Fields, Circles and the Wisdom of Immaterial Objects

I have spent some time pointing to milestones in the history of modern mathematics where a conceptual shift produces provocative new thought – as when Riemann gave a new foundation to geometry, or when Cantor brought precision to the notion of countability. Modern mathematics, partnered with physics, increasingly refines what the human mind can perceive.  In this light it is easy to understand the nineteenthcentury German physicist Heinrich Hertz (first to detect electromagnetic radiation) when he said:

One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.

Frank Wilczek found this worth repeating in The Lightness of Being when he discusses the potential implications of metric field mathematics in physics. He explains that general relativity can be described using either of two mathematically equivalent ideas: curved space-time or metric field.  He refers to curved space-time as the geometric version of the field view. But the field view, Wilczek explains, makes Einstein’s theory of gravity look more like other theories of fundamental physics and the hope is that its use will help usher in the sought after unification of the four physical forces or interactions.  But I have also thought that the role of something much simpler, something like trigonometry, can illustrate the way, as Wilczek puts it, “our concepts are not just our products but also our teachers.”

Trigonometry is grounded in the simple idea of ratios among lengths, in particular, of line segments forming right triangles.  The properties of these relationships were documented early in our history, in ancient Greek mathematics and in early Hindu astronomy. Trigonometric thoughts often start with the circle and Greek mathematicians studied the properties of the angles opposite various chords of a circle. They proved theorems that are equivalent to modern trigonometric formulas. The modern sine function was also defined in an ancient Indian treatise using the shadows cast by a sundial.

Various facts and relationships among these ratios were calculated and widely used in the analysis of the ground we walked on, as well as distances in the sky.  Most of us begin learning the fundamental ancient facts in school, using the sides of right triangles and those familiar words: opposite over hypotenuse, adjacent over hypotenuse, and opposite over adjacent.

But the growing idea of a function, a rule of correspondence between two values, which became more and more sharply defined in the seventeenth and eighteenth centuries, took hold and became the general way we looked at relations.  Trigonometric ratios are now understood as functions of real numbers, where the radian measure of an angle is paired with the sine, cosine or tangent value of that angle.  The idea is introduced in textbooks with the unit circle, where we imagine wrapping the real number line infinitely many times around the circumference of the circle.  The x,y coordinates of the points on the circle, hit by the ray which describes the angle, correspond to the cosine and sine values of that angle.  These become the y values of the sine and cosine curves when they are plotted as waves.

When the circle is drawn on the complex plane, the same kind of configuration describes the relationship between complex numbers and trigonometric ratios, and this exposes the relationship between the exponential function and trigonometric relationships.  Circles, circles, and more circles.  The fact that other functions could be represented by a trigonometric series (a sum where each term involves a trigonometric function), has also had a major effect on analysis and physics and found application in the pure mathematics of number theory.  In physics, solutions to Schrodinger’s equation, an important differential equation describing non-relativistic quantum mechanics, often involve sines and cosines.

Trigonometric ideas clearly contained more than they were initially used to describe.  Their relationship to the circle kept unfolding and their expression of periodicity, together with their analytic development, continues to support even the most modern ideas in physics.    There can be no doubt that we got more out of them than we originally put into them.  This is just a small example of the way a purely conceptual object may have its own nooks and crannies, ones that actually contain hidden jewels.  It raises intriguing questions about the nature of mathematical thought, how it happens, and how it captures more than is immediately apparent even to its authors.