What are some results which were widely believed to be false, but were later to be shown to be true, or vice-versa?
Results that were widely believed to be false but were later shown to be true
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7What does it mean for a theorem to be "shown to be true with probability $0$"? How does one determine the "probability" of a result to be true in such a way that one can *prove* that this is the probability? Certainly one can offer heuristic arguments to *suggest* that something is "likely" or "unlikely" (or "very likely" or "very unlikely"), but these are not by way of formal proofs of the 'probability of being true'. That would require some sort of probability distribution among "statements"... – 2011-06-15
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0Now, if you want to ask about conjectures that were widely (and strongly) believed to be false but were later proven true (equivalently, widely believed to be true but were later proven false), then perhaps you might want to rephrase it that way. – 2011-06-15
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2Discussing on meta regarding reopening: http://meta.math.stackexchange.com/questions/2358/regarding-theorems-which-were-shown-to-be-true-with-probability-zero. – 2011-06-15
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0I didn't really know how to phrase such a question. But Arturo Magidin's 2nd comment is what I was asking for; widely believed to be true, but proven false and vice versa. – 2011-06-16
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2This [thread on MO](http://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong) could contain some things you're after. Please do try to formulate a better and more specific version of your question along the lines that Arturo suggests then the question might be reopened. – 2011-06-16
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0@805801: Well, you can phrase it exactly like I did; "What are examples of results that were widely believed to be false, perhaps even argued heuristically that they 'ought' to be false or were 'likely' to be false, but were later proven to be true? Or, vice-versa, that were widely believed to be true but were later shown to be false?" You should be able to edit the question so that it says something along those lines. If you do, you might garner some votes on reopening. – 2011-06-16
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0See also this thread: http://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples. This is related since a frequent reason for conjectures being widely believed to be true is strong numerical evidence. It's not entirely clear whether you meant something like this by "probability $0$", or also wanted to include other reasons for conjectures being widely held to be true. I'd vote to reopen if you clarify the question and it differs from that other one. @Theo: Note that your MO post asks about results widely believed to be *proved*. – 2011-06-16
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0@joriki: I know, but that's a minor semantic/sociological difference :) – 2011-06-16
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0@Theo: Well, I think there's a subtle but important difference between a conjecture that was widely believed to be proven that turned out to be false, and a conjecture that was widely believed to be true (but acknowledged to be unproven) that turned out to be false. – 2011-06-16
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0@Arturo: No disagreement here :) I overlooked that you wrote about conjectures while the question was about theorems. I only addressed "widely believed to be true, but proven false". – 2011-06-16
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0I have edited the question to hopefully be more clear. – 2011-06-16
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2@805801: Note that the "vice-versa" question is really equivalent to the "direct" question. If a conjecture is widely believed to be false but is later proven true, then the *negation* of the conjecture is widely believed to be true but is later proven false. They are two sides of the same coin. The difference between your question and the MO thread is not that you are asking about "false later proven true", but rather that you are asking about **expectation** rather than a mistaken belief that the issue had already been settled. – 2011-06-16
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3@805: +1 for your ability to navigate through the complicated web of etiquette, and eventually pose a good question. – 2011-06-16
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0@Arturo: Oops, thanks for pointing this out. I must really be lacking on sleep. – 2011-06-16
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1It's still not clear to me what the motivation behind this question is. – 2011-06-16
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0Also, there is a large grey area, where it's not clear to me whether the answer would satisfy your criteria. For example the Shimura-Taniyama conjecture was originally formulated in very imprecise terms, more like a heuristic, which people were extremely sceptical about. When the formulation got sharper, it seems like more mathematicians were willing to believe it. – 2011-06-16
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0@Alex: Or the Moonshine Conjecture, which was so "out there" when originally posed (though I don't know if it came to the status of 'expected to be true' before Borcherds proved it). – 2011-06-16
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0@Arturo: Why did you first point out that the "vice-versa" question is equivalent, but then edited to add it? – 2011-06-16
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0See the discussion at http://mathoverflow.net/questions/35468 . – 2011-06-16
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1@Arturo: I think the Monstrous Moonshine conjecture was expected by many to be true when Frenkel, Lepowsky, and Meurman constructed an infinite dimensional graded representation of the monster with graded dimension equal to the $j$ function (around 1984). – 2011-06-16
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0@Scott: Thanks! Just to be clear, I was expressing my ignorance on the subject, not my skepticism of the status of the conjecture. – 2011-06-16
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0@joriki: I pointed out that vice-versa was logically equivalent because 805801 was identifying that as the difference between his query and that in MO (it wasn't; the difference is that the query in MO was about incorrect proofs that had been widely accepted, not about expectations). But I think "thought to be true, but a counterexample was found" than "thought to be false, but a proof was found", so just in case, I added that to the body to clarify. – 2011-06-16