>"<i>Tree calculus</i> is a minimal, modular, Turing-complete and reflective calculus."<p>This makes it a candidate for the foundation of all of Mathematics...<p>Other candidates in this space include such things as <i>Category Theory</i> (everything reduces down to a single operation called a 'composition'), <i>Lambda Calculus</i> (everything reduces down to function application), <i>Formal Rewriting Systems</i> (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...), <i>Homotopy Type Theory</i> (aka <i>HoTT</i>: 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.")<p>That's some of them, others include such things as <i>Turing Complete Finite Automata</i>, i.e., Rule 110, etc., etc.)<p>Anyway, excellent link, and we welcome <i>Tree Calculus</i> to this list!<p>Related: <a href="https://en.wikipedia.org/wiki/One-instruction_set_computer" rel="nofollow">https://en.wikipedia.org/wiki/One-instruction_set_computer</a>
Great notes!<p>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.<p>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.<p>I think his 2025 PEPM paper did a great job explaining the ideas to a regular programmer / wannabe academic like myself: <a href="https://github.com/barry-jay-personal/typed_tree_calculus/blob/main/typed_program_analysis.pdf" rel="nofollow">https://github.com/barry-jay-personal/typed_tree_calculus/bl...</a>