What is 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 necessarily also with their smooth topological structure? Is the real field necessarily distinguished as a fixed particular subfield of the complex numbers? Do we understand the complex numbers necessarily to come with their rigid coordinate structure of real and imaginary parts?

These different perspectives ultimately amount, I argue, to mathematically inequivalent structural conceptions of the complex numbers, conceptions exhibiting different symmetries and thus also different automorphism groups. The additional structural features of the richer conceptions, after all, with the topology, the particular copy of ℝ as a distinguished subfield, or with the coordinate structure, are not determined uniquely by the algebraic field structure alone, and the coordinate structure is not determined uniquely by the topology. The automorphism groups of the conceptions are vastly different—the complex field admits crazy wild automorphisms, while the complex field over ℝ has only complex conjugation as a nontrivial automorphism, and the complex plane is rigid, having no nontrivial automorphisms at all. Which of these mathematical structural conceptions do we regard ultimately as the essential structure of the complex numbers? Indeed is there any one such essential structure?

It turns out that mathematicians do not agree—they do not all have the same conception of the complex numbers. Discussing the issue in various mathematical venues, I found a range of views, split roughly equally amongst the perspectives I discuss in this paper. I should like to discuss these various conceptions and then explain how they engage with several aspects of the philosophy of structuralism in the philosophy of mathematics.

Different perspectives on the complex numbers

To my way of thinking, there are at least four natural perspectives to take on the essential structure of the complex numbers, which I shall refer to under the following slogans.

• Analytic: The complex field ℂ over ℝ

• Smooth: The topological complex field

• Rigid: The complex plane

• Algebraic: The complex field

Two of these perspectives, it will turn out, arise from what amounts to the same underlying structural conception of the complex numbers (do you see which two?), and so those two perspectives are equivalent in this sense. Ultimately, therefore, we have here only three different perspectives in play. Let me elaborate on them.

Please enjoy this extended essay on the topic of the complex numbers. We shall see how our different various conceptions of them ultimately engage with several issues concerning the philosophy of structuralism in the philosophy of mathematics.

As a special treat, toward the end of the essay we shall see how the complex numbers once made a key appearance in the midst of a contentious philosophical dispute, with many philosophers ultimately taking it decisively to refute what had seemed a promising position in the philosophy of mathematics.