"Classical" algebra dealt with "concrete objects": real or complex numbers, polynomials with complex coefficients or specific groups of transformations. "Modern" algebra replaced these concrete objects by elements of a set, whose nature is irrelevant and whose relationships to each other are specified by axioms. This new "abstract" algebra studied sets endowed with one or more operations whose properties are deduced from axioms. As a result, algebra attained unprecedented abstraction, clarity and generality.

Disciplines dealt with in modern mathematics include vector spaces, matrices and linear spaces, number theory and group theory.