Despite trying many times to make greater use of it, I've found that it often just makes a lot of actual physics work less clear, and with very little practical benefit.
There's times where it affords quite pretty notation, but often you have to actually unpeel all that notation before you actually do something with it. And what's the point of nice notation if none of your colleagues can even read it? The only time I ever really found that GA was actually a benefit to me was performing rotations.
Maybe that's why I've found it so useful when doing rigging for animation—that's the entire job!
> GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots. Hestenes honestly sounds like one a lot of the time, and I’m not really sure whether he is or isn’t. It makes sense, really.
> GA ended up appealing to a lot of fringes: people who only had undergraduate degrees, people who had dropped out of PhDs, people with PhDs from unrigorous programs, people who had been good at math but were perhaps going a bit senile, random passerbies from engineering or computer programming, run-of-the-mill circle-squarers, people who had a bone to pick with establishment mathematics and felt like all dissenting views were being unfairly suppressed
> It didn’t help that a lot of the texts by the actually-competent GA people, like the Cambridge group, tended to say things that sounded and still sound kind of crackpotty as well.
After reading the article, the main "case against geometric algebra" I could find in there was that the author does not like the people using/doing research in geometric algebra, such as the ostensibly failed academics from a Cambridge research group [1] which the article links to.
I was expecting in the "An Actual Case Against GA" section that the author would demonstrate something like "Geometric Product actually does not work if you apply it to xyz domain". Rather, the section just ended up being mostly about the type of bikeshedding you see about naming of variables in programming.
There is I guess merit to the core "there is no good general interpretation or usage for the geometric product or mixed-grade multivectors" thesis of the article but calling other academics crackpots really subtracts from that message.
For people who actually know the curriculum side: where does geometric algebra fit? Is it something that should come after Calc III / linear algebra, alongside linear algebra, or as part of a more geometric replacement for vector calculus?
https://m.twitch.tv/videos/2282548167
TLDR it is quite a bad article. One of the closest thing he has to a real argument is "I don't like it when geometric objects are identified with operators, I want those to be separate things". But this is both anti-GA and anti-Lie-Theory. As he says, he is critical of mathematics as conventionally practiced. So be warned that if you find yourself disliking GA for anything like the reasons he dislikes GA, there's a lot of other (mainstream/prestigious) fields you dislike too.
The author has completely failed to understand the meaning and the purpose of geometric algebras, though to be fair this is not entirely the author's fault, because there are a lot of bad presentations of the geometric algebra theory, many of which contain actual mathematical mistakes, as listed in an article by Eric Lengyel that is linked in the parent article.
The main correct criticism of the parent article is that the geometric product is an operation that is seldom useful in practice.
In practice, the important operations are the generalizations of the inner product and of the outer product. The inner product and the outer product have been defined by Hermann Grassmann in the 19th century and the publications of Grassmann together with the theory of quaternions by Hamilton have been the sources on which William Kingdon Clifford has created the theory of geometric algebras.
Unfortunately, today a lot of people use incorrectly the term "outer product", using it to name the product defined by Johann Georg Zehfuss, which is also called "tensor product". "Tensor product" is also not a really appropriate term, but at least it is not as ambiguous as "outer product" has become, so it should always be preferred for the Zehfuss product. For the outer product in the Grassmann sense, a non-ambiguous term is "wedge product" though it is rather meaningless.
While the geometric product does not have a practical importance, it has a great theoretical importance, because with it the geometric algebras can be defined with a small set of simple and natural axioms. Then the operations that are important in practice, i.e. the generalized inner and outer (wedge) products can be defined based on the geometric product.
The author is right that some geometric algebra proponents have tried to shoehorn the use of the geometric product in some applications for which it is not the right tool, but that has nothing to do with the theory of geometric algebras.
The theory of geometric algebras has a modest practical importance, but it has an immense theoretical importance, because it unifies many mathematical concepts that previously seemed to be unrelated and it illuminates the relationships between them and also the distinctions between things that were previously confused, even by the best mathematicians and physicists, for more than a century.
There is a high probability that the progress of physics has been delayed by many decades by the fact that both William Clifford and James Clerk Maxwell have died prematurely and almost simultaneously, before they could make order, based on the theory of geometric algebras, in the mess that was at that time the theory of vectors, complex numbers and quaternions. After their death, the theory of geometric algebras has been forgotten and a lot of mistaken theories of vectors have been created, by Josiah Willard Gibbs, Oliver Heaviside and others (because they did not understand the relationships between various physical quantities, like polar vectors, axial vectors, quaternions, complex numbers, pseudoscalars).
When I have first encountered the theory of geometric algebras, that was one of the most beautiful moments in my experience of learning mathematics, it was like turning the light on in a dark room full of previously hidden things. The only similar moments, have been when learning for the first time projective geometry, the theory of spatial symmetry groups and certain parts of topology, which are also theories that have unified a great number of seemingly unrelated concepts.
Like I have said, geometric algebras have very little importance for writing algorithms or the like, where the classic linear algebra with matrices is what matters most, but anyone who does not understand geometric algebras does not really understand physics and this lack of understanding will prevent the correct solution of many problems.
