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One of my profs mentioned that sometimes people formulate theories about some type of object, but then later realize that those objects do not exist. Can someone given me an example of such a theory? I know this is vague, but I hope it makes sense.

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    Out of the blue, and without being a scholar *at all* in these matters, what about the mathematics of string theory? So far it seems to work fine yet the physics world hasn't yet found some physical proof of strings...2012-12-07
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    Don's comment demonstrates that you should clarify whether you mean mathematical objects (i.e. the formulated theory is later shown to be inconsistent) or physical objects (i.e. the formulated theory turns out not to correspond to anything in physical reality).2012-12-07
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    I asked a precise variant of this question [here](http://mathoverflow.net/questions/115735/groups-that-do-not-exist).2012-12-07
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    In a weak sense, the set of all sets.2012-12-07
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    @DonAntonio I believe string theory is both mathematically consistent and consistent with all physical observations so far. However, the acid test for a physical theory is that it yields _new_ predictions which are testable. For example, General Relativity is much more complicated of a theory than Newtonian gravity, but it explains new and testable results (like gravitational lensing). OP's question is different from this (and math differs from physics) because it is deductive not inductive -- we don't use Occam's razor to select "correct" theories in math.2012-12-07
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    There's some widely-told horror story about a student who was up at the board, in the process of defending their thesis, when one of the examiners (Milnor, iirc?) proved that there were no nontrivial examples of whatever objects the thesis had set out to study.2012-12-07
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    Well @orlandpm, I don't think physics uses Occam's razor to *select* correct theories, either, though they could probably *propose* some by this reasoning. And the Principle of Induction could get deeply offended if it read your general affirmation that mathematics is deductive and not inductive (and in fact, not only the PI, imo). Anyway, and after reading a little at Mariano's past question and the comments and answer there, The Feit-Thompson's theorem proof could be used to give an idea about this.2012-12-08
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    There are a couple examples in this question about PhD's going horribly wrong because the examples that exist are either nonexistant or trivial: http://mathoverflow.net/questions/53122/mathematical-urban-legends2012-12-08
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    See also [this mathoverflow question](https://mathoverflow.net/questions/182006/what-is-the-most-useful-non-existing-object-of-your-field).2018-09-19

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