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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 $\neg P$?

user212223
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    Once in a while there is a group of people that believe they have proved $P=NP$ and once in a while there is a group of people that believe they have proved $P\ne NP$. Let's see who is right;) – pisoir Feb 01 '15 at 19:33
  • The only disagreements are foundational, basically what axioms and deduction is allowed ( e.g. the excluded middle ). – Mohamed Alaa El Behairy Feb 01 '15 at 19:34
  • I think, mathematicians only conjecjture a statement (or its converse), if there is at least some evidence that it is true (false). If there is no evidence for the truth and the falseness, I do not think that the truth (falseness) is actually conjectured. – Peter Feb 01 '15 at 19:35
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    Concerning, $P=NP$, the overhelming majority of mathematicans believe $P\ne NP$ – Peter Feb 01 '15 at 19:36
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    There may be some controversy about whether proofs where computers are required to check many cases are valid. Like with the four color theorem. Some people might argue we're not sure the computer is working correctly. – littleO Feb 01 '15 at 20:00
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    possible duplicate of Do mathematicians, in the end, always agree? – user26486 Feb 01 '15 at 20:02
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    Well, there are a few proponents of finitism out there, but I don't know if anyone takes them seriously. – hasnohat Feb 01 '15 at 20:07
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    There is the issue of constructivism. – Michael Hardy Feb 01 '15 at 20:14
  • I doubt it. I think disagreements among mathematicians tend to be about primacy, e.g., Newton-Leibniz on calculus: http://www.math.rutgers.edu/courses/436/Honors02/newton.html – Robert Soupe Feb 01 '15 at 20:39
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    I don't see why this question is opinion-based -- the question is (basically) whether or not there exist sizable groups of mathematicians who disagree about whether a particular proof is correct or valid, and the existence or non-existence of such groups of mathematicians doesn't seem to be a matter of opinion. – littleO Feb 01 '15 at 21:15
  • Are there any disagreements among mathematicians? - This questions is ingeniously crafted to yield a positive response: One mathematician will come and say: "No, there are no disagreements!", and then another one will automatically respond: "Yes, there are!", and thus discord is artificially created where once there ruled only peace and harmony! :-$)$ – Lucian Feb 02 '15 at 00:34
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    @littleO, I agree, this is a perfectly legitimate question. – goblin GONE Apr 07 '15 at 13:40

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