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 real / 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 reals" 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". 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):
$$\mathbb{R}(T,L,M,I,\Theta,J)$$
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 $\mathbb{R}$. (This function has the property that given two real numbers $x,y$ it will output a number $k(x,y)$ which measures how similar $x,y$ are: $0 \le k(x,y) \le 1$, where $1$ is $100\%$ similar and $0$ is $0\%$ similar. Furthermore this function is the dot product of some Hilbert space $k(x,y) = \left< x,y \right>$ and has the property that $k(ac,bc) = k(a,b)$, which makes it useful when considering change of units in meausrements.)
We can define a similarity and positive definite kernel (which has the same properties as the function $k$) 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.
thir edit:
By the reference given in the answer below, one should think about the formal sums in the Laurent polynomial ring as being in two $T,T^{-1},L,L^{-1}$ variables: https://arxiv.org/abs/0711.4276.
Best Answer
No, it seems like a valid isomorphism. But why would it be meaningful, either? You say "no one would count subjectively these as dimensions" but if it is a valid isomorphism then one would have to count these as dimensions.