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 surface structures; correspondence between deep and surface structures in only partial. (…) The power of mathematics (…) lies in its conceptual structures (…), its organized network of ideas. These ideas are purely mental objects, invisible, inaudible, and not easily accessible even to their possessor. Before we can communicate them, ideas must become attached to symbols. These have a dual status. Symbols are mental objects, about which and with which we can think. But they can also be physical objects—marks on paper, sounds—which can be seen or heard. These serve both as labels and as handles for communicating the concepts with which they are associated. Symbols are an interface between the inner world of our thoughts, and the outer, physical world.