Are there any disagreements among mathematicians?

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I would like to know if there is any genuine disagreement among mathematicians. By this I don't mean disagreement over convention (e.g. Should a ring contain $1$?). Formally, I mean is there a mathematical proposition $P$, such that a sizable group of mathematicians believe themselves to have a proof of $P$, and a opposing faction believing themselves to have a proof of $\neg P$?