TITLE: Mistakes in mathematics, false illusions about conjectures
QUESTION [95 upvotes]: Since long time ago I have been thinking in two problems that I have not been able to solve. It seems that one of them was recently solved. I have been thinking a lot about the motivation and its consequences. Mostly because people used to motivate one of them with some very interesting implications. My conclusion however, is that there is a mistake in the motivation of the problem, and that, while being a really interesting result, it does not make any sense in the setting in which is formulated. As my opinion is not relevant compared to one of the experts in the area, I do not say anything. 
My question is if you can provide me some examples of conjectures that were believed to be interesting in the mathematical community because of a specific reason, but that once having the proof, people realized that the reason to motivate the problem  was not truly related to its solution. Or in other words, the solution of the problem gives no clues about the original motivation.

REPLY [13 votes]: In knot theory, the motivation came from outside of math but it is a good example of what you discuss. It was first developed to help study atoms, following an atomic model by Thompson; this model was later completely discarded but knot theory took a life of its own and is now a thriving field of mathematics.