“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”?
Taking up Spacetime 
A related way to characterize structure is in terms of symmetry groups. The Ladyman/French school of structural realists seem to have lead the way on this idea. Here’s one way to put it:
Group Structural Realism (GSR): The existing entities described by quantum theory are organized into a hierarchy, in which a particular symmetry group occupies the most fundamental position.
As you suggest, precise definitions are important here. I worry that GSR and many other kinds of structural realism are prone to underdetermination. The reason is, a structure might itself have structure. Of course, this worry isn’t too compelling without a definition of structure. But in the case of symmetry groups, you can actually make it precise. (I once thought it through on the blogotubes, here: http://www.soulphysics.org/2009/01/group-structural-realism-part-1.html )
Bryan,
Thanks a lot for your comment, I obviously need to read your paper which is forthcoming in the British Journal for the Philosophy of Science, but also available at the PhilSci Archive.
The worry that you express is, of course, right on. But if you characterize the structure of a theory by the symmetry group of its fundamental level of entities, then you may be assuming “too much structure” of the structure already. I believe that my sketch (which really is the mathematical logicians way of characterizing structure) is more general than that and builds in as little structure into the structure as possible.
Of course, you will not be surprised to hear that I think that the worry concerning underdetermination still stands, however.
There is good reason for the structuralist to avoid a model-theoretic account of structure as relations.
Even if you don’t think there are any objects, there seem to be facts about how many ‘object-like’ structural features there are in the world. That is, we generally think we can determine the size of a set of objects like electrons (or in the case of GR say that the spacetime manifold is a continuum). So even if objects are really just manifestations of more structure, then we still out to be able to recover analogous facts about how many there are. If we treat ‘structure’ as set of relations defined on the domain A , then we cannot recover these facts–stipulating a relational structure doesn’t fix the cardinality of the domain of objects.
You can find a much more thorough treatment of this point in a paper to appear shortly in Synthese–a preprint can be found here (under ‘Ontic Structuralism’): http://www.andrew.cmu.edu/user/bjantzen/Research.html
Ben–
Thanks a lot for this post. Obviously, I also need to your your forthcoming paper.
What you say is exactly right, that’s essentially the problem I tried to highlight in my article “Challenging the spacetime structuralist” (Phil Sci 76 (2009): 1039-1051), although of course not as generally as you present it. So there are reasons for the structuralist to avoid the model-theoretic characterization of structure as a domain of objects and a set of relations that are defined over it. But should be adopted in its stead? There are different ways to characterize structure, e.g. by using tools from graph theory. But I just fail to see how you could possibly get rid of objects entirely in the sense that they can be fully reduced to manifestations of yet more structure without running into problems of individuality.
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