Originally Posted by IMSingular

Well. Some times you have to attempt to solve problems you know that there is high probabilty that you may not solve at all. Reiman's hypothesis, for example, hasn't be proved yet but many have tried to solve it and couldn't do it in their life time. There was, of course, serendipity effect, and they contributed immensly to the theory of complex variables that might have contributed to how to design aeroplane wings and solve problems in electromagnetism.

There are discoveries made by Hawking in differential geometry- he was the first one to take that approach to General Theory of Relativity- which may have implications in how to solve nonlinear programming problems in economics and finance not just in physics.

It is not necessary that you solve a problem in your life time. Mere intellectual activity is good enough. Ramanujan, for example, wrote down many a identities which neither he nor any one new how to prove and they were proved decades after his death, and some of the stuff is useful in string theory.

Some times just writing down problems coherently itself requires great intelligence, e.g. Hilbert's 24 problems that even he didn't know how to or whether they will be solved at all, but gave a road map to many a mathematicians.

Then there are discoveries left in obscurity until a need for their use arises and they are rediscovered. Gauss discovered Fast Fourier Transform but it was rediscovered by Tuckey and Cooley. Hamiton invented quaternion but Pauli reinvented them to use them in Quantum Mechanics.