There are good ideas that gain traction super quickly. GA is not that. Possibly because there's already an ok solution (dot/cross-products), the benefits of switching are pretty incremental (or negative in terms of net effort for people who use dot/cross products a few times a year).
Category Theory is an example of a unifying theory that caught on. It took time, but my brain was able to assimilate to it quite quickly. With GA none of it sticks for me - I have other tools that suffice (I work on games and have reasonably regular 3D exposure, but the traditional tools of calculus textbooks mostly suffice, and if I need fancy concepts they tend to come from differential geometry).
I remember another big war being to push gauge integrals into Calculus textbooks (as an alternative/augmentation to Riemann integrals that vastly increase the scope of what can be integrated). I assume that went nowhere, but that was a valiant struggle.
My breakthrough with the cross product was coming to understand it as a weird but practical tool rather than representing some fundamentally beautiful geometrical concept. It might be one of the most well known weird mathematical tools. The cross product of two vectors has no real meaning to me intrinsically (I don't think it has or deserves a platonic identity) but it's useful for talking about orientations/picking right-angles/planarity.
I can imagine at some point educators might become convinced about the utility of GA and things flipping, but it feels far from that tipping point still.
From your description, you have not experienced [what I consider] a breakthrough with the cross product. In 3D there is a one-to-one mapping between vectors and oriented areas (feel free to think of it as a plane along with scalar denoting an area of the plane). When I learned about vector calculus, this one-to-one mapping was used liberally and implicitly, even though the vector a×b behaves differently from the vector a and the vector b.
Even if you take the magnitude of a×b to get a scalar |a×b|, that scalar does not behave the same as e.g. the scalar a ⋅ b. The SI system cannot distinguish between a scalar torque Newton-meters and scalar work Newton-meters, though they are different.
Geometric Algebra makes the difference explicit. For me, it's not about giving a meaning to a×b after the fact, but rather to ensure it has meaning by construction.
I think it's valid to assume the operation a×b and then figure out what it means (or doesn't mean, or call it a weird tool). But Geometric Algebra gives one way to think about it more clearly. It's not about fundamental beauty for me. It's about having a consistent and clear system to operate within.
The cross product is just an infinitesimal rotation. If you rotate b around the axis a, then the derivative of the new vector is axb. So the cross product is the product in the Lie Algebra of rotations. (Thats why you have the Jacobi identity). I find this gives much more insight than labeling it a bivector.
Edit: also this view can generalize. You can ask what other reprensentation of the rotation Lie algebra exists and find e.g su(2), this leads you to spin-bundles then.
This is not very clear to me. Can you be more precise?
If we're talking about the Lie algebra of space rotations, then the elements of that are infinitesimal rotations. Those elements form a vector space, and there's also the Lie bracket, which is the cross product as you say.
I don't see the sense in saying "rotate b about a" when both a and b are to represent infinitesimal rotations. If we're taking b to be a position in space, and a to be an infinitesimal rotation, then your explanation makes sense (though in a backwards way from how I think of it. a x b is the vector you rotate a about in order to get the derivative of the tip to be b). How do you reconcile that with the Lie Algebra?
More importantly, what if your vectors don't model rotations to begin with? Then you still are left with giving a meaning to the cross product. How can you clarify anything by taking your vectors to be infinitesimal rotations, whose cross product gives a third infinitesimal rotation, which carries information about the non-commutativity of the [non-infinitesimal] rotations?
It is one thing that the cross product does, but I don't see how it explains, for example, what the cross product has to do with calculating volumes of parallelepipeds, or torques.
Geometric Algebra is certainly not the only theory to make this distinction. Most treatments of tensor algebra will cover Hodge duality and emphasize the importance of ensuring all quantities have well-behaved "types". Category theory helps to organize everything a bit more neatly, too.
Category Theory is an example of a unifying theory that caught on. It took time, but my brain was able to assimilate to it quite quickly. With GA none of it sticks for me - I have other tools that suffice (I work on games and have reasonably regular 3D exposure, but the traditional tools of calculus textbooks mostly suffice, and if I need fancy concepts they tend to come from differential geometry).
I remember another big war being to push gauge integrals into Calculus textbooks (as an alternative/augmentation to Riemann integrals that vastly increase the scope of what can be integrated). I assume that went nowhere, but that was a valiant struggle.
My breakthrough with the cross product was coming to understand it as a weird but practical tool rather than representing some fundamentally beautiful geometrical concept. It might be one of the most well known weird mathematical tools. The cross product of two vectors has no real meaning to me intrinsically (I don't think it has or deserves a platonic identity) but it's useful for talking about orientations/picking right-angles/planarity.
I can imagine at some point educators might become convinced about the utility of GA and things flipping, but it feels far from that tipping point still.