"Computers are useless. They can only give you answers." -- Picasso
"The use of a program to prove the 4-color theorem will not change mathematics - it merely demonstrates that the theorem, a challenge for a century, is probably not important to mathematics." -- PerlisOnce we can program a computer to solve a problem, that problem becomes boring.
It seems that when something is programmable, computable, it immediately loses its appeal as an endeavour for humans. Something that can be solved by an unthinking machine is no longer worthy of study by humans. (What remains to be of interest to humans are better ways to solve the same problem - write the same program better, more efficiently, more elegantly - but these seem second-order, once we have solved the problem.)
Of course this question opens a whole can of worms about what it means to be computable - e.g. does it have to be economically computable, or does it have to be computable before the end of the universe? But in general, I think that programming is interesting, because it touches upon the interesting issue of what it means to be boring.