TITLE: Are there any big controversies in contemporary mathematical research?
QUESTION [14 upvotes]: Are there any big controversies in contemporary mathematical research?
Other domains contain big controversial research topics (for example string theory in physics).
The specific nature of mathematics however, makes me suspect that there isn't much room for any serious disagreement, or at least less room in comparison..?
Is this intuition right, or completely ignorant and absurd?
--- This is not a question about mathematics per se, but more a question about the people doing it. I hope the question is still appropriate in this stackexchange. ----

REPLY [5 votes]: Maybe "controversy" isn't the word, but like the Gallic village, the constructivism continues alive and defying the mainstream mathematics.
The entry Constructive Mathematics of the Stanford Encyclopedia of Philosophy is very interesting. In the bibliography you can find examples of recent constructivist research.