I attemted to create a formal definition of dimensional analysis. Here were the results. What is good and bad about this formalization, and what problems are there?
Let $\mathbf{F}$ be a set of fundamental units. Then the unit space of $\mathbf{F}$ is the abelian group $(\mathbf{U},\cdot)$ satisfying the following axioms.
- The identity element of $\mathbb{U}$, denoted $ø$, is not in $\mathbf{F}$.
- $\forall f \in \mathbf{F}, f \in \mathbf{U}$
A dimensional space over a field $\mathbf{Q}$ and unit space $\mathbf{U}$ is a set $\mathbf{D} = \mathbf{Q} \times \mathbf{U}$ together with binary operators $+,-,*,/$ which satisfies the following axioms.
- $\mathbf{D}$ is an Abelian group under $*$, with identity $(1,ø)$.
- $\forall a,b \in \mathbf{Q}\forall c,d \in \mathbf{U}, (a,c)*(b,d) = (a*b,c\cdot d).$
- $\forall a,b \in \mathbf{Q}\forall c,d \in \mathbf{U}, (a,c)/(b,d) = (a/b,c\cdot d^{-1}).$
- $\forall a,b \in \mathbf{Q}\forall c \in \mathbf{U}, (a,c)+(b,c) = (a+b,c).$
- $\forall a,b \in \mathbf{Q}\forall c,d \in \mathbf{U},c\ne d, (a,c)+(b,d)$ is not defined.
- $\forall a,b \in \mathbf{Q}\forall c \in \mathbf{U}, (a,c)-(b,c) = (a-b,c).$
- $\forall a,b \in \mathbf{Q}\forall c,d \in \mathbf{U},c\ne d, (a,c)-(b,d)$ is not defined.
A dimensionless quantity in dimensional space D is an element $(a,b)$ of $\mathbf{D}$ such that $b = ø$.
