Alec's taxonomy of units

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  1. Membership - an equivalence relation or partition of the members, where we can speak of a measurement being in a group, or not in a group. Possibly giving a sense of equality
    • Examples: (broadly) colour, for example "green", "blue", "red" being the only categories
  2. Ordered - a total order is introduced, specifically we can now speak of [ilmath]>[/ilmath] or [ilmath]<[/ilmath] - if we can speak of equality that comes from a membership classification also being in play
    • As any total order (aka linear order) that is finite is (obviously) in some sense "isomorphic" to a subset of the natural numbers, [ilmath]\mathbb{N} [/ilmath] with their obvious ordering any ordered measure is a ranking system
  3. Additive - We get the [ilmath]+[/ilmath] and [ilmath]-[/ilmath] operators. But there is no concept of ratio, for example [ilmath]{}^\circ\text{C} [/ilmath] and [ilmath]{}^\circ\text{F} [/ilmath] - note we can't speak of "twice as hot" in these scales. it's different for each! Notice we can speak of "twice as hot" with [ilmath]{}^\circ\text{K} [/ilmath] (kelvin) though. - Kelvin is a real measure, the next category.
    • Some "real" concepts (like ratio) may bleed in. For example 2x "a" is "a+a" - in this case a measure is both real and additive
  4. Real - these have a unique concept of zero. For example "mass" - we may now speak of ratios such as "twice as heavy" or "twice as light" Additionally there is a zero (think of the real field as an example)

"Average type things"

TODO: I need a name for this? "average concepts", "middle bit" - hmm

Possibly we might be able to use "geometric mean" sensibly with real units, as it has a true [ilmath]0[/ilmath]


Do these nest? That is, is the following true:

  • [ilmath]\text{Membership} \subseteq \text{Ordered}\subseteq\text{Additive}\subseteq\text{Real} [/ilmath]

Additionally, take the "membership" concept on a real scale. Sure things can be of equal mass, but we can't measure that in practice. What if we measure only to the nearest 0.1kg? So forth.

Should the practicality of measures bleed in here?

There are also questions about what can dimensional analysis yield as clues? For example can we have a not-dimensionless unit that is just additive?


Unit/measurement Membership Ordered Additive Real
Mass units
Mass (Kg)  ???
Mass (kg)
Temperature units
Kelvin  ???
Kelvin [ilmath]\mathbf{ {}^\circ\text{K} } [/ilmath]
Celsius  ???
Celsius [ilmath]\mathbf{ {}^\circ\text{C} } [/ilmath]
Fahrenheit  ???
Fahrenheit [ilmath]\mathbf{ {}^\circ\text{F} } [/ilmath]

To be classified

  • Time - iffy, there might be units for differences, where we can speak of "1 hour ..." - this is real as it has a 0, and "twice as long" for a duration as 1 hour is 2. However dates "instants of time" are not unique. For example noon being [ilmath]t\eq 0[/ilmath] makes no sense compared to any other "zero point" - thus that sense of time is only additive