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ORIENTATION

## NEW PAGE: This page has been superseded by a new Riemann site. Please visit: science.larouchepac.com/riemann.

Lyndon LaRouche, in almost all of his larger writings on economics, has referenced the importance of his understanding of Riemann's work in making his own breakthroughs in economic method. Usually considered a mathematician in popular literature, Bernhard Riemann's work sets out the limits of mathematical reasoning, demonstrating that even such seemingly mathematical questions as the nature of space, are properly the province of physics. This pedagogy package unites LPAC material on Riemann in one location. For this, there are three main parts of Riemann's work.

## Habilitation Dissertation

In universities in many parts of the world (including Germany in Riemann's day), a certification, known as *habilitation*, beyond the doctorate was required to become a university instructor. On June 10, 1854, Riemann delivered his habilitation lecture at Göttingen, *Über die Hypothesen welche der Geometrie zu Grunde liegen* (On the Hypotheses that Underlie Geometry). It was a shocking presentation that demonstrated the workings of curved spaces, in a manner similar to curved surfaces. Among the many possible curvatures of space, the angles in a triangle could be less than, equal to, or greater than 180 degrees in different locations. Although the flatness of space was generally considered as a given, Riemann showed that this was actually a *hypothesis*. Among possible spaces, the only way to choose (or conceptualize) the right one would come not from mathematical theorizing, but from hypothesizing the physical principles whose interrelation *is* space. The basis of geometry is not mathematics, but rather physics.

Here, we have:

- Video: Presentation on Riemann's lecture
- Text: Riemann's original 1854 report
- Video: Interlude announcing the then-upcoming video on the habilitation dissertation.

## Abelian Functions

In 1857, Riemann published a report on the theory of Abelian functions, in which report he saw the opportunity to defend the existence of true ideas (metaphor) in science, despite the tendency of mathematical language to kill such thinking. Using Dirichlet's principle, he was able to transform complex functions from a mathematical subject into a physical one, and saw in Abel's series of functions, an analogy to the continuously developing universe we inhabit and shape.

- Video: Potential and Metaphor
- Video: Abelian Functions
- English translation of Riemann's
*Theory of Abelian Functions*: Part I · Part II

## Series of Four Presentations on Non-Quantitative Change

In his habilitation dissertation, Riemann hints at geometries in which quantitative measurements are impossible. The *topological* approach he takes in treating Abelian functions is an example of higher complexities rather than larger magnitudes. This makes it a necessary field to study for economics, where non-quantitative change is the essence of economic progress. This four-part series of presentations introduces non-quantitative changes, and goes on to develop the nature of complex functions, basic topology, and Riemann's application of topology and Dirichlet's principle to create a truly free means of dealing with complex functions, based on the space they implicitly live in, rather than as concatenations of mathematical symbols:

- Video: Georg Cantor and Non-Quantitative Change – are all comparisons quantitative? How can we represent qualitative changes?
- Video: Complex Functions – introduction to complex magnitudes and the
*field*of economics. (extras) - Video: Riemann Surfaces – how singularities determine spaces (extras)
- Video: Combining Topology and Dirichlet – putting it all together: Riemann's treatment of Abelian functions, and an overview of
*metaphor*, the means by which we determine the principles that shape reality

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