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What are some results which were widely believed to be false, but were later to be shown to be true, or vice-versa?

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What 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"... –  Arturo Magidin Jun 15 '11 at 19:42
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Discussing on meta regarding reopening: meta.math.stackexchange.com/questions/2358/…. –  Yuval Filmus Jun 15 '11 at 22:49
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This thread on MO 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. –  t.b. Jun 16 '11 at 2:45
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@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. –  Arturo Magidin Jun 16 '11 at 3:48
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@805: +1 for your ability to navigate through the complicated web of etiquette, and eventually pose a good question. –  The Chaz 2.0 Jun 16 '11 at 3:49

5 Answers 5

Shing Tung Yau describes that there was a general skepticism among mathematicians about the Calabi conjecture. He presented a proof that it was false to an informal audience which included Eugenio Calabi. On being contacted by Calabi to write him the arguments. Yau tried to make his assertions rigorous, found a mistake in his own proof, and in trying to correct it, ended up proving it.

This is described in detail by Yau in his book The Shape of Inner Space

On why Yau and others were skeptical. pp. 103-104

... but in the early 1970s, I (among many others) still needed some convincing that it was more than a molehill. I didn’t buy the provocative statement he’d put before us. As I saw it, there were a number of reasons to be skeptical. For starters, people were doubtful that a nontrivial Ricci-flat metric—one that excludes the flat torus— could exist on a compact manifold without a boundary. We didn’t know of a single example, yet here was this guy Calabi saying it was true for a large, and possibly infinite, class of manifolds.

[...]

I was also wary for some additional technical reasons. It was widely held that no one could ever write down a precise solution to the Calabi conjecture, except perhaps in a small number of special cases. If that supposition were correct— and it was eventually proven to be so—the situation thus seemed hopeless, which is another reason the whole proposition was deemed too good to be true.

On proving it. pp 106

Calabi contacted me a few months later, asking me to write down the argument, as he was puzzled over certain aspects of it. I then set out to prove, in a more rigorous way, that the conjecture was false. Upon receiving Calabi’s note, I felt that the pressure was on me to back up my bold assertion. I worked very hard and barely slept for two weeks, pushing myself to the brink of exhaustion. Each time I thought I’d nailed the proof, my argument broke down at the last second, always in an exceedingly frustrating manner. After those two weeks of agony, I decided there must be something wrong with my reasoning. My only recourse was to give up and try working in the opposite direction. I had concluded, in other words, that the Calabi conjecture must be right, which put me in a curious position: After trying so hard to prove that the conjecture was false, I then had to prove that it was true. And if the conjecture were true, all the stuff that went with it—all the stuff that was supposedly too good to be true—must also be true.

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$|\mathbb{R}|=|\mathbb{R}^2|$, i.e. there exists a bijection from the real line to the plane.

Also, it was believed that there don't exist wild embeddings $\mathbb{S}^2\hookrightarrow\mathbb{R}^3$ until Alexander found his horned sphere.

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This may be a bit tangential to your question, but Gödel's Incompleteness Theorem probably deserves mention here. It had been widely believed since at least the beginnings of Hilbert's program that a decision procedure for all mathematical questions could be created. Gödel showed that this is impossible, a big surprise at the time.

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I'd think any statement classified as a "paradox" would qualify here. The paradoxes of material implication, Russell's paradox, the Banach-Tarski paradox, the cardinality of R exceeding that of N and Q, etc.

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This is basically what "paradox" means: "para" is "contrary to" and "dox" is "opinion". etymonline.com/?term=paradox –  Najib Idrissi Jun 6 '13 at 20:38

Complexity theory is full of such nice things. IP=PSPACE was very surprising for its time as everyone believed IP is not as strong as PSPACE.

Barrington's theorem was very surprising as it was believed branching programs would admit much stronger lower bounds.

The Immerman–Szelepcsényi theorem of an equality regarding space complexity (NL=coNL) that was believed to be false (because of our intuition regarding time complexity, where we believe NP to be different than coNP).

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