Mathematical Abstraction as Unstable Translation Between Concrete Presentations

Abstract

This paper suggests that mathematical abstraction be conceived not as the construction of some mental or ideal object, but as a practice of translation between various symbolic systems. This view aims to capture the intangibility of abstraction by framing it as an imperfect and somewhat unstable translation between symbolic systems. The imperfection and instability of the translation means that it does not establish an invariant core, but rather relates aspects of various symbolic systems in a dynamic and possibly even inconsistent manner. I support my argument with examples from mathematics education and the history of mathematics.