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> baking in a lot of assumptions about the setting that fail to generalize That’s the point! That’s the entire point! Mathematicians want the most general, mo

> baking in a lot of assumptions about the setting that fail to generalize That’s the point! That’s the entire point! Mathematicians want the most general, most abstract approach. They want to generalise to a wide range of problems and not be painted into any one specific example. Physics theories have an opposite goal to this: the ideal theory ought to take no parameters, and produce “reality” as the one and only possible outcome. The ideal theory ought not generalise to un-physical models. For example, the mathematics of general relativity have excess degrees of freedom that must be constrained through additional restrictions. Similar issues turn up almost anywhere matrices are used: they have too many degrees of freedom. Geometric Algebra is typically a better fit for what actually goes on in physics. For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated. Rotations implemented using GA have none of these issues.

nyssos 3y ago
> The ideal theory ought not generalise to un-physical models. Sure - but a theory that fails to generalize to physical models is a bad one. A good classical theory should be a straightforward deformation of the corresponding quantum and/or relativistic theory. In this respect the best versions of classical mechanics are the standard Lagrangian and Hamiltonian approaches. > For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated. No physicist was ever under the impression that euler angles were any realer than any other way of parameterizing SO(3), that matrices were the linear transformations they represent, that manifolds are their charts, or any other trivial map-territory confusion. Paying careful attention to the distinction between real physical objects on the one hand and their representations on the other is the central theme of the last century of physics: that's basically all a gauge theory is! This is exactly what I'm talking about with geometric algebra advocates only ever comparing it to the worst sort of high school coordinate-bashing imaginable. The argument always goes - Look at this horrible vector algebra with explicit charts and numbers all over the place - Now look at this nice coordinate-free geometric algebra construction - Therefore geometric algebra is the right setting for physics but it's a complete non sequitur, because the coordinate-freeness is doing all the heavy lifting. But everyone already works without coordinates wherever it's practical to do so! The question isn't whether you should work in terms of abstract objects or explicit coordinates, it's which abstract objects you should work with.

nyssos 3y ago

> The ideal theory ought not generalise to un-physical models. Sure - but a theory that fails to generalize to physical models is a bad one. A good classical theory should be a straightforward deformation of the corresponding quantum and/or relativistic theory. In this respect the best versions of classical mechanics are the standard Lagrangian and Hamiltonian approaches. > For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated. No physicist was ever under the impression that euler angles were any realer than any other way of parameterizing SO(3), that matrices were the linear transformations they represent, that manifolds are their charts, or any other trivial map-territory confusion. Paying careful attention to the distinction between real physical objects on the one hand and their representations on the other is the central theme of the last century of physics: that's basically all a gauge theory is! This is exactly what I'm talking about with geometric algebra advocates only ever comparing it to the worst sort of high school coordinate-bashing imaginable. The argument always goes - Look at this horrible vector algebra with explicit charts and numbers all over the place - Now look at this nice coordinate-free geometric algebra construction - Therefore geometric algebra is the right setting for physics but it's a complete non sequitur, because the coordinate-freeness is doing all the heavy lifting. But everyone already works without coordinates wherever it's practical to do so! The question isn't whether you should work in terms of abstract objects or explicit coordinates, it's which abstract objects you should work with.