Why do pure mathematicians keep proving new theorems but don’t know their applications in real life?

My answer to a question in Quora: Why do pure mathematicians keep proving new theorems but don’t know their applications in real life?

I am a pure mathematician, but I have some experience of solving raw engineering problems, that is, problems where engineers have no vaguest idea what kind of mathematics could be applied. In one peculiar case I was using in my solution the classification of finite simple groups, the one of the most notorious achievements of the 20th century algebra — its original solution was spread over 100+ journal papers of about 10,000 pages in total (and contained gaps and holes which required additional hundreds of pages of proofs). It is hard to imagine anything more pure and remote from the so-called “real life” then the classification of finite simple groups.

I already said somewhere on Quora that

mathematics can be useful, but what makes it useful is not the same as what makes it mathematics.

Mathematics is a living organism which has to meet its own needs just to stay alive. It can be usefully compared with a cow. Cow is useful, for example, she gives us milk. Her udder definitely belongs to applied mathematics. But let us accept that the whole remarkably useful cow is applied mathematics.

The cow needs food and water, and air for breathing, etc. For production of milk, once in a year cow needs a rendezvous with a bull. We may perhaps compare pure mathematics with this bull.

Then I had commented a someone else’s answer:

Someone’s answer: There was a mathematician who believed math to be irrelevant if it had real world applications, so he devised a branch of mathematics using only on/off states. His name was George Boole, and the Boolean math he created is the basis of most of the electronic devices in use today. You don’t know what will become useful, but the math will be there for it.

My comment:

I disagree with your assessment of Boole. He designed what is now known as Boolean algebra as a way of checking and resolving logical arguments by some kind of arithmetic with True and False being values used in place of numbers. Prior to 19th century (when Boole lived) the use of logic was confined to Theology and Law, and Aristotelian logic taught to students (as a rule, of privileged classes) who were supposed to become priests or lawyers. As it is clear from his writing, Boole had aim to create logic for the masses, mental tools assisted by calculation on paper, for ordinary working people to understand, for example, court proceedings. He was, in nowadays terminology, a social activists: fought for improvement of working conditions for shopworkers, founded schools, credit unions, etc. The purpose of logical calculus that he invented was very pragmatic.