The Wikipedia article on dimensional analysis says:
the dimensions form an abelian group under multiplication
This is used to justify the manipulation of ratios of incommensurable quantities. My question is, how do we know that the (physical) dimensions form an abelian group under multiplication? Is it just that at the outset we decided that abelian groups are easy to deal with, and so designed the dimensional system explicitly to form an abelian group, or is the abelian-groupness of physical dimensions derived from more basic properties?