| | Hi Daniel,
First: your summary of Rand's metaphysics and epistemology seems right. Essentially, if one were to try to case Rand's system into "worlds", it boils down to: there are two ways to sense reality-- the five senses which give us material information, and introspection, which tells us about non-physical existents.
Next: Your summary of what you think is going on with my conception of the measurement problem helps, too.
Now imagine they both look at their "plain ruler" - and sure enough, there they are, clear as day, the 1mm and 2mm markings. So regardless of the vagaries of exactly establishing (say) the microscopic point 1.474mm between them - the impossibility of which is the very reason Rand suggests this in-between solution - she can always say you have established your measurement with "absolute precision" by saying it is somewhere between those two exact markings. Problem of "perfect exactness" solved! Well, the real point she was trying to get across is that "exact measurement" isn't possible. Hence, this range idea, which is the next best thing, is what we ought to describe as precise, since it's the only technique of measurement of this kind possible to man. Also, in this paragraph:
Except for one slight detail...*by what method are the 1mm and 2mm points - the start and end points of your range - "exactly" established on your plain ruler*? It turns out to be the same method by which the point 1.474 is established. That is, they are *approximated* too (first in creating the standard block, where the points are estimated between other points by breaking the meter into 1000 approximately equal parts, then by making however many *approximate copies* - or "plain rulers"). First, 1mm and 2mm do not correspond to "points" on a ruler-- that would be improperly reifying the real line. The numbers preceding the "mm" correspond to the number of millimeters it takes to get just past the length of an object. 1.474mm is just shorthand for 1474 micrometers.
Next, how we put the tics on a meterstick is the problem of subdivision I alluded to earlier in our discussion. One doesn't subdivide a ruler by eyeing it-- one finds a unit (or calibrates an instrument) in such a way so that, say, 100 of these new subdivisions correspond to the original unit you were given.
(My guess is that some kind of mechanical device is used which prints a tic, translates a prespecified distance that can be set by the machine, and then prints another tic, etc. However, I don't know exactly how it's done.)
As for the approximate copies, I've already shown how one can verify how closely a copy unit deviates from an original unit.
If you have meter as your unit, your job of refining units is not done-- you have to actually find something which corresponds to 1cm (read: a unit 100 of which has the same length as a meter), otherwise your "subdivisions" are just blind guesswork, and hence aren't valid.
So I suppose since everyone practically uses copies of units, you could call measurement an "approximation of an approximation", if you wanted. But's that's as far as the error goes, and it's determinable.
Nate T.
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