The case against geometric algebra (2024)

I found this article pretty confusing.

And my comment ended up being pretty long, so I will TL;DR it:

1. The social critique doesn’t match my experience and seems under-supported?

2. The technical critique is interesting, looks like a mix of good points, and some that need more work put into it. I think GA is legitimately cool in my opinion, but if there are better abstractions, we should find/define them and use them.

Longer version:

I hear people bring up the conspiracy/crackpot side of GA a lot, but I learned about Geometric Algebra a few years ago and am currently learning it alongside standard linear algebra.

I think GA is pretty cool. The author seems to have some decent points about its limitations and some ontological smells (like, maybe there is a cleaner representation hiding somewhere). But a lot of the criticism is aimed at the social side of the movement, and maybe I am just blind to it, but I have not really run into that much.

The author says things like:

    Basically, GA is considered a kooky, crackpotty sideshow. And because it is so dubious and un-self-aware, the movement ends up alienating most people, except for a particular type of… zealous individual… who write about it with a sort of pseudoreligious zeal, and are prone to conspiracy, as if the only reason GA is not mainstream is that they are being oppressed by close-minded traditionalism.

and:

   In practice GA always refers to the particular platform and social movement which descends from the work of David Hestenes from the 1960s. It specifically does not refer to the underlying material of Clifford Algebras

Maybe this is true in some parts of the internet or in some older discourse, but from the material I have read, people seem pretty explicit about the roots of Geometric Algebra.

Trying to build a unifying framework seems pretty normal to me. Lots of math is trying to expose common structure across different domains. Category theory, abstract algebra, topology, and, to a much bigger extent, the Langlands program all have that flavor. Obviously some unifications are more successful than others, but “this gives a unified language for a bunch of things” does not seem like a red flag by itself.

Some of the actual technical criticisms of GA are interesting, e.g. the proliferation of operations, but at this point I'm more interested in a formal accounting of the complexity of both theories rather than opinions or vibes. It would be nice to have description-length / complexity-accounting comparison of the formalisms.

Disclaimer: I have not read Hestenes’s original work, so maybe I am missing some of the historical baggage. But the modern resources I have seen seem mostly grounded in their claims.

I'm also learning both GA and linear algebra at the same time, GA has definitely helped me understand the linear algebra more deeply. In my opinion, alternative representations like GA gives your brain more structure to grab onto, even if they aren't perfect.

Also... math pedagogy does have a lot of inertia that hurts students. Doesn't Lockhart's Lament famously resonate with anyone who fell in love with math?

[PDF Warning] https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%...

I get why it is interesting and useful to write complex numbers in '+' notation rather than the conventional way to denote a 2d vector, like a tuple of components.

The benefit is that multiplication and distributive property is a beauty in the '+' notation, no special rules need to be memorized for multiplying 2d vectors, i*i = -1 takes care of it.

On the other hand I never understood what the benefit, of writing the tuple of wedge and dot products in '+'notation, is.

Perhaps I am not being fair, that it is the same idea and I have not used it as much as I have used complex numbers.

Crackpot really has connotations like "flat earther" and "aliens built the pyramids". It's one thing to say "I believe GA proponents' claims regarding the usefulness of the geometric product are overstated". It's another to say "GA proponents are crackpots".

The basic issue with geometric algebra is that geometric vectors generally do not have a distinguished notion of unit magnitude (is unit magnitude 1 meter? 1 mile? 1 inch?), so it is silly to work in a framework that requires pretending they do (since the definition of the geometric product of two vectors is dependent upon this choice). Dimensional analysis (a very handy way of tracking mathematical symmetries and thus sanity checking results) goes out the window when working with mixed grade multivectors.

This is not an issue when working with non-mixed-grade multivectors, for which dimensional analysis works just fine in the ordinary way. As the linked article notes, exterior algebra/the wedge product is great. Thinking about exterior powers of vector spaces is great. It's the further move of forcing everything into a Procrustean bed of Clifford algebra that is misguided for almost any application other than some spinor stuff.