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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?

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Dennis
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  • There is this quabble about the uniqueness of the algebraic closure of a field. – Pedro Nov 01 '14 at 22:12
  • @Pedro: what quibble? It's a theorem that everyone agrees on that you have uniqueness in the presence of the axiom of choice. – Qiaochu Yuan Nov 02 '14 at 00:15
  • @QiaochuYuan I remember Mariano and Martin getting into one some months ago. Cannot remember the details. – Pedro Nov 02 '14 at 01:16

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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.

Jürgen Sukumaran
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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.

Matt Samuel
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  • At least one out of two ... – Mark Bennet Nov 01 '14 at 22:10
  • In mathematics, the "at least" is implied :-p – Matt Samuel Nov 01 '14 at 22:12
  • This is simply not true. There can be disagreements about which axioms to take and different axioms can obviously lead to very different theories. One example is the axiom of choice, in which case if you take the axiom and I don't then we can argue about the mathematical content involving cutting spheres... – Jürgen Sukumaran Nov 01 '14 at 22:13
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    Disagreeing about which axioms to use is not a disagreement about mathematical content. Each person using the different axioms can evaluate each other's work and verify that it is correct given the axioms. – Matt Samuel Nov 01 '14 at 22:14
  • Rational people could still disagree if the evidence available to them (extant mathematics) isn't sufficient to settle the issues. Or, like @Tony S.F. just pointed out, if they disagree over foundational matters like the choice of axioms. – Dennis Nov 01 '14 at 22:14
  • I do really like the computer proofs example, though, I had forgotten that dispute. – Dennis Nov 01 '14 at 22:15
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    I see what you mean @MattS but I would still make the argument that the question of which axioms to choose falls under mathematics and rational people can disagree over this question. – Jürgen Sukumaran Nov 01 '14 at 22:19
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    The verifiability is true only in principle, and might be very difficult, essentially functionally impossible, in practice. – paul garrett Nov 01 '14 at 22:24
  • I'm thinking that, e.g., the dispute over the truth of CH in ZFC prior to the discovery of the independence proofs was a rational one. Just because it ultimately turned out to have an unexpected answer doesn't mean the disputants were irrational. Based on extant mathematical knowledge, it was an open question whether on not CH was true in ZFC which could be rationally disputed. – Dennis Nov 01 '14 at 22:27
  • Another example where rational people eventually agreed, in this case that CH was undecidable in ZFC. – Matt Samuel Nov 01 '14 at 22:53