Mathematicians study a wide range of mathematical structures, but the most commonly associated with mathematics are number systems like integers or real numbers. However, mathematics can be done without explicit reference to numbers, as demonstrated by the ancient Greeks, who used geometric operations as substitutes for arithmetic operations in Euclidean geometry. In modern physics, physical quantities are measured and manipulated using the real number system or related systems. The use of units and dimensional analysis has been effective in physics, but does it have a solid mathematical foundation? Different approaches, such as the parametric and abstract models, aim to formalize dimensionful quantities in mathematics. The limitation of performing dimensionally inconsistent operations helps catch errors and provides conceptual clarity.
https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/