Not just Philosophers, Mathematicians too

Philosophers sometimes work on idiosyncratic problems which members of award and grant committees fail to see the value of. What about this for a grant proposal? "I would like to spend the next two semesters examining whether one can construct a feasible version of Lewis' (1968) counterpart theory but without postulate P2". I would immediately give money to this project but for obvious reasons I do not recommand that you submit it as your next grant proposal (unless you know I am on the committee). Speaking of failure to see the value of others' projects, here are some projects in mathematics which I bet many grant committees would turn down (or would have turned down). The projects are described in more detail here.

In 1896 Hadamard and de la Vallee Poussin proved that the chance that a random number nearby some large number n is prime is about 1 / ln(n), where ln(n) denotes the natural logarithm of n.

In 1930, L.G. Schnirelmann proved that every even number n greater than or identical to 4 can be written as the sum of at most 300,000 primes (this is my favorite).

In 2002 Liu Ming-Chit and Wang Tian-Ze proved that every odd number n greater than 2 x 10^{1346} is the sum of three primes. Meanwhile Oliveira e Silva is running a distributed computer search that has verified the weak Goldbach conjecture, viz. that all odd numbers greater than 9 are the sum of three odd primes, up to n > 4 x 10^{17}. So the gap is closing. Perhaps in a few years (with the faster computers the future will bring) the gap will close, and the conjecture will be proven (if the grant committees are willing to chip in).