That makes me imagine a future where various languages import their core data structures and algorithms from a common, language-agnostic "standard library" of mathematically rigorous foundation.

Indeed, what I'd like is formal mathematics proving language that could be viewed as an outline in that natural language math proofs are structured as outlines. The reader would be able to choose what to view - definitions, theorems, lemmas, examples, key-steps, all steps etc.

Ideally, a view would let a reader expand into not just the proof at hand but the various theories in the repository.

A reader could even tell their viewing preferences and it might strip down the UI for those purposes.

You might be thinking of the Langlands program

We research mathematicians are still learning how to use AI in research and how to deal with the output. I drafted eight papers last weekend. This week I advanced on none. AI still gets stuck at the key point on a problem where there isn't already a lot of work done. What falls is where techniques not known to the researcher are available and profitably employed - ai brings them into the fray fast - and someone with experience immediately sees where to go with it - and the dominos fall.

So far AI still doesn't make definitions. It tries but it doesn't really see structure - only patterns on which it can impose (by brute force) known structures.

What's not clear is how the professional infrastructure will survive. Journals seem likely to die. The arxiv is swamped. A human can't assess the volume of the output. Serious conjectures have fallen and serious people have checked them.

We are looking for constructive and productive and socially useful ways forward. One reflection is that a lot of what we have been doing was maybe not so interesting as we thought.

> in the same way much of the slop LLMs write in spoken language is really for the consumption of LLMs.

Is it though? We're not benchmarking how well LLMs 'like' it, and slop is characterised by an unnatural absence of useful information.

It seems mostly designed to mislead humans into believing it was written by someone knowledgeable on the subject.

This is irrelevant. The theorems Avigard, Buzzard, et al. are talking about all have natural language proofs already that in theory should be formalizable.

They are not claiming all possible theorems are formalizable. They are claiming to try to formalize all already proven theorems in the mathematics canon.

To the extent the theorems in question are impossible to formalize this means the original proof must be in error.

If a theorem has a published proof, it is either formalizable or incorrect.

Do you honestly Jeremy Avigad (to pick one of the authors) isn't intimately familiar with the Incompleteness Theorems? For starters, Jeremy is chair in logic and philosophy of maths at CMU. https://www.andrew.cmu.edu/user/avigad/

Perhaps you'd like to explain how you think the Incompleteness theorems affect this project?

Also Chaitin would like a few words.

I understand that, even taking Gödel's results into account, formalization helps us analyze mathematics and can lead to new (non-mechanical) ideas and results.

Yeah which is why I don't have much sympathy for folks like Tao either

The ability to conjure math is practically as endless as time

At some point the outcome is just generation of new visual art in the form of syntax

We can't physically observe the past or future so we can only theorize and simulate; IMO that's the physical phenomena that belies Incompleteness Theorem; from Earth in the relative year of 2026 my view of the universe is fundamentally incomplete. As such all language describing too far either is incomplete always.

It makes any kind of longtermism, religious or agnostic framework espoused by kids of rich academics, empty. We all gonna stick around 10 more generations just to make sure?

Whole lot of educated people are just peddling euphemism for religious conviction they steeped in as kids but hid behind atheist semantics like "Intelligent Design."

The thing is that logicism and formalism were both essentially successful - Godel might thrown a wrench into Hilbert's plans but correcting for the incompleteness problem, a formal process can go forward. Intuitionism was a failure in the sense that just the existence of computers shows

The thing is, I think mathematics as a whole has reached a bit of the sort of crisis that mathematical foundations reached after Godel's theorem. I think Raymond Smullyan said something like (paraphrasing)"We know the continuum hypothesis is independent, it can't be proven or disproven by other hypotheses, we're not trying to do that. But if it can be stated in such a way that it is as intuitively clear as the other other axioms of set theory, then it will likely be accepted".

Mathematicians now can cruise through a huge area of "what can be mechanically deduced/mechanically constructed" and it seems like the task becomes highlighting what matters to us (as humans or mathematicians or some community) within that realm.

Oh how interesting, they're using a fork of Agda called Mikan, not Lean as I imagined before visiting the link. https://1lab.dev/Mikan.html