{"slug": "metamath", "name": "Metamath", "category": "other", "category_name": "Other software", "description": "", "recommenders": 6, "commenters": 10, "warners": 1, "score": 5.218, "rising": -1.96, "first": "2025-11-03", "latest": "2026-08-05", "why": "When I first heard about metamath I was blown away at how I could drill down to the base axioms", "why_is_quote": true, "has_verdict": false, "sentiment": {"recommend": 6, "neutral": 3, "compare": 0, "warn": 1}, "spark": [1, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0], "listed": true, "loved": [], "caveat": "", "short_quote": "When I first heard about metamath I was blown away at how I could drill down to the base axioms", "kind": "", "icon": true, "domain": "metamath.org", "github_repo": null, "official_url": "https://us.metamath.org", "pricing": "", "aliases": ["Metamath"], "verdict": null, "praise": [{"label": "", "summary": "", "quotes": [{"id": 786203617, "item_id": 49137726, "url": "https://news.ycombinator.com/item?id=49137726", "by": "michaelfm1211", "date": "2026-08-01", "stance": "recommend", "quote": "Stuff like this just wouldn't happen in Metamath.", "excerpt": "Reminds me of this: https://mathoverflow.net/questions/513742/are-we-stuck-with-lean I know this is an implementation bug not a meta-theory bug, but I'd almost consider the fact soundness bugs are possible as a bug in the ideology, or at least a severe drawback. Stuff like this just wouldn't happen in Metamath. In a future where AI is autogenerating formalizations, why not have the AI use a harder but airtight system", "context": "", "use_case": "Formal proof system", "switched_from": "", "story": "Postmortem for Kernel Soundness Bug #14576"}, {"id": 785789632, "item_id": 49111852, "url": "https://news.ycombinator.com/item?id=49111852", "by": "7373737373", "date": "2026-07-30", "stance": "recommend", "quote": "My favorite minimalistic example of the Metamath base language", "excerpt": "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", "context": "", "use_case": "Formal proof verification", "switched_from": "", "story": "Are We Stuck with Lean?"}, {"id": 783986977, "item_id": 48999186, "url": "https://news.ycombinator.com/item?id=48999186", "by": "YoshiRulz", "date": "2026-07-21", "stance": "recommend", "quote": "https://us.metamath.org", "excerpt": "From the article: https://mathlib.org Also https://1000-plus.github.io and https://us.metamath.org", "context": "", "use_case": "Formal math proofs", "switched_from": "", "story": "Human mathematicians are being outcounterexampled"}, {"id": 766835361, "item_id": 47927210, "url": "https://news.ycombinator.com/item?id=47927210", "by": "raphlinus", "date": "2026-04-27", "stance": "recommend", "quote": "I think it's worthy of study and understanding.", "excerpt": "People interested in alternatives to Lean should also look at Metamath. It has nowhere near the adoption that Lean is getting, but is holding its own in [100] theorems results. It has some advantages and compelling properties, not least of which is that it's very simple, so much so that there are many implementations of checkers; most other proof systems are ultimately defined by a single implementation. It's also as", "context": "", "use_case": "Formal proof verification", "switched_from": "", "story": "\"Why not just use Lean?\""}, {"id": 762158912, "item_id": 47634932, "url": "https://news.ycombinator.com/item?id=47634932", "by": "smj-edison", "date": "2026-04-03", "stance": "recommend", "quote": "When I first heard about metamath I was blown away at how I could drill down to the base axioms", "excerpt": "I feel the same. When I first heard about metamath I was blown away at how I could drill down to the base axioms (I had only tried Lean before). Lean also feels too magical for my taste, and I dislike that I don't have a good mental model of its execution under the hood. I care a lot about execution speed as well, and Lean... isn't always fast. It's another reason metamath's design really speaks to me. You might find", "context": "", "use_case": "", "switched_from": "", "story": "Automatic Textbook Formalization"}]}], "caveats": [{"label": "", "summary": "", "quotes": [{"id": 786204672, "item_id": 49137792, "url": "https://news.ycombinator.com/item?id=49137792", "by": "derdi", "date": "2026-08-01", "stance": "warn", "quote": "Took me about 90 seconds to find a Metamath implementation bug", "excerpt": "Took me about 90 seconds to find a Metamath implementation bug that apparently allowed proving something that shouldn't be provable: https://github.com/metamath/metamath-exe/issues/184", "context": "", "use_case": "", "switched_from": "", "story": "Postmortem for Kernel Soundness Bug #14576"}]}], "who": [], "tags": [], "use_cases": ["formal proof verification", "formal proof system", "formal math proofs", "formal math proof system"], "compared": [{"slug": "lean", "name": "Lean", "co": 4, "switched_from": 0, "switched_to": 0}, {"slug": "isabelle", "name": "Isabelle", "co": 2, "switched_from": 0, "switched_to": 0}], "makers": [{"id": 785775632, "item_id": 49110977, "url": "https://news.ycombinator.com/item?id=49110977", "by": "ux266478", "date": "2026-07-30", "stance": "recommend", "quote": "", "excerpt": "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", "context": "", "use_case": "", "switched_from": "", "story": "Are We Stuck with Lean?"}, {"id": 785767024, "item_id": 49110439, "url": "https://news.ycombinator.com/item?id=49110439", "by": "dwheeler", "date": "2026-07-30", "stance": "recommend", "quote": "Metamath contributor here! Each proving tool has its pros and cons, but always happy to see Metamath noted", "excerpt": "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: htt", "context": "", "use_case": "", "switched_from": "", "story": "Are We Stuck with Lean?"}], "threads": [{"url": "https://news.ycombinator.com/item?id=49108678", "title": "Are We Stuck with Lean?", "mentions": 3}, {"url": "https://news.ycombinator.com/item?id=49137060", "title": "Postmortem for Kernel Soundness Bug #14576", "mentions": 2}, {"url": "https://news.ycombinator.com/item?id=49355968", "title": "Palomar: A registry of Lean verified mathematics", "mentions": 1}, {"url": "https://news.ycombinator.com/item?id=49123231", "title": "Why is it all in the kernel?", "mentions": 1}, {"url": "https://news.ycombinator.com/item?id=48983382", "title": "Human mathematicians are being outcounterexampled", "mentions": 1}, {"url": "https://news.ycombinator.com/item?id=47922079", "title": "\"Why not just use Lean?\"", "mentions": 1}], "sentiment_by_month": {"2025-11": {"recommend": 1, "neutral": 1}, "2025-12": {}, "2026-01": {}, "2026-02": {"neutral": 1}, "2026-03": {}, "2026-04": {"recommend": 2}, "2026-05": {}, "2026-06": {}, "2026-07": {"recommend": 2}, "2026-08": {"recommend": 1, "warn": 1, "neutral": 1}, "2026-09": {}, "2026-10": {}}, "quotes": [{"id": 789715296, "item_id": 49357206, "url": "https://news.ycombinator.com/item?id=49357206", "by": "dwheeler", "date": "2026-08-18", "stance": "neutral", "quote": "The metamath community tends to centralize results", "excerpt": "Very cool. The metamath community tends to centralize results, so its equivalent is simply: https://us.metamath.org/", "context": "", "use_case": "", "switched_from": "", "story": "Palomar: A registry of Lean verified mathematics"}, {"id": 786903376, "item_id": 49181461, "url": "https://news.ycombinator.com/item?id=49181461", "by": "xelxebar", "date": "2026-08-05", "stance": "neutral", "quote": "Intriguingly, Metamath handles recursive definitions by factoring out the recursion into a single higher-order function", "excerpt": "Intriguingly, Metamath handles recursive definitions by factoring out the recursion into a single higher-order function: https://us.metamath.org/mpeuni/df-rdg.html Essentially, it's just doing a lazy fixpoint a la Haskell's fix function. The definition is a little more general, though, to make it work for both transfinite and well-founded recursions as well. This chashed out nicely in a sequence builder: https://us.m", "context": "", "use_case": "", "switched_from": "", "story": "Why is it all in the kernel?"}, {"id": 786204672, "item_id": 49137792, "url": "https://news.ycombinator.com/item?id=49137792", "by": "derdi", "date": "2026-08-01", "stance": "warn", "quote": "Took me about 90 seconds to find a Metamath implementation bug", "excerpt": "Took me about 90 seconds to find a Metamath implementation bug that apparently allowed proving something that shouldn't be provable: https://github.com/metamath/metamath-exe/issues/184", "context": "", "use_case": "", "switched_from": "", "story": "Postmortem for Kernel Soundness Bug #14576"}, {"id": 786203617, "item_id": 49137726, "url": "https://news.ycombinator.com/item?id=49137726", "by": "michaelfm1211", "date": "2026-08-01", "stance": "recommend", "quote": "Stuff like this just wouldn't happen in Metamath.", "excerpt": "Reminds me of this: https://mathoverflow.net/questions/513742/are-we-stuck-with-lean I know this is an implementation bug not a meta-theory bug, but I'd almost consider the fact soundness bugs are possible as a bug in the ideology, or at least a severe drawback. Stuff like this just wouldn't happen in Metamath. In a future where AI is autogenerating formalizations, why not have the AI use a harder but airtight system", "context": "", "use_case": "Formal proof system", "switched_from": "", "story": "Postmortem for Kernel Soundness Bug #14576"}, {"id": 785789632, "item_id": 49111852, "url": "https://news.ycombinator.com/item?id=49111852", "by": "7373737373", "date": "2026-07-30", "stance": "recommend", "quote": "My favorite minimalistic example of the Metamath base language", "excerpt": "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", "context": "", "use_case": "Formal proof verification", "switched_from": "", "story": "Are We Stuck with Lean?"}, {"id": 783986977, "item_id": 48999186, "url": "https://news.ycombinator.com/item?id=48999186", "by": "YoshiRulz", "date": "2026-07-21", "stance": "recommend", "quote": "https://us.metamath.org", "excerpt": "From the article: https://mathlib.org Also https://1000-plus.github.io and https://us.metamath.org", "context": "", "use_case": "Formal math proofs", "switched_from": "", "story": "Human mathematicians are being outcounterexampled"}, {"id": 766835361, "item_id": 47927210, "url": "https://news.ycombinator.com/item?id=47927210", "by": "raphlinus", "date": "2026-04-27", "stance": "recommend", "quote": "I think it's worthy of study and understanding.", "excerpt": "People interested in alternatives to Lean should also look at Metamath. It has nowhere near the adoption that Lean is getting, but is holding its own in [100] theorems results. It has some advantages and compelling properties, not least of which is that it's very simple, so much so that there are many implementations of checkers; most other proof systems are ultimately defined by a single implementation. It's also as", "context": "", "use_case": "Formal proof verification", "switched_from": "", "story": "\"Why not just use Lean?\""}, {"id": 762158912, "item_id": 47634932, "url": "https://news.ycombinator.com/item?id=47634932", "by": "smj-edison", "date": "2026-04-03", "stance": "recommend", "quote": "When I first heard about metamath I was blown away at how I could drill down to the base axioms", "excerpt": "I feel the same. When I first heard about metamath I was blown away at how I could drill down to the base axioms (I had only tried Lean before). Lean also feels too magical for my taste, and I dislike that I don't have a good mental model of its execution under the hood. I care a lot about execution speed as well, and Lean... isn't always fast. It's another reason metamath's design really speaks to me. You might find", "context": "", "use_case": "", "switched_from": "", "story": "Automatic Textbook Formalization"}, {"id": 751075744, "item_id": 46942234, "url": "https://news.ycombinator.com/item?id=46942234", "by": "LegionMammal978", "date": "2026-02-08", "stance": "neutral", "quote": "I've recently been dabbling a fair bit with Metamath", "excerpt": "Thank you for that information, all I could find on the website was that \"The source code of the Mizar verifier and accompanying tools is available to the members of SUM\" [0], which of course does not reflect the newer status quo. > Mizar is indeed \"scarcely advertised\", but all the information is publicly available, who wants to know knows. Yes, there indeed seems to be a good bit of information available, especiall", "context": "", "use_case": "Formal math proof system", "switched_from": "", "story": "First Proof"}, {"id": 736291280, "item_id": 46018205, "url": "https://news.ycombinator.com/item?id=46018205", "by": "librexpr", "date": "2025-11-22", "stance": "neutral", "quote": "plenty of computer-verified mathematics done in Metamath/set.mm and Isabelle/ZF", "excerpt": "> Say I want natural numbers, I need to choose a concrete implementation in set theory e.g. Von Neumann, but there are multiple choices. You don't need to choose a concrete implementation. If you don't want to choose a construction, you can just say something like \"let (N, 0, +, *) be a structure satisfying the peano axioms\" and work from there. > For all good definitions, so get Peano arithmetic and can work with, b", "context": "", "use_case": "", "switched_from": "", "story": "Set theory with types"}, {"id": 732898992, "item_id": 45806187, "url": "https://news.ycombinator.com/item?id=45806187", "by": "DoctorOetker", "date": "2025-11-03", "stance": "recommend", "quote": "Check out metamath.org", "excerpt": "Check out metamath.org I can't fathom any valid argument that couldn't be formalized to mathematical statements. There would still be disagreements on axioms and physical postulates, especially where there are conflicts of interest.", "context": "", "use_case": "formal proof verification", "switched_from": "", "story": "Robert Hooke's \"Cyberpunk\u201d Letter to Gottfried Leibniz"}]}