“Structure” follows me in my dreams these days: my paper on spacetime structuralism has just appeared, I am preparing to teach a graduate seminar on structure in mathematics, philosophy, and physics, and I have been asked to referee two articles on structural realism in the last two weeks. The question that haunts me most is the issue of defining what structure is. Many authors simply behave as if it were entirely clear what it is. Well, it isn’t–at least not to me.
In my paper, I simply said that a structure is an ordered pair of a set of (physical) objects and a set of (concrete) relations defined on them. I discovered that in mathematical logic, a structure is sometimes defined as an ordered triple (A, tau, I) consisting of a domain of objects (or universe) A, a “signature” tau, and an interpretation function I of tau. The signature essentially consists of relation and function symbols defined on A. The job of I then is to link the relation and function symbols to relations and functions defined on A. To put things a bit simpler, then, a structure, as far as a mathematical logician is concerned, consists of a set of objects and sets of functions and relations defined on the set of objects. Apparently, if the set of functions is empty, as I assumed it to be in my paper, then one speaks of a relational structure.
OK, fair enough. I assume that we mostly talk about relational structures then when we say “structure” in the foundations of physics. Does this sound right? Any alternative ways of characterizing “structure”?