Mathematics is immensely useful but what makes mathematics mathematics is not the same as what makes it useful.
Mathematics has its own intrinsic needs that must be addressed for it to stay alive.
Let us compare mathematics with a cow.
The cow is useful, it gives us milk (whole, low-fat, skim, fat-free, organic …), cream (single, double, soured, clotted …), butter (unsalted, salted, spreadable …), a variety of cheeses — 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 roast it to taste. What is a piece of cow? Mathematicians. Financial industry, security sector, etc. are connoisseurs of a good steak. NSA advertises itself as the biggest employer of mathematicians in the USA.
Pure mathematicians are sometimes accused on focusing on “useless” 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 the effort of generations of mathematicians on development of methods of proof in particular areas of mathematics, they are drosophilas of mathematics. In some cases (and the Riemann Hypothesis is the archetypal case) they have exceptional importance for mathematics as a whole. Дети, любите корову – источник мяса!