cubicaltt VS Coq-HoTT

- cubicialtt a programming language based on cubical type theory in which univalence from homotopy type theory isn't an axiom but a theorem

If you’re interested in programming languages specifically, the current state of the art is called Cubical Type Theory. CuTT has lots of flavours and the community hasn’t coalesced around a single design. The paper I personally found easiest to digest was the “ABCFHL” paper, but I’d recommend reading it alongside the original CCHM paper. None of the publications made an ounce of sense to me until after I’d digested Favonia’s YouTube channel, Mortberg’s lecture notes and this other series of lectures from Harper (particularly the final one).

I updated the blog post to include some people in academia who contributed. I've been interacting with Kent Palmer and Sylvester James Gates, Jr. The latter held lectures about the philosophy of mathematics. I've been using work inspired by Vladimir Voevodsky, e.g. cubicaltt (https://github.com/mortberg/cubicaltt), which is also performed by academics.

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... :

> * Today, Zermelo–Fraenkel set theory [ZFC], with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common* foundation of mathematics.

Foundation of mathematics: https://en.wikipedia.org/wiki/Foundations_of_mathematics

Implementation of mathematics in set theory:

> The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU, the version of Quine's New Foundations shown to be consistent by R. B. Jensen in 1969 (here understood to include at least axioms of Infinity and Choice).

> What is said here applies also to two families of set theories: on the one hand, a range of theories including Zermelo set theory near the lower end of the scale and going up to ZFC extended with large cardinal hypotheses such as "there is a measurable cardinal"; and on the other hand a hierarchy of extensions of NFU which is surveyed in the New Foundations article. These correspond to different general views of what the set-theoretical universe is like

IEEE-754 specifies that float64s have ±infinity and specify ZeroDivisionError. Symbolic CAS with MPFR needn't be limited to float64s.

HoTT in CoQ: Coq-HoTT: https://github.com/HoTT/Coq-HoTT

leanprover-community/mathlib4//

For those interested in formalisation of homotopy type theory, there are several (more or less) active and developed libraries. To mention a few:

UniMath (https://github.com/UniMath/UniMath, mentioned in the article)

Coq-HoTT (https://github.com/HoTT/Coq-HoTT)

agda-unimath (https://unimath.github.io/agda-unimath/)

cubical agda (https://github.com/agda/cubical)

All of these are open to contributions, and there are lots of useful basic things that haven't been done and which I think would make excellent semester projects for a cs/math undergrad (for example).

HoTT is somewhat independent of the choice of proof assistant.

Coq: https://github.com/HoTT/HoTT

Lean: https://github.com/gebner/hott3

idk what you mean by "blue screened", or results being on the way. afaict most of the non-foundational work present (what I assume you mean by "results") in these libraries are basic properties of basic mathematical concepts being rebuilt on HoTT.

The major one for me is version control (git). Imaging having to write a book like HoTT without having revisions and easy way to work on your changes without interfering with anyone else's work and then easily merging everything together.