I once complained that my student Brendan Fong said ‘semantics’ too much. You see, I’m in a math department, but he was actually in the computer science department at Oxford: I was his informal supervisor. Theoretical computer scientists love talking about syntax versus semantics—that is, written expressions versus what those expressions actually mean, or programs versus what those programs actually do. So Brendan was very comfortable with that distinction. But my other grad students, coming from a math department didn’t understand it… and he was mentioning it in practically ever other sentence.
In 1963, in his PhD thesis, Bill Lawvere figured out a way to talk about syntax versus semantics that even mathematicians—well, even category theorists—could understand. It’s called ‘functorial semantics’. The idea is that things you write are morphisms in some category while their meanings are morphisms in some other category There’s a functor which sends things you write to their meanings. This functor sends syntax to semantics!
But physicists may not enjoy this idea unless they see it at work in physics. In physics, too, the distinction is important! But it takes a while to understand. I hope Prakash Panangaden’s talk at the start of the Simons Institute workshop on compositionality is helpful. Check it out: