Very interesting, on many levels: first, the raw additional compute / search harness is worth reading about; huge numbers of Lean 4 theorems, thousands of vCPUs available for spreading out search, embedding databases of proofs, all very interesting.
Second, the proofs -- I understand the Lean 4 proofs to be refereed by Fable, and generated by Chat 5.6 Sol. Unlike the leaked proof of the Cycle Double Cover Conjecture last week which had a very nicely readable nearly humanlike writeup, the proof summaries (from Fable) read like Claude tends to read to me these days - real difficulty with the theory of mind of the reader, they are filled with technical phrases, acknowledgment of hard bits and oblique reference to solutions. In short, they suck. I didn't see the word load-bearing, but I bet it's there.
That said, a Lean 4 proof is a pretty compelling output artifact. I find it interesting that it's an additional type of effort to turn these into human readable / appreciable / beautiful / non-shitty proofs.
To those who say who cares -- indeed. But. One of the major reasons things like the Erdos problems are valuable is that they can at times spur new techniques and concepts. The best of these concepts are applied elsewhere, advancing the frontier. While we gain a lot from solving these problems, we'll gain even more from that next step of distillation / explanation into something humans and computers can grok together. I'd hope that with so many tentatively marked 'solved' we will see some new techniques / ontology / concepts. If not, still pretty amazing.
This is great feedback (thank you for taking the time), & you especially bring up a fair point on the writeups needing to be more human readable. I'll work on that
This reminds me of certain simple but addictive video games: "What are these virtual coins good for?" "You can buy better equipment" "Why do you need this equipment?" "To get more virtual coins of course!"
I also had this sort of thoughts when finishing my master's degree. I guess what breaks the cycle is that proofs (like other artefacts in other human activities) deliver aesthetic bliss.
There still seems to be a difference between useless pure math research and useless science or useless philosophy. Science, even useless science, still has a subject matter that is relevant to us independently of science, the real world. And philosophy studies concepts (like "knowledge") that occur in natural language and thought, and those concepts are relevant to us independently of philosophy. But pure math is entirely self-referential. Pure math abstractions are used only in pure math. Pure mathematics is relevant exactly to pure mathematics and those who study it.
"useless" pure math often turns out to have scientific applications.
And surely you don't mean all pure math, so your argument ends up being circular --- useless math is useless. You happen to think that this math is useless, but you might be, or turn out to be, wrong.
Also, you're misusing the term "self-referential". You seem to mean that it's a closed system ... but it's not, since mathematicians interact with it.
Finally: so what? We do all sorts of enjoyable activities with no benefit other than the enjoyment.
My mouth is agape at the fact that this project
is basically what I have been working on non-stop
for the last three weeks and just yesterday gotten
to the point of evaluating; hats off... I only have
one novel proof (non-Erdos) and 13 first-time
formalizations thus far.
I still like doing maths by pen and paper, but
this is fun too.
When you say "working on" what is your actual contribution? Like, what should I imagine you do? For most people who tell AIs what to do and are proud of it, it's sadly mostly sitting around and staring at "thinking" output, and steering a bit, so I'm curious what the work looks like.
Thank you for the kind words! I agree, it's exciting that we can now build advanced AI systems for solving novel math (but i still love pen & paper too)
I was studying Erdos problems by only taking ChatGPT 5.5 outputs and just asking it to keep on attempting to solve it by asking it to go further. I haven't started doing this with chatgpt 5.6 I have some partial results here https://chatgpt.com/g/g-p-69f03400f420819192418b18ca90ffee-d...
What was really interesting is that during the process it was able to find lemmas or theorems that might be related or relevant to be published.
While I was doing that I was also trying to use Aristotle to do the Lean formalization and I have a WIP system to do that at https://github.com/aconsapart/thesisus/
I own a dedicated 48vCPU with even 160GB RAM.. its not that expensive, check ebay, maybe now with mem prices it will be a bit more steep but as a hobby it's not crazy to think one owns such a piece of hardware.
My dual GPU setup was more expensive I think.
When I looked into this a year ago, it was like €60/mo through Hetzner auction. Might be more now but even if it's double or triple it's not that crazy for a hobby.
If you built yourself out of used parts you could do it for under a grand back then too.
No, the people growing their food built modern civilization.
(Or millions of disconnected stakeholders with different incentives collectively built modern civilization, but who wants to put that on a bumper sticker)
As the tools for AI assisted proof become better and mathematicians make it mainstream (could take a while), we're going to be seing some pretty crazy shit. I can't think of a discipline that has more impact on our current toolchains
Very cool! It seems you've got a great setup. An addition that would be very convincing is going the extra mile and making a comparator setup for your Lean proofs. (https://github.com/leanprover/comparator) This ensures that the AI is not, in any way, modifiying the Lean context in ways that could lead to unsoundness.
Poor wording on my end, thanks for flagging. I pull the OAuth refresh token from each Codex account into a custom broker, which mints short-lived access tokens per request and load-balances across the pool.
I've been wanting to experiment with using AI to prove math theorems, but compute is obviously a massive limiting factor here. Are there any plans to open source this?
I didn't know people could just have GPT running on their own hardware. How does one...do that? Do you have a special relationship with OpenAI and they lock down your servers or something?
Isn't this sucking the fun out of math? It's not like we're going to get any tangible benefit out of them, so why not let mathematicians keep their jobs?
The thing about math is we don't usually know what is pure fancy and what is civilization altering until far after the discovery. Once in a while it's a real targeted crack at something practical but most often it's collecting things which seem trial until you use them together and suddenly you have computers running LLMs.
If it were really just about funding people who like math to have fun then it's easy to do forever: just don't have them look at the results and keep paying.
Mathematicians will be the ones who can tell us if the computer theorems are decent or not.
Otherwise they’ll be the ones like Erdős who pose the questions in the first place.
Either way it will always be humans who decide what matters. AI is speaking our languages, not the other way around. We’re in charge. It’s impossible for us not to be, unless we can train an AI from dolphin data or other natural phenomenon.
The AIs intelligence is tuned to us and in 300 years we’ll need new training runs for the update from human zeitgeist language and the 2200 century famous mathematicians.
> Either way it will always be humans who decide what matters. AI is speaking our languages, not the other way around. We’re in charge. It’s impossible for us not to be, unless we can train an AI from dolphin data or other natural phenomenon.
AI companies are accruing power by virtue of its knowledge and ability to do work. If endowed with agency, which seems likely at this rate, it is the AI itself that will be powerful. And we'll be in charge because AI is trained on human language? I can't fathom the logic behind this.
In undergraduate math it doesn't matter if someone else did prove a result a hundred years before you.
You still need to write your own proof and deeply understand it.
Maybe in the most advanced PhD math it can have some impact, but these proofs are becoming intractable by humans alone.
I don't envy the talented young research mathematicians. While there's still space to distinguish yourself (inventing completely new mathematics), the path to status is narrower.
Very interesting, on many levels: first, the raw additional compute / search harness is worth reading about; huge numbers of Lean 4 theorems, thousands of vCPUs available for spreading out search, embedding databases of proofs, all very interesting.
Second, the proofs -- I understand the Lean 4 proofs to be refereed by Fable, and generated by Chat 5.6 Sol. Unlike the leaked proof of the Cycle Double Cover Conjecture last week which had a very nicely readable nearly humanlike writeup, the proof summaries (from Fable) read like Claude tends to read to me these days - real difficulty with the theory of mind of the reader, they are filled with technical phrases, acknowledgment of hard bits and oblique reference to solutions. In short, they suck. I didn't see the word load-bearing, but I bet it's there.
That said, a Lean 4 proof is a pretty compelling output artifact. I find it interesting that it's an additional type of effort to turn these into human readable / appreciable / beautiful / non-shitty proofs.
To those who say who cares -- indeed. But. One of the major reasons things like the Erdos problems are valuable is that they can at times spur new techniques and concepts. The best of these concepts are applied elsewhere, advancing the frontier. While we gain a lot from solving these problems, we'll gain even more from that next step of distillation / explanation into something humans and computers can grok together. I'd hope that with so many tentatively marked 'solved' we will see some new techniques / ontology / concepts. If not, still pretty amazing.
This is great feedback (thank you for taking the time), & you especially bring up a fair point on the writeups needing to be more human readable. I'll work on that
Can you explain what you're using the local compute for vs. the API-based frontier models? That was entirely unclear to me.
Are you running tool calls that include inference with local fine tunes? And fast math packages? Controlled by the frontier model agents?
Is there a way folks can contribute to this?
Thank you for the questions echelon.
1) As far as the AI models go, we used GPT 5.6 Sol, Fable 5, and Gemini-2-embeddings across the system
2) Yes, the agents are given bash tools that allows them to interact with the preinstalled mathematics packages/dependencies that are on the VMs
3) This was a setup as a relatively quick project without much thought for future contributions, I will spend some time thinking about that.
This reminds me of certain simple but addictive video games: "What are these virtual coins good for?" "You can buy better equipment" "Why do you need this equipment?" "To get more virtual coins of course!"
Which is a metaphor for life.
I also had this sort of thoughts when finishing my master's degree. I guess what breaks the cycle is that proofs (like other artefacts in other human activities) deliver aesthetic bliss.
There still seems to be a difference between useless pure math research and useless science or useless philosophy. Science, even useless science, still has a subject matter that is relevant to us independently of science, the real world. And philosophy studies concepts (like "knowledge") that occur in natural language and thought, and those concepts are relevant to us independently of philosophy. But pure math is entirely self-referential. Pure math abstractions are used only in pure math. Pure mathematics is relevant exactly to pure mathematics and those who study it.
But the thing with "pure" math, is that it can unexpectedly get adulterated by debased concerns such as enabling cryptography for the world economy.
"useless" pure math often turns out to have scientific applications.
And surely you don't mean all pure math, so your argument ends up being circular --- useless math is useless. You happen to think that this math is useless, but you might be, or turn out to be, wrong.
Also, you're misusing the term "self-referential". You seem to mean that it's a closed system ... but it's not, since mathematicians interact with it.
Finally: so what? We do all sorts of enjoyable activities with no benefit other than the enjoyment.
My mouth is agape at the fact that this project is basically what I have been working on non-stop for the last three weeks and just yesterday gotten to the point of evaluating; hats off... I only have one novel proof (non-Erdos) and 13 first-time formalizations thus far.
I still like doing maths by pen and paper, but this is fun too.
When you say "working on" what is your actual contribution? Like, what should I imagine you do? For most people who tell AIs what to do and are proud of it, it's sadly mostly sitting around and staring at "thinking" output, and steering a bit, so I'm curious what the work looks like.
Thank you for the kind words! I agree, it's exciting that we can now build advanced AI systems for solving novel math (but i still love pen & paper too)
I was studying Erdos problems by only taking ChatGPT 5.5 outputs and just asking it to keep on attempting to solve it by asking it to go further. I haven't started doing this with chatgpt 5.6 I have some partial results here https://chatgpt.com/g/g-p-69f03400f420819192418b18ca90ffee-d...
What was really interesting is that during the process it was able to find lemmas or theorems that might be related or relevant to be published.
While I was doing that I was also trying to use Aristotle to do the Lean formalization and I have a WIP system to do that at https://github.com/aconsapart/thesisus/
This looks interesting. I am not really familiar with lean, etc... Could I use this to formalize/verify a proof from a paper?
I haven't played around with Aristotle at all, thanks for bringing it up & (also your codebase, thesisus, is very solid!)
Who is funding this? Sounds like a fun experiment but that’s a huge amount of compute if I understand correctly.
According to a quick google search:
"He is currently CTO at Xinobi AI, a Japan-based startup developing personal AI agents."
This is a self funded weekend project for me. It's not associated with any employer (:
> dedicated 60-vCPU server
How many of these are you paying for out of pocket??
I own a dedicated 48vCPU with even 160GB RAM.. its not that expensive, check ebay, maybe now with mem prices it will be a bit more steep but as a hobby it's not crazy to think one owns such a piece of hardware. My dual GPU setup was more expensive I think.
When I looked into this a year ago, it was like €60/mo through Hetzner auction. Might be more now but even if it's double or triple it's not that crazy for a hobby.
If you built yourself out of used parts you could do it for under a grand back then too.
Post-money people with side interests are what built the current western civilization.
No, underpaid nerds have built modern civilization.
No, the people growing their food built modern civilization.
(Or millions of disconnected stakeholders with different incentives collectively built modern civilization, but who wants to put that on a bumper sticker)
No people procreating built civilization.
Nature created people and the metaphysics in their heads.
As the tools for AI assisted proof become better and mathematicians make it mainstream (could take a while), we're going to be seing some pretty crazy shit. I can't think of a discipline that has more impact on our current toolchains
Very cool! It seems you've got a great setup. An addition that would be very convincing is going the extra mile and making a comparator setup for your Lean proofs. (https://github.com/leanprover/comparator) This ensures that the AI is not, in any way, modifiying the Lean context in ways that could lead to unsoundness.
I haven't come across this before. I will spend time on comparator. thank you very much for the suggestion.
What kind of harness does the exploration? Where did the corpus of Lean proofs come from? Is the code backing Ton 618 open source?
To the author: for the absolute Galois of Q_p problem, the link is wrong.
thank you very much for catching this. just fixed
I'm not sure how to interpret this part: "each running its own GPT-5.6 instance".
GPT-5.6 is a closed source model and this seems to be a personal project and not something done by OpenAI.
yeah, is it API or codex?
Poor wording on my end, thanks for flagging. I pull the OAuth refresh token from each Codex account into a custom broker, which mints short-lived access tokens per request and load-balances across the pool.
I've been wanting to experiment with using AI to prove math theorems, but compute is obviously a massive limiting factor here. Are there any plans to open source this?
Have people tried these on Millenium problems.. letting it run all night? You never know.
yeah, im currently running the system on navier-stokes (making real progress).
Unfortunately P vs NP, on the other hand, is going to have to wait for GPT 7
solve p=np make no mistakes
n=1 or p=0
Have been having an awful day today, this really cheered me up. Thank you so much for sharing your dumb joke!
It's hard to solve P?=NP due to P!=NP.
I didn't know people could just have GPT running on their own hardware. How does one...do that? Do you have a special relationship with OpenAI and they lock down your servers or something?
I think they meant they just ran a different context per invocation, not that they hosted the model themselves.
surely it's not copy pasting answers from some obscure polish forum right bros
Isn't this sucking the fun out of math? It's not like we're going to get any tangible benefit out of them, so why not let mathematicians keep their jobs?
The thing about math is we don't usually know what is pure fancy and what is civilization altering until far after the discovery. Once in a while it's a real targeted crack at something practical but most often it's collecting things which seem trial until you use them together and suddenly you have computers running LLMs.
If it were really just about funding people who like math to have fun then it's easy to do forever: just don't have them look at the results and keep paying.
What is their pay going to be justified by once computers start conjecturing and proving theorems on their own?
Mathematicians will be the ones who can tell us if the computer theorems are decent or not.
Otherwise they’ll be the ones like Erdős who pose the questions in the first place.
Either way it will always be humans who decide what matters. AI is speaking our languages, not the other way around. We’re in charge. It’s impossible for us not to be, unless we can train an AI from dolphin data or other natural phenomenon.
The AIs intelligence is tuned to us and in 300 years we’ll need new training runs for the update from human zeitgeist language and the 2200 century famous mathematicians.
> Either way it will always be humans who decide what matters. AI is speaking our languages, not the other way around. We’re in charge. It’s impossible for us not to be, unless we can train an AI from dolphin data or other natural phenomenon.
AI companies are accruing power by virtue of its knowledge and ability to do work. If endowed with agency, which seems likely at this rate, it is the AI itself that will be powerful. And we'll be in charge because AI is trained on human language? I can't fathom the logic behind this.
This is kind of insane reasoning. It's basically asking "what is their pay going to be justified by once their pay isn't justified?"
I think you're trying to say their pay won't be justified? You are not being clear.
Attending department faculty meetings
In undergraduate math it doesn't matter if someone else did prove a result a hundred years before you. You still need to write your own proof and deeply understand it. Maybe in the most advanced PhD math it can have some impact, but these proofs are becoming intractable by humans alone.
I don't envy the talented young research mathematicians. While there's still space to distinguish yourself (inventing completely new mathematics), the path to status is narrower.
Isn't the pursuit of knowledge alone good enough?
That’s the problem, the coupling of work with the right to survive
The job of a mathematician is to study mathematics, not to create proofs.
An automatic proof solver doesn't make mathematicians obsolete any more than the excel sheet made accountants obsolete.
Conjectures and proofs are the fruit of the understanding. Nobody gets paid to think without producing anything.
How about philosophers?
They make cab rides more enjoyable with their deep conversations.
They get paid to produce confusion.
They're getting paid?
It's probably more akin to changing where the fun is in maths.
or get those bright minds out of academia daycare and back to more actionable needs such as steering agents
| your brain on neocapitalism |
This will keep happening until we stop people from doing it.
Get the looms while you're at it