Thursday, April 28, 2011

A gauge theory of linguistics?

Here is a bit of idle speculation. In my reading on theoretical physics, I have learned something about gauge theories. A concise description is found, naturally, on the Wikipedia page, where it is explained that the root problem addressed by gauge theory is the excess degrees of freedom normally present in specific mathematical models of physical situations. For instance, in Newtonian dynamics, if two configurations are related by a Galilean transformation (change of reference frame), they represent the same physical situation. The transformations form a symmetry group, so then a physical situation is represented by a class of mathematical configurations which are all related by the symmetry group. Usually the symmetry group is some kind of Lie group, but it need not be a commutative (abelian) group. A gauge theory is then a mathematical model that has symmetries (there may be more than one) of this kind. Examples include the Standard Model of elementary particles.

So far so great, but what does this have to do with linguistics? Well, it seems to me that mathematical models of language are often encumbered by irrelevant detail or an overly rigid dependence on conditions that are in reality never fixed. A simple example would be that a typical generative grammar of a language (in any theory) depends critically on the vocabulary and the categories assigned to it. In reality, different speakers have different vocabularies, and even different usages assigned to vocabulary items, although they may all feel they are speaking a common language. There is a sense in which we could really use a mathematical model of a language that is flexible enough to allow for some "insignificant" differences in the specific configuration assigned to the language. There may even be lurking a useful notion of "Galilean transformation" of a language.

This idea is stated loosely by Edward Sapir in his classic Language. He applies it to phonology, where he explains his conviction that two dialects may be related by a "vowel shift" in which the specific uses or identities of vowels are changed, but (in a sense that is left vague) the "phonological system" of the vowels in the two dialects is not fundamentally different. This idea may help to explain how American English speakers from different parts of the country can understand one another with relative ease even though they may use different sets of specific vowel sounds.

This is all a very general idea, of course. Gauge theory as applied in physics is really quite intricate, and I do not know yet if the specifics of the formalism can be "ported" to the particular problems of linguistic variation in describing a "common system" for a language. But what better place than a blog to write down some half-baked ideas?


  1. This comment has been removed by the author.

  2. Hi,

    I know precious little about physics and I would not call myself a mathematical linguist either, but I think that something close to what you are suggesting has been tried by Keenan and Stabler in their book "Bare Grammar":

    I have not studied it in depth, but if I understand correctly they aim at characterizing natural languages in relatively simple, theory neutral algebraic terms, in order to capture real, structural properties of the object of study as opposed to artifacts of this or that framework.

  3. Yes, you're right about that. I am aware of this book, but never really read it and so I forgot about it. But I think it may be on the track I'm suggesting here. I worked with Keenan and Stabler when I was a student, and they were working on this book at that time. I think I'll have to read it properly now.

  4. I don't know much about physics and sadly have not had the chance to read Keenan and Stabler properly. But i like your "half baked ideas" and see how the concept us useful for describing the relationships between dialects or idiolects. It may also have bearing on what K & S called "structural invariants" of the grammar or those elements which always map to themselves under any structure preserving transformations. Thanks for the insight and bringing gauge theory to my attention.