3 ms·

Perhaps I’m stating the obvious, but doesn’t that poke a pretty big hole in Tao’s proposed formalisms?

r0uv3n 3y ago
I don't think this distinction is often made in practice, even though it might be useful. As far as I can tell, combining Tao's approach here with a Clifford algebra approach (where energy is a scalar and torque a bivector) would be entirely possible and solve this problem.
- mikhailfranco 3y ago
  Yes, this is the correct answer. The Gibbs formulation of 3D vector analysis/calculus was certainly useful, but forgets most of the spatial structure. Using Clifford Algebra (Geometric Algebra) fixes everything.
  - ajkjk 3y ago
    daily chance to inject my opinion on GA: Almost all of the value in GA comes from the use of multivectors and the wedge product. It's very hard to find a justification for the "geometric product", and there's no real geometric interpretation of it. Therefore treating it as fundamental is a difficult-to-justify choice. "Exterior Algebra" on its own has everything you want without the objectionable parts.
contravariant 3y ago
Dimensional analysis doesn't need to be perfect, it just needs to rule out some options. You can convert all units to mass and still get useful results, but the more diverse you can keep the units the better. I suppose you could even go through the relevant laws of physics and only identify the units for quantities that need to be added together (or to make the quantities inside more complex functions unit-less). In the case of torque and energy you basically run into the problem that torque times angle is an amount of work and angle is traditionally unit-less. You could however give angle a unit by introducing some 'unnecessary' conversion factor like 'degrees per radian', which is only used for trigonometry. In that case you could reformulate torque as 'energy per angle'.
- cb321 3y ago
  > even go through the relevant laws of physics Possibly of interest, as mentioned in Appendix C of https://www.nist.gov/publications/architecture-software-assisted-quantity-calculus https://www.nist.gov/publications/architecture-software-assi... there is a start to this in https://en.wikipedia.org/wiki/ISO/IEC_80000 https://en.wikipedia.org/wiki/ISO/IEC_80000

I don't think this distinction is often made in practice, even though it might be useful. As far as I can tell, combining Tao's approach here with a Clifford algebra approach (where energy is a scalar and torque a bivector) would be entirely possible and solve this problem.

mikhailfranco 3y ago
Yes, this is the correct answer. The Gibbs formulation of 3D vector analysis/calculus was certainly useful, but forgets most of the spatial structure. Using Clifford Algebra (Geometric Algebra) fixes everything.
- ajkjk 3y ago
  daily chance to inject my opinion on GA: Almost all of the value in GA comes from the use of multivectors and the wedge product. It's very hard to find a justification for the "geometric product", and there's no real geometric interpretation of it. Therefore treating it as fundamental is a difficult-to-justify choice. "Exterior Algebra" on its own has everything you want without the objectionable parts.

contravariant 3y ago

Dimensional analysis doesn't need to be perfect, it just needs to rule out some options. You can convert all units to mass and still get useful results, but the more diverse you can keep the units the better. I suppose you could even go through the relevant laws of physics and only identify the units for quantities that need to be added together (or to make the quantities inside more complex functions unit-less). In the case of torque and energy you basically run into the problem that torque times angle is an amount of work and angle is traditionally unit-less. You could however give angle a unit by introducing some 'unnecessary' conversion factor like 'degrees per radian', which is only used for trigonometry. In that case you could reformulate torque as 'energy per angle'.

cb321 3y ago
> even go through the relevant laws of physics Possibly of interest, as mentioned in Appendix C of https://www.nist.gov/publications/architecture-software-assisted-quantity-calculus https://www.nist.gov/publications/architecture-software-assi... there is a start to this in https://en.wikipedia.org/wiki/ISO/IEC_80000 https://en.wikipedia.org/wiki/ISO/IEC_80000