tree-calculus

page. It led me to a kind of branching, graph-like manner of learning, where I open pages on categories, products, morphisms, uniqueness, equality - filling holes in my understanding as needed, then going back to more specific ideas built from those general foundational concepts. An interactive textbook as a computational medium I feel has a lot more potential for usefulness and expression, to support an autodidactic and exploratory style of learning. It relates to topics raised elsewhere in this thread, like a "proper IDE for math" (integrated development environment for mathematics) and a language-agnostic "standard library" of data structures, algorithms, definitions, theorems, etc., where programming languages can import from a shared corpus of mathematically rigorous primitives. > how to use dependent types to explore, teach, and formalize mathematics in a pro-human way I find it meaningful how you phrased this - at this point in history with the state of technology that can overwhelm and devalue human intelligence, it's more important than ever to focus on how we teach, learn, communicate and cultivate understanding. On a tangent, one of my interests is the book _Reflective Programs in Tree Calculus_² by Barry Jay, its proofs in Rocq, his more recent article "Typed Program Analysis without Encodings"³, and the work of his main collaborator⁴. I lack the relevant academic background to fully grasp it yet, so I've been chipping away at learning the necessary basics. So the link to 1lab is appreciated, it's a resource I'll be digging into over time. --- ¹ With Category Theory, Mathematics Escapes From Equality https://www.quantamagazine.org/with-category-theory-mathemat... ² https://github.com/barry-jay-personal/tree-calculus/blob/mas... (PDF) ³ https://github.com/barry-jay-personal/typed_tree_calculus/bl... (PDF) ⁴ https://github.com/lambada-llc/tree-calculus

----------------------------------- [0] : Quipu: The Ancient Computer of the Inca Civilization https://www.peruforless.com/blog/quipu [1] : magnetic bubble memory : https://www.youtube.com/watch?v=DbfrsnsWmhk [2] : magnetic bubble memory storage on a heathkit h8 : https://www.youtube.com/watch?v=P_miv4wYzvg&t=38s= [3] : https://github.com/prathyvsh/lambda-calculus-visualizations [4] : https://github.com/barry-jay-personal/tree-calculus/blob/mas... [5] : https://writings.stephenwolfram.com/2025/09/the-ruliology-of... [6] : https://en.wikipedia.org/wiki/DNA_barcoding

ulus can express a type system, here’s a recent ACM paper: https://dl.acm.org/doi/pdf/10.1145/3704253.3706138 Personal context, Barry Jay is a respected academic and researcher who’s collaborated with people like Simon Peyton Jones and Eugenio Moggi. His PhD advisor was Joachim Lambek (from the Curry-Howard-Lambek correspondence). He’s not a random professor with a neat toy, Barry’s been working with many of the best minds on the foundations of computation long before most of us knew category theory existed. He’s been formalizing and defining pattern matching, higher-ordered patterns and has been searching/separating what is truly essential from what is not essential for decades. Seriously, look at his research history on Google Scholar. I think it will take the rest of us a while to understand and unpack the insight he’s already imbued into such a small and simple calculus.

f us actually paying attention.

e allowed to act on the syntax tree of a program, i.e. its quotation. [...] When quotation lies outside of the formal calculus then interpretation is separated from computation proper, so that some sort of staging is required. A demo of a size function is given here [1], implemented directly in the tree calculus: size = \x (y $ \self \x compose succ $ triage id self (\x \y compose (self x) (self y)) x) x 0 [0] https://github.com/barry-jay-personal/tree-calculus/blob/mas... [1] https://treecalcul.us/live/?example=demo-program-optimizatio...

he notion of homoiconicity as in Lisp. [0] https://en.wikipedia.org/wiki/SKI_combinator_calculus [1] https://github.com/barry-jay-personal/tree-calculus/blob/mas... [2] https://treecalcul.us/specification/ [3] https://sci-hub.se/https://dl.acm.org/doi/abs/10.1016/j.tcs....

notion of homoiconicity as in Lisp, but probably more precisely or even strongly stated. > Plenty of programming languages with both macros and first class function objects (those that can be passed around and thus have data representations). A language may have first class function objects, but its actual structure may be opaque and not open to reflection or manipulation (beyond of course just munging the source code as plaintext). You can maybe create a function literal, pass the function around and to higher-order functions, but you can't inspect or modify its internal structure, or decide program equality (based on either exact structure, or one program reducing to another according to the reduction rules of the calculus). Lastly the tree calculus would also appear to differ from the lambda calculus in that programs are stable and won't reduce infinitely, instead converging on some normal form of irreducible terms. [1] [0] https://github.com/barry-jay-personal/tree-calculus/blob/mas... [1] https://sci-hub.se/https://dl.acm.org/doi/abs/10.1016/j.tcs....