Communicating Mathematics: Surface Structures and Deep Structures

Abstract

A distinction is made between the surface structures (syntax) of mathematical symbol-systems and the deep structures (semantics) of mathematical schemas. The meaning of a mathematical communication lies in the deep structures — the mathematical ideas themselves, and their relationships. But this meaning can only be transmitted and received indirectly, via the structures; correspondence between deep and surface structures is only partial. Some resulting problems of communicating mathematics are discussed, and some remedies suggested.