home.social

#categorytheory — Public Fediverse posts

Live and recent posts from across the Fediverse tagged #categorytheory, aggregated by home.social.

  1. Propositions As Types Analogy • 1
    inquiryintoinquiry.com/2013/01

    One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

    Proof Hint ∶ Proof ∶ Proposition

    Untyped Term ∶ Typed Term ∶ Type

    or

    Proof Hint ∶ Untyped Term

    Proof ∶ Typed Term

    Proposition ∶ Type

    See my working notes on the Propositions As Types Analogy —
    oeis.org/wiki/Propositions_As_

    #Mathematics #CategoryTheory #ProofTheory #TypeTheory
    #Logic #Analogy #Isomorphism #PropositionalCalculus
    #CombinatorCalculus #CombinatoryLogic #LambdaCalculus
    #Peirce #LogicalGraphs #GraphTheory #RelationTheory

  2. Survey of Precursors Of Category Theory • 6
    inquiryintoinquiry.com/2025/05

    A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. A Survey of resources on the topic is given below, still very rough and incomplete, but perhaps a few will find it of use.

    Background —

    Precursors Of Category Theory
    oeis.org/wiki/Precursors_Of_Ca

    Propositions As Types Analogy
    oeis.org/wiki/Propositions_As_

    Blog Series —

    Notes On Categories
    inquiryintoinquiry.com/2013/02

    Precursors Of Category Theory
    1. inquiryintoinquiry.com/2024/05
    2. inquiryintoinquiry.com/2024/05
    3. inquiryintoinquiry.com/2024/05
    4. inquiryintoinquiry.com/2024/05
    5. inquiryintoinquiry.com/2024/05
    6. inquiryintoinquiry.com/2024/05

    Precursors Of Category Theory • Discussion
    1. inquiryintoinquiry.com/2020/09
    2. inquiryintoinquiry.com/2020/09
    3. inquiryintoinquiry.com/2020/09

    Categories à la Peirce —

    C.S. Peirce • A Guess at the Riddle
    inquiryintoinquiry.com/2012/03

    Peirce's Categories
    1. inquiryintoinquiry.com/2015/10
    2. inquiryintoinquiry.com/2015/10
    3. inquiryintoinquiry.com/2015/11
    •••
    19. inquiryintoinquiry.com/2020/05
    20. inquiryintoinquiry.com/2020/05
    21. inquiryintoinquiry.com/2020/06

    C.S. Peirce and Category Theory
    1. inquiryintoinquiry.com/2021/06
    2. inquiryintoinquiry.com/2021/06
    3. inquiryintoinquiry.com/2021/06
    4. inquiryintoinquiry.com/2021/06
    5. inquiryintoinquiry.com/2021/06
    6. inquiryintoinquiry.com/2021/06
    7. inquiryintoinquiry.com/2021/07
    8. inquiryintoinquiry.com/2021/07

    #Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
    #Abstraction #Analogy #CategoryTheory #FunctionalLogic #RelationTheory
    #PrecursorsOfCategoryTheory #PropositionsAsTypes #Semiotics #TypeTheory

  3. We've computationally verified that Peano arithmetic emerges naturally from just two operators: Δ (distinction/branching) and Σ (connection/composition).

    This isn't just coding — it's evidence for the Δ–Σ Turing Completeness Theorem: a system is Turing-complete iff it can be represented through Δ and Σ.

    Code implements the proofs: github.com/muskin88/delta-sigm
    Formal statement: zenodo.org/records/17895986
    (Theorem 3)

    The implications are ontological: these operators appear inevitable for any non-trivial reality. The framework unites computation, mathematics, and fundamental ontology.

    #CategoryTheory #FoundationsOfMath #Computation #Ontology #FormalMethods #TypeTheory #PeanoArithmetic #TuringCompleteness #MathematicalPhilosophy

  4. I've been on a longer hiatus from livestreaming than I originally intended, but you can see me give a seminar talk this evening at the The New York City Category Theory Seminar:

    sci.brooklyn.cuny.edu/~noson/S

    I'll be talking about the invariant theory part of my thesis (arxiv.org/abs/2402.18063) at 7PM, New York time. I'll discuss how I found that every (positive) property of finite structures can be checked by counting small* substructures.

    *Terms and conditions may apply. Small is constrained by the logical complexity of a property and may not conform to mundane notions of smallness in bad cases.

    #CategoryTheory #combinatorics #logic #Bourbaki #algebra #AbstractAlgebra

  5. I came across a diagram illustrating transfer learning in the ML textbook I'm currently in the middle of, and it looked an awful lot like diagrams for natural transformations. I bet I'm not the first person to have noticed that, or the tenth. Any interesting papers I ought to read?

    #MachineLearning #TransferLearning #CategoryTheory

  6. I came across a diagram illustrating transfer learning in the ML textbook I'm currently in the middle of, and it looked an awful lot like diagrams for natural transformations. I bet I'm not the first person to have noticed that, or the tenth. Any interesting papers I ought to read?

    #MachineLearning #TransferLearning #CategoryTheory

  7. I came across a diagram illustrating transfer learning in the ML textbook I'm currently in the middle of, and it looked an awful lot like diagrams for natural transformations. I bet I'm not the first person to have noticed that, or the tenth. Any interesting papers I ought to read?

    #MachineLearning #TransferLearning #CategoryTheory

  8. I came across a diagram illustrating transfer learning in the ML textbook I'm currently in the middle of, and it looked an awful lot like diagrams for natural transformations. I bet I'm not the first person to have noticed that, or the tenth. Any interesting papers I ought to read?

    #MachineLearning #TransferLearning #CategoryTheory

  9. I have a new(ish) preprint on the arXiv! You can find "Invariants of structures" at arxiv.org/abs/2402.18063. This is a somewhat embellished version of one half of my PhD thesis. A talk which I gave about this subject in the fall of 2022 is available at youtube.com/watch?v=5TeGZZ_mep.

    In this new version, I have finally added an explicit description of something I've been telling people for years: My main result shows that any first-order property of finite structures can be computed by counting small substructures. Perhaps surprisingly, this comes as a result of synthesizing a categorical treatment of Bourbaki's notion of mathematical structure with Hilbert's classical result on symmetric polynomials.

    #CategoryTheory #combinatorics #algebra #AbstractAlgebra #logic #Bourbaki

  10. Just posted this talk (youtu.be/5TeGZZ_mepc) I gave on the categorified invariant theory part of my PhD thesis last fall! You can also find my thesis itself online now at aten.cool/documents/thesis.pdf if you'd like to see more.

    Part of the reason I waited so long to post this is because I kind of flubbed the last example after the main part of the talk due to having not looked at this stuff for a while before giving the lecture. I thought I'd cut that last part out once I had more time, but enough time has passed and it no longer bothers me.

    I actually wrote most of this part of my thesis in 2020, so I waited a long time to advertise this work.

    #math #thesis #algebra #AbstractAlgebra #CategoryTheory #Bourbaki #combinatorics