typed_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

is a ternary operator)..." I wonder if Tree Calculus (and/or binary trees in general!) could be related to, and/or applied to the Ternary Tau and/or the Balanced Phinary number systems: https://neuraloutlet.wordpress.com/tag/ternary-tau-system/ https://en.wikipedia.org/wiki/Balanced_ternary

ges for over a decade (see e.g. https://lambada.pages.dev/). In that work I started noticing that the Iota combinator (https://en.wikipedia.org/wiki/Iota_and_Jot) is not a combinator in the most traditional sense: It is usually defined as behaving like "\x x S K", but like, how can we just refer to a lambda in the definition of a logic? One could write reduction rules such as "Iota Iota x -> x", but now Iota appears on the left hand side, in argument position! Doesn't that allow reflecting on arguments? Horror! I just started realizing that, while SK is perfectly Turing complete and sane, forcing down the number of combinators from 2 to 1 magically forces some amount of intensionality! Curious, isn't it? And that's when I came across Barry's work and book! A calculus that embraces intensionality. So I reached out to see if I can help. How can I be most useful? Barry has been figuring out theory, professionally, before I could think. So I felt like helping spread the word, maybe to a less academic developer crowd, would be a more effective way to contribute. I spent my entire career building developer tools, so I have some ideas what could be useful and what might be necessary to convince various kinds of developers. That's how the website came to be, and it is very much a work in progress. We sync regularly, for instance Barry had excellent ideas for demos and I suggested slightly tweaked reduction rules at some point (not more powerful in theory, just more convenient in practice), see "triage calculus" in typed_program_analysis.pdf