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« on: February 01, 2007, 01:16:07 pm »
Another consequence of the contemporary approach, attributable in large measure to the Procrustean bed represented by Bourbakiste axiomatization trying to complete the work of David Hilbert, is to create winners and losers. The Ausdehnungslehre (calculus of extension) of Hermann Grassmann was for many years a mathematical backwater, competing in three dimensions against other popular theories in the area of mathematical physics such as those derived from quaternions. In the shape of general exterior algebra, it became a beneficiary of the Bourbaki presentation of multilinear algebra, and from 1950 onwards has been ubiquitous. In much the same way, Clifford algebra became popular, helped by a 1957 book Geometric Algebra by Emil Artin. The history of 'lost' geometric methods, for example infinitely near points, which were dropped since they did not well fit into the pure mathematical world post-Principia Mathematica, is yet unwritten. The situation is analogous to the expulsion of infinitesimals from differential calculus. As in that case, the concepts may be recovered by fresh approaches and definitions. Those may not be unique: synthetic differential geometry is an approach to infinitesimals from the side of categorical logic, as non-standard analysis is by means of model theory.
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