Thanks for the reference and definition. Though I'm inclined to side with Merlin's position. As I understand it, he objects to calling an arbitrary or coincidental mathematical correlation a measurement. As a matter of aesthetics, I immediately agree. But I should identify and define why.
I gather from "Foundations..." the authors have a quite natural abstract/material, analytic/synthetic premise. That mathematic abstractions have non-material origin. Which I suppose is appropriate enough, since the work doesn't address philosophy.
The Objectivist position is numbers, mathematical functions and abstractions, do not exist apart from concretes. And the action preformed by our brains (computers) is to *act* on (external or internal) perceived differences in physical concretes, according to our values and volition.
Our brains have a metaphor, a model, of what we are measuring. We observe a spring and mass bouncing, and in our minds, neurons fire in synchrony and we learn what to expect, how to derive units, how to model, and what ratio of units is appropriate to create measures of analogous phenomena.
The act of measurement is a back-reference, a post-hoc observation, an affirmation of identity, according to similar truths we've already learned. We can measure a volt here or there, and its the same, because an electron here or there is the same, because charge, mass, space and time are the same here or there. What is different is time, location and the flow of energy in its fields.
No doubt physical laws are the universals in existence which enable us to recognize the differences in the fields, particles and energy, which vary through (I should say actually are) space and time.
I will change my definition of measurement, "a ratio between differentiable units". Returning to my thoughts on the nature of integer numbers, is an integer anything different than the observation of the ratio of one complete cycle between different units? Why do you have to use "one" cycle as a unit standard? Why not 1/3, or Pi/4?
We don't have to make life hard, expecting integers to be everywhere just because we don't have webbed fingers and toes!
Thanks for reminding me about apparent power Merlin. Its another rant, but I've always had a problem with imaginary numbers and power.
Power is energy * time. Well, in a loss-less circuit, no "real power" is being dissipated. To an inductor, a capacitor discharging through it does "work" creating flux. And if to a capacitor, an inductor sourcing current is doing "work", giving it energy over time, charging it up. It only from the perspective of a resistive load that power is removed from the system. Transduced.
So "real" or "imaginary" is a kind of arbitrary choice of perspectives. And "i" is only used in the context of complex (two - component) quantities. Saying there is such a thing as a negative scalar or single-valued square-root seems to me context dropping, and a violation of the nature of (single-valued) multiplication. But I'm no mathematician, so I better shut-up now.