Professor Thurston,
In my own study, I've struggled with giving meaning to mathematical objects for many years. It's likely I've learned more slowly than most, because I find it difficult to move on unless I've found something I can rely on. Usually, when I try to explain mathematics, I will give the person I'm speaking to a simple problem to give him or her the experience of doing mathematics. Almost always, I will ask to show that $\pi > 3.$ The experience is usually familiar. One person has asked me, "how is this different from finding the answer to any question?"
Another example I like to give people is one of the several graphical proofs of the Pythagorean theorem. The one I present is #9 from this site, since it seems to be the simplest:
http://www.cut-the-knot.org/pythagoras/index.shtml
I've always had trouble explaining things like group without giving simple geometric examples, like the dihedral groups. That the turns of the Rubik's cube is offered as an example of a group is well known. Conjugation, for instance, immediately makes sense. Elohemahab Solomon offered this video
Does A "Connections" Blog/Podcast exist for Math?
where Mr. du Sautoy, (at about 12:00) in the context of groups of symmetry, compares the founding of the concept of group with that of number. To summarize Mr. du Sautoy, the symmetry a group measures is analogous to the quantity number measures. For example, we have chairs and tables, which are different, but if we have 3 of each, the quantity is the same. If we have two walls of the Alhambra with different looking designs, but have the same symmetry, then one group acts on the designs. He discusses $S_3$ and $\mathbb Z_6.$
I accepted complex numbers for a long time, but after experience with math past calculus, and the requirement for rigor slowly pervaded my thinking, I couldn't accept them as more than ad hoc constructions. Finally, I was satisfied with their construction as elements of the field $\mathbb R[x]/(x^2+1).$ I talked about this in my answer to the question:
I now think of complex numbers as an abstract construction from things I have intuition about, that is, real numbers and polynomials. The usual interpretations, as vectors in the complex plane, vectors in $\mathbb R^2$ that one can multiply, are still of course, extremely useful in understanding them.
It's likely you know this already: it looks like Hilbert coined the term "ring" from Mr. Gleason's interpretation of cyclic groups.
http://en.wikipedia.org/wiki/Ring_(mathematics)#History
A distribution is something, with whose definition I've worked successfully, but still have little sense of. I know they are generalized functions, but my lack of understanding might just be lack of experience.
