Customer Reviews: Extension Theory (History of Mathematics, 19.)
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on July 27, 2004
Grassmann's (1862) Ausdehnunglehre introduces the reader to vectors, spaces and subspaces, bases, dimensions, exterior products, complementation, inner products, and a host of geometric algebra. The linear algebra taught to undergraduates - and a lot of simple, practical mathematics that isn't taught to undergraduates largely for historical reasons (Hamilton bound vectors to 4 dimensions, Gibbs later bound them to 3 dimensions, failing to emphasize the generality of Grassmann's invention) - is presented simply and cleanly.

Dirac's famous braket notation, quantum logic (introduced by Birkhoff and von Neumann in 1935), the modern definitions of dimension, orthogonality, ordering, and the meets and joins of lattice theory, can all be found here. Many of these ideas are written in the same notation and language we use today, and could be found stated identically in a modern text on geometric algebra.

Only last year I published results which used these simple operators to improve information retrieval in a search engine, claiming that I'd made a logical breakthrough. I had certainly made a contribution - but I'd have made it much more quickly if I'd read Grassmann's simple instructions first. The Extension Theory is a treasure trove of techniques which can be applied to logic, information, robotics and virtual reality ... the list goes on, and we're only just beginning to reap the benefits. If you're looking to for successful and useful research opportunities, try reading Grassmann.
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