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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... – DonAntonio Dec 7 '12 at 19:08
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). – joriki Dec 7 '12 at 19:17
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I asked a precise variant of this question here. – Mariano Suárez-Alvarez♦ Dec 7 '12 at 19:21
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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. – orlandpm Dec 7 '12 at 20:04
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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. – Aaron Mazel-Gee Dec 7 '12 at 20:33
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