The Dangers of Visualizing Abstract Things

To follow on to my last post, a big part of visualizing abstract mathematics is, ironically, accepting that you can’t completely visualize it. Sometimes you have to settle with what you know. For instance, you can imagine points in L^2, which is the vector space of square-integrable functions. You just imagine E^2 or E^3 (the Euclidean spaces humans are most familiar with) and pretend that each point in that space represents a function. This isn’t technically correct, since the inner product for L^2 is quite different from the Euclidean dot product. But it is a useful mnemonic.

Maybe I’m just a visual person, but if I don’t have a picture.. any picture.. I just can’t understand anything. It’s much better, in my opinion, to have a picture that demonstrates or represents a single property of some abstract thing—even if all the other properties don’t really match—than to have no picture whatsoever.

Let’s enter rant mode for a second, though. Because I’m somewhat skeptical of the idea that these kinds of informal visualizations should be widely popularized à la Discovery/History Channel. I absolutely hate it when science popularization does not sufficiently caveat a technically incorrect analogy (which is not terribly difficult to do, I don’t think). It’s especially insidious coming out of textbooks. There’s a level of understanding required to fully grasp certain analogies and ways of thinking about things. Without that knowledge, you’re only getting a hint of a hint of what’s going on, at best. At worst, you get quantum-wave-energy-photonic-crystal-tao-healing.

Now, ‘technically incorrect’ could mean a lot of things as you attempt to make such a caveat. I imagine it’s real easy to be a snit about what constitutes ‘correct’, especially as you get to more and more complicated topics with very rigorous definitions and so forth. There’s a popular way of making such caveats that’s always satisfied me: “It’s correct to a first approximation”.

At the end of the day, asking the History Channel to do this sort of thing is probably a waste of time. Between all the science shows are the Jesus-UFO-Conspiracy shows featuring Ralph the Overly Dramatic Narrator. And you know they ain’t gettin’ no caveat. And even if they did, the only one apropos would be “I’M IN UR TV, MAKIN U DUMB”.

Which should literally pop up every time you turn to the History Channel, in place of the H. And then there should be a little animation of a guy pointing and laughing on an infinite loop, you mindless cretin.

Shame on you, forever.

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