Has anything in mathematics ever been proven wrong?

  • Thread starter Holocene
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In summary, many widely accepted beliefs and proofs in mathematics have later been proven wrong or invalid. Examples include Von Neumann's proof in quantum mechanics, Sylvester's proof of the four color theorem, and the belief that Euclid's parallel postulate could be derived. Additionally, Godel's work showed that no mathematical system strong enough to do arithmetic can be proven consistent or complete. This highlights the importance of constantly checking and reevaluating established theories and beliefs in mathematics.
  • #1
Holocene
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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?
 
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  • #2
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.
 
  • #3
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".
 
  • #4
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".
 
  • #5
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)
 
  • #6
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.
 
  • #7
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?
 

1. Has mathematics ever been proven wrong?

Yes, there have been instances where mathematical theories or concepts have been proven wrong. This can happen when a flaw is found in a proof or when new evidence or discoveries contradict previous mathematical ideas.

2. What is an example of a mathematical theory that was proven wrong?

One famous example is the Euclidean parallel postulate, which stated that parallel lines never intersect. This was proven wrong by mathematician Nikolai Ivanovich Lobachevsky, who developed non-Euclidean geometry, where parallel lines can intersect.

3. How does mathematics correct errors or mistakes?

Mathematics is a self-correcting field, meaning that errors or mistakes are usually caught and corrected through peer review and collaboration among mathematicians. If a mistake is found in a proof or theory, it is revised or discarded in favor of a more accurate explanation.

4. Can mathematical proofs be wrong?

Yes, mathematical proofs can be wrong. Even the most celebrated mathematicians have made mistakes in their proofs, and it is important for other mathematicians to carefully review and critique them to ensure their accuracy.

5. How do mathematicians ensure the accuracy of their work?

Mathematicians use rigorous methods, such as logic and deductive reasoning, to ensure the accuracy of their work. They also rely on peer review and collaboration to catch and correct any errors or mistakes in their proofs or theories.

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