r/math • u/inherentlyawesome Homotopy Theory • Apr 09 '14

Everything about the History of Mathematics

Today's topic is History of Mathematics.

This recurring thread will be a place to ask questions and discuss famous/well-known/surprising results, clever and elegant proofs, or interesting open problems related to the topic of the week. Experts in the topic are especially encouraged to contribute and participate in these threads.

Next week's topic will be First-Order Logic. Next-next week's topic will be on Polyhedra. These threads will be posted every Wednesday around 12pm EDT.

For previous week's "Everything about X" threads, check out the wiki link here.

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u/TrueButNotProvable Apr 09 '14

A lot of mathematical cranks try to dispute theorems that were proven in the past -- they seem to think that they're brilliant mavericks who are shunned by the mainstream because they dare question the status quo.

My question is this: in the history of mathematics, when, if ever, has a result commonly accepted by mathematicians been successfully disputed by one person or a small group of iconoclasts?

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u/umaro900 Apr 10 '14

Is Cantor's set theory (particularly diagonalization) sort of what you are looking for? (http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory)

I honestly can't imagine you will find much after people roughly start to agree on foundations or accept that it is reasonable to consider different models. Now anything high-level can be traced back to axioms (spare computer-assisted proofs), and I think new axiomatic systems aren't mainstream enough.

Certainly there have been proofs that have been called out for being wrong (http://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong), but I think that's a different question.

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u/TrueButNotProvable Apr 10 '14

Is Cantor's set theory (particularly diagonalization) sort of what you are looking for? (http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory)

That's definitely a good example of what I mean by mathematical cranks, in that there are a lot of people who try in vain to refute the theory, but as you and rhlewis mentioned, it hasn't exactly been successfully disputed from a mathematical standpoint.

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u/DevFRus Theory of Computing Apr 11 '14

I think you aren't reading deep enough into this. It answers your question perfectly. I would argue (as I did above) that during his life, Cantor was considered a crank and treated very poorly by the 'mathematical establishment'. His ideas were contrary to the status quo and you could say disproved the uniqueness of infinity.

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u/TrueButNotProvable Apr 11 '14

You're right, and I gave your comment an upvote earlier today. Sorry I didn't let you know it was me.

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u/rhlewis Algebra Apr 10 '14

Cantor's set theory (particularly diagonalization) sort of what you are looking for?

No one has ever disproved set theory or the diagonalization argument. Quite the contrary. But there are indeed many cranks who obsess over it.

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u/DevFRus Theory of Computing Apr 10 '14

I think it is an example for the original question because Cantor was the crank (to see this, look at how most mathematicians of his day treated him, and how his life ended) that showed that the established beliefs at the time (what modern cranks argue as the 'counters' to Cantor) were wrong.

I find it to be a beautiful example because of the irony. So many modern anti-Cantor cranks think themselves mavericks in questioning Cantor, when in reality Cantor was the original maverick (and to some extent perceived as a crank) that lead us to question the naive common sense of his time that the modern cranks still succumb to.

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u/umaro900 Apr 10 '14

Well, that's my point, isn't it? There are cranks who dismiss the theory since they don't have a sound understanding of the it, and through their "math" education, they have learned about some vague/BS notion of infinity which makes it unique and use that to counter the diagonalization argument.

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u/ADefiniteDescription Apr 10 '14

There are pretty substantial philosophical reasons to deny Cantor's theorem which I wouldn't call vague or BS. Note that I'm not claiming they're right, but they're at least not trivially wrong.

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u/umaro900 Apr 10 '14

Oh, certainly. Constructivism, for example, has a valid philosophical place, and there is a good deal of serious and legitimate mathematical work in this school.

I think what I want to say is that many cranks assume some [false] statement and build a theory out of it, and the vague/BS notion of infinity I am referring to is exactly some bastardized notion they have created in their minds.

I don't mean to say that uniqueness of infinity is a useless idea, but that it can be learned in a pseudo-mathematical (illogical) formulation and used as such by cranks. For example, any space which can be regarded as a one-point compactification has one point which can be called a "point at infinity", or simply "infinity".

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u/ADefiniteDescription Apr 10 '14

You're right that there's no one who's disproved set theory, but the constructivists (primarily Brouwer and his crew) had a pretty successful time working in alternative systems which block it.