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Linguistics Using Category Theory (2018) (utexas.edu)
108 points by jesuslop 26 days ago | hide | past | web | favorite | 30 comments

Even 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).

In your particular example, I like the cube rule (e.g. to answer 'is a hotdog a sandwich?')

EDIT: http://cuberule.com/

Then 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.

Up to an isomorphism.

I'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?

If 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).

Jean-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.

Here'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.

>> -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.

Hmmm, I have a strange sense of deja vu in replying to this comment.

There is https://en.wikipedia.org/wiki/Tekt%C5%8Dn which broadly supports your first paragraph.

On the other hand, in architecture, tectonics is usually contrasted with stereotomics. Tectonics is concerned with framing, stereotomy with compressive masses. The former is paradigmatically a branch of carpentry, the latter part of stone-masonry. This is how the words are used in architectural history. See e.g. https://www.campobaeza.com/wp-content/uploads/2016/11/1996_0...

I'm an architectural historian, so I tend to care most about the way the term tectonics functions within architectural discourse. But I think I would translate tecton as carpenter anyway, because of the strong older biblical tradition of doing so, and (independently) the etymology of the word.

'* teks- Proto-Indo-European root meaning "to weave," also "to fabricate," especially with an ax," also "to make wicker or wattle fabric for (mud-covered) house walls."'


NB the distinction in German between Wand and Mauer. Both mean "wall", but Wand corresponds to the light, woven wattle partition, while Mauer is associated with massive masonry.

One thing that would convince me to drop the architectural association of tectonics with carpentry would be a classical Greek text describing fortifications or other heavy stone masses being built by people referred to as "tecton", since that scenario would strongly suggest that they were not carpenters. The revisionist biblical concerns mentioned on the Wikipedia page seem less relevant to the connotations of the Greek word—they have more to do with recovering the connotations of the text in its original language, before translation to classical Greek.

Re the deja-vu. Maybe I have made the same comment on "τέκτων" before. Sorry, I can't remember. I write way too much on HN.

The wikipedia page you link favours the "wood-worker" translation, so you are probably right and I'm wrong, but I have to say that Greek _is_ my language and "τέκτων" just doesn't sound anything like "wood-worker" to me. It's probably just my modern ears. Thanks for correcting me and apologies for opining on something I don't understand that well after all (the etymology of the word).

No, in the article I'm referencing he goes further, to concepts. It's here: https://www.amazon.com/What-Category-Theory-Giandomenico-Sic...

I had confused in my mind -tectonic with a phrase from another book I admire: https://www.amazon.com/Between-Two-Ages-Americas-Technetroni.... I guess that's what I get for not double-checking; and you get a random book link :)

Yes, I did (later on) see his "stretch" version of the slogan at the end of the article, but I don't buy it. Everyone knows there are operations that can't be encoded in category theoretic terms, for example a lot of things in analysis. It's silly to claim that it subsumes all concepts.

He also alludes in that final paragraph to Kant (who used the term architectonic in his philosophy) so that's a hint that he has a Kantian perspective in mind. IMO that makes that final paragraph too speculative and frankly just out of scope if we're trying to talk about the design of software. I know a bit about Kant, and it seems to me that he doesn't really have anything to say about what a theory of the design of abstractions would look like. It's all too much of a stretch. Combining philosophers and category theory (take Zalamea for example) seems to produce "architecture astronauts,".

However thanks for not using referral/affiliate links. :)

Is there anything more fundamental than mathematics though? If category theory represents the mathematical aspect of design and mathematics is fundamental to the universe then perhaps category theory is a fundamental theory of design.

There's got to be counter-examples of good system design where category cannot be applied.

> Jean-Pierre Marquis, in his contribution to the volume "What is Category Theory", wrote: 'Category theory is the Architectronic of Concepts'.

As a mathematician, I feel that I have a reasonable understanding of category theory, but have never heard the word 'architectronic' (and Google, or at least the first page of Google results, knows it only as the name of a Red Snapper song). What does it mean?

EDIT: Ah, notfashion (https://news.ycombinator.com/item?id=20772313) clarifies that it is architectonic (just one 'r'): https://en.wiktionary.org/wiki/architectonics#English .

You wouldn't happened to have a copy I could trouble you for would you?

It 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/

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:



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

[0] Not freely available, but some slides are at 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-p...

But the best introduction to category theory I have read is Leinster's book, 'Basic Category Theory': https://arxiv.org/abs/1612.09375

And as you might have guessed, I do agree with your statement!

> It turns out that the usual categorical semantics for linguistics shares a lot with the categorical semantics for quantum mechanics


One 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.

Type 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.

I'm showing 1M and 200K Google results for those phrases, respectively.

Those exact phrases? In quotation marks?

No, those result numbers do not include quotation marks. With the quotes I show the same results as you.

Not to be glib; it depends what you mean by 'design' and 'abstraction', doesn't it?

The 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.

Chris 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.

That 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.

A post on Category Theory from UT that doesn't mention Steedman or Baldridge?!

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