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aronwall comments on Checklist of Rationality Habits - Less Wrong
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Taking logs of a dimensionful quantity is possible, if you know what you're doing. (In math, we make up our own rules: no one is allowed to tell us what we can and cannot do. Whether or not it's useful is another question.) Here's the real scoop:
In physics, we only really and truly care about dimensionless quantities. These are the quantities which do not change when we change the system of units, i.e. they are "invariant". Anything which is not invariant is a purely arbitrary human convention, which doesn't really tell me anything about the world. For example, if I want to know if I fit through a door, I'm only interested in the ratio between my height and the height of the door. I don't really care about how the door compares to some standard meter somewhere, except as an intermediate step in some calculation.
Nevertheless, for practical purposes it is convenient to also consider quantities which transform in a particularly simple way under a change of units systems. Borrowing some terminology from general relativity, we can say that a quantity X is "covariant" if it transforms like X --> (unit1 / unit2 )^p X when we change from unit1 to unit2. Here p is a real number which indicates the dimension of the unit. These things aren't invariant under a change of units, so we don't care about them in a fundamental way. But they're extremely useful nevertheless, because you can construct invariant quantities out of covariant ones by multiplying or dividing them in such a way that the units cancel out. (In the concrete example above, this allows us to measure the door and me separately, and wait until later to combine the results.)
Once you're willing to accept numbers which depend on arbitrary human convention, nothing prevents you from taking logs or sines or whatever of these quantities (in the naive way, by just punching the number sans units into your calculator). What you end up with is a number which depends in a particularly complicated way on your system of units. Conceptually, that's not really any worse. But remember, we only care if we can find a way to construct invariant quantities out of them. Practically speaking, our exprience as physicists is that quantities like this are rarely useful.
But there may be exceptions. And logs aren't really that bad, since as Kindly points out, you can still extract invariant quantities by adding them together. As a working physicist I've done calculations where it was useful to think about logs of dimensionful quantities (keywords: "entanglement entropy", "conformal field theory"). Sines are a lot worse since they aren't even monotonic functions: I can't imagine any application where taking the sine of a dimensionful quantity would be useful.
I think it'd be obvious how to take the log of a dimensional quantity.
e^(log apple) = apple
Right, but then log (2 apple) = log 2 + log apple and so forth. This is a perfectly sensible way to think about things as long as you (not you specifically, but the general you) remember that "log apple" transforms additively instead of multiplicatively under a change of coordinates.
Isn't the argument to a sine by default a quantity of angle, that is Radians in SI? (I know radians are epiphenomenal/w/e, but still)
Machine learning methods will go right ahead and apply whatever collection of functions they're given in whatever way works to get empirically accurate predictions from the data. E.g. add the patient's temperature to their pulse rate and divide by the cotangent of their age in decades, or whatever.
So it can certainly be useful. Whether it is meaningful is another matter, and touches on this conundrum again. What and whence is "understanding" in an AGI?
Eliezer wrote somewhere about hypothetically being able to deduce special relativity from seeing an apple fall. What sort of mechanism could do that? Where might it get the idea that adding temperature to pulse may be useful for making empirical predictions, but useless for "understanding what is happening", and what does that quoted phrase mean, in terms that one could program into an AGI?
"units are a useful error-checking homomorphism"
I don't think "homomorphism" is quite the right word here. Keeping track of units means keeping track of various scaling actions on the things you're interested in; in other words, it means keeping track of certain symmetries. The reason you can use this for error-checking is that if two things are equal, then any relevant symmetries have to act on them in the same way. But the units themselves aren't a homomorphism, they're just a shorthand to indicate that you're working with things that transform in some nontrivial way under some symmetry.
The map from dimensional quantities to units is structure-preserving, so yes, it is a homomorphism between something like rings. For example, all distances in SI are mapped into the element "meter", and all time intervals into the element "second". Addition and subtraction is trivial under the map (e.g. m+m=m), and so is multiplication by a dimensionless quantity, while multiplication and division by a dimensional quantity generates new elements (e.g. meter per second).
Converting between different measurement systems (e.g. SI and CGS) adds various scale factors, thus enlarging the codomain of the map.