Justus Grassmann's contributions to the foundations of mathematics: Mathematical and philosophical aspects (Q1975528)

It is a commonplace that the Grassmann brothers Robert and Hermann (Günther), pioneers in the development of abstract mathematics, vector analysis and lattice theory, were much influenced by their father Justus, mathematics teacher at the Gymnasium in Stettin and crystallographer. It is also known that via \textit{Justus Grassmann} a relation can be established between the younger Grassmanns and the idealistic philosopher \textit{Friedrich Wilhelm Joseph Schelling}. Interest on Justus Grassmann's work was hitherto focused on his crystallography [``Zur physischen Krystallonomie und geometrischen Kombinationslehre'' (Stettin 1829)], being an expression of his preference for combinatorial geometry. His ideas on the foundations of mathematics, however, were widely ignored. The paper under review closes the gap. Grassmann's foundational program as exposed in the school program pamphlet ``Ueber Begriff und Umfang der reinen Zahlenlehre'' [Stettin 1827] is in the center of its presentation. Grassmann's foundational conception is placed into its didactic context (Humboldt's education reform) and related to its philosophical roots, i.e. Schelling's philosophy of mathematics as opposed to the influential Kantian doctrine and based on the concept of construction. This concept influenced Grassmann's definition of mathematics as synthesis according to external relations of equals or unequals (p. 18). The synthesis of equals (which might be a better term than the author's ``synthesis as equal'') produces magnitudes, whereas the synthesis of unequals leads to combinations (p. 19). These kinds of synthesis are thoroughly discussed in the paper, especially their role in the foundation of natural numbers, the special case of negative integers, the role of multiplication in the foundational system, and the status of algebra. The illuminating paper closes with a short comparison of Justus Grassmann's conceptions with the ideas of Hermann Grassmann, especially with the latter's influential general theory (or doctrine) of forms (``allgemeine Formenlehre'') which opens the seminal ``Lineale Ausdehnungslehre'' [Leipzig 1844].