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Poor Foundations in Geometric Algebra

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Re: Poor Foundations in Geometric Algebra

#101

Post by ndriscoll »

koolala wrote: post #96
nyssos wrote: post #89

Earlier quoted context omitted.

Imaginary numbers aren't a field, so there's no such thing. Clifford algebras over the complex numbers work fine, but it's usually not what the people talking about "geometric algebra" are doing.

How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)

Typically GA people are working with real algebras, meaning the coefficients are real, and things like a square root of -1 appear as some object in the algebra (like a 2-blade). But you could also have a Clifford algebra with coefficients in e.g. the complex numbers or fields of finite characteristic.

In fact using different coefficient rings is one way to write a compact recursive definition of real Clifford algebras:

http://blog.sigfpe.com/2006/08/geometric-algebra-for-free_30...

Re: Poor Foundations in Geometric Algebra

#102

Post by kkylin »

nyssos wrote: post #76
jmount wrote: post #62

A lot of the (valid) criticism can be summarized by the "No scalar product necessary. Get rid of it" point (early in one of the tables). The next point on inner product is good, and related. The point is: there is no natural or distinguished isomorphism from a finite vector space to the vector space of linear functionals over it. We know they are isomorphic- but there are many such isomorphisms with little to disting…

There's no mathematically natural choice. In physical settings, on the other hand, you almost always want the one induced by the metric.

Not a physicist, but this seems to depend on the physical setting? Seems there are usually metric with physical meaning in continuum mechanics, e.g., elasticity or GR, but not so much if one is working in say geometric mechanics — one can define a Hamiltonian flow on a symplectic manifold without a metric.

Re: Poor Foundations in Geometric Algebra

#103

Post by ajkjk »

gugagore wrote: post #92
ajkjk wrote: post #75

Earlier quoted context omitted.

Those concepts are from exterior algebra which GA is built on. You do not need GA (the geometric product, etc) for them.

That's fine. That's still a step beyond just "linear algebra".

Oh yes, strongly agree. They should be taught in linear algebra really.

Re: Poor Foundations in Geometric Algebra

#104

Post by gugagore »

ajkjk wrote: post #103
gugagore wrote: post #92

Earlier quoted context omitted.

That's fine. That's still a step beyond just "linear algebra".

Oh yes, strongly agree. They should be taught in linear algebra really.

Happy to have agreement that this goes beyond "traditional linear algebra".

But, pedagogical treatment is a separate question from what is linear algebra.

Should these all be the same wikipedia page?

- https://en.wikipedia.org/wiki/Exterior_algebra

- https://en.wikipedia.org/wiki/Multilinear_algebra

- https://www.georgehart.com/research/multanal.html (okay, not mainstream enough to have a relevant page, but it is extremely relevant in any engineering practice of linear algebra)

Re: Poor Foundations in Geometric Algebra

#105

Post by ajkjk »

gugagore wrote: post #104
ajkjk wrote: post #103

Earlier quoted context omitted.

Oh yes, strongly agree. They should be taught in linear algebra really.

Happy to have agreement that this goes beyond "traditional linear algebra". But, pedagogical treatment is a separate question from what is linear algebra. Should these all be the same wikipedia page? - https://en.wikipedia.org/wiki/Exterior_algebra - https://en.wikipedia.org/wiki/Multilinear_algebra - https://www.georgehart.com/research/multanal.html (okay, not mainstream enough to have a relevant page, but it is ext…

Well by "taught in linear algebra" I mean "taught to undergraduates in their first or second year". It is totally bizarre that people learn about determinants, matrix minors, curl, magnetic fields, angular momentum, etc... but not about wedge products, in which all of these things are much easier to understand.

Re: Poor Foundations in Geometric Algebra

#106

Post by koolala »

ndriscoll wrote: post #101
koolala wrote: post #96

Earlier quoted context omitted.

How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)

Typically GA people are working with real algebras, meaning the coefficients are real, and things like a square root of -1 appear as some object in the algebra (like a 2-blade). But you could also have a Clifford algebra with coefficients in e.g. the complex numbers or fields of finite characteristic. In fact using different coefficient rings is one way to write a compact recursive definition of real Clifford algebra…

As this shows, ei and ej are imaginary numbers. This confusion is the biggest issue in GA to me. Typically Linear Algebra users don't use imaginary numbers either. It's all connected when doing geometry though and its pointless to draw invisible lines between these concepts.

We have no way to talk about more general types of complex / imaginary numbers besides rigid math lingo that provides no geometric intuition or grace for geometric imagination.

Re: Poor Foundations in Geometric Algebra

#107

Post by koolala »

mkl wrote: post #98
koolala wrote: post #96

Earlier quoted context omitted.

How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)

You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.

I'm glad you seem to know what I mean. I love imaginary numbers like the Square Root of -1 or the Square Root of 0. They can only be used like:

nil²=0

i²=-1 j²=-1

And combining them into complex forms like:

(i + j)

(nil + i)

What can I call this general idea of using imagination to determine new number rules and combining them together?

If I call this GA or Clifford Algebra in a math community will it trigger rigor admins to ban me from talking because I'm not using their terms?

I wish we had artistic imaginary math communities for exploring Geometry without rigor turing everything into Semantics. Geometry literally doesn't need semantics if you agree on points, lines, planes ect. Algebra to me should just be simple maps from clifford numbers to examples of intuitive geometry / physics.

Re: Poor Foundations in Geometric Algebra

#108

Post by Certhas »

koolala wrote: post #81

Squaring and Cubing should be geometric ideas - not just another way to do 1D multiplication. We need new operators for transforming a number into a higher dimention.

As mentioned elsewhere, that's the wedge product of exterior algebra. Standard stuff first introduced in 1844 that has been foundational to a ton of modern math.

https://en.m.wikipedia.org/wiki/Exterior_algebra

Re: Poor Foundations in Geometric Algebra

#109

Post by gugagore »

ajkjk wrote: post #105
gugagore wrote: post #104

Earlier quoted context omitted.

Happy to have agreement that this goes beyond "traditional linear algebra". But, pedagogical treatment is a separate question from what is linear algebra. Should these all be the same wikipedia page? - https://en.wikipedia.org/wiki/Exterior_algebra - https://en.wikipedia.org/wiki/Multilinear_algebra - https://www.georgehart.com/research/multanal.html (okay, not mainstream enough to have a relevant page, but it is ext…

Well by "taught in linear algebra" I mean "taught to undergraduates in their first or second year". It is totally bizarre that people learn about determinants, matrix minors, curl, magnetic fields, angular momentum, etc... but not about wedge products, in which all of these things are much easier to understand.

Totally agree! And this is why I sympathize with GA fanatics. Although it might not really make sense to do the rebranding or be maximalist about the geometric product, I do think there is a shared goal of making these things easier to understand.

Re: Poor Foundations in Geometric Algebra

#110

Post by gugagore »

ajkjk wrote: post #75
gugagore wrote: post #63

Earlier quoted context omitted.

What other way is there around avoiding polar vs axial vectors looking the same but behaving differently? Or normal vs tangent vectors transforming differently?

Those concepts are from exterior algebra which GA is built on. You do not need GA (the geometric product, etc) for them.

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

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.

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