I wondered if there are any theoretical backgound or "formalization" of dimensional analysis. I had an attempt on doing this, by providing some axioms and then "deriving" how dimensional analysis works. In my attempt I wondered if I was somewhat successful, or the reasoning is just circular or unconvincing/unnessecarily complicated. Also: is axiom II nessecary? Are there better axioms that I could have made?
Axioms:
I. We may freely create any unit we want and attach a meaning to it.
II. There is a unit for unitless quantities; the unitless unit.
III. A "measurement" is something that can be expressed as a product between a mathematical object ($M$) and a unit ($u$): $M\cdot u$
Now by axiom we can create a unit called seconds and attach the meaning to it that a secound is the same as "taking the fixed numerical value of the caesium frequency, ΔνCs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to $s^{−1}$"
Def 1: A combined unit is a unit that is defined by a combination (and/or scaling) of one or more units.
Def 2: A base unit is a unit that can not be expressed in terms of other units, in our current units system.
Def 3: A unit system is a set of units that can be used to express measurements.
Now we may say that the meaning of the unit "day" is that it represents "roughly 86400 s". And that the meaning of the unit $m/s$ is that it represents the unit $m$ (meter) devided by the unit $s$.
What we could do is use "day" as a base unit and then make "second" a combined unit. We could for example say that the meaning of day is "the time from sunrise to sunrise" and that a second is 1/86400 of a day. Thus our choise of base units is in some sense arbitrary.(*)
Let us use axiom III. to study units and develop the rules of dimensional analysis. Let $M_1, M_2, M_3$ be mathematical objects (numbers, vectors, tensors, etc.) and $u_1, u_2, u_3, ...$ be units.
Theorem 1: We can add two measurements with the same unit, resulting in a new measurement with the same unit.
Proof: Suppose we have $M_1\cdot u_1$ and $M_2\cdot u_1$. Then we have:
$$ M_1 \cdot u_1 + M_2 \cdot u_1 = (M_1+M_2)\cdot u_1 = M_3\cdot u_1 $$
"Collary 1": If we add two measurements with different units, the result is not a measurement.
Proof: If we have $M_1 \cdot u_1 + M_2\cdot u_2$ we can not factor this in a way such that we get $M_3 \cdot u_3$, where $M_3$ is a pure mathematical object and $u_3$ is a unit that can be expressed in terms of $u_1$ and $u_2$.
Theorem 2: multiplying measurements is well defined and results in a new measurement whose unit is the units of the measurement factors multiplied.
Proof: $(M_1\cdot u_1) \cdot (M_2 \cdot u_2)= M_1 \cdot M_2 \cdot u_1 \cdot u_2 = (M_1\cdot M_2)\cdot (u_1 \cdot u_2) = M_3 \cdot u_3$
Collary 2: If we devide a measurement by another measurement, the resulting measurement will have units of the dividend inversly multiplied by the divisor.
Proof: Same as Theorem 2, but with $\frac{1}{M_2\cdot u_2}$ instead of $M_2 \cdot u_2$.
Also, it follows from II. that any mathematical object can be expressed as a unit. Here is why: Let $\varnothing$ be the unitless unit. Then we have that $M\cdot \varnothing = M$.
What do you think? I think the axioms may be poorly formulated, but otherwise I think it kind of works. If we assume these three things, then we get out the usual rules of dimensional analysis. I am also a little uncertain if axiom II is needed, but it provides a great parallell to the identity element in vector spaces allowing us to adapt a sort of view of the units as unit vectors in a vector space.
The point (*) can be elaborated: if we have units that sort of form a "vector space", we may chose a set of base units that spans this vector space. For example: we have the units $m$ and $s$. We may also chose the base units $m/s$ and $s$, making $m=(m/s)\cdot s$. If, however, we only chose $m/s$ as a base unit, then this unit system will not be able to express the same measurements as the unit system consisting of $m$ and $s$ as base units. Thus, the two unit systems would be different.