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September 11, 2010

Categories and Information in Oxford

Posted by John Baez

What branch of mathematics is sufficiently general to apply both to quantum physics and linguistics? You guessed it:

ANNOUNCEMENT

The Quantum and Computational Linguistics groups of the Oxford University Computing Laboratory will host a three-day workshop on the interplay between algebra and coalgebra that can be thought of as information flow, and its applications to quantum physics and linguistics.

TOPIC

The aim of the workshop is to bring people together from the fields of quantum groups, categorical quantum mechanics, logic, and linguistics, to exchange talks and ideas of a (co)algebraic nature, about the interaction between algebras (monoids) and coalgebras (comonoids) that can be thought of as “information flow”. 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.

PROGRAMME

The precise schedule is not yet fixed, but speakers include:

* Samson Abramsky (Oxford)

* Daoud Clark (Hertfordshire)

* Stephen Clark (Cambridge)

* Bob Coecke (Oxford)

* Lucas Dixon (Edinburgh)

* Bertfried Fauser (Birmingham)

* Mai Gehrke (Nijmegen)

* Helle Hansen (Eindhoven)

* Bart Jacobs (Nijmegen)

* Shahn Majid (London)

* Michael Moortgat (Utrecht)

* Michael Müger (Nijmegen)

* Peter Hines (York)

* Alessandra Palmigiano (Amsterdam)

* Benjamin Piwowarski (Glasgow)

* Anne Preller (Montpellier)

* Stephen Pulman (Oxford)

* Keith van Rijsbergen (Glasgow)

REGISTRATION

Registration is free, but for logistic purposes, please inform the organizers, Chris Heunen and/or Mehrnoosh Sadrzadeh, if you plan to attend.

TRAVEL AND ACCOMODATION

For more information, please see the website, which is continually updated.

SPONSORS

The workshop is made possible by funds of the British Council and Platform Beta Techniek, and cooperation with the Institute for Logic, Language and Computation at the University of Amsterdam, the University of Utrecht, and the Radboud University Nijmegen.

BONUS

And if you’ve read all this way, why don’t you go to the website and hover your mouse cursor over the title?

Posted at September 11, 2010 6:53 AM UTC

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Re: Categories and Information in Oxford

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?

Posted by: Urs Schreiber on September 11, 2010 9:09 PM | Permalink | Reply to this

Re: Categories and Information in Oxford

Urs wrote:

What might be an application of category theory in linguistics?

I’m not an expert on this, but maybe my feeble reply will prompt experts to say more.

In 1956, the famous linguist Noam Chomsky described a hierarchy of formal grammars: that is, rules for forming strings of symbols. Besides its importance in computer science, this ‘Chomsky hierarchy’ was a preliminary attempt to size up the problem of how humans can generate grammatical sentences.

Anyone with a drop of category-theoretic blood in their veins need only take one look at the Chomsky hierarchy, and they’ll wish it was described in the language of category theory.

Indeed, the grammars in the lowest level in the Chomsky hierarchy, the regular grammars, are in one-to-one correspondence with nondeterministic finite-state machines. The grammar generates exactly the language that the corresponding machine accepts.

Later, Samuel Eilenberg, one of the fathers of category theory, shocked a lot of people by leaving algebraic topology and devoting a years to work on finite-state machines from a category-theoretic perspective:

  • Samuel Eilenberg, Automata, Languages and Machines, Vol. A, Academic Press, New York, 1974.
  • Samuel Eilenberg, Automata, Languages and Machines, Vol. B, Academic Press, New York, 1976.

Then, later, Joachim Lambek did a lot of work on category theory and grammar

… but I’ve got to go make breakfast now!

Posted by: John Baez on September 12, 2010 5:00 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

Anyway, Lambek’s work on what people call the “Lambek calculus” was apparently quite influential, and helped initiate a subject called categorial grammar. Here’s an introduction to some aspects of that:

  • Michael Moortgat, Categorial type logics, Chapter Two in Handbook of Logic and Language, eds. J. van Benthem and A. ter Meulen, Elsevier, to appear.

and here’s his summary of the goal:

The central objective of the type-logical approach is to develop a uniform deductive account of the composition of form and meaning in natural language: formal grammar is presented as a logic — a system for reasoning about structured linguistic resources. In the sections that follow, the model-theoretic and proof-theoretic aspects of this program will be executed in technical detail. First, we introduce the central concept of ‘grammatical composition’ in an informal way. It will be useful to distinguish two aspects of the composition relation: a fixed logical component, and a variable structural component. We discuss these in turn.

I don’t understand this stuff very well, but it seems he’s trying to develop a formal logic suitable for natural language: you’ll see analyses of phrases like ‘the mathematician whom Kazimierz talked to’, and so on.

I don’t at all understand the new stuff that Mehrnoosh Sadrzadeh is doing, so someone else will have to comment on that.

Posted by: John Baez on September 12, 2010 6:18 AM | Permalink | Reply to this

Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

George Kenneth Berger emailed me to say:

“Jonathan: A bit of history. Lambek did not help “initiate” categorial grammar. Starting around 1844, it’s implicit in the work of Bolzano, but was quickly forgotten. Before 1910 Husserl resurrected the basic notion, after he had read Bolzano on it. Then Lesniewski discussed it in his classes, inspired by Husserl. In 1936, Ajdukiewicz published the first complete such grammar in a classical paper. (Lesniewski also mentioned the notion in at least one later paper.) Ajdukiewicz’s paper has an important technical flaw, but the notion is fully explicit. Later on, Bar-Hillel wrote papers about Bolzano and Husserl. A bit later, the Swedish philosopher-logician Jan Berg wrote and published an Uppsala Doctoral Dissertation on ‘Bolzano’s Logic,’ and Rolf George and van Benthem wrote on Bolzano. I know nothing about the Lambek Calculus itself, but I do know that it was developed decades after Ajdukiewicz. My best teacher in philosophy was Ajdukiewicz’s last pre-War student, who told me much of this. It’s all been documented. I don’t know if Lambek mentions any of this anywhere, but this material ought to be better know than it is. Polish logic is one of Poland’s most important cultural achievements, it’s members were familiar with Husserl and Bolzano, and thanks to the influence of Brentano they were very careful philosophers and logicians who lacked the revolutionary spirit of much of the Vienna Circle.”

Posted by: Jonathan Vos Post on September 13, 2010 6:35 PM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

But what does any of this history, interesting though it is, have to do with categorial grammar, i.e., grammar studied with tools and concepts of category theory?

Posted by: Todd Trimble on September 13, 2010 10:13 PM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

Terminologically, in a sense, nothing: the ‘category’ in ‘categorial grammar’ is generally understood to use ‘category’ as a fancy word for ‘part of speech’. But there is a category-theoretic take on it, for example you can view semantic interperetation as a kind of functor. Lambek might be the one who worked out the connections.

Posted by: Avery Andrews on September 13, 2010 10:38 PM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

Really? That’s interesting! So people were using the phrase “categorial grammar” (or the equivalent in Polish, perhaps) before any connections with category theory were made?

Even so, I understood John’s use of the phrase as referring to category theory, and was saying that the category-theoretic study of grammar more or less began with Lambek (who was studying aspects of grammar in terms of monoidal biclosed categories; at the time he called them “residuated categories”). That claim wouldn’t make seem to make as much sense if John actually meant “categorial” in this other way you mention.

Posted by: Todd Trimble on September 14, 2010 2:12 AM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

Yes, categorial grammar started up in the 30s. I think John was referring to category-theoretic connections, which do exist, although linguists on the whole know very little about them.

I’m interested, but whether can transform the interest into any useful result remains to be seen.
Here’s an effort.

Posted by: Avery Andrews on September 14, 2010 2:52 AM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

Todd wrote:

So people were using the phrase “categorial grammar” (or the equivalent in Polish, perhaps) before any connections with category theory were made?

Avery wrote:

Yes, categorial grammar started up in the 30s.

Hmm! Like I said, I know very little about this stuff. I knew the category theorist Lambek had done a lot of work on grammar, so I looked him up on Wikipedia and saw a reference to an article on the Lambek calculus, which said:

Joachim Lambek proposed the first noncommutative logic in his 1958 paper “Mathematics of sentence structure” to model the combinatory possibilities of the syntax of natural languages. His calculus has thus become one of the fundamental formalisms of computational linguistics.

but referred to a ‘main article’ on categorial grammar. This article said:

A categorial grammar shares some features with the simply typed lambda calculus. Whereas the lambda calculus has only one function type ABA \to B, a categorial grammar typically has more.

and it gave one online reference, namely to Moortgat’s paper Categorial type logics.

Since Lambek is famous for noting that the simply typed lambda calculus is ‘just the theory of cartesian closed categories’ (a bit of an exaggeration, but still a great idea), and type theory has been deeply embraced by category theorists, I jumped to the conclusion that ‘categorial grammar’ has something to do with categories.

It obviously does have something to do with categories, but now I’m getting the feeling that this connection hasn’t been fully exploited. For example, now that I look at it more carefully, I see that Moortgat’s paper uses the language of proof theory very heavily, but not the language of category theory.

So, there should be a big opportunity here for someone with the right background! All this stuff looks ripe for an explicitly category-theoretic treatment — and such a treatment usually makes some things obvious that weren’t obvious before.

Somebody should jump in and try it! Why should Lambek have all the fun?

Posted by: John Baez on September 14, 2010 6:09 AM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

Indeed, the connections between categorial grammars and category theory are, so far as I’m aware, largely untapped. The goal for my dissertation is to tap it :) I’ve already started some work on a modification to CCG for supporting free constituent order (non-formal presentation) but I haven’t finished categorifying it yet. Part of the problem I’m running into is that it’s similar to too many things, so it’s taking a while to nail down exactly what it is.

Posted by: wren ng thornton on September 14, 2010 6:33 AM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

Well that’s pretty much my motive for looking in here every now and again, although I don’t have the right background. Carl Pollard made a bit of an attempt to use topos theory in his “Higher Order Grammar” theory a few years ago, but has stopped talking about it, for some reason. Chung-chieh Shan wrote a paper about using monads for natural language semantics, but seems to stopped talking about them to linguists, although he does still talk about using continuations.

Maybe Bob’s group will be the beginning of it, they are getting into a fair range of linguistic structures now, although I don’t know how they’d go with some of the more hardcore syntax problems such as case, agreement and word-order in Icelandic.

Posted by: Avery Andrews on September 14, 2010 6:49 AM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

The connection between categorial grammars and categories was already discussed by Lambek at a conference in Tuscon in 1985 and published in 1988 in a volume edited by Dick Oehrle, Emmon Bach, and Deirdre Wheeler. The basic ideas are that

(1) a Lambek-style syntactic calculus (essentially an intuitionistic propositional logic with no structural rules) can be reified as a biclosed monoidal category;

(2) the space of meanings expressed by linguistic expressions can be thought of as a cartesian closed category, more specifically a topos; and

(3) semantic interpretation of linguistic expressions is a biclosed monoidal functor from the first category to the second. This way of thinking of things reveals the underlying naturalness and elegance of the logical semantics for natural language proposed by Montague in the late 1960s.

It is true that I have not talked much about categorical semantics of natural language recently, because there still are not many linguists who are familiar with categories. However, there does seem to be increasing interest in applying categorical ideas in linguistic theory in the past year or so and so I have started talking about it again!

Posted by: Carl Pollard on June 23, 2011 7:44 AM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

Lambek’s seminal ‘58 paper is also credited for the first appearance of Linear Logic. That’s what makes it a great paper, it’s influence goes well beyond it’s scope. The craziest thing is that 50 years after he completely revises his views and proposes pregroups for grammar; categories with duals!

Posted by: bob on September 14, 2010 1:59 PM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

There’s a essay about some of this by Claudia Casadio in E. B. R.T. Oehrle and D. Wheeler (Eds.), Categorial Grammar and Natural Language Semantics. Bob Meyer told me once that he reckoned that substructural logic basically came from Poland too (presumably via the Polish logician whose name I can’t remember who introduced Meredith to the logic of implication). Iirc one thing that Lambek did was to make the logic of CG noncommutative, so that you use to express facts about word-order.

Posted by: Avery Andrews on September 13, 2010 10:32 PM | Permalink | Reply to this

Re: Bolzano, Husserl, Lesniewski, Ajdukiewicz; Re: Categories and Information in Oxford

For the connection between and an historical account on Lambek calculus, Ajdukiewicz–Bar-Hillel calculus, Grishin monoids and Lambek pregroup calculus (categorically: closed monoidal categories, ?, *-autonomous categories, compact closed categories) a nice survey is chapter 4.7 of Lambek’s paper Type Grammar Revisited in which he introduced pregroups.

Posted by: bob on September 14, 2010 6:45 PM | Permalink | Reply to this

Re: Categories and Information in Oxford

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.

Posted by: Avery Andrews on September 12, 2010 8:45 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

Thanks, Avery and John!

Okay, so one can use category theory tools generally for speaking about formal languages.

But from the conference abstract, it seems to me that the reason we see quantum theory meet linguistics here is specifically that somebody noticed that the tools used in the description of quantum mechanics in terms of dagger-compact categories turn out to be applicable to certain

vector space models of natural language

I know everything about the former. I would like to get an idea about how it is claimed to usefully apply to the latter.

Posted by: Urs Schreiber on September 12, 2010 9:41 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

Can’t help you there, but Daoud Clark’s thesis appears to address the issue, although I don’t understand it. The syntax chapter at the end mentions quantum mechanics, but the grammatical power of the actual model is only context free, it says, which means there’s lots of important stuff that it can’t do.

There’s sort of a division in the participants, with Preller and Moortgaat resembling Lambek in doing something fundamentally similar to Chomskian linguistics but with different mathematical means, and the others being more into computer science and informational retrieval, which is very different. Daoud seems to me to be a combination of both.

I’m a bit surprised not to see anybody like Philippe de Groote or Christian Retore etc on the program, because they’ve been doing linear logic/category theory influenced linguistic work for quite a long time.

Posted by: Avery Andrews on September 12, 2010 10:12 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

This might be helpful:

Quantum logic unites compositional functional
semantics and distributional semantic models

Posted by: Avery Andrews on September 12, 2010 11:36 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

The place where compact categories and grammar find each other is in Lambek’s relatively recent pregroup grammars, which stands to the more traditional Lambek grammars as compact closed categories stand to monoidal closed categories e.g see here or Lambek’s recent book on it. (Given that these categories are non-symmetric, calling them compact closed is probably somewhat abusive.) Steve Clark, Mehrnoosh Sadrzadeh and I exploited the correspondence of pregroups and the category of vector spaces, i.e. both being compact closed, to propose a manner for computing the meaning of a sentence from the meaning of its words within the distributional model. Here is the paper on it entitled Mathematical Foundations for a Compositional Distributional Model of Meaning and here is a talk on it from our talks archive. We have been talking to Daoud Clark to see the connections with his field-theory inspired stuff, and maybe some more gets done there at the workshop. As Tom says, things will probably be filmed if whiskey club doesn’t run out of hand the day before.

Posted by: bob on September 12, 2010 11:38 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

Thanks the the links. Maybe I find time to look at them, until then, a question:

propose a manner for computing the meaning of a sentence from the meaning of its words

Where does meaning take values in? What is it you are actually computing here?

Posted by: Urs Schreiber on September 12, 2010 12:34 PM | Permalink | Reply to this

Re: Categories and Information in Oxford

This is in principle variable, depending of what you want `meaning’ to be modeled by, but in practice (cf Google et al) this is a vector in a vector space. For the case of meaning of words, one takes a basis of (lots) of reference words |k>, and then takes the number of times an arbitrary word Ix> appears in the context of these basis within a huge corpus of texts as the coordinate , after normalization. The inner-product then turns out to be a very adequate measure of similarity of words Ix> and |y>, and there are lots of other tricks one can then do. The Preller-Prince paper considers a rather logical model of meaning, staying closer to the Montague tradition.

What makes the above interesting for me is that once you have your theory you have to start doing experiments to see how things work out `in the real world’ with `real texts’. My student Ed Grefenstette and Mehrnoosh Sadrzadeh are in the process of doing so; there should be a paper about this soon.

Posted by: bob on September 12, 2010 9:30 PM | Permalink | Reply to this

Re: Categories and Information in Oxford

what you want `meaning’ to be modeled by, but in practice (cf Google et al) this is a vector in a vector space.

Ah, thanks, that helps. This gives me finally a feeling for what this is all about.

For the case of meaning of words, one takes a basis of (lots) of reference words |k>, and then takes the number of times an arbitrary word Ix> appears in the context of these basis within a huge corpus of texts as the coordinate , after normalization.

I see. So what about the Frobenius algebras from the conference abstract now: they act on these vector spaces? What do they serve to encode? The basis (aka “classical structure”)? Is that what’s going on?

Posted by: Urs Schreiber on September 13, 2010 7:55 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

Hi Urs, here’s a quick-and-dirty diagram from one of Mehrnoosh’s talks that shows how duality (“quantum stuff”) string diagrams feature in linguistics:

john likes mary pic

Do you get it? The verb “likes” needs to have a subject and an object. A bit of sweat and you transform this into the statement “we need left and right duals here, people.”. There’s nonlocality going on. String diagrams for linguistics, pretty gnarly! (This is just the basic-basic-basic stuff).

Posted by: Bruce Bartlett on September 13, 2010 9:55 PM | Permalink | Reply to this

Re: Categories and Information in Oxford

Avery wrote:

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

This group are usually diligent about filming talks and putting them on the web, so you may be able to simulate a fly-like status.

Posted by: Tom Leinster on September 12, 2010 9:43 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

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.

Posted by: Rod McGuire on September 13, 2010 5:00 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

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).

Posted by: Avery Andrews on September 13, 2010 6:00 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

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

Posted by: M. on September 13, 2010 11:58 AM | Permalink | Reply to this

Re: Categories and Information in Oxford

Btw, for the nLab database project, I’ve implemented the display of entries as fully variablized Attribute Value Matrices from Functional Linguistics. However most hard core mathematical linguists don’t work in that sub field.
Posted by: Rod McGuire on September 13, 2010 2:59 PM | Permalink | Reply to this

Re: Categories and Information in Oxford

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.

Posted by: Rajesh Kasturirangan on September 13, 2010 5:20 PM | Permalink | Reply to this

Re: Categories and Information in Oxford

I think there’s enough almost category-theory-ready material lying around in linguistics so that you wouldn’t have to construct any grand conceptual framework, ‘just’ solve a problem that people are interested in.

Posted by: Avery Andrews on September 14, 2010 3:40 AM | Permalink | Reply to this

Language of Thought Hypothesis; Re: Categories and Information in Oxford

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.”

Posted by: Jonathan Vos Post on September 23, 2010 6:51 PM | Permalink | Reply to this

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