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Triangulation Conjecture Disproved
- tunesmith 12y agoAre 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. :)
- Xcelerate 12y agoPretty cool. I was actually wondering the other day if, for a given manifold, there is a general algorithm to randomly choose a point on it such that each point has an equal probability of being chosen.
- wolfgke 12y agoThe probability of some point to be chosen is 0 for any probability measure that has a density.
- teraflop 12y agoTechnically true, but not useful. The same is true for, say, choosing a real number uniformly at random on the interval [0,1), but nobody thinks twice about treating that as a well-defined sampling problem.
- wolfgke 12y ago> but nobody thinks twice about treating that as a well-defined sampling problem. This is what I wanted to show the OP: That it's not a well-defined sampling problem.
- colanderman 12y agoWell, obviously. Even a slightly generous reading of the GP's question would indicate that pretending he asked about an algorithm "such that, for subregions a and b of equal volume, P[a]/P[b] approaches 1 as the subregions' volume approaches 0" would give him the answer he was intuitively searching for.
- Xcelerate 12y agoThank you for clarifying that haha. I forgot that on HN one has to articulate all ideas as formally and rigorously as possible in order to prevent the pathological cases from becoming the centerpiece of a conversation.
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- Retric 12y agoSeems odd that they did not provide a counter example. My suspicion there is an incorrect assumption for non fractal structures. PS: This is really one of those cases where a paper does not mean much. You really need to dig through http://arxiv.org/abs/1303.2354 http://arxiv.org/abs/1303.2354 and find the flaw as there is no peer review. Though it supposedly is going to show up in Journal of the AMS so it's not junk and probably worth diging into.
- panic 12y agoHere is the actual paper: http://arxiv.org/pdf/1303.2354v4.pdf http://arxiv.org/pdf/1303.2354v4.pdf. It looks like this is their family of counter examples: A specific non-orientable five-dimensional manifold M5 with Sq1 ∆(M) ̸= 0 is constructed in [25]. Hence M5 is non-triangulable. To get non-orientable examples of non-triangulable manifolds in dimensions n > 5 we can take the product of M5 with the torus Tn−5. To get an orientable example in dimension 6 we can consider the non-orientable S1-bundle over M given by M ̃ ×Z/2 S1, where M ̃ → M is the oriented double cover; the total space of this bundle is orientable. To get orientable examples in dimensions n > 6, we can then take products with Tn−6.
- Grue3 12y agoIs there a reason why previously found 4-dimensional counterexample couldn't have been used as the basis of induction?