[Math] Results that were widely believed to be false but were later shown to be true

Question

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

Answer 1

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.