#categorytheory — Public Fediverse posts

New paper: "Valuation Without ℝ: A Category-Theoretic Foundation for Finite Measurement"

Can measurement be formalized independently of real numbers AND set theory?

We axiomatize "valuation" as a graded distinguishability map satisfying the ultrametric inequality. Dimension emerges as the growth exponent of the distinguishability graph.

Full paper: https://doi.org/10.5281/zenodo.21795656

#pAdic #QuantumFoundations #CategoryTheory #Measurement

Readings shared April 4, 2026. https://jaalonso.github.io/vestigium/posts/2026/04/04-readings_shared_04-04-26 #AI #AI4Math #ATP #Agda #AlphaProof #Autoformalization #CategoryTheory #CoqProver #FunctionalProgramming #ITP #IsabelleHOL #LLMs #LambdaCalculus #LeanProver #Lisp #Logic #LogicProgramming #LLMs #Math #Physics #Programming #Prolog #Racket #RocqProver #Vampire

Readings shared March 09, 2026. https://jaalonso.github.io/vestigium/posts/2026/03/10-readings_shared_02-09-26 #AI #AI4Math #CategoryTheory #CoqProver #Emacs #FunctionalProgramming #Haskell #ITP #IsabelleHOL #LambdaCalculus #LeanProver #Lisp #Math #Physics #RocqProver

Emily Riehl: How I became seduced by Univalent Foundations

[tbh I think shes only disclosing her theoretical motivations her; I recall she was posting questions about Linux distros a few years ago, and if we're honest being a nerd is the actual reason 99% of people get into HoTT]
https://www.youtube.com/watch?v=XIYoI5j5Flo&t=486s

#hott #mathematics #categorytheory #theoremproving

Differential Propositional Calculus • 10

Special Classes of Propositions (cont.)

Let’s pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general.  We can do this by recruiting our visual imaginations and drawing up a sufficient budget of venn diagrams for each family of propositions.  The case for 3 variables is exemplary enough for a start.

Linear Propositions

The linear propositions, may be written as sums:

One thing to keep in mind about these sums is that the values in are added “modulo 2”, that is, in such a way that

In a universe of discourse based on three boolean variables, the linear propositions take the shapes shown in Figure 8.

At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following three forms.

Next are the venn diagrams for the three linear propositions of rank 2, which may be expressed by the following three forms, respectively.

Next are the three linear propositions of rank 1, which are none other than the three basic propositions,

At the bottom is the linear proposition of rank 0, the everywhere false proposition or the constant function, which may be expressed by the form or by a simple

Resources

• Logic Syllabus
• Survey of Differential Logic

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

#Amphecks #Animata #BooleanAlgebra #BooleanFunctions #CSPeirce #CactusGraphs #CategoryTheory #Change #Cybernetics #DifferentialAnalyticTuringAutomata #DifferentialCalculus #DifferentialLogic #DiscreteDynamics #EquationalInference #FunctionalLogic #GraphTheory #Hologrammautomaton #IndicatorFunctions #InquiryDrivenSystems #Leibniz #Logic #LogicalGraphs #Mathematics #MinimalNegationOperators #PropositionalCalculus #Time #Topology #Visualization

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 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