The idea goes back to the 1930s when algebraic geometers understood that points
of an algebraic variety "are" maximal (or prime) ideals of the ring of regular functions
on it. The counterpart of this in analysis is the theory of commutative Banach algebras of Gelfand. Later Grothendieck revised  the foundations of the whole algebraic geometry based on this idea, and it spread to many other areas of mathematics.