5

There appears to be a cult following of geometric algebra/geometric calculus (GA/GC) as developed by David Hestenes. Many questions on stack exchange regarding this. I wanted to make this question different than others, this question hopes to bring together many mathematicians whom can contribute to the discussion of the validity of the claims and soundness of GA/GC. I want to ask the mathematicians here on stack exchange what they know and what they think about GA/GC.

The claims are always as follows, GA/GC provides a unified formalism for physics and mathematics. It provides a framework which unifies many of the concepts in fields of mathematics such as differential geometry, algebra and many more.... Why are theoretical physicists/mathematicians not rushing into this field? If it was so universal and apparently simpler to learn (also as claimed), then why is it not as widespread?

Now most textbooks and articles on this subject give the answer that Clifford developed this formalism concurrently with Gibbs standard formalism for vector calculus. Gibbs was a famous physicists/mathematician and worked at Princeton and so it is the reason his approach was so widespread and Cliffords approach was swept under the rug. Not until David Hestenes revived it, and now there are many descendants/followers of his whom repeat what I just said in the introduction or preface to their book or article. A prolific follower and firm believer of GA/GC is Alan MacDonald a professor of mathematics at Luther College in Decorah, IA. He wrote the following article (there are MANY more by other authors).

https://www.astro.umd.edu/~jph/GAandGC.pdf

Just read the introduction paragraph and this is generally the type of introduction which comes from most articles and books on GA/GC. It is basically a rephrasing of what Hestenes said in his pioneering work and books on GA/GC, such as his book Space-Time Algebra.

I need honest opinions from reputable mathematicians all over. It is important for us to get feedback from mathematicians outside the field of GA/GC because it seems the only people HIGHLY RECOMMENDING using GA/GC are only those who are using it exclusively. It would be good to get many opinions from mathematicians in fields related to GA/GC, such as Differential Geometry, Mathematical Physicists who make extensive use of Differential Geometry, Modern Algebraic Geometers, algebraists....

I know it is difficult for someone to comment on other mathematicians work, but when something makes such big claims as Unifying much of mathematics and physics, it is important for the community to discuss this, so that opinions and ideas are shared and cult following is either justified or unjustly spreading.

I am not against GA/GC, I am curious and want to know what other professional mathematicians think of this field and their claims. Please spread this article around so that others can make their comments and we can either begin to embrace GA/GC or refute its claims.

Thank you.

kiwani
  • 297
  • 2
    Sorry, matters of opinion are not for this site. –  May 04 '20 at 18:45
  • 1
    I figure if people can try justifying their opinion using mathematical arguments and rigour, then we should be able to consider a logical examination of an opinion as similar to mathematical expositions. @YvesDaoust In this sense, why aren't matters of opinions not for this site? Maybe I am missing something from the code of conduct, but so far my argument sounds reasonable. If my question doesn't get the attention I had hoped for, which website(s) can you suggest? – kiwani May 04 '20 at 18:54
  • 3
    I think it's not just a matter of opinion though -- there is a real question of whether geometric algebra offers something new, or if it is merely a different vocabulary or a rephrasing to describe what is already done with, say, differential forms. Why is geometric algebra not more mainstream? It must be because somehow mainstream mathematicians and engineers don't need it, and I'd like to hear why specifically they don't need it. Do they already have something in their toolkit that can do everything geometric algebra can do? – littleO May 04 '20 at 18:58
  • Myself, I never hear d of this and never used it, but, it superficially seems like a discussion/application of Clifford algebras by people uncomfortable with smooth manifolds. Clifford algebras as such are a pretty standard tool in some areas of geometry and topology. – Moishe Kohan May 04 '20 at 18:59
  • 1
    @KayBei: https://math.stackexchange.com/help/dont-ask –  May 04 '20 at 19:02
  • @MoisheKohan I hear this a lot. Many mathematicians are not aware of GA/GC. This is what worries me considering the claims. There are many textbooks on this subject now and here a few so you can see that it is a legitimately studied subject. https://www.amazon.com/Geometric-Algebra-Physicists-Chris-Doran-ebook/dp/B00AKE1Q0O , https://www.springer.com/gp/book/9780817682828 , https://link.springer.com/book/10.1007/978-1-84628-997-2 – kiwani May 04 '20 at 19:03
  • 2
    See https://math.stackexchange.com/q/2210804/29335 and https://math.stackexchange.com/q/1852691/29335 . The short answer is that it doesn't really bring anything new unless you count " a new viewpoint." That can be valuable, but it does not appear to be "all that" compared to other approaches. – rschwieb May 04 '20 at 19:03
  • 3
    Actually the whole situation is very much like the old quaternion-vs-vector-algebra battle that broke out originally over a hundred years ago. It is a bit embarrassing what happened, because it is moot which side is more valuable. There didn't have to be a winner, and I feel like since vector algebra "won" we've been slowly seeing examples of how the other viewpoint is sometime useful. – rschwieb May 04 '20 at 19:06
  • 1
    Why does it worry you? As I said, it is a form of Clifford formalism; if it helps people who use it, fine. As a mathematician, if I see some theorems of interest to me that are proven using this formalism and that I cannot prove otherwise, I would look seriously into it. Otherwise, I will be happily working on may area of math and not care about the books that these people publish. – Moishe Kohan May 04 '20 at 19:14
  • I checked this relevant discussion in PSE. It seems that there is a push from some on physics side to replace linear algebra with geometric algebra when teaching undergraduate students. I would strongly advise against doing so. – Moishe Kohan May 04 '20 at 19:46
  • Clifford algebras are a fairly standard object in quadratic form theory, and have their usefulness. I know they also have found some applications in physics, mainly using the Minkowski metric as the quadratic form. I don't see how it would "replace linear algebra" or "replace vector calculus". It would be like saying we should replace arithmetic by the Hurwitz quaternions. It is just one construction, which is sometimes useful and sometimes not. – Captain Lama May 04 '20 at 20:46

0 Answers0