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A comment on Poor Foundations in Geometric Algebra.

Re:Poor Foundations in Geometric Algebra by nkhodyunya on Tuesday August 27, 2024 @02:44AM (#41365087)
The claim that geometric algebra is not rigorous made me confused. But the lack of proper references is much more confusing to me. Back in the days when I was doing research in Chevalley groups the definitive and most cited source was (and still is) Geometric algebra by E. Artin. And there are some more recent treatments recognized in math community, e.g. classical groups and geometric algebra by L. Groove and the geometry of the classical groups by D. Taylor. At least this books made everything rigorous and clear. But may be there is some other field with the same title?

I've heard about book by Doran and always thought that it's more about applications.

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Re:Poor Foundations in Geometric Algebra by auggierose on Tuesday August 27, 2024 @04:12AM (#41365467)
They are talking about different stuff. Geometric algebra is here the name used for the popularisation of a brand of Clifford algebra. The popularisation books are often written by physicists and engineers, and their maths is shaky, because they care more about applications than about proofs and rigour.

There are exceptions, for example I like the two books by Alan Macdonald about the topic:

http://www.faculty.luther.edu/~macdonal/

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Re:Poor Foundations in Geometric Algebra by elengyel on Tuesday August 27, 2024 @07:41PM (#41374480)
I think Macdonald's book is very concise and clean compared to others, but it does have some of the same issues that I wrote about in my post. In particular, Definition 6.15 gives one of the problematic definitions of the inner product, and Definition 6.23 gives the same broken definition of dual that has an inconsistent orientation and fails to extend to the degenerate metrics of projective algebras. Has also says, at the bottom of page 111, that the Hodge dual can't be defined directly in the exterior algebra, which is not correct.
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Re:Poor Foundations in Geometric Algebra by auggierose on Wednesday August 28, 2024 @06:35AM (#41378042)
Thank you for this analysis, will check it out!
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