IMO mathematics benefits from being highly generic compared to normal code, so much so that the implementation itself doesn't really matter (and it's entirely normal in mathematics to skip details, unlike in software). So then to me it makes sense to have a clean separation between basically header files and implementation files like in C and C++. Then something like mathlib should be a pristine collection of what belongs in headers: data structure definitions, instances, theorem statements, macros/tactics for common techniques, etc. So basically it is a curated set of "axioms" that can be used for other math and remain understandable, indexable, etc. Maybe a companion mathlib-full that can actually compile the entire theory.
I suppose this could be done with just tooling to generate a skeleton/index of a giant repository of all mathematics if you decided that you're going to index all definitions and theorems but not lemmas or examples or something like that.
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.
The Langlands program is about correspondences between analytic number theory and harmonic analysis. You could think about it as a unification of parts of mathematics that seem to have no reason to join up but turn out to have deep and non-obvious connections. It doesn't have anything to do with formalization.
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.
>To the extent the theorems in question are impossible to formalize this means the original proof must be in error.
Theorems by definition must be formalizable. A theorem is not simply a true proposition, it's a provable proposition and a proof is necessarily a formal/syntactic property of a proposition.
That said, I agree with the sentiment that dismissing the entire article on the basis of the incompleteness theorems is shallow and the article is not about formalizing "all" of mathematics in the strict sense. It's mostly about introducing formal tools to the everyday mathematics.
As the silliest example of this: the Calculus of Inductive Constructions (the core calculus implemented in Lean) has a published proof of its soundness (that you cannot derive False in it). That proof can (probably, thanks to Gödel) not be formalized in Lean or any formal system of axioms weaker than Lean.
If what you were trying to do was define what it means for something to be a correct proof, then you must equally admit that there can't be a correct proof of the fact that Lean can't prove False. And if you find that world a happy world to live in while we delegate all concerns about truth to Lean, then I have an infinite hierarchy of monotonically growing bridges to sell you.
In principle, a computer can check a nonconstructive proof. I don't know if that's possible in Lean. The interpretation of proofs still assumes the consistency of their axioms.
Any proof which is correct can be formalized. That is what is means for a proof to be correct.
I’m not sure why you are restricting the scope of our discussion to what _Lean_ can(not) formalize. Even if we do choose to do that for some reason, Lean is not a formal system of fixed axioms; it supports optional additional axioms.
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?
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."
I like the pragmatic take on AI in mathematics. Running off and proving/formalizing things in incomprehensible ways is marginally useful, and probably harmful for any work that needs to move beyond it. But, there's value in what AI is capable of doing for working mathematicians, when used as a partner.
The genie is out of the bottle though. It's hard to see how there won't be weird and inscrutable branches of mathematics that LLMs build and use effectively _for themselves_, in the same way much of the slop LLMs write in spoken language is really for the consumption of LLMs.
Surely there will be a new kind of research mathematician that's investigating this output in the same way, say, academics in literature read.
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.
I suppose this could be done with just tooling to generate a skeleton/index of a giant repository of all mathematics if you decided that you're going to index all definitions and theorems but not lemmas or examples or something like that.
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.
Here's a really good and accessible intro to Langlands https://www.youtube.com/watch?v=4dyytPboqvE
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.
Theorems by definition must be formalizable. A theorem is not simply a true proposition, it's a provable proposition and a proof is necessarily a formal/syntactic property of a proposition.
That said, I agree with the sentiment that dismissing the entire article on the basis of the incompleteness theorems is shallow and the article is not about formalizing "all" of mathematics in the strict sense. It's mostly about introducing formal tools to the everyday mathematics.
As the silliest example of this: the Calculus of Inductive Constructions (the core calculus implemented in Lean) has a published proof of its soundness (that you cannot derive False in it). That proof can (probably, thanks to Gödel) not be formalized in Lean or any formal system of axioms weaker than Lean.
If what you were trying to do was define what it means for something to be a correct proof, then you must equally admit that there can't be a correct proof of the fact that Lean can't prove False. And if you find that world a happy world to live in while we delegate all concerns about truth to Lean, then I have an infinite hierarchy of monotonically growing bridges to sell you.
Any proof which is correct can be formalized. That is what is means for a proof to be correct.
I’m not sure why you are restricting the scope of our discussion to what _Lean_ can(not) formalize. Even if we do choose to do that for some reason, Lean is not a formal system of fixed axioms; it supports optional additional axioms.
Perhaps you'd like to explain how you think the Incompleteness theorems affect this project?
https://ia801804.us.archive.org/35/items/A-Z_Library_G-K_Jos... (PDF)
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 genie is out of the bottle though. It's hard to see how there won't be weird and inscrutable branches of mathematics that LLMs build and use effectively _for themselves_, in the same way much of the slop LLMs write in spoken language is really for the consumption of LLMs.
Surely there will be a new kind of research mathematician that's investigating this output in the same way, say, academics in literature read.
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.
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.
1) What is Mathematics? - https://sci-hub.reatk.com/10.2307/2320582
2) The Three Crises in Mathematics: Logicism, Intuitionism and Formalism - https://sci-hub.reatk.com/10.2307/2689412
AI is the fourth crisis in Mathematics!
Formalism has been so successful that the General Public thinks that is all there is to Mathematics. And now AI has made this trivial.
But hopefully, Mathematicians are looking at more foundational and philosophical underpinnings.