Galileo spoke of the universe as a book, written in the language of mathematics. Following his metaphor, we ask – why does the language of physics include so many different kinds of math, some simple, others extremely complex? Like any language, this one has both a formal structure and a web of semantic relationships, that give contexts of meaning to its terms and expressions. In physics the formal, mathematical structures have been studied in depth, and despite their difficulties, this aspect of the language is quite well understood. The semantic aspect of physics, though, remains unexplored. We tend just to take it for granted that the many terms that appear in the equations are physically meaningful – terms like space and time, mass and charge, etc. Yet none of these is observable or even definable by itself, apart from the contexts given by other terms in the language. Each physical variable and constant appears in many key equations, which together define its meaning in relation to other terms. This essay considers what it takes for a language to do this – to make all its expressions meaningful in terms of each other. This kind of semantic self-sufficiency is unique to physics, since in any other language, expressions have meaning primarily by referring to things beyond the language itself. But the language of the physical world is fundamental; there’s no deeper level of meaning to which it can refer. We consider the diverse mathematics involved in atomic structure to illustrate how physics is able to give meaning to the complex variety of facts and regularities on which everything else in the universe depends.
Conrad Dale Johnson