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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?
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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 inte…
> It turns out that the usual categorical semantics for linguistics shares a lot with the categorical semantics for quantum mechanics so?
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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 co…
"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.
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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 co…
>> -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…
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."'
https://www.etymonline.com/word/*teks-?ref=etymonline_crossr...
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.
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> It turns out that the usual categorical semantics for linguistics shares a lot with the categorical semantics for quantum mechanics so?
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.
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.
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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”,…
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.
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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”,…
I'm showing 1M and 200K Google results for those phrases, respectively.