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Metamath contributor here! Each proving tool has its pros and cons, but always happy to see Metamath noted :-).
One thing that's cool about Metamath is that the axioms are not built-in. It's true that the most-used system is based on classical logic and ZFC set theory https://us.metamath.org/mpeuni/mmset.html ... but you don't have to use that system. There's a well-maintained database using intuitionistic logic: https://us.metamath.org/ileuni/mmil.html ; on the so-called "New Foundations" (a many-sorted system): https://us.metamath.org/nfeuni/mmnf.html ; on HOL https://us.metamath.org/holuni/mmhol.html ; and you can make your own if you want to.
In Metamath the proofs hide absolutely nothing. There's no hand-waving "it's obvious that". Every step in a proof must be rigorously and directly proven by some axiom or a previously-proven theorem with absolutely no exceptions. This also means that while finding proofs can be hard, verifying proofs is fast. I just ran a proof verification run of over 47,000 theorems in 6.35 seconds. In the Metamath Proof Explorer / set.mm database (the one with classical logic and ZFC), we routinely run multiple provers by different people on every proposed change. So not only is the kernel small, it's implemented by multiple different programs, making it extremely unlikely we'll accept an invalid proof.
This video I made years ago summarizes Metamath: https://www.youtube.com/watch?v=8WH4Rd4UKGE
Metamath is interesting for education in the principles of formal proofs, because:
1. The Metamath Book is self-contained well written, simple introduction.
2. Everything is explicit, nothing is hidden.
3. Because of extremely simple syntax and semantics it's easy write own proof verifier.
The drawbacks are:
1. Non-standard approach to free and bound variables.
2. Very weak logic in comparison to Lean. The logical syntax rules and logical semantics rules used for proving mathematical statements in Metamath, have to be explicitly specified in the Metamath database. In Lean they are part of the Lean kernel. An bug in this Metamath specification is equivalent to a bug in Lean kernel.
3. Almost no proof automation. No tactics. Long, hard to read proofs.
I would characterize Metamath as the assembly language of proof assistants, very low level and detailed way to write proofs.
What semantics do you use for your HOL library? I scanned around but documentation on that page is a bit sparse. The github repo goes to a random user's page, and all I could find there was this unrelated repo: https://github.com/digama0/HOL
Yeah, that github URL should be fixed. The HOL database is here https://github.com/metamath/set.mm/blob/develop/hol.mm
Reading `hol.mm` made me grateful for Lean 4's general purpose programming.
I get the string substitution focus, and respect what metamath has achieved, but the bridge Lean 4 makes wrt systems programming has left quite an impression.
I wonder how many people mostly see mathlib4 & think that's the one prescribed route (ala Rust) when one of Lean 4's under-documented super powers is the ease with which you can roll your own light-weight low to zero overhead domain-specialized constructs that are also trivial to prove because of the dependent type system.
(Perhaps more for verified functional systems than deep math.)
There are other forms of logic? is intuitionistic logic as rigorous? fascinating
edit: the link says it is a weakening. if it is weakened, how can you prove the same stuff? i am a bit confused but i can see how it is useful for smarter people than me!
> if it is weakened, how can you prove the same stuff?
Sometimes, you can't. In particular, so-called "non-constructive" proofs don't work in intuitionistic logic. Some mathematicians like to work in intuitionistic logic: for philosophical reasons, pragmatic technical considerations, or just because they think it's interesting.
This is not entirely true: non-constructive proofs can be fully understood within intuitionistic logic as proofs of negative statements, viz. statements about what isn't constructively true or doesn't constructively exist. From this point of view, intuitionistic logic can be seen as stronger and more powerful than classical logic, because it can endow positive statements (particularly statements about logical disjunctions or existentials) with computational content whenever they have a direct proof.
Classical logic doesn't bother with this; positive and negative statements are completely dual to one another, and this duality is elegant in its own way (as people who are committed to classical logic will readily point out), but one needs linear logic in order to combine it with an awareness of constructive statements.
Intuitionistic logic can prove less than classical logic, but what you gain is that proofs are constructive. Also you can use it to reason about things for which law of excluded middle doesn't hold (typically types).
Having a weaker base system means you can distinguish more fine grained between statements.
For example, in an intuitionistic setting there is a difference between a set being non-empty and a set having an element.
I find it really strange that people who don't use lean don't just get on and use the alternatives rather that trying to get everyone who is using lean to use something else. It feels exactly like if all the emacs users in the world tried to force all vim users to use emacs.
It's important to meet reality head on: Every mathematician is not going to collaborate on the same tooling (as wonderful as that might seem on the surface to be as an outcome) human beings are different and want different things, and people are productive in different environments. In particular, people who want to formalize results within the standard framework (including zfc) are never really as a group going to care that much that lean4 doesn't let them formalize results outside of zfc.
> It feels exactly like if all the emacs users in the world tried to force all vim users to use emacs.
I don't know about forcing, but there's certainly a long history of emacs users trying to convince vim users that they're wrong. Without loss of generality - the same is true in the other direction of course.
Software devs are more than used to having a multitude of tools at our disposal (text editors, programming languages, Linux distros...), having strong preferences, and engaging in the delicate balancing act of advocating for our favourites while living-and-letting-live as the cosmic ballet goes on. It's part of our culture; I don't think that culture necessarily exists outside. For instance, a common layman question to ask is, "Why are there so many programming languages? Why doesn't everyone just use the same one?" And I don't think mathematicians are immune to this blind spot.
The nearest equivalent to "tooling" in math, traditionally, would be notation I suppose, and mathematicians are accustomed to eventually coalescing around a single standard notation.
> It feels exactly like if all the emacs users in the world tried to force all vim users to use emacs.
That's why emacs has a vim mode[1].
I have zero experience with formal verification, just speculating here - but could it be because of the network effects?
More users on your preferred tool means a more comprehensive database of existing proofs, and that makes writing new proofs easier, right?
Absolutely that's the reason. But the point is the big communities (I'm thinking https://leanprover-community.github.io/ , Kevin Buzzard and all the stuff he's got going at imperial college in the uk etc, the analogous efforts around roq, agda etc which I'm not as familiar with) have made their respective choices and are just getting on with formalising maths. Then there is a vocal minority who want to sit on the sidelines and say they want all these people to use something different from what they have already decided to use. Seems weird.[1]
But yeah it is definitely easier if you just need to formalise the piece you are working on and not invent the whole universe just to bake an apple pie.
[1] And I know it's exactly the same as the people on here and other forums who say other people should down tools on project X and rewrite it in go/rust/zig/whatever. I find that weird also. Like if you want to rewrite a thing in a different language go do that by all means. But saying someone else who develops something in their own time should instead develop a different thing or use a different language is just weird.
I too know nothing about formal verification but I know that mathematical proof is different individual tooling. Informal statements, Lean and whatever other formal provers are all language and languages are social so what someone uses will impact you and so it's legitimate to have an opinion.
Agreed. In addition I think a big reason why we are discussing formalization so much at the moment is that it has only recently become viable to do large scale formalization of mainstream mathematics, using LLMs. These same LLMs will make it much easier to translate from one language to another and port even larger codebases over.
My prediction would therefore be that the LLMs will let us work more-or-less using standard mathematical prose that then gets to codified to some machine-readable and checkable language, but what the language used by the proof checker actually is will be more of a technical detail. In particular, I suspect people will not care too much about the language used for proofs themselves, which means that we might allow for more boilerplate if it is faster to elaborate/compile, unlike current interactive theorem provers which are meant to ease the work of humans. How the language looks like for the statements of the theorems and definitions is probably more important however, since humans will want still to be able to check that what is being formalized corresponds to what they had in mind.
This is horseshit. Mathlib3 and mathlib4 all existed prior to LLMs. Unimath of Agda, mathematical components of Rocq, the list goes on.
LLMs have done nothing for "making large scale mechanization viable." They have been viable. The only thing has changed is the perception of the random developer who never wanted to put the effort into learning what actually needed to be learned and are instead happy to spit our complete garbage, spec and all, and say it's a proof of something.
It's not shocking at all that the people who seem to get any benefit out of LLMs in the proof assistant space are the ones who could have just don't it themselves anyway.
I actually noticed after I posted that I should have said 'by individuals at pace not too different from writing down a standard prose proof' or something similar to address this point. Mathlib for instance is of course a phenomenal project, but it was written by a large amount of people over many years. Granted their goal was not so much speed as it was elegance, but by 'viable' I had in mind something that a research mathematician could actually use in real time along actual math research. The content in Mathlib still falls very much short of being enough for formalizing much of the actual research being done in mathematics, but people are now autoformalizing non-trivial extensions on top of that, both individually and collaboratively (such as the recent Tau Ceti project).
Anyway, what do you mean by 'say it's a proof of something'? If you mean this in technical sense, there are plenty of examples around where single individuals or small teams have autoformalized theorems with tens or hundreds of thousands of lines Lean code that pass a comparator challenge (so the theorem is correct). If you mean this in the sense that the proof is also a proof in the eyes of humans, so that someone has actually read or understood the proof, I'm willing to acknowledge that in this area there is much work still to do. In my own experience the current LLMs are already very strong at formally proving theorems (with correct semantics), but they are still lacking in writing human-readable math prose based on these formalizations, for instance.
I think it’s a lot like R vs SAS or Python and not any departure from the norm.
Mathematicians are not highly technical users that generally like programming. If you give them a path of importing someone else’s work they’ll do that instead of redoing it in their theorem prover of choice.
See the old mathematician joke:
A mathematician is asked to make tea. They take the kettle from the shelf, fill it with water, put it on the stove, turn on the heat and boil the water, then pour it into a cup.
The next day someone asks the mathematician to make tea again. The kettle is still on the stove from yesterday and there’s some leftover water in it, enough for tea.
The mathematician dumps the kettle out and puts the it away on the shelf. The mathematician says the system is in a known previously solved state and therefore trivially solved.
Now substitute 1500 lines of coq or lean for the kettle. As far as I know the outputs aren’t interchangeable like a vim or eMacs text file.
> lean4 doesn't let them formalize results outside of zfc.
A small correction: Lean 4's native semantics are DTT, not ZFC. Formalizing results for TT is arguably easier, it's the default. But you can use it for any foundation, so long as you have an implementation.
Presumably because of potential network effects that would result from everyone being able to work in a single system. I don't think it's strange or implausible for people to want that.
Metamath's Python verifier - its trusted kernel - is just 700 lines of Python short: https://github.com/david-a-wheeler/mmverify.py/blob/master/m...
Metamath Zero's Haskell implementation 700, and the C implementation 1000 lines (or 1800 overall) https://github.com/digama0/mm0
How do other proof systems compare?
Some bug counts: https://tristan.st/blog/in_search_of_falsehood
According to [0], "Rocq’s kernel spans approximately 41K lines of OCaml code, while Lean’s kernel consists of approximately 8K lines of C++ code."
My favorite minimalistic example of the Metamath base language (which higher level languages can compile down to), which, saved as, say, prop.mm can be verified with the verifier:
$c wff $. $( we use this $constant as a type of formula (well formed formula) $)
$c ( ) ! -> $. $( brackets, negation, implication $)
$v A B C $. $( $variables to be used in formulas $)
wa $f wff A $. $( $floating hypothesis "wa" which says, that A is a well-formed formula $)
wb $f wff B $.
wc $f wff C $.
$( The following assertions define the rules to create formulas $)
$( In Metamath (unlike Metamath Zero), definitions also use the $axiom statement type $)
wng $a wff ! A $. $( "not A is a well-formed-formula" - mandatory hypotheses are (wa) $)
wim $a wff ( A -> B ) $. $( "A implies B is a well-formed-formula" - mandatory hypotheses are (wa, wb) $)
$c |- $. $( this $constant will be used as a type of provable formula $)
$( Schemes of $axioms of propositional logic $)
a1 $a |- ( A -> ( B -> A ) ) $.
a2 $a |- ( ( A -> ( B -> C ) ) -> ( ( A -> B ) -> ( A -> C ) ) ) $.
a3 $a |- ( ( ! A -> ! B ) -> ( B -> A ) ) $.
$( Definition of the Modus Ponens rule of inference in new scope; otherwise $essential hypotheses (inputs) mp1, mp2 will become mandatory for all upcoming assertions $)
${
mp1 $e |- A $.
mp2 $e |- ( A -> B ) $.
mp $a |- B $. $( mandatory hypotheses of "mp" are (wa, wb, mp1, mp2) $)
$}
$( A $proof states the string of symbols to be proven, followed by a list of labels used by the stack machine $)
$( $floating and $essential hypotheses are pushed onto the top of the stack, $axioms and $proofs transform it by using the top of the stack as inputs $)
$( When the stack is empty at the end, the proof is successful, and the proved statement can be reused in further proofs using its label $)
$( " A implies ( B implies C) is a well-formed-formula" $)
formula1 $p wff ( A -> ( B -> C ) ) $= wa wb wc wim wim $.
$( "( A -> A ) -> ( A -> A ) is true (follows from the axioms)" $)
formula2 $p |- ( ( A -> A ) -> ( A -> A ) ) $= wa wa wa wim wim
wa wa wim wa wa wim wim
wa wa a1
wa wa wa a2
mp $.Fair to say that perhaps isn't selling it as much as you may think. It looks like perl that has been written by someone who is in the process of having a stroke.
https://github.com/Syzygies/Compare
I spent far too long comparing dozens of languages, only some of which are part of the above project, before switching from Haskell to Lean for a new phase of my math research. Lean wins for me without considering or using dependent types (for now). It is simply a better programming language than any I have seen.
This is a critical advantage for those interested in proof. One codes tactics using the same language as for proof.
https://github.com/Syzygies/Compare/blob/main/source/lean/Na...
I do have a future interest in proof, giving Lean an edge for me. I prefer my symbolic reasoning in visual form. I anticipate a future where we draw and view AI drawings, and see any printing press derived notation as antiquated as cuneiform. The above image is a possible language for representing the first Lean proof in "The Natural Number Game". About one in ten mathematicians that I show this to can grasp it much faster than the Lean notation. The other nine imagine a visual programming language to be something like a PowerPoint slide or a children's graphics language, and don't see the point.
Reminds me of the idea of Radical Monopolies from Ivan Illich in a way. If a technology or service becomes so wide spread within society, even though many different versions of the technology or service may exist, a Radical Monopoly means that non users will suffer for their non use. Cars and non drivers in cities are the typical example. And I wonder, whether mathematicians who don't user theorem provers will soon suffer under the tyranny of the theorem provers, whether it be Lean or one of the others.
> Cars and non drivers in cities are the typical example.
Have a look at Dutch cities for how to avoid this.
I immediately thought that’s not totally fair due to the size of the Netherlands vs other countries.
I asked Mistral to do an analysis: nearly zero R^2 for car ownership vs log country area, and it’s the same with proportion of urban population in OECD countries.
Netherlands isn’t very different from peers in car ownership, they just treat cyclists very well it seems.
This is a total tangent, just found it interesting.
So that link(https://infosec.exchange/@0xabad1dea/117002106099986943) buried in the comments of comments sounds pretty damning.
https://lipn.info/@mevenlennonbertrand/116997927457012577 , you and they might be jumping to conclusions
My takeaway was not the involvement of the LLMs, which I consider to be irrelevant; the core was that there was an exploit of a flaw in the verification engine that allowed an incorrect proof to be validated. That is not great.
In addition to their use as tools for pure math, they are also used for software verification, where adversarial examples could have real-world applications in verifiable supply chain attacks.
Metamath (and specifically Metamath Zero) is formally verified.
Recent posts on formal proofs usually talk about Lean, Rocq, Isabelle and (due to this post) Metamath.
What do people think of F* [1]? At least, for non-mathematics projects, doesn't it seem to be a more appropriate option [2]? It seems even the CS community is gravitating towards Lean.
[2] https://fstarlang.github.io/lowstar/html/Introduction.html
Isn't it today/wouldn't it be in the close future relatively trivial to port most of the already formalized results between languages with help of LLMs?
Was genuinely expecting an article about manufacturing methodologies! Very interesting teachable moment for me, I've always wanted to learn more about Set Theory.
Haven't seen much about SPARK and why3 recently, combined with frontier LLMs. It would seem easier to progressively prove properties from an actual implementation (going from absence of runtime errors to partial functional proof to full functional proof if you can beau the cost) and only focus the Lean effort on places where why3 (and its menagerie) of SMT provers give up ?
It would also seem highly agent-able since verification is very modular in SPARK.
> Metamath is based on set theory, and would therefore address some concerns one might have with the propositions-as-types philosophy used by Lean
Isn't the entire point mathematicians adopted Lean where they spurned Haskell is because of the batteries-included ZFC object language in the former? Metamath implements a set theory object language just the same, it's not based on it at all in this sense. You just changed one metalanguage for another.
I don't think Haskell would have been useful anyway? You'd want a dependently typed language for this, not Haskell. Haskell's types can't really express anything non-trivial.
Correct. There are awful tricks to write [1] dependent Haskell but even then it isn't powerful enough and has a significantly worse user experience then a proper dependently typed proof checker (as bad as the UX is on those!).
That said there are other languages such as Agda, Idris, and Rocq that would be fantastic replacements to Lean, especially if you care about staying constructive.
1. https://homepages.inf.ed.ac.uk/slindley/papers/hasochism.pdf
That's not quite right, Haskell's type system is fully turing complete. Strictly speaking, it can encode anything any other program can. If we put aside language extensions, I can see a pragmatic and ergonomic limitations obviously, but that's not an expressibility problem, and I wouldn't really equate what remains practical with triviality. With language extensions, you can just kind of do whatever as shown by Liquid Haskell.
I can see the real pain point more being the fact that it's not like a typical proof assistant, and utilizing it as one is going to be unintuitive and strange.
I had not been familiar with the term "Dumey microsecond".
For those who found it interesting, perhaps you will also find this term enjoyable https://en.wikipedia.org/wiki/Kairos for its similarity.
No programming language will ever be perfect. But it's still better for everyone to focus on improving one language than to recreate the same ideas across dozens of different languages, like a Tower of Babel of duplicated effort.
Are you programming in Fortran?
That's a bad analogy. LEAN is a leading bleeding edge proof verification tool.
I don't see what the issue is. Isn't the entire point of LLMs that we can easily translate between languages? What better way to cross check the correctness of a proof than to run it on several different systems simultaneously.
Recently I was playing around in this space and came across Dafny, rather I should confess that Claude code pointed be to it.
For software projects it seemed very approachable, even for a complete newbie like me. Probably not for Math though.