6 ms·

Applied Compositional Thinking for Engineers

84 points by pawsed 6y ago

tacon 6y ago
Category theory jumps out of its software silo and is applied to ... well, everything else. This looks like a fun course!
eli_gottlieb 6y ago
Nice to see the ACT field moving forward :-).
Koshkin 6y ago
Dr. David Spivak, a guest speaker, is one of the well-known researchers and educators in this area.
gugagore 6y ago
I had (maybe unreasonably) hoped that this course would provide a glimpse into how CT can be applied to organizing and processing data in the sense of keywords like "knowledge graphs", "graph databases", "ontologies", "model-based engineering".... And on top of that, representing operations to do meaningful (semantic) version control on these representations (e.g. [1, 2]), and bidirectional transformations [3] between structured representations (e.g. [3, 4] and "Triple Graph Grammars"). I have the sense that there are dozens of disparate concepts and that category theory offers some unifying power. I hope that there simply hasn't been enough work done to do a category-theoretic treatment of all of these topics, and that perhaps even more category theory itself needs to be developed so that there are good ways to talk about concepts that are almost-but-not-quite-entirely described or subsumed by category theory. The alternative is that I'm painfully wrong about what applied category theory aims to be, and that I have a ton of application-specific terms to learn about and won't find a formalization of the sense in which all of these concepts relate. [1] https://en.wikibooks.org/wiki/Understanding_Darcs/Patch_theory https://en.wikibooks.org/wiki/Understanding_Darcs/Patch_theo... [2] https://github.com/trailofbits/graphtage https://github.com/trailofbits/graphtage [3] http://bx-community.wikidot.com http://bx-community.wikidot.com [4] https://github.com/grammarware/bx-parsing https://github.com/grammarware/bx-parsing [5] https://en.wikipedia.org/wiki/QVT https://en.wikipedia.org/wiki/QVT [6] http://graphdatamodeling.com/Graph%20Data%20Modeling/GraphQL/GraphQL.html http://graphdatamodeling.com/Graph%20Data%20Modeling/GraphQL... [7] https://neo4j.com/developer/guide-data-modeling/ https://neo4j.com/developer/guide-data-modeling/ [8] https://web-cats.gitlab.io/#some-of-the-cats-we-come-across https://web-cats.gitlab.io/#some-of-the-cats-we-come-across [9] http://pauillac.inria.fr/~pilkiewi/papers/boomerang-tr.pdf http://pauillac.inria.fr/~pilkiewi/papers/boomerang-tr.pdf
- bcheung 6y ago
  CT is universal in the sense that if you understand how it is applied in one domain you can easily apply it to others. The problem with CT education IMO is that it cannot be taught at the abstraction level only -- there are too many floating abstractions that people can't anchor to any existing knowledge. This means CT can only REALLY be understood once you apply it to a domain. The problem is that not many people outside of mathematicians understand the domains that CT is traditionally taught with. CT for Engineers, CT for Programmers, CT for XYZ, is probably the only viable way CT is going to see wider adoption.
  - Koshkin 6y ago
    In CT terms, "CT has a universal property..."
spekcular 6y ago
I always groan when I see posts on HN with grandiose claims about category theory, like this one. I think it is actively harmful to propagate pseudo-mathematical claims like the those, for example, found in the slides of Guest Lecture 1: >>> It’s touched or greatly influenced all corners of mathematics. >>> It’s become a gateway to learning mathematics. And from the audio of the lecture (paraphrasing): >>> Category theory is the stem cell that differentiates into and lies at the root of all pure mathematics. ["All forms of pure math" is also written on the slides.] These statements are simply false. The vast majority of pure mathematics research done today does not involve category theory at all, and does not benefit from it. An even greater majority (like 99%+) of mathematics done in industry and in national labs does not involve category theory. Numerical analysis, probability, statistics, partial differential equations, dynamical systems, harmonic analysis, even lots of modern differential geometry – no category theory to be seen! Want proof? Pick up any introductory graduate textbook, or any major journal in these fields. Now, if you want to do research in number theory, or algebraic topology, or algebraic geometry – sure, you likely would benefit from categorical thinking. But those fields hardly have a monopoly on pure mathematics. Even in, for example, Hatcher's introductory graduate text on algebraic topology (perhaps the most widely used), category theory is stuck in a small appendix and you can read the entire thing without it, with no real loss [1]. I don't have the energy right now to explain why I find the lecture series misguided more generally, but I want to at least flag these obviously incorrect statements and urge caution. [And before anyone grabs a PDE book and tells me the use of cohomology groups in certain places means PDE uses category theory, please note that e.g. homological algebraic and category theory are different things.] [1] OK, I guess you need to know what e.g. a natural transformation is to read some parts of the last chapter, but no one does that in an introductory course anyway. A motivated teacher could easily present the material in such a way that this didn't matter.
- Koshkin 6y ago
  Personally, I wouldn't be that skeptical. By its very nature CT as a foundational theory and is relevant to, and has at least in that sense indeed touched, all corners of mathematics. (Mathematicians had the same skepticism about Set Theory when it first appeared.) Especially the "theoretical" (pure) math. So, sure, "you can read the entire thing without it, with no real loss", but this only says something about the particular textbook and not the subject itself. I assure you, the actual loss, whether you realize it or not, will be very real. Books like Aluffi's Algebra: Chaper 0 have a very good reason behind them. Category Theory is the chapter zero of the modern understanding of mathematics (and not only).
madhadron 6y ago
I followed the first two thirds of this, and stopped because I felt that I wasn't getting much. A lot of names for things I already knew, but nary a theorem or meaty result that gave me any new insight. I think my greatest benefit was no longer feeling like MacLane's book needs to be on my reading list. Also, I discovered that there are cranks even in category theory! There was this guy Robert Rosen who tried to apply it to biology, got totally confused, wrote three giant books, and now has a posthumous following. That wasn't something the lecturers were pushing, but it came up among the students.
aWidebrant 6y ago
Is this just the second coming of cellular automata?
motohagiography 6y ago
The way that CT is explained to engineers here is what tech architects do every day, and with the rigour of formalisms that would help clarify a lot of the muddled thinking some architects suffer from. Arguably, an architect is someone who uses categories and relationships between them to solve and optimize for aggregate behaviour and outcomes. I watched the first guest lecture, which was very good. I'm not a mathematician, engineer, or a category theorist, but I can apply the formalisms to system architecture instantly.
- skybrian 6y ago
  I guess, but I'm reminded of the promises made for UML, where in the end it just introduced some standard conventions for whiteboard diagrams. Abstract, domain-independent formalisms can make ideas harder to understand than domain-specific, concrete examples. With category theory, I'm not seeing examples of the formalism paying off that would justify the endeavor.
  - zozbot234 6y ago
    UML did not have a categorical foundation in the way that, e.g. commutative diagrams, tensor networks or proof nets do, though. Categorical foundations help define some implied properties and allowed operations (e.g. "diagram chasing" as a diagrammatic representation of composition) that have no equivalent in ad-hoc modeling notations like UML.
    - skybrian 6y ago
      Sure, it's real math, but is that a difference that makes a difference? The real-world applications could still be overly hyped.

84 points by pawsed 6y ago

Category theory jumps out of its software silo and is applied to ... well, everything else. This looks like a fun course!

eli_gottlieb 6y ago

Nice to see the ACT field moving forward :-).

Koshkin 6y ago

Dr. David Spivak, a guest speaker, is one of the well-known researchers and educators in this area.

gugagore 6y ago

I had (maybe unreasonably) hoped that this course would provide a glimpse into how CT can be applied to organizing and processing data in the sense of keywords like "knowledge graphs", "graph databases", "ontologies", "model-based engineering".... And on top of that, representing operations to do meaningful (semantic) version control on these representations (e.g. [1, 2]), and bidirectional transformations [3] between structured representations (e.g. [3, 4] and "Triple Graph Grammars"). I have the sense that there are dozens of disparate concepts and that category theory offers some unifying power. I hope that there simply hasn't been enough work done to do a category-theoretic treatment of all of these topics, and that perhaps even more category theory itself needs to be developed so that there are good ways to talk about concepts that are almost-but-not-quite-entirely described or subsumed by category theory. The alternative is that I'm painfully wrong about what applied category theory aims to be, and that I have a ton of application-specific terms to learn about and won't find a formalization of the sense in which all of these concepts relate. [1] https://en.wikibooks.org/wiki/Understanding_Darcs/Patch_theory https://en.wikibooks.org/wiki/Understanding_Darcs/Patch_theo... [2] https://github.com/trailofbits/graphtage https://github.com/trailofbits/graphtage [3] http://bx-community.wikidot.com http://bx-community.wikidot.com [4] https://github.com/grammarware/bx-parsing https://github.com/grammarware/bx-parsing [5] https://en.wikipedia.org/wiki/QVT https://en.wikipedia.org/wiki/QVT [6] http://graphdatamodeling.com/Graph%20Data%20Modeling/GraphQL/GraphQL.html http://graphdatamodeling.com/Graph%20Data%20Modeling/GraphQL... [7] https://neo4j.com/developer/guide-data-modeling/ https://neo4j.com/developer/guide-data-modeling/ [8] https://web-cats.gitlab.io/#some-of-the-cats-we-come-across https://web-cats.gitlab.io/#some-of-the-cats-we-come-across [9] http://pauillac.inria.fr/~pilkiewi/papers/boomerang-tr.pdf http://pauillac.inria.fr/~pilkiewi/papers/boomerang-tr.pdf

bcheung 6y ago
CT is universal in the sense that if you understand how it is applied in one domain you can easily apply it to others. The problem with CT education IMO is that it cannot be taught at the abstraction level only -- there are too many floating abstractions that people can't anchor to any existing knowledge. This means CT can only REALLY be understood once you apply it to a domain. The problem is that not many people outside of mathematicians understand the domains that CT is traditionally taught with. CT for Engineers, CT for Programmers, CT for XYZ, is probably the only viable way CT is going to see wider adoption.
- Koshkin 6y ago
  In CT terms, "CT has a universal property..."

spekcular 6y ago

I always groan when I see posts on HN with grandiose claims about category theory, like this one. I think it is actively harmful to propagate pseudo-mathematical claims like the those, for example, found in the slides of Guest Lecture 1: >>> It’s touched or greatly influenced all corners of mathematics. >>> It’s become a gateway to learning mathematics. And from the audio of the lecture (paraphrasing): >>> Category theory is the stem cell that differentiates into and lies at the root of all pure mathematics. ["All forms of pure math" is also written on the slides.] These statements are simply false. The vast majority of pure mathematics research done today does not involve category theory at all, and does not benefit from it. An even greater majority (like 99%+) of mathematics done in industry and in national labs does not involve category theory. Numerical analysis, probability, statistics, partial differential equations, dynamical systems, harmonic analysis, even lots of modern differential geometry – no category theory to be seen! Want proof? Pick up any introductory graduate textbook, or any major journal in these fields. Now, if you want to do research in number theory, or algebraic topology, or algebraic geometry – sure, you likely would benefit from categorical thinking. But those fields hardly have a monopoly on pure mathematics. Even in, for example, Hatcher's introductory graduate text on algebraic topology (perhaps the most widely used), category theory is stuck in a small appendix and you can read the entire thing without it, with no real loss [1]. I don't have the energy right now to explain why I find the lecture series misguided more generally, but I want to at least flag these obviously incorrect statements and urge caution. [And before anyone grabs a PDE book and tells me the use of cohomology groups in certain places means PDE uses category theory, please note that e.g. homological algebraic and category theory are different things.] [1] OK, I guess you need to know what e.g. a natural transformation is to read some parts of the last chapter, but no one does that in an introductory course anyway. A motivated teacher could easily present the material in such a way that this didn't matter.

Koshkin 6y ago
Personally, I wouldn't be that skeptical. By its very nature CT as a foundational theory and is relevant to, and has at least in that sense indeed touched, all corners of mathematics. (Mathematicians had the same skepticism about Set Theory when it first appeared.) Especially the "theoretical" (pure) math. So, sure, "you can read the entire thing without it, with no real loss", but this only says something about the particular textbook and not the subject itself. I assure you, the actual loss, whether you realize it or not, will be very real. Books like Aluffi's Algebra: Chaper 0 have a very good reason behind them. Category Theory is the chapter zero of the modern understanding of mathematics (and not only).

madhadron 6y ago

I followed the first two thirds of this, and stopped because I felt that I wasn't getting much. A lot of names for things I already knew, but nary a theorem or meaty result that gave me any new insight. I think my greatest benefit was no longer feeling like MacLane's book needs to be on my reading list. Also, I discovered that there are cranks even in category theory! There was this guy Robert Rosen who tried to apply it to biology, got totally confused, wrote three giant books, and now has a posthumous following. That wasn't something the lecturers were pushing, but it came up among the students.

aWidebrant 6y ago

Is this just the second coming of cellular automata?

motohagiography 6y ago

The way that CT is explained to engineers here is what tech architects do every day, and with the rigour of formalisms that would help clarify a lot of the muddled thinking some architects suffer from. Arguably, an architect is someone who uses categories and relationships between them to solve and optimize for aggregate behaviour and outcomes. I watched the first guest lecture, which was very good. I'm not a mathematician, engineer, or a category theorist, but I can apply the formalisms to system architecture instantly.

skybrian 6y ago
I guess, but I'm reminded of the promises made for UML, where in the end it just introduced some standard conventions for whiteboard diagrams. Abstract, domain-independent formalisms can make ideas harder to understand than domain-specific, concrete examples. With category theory, I'm not seeing examples of the formalism paying off that would justify the endeavor.
- zozbot234 6y ago
  UML did not have a categorical foundation in the way that, e.g. commutative diagrams, tensor networks or proof nets do, though. Categorical foundations help define some implied properties and allowed operations (e.g. "diagram chasing" as a diagrammatic representation of composition) that have no equivalent in ad-hoc modeling notations like UML.
  - skybrian 6y ago
    Sure, it's real math, but is that a difference that makes a difference? The real-world applications could still be overly hyped.