Comment by peter_d_sherman
5 hours ago
>"Tree calculus is a minimal, modular, Turing-complete and reflective calculus."
This makes it a candidate for the foundation of all of Mathematics...
Other candidates in this space include such things as Category Theory (everything reduces down to a single operation called a 'composition'), Lambda Calculus (everything reduces down to function application), Formal Rewriting Systems (aka Symbol Substitution / Post Canonical System / Markov Algorithm -- everything reduces to a single operation: string rewriting, matching a pattern of symbols and replacing it with another. Turing Machines, for example, exist within this space...), Homotopy Type Theory (aka HoTT: Paths as Transformations, statements of equality (a = b) are not static truth values; they are paths (or continuous transformations) living in a higher-dimensional space. Logical proofs, algebraic manipulations, and geometric deformations are all unified under the concept of "path induction." Proving that two mathematical structures are equivalent is equivalent to finding a continuous path of transformation between them.")
That's some of them, others include such things as Turing Complete Finite Automata, i.e., Rule 110, etc., etc.)
Anyway, excellent link, and we welcome Tree Calculus to this list!
Related: https://en.wikipedia.org/wiki/One-instruction_set_computer
Great notes!
I think it’s also worth noting that Barry Jay has a particularly strong vantage point for unearthing this particular foundational candidate. First, his advisor was Joachim Lambek, who’s responsible for adding Category Theory in the Curry-Howard-Lambek correspondence.
If one looks at Barry’s body of work over his lifetime, it’s no surprise it would culminate into an elegantly beautiful system capable of self-reflection.
I think his 2025 PEPM paper did a great job explaining the ideas to a regular programmer / wannabe academic like myself: https://github.com/barry-jay-personal/typed_tree_calculus/bl...