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.
Poor Foundations in Geometric Algebra
81–90 of 131 posts
Re: Poor Foundations in Geometric Algebra
#82As a former mathematician from one of the more elitist subfields of math, I have to say this is the first time I’ve heard anyone claim geometric algebra lacks rigor.
This person is selling "rigor"...
Re: Poor Foundations in Geometric Algebra
#83Earlier quoted context omitted.
In the linked blog, the author describes himself: “I am not a mathematician or physicist and am definitely not credible at all”.
The author responded to this exact statement at this Reddit comment: https://old.reddit.com/r/math/comments/1b5s32x/the_case_agai... You may also read other discussions by the author and others at the Reddit post: https://old.reddit.com/r/math/comments/1b5s32x/the_case_agai... A few Reddit comments appear to agree at that non-mathematicians (e.g. amateurs and game programmers) use geometric algebra non-rigorously. I…
What's it even mean to use Imaginary Numbers or Quaternions "rigorously"? It's Geometry. The point is points, lines, applying transformations. If anything the field needs less rigor and more friendly intuition.
Re: Poor Foundations in Geometric Algebra
#84Earlier quoted context omitted.
(I know you it doesn't let you, unless you have an alt account.) I must say that I am a little surprised that this is something you have not come across before and I take it you are surprised as well. Apparently foundations are not taught in every school. When we are talking about foundations we are usually concerned with something that prompted Euclid to write a very long time ago. In his Book 2 he defines geometric…
If your thought experiment involving pieces of paper has helped you to think of useful or interesting things that you wouldn't otherwise have thought of, that's great. It hasn't so far done anything of the kind for me. The fault could of course be mine. What I said was not (as you claimed in another comment in this thread) that I don't see how constructivism is relevant -- though in fact I don't think constructivism…
"What can we do with the black-on-white paper construction to extract information from the system?"
I will need to leave this discussion with you here. I only mean to hold open the door for you and to give you two more variations of the tools that were used to define this long tradition of rational inquiry.
Re: Poor Foundations in Geometric Algebra
#85The claim that geometric algebra is not rigorous made me confused. But the lack of proper references is much more confusing to me. Back in the days when I was doing research in Chevalley groups the definitive and most cited source was (and still is) Geometric algebra by E. Artin. And there are some more recent treatments recognized in math community, e.g. classical groups and geometric algebra by L. Groove and the geometry of the classical groups by D. Taylor. At least this books made everything rigorous and clear. But may be there is some other field with the same title?
I've heard about book by Doran and always thought that it's more about applications.
Re: Poor Foundations in Geometric Algebra
#86Earlier quoted context omitted.
> Yet, I don't think Gunn should be criticized for it at all. The article dedicates itself to critiquing the techniques, but spends no time that I can remember talking about the human beings and their feelings. That’s how I write professionally, and I’ve found it can really upset people who infer that critique of some thing is therefore critique of some person — even when no such inference is intended by the author.…
He repeatedly makes assertions that "the authors" have "no idea what they're talking about". The difference between flaming an individual person and "the authors" is thin at best, arguably only different in that it flames more than one person at once. And honestly, to the extent he's correct, I think that's fine. But let's not kid ourselves about what we're reading.
Pot. Kettles. Not-cracked but Blackened by hellfire.
Re: Poor Foundations in Geometric Algebra
#87Earlier quoted context omitted.
> Yet, I don't think Gunn should be criticized for it at all. The article dedicates itself to critiquing the techniques, but spends no time that I can remember talking about the human beings and their feelings. That’s how I write professionally, and I’ve found it can really upset people who infer that critique of some thing is therefore critique of some person — even when no such inference is intended by the author.…
Well, he describes some folks' work as "crackpot-level bullshit" and repeatedly states that other authors "have no idea what they're talking about". These seem to me to be personal critiques. Generally, I'm not sure the distinction you draw is quite so clear cut -- if I say somebody is great but all their work is hot garbage, that sounds fairly personal.
Mathematics and Namecalling is a major ego problem. I agree 100%.
Re: Poor Foundations in Geometric Algebra
#88Where are the mathematicians who could be describing Geometric Algebra correctly? For a professional working in linear algebra, supervising a graduate student, it should be straightforward to translate these "higher order" linear algebra objects into the geometric algebra model, without publishing technically incoherent mistakes.
"Geometric algebras" are Clifford algebras over the reals. Differential geometry is probably the field where you're most likely to see mathematicians discuss them, though they also come up in certain (closely related) areas of mathematical physics via spinors.
What about Clifford algebras over the imaginaries? Is that included in your idea of reals?
Re: Poor Foundations in Geometric Algebra
#89Earlier quoted context omitted.
"Geometric algebras" are Clifford algebras over the reals. Differential geometry is probably the field where you're most likely to see mathematicians discuss them, though they also come up in certain (closely related) areas of mathematical physics via spinors.
What about Clifford algebras over the imaginaries? Is that included in your idea of reals?
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.
Re: Poor Foundations in Geometric Algebra
#90The claim that geometric algebra is not rigorous made me confused. But the lack of proper references is much more confusing to me. Back in the days when I was doing research in Chevalley groups the definitive and most cited source was (and still is) Geometric algebra by E. Artin. And there are some more recent treatments recognized in math community, e.g. classical groups and geometric algebra by L. Groove and the ge…
They are talking about different stuff. Geometric algebra is here the name used for the popularisation of a brand of Clifford algebra. The popularisation books are often written by physicists and engineers, and their maths is shaky, because they care more about applications than about proofs and rigour.
There are exceptions, for example I like the two books by Alan Macdonald about the topic: