#categorytheory — Public Fediverse posts

This is getting complicated! #haskell #categorytheory

Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/

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 —
• https://oeis.org/wiki/Propositions_As_Types_Analogy

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

Survey of Precursors Of Category Theory • 6
• https://inquiryintoinquiry.com/2025/05/05/survey-of-precursors-of-category-theory-6/

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
• https://oeis.org/wiki/Precursors_Of_Category_Theory

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Blog Series —

Notes On Categories
• https://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/

Precursors Of Category Theory
1. https://inquiryintoinquiry.com/2024/05/25/precursors-of-category-theory-1-a/
2. https://inquiryintoinquiry.com/2024/05/26/precursors-of-category-theory-2-a/
3. https://inquiryintoinquiry.com/2024/05/27/precursors-of-category-theory-3-a/
4. https://inquiryintoinquiry.com/2024/05/28/precursors-of-category-theory-4-a/
5. https://inquiryintoinquiry.com/2024/05/29/precursors-of-category-theory-5-a/
6. https://inquiryintoinquiry.com/2024/05/30/precursors-of-category-theory-6-a/

Precursors Of Category Theory • Discussion
1. https://inquiryintoinquiry.com/2020/09/13/precursors-of-category-theory-discussion-1/
2. https://inquiryintoinquiry.com/2020/09/21/precursors-of-category-theory-discussion-2/
3. https://inquiryintoinquiry.com/2020/09/25/precursors-of-category-theory-discussion-3/

Categories à la Peirce —

C.S. Peirce • A Guess at the Riddle
• https://inquiryintoinquiry.com/2012/03/21/c-s-peirce-a-guess-at-the-riddle/

Peirce's Categories
1. https://inquiryintoinquiry.com/2015/10/30/peirces-categories-1/
2. https://inquiryintoinquiry.com/2015/10/31/peirces-categories-2/
3. https://inquiryintoinquiry.com/2015/11/04/peirces-categories-3/
•••
19. https://inquiryintoinquiry.com/2020/05/13/peirces-categories-19/
20. https://inquiryintoinquiry.com/2020/05/14/peirces-categories-20/
21. https://inquiryintoinquiry.com/2020/06/25/peirces-categories-21/

C.S. Peirce and Category Theory
1. https://inquiryintoinquiry.com/2021/06/23/c-s-peirce-and-category-theory-1/
2. https://inquiryintoinquiry.com/2021/06/24/c-s-peirce-and-category-theory-2/
3. https://inquiryintoinquiry.com/2021/06/27/c-s-peirce-and-category-theory-3/
4. https://inquiryintoinquiry.com/2021/06/28/c-s-peirce-and-category-theory-4/
5. https://inquiryintoinquiry.com/2021/06/29/c-s-peirce-and-category-theory-5/
6. https://inquiryintoinquiry.com/2021/06/30/c-s-peirce-and-category-theory-6/
7. https://inquiryintoinquiry.com/2021/07/01/c-s-peirce-and-category-theory-7/
8. https://inquiryintoinquiry.com/2021/07/02/c-s-peirce-and-category-theory-8/

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

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: https://github.com/muskin88/delta-sigma-peano/blob/main/Peano_from_deltasigma.py
Formal statement: https://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

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:

https://www.sci.brooklyn.cuny.edu/~noson/Seminar/index.html

I'll be talking about the invariant theory part of my thesis (https://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

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

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

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

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

I have a new(ish) preprint on the arXiv! You can find "Invariants of structures" at https://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 https://www.youtube.com/watch?v=5TeGZZ_mepc.

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

Just posted this talk (https://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 https://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