What is a valid unit of measurement?

Question

Ok, so this is a bit of an odd question. It is sort of a physics question, but I felt it was more appropriate here. Feel free to suggest another venue.

Let's just look at a simple example. We'll use kilograms as an example unit of measurement. Let's say we have three objects, A, B, and C with masses 11 kg, 12 kg, and 23 kg. It is clear there is an additive relationship here that has a real physical meaning. The third object really does have the equivalent amount of matter as the first two combined. This real relationship is played out with how physical laws play out. We can go down the philosophical rabbit hole here, but I don't think it is productive.

From the wikipedia article, a measurement in a given unit must be equivalent to that number times the unit, i.e. $11\ kg = 11 \times (1\ kg)$. Of course, multiplication can be defined in terms of addition. So we actually require that $$11\ kg = \underset{\text{11 terms}}{\underbrace{1\ kg + 1\ kg + \cdots 1\ kg}}.$$ In a real physical sense, the amount of stuff in eleven objects each with 1 kg mass really is equivalent to the amount of stuff in one object with 11 kg of mass.

So here is the main question: Is number of kilograms beyond 10 kilograms a valid unit of measurement? Let us denote it $kg_{10}$

Now our objects have "masses" $1\ kg_{10}$, $2\ kg_{10}$, and $13\ kg_{10}$. These aren't actually masses, they are "masses beyond 10kg". The additive relationship from our original unit of kg does not carry over to our new "unit" of kg$_{10}$.

Can we even say that $1\ kg_{10}$ + $1\ kg_{10}$ = $2\ kg_{10}$? What are we actually adding here? We are not adding kilograms! If I have an object that has mass 11 kg ($1\ kg_{10}$) and I increase its mass by 1 kg, it now has mass 12 kg ($2\ kg_{10}$) so it would seem that $1\ kg_{10}$ + $1\ kg$ = $2\ kg_{10}$.

So, maybe this is a stupid thing to worry about, but it just seems that we should have rigorous set of rules about what constitutes a valid unit of measurement. Those rules would include something about additivity and the relation to the thing being measured and not being able to just arbitrarily set zero anywhere.

I would argue that kg$_{10}$ is not technically a valid unit of measurement, and that this is reasonable since the unit of measurement is actually kilograms, which is a true unit of measurement (arithmetic operations on it make sense in terms of the physical thing it is actually measuring), but for whatever reason, we might be interested only in kilograms beyond 10.

Answer 1

I'm not sure what you mean by "valid" unit of measurement, but it is certainly impractical because, as you correctly identified, you cannot add properly. You could obviously define $1\ kg_{10} + 1\ kg_{10} = 12 \ kg_{10}$, but this gets clumsy at best...

Answer 2

The International Vocabulary of Metrology (VIM) states that a measurement unit should be a

real scalar quantity, defined and adopted by convention, with which any other quantity of the same kind can be compared to express the ratio of the two quantities as a number

So by the BIPM ( International Bureau of Weights and Measures ) the number of kilograms beyond 10 kilograms cannot be defined to be a unit, since it's not a ratio.

And quantity is expressed by the VIM as

property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference.

So it has to be connected to nature itself. Like the meter which was first defined as as one ten-millionth of the distance from the equator to the North Pole. But changes between one unit and other are still aceptable and must be possible. In the VIM there is a table of all the principals quantities.

I think that occurs because of the way we do math, you have state $1\ kg_{10} + 1\ kg_{10} = 2\ kg_{10}$ that wouldn't be true if we try to convert the numbers to kilograms, since $2\ kg_{10} = 12\ kg_{10} \neq 22\ kg = 1\ kg_{10} + 1\ kg_{10}$. So a unit should always respect our math principles and that would leave us only with ratio units, which are easily convertible, opposite to your created unit. But I guess any unit is possible to make, as long as you invent a different math to it.

Answer 3

If I've understood your system correctly, we do this all the time with temperatures: absolute temperature is measured in kelvins but degrees Celsius are effectively "kelvins beyond 273.16". (Loosely, the freezing point of water.) We routinely use them all the time to compare temperaures, but things go awry if we want to say that one thing is twice as hot as another.

Note that 30°C isn't 20°C plus 10°C unless we use $x$°C to mean two different things, namely a temperature of $x$ temperature steps above freezing, and an increment of $x$ temperature steps.

You might argue that we do something similar (but inconsistently) with weights when we treat them like mass: we're neglecting the buoyancy effect in air so there's a slight offset. This actually has to be taken into account when dealing with very precise mass measurenents.

And decibels are in fact meaningless on their own as a unit: a decibel is a power ratio of $\sqrt[10]{10}$ so the actual measurement unit is dBA, ie decibels above the threshold of hearing. On its own, a decibel is simply the number $\sqrt[10]{10}$ treated as a multiplication factor.