Reforming the Mathematical Language of Physics (2002) [pdf]

I think the author proposes geometric algebra for widespread use in teaching fundamental physics. That's a good idea, it unifies a lot of things and brings new insights! Hestens seems to be quite popular with his opinion, see for instance https://link.springer.com/chapter/10.1007/978-94-011-5036-1_... and https://en.wikipedia.org/wiki/Geometric_algebra

Very interesting! But it requires really a skilled lecturer to make students understand the concepts. I've heard advanced mathematical physics by a, well, bad lecturer, who taught everything on some abstract level which I never heard from previously (and after). I had to learn differential geometry and all that later on my own in order to graduate in some topic in general relativity.

I recently read John Vince's Geometric Algebra for Computer Graphics, which seemed excellent – written, the author explains, because he realized you had to virtually be a mathematician to understand the books in the field, and he wanted to write something much more accessible. It sounded great until Chapter 11, Conformal Geometry..

  Various arguments have been proposed for adopting this model such as, conformal space:
  • is homogeneous and consequently removes the origin as being a special point
  • supports points and lines at infinity
  • provides a single geometric mechanism for representing lines, circles, planes and spheres, which introduces a rare quality of elegance to problem solving
  • enjoys all the normal features of Euclidean space.
  A model of space that possesses such a range of positive features sounds too good to be true! But remember, in mathematics there is no such thing as a free lunch. So what price must we pay for this model?

Then Vince explains that the conformal model, "an alternative space to Euclidean space for solving 3D geometric problems", is covered by a patent..of Hestenes, US Patent 6,853,964 System for encoding and manipulating models of objects. Vince says "Licences to employ this system for commercial purposes are available through the patent holders, although there are no restrictions for academic research and educational use. All of the above may be a small price to pay for the benefits associated with the conformal model, and I will do my best to explain the model in this chapter."

Uh what?! So it actually costs money to use... That rather put me off Hestenes' noble mission. It seems...bizarre that what seems a whole subfield of mathematics is patented, rather different to patenting a particular algorithm or technique. Very strange, after reading so much of Hestenes' wondering why the superior GA isn't used everywhere. (Is it to make the world a better place..or to make Hestenes rich?..) I stopped reading at that point.

As far as using Geometric Algebra in a Euclidean setting goes, most of my programming is in 2D, but when I do 3D next I will consider using it, it seems very cool. Most of the features/differences/advantages over the usual vector/matrix geometry only appear in 3 or more dimensions. Vince's book is great at explaining it, highly recommended.

Hestenes on Victor's radar, that seems promising! :)

I like to mention on such threads:

"Geometric Algebra for Electrical and Electronic Engineers"

> In this paper, we explicate the suggested benefits of Clifford's geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate an additional structural check provided by GA for formulas in addition to the usual checking of physical dimensions. Naturally, there is an initial learning curve when applying a new method, but it appears to be worth the effort, as we show significantly simplified formulas, greater intuition, and improved problem solving in many cases.

https://ieeexplore.ieee.org/document/6876131

Hestenes; The Genesis of Geometric Algebra: A Personal Retrospective (2017)[1]

[1] https://link.springer.com/content/pdf/10.1007/s00006-016-066...

This line on page 6 confuses me a bit --For example, for applications to rotations, quaternions are demonstrably more efficient than the vectorial and matrix methods taught in standard physics courses

And then claims that aerospace engineers mostly use quaternions. It's from 2002, so it's sense of what's in vogue might be dated, but I'd understood at least in the graphics manipulation world, the vector/matrix methods were dominant. Those are separate industries, so both statements could be true, anybody have opinions?

First I thought it would be about dumbing down the existing maths so more people can understand.

Some 10 pages in I realized it's explaining the concepts I've internalized years ago in a way I can't comprehend at all.

There is some very good propositions in this paper. Physics should definitively have less redundant mathematics formulations of problems. Hamiltonian physics is quite close to Newtonian in comparison quantum physics.

Good that people sometimes remember that a more modern base for physics would be nice.

Doing electrodynamics without using alternating products and differential forms feels so wrong...

(Disclaimer: haven't read the whole thing) This just seems like one of these situations: https://xkcd.com/927/

The fact that the same physical system can be described by very different kinds of mathematics is often very illuminating and interesting, rather than confusing. It's one of the reasons that physics can inspire mathematics, rather than maths just being a tool that physicists use. Different types of mathematics often allow you to emphasise different properties of the same system, so being comfortable in more than one language brings you freedom and new insights.