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Are there historical examples of conjectures widely believed to be true and then proven to be false? I suppose Gödel's incompleteness theorems and Hilbert's se
by tunesmith 12y ago
Are there historical examples of conjectures widely believed to be true and then proven to be false?
I suppose Gödel's incompleteness theorems and Hilbert's second problem is along those lines, but I'm not sure Hilbert's problem was regarded as a conjecture.
- jonnybgood 12y agoThere is the two sentence counterexample to Euler's conjecture. http://www.openculture.com/2015/04/shortest-known-paper-in-a-serious-math-journal.html http://www.openculture.com/2015/04/shortest-known-paper-in-a...
- tunesmith 12y agoThanks, that's awesome!
- aaron695 12y agoI'd go with no. I think there's some classic examples people pull out, but whether the relevant community actually thought them to be correct but unproven, I'd like to know. I find this an interesting topic, the concept of we 'know' this is true but can't prove it yet. I wonder if the idea of 'absolutes' sets back Mathematics sometimes. But maybe there are whole fields that rely on yet unproven but assumed true ideas? Complex engineering for instance relies on 'this is probably true'. They make catastrophic mistakes, but they just pick themselves up and move on. Unless they are NASA I guess.
- namlede 12y agoOne field that relies on unproven but assumed true mathematical conjectures is cryptography. Provably secure crypto usually assumes that P != NP (it also incorporates further assumptions such as the intractability of integer factorization).
- ccvannorman 12y agoNewtonian physics dominated the Universal model until Einstein and relativity (I believe?)
- ajkjk 12y agoNot the same. Physical models don't really live in the same space as mathematical conjectures. They are expected to be approximations, valid within some domain. Newtonian physics is still 'correct', in that is a valid approximation, because that's all a model ever is (except for the imagined 'theory of everything').
- vanderZwan 12y agoWhile it's true that this is the modern interpretation of what physics is, I'm not sure if that always was the case. It might be precisely these repeated breakthroughs (and I'd guess relativity and quantum mechanics especially) that are the cause of this attitude. I recall being taught by my physics professor that Galileo never actually dropped two lead balls to test his theory - he reasoned that "heavier objects fall faster" made no sense because it implies that connecting two objects would make both fall faster.
- BioGeek 12y agoSeveral good answers can be found on http://mathoverflow.net/q/95865 http://mathoverflow.net/q/95865 and http://math.stackexchange.com/q/45594/14018 http://math.stackexchange.com/q/45594/14018
- CurtMonash 12y agoThe most famous example is probably Euclid's Fifth Postulate, aka the Parallel Postulate, which states that given a line and a point, there's a single line through the point parallel to the first line. http://en.wikipedia.org/wiki/Parallel_postulate http://en.wikipedia.org/wiki/Parallel_postulate For 2000 years, it was conjectured that it followed from Euclid's other postulate. Lobachevsky, however, disproved it. (As with many 19th Century results, there are questions of precedence -- in part because Gauss claimed to have thought of everything first himself -- but his name is most commonly attached to the result.) Riemannian Geometry depends on the Parallel Postulate being false. General Relavity depends upon Riemannian Geometry.
- CurtMonash 12y agoAlso, Mumford's series of papers on Pathologies probably disproved some conjectures along the way. I'm not clear on the details.
- CurtMonash 12y agoThe best example after the Parallel Postulate may be a conjecture by, of all people, Fermat. He thought all numbers of the form 2^(2^n))+1 were prime. http://en.wikipedia.org/wiki/Fermat_number http://en.wikipedia.org/wiki/Fermat_number However, I'm not aware that he ever claimed to have proved the result, "wonderfully" or otherwise. :)