When Ciro Santilli first learnt the old Zermelo-Fraenkel set theory and the idea of formal proofs, his teenager mind was completely blown.
Finally, there it was: a proper and precise definition of mathematics, including a definition of integers, reals and limits!
Theorems are strings, proofs are string manipulations, and axioms are the initial strings that you can use.
Once proved, press a button on your computer, and the proof is automatically verified. No messy complicated "group of savants" reading it for 4 years and looking for flaws!
There are a few proof assistant systems with several theorems in their Git tracked standard library. The hottest ones circa 2020 are:
- github.com/HOL-Theorem-Prover/HOL
- github.com/seL4/isabelle. Rumours have it that this is "uncompilable" from source without blobs. It does however offer a very rich IDE.
- github.com/coq/coq
- Metamath this one is likely an older and less powerful system, but the web presentation and tutorial are very good! Source: github.com/metamath/metamath-exe Here is a proof that 2 + 2 equals 4: us.metamath.org/mpeuni/2p2e4.html
- Lean
- www.bookofproofs.org/branches/fpl-formal-proving-language/ from BookofProofs
And here are some more interesting links:
- github.com/awesomo4000/awesome-provable an awesome list of formal stuff
- devel.isa-afp.org/ Isabelle Archive of Formal Proofs. A curated list of Isabelle proofs, with minimal web UI. This is almost what we need, but without the manual curation, and with a better web UI.
- www.cs.ru.nl/~freek/100/ list of how many of the "arbitrarily" selected the Hundred Greatest Theorems by Paul and Jack Abad (1999) had been proved in several formal systems, serving therefore as a benchmark of sorts
However, as expressed by the QED manifesto, is unbelievable that there isn't one awesome and dominating website, that hosts all those proofs, possibly an on the browser editor, and which all mathematicians in the world use as the one golden reference of mathematics to rule them all!
Just imagine the impact.
Standard library maintainers don't have to deal with the impossible question of what is "beautiful" or "useful" enough mathematics to deserve merged: users just push content to the online database, and star what they like!
We then just use GitHub-like namespaces for each person's theorem, e.g. "cirosantilli/fundmaental-theorem-of-calculus" or "johndoe/fundmaental-theorem-of-calculus" so that each person owns their own preferred definition IDs, which others can reuse.
No more endless bikeshedding over what insane level of generality do your analysis theorems need to be (Ciro Santilli attended at talk about Lean where the speaker mentioned this was a problem)!
This would move things more out of the "pull request and Git tracked code" approach, into a more "database with entries" version of things.
Furthermore, it is just a matter of time until the "single standard library" approach starts to break down, as the git clone becomes impossibly large. At this point, people have to start publishing separate packages. And when this happens, you would need to retest every package that you add to your project. This is why a centralized database is just inevitable at some point, it just scales better.
Interested in a conjecture? No problem: just subscribe to its formal statement + all known equivalents, and get an email on your inbox when it gets proved!
Are you a garage mathematician and have managed to prove a hard theorem, but no "real" mathematician will read your proof because your unknown? Fuck that, just publish it on the system and let it get auto verified. Overnight fame awaits.
Notation incompatibility hell? A thing of the past, just automatically convert to your preferred representation.
Such a system would be the perfect companion to OurBigBook.com. Just like computer code offers the backbone of Linux Kernel Module Cheat Linux kernel tutorials, a formal proof system website would be the backbone of mathematics tutorials! You know what, if OurBigBook.com becomes insanely successful, Ciro is going to add this to it later on.
Furthermore, it would not be too hard to achieve this system!
All we would need would be something analogous to a package registry like PyPI or NodeJS' registry.
Then, each person can publish packages containing proofs.
Packages can rely on other packages that contain pre-requisites definition or theorem.
Packages are just regular git repos, with some metadata. One notable metadata would be a human readable description of the theorems the package provides.
The package registry would then in addition to most package registries have a CI server in it, that checks the correctness of all proofs, generates a web-page showing each theorem.
All proofs can be conditional: the package registry simply shows clearly what axiom set a theorem is based on.
This is a close as we can get to Erdős' book.
Maybe Ciro will just stuff this into OurBigBook.com once that takes over the world.
This project could be seen as a more automated/less moderated version of ProofWiki.
Bibliography:
- The Math Genome Project has very similar end goals. Apparently it will run proofs on server against the stdlib, but not allow one proof to depend on another, so in the end you still have to pull request everything back. Also there may be moderation forever, unclear. Ciro tried to create a dummy lolol theorem without any correct syntax and it just became private. Also apparently every single proof needs corresponding LaTeX manually written to be accepted. Cowards!
- math.stackexchange.com/questions/1767070/what-is-the-current-state-of-formalized-mathematics/3297536#3297536
- math.stackexchange.com/questions/2747661/why-is-there-not-a-system-for-computer-checking-mathematical-proofs-yet-2018
- stackoverflow.com/questions/19421234/how-do-i-generate-latex-from-isabelle-hol
- stackoverflow.com/questions/30152139/what-are-the-strengths-and-weaknesses-of-the-isabelle-proof-assistant-compared-t
- arxiv.org/abs/2102.03044 SPIRG, a decentralized version of this
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