There is also Weilandt's characterisation of the gamma function, it is the unique meromorphic function which is bounded in vertical strips.

I think that the first 2 paragraphs of this answer are truly excellent.

@Yemon: Several answers can provide insight, I'm uncomfortable with singling one out and labeling it as the accepted answer. These 2 answers are not what I had in mind, but I don't have enough understanding to comment on them.

I think Gosper's algorithm is worth mentioning here.

@Yemon: Thanks for resurrecting my questions.

@Scott: Do you hate all of my questions?

Are you kidding me? Closed because it is not a real question?

@KConrad: I think in terms of symbols rather than visually so that may be another reason for my difficulties. Another example for me is the tensor product: taking linear combinations and then defining an equivalence relation I can understand; but every bilinear map can be factored uniquely is much less clear to me. I by no means meant to criticize the language, I was trying to gain some understanding of what it was doing. From the answers, I gather that I haven't seen sufficiently advanced mathematics to appreciate this view.

It's difficult to be certain with Ramanujan - most of his methods are completely unknown.

I certainly don't want to be closed minded. In contrast to metric spaces I find topological spaces extremely hard to understand, I can see the trees but I can't see the forest; I am seeing the definitions and the theorems but I am not seeing the ideas that are running underneath them. Perhaps the reason for this is that I don't have enough experience with non-metrizable topologies. I was looking for a heuristic that was more precise than "topological spaces generalize metric spaces".

edit: should be "As I understand it, he means something like: ..."

@Ryan: As I understand it, he is not referring to something like: imagine a world where we knew the fundamental theorem of calculus, but we didn't know how to find anti-derivatives. Homotopies corresponds with anti-derivatives, but I don't know what corresponds to the fundamental theorem of calculus.