"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
What notions are used but not clearly defined in modern mathematics?
To clarify further what is the purpose of the question following is another quote by M. Emerton:
"It is worth drawing out the idea that even in contemporary mathematics there are notions which (so far) escape rigorous definition, but which nevertheless have substantial mathematical content, and allow people to make computations and draw conclusions that are otherwise out of reach."
The question is about examples for such notions.
The question was asked by Kakaz