Palomar: A registry of Lean verified mathematics
terrytao.wordpress.com
terrytao.wordpress.com
I find this kind of endearing, how he almost makes it sounds like "If even I can do it, you can too!", disregarding he's probably the most prolific mathematician currently alive.
Some years ago I suggested it might be interesting to have a kind of blockchain of formally verifiable mathematical proofs, but was quickly shutdown by people telling me it was a useless endeavor due to Goedels incompleteness theorems.
I think the larger issue of computationally creating a library of math proofs is still that one might come up with an infinite amount of useless theorems that are trivial to prove, but I suppose this registry is manually vetted. Theres a strong inductive bias in maths in that humans still decide what axiomatic systems, theorems, definitions etc are interesting to us.
But I think the method of proving things will soon go out style as an endeavor to devote perhaps years of your life to.
How does a blockchain help here?
> but was quickly shutdown by people telling me it was a useless endeavor due to Goedels incompleteness theorems.
I'm curious about those arguments ... what does incompleteness have to do with this?
> But I think the method of proving things will soon go out style as an endeavor to devote perhaps years of your life to.
I'm not in that camp. With the new tools available, it might change how this endeavor exactly works, but there will always be claims at the edge of our understanding for which we seek explanations and proofs are just that. The ones that can be easily and autonomously discovered by a machine will not be subject to human attention for that purpose, but there will still be a frontier that needs (and will get) attention.
Mind you that not all math activity happens at that frontier. Generations of aspirarional young math fans have re-proven countless foundational results just for fun, because it's a satisfying and eye-opening activity, and that's how you learn. That will not change fundamentally even though maybe the tools they use will.
It's also not new that the setting for frontier research changes. Not too many generations ago, proofs were written with literal pen on paper and communicated by letter to dear math colleagues in distant countries. That's very different today, but has not killed math as a profession. To the contrary. There are far more mathematicians today than there were 300 years ago.
edit: Yes I think this should be possible, using a "recursive SNARK".
You shouldn't have listened to them, "incompleteness" only applies when you try to encode the language and meta-language in the same encoding—which no one actual does. (For example, programming language syntax is complete.)
Either Terry Tao is - imprecisely and inaccurately - using the name of the popular git repository hosting service "Github" as a synonym for the open-source version control system "git", likely because he has some familiarity with software version control but is not himself a specialist in computer programming; or he does fully understand the distinction and Palomar was written in such a way that it only works for git repositories hosted on GitHub specifically and Tao is accurately describing that. Either possibility is unfortunate. I do not like GitHub's de-facto mindshare monopoly on decentralized version control system hosting.
And what a strange decision indeed.
This is not the way Palomar works. From the About page (https://palomar-registry.org/about):
> Palomar does keep a public preservation fork of every registered source, solely as a backup for the registry in the event that the original repository disappears.
The decision to limit git sources to Github is likely in order to be able to use Github's fork mechanism. Palomar could still offer to take a copy of the relevant commit of non-Github repositories.
Besides, it isn't "self-hosted basement Git server VS Github". There are plenty of other large and reliable forges out there, such as GitLab, Sourcehut, Bitbucket, or Codeberg. Considering how many people - from individual devs to major open-source projects - are abandoning Github, launching a new project in 2026 which only works with Github is a rather odd choice.
It doesn't even make sense from a technical perspective. Git has a standard protocol it uses for cloning repos. You have to go out of your way to make it not work with other forges. And if reliability is critical, surely you'd just mirror it locally? Heck, why not make use of Git's inherent distributed nature and allow defining multiple upstream sources? Integrity is already handled by the commit hash itself, so it doesn't matter if you fetch that commit from Github, GitLab, or some guy's basement server - just take whichever one happens to respond the fastest.
If I was a mathematician looking for a simple solution, I would go for GitHub too. Software engineering concerns don’t really apply here.
That is a terrible metric to look into. Github being down is news, but on a long enough time horizon Gitlab or other provider isn't significantly better. It's just they are in news less often.
I suspect the same. But do you have any evidence for this? The status pages of Github and Gitlab (more precisely their history pages) don't seem to be a good starting point for comparisons. Anything I find online are people reporting their own experiences, and it's difficult to tell how accurate they are, how many users were truely affected etc. The only thing I can vouch for is that Github has got more unreliable - adding to the hearsay myself...
https://unlocked.microsoft.com/ai-anthology/terence-tao/
He clearly loves big industry, as unfortunately also too many software engineers do.
Tangentially: Although many of the creators, maintainers and board members of Palomar have a background in Lean, the project welcomes alternative proof assistants, see "What about other proof assistants?" on the about (https://palomar-registry.org/about) page. From what I can tell, many in the mathematical community lament the predominance of Lean, but it reached some sort of critical mass (ecosystem, size of library) that makes it very hard to compete with - e.g. find someone who volunteers to support an alternative on Palomar, with all that this entails.
And since the bar for validation is expressly stated to be rather weak, I guess this would be best conceptualized as a specialized search engine that can weed out the <80th percentile of mediocre formalized proofs, with a target audience of professional mathematicians who have the ability to make determinations on the final 20% themselves.
Maybe also something I'd wanna skim over at some point, as a non-mathmatician who just thinks lean is neat. Its not useful to me but its fun to learn about.
Isn’t this kind of a recursive problem where you now have to prove your proof that their proof proves their claimed statement?
There will always be a leap from the real world to the formal world.
My man Terrance Tao, I hope to contribute to your symphony of progress.
If the interrelationships of this are also exposed and searchable, if it can build many bridges inside itself, then this is truly the cipher key to all that can be known.
After that, we can do statistics to see which ideas in math are most used, and perhaps mine for unrecognized patterns, refactoring math to find new abstractions.
Seems to be doing exactly the same?
Theoremdb is just a slop farm.