The Math of Language

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My cousin Taimi is a linguist, and at our Big Family Reunion last month she told me that some people tried to develop a mathematical symbolism for language (language in general, regardless of the actual language spoken) and it worked well for speech that was meant to be informative. For speech patterns that were social and culturally based, though (when and how to thank someone, for example), the math language of language fell apart. It just couldn’t be applied universally correctly.

So when we returned to New York I looked up what this might be and found myself, well, confused. The closest example that I could find (which may or may not be what Taimi was referring to) was in a Keith Devlin column from 1996. He used an example from X-bar theory. (Doesn’t X-bar sound like a drinking establishment? In fact, there is such a place in Los Angeles, although it looks geared more towards Gen X than mathematicians).

In linguistics, X-bar theory seems to be a way of describing all sorts of phrases in a recursive fashion. So if you want to talk about the noun cat, you might add on some descriptions like gray or fabulous. The grammatical rules allow you to change X (a noun, a verb, a proposition, an adjective, etc.) into something that’s modified with things called complements and adjuncts, and the result is called X-bar and should be written \overline{X} but is often written just as X’ because it’s hard to typeset the whole “bar” thing. So a noun-bar (N’) would be fabulous gray cat (instead of just cat) or food bowl (instead of just bowl)

Then you can move up to X Phrases. An X Phrase is an optional specifier, followed by an X’, and then maybe some Y Phrases. This is written along the lines of
XP→(specifier)X’YP*

As an example a noun phrase (NP) would be something like “the fabulous gray cat” where the is the specifier and fabulous gray cat is N’. Another noun phrase is “the food bowl” where the is a specifier and food bowl is N’.

From this you can form a verb phrase (VP) “sees the food bowl”. Here the V’ is just the single verb sees but it’s followed by the noun phrase the food bowl. In other words, VP→(specifier)V’NP*. This is where the recursive part comes in, using Y phrases to build X phrases.

Presumably the next step would be to combine “the fabulous gray cat” and “sees the food bowl” into an actual sentence, but building sentences involves a whole other set of rules.

In many ways this reminds me of diagramming sentences, except that instead of starting with the sentence and breaking it down, the rules have to be developed in such a way that they can be described regardless of the actual sentence or even actual language.  Because the beauty and complication of this is that all of the X-bar rules apply regardless of whether your noun phrase is “the fabulous gray cat” or “el gato gris fabuloso” or “η μυθική γκρίζα γάτα”.

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