[jiggawatts]

To be honest I was struggling to phrase my argument in a cohesive narrative without it turning into a ten page blog post.

The point I’m trying to make is that there are necessarily complexities inherent in all areas of study, and there are incidental complexities because of historical reasons, “culture” within certain fields, or juniors putting out their fields’ equivalent of spaghetti code.

Geometric Algebra sweeps away a lot of the rather messy parts of now century-old physics, but the work of doing that substitution is decidedly non-trivial and thankless, so other than Hestenes, nobody seems to be pushing for it.

It’s like the 2pi versus tau fad on the internet.

Mathematicians argue that they’re “the same”, so it doesn’t matter, and ramble on about their equivalent of “learn the Latin to be smart like me”.

No. It’s stupid. It was an error. Tau is the correct circle constant and eliminates magic constants that don’t belong from literally hundreds of famous formulas!

I and many others simply failed to understand radians until I learnt to treat 2pi as a single ligature instead of “two of something”.

[SonOfLilit]

I tried to make it clear that I wasn't arguing against your main point, that was made very clearly, just against a comparison you used that I think was a bit slanderous (tongue in cheek). Yes, obviously Tau is correct, and that's a better comparison to use.

Having dived deeper into the essay, author claims that some of the new notation is obviously better (clifford algebras) and the rest is overzealous unification that obscures rather than clarifies because it mixes types in a weird way (geometric product).

I've never heard of any of this before, but author's second point looks rather convincing. Can you give counterexamples, ideas that are much clearer to think about once represented using GP? I'd love to dive a bit deeper.

[jiggawatts]

I'm a bit pressed for time, but one annoyance I've had with the classic "greek" physics notation is that they represent things from "both ends" of a graded vector space. So for example, they start with a scalar, then a vector, then ... pseudoscalar-1, and finally the pseudoscalar.

It's a shortcut useful only if you need to scribble on paper and your wrists hurt from writing too much, but it obscures the underlying physics.

The programming equivalent is putting abbreviations in identifiers where, sure, it's fewer characters, but then the reader needs to a track a mental lookup table to translate back to the intended meaning.

Pushing things like this too far results in meaningful aspects of the equations getting squeezed out entirely. For example, the generality of GA means that you have to (correctly) track negative signs and multiplications by pseudoscalars such that your formulas work in all dimensions. In traditional vector algebra it's all too tempting to eliminate certain products because in "your chosen dimension" they multiply to 1 or -1 or whatever and just... disappear due to traditional algebraic simplification conventions. But then if you need to work in 4D SR or curved spaces, you can't, because you threw away something essential while "optimising for characters on a page".

You have then "start over", typically reaching for a partial and incomplete subset of GA, reinventing that wheel over and over.

Hence the push for unification onto GA, to break this cycle.