Magnitudes, Scalable Monoids and Quantity Spaces

Abstract

In ancient Greek mathematics, magnitudes such as lengths were strictly distinguished from numbers. In modern quantity calculus, a distinction is made between quantities and scalars that serve as measures of quantities. It can be argued that quantities should play a more prominent, independent role in modern mathematics, as magnitudes earlier. The introduction includes a sketch of the development and structure of the pre-modern theory of magnitudes and numbers. Then, a scalable monoid over a ring is defined and its basic properties are described. Congruence relations on scalable monoids, direct and tensor products of scalable monoids, subalgebras and homomorphic images of scalable monoids, and unit elements of scalable monoids are also defined and analyzed. A quantity space is defined as a commutative scalable monoid over a field, admitting a finite basis similar to a basis for a free abelian group. The mathematical theory of quantity spaces forms the basis of a rigorous quantity calculus and is developed with a view to applications in metrology and foundations of physics.