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Linguistics Using Category Theory (2018)
- crimsonalucard 7y agoI'm starting to study category theory and I'm realizing if there exists a formal theory around 'design' and 'abstraction' category theory is it. I could be wrong though. What do you guys think?
- autopoiesis 7y agoIt sounds like you would be interested in the book / course '7 Sketches in Compositionality' by David Spivak and Brendan Fong, which studies precisely those ideas categorically, with a particular focus on systems that you might broadly call 'computational': http://math.mit.edu/~dspivak/teaching/sp18/ http://math.mit.edu/~dspivak/teaching/sp18/ It has been discussed on Hacker News at least a couple of times previously -- fairly recently, even. You might be interested to look at these discussions: https://news.ycombinator.com/item?id=20376325 https://news.ycombinator.com/item?id=20376325 https://news.ycombinator.com/item?id=19701767 https://news.ycombinator.com/item?id=19701767 Edit to add: You might also be interested to learn that categorical approaches to linguistics typically take as their starting point monoidal categories, in which there are notions of 'parallel' as well as 'sequential' composition. It turns out that the usual categorical semantics for linguistics shares a lot with the categorical semantics for quantum mechanics: roughly, meanings are vectors, like quantum states. You can read more about doing (finite-dimensional) quantum mechanics entirely using string diagrams (the formal diagrammatic calculus of monoidal categories) in the work of Bob Coecke, who also played a large part in originating these approaches to linguistics. For example, on the quantum side, an excellent book is 'Picturing Quantum Processes' [0]. And on the linguistics side, the paper linked in the article is a good start: https://arxiv.org/abs/1003.4394 https://arxiv.org/abs/1003.4394 [0] Not freely available, but some slides are at https://www.cs.ox.ac.uk/ss2014/programme/Bob.pdf https://www.cs.ox.ac.uk/ss2014/programme/Bob.pdf Edit, again: There is also of course Bartosz Milewski's book / blog series 'Category Theory for Programmers', which introduces category theory from the perspective of Haskell and C++ programming: https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ https://bartoszmilewski.com/2014/10/28/category-theory-for-p... But the best introduction to category theory I have read is Leinster's book, 'Basic Category Theory': https://arxiv.org/abs/1612.09375 https://arxiv.org/abs/1612.09375 And as you might have guessed, I do agree with your statement!
- igravious 7y ago> It turns out that the usual categorical semantics for linguistics shares a lot with the categorical semantics for quantum mechanics so?
- notfashion 7y agoOne other thing that has a lot in common with those two is algebraic data types. Products and sums crop up in all these areas. Maybe it's enough to say that with category theory, we feel like we are revealing the "elementary particles" (or rules) of all of these systems.
- igravious 7y agoType theory has also been applied to both linguistics and quantum mechanics. What does it mean that both category theory and type theory have been applied to both linguistics and quantum mechanics? "categorical semantics for linguistics" gives 0 hits in Google btw. "categorical semantics for quantum mechanics" gives 5 hits all of which reference the same paper by Bob Coecke titled “Strongly Compact Closed Semantics”, which uses the phrase only once.
- codekilla 7y agoJean-Pierre Marquis, in his contribution to the volume "What is Category Theory", wrote: 'Category theory is the Architectronic of Concepts'. I think this is the best way I have heard it put.
- notfashion 7y agoHere's the quote: "Once category theory was developed and used, in particular when the central theoretical role played by adjoint functors was understood, a fascinating process of reversal of perspective, a gestalt switch, took place: what was seem as a useful tool in organizing and guiding mathematical thought became a theoretical framework that revealed the basic or fundamental principles underlying mathematical concepts, theories and theorems. Thus, the Stone duality theorem is indeed more perspicuously presented in the context of categories and functors—it is organized nearly and the basic consequences of the result are transparent—but once it is seen as a special case of a very general adjoint situation, a theoretical understanding of the phenomenon becomes available. Category theory is not applied to Stone's theorem, it is the latter that becomes a specific instance of a general, universal conceptual situation. Although it might in the end be more obscure than what I have said so far, I dare at this stage put forward a slogan that, I believe, sums up the core of what I have been presenting: category theory is the architectonic of mathematics. Category theory is, indeed, as in the philosophical sense of the expression "architectonic", the systematization of mathematical knowledge. Mathematical knowledge is systematic. Mathematics is a conceptual system. That much is indubitable." Note that it's -tectonic (from the Greek for carpenter) not -ectronic (from electron). And it's just "the architectonic of mathematics", not concepts in general. There are obviously architectonics of other things as well, like (spatial) architecture itself, or music, or cooking. So while perhaps CT is the ultimate theory of algebraic abstractions, but it's not the only kind of design system that exists. It represents the mathematical aspect of design.
- YeGoblynQueenne 7y ago>> -tectonic (from the Greek for carpenter) "Tecton" ("τέκτων") is probably best translated as "mason". It has the connotation of a builder who works with stone, so a stone-mason. It is also the more official name of freemasons (colloquially known as "μασώνοι", a Greek transliteration of "masons"). Grammatically, "τέκτων" is the gerund of the verb "τίκτω", meaning "to give birth". So a more literal interpretation is "he who gives birth", or, more reasonably, "he who creates", a "creator". I'm not sure what's (older) Greek for carpenter, but in modern greek it is "ξυλουργός"- "wood-crafter". I think the connotation of stone-masonry is closer to the meaning of architecture- and a better representation of the idea that category theory is the lore of the building blocks of mathematics. After all, the most impressive monumental architecture, indeed most architecture, is made of stone.
- igravious 7y agoNot to be glib; it depends what you mean by 'design' and 'abstraction', doesn't it?
- crimsonalucard 7y agoThe design and abstraction of systems. So usually we "design" one possible system out of many to solve a problem in a solution space we do not completely understand. Once a formal axiomatic theory is in place it could very well become "deriving" the best system to solve a problem in a solution space we completely understand. Of course it's likely not that simple and likely there is no singular "best" solution but a theory, in my opinion, would quantify all possible tradeoffs between a subset of "best" solutions out of all possible solutions. Category theory seems to be the closest thing to such a theory I have found. I'm a beginner in learning this stuff though. So far I'm guessing that it's more than likely that there are no algorithms within category theory that can be used to optimize systems or even derive one... but it is the closest thing to such a theory that I have uncovered? My question in the initial post was asking whether or not anyone has found anything better.... Some posters have remarked that the movement of category theory into engineering (a field which is largely involved with designing solutions more-so than deriving solutions) is starting to happen.
- proc0 7y agoIf we define abstraction as hiding the unnecessary in order to reveal the essentials, and we take it to an extreme such that we reduce everything to a point and only talk about relationships of the point, then yes I completely agree. CT is the math of abstraction, from which I think composition arises (since you cannot look inside an object).
- willis936 7y agoEven if a mathematical definition of a sandwich is presented, it is very unlikely to have majority support in localities of any size (down to two individuals).
- proc0 7y agoUp to an isomorphism.
- rgoulter 7y agoIn your particular example, I like the cube rule (e.g. to answer 'is a hotdog a sandwich?') EDIT: http://cuberule.com/ http://cuberule.com/
- willis936 7y agoThen subs are tacos and not sandwiches? I would disagree. The definition I like for sandwich is “A set of materials in the configuration ABA, mated along their longest side”. It holds up in contexts outside of food, that the word “sandwich” is sometimes used in, and is valid in dimensionalities other than 3. The sub issue remains though. Semantically, it could be argued that subs are sandwiches that have not had their end fully cut.
- haxterstockman 7y agoA post on Category Theory from UT that doesn't mention Steedman or Baldridge?!
- WaxProlix 7y agoChris Barker of NYU does a lot of category theory and linguistics work. I remember a colloquium or two which did good work in talking about modeling certain semantic structures in monad terms. I remember it because at the time I liked the idea of keeping a monadically updated 'context' node (at or above C) in an otherwise fairly orthodox Chomskyan x-bar framework to start modeling the syntax-semantics interface, but never really pursued linguistics as the money was just so bad. Good times, formal linguistics could use some of the rigor and computational methods that computer science has offered for a few decades.
- jesuslop 7y agoThat colloquium looked nice, I assume no slides are available? Barker is anouncing another seminar on deep learning and semantics, that looks intriguing from the attached public materials. I think that compsci furnishes ideas from formal language parsing that can be toy models of fragments of natural languages, Montague made a case in favour of formalizability. Wadler investigated monadic parsing and this can be an entry point for what is happening in linguistics. I have as reference arXiv:cs/0205026v1 added: on Barker seminar day 4 I see CCG derivations that are a close formalism of pregroup grammars used in the original post, nice to see this concurrently with nnets talk.