The ideas of Hermann Grassmann in the context of the mathematical and philosophical tradition since Leibniz

Abstract

The only public recognition that the outstanding mathematician and linguist H. Grassmann (1809–1877) had among amthematicians in his lifetime was the award of the prize of the “Fürstlich Jablonowskischen Gesellschaft” for his work of 1846, Geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik. In his book he develops ideas that Leibniz had expressed in a letter of September 1679 to Huygens concerning the establishment of a calculus directly applicable to geometric situations. Leibniz had criticized both Euclid and Descartes because of their purely constructive and synthetic approaches to geometrical questions. Leibniz sought to develop a genuine analytic method that on the one hand should be really universal and on the other should operate through specific geometrical relations rather than algebraic equations. J. Echeverría has criticized the now commonly accepted view that Grassmann's Geometrische Analyse be legitimately seen as an extension of Leibniz's ideas. This disagreement provides a new opportunity to present the issue within the framework of broader mathematical and philosophical concerns. What had in fact changed was that the ontological foundation of classical epistemology was no longer valid during the 19th century. The conviction that thinking directly understands being itself, no longer existed. Scientific thinking now either is committed to positivistic empiricism, or insists on the “theory ladenness” of observation as well as of intuition, while at the same time trying not to fall back into classical ontologism.