Grassmann ’ s epistemology : multiplication and constructivism 1

Grassmann’s epistemological insights have attracted less attention than his mathematical results. Mathematicians have considered him as a precursor of modern mathematical theories – vector algebra, exterior algebra, Clifford algebra, and geometric algebra –, paying attention to the development of mathematical results rather than to the philosophical differences between Grassmann’s own project and modern mathematical reconstructions [cf. Cartan 1908; Hestenes 1986]. Historians of mathematics have analyzed the reception and influence of Grassmann (see for example the papers collected in [Schubring 1996]), and philosophers’ attention has been driven especially to the Introduction to the 1844 edition of Ausdehnungslehre, or to a comparison with other philosophers. This paper will follow a different path, namely an inquiry on the reciprocal influences of Grassmann’s mathematical and philosophical insights, approaching Grassmann’s epistemology from the perspective of his mathematical work. The starting point will be an analysis of the notion of product between extensive magnitudes, to which the philosophical literature has paid insufficient attention. Several works of Grassmann will be examined, including the two editions of Ausdehnungslehre [A1, A2], Theorie der Ebbe und Flut [EBBE], Kurze Uebersicht über das Wesen der Ausdehnungslehre [Grassmann 1845], Geometrische Analyse [PREIS], Sur les différents genres de multiplication (SD) [Grassmann 1855 seq.], and Der Ort der Hamilton’schen Quaternionen in der Ausdehnungslehre (HQ) [Grassmann 1877]. The second section of the paper will be devoted to a comparison with three vectorbased systems: vector analysis, exterior algebra, and geometric algebra. Considering similarities and differences with respect to these mathematical theories, the attention will be directed to some philosophical issues, such as the question of the homogeneity of the elements, the consideration of the domain as having a fixed or a variable number of dimensions, and the general properties of multiplicative operations. I will claim that there is a certain continuity in Grassmann’s epistemological program – contrary to the common tendency in the literature to emphasize the elements of discontinuity between A1 and A2. In the last section of the paper some aspects of Grassmann’s epistemology – the difference between numbers and magnitudes, the relation between geometry and extension