I don’t think it’s actually possible to answer the question in the title of this post, but I still believe it’s worth asking. We’ve thought of things ‘hidden under a microscope,’ or obscured by great distances, but in mathematics when something is hidden, it’s because we haven’t been able to imagine it yet. And when we do, we often get there by following the implications of previously imagined possibilities. The way that mathematics can direct our imagination is certainly demonstrated in the history of the complex number. It is worth looking at how they were first imagined and then how far from this came their significant impact.

Let’s start at the end, so to speak, with what Riemann had to say about complex numbers:

The reason and the immediate purpose for the introduction of complex quantities into mathematics lie in the theory of uniform relations between variable quantities which are expressed by simple mathematical formulas. Using these relations in an extended sense, by giving complex values to the variable quantities involved, we discover in them a hidden harmony and regularity that would otherwise remain hidden.

This I took from the book Riemann, Topology and Physics by Michael I Monasyrsky.

Now lets go to the beginning. As a purely numerical manipulation, the square root of a negative number was first written down to demonstrate the impossibility of splitting the number 10 into two parts, the product of which was 40 (shown to be 5 + the square root of -15 and 5 – the square root of -15). However, after it was written down, Italians of the Renaissance found that they could actually use these strange expressions to find real solutions to cubic equations.

As late as the early 19^{th} century, many mathematicians could not accept the use of these numbers. The general sense was that mathematics was not the study of conceptual truths but of real truths, ones that could be thought of as constructions of intuition. In this regard, Gauss defended the complex number by saying “the arithmetic of the complex number can be given a most intuitive representation.” (See, among other histories, The Shaping of Arithmetic by Catherine Goldstein).

There is a very nice animation of the way the value or meaning of the complex number can be hidden here.

Gauss proposed a more abstract theory of magnitudes where one measured not substances but relations among substances. Riemann’s study of functions of a complex variable led him to describe what we now call a Riemann Surface and likely motivated Riemann’s most general notion of a manifold. Riemann’s notion of manifold also gave new definition to the ideas of space, measurement, and geometry. Many would agree, his reworked foundations for geometry set the course for modern mathematics. And all of these realizations are clearly manifest in the significant role that the complex number plays in physics and engineering.

This is the story of the relentless unwrapping of an imagined object, and finding what was yet hidden – like the complex roots of a polynomial, or the properties of functions of a complex variable and their relationship to real valued functions, or the Riemann Surface and the shapes that space can take.

The relationship between the simple idea of number, that we all grasp quickly, and these unfamiliar but powerful conceptual worlds is completely unexpected. We are digging into a cognitive invention and finding more than we seem to have put there. Where was the hidden hidden? This must have something to say about the nature of what we call cognition.

Great blog post. Really Great.