Widespread, persistent mathematical disagreement? My question is related to this one about whether mathematicians always ultimately agree, with a slight variation.
I'm curious not whether mathematicians always ultimately agree, but whether there are examples of widespread disagreement in mathematics. All of the examples of disagreement I can think of concern relatively minority positions rejecting portions of mathematics for philosophical reasons (e.g., finitists, ultra-finitists, constructivists/intuitionists). 
But all of these positions are (with the possible exception of constructivism?) held by only a small portion of mathematicians and so I wouldn't consider these disagreements widespread.
My question is more whether, within the community of classical mathematicians, there are examples of widespread, persistent disagreement over, e.g., the validity of some proof technique, the legitimacy of some new branch/theory, and so on.
I can think of some examples (perhaps transfinite induction?) that provoked widespread disagreement initially. But in every such case I'm aware of the disagreements seem to have gone away over (a relatively short period of) time, and so fail to be persistent.
By comparison, moral disagreement often seems widespread and persistent. American views on the permissibility of abortion have remained split relatively evenly since at least 1975. So, the disagreement over abortion in America is an example of what I'd call widespread and persistent.
Are there disagreements like this in mathematics? Disagreements that are both persistent and widespread?
 A: How about the disagreement regarding frequentist interpretation of statistics and the bayesian interpretation? One school of thought believes that parameters are fixed and if we could sample the entire population we would know exactly the parameter. The other school of thought claims that parameters are random. This has very important implications to conditional probability. There was even a third school of thought on the matter, fiducial probability or faith based probability, but it was quickly ruled out to be unsatisfactory.
A: There is disagreement as to whether proofs requiring a computer should be accepted by the community because they can't be directly checked by hand. Rational people cannot disagree about the actual content of mathematics because one out of any two people who disagree about mathematical content is wrong. Even if two mathematicians disagree about which axioms to use, each mathematician can verify another mathematician's work is correct given the axioms that they used.
