Syntax, Semantics, and the Problem of the Identity of Mathematical Items

Abstract

The items of mathematics, such as the real line, the triangle, sets, and the natural numbers, share the property of retaining their identity while receiving axiomatic presentations which may vary radically. Mathematicians have axiomatized the real line as a one-dimensional continuum, as a complete Archimedean ordered field, as a real closed field, or as a system of binary decimals on which arithmetical operations are performed in a certain way. Each of these axiomatizations is tacitly understood by mathematicians as an axiomatization of the same real line. That is, the mathematical item thereby axiomatized is presumed to be the same in each case, and such an identity is not questioned. We wish to analyze the conditions that make it possible to refer to the same mathematical item through a variety of axiomatic presentations.