On Grassmann's regressive product
Grassmann's planwas to develop a purely formal algebra tomodel natural operations on geometric objects: flat or linear pieces of space of all possible dimensions. His approach was to be synthetic, so that the symbols in his algebra would denote geometric objects themselves, not just numbers (typically, coordinates) that could be derived from those objects by measurement. His was not to be an algebra of numerical quantities, but an algebra of pieces of space.
Crapo, H. (2011)., On Grassmann's regressive product, in S. Russ & J. Liesen (eds.), From Past to Future: Graßmann's Work in Context, Dordrecht, Springer, pp. 261-274.
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