Full Idea
Mathematical structures are characterised axiomatically (as implicit definitions), or they are defined in set theory.
Gist of Idea
Mathematical structures are defined by axioms, or in set theory
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 8.3)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.255
A Reaction
Presumably earlier mathematicians had neither axiomatised their theories, nor expressed them in set theory, but they still had a good working knowledge of the relationships.