Mathematicians disagree on the essential structure of the complex numbers

I discuss several commonly held perspectives on the complex numbers and explore how their differences engage with several aspects of structuralism in the philosophy of mathematics.

Introduction

How are we to think of the complex numbers? What I mean is, with what fundamental structure at bottom do we take the complex numbers to be endowed? In short, what is the essential structure of the complex numbers? They form the complex field, of course, with the corresponding algebraic structure, but do we think of the complex numbers necess…