lean-liquid VS adjoint

It's a long way away from research-level, but there is real progress; now, I don't see how things like topology and such like will be formalized soon, having to write down explicit transformations for things instead of just saying "this is clearly such a transformation", but I've been impressed.

Re your second para: you make speculations, but here are the facts. There certainly is a bunch of interest right now in formalising mathematics, there are fully paid-up mathematicians like myself hanging out on the Lean chat, working on stuff like this -- undergraduates, PhD students, post-docs and permanent staff. The repo is here and it's coming along nicely. Movement is happening. But it will be a while before we can convince the "generic mathematician" that these tools are useful. The Scholze project linked to in those links above is just another data point, but I fear we will need many more.

This idea shows up in essentially all scientific fields. It’s the idea of adjointness. Together with norm, they give you the idea of fixed points, (invariants, spectra, embeddings, braids etc).I'm

Lawvere's fixed point theorem is I think the best formulation of the idea

https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore...

I've been putting together a brain dump on the topic

https://github.com/adamnemecek/adjoint/

Join the discord https://discord.gg/mr9TAhpyBW

I’ve always thought that the completeness theorem was more interesting.

Also I really like approaching this from Lawvere’s fixed point theorem. https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem

I have been thinking about some of this stuff

https://github.com/adamnemecek/adjoint

Yeah I have been realizing this lately as well. I have written up a bit on this

https://github.com/adamnemecek/adjoint/

Check the raw source there’s a bunch of links.

Fixed points, diagonalizations and eigenshit are all the same thing.

I think that there are like three concepts in math or the world, the concepts being the adjoint, norm and fixed point.

https://github.com/adamnemecek/adjoint

The concept of a fixed point is so ubiquitous.

Game theory, just liek essentially everything in math, physics and probability, and cs is about adjoints, norms, and fixed points https://github.com/adamnemecek/adjoint/

Nash equilibrium is a fixed point.

For me it was the idea of adjoint functors which is the central idea of category theory.

I wrote up a bit on it here https://github.com/adamnemecek/adjoint