The Ground Riemann Broke

Even students of mathematics rarely have the opportunity to explore the kind of thinking that leads to ground-breaking achievements in their discipline.  I was struck, very recently, by how students in my calculus class would not likely reflect on how it was possible that the tedious arithmetic they were doing (solving equations involving clumsy fractions and roots), which could appear to have nothing to do with anything,  led them to a conclusion about how much human labor and capital investment would maximize the production of a particular product.  Their small calculations were somehow giving them information about very worldly activity modeled by solids and curves on a grid.  But this is not directly related to Riemann (perhaps Descarte and certainly LaGrange).

Yet I have often been excited by the kind of thoughtfulness that lead Bernhard Riemann to peel apart ideas that, before his approach, had no parts.  We first learn about the coordinates of a space as being the way to locate a point in that space.  On the plane, given the location of some central point, we can locate any other point by knowing its horizontal and vertical distance from that center.    The Cartesian coordinate system definitively connects the geometric properties of objects drawn on the plane with algebraic equations that represent those objects.  The relentless exploration of this mathematical space and the physical spaces of our lives led to increasingly generalized notions of space, object and dimension.  There is no one line of thinking one can follow.  But Riemann broke one of our most stubborn habits.

In my general searching for accounts of imaginative departures from the consensus, I found a few publications by a historian of math and science, Jose Ferreiro. In one paper in particular he says:

Many of the investigations about geometry in the 19th century, and especially on non-Euclidean geometry, were of a foundational character. Not so with Riemann: his main aim was not to axiomatize, nor to understand the new ideas on the basis of established geometrical knowledge (say, projective geometry), nor to analyze questions of independency or consistency – rather, he aimed to open new avenues for physical thought.

He is making a distinction here between the investigation of axioms in geometry like whether Euclid’s fifth postulate was independent of the others or, as had been determined, whether a different set of axioms could establish a valid non-Euclidean geometry.  Instead, one might say that what Riemann did was distill geometry into its component parts.

Riemann began by criticizing traditional geometry, grounded in axioms, finding the definitions of space and constructions in space and their relations “in darkness.”  He demonstrated that one could embed physical space into a more general concept which is now referred to as a manifold.  This is a mathematical space that can be made to resemble a Euclidean space of a particular dimension but, in its most general form, opens the door to less restricted ideas and rich mathematical landscapes.  The generality is a manifold extended by any number of components, the dimension of which is that number minus one.  Euclidean examples are the line and the circle, each a 1-dimensional manifold embedded in a 2-dimensional space (the Cartesian plane) and the plane and the sphere, each 2-dimensional manifolds in a 3-dimensional space.  But one can imagine a purely abstract space determined by all possible values a variable might take within certain constraints.

About these Riemann says:

It will follow from this that a [n-dimensional manifold] is capable of different measure-relations, and consequently that space is only a particular case of a triply extended manifolds. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions …, but that the properties which distinguish space from other conceivable triply extended manifolds are only to be deduced from experience.

It was an extraordinary insight with far-reaching consequences.  My interest here is to draw attention to the unlikely yet amazingly productive impulse to extrapolate the notions of position and measure or magnitude from their geometric origin (which has its own roots in spatial experience) and in this way plant the seeds for the revolutionary ideas in both math and physics (Relativity being one among others).

A translation of Riemann’s famous address can be found here. A comprehensive look at Riemann can be found in a book by Detlef Laugwitz. I’ll end with a statement from another paper by Ferreiros:

Riemann consciously avoided the image of Reason as the a priori source of knowledge. In his view, all knowledge arises from the interplay of experience broadly conceived (Erfahrung) and ìre ectionî in the sense of reconceiving and rethinking (Nachdenken); it begins in everyday experiences and proceeds to propose conceptual systems which aim to clarify experience going beyond the surface of appearances. Reason in the old sense is found nowhere…