t an image – and even Hacker concedes that math can be used to describe much of the world. For most scientists the discussion stops at this point. The verdict in my filter bubble in unanimous: mathematics is the language of nature, and if schools teach one thing, that’s what they should teach.I agree with that of course. And yet, the argument that math is the language of nature preaches to the converted. For the rest it’s meaningless rhetoric, countered by the argument that schools should teach what’s necessary: necessary to fill in a tax return, calculate a mortgage rate, or maybe estimate how many bricks you need to build a wall along the US-Mexican border.
School curriculums have to be modernized every now and then, no doubt about this. But the goal cannot be to reduce a subject of education based on the reasoning that it’s difficult. Math is the base of scientific literacy. You need math to understand risk assessments, to read statistics, and to understand graphs. You need math to understand modern science and tell it from pseudoscience. Much of the profusion of quack medicine like quantum healing or homeopathy is due to people’s inability to grasp even the basics of the underlying theories (or their failure to notice the absence thereof). For that you’d need, guess what, math.
But most importantly, you need math to understand what it even means to understand. The only real truths are mathematical truths, and so proving theorems is the only way to learn how to lead watertight arguments. That doesn’t mean that math teaches you how to lead successful arguments, in the sense of convincing someone. But it teaches you how to lead correct arguments. And that skill should be worth something, even if Hacker might complain that the arguments are about nothing.
I thought of this recently when my daughters had their school enrollment checkup.
One of the twins, Lara, doesn’t have stereo vision. We know this because she’s had regular eye exams, and while she sees well on both eyes separately, she doesn’t see anything on the 3d test card. I’ve explained to her why it’s important she wears her eye-cover and I try to coax her into doing some muscle building exercises. But she doesn’t understand.
And how could she? She’s never seen 3d. She doesn’t know what she doesn’t see. And it’s not an obvious disability: Lara tells distances by size and context. She knows that birds are small and cars are large and hence small cars are far away. For all she can tell, she sees just as well as everybody else. There are few instances when stereo-vision really makes a difference, one of them is catching a ball. But at 5 years she’s just as clumsy as all the other kids.
Being math-blind too is not an obvious disability. You can lead a pleasant life without mathematics because it’s possible to fill in the lack of knowledge with heuristics and anecdotes. And yet, without math, you’ll never see reality for what it is – you’ll lead your life in the fudgy realm of maybe-truths.
Lara doesn’t know triangulation and she doesn’t know vector spaces, and when I give her examples for what she’s missing, she’ll just put on this blank look that children reserve for incomprehensible adult talk, listen politely, and then reply “Today I built a moon rocket in kindergarten.”
I hear an echo of my 5 year old’s voice in these essays about the value of math education. It’s trying to tell someone they are missing part of the picture, and getting a reply like “I have never used the quadratic formula in my personal life.” Fine then, but totally irrelevant. Rather than factoring polynomials, let’s teach kids differential equations or network growth, which is arguably more useful to understand the world.
Math isn’t going away. On the very contrary it’s bound to dramatically increase in significance as the social sciences become more quantitative. We need that precision to make informed decisions and to avoid reinventing the wheel over and over again. And like schools teach the basics of political theory so that children understand the use of democracy, they must teach mathematics so that they understand the use of quantitative forecasts, uncertainties, and, most of all, to recognize the boundary between fact and opinion.