Sometimes I start with a geometric idea in 2D and see if it generalizes.
Other times I think of a 2D caricature that encapsulates some important operation but doesn’t necessarily relate to the actual idea. It's more like a heuristic visual than anything. A simple example might be if I’m thinking about projecting onto a linear subspace. The subspace is some convex blob in the plane in my mind’s eye, even though that’s not a linear subspace, it’s easier for me to conjure and the important part, the projection, remains intact. If the linear part was also important then I’d probably make it a line but I can’t think of a time I’ve done that.
Some concepts only made sense to me when, as you said, I stopped trying to visualize them. The most notable one to me was quaternions, because those are 4D, but even complex numbers only made sense when I stopped looking for physical intuition and realized they’re just a description of certain operations for points in the plane.
Now, I can suffer through a lot of abstract nonsense, but the worst classes I took were heavy on the abstract nonsense and light on answers to “why do we need all this machinery at all?” If I couldn’t imagine an application I cared about, I had a hard time figuring out what were the important concepts and how to fit them together.
Sometimes I start with a geometric idea in 2D and see if it generalizes.
Other times I think of a 2D caricature that encapsulates some important operation but doesn’t necessarily relate to the actual idea. It's more like a heuristic visual than anything. A simple example might be if I’m thinking about projecting onto a linear subspace. The subspace is some convex blob in the plane in my mind’s eye, even though that’s not a linear subspace, it’s easier for me to conjure and the important part, the projection, remains intact. If the linear part was also important then I’d probably make it a line but I can’t think of a time I’ve done that.
Some concepts only made sense to me when, as you said, I stopped trying to visualize them. The most notable one to me was quaternions, because those are 4D, but even complex numbers only made sense when I stopped looking for physical intuition and realized they’re just a description of certain operations for points in the plane.
Now, I can suffer through a lot of abstract nonsense, but the worst classes I took were heavy on the abstract nonsense and light on answers to “why do we need all this machinery at all?” If I couldn’t imagine an application I cared about, I had a hard time figuring out what were the important concepts and how to fit them together.