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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 t

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.

notfashion 7y ago
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.
- 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.
  - notfashion 7y ago
    Hmmm, I have a strange sense of deja vu in replying to this comment. There is https://en.wikipedia.org/wiki/Tekt%C5%8Dn 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_02_THE-BUILT-IDEA_FROM-THE-CAVE-TO-THE-HUT.pdf 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_crossreference 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.
    - YeGoblynQueenne 7y ago
      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).
- crimsonalucard 7y ago
  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.
- codekilla 7y ago
  No, in the article I'm referencing he goes further, to concepts. It's here: https://www.amazon.com/What-Category-Theory-Giandomenico-Sica/dp/8876990313 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-Technetronic/dp/0313234981 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 :)
  - fapjacks 7y ago
    However thanks for not using referral/affiliate links. :)
  - notfashion 7y ago
    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,".
jdc 7y ago
You wouldn't happened to have a copy I could trouble you for would you?
- notfashion 7y ago
  https://zenodo.org/record/584073#.XV8FKmko80M https://zenodo.org/record/584073#.XV8FKmko80M
JadeNB 7y ago
> 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 https://news.ycombinator.com/item?id=20772313) clarifies that it is architectonic (just one 'r'): https://en.wiktionary.org/wiki/architectonics#English https://en.wiktionary.org/wiki/architectonics#English .

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.

notfashion 7y ago

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.

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.
- notfashion 7y ago
  Hmmm, I have a strange sense of deja vu in replying to this comment. There is https://en.wikipedia.org/wiki/Tekt%C5%8Dn 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_02_THE-BUILT-IDEA_FROM-THE-CAVE-TO-THE-HUT.pdf 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_crossreference 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.
  - YeGoblynQueenne 7y ago
    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).
crimsonalucard 7y ago
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.
codekilla 7y ago
No, in the article I'm referencing he goes further, to concepts. It's here: https://www.amazon.com/What-Category-Theory-Giandomenico-Sica/dp/8876990313 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-Technetronic/dp/0313234981 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 :)
- fapjacks 7y ago
  However thanks for not using referral/affiliate links. :)
- notfashion 7y ago
  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,".

jdc 7y ago

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

notfashion 7y ago
https://zenodo.org/record/584073#.XV8FKmko80M https://zenodo.org/record/584073#.XV8FKmko80M

JadeNB 7y ago

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