Comment by generationP
1 year ago
This is exactly the "specification by universal property" that the author is talking about. In your case, it's saying "a 3-dimensional vector space is a vector space with three chosen vectors e, f, g and three linear maps x, y, z such that each vector v equals x(v) e + y(v) f + z(v) g". But as the author points out, not everything follows from universal properties, and when it does, there is often several universal properties defining the same object, and that sameness itself needs to be shown.
Yes, I know it's the meaning of it, but I'm saying that the presence of "units" allows you to sort of... operationalize it? In a way that removes the ambiguity about what's going on. Or like, in theory. I dunno it's a comment on the internet lol.
Units help with some common cases, but units still don't allow you to distinguish between, say, energy, work and torque.
I think that is due to insufficient imaginativeness. For example, energy and work are the same units but energy is an absolute quantity while work is a delta. So arguably work should be a sort of tangent element of the energy, rather than the same thing. There's no distinction in flat space but if you e.g. changed coordinates such that energy was on the surface of a sphere, then work is a spherical displacement instead, which is a totally different class of objects.
Likewise, torque is only in the same units because we don't regard radians as a unit, but we should. They are distinctly different.
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