Horizontal and vertical

Learning math involves unlearning many obvious things.

Our tutor had a Geometry student this past week, working on a problem set that involved angles built out of a straight line.  If you add all of them that lie one one side, you should get 180 degrees.  That’s a fairly basic fact, and it doesn’t even change in the alternative geometries invented in the nineteenth century.  This student seemed to have assimilated the idea, but then asked: is it the same when the line is vertical?  The first series of problems all had the line horizontal, and he wasn’t sure it stayed the same.  Our tutor did not say, “of course!” or “obviously!”  It was a serious question, and he answered it seriously.  Moreover, in a very important sense it is not obvious at all.

The eye-brain system is a very complicated thing, and works in sometimes mysterious ways.  One of the things that has been established, however, is that there are neurons that fire when an object is horizontal, and different ones when an object is vertical.  Orientation is something of basic importance.  In the world outside the pages of a Geometry book, very few things look or act the same lying down as they do standing up.  In that sense, mathematics is unnatural.

Of course when counting one’s sheep or the fish in one’s net, things behave as numbers do.  Five in this field added to three in that field total eight sheep.  But take another step, and unlearning comes into play.  The hay field that measures fifty yards along the river and twenty away from it is certainly not the same as one with twenty yards of river frontage and a depth of fifty; it’s more well-watered, for one thing.  It takes a step of abstraction to say that twenty times fifty gives the same number of square yards as fifty times twenty.  Even our tutor has students who forget, from time to time, that order doesn’t matter in multiplication.

(And in some parts of mathematics it does matter.  Place a book on your desk, top to the north, opening to the east.  If you rotate it counter-clockwise ninety degrees (which is a sort of multiplication) and then rotate the south-facing side to be vertical, it’s in a different position than if you rotated the south-facing side vertical first and then turned it ninety degrees counterclockwise.)

Most students, most people, have no trouble eventually separating the world of mathematics found in their textbooks from the world they live in.  (This is in spite of the attempts of modern textbook writers to make homework problems more relevant, something we’ve noticed as being much more prominent than in our student days.)  But often a mathematician invents or discovers something that seems absolutely divorced from any kind of connection with our world, like those alternative geometries we mentioned, only to find that physicists immediately put it to use.  And that’s really strange.