Erdős Problem #1026
terrytao.wordpress.com
terrytao.wordpress.com
I mean it should be obvious that making executive decisions about what the code should do exactly should only be left to a RNG powered model if the choices made are unimportant.
Low level, automated theorem providing is going to fall way quicker than most expected, like AlphaGo, precisely because an MCTS++ search over lean proofs is scalable/amendable to self play/relevant to a significant chunk of professional math.
Legit, I almost wish the US and China would sign a Formal Mathematics Profileration Treaty, as a sign of good will between very powerful parties who have much to gain from each other. When your theorem prover is sufficiently better than most Fields medalists alive, you share your arch/algorithms/process with the world. So Mathematics stays in the shared realm of human culture, and it doesn't just happen to belong to DeepMind, OpenAI, or Deepseek.
The workflow I'm envisioning here is the plan document we're all making nowadays isn't being translated directly into code, but into a TLA+/Alloy/... model as executable docs and only then lowered into the code space while conformance is continuously monitored (which is where the toil makes it not worth it most of the time without LLMs). The AI literature search for similar problems and solutions is also obviously helpful during all phases of the sweng process.
I'm sure we've agreed on this before, but I agree again ;) There are dozens of us at least, dozens! There's also a recent uptick in posts with related ideas, for example this hit the front-page briefly ( https://news.ycombinator.com/item?id=46251667 ).
I was tempted to start with alloy/tla for my own experiments along these lines due to their popularity, but since the available training data is so minimal for everything in the space.. I went with something more obscure (MCMAS) just for access to "agents" as primitives in the model-checker.
Haven't tried anything other than Alloy yet, but I've got a feeling Anthropic has employed some dark arts to synthesize either Alloy models or something closely related and trained Opus on the result - e.g. GPT 5.1 commits basic syntax errors, while Opus writes models like it's just another day at the office.
I'm curious if there is a scenario in which a large automated proof is achieved but there would be no practical means of getting any understanding of what it means.
I'm an engineer. Think like this: a large complex program that compiles but you don't understand what it does or how to use it. Is such a thing possible?
That's true though of Lean code written by a human mathematician.
AI systems are capable (and generally even predisposed to) producing long and roundabout proofs which are a slog to decipher. So yes the feeling is somewhat similar at times to an LLM giving you a large and sometimes even redundant-in-parts program.
Sometimes the original proof is compleyely replaced, bit by bit, until there is an easy to understand version.
"10 or 20" -> "10 or 20 years"
If curl developers are overwhelmed by AI PRs, imagine how mathematicians will feel verifying a huge backlog of automated proofs.
Or isn't there such a thing? Can someone make a very complicated automated proof that ultimately reveals itself to be useless?
There are a few "solutions" of conjetures that may be correct, like https://en.wikipedia.org/wiki/Abc_conjecture I'm not sure about the current state. There may be a few mathematicians trying to read some parts, or perhaps no. Perhaps in a few years the easy parts will be refactored and isolated, and published as special cases. And after a while, it may be verified or someone will find a gap and perhaps fix it. Just wait a few decades.
> Can someone make a very complicated automated proof that ultimately reveals itself to be useless?
It depends, on what you consider insightful. Take a look at "Determination of the fifth Busy Beaver value" https://news.ycombinator.com/item?id=45273999 in particular the first comment. Is that an heroic result that opens a lot of paths or a useless combination of tricks that no one will ever understand? (In my opinion a proof is a proof [standing applause emoji].)
Mathematicians are obviously not ignoring automated proofs. Terry's post is an evidence of that.
Consider LK99 instead of crackpot P vs NP proofs. That wasted a lot of academia time.
It seems that it could happen to math.
But see you neglect something important: it's the programmer that is establishing the rules of the game, and as Grothendieck taught us already, often just setting up the game is ALL of the work, and the proof is trivial.
Because AlphaGo can only do one.
AI could very well be better at formal theorem proving than fields medalists pretty soon. It will not have taste, ability to see the beauty in math, or pick problems and set directions for the field. But given a well specified problem, it can bruteforce search through lean tactics space at an extremely superhuman pace. What is lacks in intuition and brilliance, it will make up in being able to explore in parallel.
There is a quality/quantity tradeoff in search with a verifier. A superhuman artificial theorem prover can be generating much worse ideas on average than a top mathematician, and make up for it by trying many more of them.
It's Kasparov vs DeepBlue and Sedol vs AlphaGo all over.
It's also nowhere near AGI. Embodiment and the real world is super messy. See Moravec's paradox.
Practical programs deal with the outside world, they are underspecified, their utility depends on the changing whims of people. The formal specification of a math problem is self contained.
right, but because math is not well specified and formalized yet, it could be a problem, and that's where humans with intuition and more rigid reasoning still necessary.
{-2, 1, -1, 1/2, -1/2, 1/3, -1/3, 1/4, … -1/(k/2)} is a sequence of {k^2+1} distinct real numbers, but the longest increasing or decreasing subsequences are of length 2, not k+1.
What am I missing?
a(n_1), a(n_2), a(n_3), ... is a subsequence of a_n and is denoted a(n_k).
So the indexes don't need to be contiguous, just increasing.
So in your example 2, 1, 1/2, 1/3, ... is a decreasing subsequence.
edit: changed to using function-style notation because the nested subscript notation looks confusing in ascii
a
n
k
It relies on a combination of Humans, LLMs ('General Tools'), Domain-Specific Tools, and Deep Research.
It is apparent that the static data encoded within an LLM is not enough; one must re-fetch sources and digest them fresh for the context of the conversation.
In this workflow, AlphaEvolve, Aristotle, and LEAN are the 'PhDs' on the team, while the LLM is the Full Stack Developer that glues them all together.
[1] If one likes pompous terms, this is what 'AGI' will actually look like.
Literally not AGI.
[from the Aristotle paper]
> Aristotle integrates three main components: a Lean proof search system, an informal reasoning system that generates and formalizes lemmas, and a dedicated geometry solver.
[from elsewhere on how part 2 works]
> To address IMO-level complexity, Aristotle employs a natural language module that decomposes hard problems into lists of informally reasoned lemmas. This module elicits high-level proof sketches and supporting claims, then autoformalizes them into Lean for formal proving. The pipeline features iterative error feedback: Lean verification errors are parsed and fed back to revise both informal and formal statements, iteratively improving the formalization and capturing creative auxiliary definitions often characteristic of IMO solutions.
Is this an example of the 4 minute mile phenomenon or did the AI proof provide key insights that Chan was able to use in their proof?