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Poor Foundations in Geometric Algebra

terathon.com

11–20 of 131 posts

Re: Poor Foundations in Geometric Algebra

#11

This explains very nicely why despite working through lots of GA resources, I never quite grasped it. It’s logically incoherent!

The author has books on Geometric Algebra so it makes no sense to assume that he is going against GA as a whole.

Calculus had poor foundation and was thus logically incoherent from Newton/Leibniz' discovery to roughly the middle of 19th Century. None-the-less it was a powerful tool and most of the key theorems were discovered then.

The basic situation, I think, is a set of tools can be consistent in the way mathematicians use them but in the way the mathematicians explain them. And the tools can be very useful despite this.

So my guess is saying "it has poor foundations" isn't saying "I'm against it, it's worthless"

Re: Poor Foundations in Geometric Algebra

#12

Note don't to be confused with Algebraic Geometry they are different; When I first came across this topic it was eye opening, especially the fact that you could squeeze Maxwell's equations into one and the fact that pseudovector create by cross product from physics is just a bivector which in 3d could be represented like a vector orthogonal to the plane created by the two vectors in the product Primer on the topic ht…

Hah! I knew Sudgy would be your first link. There's relatively little content about GA, but once you're hooked you begin to consume it all.

It's like you never got to study Relativity, conformal geometry, electron spins, quaternions, etc then someone comes with a simple cheat code which introduces you to these topics gently. It's like what category theory wants to do for mathematics, but simple.

Here are additional resources:

GA playground: https://enkimute.github.io/ganja.js/examples/coffeeshop.html...

A physics engine in 100 lines (Gravity, Hook, and damping laws are just one line each!), and you can go from 2D to 3D to 4D by changing a single parameter: https://enki.ws/ganja.js/examples/pga_dyn.html

Other resources: https://bivector.net/

And anything by David Hestenes: https://worrydream.com/refs/Hestenes_2002_-_Reforming_the_Ma...

Re: Poor Foundations in Geometric Algebra

#14
post #10

[flagged]

You were not in the target audience. This was written for people already familiar with the theory of geometric algebras.

Like any other mathematics paper, it cannot include the content of a whole textbook inside the paper, to provide context for non-mathematicians or for mathematicians with a different specialization.

Here on HN there are every day links to "walls of text" that are trivial to understand for those familiar with computer programming, but which would be completely impossible to understand for a non-programmer.

Re: Poor Foundations in Geometric Algebra

#15

This explains very nicely why despite working through lots of GA resources, I never quite grasped it. It’s logically incoherent!

Yeah I’m pretty well-versed in free constructions of various algebraic objects and how this would interact with things like a quadratic form, etc, but couldn’t sort out GA (I think the authors of “GA4CS” had a very different sort of computer scientist in mind). When I saw the geometric algebra constructions clash with constructions I was already familiar with, I generally got suspicious and lost interest.

I’m actually quite interested in checking out Lengyel’s book. It looks rock solid.

Re: Poor Foundations in Geometric Algebra

#16

Note don't to be confused with Algebraic Geometry they are different; When I first came across this topic it was eye opening, especially the fact that you could squeeze Maxwell's equations into one and the fact that pseudovector create by cross product from physics is just a bivector which in 3d could be represented like a vector orthogonal to the plane created by the two vectors in the product Primer on the topic ht…

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Re: Poor Foundations in Geometric Algebra

#17

So much education content is so very poor. And even the best of it... A first-tier physics professor at a munch was delighted - he regaled believing he had found an error in a highly-regarded introductory textbook! But, upon many day's of thought, and several close reads of the text, he had realized it had been very carefully worded so as to be not incorrect. And so he was so delighted - yay! He thought of this as a…

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Re: Poor Foundations in Geometric Algebra

#20
post #5

> very real toxicity within the geometric algebra community. I can’t do much about I was hoping the article would be about this instead. OP wondering if you have any elaborations for us to hear.

I'm not the OP, I'm not a part of this community, and I don't know if the thing I'm about to complain about is what the author was thinking of with this comment, but as someone who was trained as a mathematician and who has read some of the popularizations of geometric algebra that sometimes get posted to HN, there is a tone that some (though probably a minority) of them take that I find pretty obnoxious.

These pieces are the ones that take the position that geometric algebra is this super secret anti-establishment mathematical samizdat that *they* don't want you to know about. They'll pit themselves against "mainstream mathematics" and say things like, "in differential geometry you do X, but you shouldn't do differential geometry; you should do geometric algebra where we do Y, which is so much better than X."

My reaction is always, "My friend, you are doing differential geometry!" Clifford algebras --- the objects that the geometric algebra people study --- are firmly within the "mainstream" of mathematics; there's simply no conflict here, at least not of the sort that these writers often seem to be imagining. It's great that people are enjoying learning about Clifford algebras. I think Clifford algebras are really fun! But we can all just come together and enjoy them together, and I think this "join me in taking down the cabal of gatekeepers who are suppressing the truth" attitude is unnecessary and turns off a lot of people who might otherwise be fun to engage with.

If you're into this stuff and feel like this doesn't describe you or the people you know, then that's great, keep doing what you're doing! But it does exist and I wish it didn't.

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