Lynet writes about how she tried explaining the Poincaré Conjecture to a literary theorist and a historian. The bone of contention wasn’t really the conjecture, which the media dumbed down just enough so that non-mathematicians could understand a statement vaguely resembling what Perelman actually proved. Rather, it was how mathematicians could visualize a four-dimensional world.
“Can mathematicians actually picture four dimensional space?”
“Roger Penrose says he did it – briefly – once,” I said, grinning .
“No, but are there people out there who can actually…”
“Not that I know of.”
My historian friend was relieved. My literary theorist friend was confused. “If you can’t picture it,” he asked, “how could you have any intuition about it? I mean, you could just say whatever you wanted about it and no-one would be able to refute you.”
I’ll leave it up to you to make an appropriate snarky comment about literary theorists. I’d reply to Lynet’s friend by noting that at least the simpler intuitions, namely, those relating to the vector space structure, are easily generalizable.
I can view points in the plane as pairs of coordinates (x, y), and work out things like angles between lines, lengths of lines, tangent lines to curves, functions on the plane, and so on. Regarding the point (x, y) as a vector from (0, 0) to (x, y), I can add vectors pointwise by (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) and multiply them by scalars by k(x, y) = (kx, ky). I can look at linear transformations of the plane, or even affine transformations.
All of those are intuitively thought of using very concrete notions: length and angle are measurable concepts; tangent lines touch curves at only one point; vector addition consists of walking from (0, 0) to (x1, y1) and then along the same direction and for the same distance as from (0, 0) to (x2, y2); linear transformations are combinations of rotations, reflections, shears, stretches, and compressions, while affine transformations add translations.
But in higher mathematics, they’re considered abstractly, in order to generalize as much as possible. Length is defined using Pythagoras’s theorem, and angle is defined using inner products. Linear transformations are defined by the more easily generalized property that T(v1 + v2) = T(v1) + T(v2) and T(kv) = kT(v), which coincides with the more concrete definition in two dimensions.
Not coincidentally, for two or three years of university, students only ever see algebraic arguments, which don’t involve visualizing anything. Later some geometric aspects return, but even they are typically schematic; people who draw a line with a loop in it in algebraic geometry only look at very general aspects, like having a point with two tangent lines and not being decomposable into two lines.