Excuse my naive question and please let me explain it:

In everyday life we experience 3 spatial "dimensions" + time etc. Usually the 3 dimensions are represented by a coordinate system and mathematically as the vector space $R^3$. In constrast, the word "dimension" in dimensional analysis has a more objective realism which is based on physical measurements. For instance let $l$ be a transcendental number.

Then $1,l,l^2,l^3,\cdots$ will be *linearly independent* over the rational numbers, and hence can serve as a basis in a vector space. For instance, instead of saying the point in spatial dimension has coordinates $(x,y,z)$ we might as well write:

$$x+y \cdot L + z \cdot L^2$$

where $L$ represents a "transcendental number over the rationals" and is associated with meter or kilometer etc.

Take for instance the derived unit $s/t^k$ for $k=0,1,2$ where $s$ denotes length and $t$ denotes time. Then $s$=length, $s/t =$ velocity, $s/t^2=$ acceleration. If we view these as a basis of a vector space, then this vector space has 3 dimensions. But no one would count subjectively these as dimensions.

On the other hand the seven basic SI-units (or 6 if you do not want to count $mol$), could be seen as a transcendental numbers over the rationals or reals, and hence powers of those transcendental numbers could give a basis for vector spaces. Monomials of these transcendent numbers would correspond, as is done in dimensional analysis, to derived units. Adding and subtracting for instance $LT^{-1}+M$ would give a point in the "space of physical quantities" (**Edit**: and indicate the measured velocity and mass of an object). As a vector space over the reals this space has infinite dimension but has transcendence degree of $7$. Every derived / basic physical quantity measured by SI-Units would correspond to a point in this space / field of transcendence degree $6$ (or $7$ if you count $mol$ which I will not do):

$$\mathbb{Q}(T,L,M,I,\Theta,J)$$

Excuse my naive question: Is there any reason from physics to discard this point of view?

**Edit**:
It seems that the main idea of this question can be implemented through the ring (Laurent polynomial ring) $L:=\mathbb{R}[T,T^{-1},L,L^{-1},M,M^{-1},I,I^{-1},\Theta, \Theta^{-1},J,J^{-1}]$

**second Edit**:
I was asked to give at least one application of this idea, which I would like to do:

Application:

Let $k(x,y) = \frac{xy}{x^2+y^2-xy}$ for $x \neq 0,y \neq 0, x,y \in \mathbb{R}$ be a Jaccard-Similarity / positive definite kernel defined on $\mathbb{R}$.

We can define a similarity and positive definite kernel on the Laurent polynomial ring $L$ as :

$$K(x,y) := \frac{1}{N_x + N_y - N_{xy}} \sum_{X_i^{\alpha_i}=Y_j^{\beta_j}} k(a_i,b_j)$$

for $x = \sum_{i} a_i X^{\alpha_i},y = \sum_{j} b_j X^{\beta_j}$ and $X = (T , L, M, I, \Theta,J)$, $\alpha_i, \beta_j \in \mathbb{Z}^6$ and $X^{\alpha_i},X^{\beta_j}$ are multinomials, and $N_x = $ number of nonzero $a_i$, $N_y =$ number of nonzero $b_j$, $N_{xy} =$ number of $(X^{\alpha_i} = X^{\beta_j})$.

Since $k(ca,cb) = k(a,b)$ for all $a,b,c \neq 0$, we deduce that $K(c \cdot x,c \cdot y) = K(x,y)$ for all $x,y \in L$, $c \neq 0, c \in \mathbb{R}$. Hence this (or any other similarity and positive definite kernel $k$ with $k(ca,cb) = k(a,b)$. This is to make sure, that rescaling of physical units, does not change the similarity between objects.) gives us a possibility to measure the similarity / inner product of two physical objects $A,B$ each of which is defined through measurements $x = \sum_{i} a_i X^{\alpha_i}$ and $y = \sum_{j} b_j X^{\beta_j}$.

Since there are different possibilities to measure similarites / define positive definite kernels, there should be different possibilities to define equality / similarity between two physical objects $A$ and $B$.

Hence aposteriori $L$ is a Hilbert space by the Aaronszajn-Kolmogorov theorem.

Example:

The meaning of this kernel is to compare two physical objects. For instance $A = 10 m/s + 1 kg$, $B = 9 m/s + 2kg$, $C = 1 m/s^2+10kg$. Then $K(A,B) = 226/276 = 0.8278$, $K(A,C) = 10/91= 0.10989$, $K(B,C) = 5/21 = 0.2381$. Hence $A$ is most similar to $B$, $B$ is most similar to $A$, $C$ is most similar to $B$, and $A,C$ are the most dissimilar physical objects in this list.

notlinearly independent over the reals. So I stopped reading. $\endgroup$7more comments