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On the other hand, the Turing machine (a machine capable of universal computation.. ie a computer) was invented as a method of solving a highly theoretical, seemingly useless mathematics problem called the Entscheidungsproblem, in Hilbert's list:

10. Determination of the solvability of a Diophantine equation. Given a Diophantine equation with any number of unknown quantities and with rational integral coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers. The Entscheidungsproblem is solved when we know a procedure that allows for any given logical expression to decide by finitely many operations its validity or satisfiability ... The Entscheidungsproblem must be considered the main problem of mathematical logic.

Computers may seem obvious in hindsight but it was formally defined to solve a theoretical mathematics problem by the mathematician Alan Turing.