Let us first compare mathematics with art. As the historian and philosopher R. G. Collingwood said in his The Principles of Art, 1938
[it is not true that] … ‘all art is quite useless’, for it is not; a work of art may very well amuse, instruct, puzzle, exhort, without ceasing to be art, and in this ways it may be very useful indeed.
what makes it art is not the same as what makes it useful.
Similarly, mathematics is useful but what makes it mathematics is not the same as what makes it useful.
Mathematics has its own intrinsic needs that have to be addressed for it to stay alive.
Now let us compare mathematics with something unambigously useful: a cow. The cow is useful indeed, it gives us cream (single, doubled, clotted), butter, cheese, milk and skimmed milk — the list can be continued.
Applied mathematics can be compared with the cow’s udder — it produces milk.
Some branches of pure mathematics are best described as the cow’s immune system — they keep the cow alive.
The cow of course has other uses. To make a steak, it suffices to take a piece of cow and gently fry it to taste. What is a piece of a cow? A mathematician.
Financial industry, security sector, etc. are connoisseurs of a good steak. NSA advertises itself as the biggest employer of mathematicians in the USA.
Some people claim that pure mathematicians’ focus on “useless” artificial problems “they invent for themselves”. But let us look at geneticists’ obsession with a pretty useless creature: Drosophila melanogaster.
An article in Wiki devoted to it says:
The species is known generally as the common fruit fly or vinegar fly. Starting with Charles W. Woodworth’s proposal of the use of this species as a model organism, D. melanogaster continues to be widely used for biological research in studies of genetics, physiology, microbial pathogenesis and life history evolution. It is typically used because it is an animal species that is easy to care for, has four pairs of chromosomes, breed quickly, and lays many eggs”.
Very frequently, “famous” mathematical problems are means of concentrating effort of generations of mathematicians on development of methods of proof in particular areas of mathematics, they are drosophilas of mathematics. The Last Fermat Theorem is perhaps the most famous example. In some cases (and the Riemann Hypothesis is the archetypal case) they, however, have exceptional importance for mathematics as a whole.
Дети, любите корову – источник мяса!
[Children, love the cow – a source of meat!]