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komponisto comments on Inverse Speed - Less Wrong
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Who said I think of relations as one-way functions? I think of them as what they are, namely subsets of the Cartesian product.
As for division, I'm very happy to trade it in for an intuitive understanding of the canonical monomorphism
(which, in concrete terms, means the ability to view something labeled "mph" as a linear map from the space of times to the space of distances).
OK, but it's still important to understand how this plays out in the 1-dimensional case. These aren't incompatible, one's just a special case. Though I'm not seeing the relevance of that particular isomorphism here, as I don't see just what it is here that would naturally be interpreted as an element of that first space in the first place?
Well, yes! That's what I seek to do, as opposed to regarding the 1-dimensional case as a separate magisterium, compartmentalized away from the general case.
Here V is distances, and W is times. If something has the label "distance", it's an element of V; if it has the label "time", it's an element of W; and if it has the label "time^-1", it's an element of W*. Something with the label "distance/time" is then an element of
.
Oh, OK. For some reason I was thinking the scaling was wrong for that to work. Of course, if you travel 3 miles in 2 hours, that's 3 mi \otimes 1/2 h^-1, not 3 mi \otimes 2 h^-1...
That's right: (1/2)h^-1 is the map that takes a time and gives its coordinate with respect the basis {2h}, which is the one being used here to define the speed.
(General rule: a/b means you input b to get a. So, since our coordinate-computing map should input 2h and output 1, it is written 1/(2h), or (1/2)h^-1.)