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Once I had a job ghostwriting for some mathematicians who wanted to make their work more accessible to the general public, and especially to children. We made up puzzles and stories that tangled with infinity, and invented playground games with mathematical strategies underneath. We tried them out in workshops and school visits and overall had that experience of learning more from the kids than we had ever imagined.
One of the things that was fun was the way kids would engage so directly and creatively with questions like, How do you know if you should believe what someone says is true? Or, What do you do with your mind when you are trying to figure something out? It seems like children spend a lot of years working hard to put together a Theory of Everything and they often give esoteric questions a lot of thought.
One question I enjoyed putting to children and mathematicians alike was whether math itself is visible or invisible.
The math in my head is invisible, even if you cut it open. What about the Eiffel Tower? Can you see the math in that?
All that math in math books is certainly visible. If you say 2 + 2 = 4 in your mind, that’s wouldn’t be visible, but if you picture it in your mind, is it a little bit visible, more so than say, something in the 8th dimension which you really can’t picture at all?
Or is it that all the math in math books is written-down stuff about math, but the math-math isn’t the writing, but the understanding of what the writing is all about. That would make math completely invisible, which makes sense, because blind people can do it.
They always say math is everywhere. But if math is invisible and everywhere, isn’t that a little creepy?
If I can look at two collections of pebbles and say one has three pebbles and the other has four, there’s something very visible about that. Would it be most correct to say that math is a whole collection of abstractions, invisible ideas that bear a relation to the visible world? Except when they don’t.
It ends up to be hard to say anything about math and its visibility and invisibility without first articulating what math is. And then it turns out that in order to make sense, you might have to explain what you mean by visible and invisible as well—if you can. Overall, I found that kids like to talk about those things for more than mathematicians do.
Here’s another mathematical tangle of visible and invisible. I know a math professor named Larry Copes who would ask his students on the first day of calculus class to picture this: a perfect mathematical forest. The trees have been planted in a grid, one foot apart in rows and columns that extend to infinity in every direction. The mathematical trunks of the mathematical trees reach infinitely high, but they have no width nor depth. The tree at the very center of the forest has been cut down, and you are standing on the stump. When you look out through the infinite forest around you, do you see daylight?
I dunno. Do you? What do you think? Can you see through the gaps between an infinite number of tree trunks that don’t have any width?