I was recently reminded (thanks to the comment from Emilio on my previous post) of some great work that was done by my former professors Ed Keenan and Ed Stabler, that I first learned about when I was a PhD student in the 1990s. The key publication for reference is their book Bare Grammar, published by CSLI in 2003. It is actually rather embarrassing that I had largely forgotten about this work while preoccupied with my own efforts in learning theory the past several years, since I should have used some of their ideas to further my own.
Bare Grammar is a largely atheoretical framework for describing syntactic structure. I won't present any details here while trying to highlight the major points. Keenan and Stabler begin by pointing out that human languages always have lexicons in which most elements are grouped into classes (like parts of speech) whose members are in some sense intersubstitutable in the "same" positions within the "same" structures. A simple example is afforded by the English sentences:
It is observed that Trevor can substitute for Nigel (and vice versa) without "changing structure," yielding equally grammatical sentences. The same can be said of the verbs laughed and cried. This means that the two sentences "have the same structure," in that they are each obtainable from the other by a sequence of "structure-preserving transformations." Hmm, this is starting to remind me of my previous post where I ruminated about gauge theory, as was suggested by the comment there.
Being mathematical linguists, Keenan and Stabler formalize all this. A structure-preserving map is defined as an automorphism of the grammar. An automorphism h can have fixed points x, such that h(x) = x. The syntactic invariants of the grammar are fixed points of every automorphism. In the lexicon, these are identified with the function words---words that cannot be substituted without changing structure. By using some fancier mathematical operations (power lifting and product lifting), it is shown how certain properties of higher order can also be fixed points, which correspond to such things like invariant properties of expressions, and invariant relations and functions. Much of the book is engaged in evaluating potential invariants of these kinds.
A major point is that natural languages all have such invariants---an interesting fact, to be sure, which is not at all necessary from the basic considerations of formal language theory. It would be easy to devise a formal language that didn't have anything like a function word, yet natural languages all have such things.
I hope to spend some more quality time with this book in the near future, and may have some more posts about it. In the meantime, I recommend it to anyone interested in a novel theoretical approach to a range of interesting linguistic facts that are not seriously dealt with elsewhere.