OK, but it’s still important to understand how this plays out in the 1-dimensional case
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.
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?
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
 .
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*.
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.)
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.)