  A rigorous mathematical theory of dimensional analysis, systematically
accounting for the use of physical quantities in science and engineering,
perhaps surprisingly, was not developed until relatively recently. We claim
that this has shaped current mathematical models of theoretical physics, which
generally lack any explicit reference to units of measurement, and we propose a
novel mathematical framework to alleviate this. Our proposal is a
generalization of the usual categories of algebraic structures used to
formulate physical theories (groups, rings, vector spaces...), herein dubbed
dimensioned, that can naturally articulate the structure of physical dimension.
Our goal in the present work is not so much to define an algebraic theory of
physical quantities - this has already been done - but to define a theory of
algebra informed by how physical quantities are used in practice. We conclude
by studying the dimensioned analogue of Poisson algebras in some detail due to
their relevance in Jacobi geometry and classical mechanics. These topics are
further explored in sequel papers by the author.
