Grassmann’s Vision

Hermann Grassmann is cast as a pivotal figure in the historical development of a universal geometric calculus for mathematics and physics which continues to this day. He formulated most of the basic ideas and, to a remarkable extent, anticipates later developments. His influence is far more potent and pervasive than generally recognized. After nearly a century on the brink of obscurity, Hermann Grassmann is widely recognized as the originator of Grassmann algebra, an indispensable tool in modern mathematics. Still, in conception and applications, conventional renditions of his exterior algebra fall far short of Grassmann’s original vision. A fuller realization of his vision is found in other mathematical developments to which his name is not ordinarily attached. This Sesquicentennial Celebration of Grassmann’s great book, Die Lineale Ausdehnungslehre [1], provides the opportunity for a renewed articulation and assessment of Grassmann’s vision. Grassmann is a pivotal figure in the historical evolution of a mathematical language to characterize human understanding of the physical world. Since quantitative concepts of space and time are fundamental to that understanding, the language is fundamentally geometrical and can best be described as a geometric calculus. For the most part the evolution of geometric calculus has been tacit and piecemeal, with many individuals achieving isolated results in response to isolated problems. Grassmann is pivotal in this historical process because he made it explicit and programmatic. In no uncertain terms, he declared the goal of his research on extension theory [1] as no less than the creation of a universal instrument for geometric research. Leibniz had clearly articulated that goal long before and blessed it with the name “geometric calculus.” However, Grassmann was the first to see clearly how the goal can be reached, and he devoted the whole of his mathematical life to the journey. When Möbius created the opportunity for Grassmann to align his own vision with Leibniz’s, Grassmann responded eagerly with an elaboration of his extension theory to show how their common vision can be realized [2]. Unfortunately, this seminal work failed to attract the recognition and attention he deserved, so Grassmann had to continue his mathematical journey alone, with only an occasional mathematician — Möbius, Hankel, Clebsch — joining him for a few steps along the way.