Alex Kritchevsky

The baseless log here is just a torsor [0]!

Lots of things are torsors: position, currency values, calendar dates etc. the vales themselves are arbitrary, and translating/scaling them by some value doesn't make a functional difference. Torsors let us talk about these things without needing to make such an arbitrary choice a priori.

In the case of baseless logs, the underlying set is "information units", i.e. log 2 is bits, log e is nats, log 10 is digits, etc. The conversion factors give us the torsor's group, and picking a privileged unit is just a trivialization of the torsor.

The vector division notation is, similarly, encoding a g-torsor in precisely the same way as length units are.

The examples so far are all torsors with abelian groups, but specifying position both requires choosing an origin and a length unit. The group of this torsor is a suitable semidirect product between translation and scaling, which gives a non-abelian group.

Most of the time we just implicitly choose a trivialization, which often causes confusion because it identifies objects with operations on them, e.g. conflating vectors as positions with vectors as translations. The author's treatise on problems with geometric algebra [1] even brings up this point!

[0]:https://math.ucr.edu/home/baez/torsors.html

[1]:https://alexkritchevsky.com/2024/02/28/geometric-algebra.html

By the way, I love your blog post https://alexkritchevsky.com/2024/02/28/geometric-algebra.html

I didn't understand this part:

> I strongly believe that if GA would make this distinction they would lose a lot fewer people. It is a completely interesting and useful thing to talk about “a representation of a particular class of operations that makes composition and inversion easy”, and completely offputting when you blur the distinction between operators and geometric objects themselves, and write every operation in terms of the geometric product when only a few of them are really compositions of operators.

I can't tell what "a few of them" refers to. What is this potential distinction between operators and geometric objects? Sounds like the the distinction between a group action and a group object?

I am willing to believe that GA is an unnecessary renaming of other simpler things, and also that it has these kind of culty vibes, but I'm focusing on the claim that (I understood as) "unifying the operators and geometric objects" is a bad feature rather than a good feature.

An article that argues against GA: https://alexkritchevsky.com/2024/02/28/geometric-algebra.html

TLDR:

- GA tends to attract a lot of crackpot. In fact most mathematicians avoid the name Geoemtric algebra and call it Clifford algebra to disassociate with them.

- Most of the usefulness of GA comes from just exterior algebra and exterior product/wedge product is more important than geometric product.

- It shows up in spinor field in physics but this does not concern most people that do not work in quantum physics.

My rudimentary view towards GA:

- It is useful in game physics since rotors can represent n-dimensional rotation in 2^{n-1} numbers instead of n^2 numbers as 2^{n-1} < n^2 when n <= 6. You can use PGA if you want to add translation as well. It is also better in interpolation.

- Outside of this you should just probably just learn exterior algebra instead.

> "What are recent developments in linear algebra?"

Have you heard of geometric algebra? It takes a while to wrap your head around, but unifies a lot of mathematics in a very elegant way. For instance, you can view Euclidean geometry as a special case, and concepts like quaternions and rotations can be explained in a very natural way. There are a few good resources below if you're interested.

[0]: http://geometry.mrao.cam.ac.uk/2020/06/euclidean-geometry-and-geometric-algebra/

[1]: https://www.youtube.com/watch?v=ipqKvNHnABs

[2]: https://alexkritchevsky.com/2020/10/15/ea-operations.html

[3]: https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K

[4]: https://crypto.stanford.edu/~blynn/haskell/ga.html

[5]: https://bivector.net/