What interesting difference is there between a cup and a plate? I claim approximately the only nontrivial difference in their shape is how much water the differ

What interesting difference is there between a cup and a plate? I claim approximately the only nontrivial difference in their shape is how much water the different shapes can hold. One could surely define a notion of usefulness as a water container (a teapot shape would then be similar to a cup and not to a plate), but I think that would largely be the end of it. It turns out that the reason theories like topology exist is not for their usefulness in distinguishing cups from plates but for their being interesting to mathematicians (and secondarily for their applications to anther things that are interesting mathematicians). A final question: is a bowl like a cup or a plate? Somewhere in between?