Has anything in mathematics ever been proven wrong?

[Holocene]

Not something random like 2 + 2 = 5. I mean somethung that was once widely accepted?

Lots of things in science have later been proven wrong. What about math?

 

[quasar987]

Often times, a conjecture is made and believed to be true by a vast group of experts. But then a counterexample is found that proves the conjecture to be wrong.

 

[Mute]

I think Von Neumann once wrote down a proof - or something, I can't quite remember - to do with Quantum Mechanics and entanglement, which John Bell showed was in error when he derived what is now known as Bell's Theorem. Von Neuman was a pretty big mathematician by then, so assuming him to be wrong was not the default position one usually took, hence the tale has become one of those "don't just assume the experts are always right - you have to check their work too!" tales. It doesn't say much about it in the Bell's theorem wikipedia article. I think I might have read about it in the book "Entanglement".

 

[HallsofIvy]

A few months after the "four color theorem" was given to the Royal Society, the mathematician Sylvester gave a proof. It was 10 years later that it was shown that his proof was invalid- one 'fact' he used in the proof was not true. Of course the four color theorem itself is true- proven about 100 years after Sylvester's "proof".

 

[LukeD]

And who could forget the once fairly widely held belief that Euclid's parallel postulate could be derived from the rest? (or at least that it was obviously true given the rest of the postulates and that no system could be conceived where it was false)

 

[mathman]

In 1900 Hilbert gave a famous lecture listing 20(?) major problems for mathematicians to work on. One was to prove the consistency of mathematics (or smething like that). In any case during the 1930's Godel showed that any mathematical system strong enough to do arithmetic could not be proven consistent or complete.

 

[CRGreathouse]

OK, so far we have examples of mistaken proofs (Sylvester's 'proof' of the 4-color theorem), widely-accepted conjectures that were false (Mertens'?), and widely-believed open problems that were undecidable (parallel postulate, decidability for 'strong' systems, AC). Are there any examples of formal systems that have been seriously studied, but later shown to be inconsistent?