I don’t think that’s really a contradiction. Indeed I think that sort of language would be quite a normal thing for a mathematician to colloquially say when des

I don’t think that’s really a contradiction. Indeed I think that sort of language would be quite a normal thing for a mathematician to colloquially say when describing the problem. Probably they would say smooth rather than “curvy” but they mightn’t say anything at all. In plenty of contexts, “loop” means “map from S1 to your space” and “map” means “continuous function” (or maybe a pointed continuous function in some contexts). Usually I would expect such problems to be introduced with a brief description and some examples and nonexamples. If the proof had some extra reasonable requirements on the curve, one might just say that the curve has to be “well-behaved in some sense” and omit details which aren’t particularly interesting.