The n-Category Café

What branch of mathematics is sufficiently general to apply both to quantum physics and linguistics?

What might be an application of category theory in linguistics? I read in the announcement:

Many such structure have been found useful across these fields, such as Frobenius algebras and bialgebras such as Hopf algebras. They have also showed up in grammatical and vector space models of natural language to for example encode meaning of verbs and logical connectives.

Ah, Frobenius algebras. Can anyone indicate briefly how bialgebras help to “encode meaning of verbs”? And how category theory helps them in helping?

This stuff has a somewhat peculiar status in linguistics … most linguists don’t understand very much about it either. Lambek’s original ideas were ignored by Chomsky and his students but taken on by Richard Montague, and introduced to many linguists in the early 70s as ‘Montague Grammar’ and its variants, where it become the basis of ‘formal semantics’ (semanticsarchive.net and quite a number of textbooks). But the Montague/Lambek semantics is usually combined with some more ‘mainstream’ form of theory of syntax (Kratzer and Heim, _Semantics in Generative Grammar_ is the most popular approach, Lexical-Functional Grammar + ‘glue semantics’ (an application of what Valeria de Paiva calls ‘rudimentary linear logic’, lacking pretty much everything that you guys consider interesting) is what I like). But the categorial and type-logical grammarians have been working away on variants of Lambek’s approach the whole time, too.

The Pregroup Grammars that Sadrzadeh is working on are Lambek’s successor to the original Categorial Grammar. In practical terms, the problem with both old and new Lambek is that there’s too much ‘math up front’, whereas the mainstream theories can be appreciated on a relatively intuitive basis, and maybe some math comes later, but at any rate the intuitive grasp is enough to be helpful with field work on previously un- or little- described languages. More generally, there are lots of people who can notice interesting things about language structure and say something sensible about it using one of the mainstream theories, but who probably can’t manage to learn much category theory.

But, still, I’d like to be a fly on the wall at this conference.

It could be an interesting get together.

I used to closely follow mathematical linguistics but I stopped because the field seemed to be rehashing the same old facts in yet a new formalism. These are my impressions from a good while ago and I hope things have changed.

A general writer may be interested in meaning and grammar, as does a linguist doing fieldwork to describe the language of a foreign culture.

In (then) mathematical linguistics, meaning or semantics had come to be equivalent to truth conditional semantics where say an adjective like “beautiful” or “constipated” has no internal meaning and what only matters is how it maps individuals to truth values, maybe in possible worlds.

Likewise syntax and grammar in hard core mathematical linguistics is usually only concerned with those aspects relevant to truth conditional semantics. The tricky thing there is quantifiers like “every” and “a” as in the sentence “every boy kissed a girl”. There are all sorts of clever schemes, often involving generalized quantifiers where one takes a sentence given as a list of words with each word having a pre-defined operator. Then with practically no control mechanism, the words’ operators can operate on their neighbors to build new operators that operate on their neighbors and so on. In the end these operators build up a parse tree and move a quantifier that was somewhere internal to the tree to the appropriate position outside of predicates representing meaning, such as kiss(x,y).

So my impression of mathematical linguistics is that it is mainly interested in quantifiers and mechanisms to relate their position in a sentence to truth value mappings. In the actual language that real people speak quantifiers are rare and the tricky cases that MLs like to analyzing involving 2 or more are very very rare and when spoken by real people often wind up being technically wrong.

So yes ML can be very interesting as clever BUT they had restricted the domain of sentences they are interested in to the point that real people don’t say them and they are mostly made up.

I hope there are new developments in ML that are not centered around quantifiers.

There are, although quantifiers are still pretty big. Lisa Matthewson’s work might strike you as interesting (how various semantic things work out in Salish languages).

Interesting to see advanced math being applied in linguistics. A workshop a decade ago (link below) highlighted “function dynamics” as a promising tool in the study of the origin of life, evolution and construction of artificial life. The idea was that in math you have (a) dynamical relations between objects and (b) structure that reflects rules for this dynamics, a familiar distinction for many here.

The idea is that life arise from the interaction of simple objects following simple rules, like chemistry in the primordial ocean, but that life then defines new symbolic states and/or stable attractors, which in turn control or are controlled by other symbolic states. So (a) and (b) symbols and rules, gives rise to each other.

According to the scientists at the time, there wasn’t math around to handle this. (An outrageous claim perhaps?) Anyway, there may or may not be a shared ground with linguistics, with their symbols and rules?

Refering to p368 here:
http://mitpress.mit.edu/journals/ARTL/Bedau.pdf

There are two issues here about the modeling of language using category theory.

The first is whether quantifier scope, verb argument structure or some other semantic phenomenon in natural language can be modeled using category theory. Even better if category theoretic models are superior to older mathematical logic based (say, model-theoretic) approaches. Here, one can think of phenomena like genericity that are hard to capture using older approaches. For example, I might say:

Politicians are thieves.

The above statement is about the whole class of politicians. However, if I say:

All politicians are thieves.

you might disagree, saying, what about that Obama, he seems pretty straight to me. However, I might be particularly pissed off with politicians as a class, and respond:

No, all of them are thieves!

knowing perfectly well that some politicians are honest and acting in the public interest.

In these situations, I think the subtle interplay between quantification, genericity and scope should be better modeled using category theoretic approaches.

That said, the second and deeper problem with mathematical linguistics is not whether category theory is better than model theory, but with the fact the most influential theories of language (say, Chomskian Linguistics or its bete noire, Cognitive Linguistics) are both theories of language as a mental faculty. Cognitive and Generative linguists are less interested in whether quantifiers or verbs can be modeled in some mathematical formalism and more interested in how quantifiers and verb argument structure arises from the underlying principles of mental organization.

For category theory to make a real impact in the study of language (like formal grammars did in the early days of generative linguistics) one would have to combine a category theoretic meta-language with a substantive account of how certain (not all) category theoretic principles are good theories of how the mind organizes language.

Newly revised (and how can this be extended with Category Theory?):

The Language of Thought Hypothesis (LOTH)

“The Language of Thought Hypothesis (LOTH) postulates that thought and thinking take place in a mental language. This language consists of a system of representations that is physically realized in the brain of thinkers and has a combinatorial syntax (and semantics) such that operations on representations are causally sensitive only to the syntactic properties of representations. According to LOTH, thought is, roughly, the tokening of a representation that has a syntactic (constituent) structure with an appropriate semantics. Thinking thus consists in syntactic operations defined over such representations. Most of the arguments for LOTH derive their strength from their ability to explain certain empirical phenomena like productivity and systematicity of thought and thinking.”