I'm currently in a class on formal mathematics/formal logic/axiomatic set theory. Someone asked, "At the end of the day, as mathematicians, why do we care about rigor?" My professor gave an example of a mathematician who did not believe in the importance of rigor (implicitly refering to a time in which mathematics was rigorous by modern standards) and developed a number of results which others took up, following in his tradition of non-rigor, which eventually formed and became recognized as a branch or subfield of mathematics on its own. He said that the whole field was non-rigorous, and when outside mathematicians eventually bothered to look at their results rigorously it was found that the whole field was unfounded due to inconsistencies in some founding axioms. I asked what the name of the field or of the mathematician were, but he did not remember details. He only said that he thought it was a group of french mathematicians in the early 1900's studying a subfield of topology.
My question is this: Does anyone else recognize this example, and if so, could you provide me with details/references? I have already dome some searching using a variety of search terms, as well as poked around on various mathematics-related SE sites and didn't find anything that looked like what my professor was talking about.