Towards collaborative, open science in Maths - with the help of computers

Open science, an old idea that has resurfaced as an obvious choice…. Even though its benefits are still not evident.
4 years after the death of Vladimir Voevodsky, I would like to talk to you about how this brilliant mathematician took part in the establishment of one of the most dazzling open science projects while, contrary to what you might expect, high-level mathematics had a tendency to be extremely closed and resistant to reproducibility.
And I am not talking about any old project: it involved nothing less than re-writing the foundations of mathematics, including automated theorem proving.
We will look at how the practices put in place, inspired by open source software, engendered this success and paved the way for greater openness in high-level maths. And how open and collaborative science is a major gain for human knowledge.

Transcript

1. francois@rudder.io @fanf42 Towards collaborative, open science in Maths - with the help of computers - 2021-11
2. Hi! secops automation/compliance app manage ten of thousands servers 2 François ARMAND CTO Free Software Company “Stay Up” Stand B14
3. Mathematics proofs are hard.
4. Mathematics proofs are hard. Today proofs are hundreds of pages long. And because of high specialization, there is routinely less than 10 people on Earth able to barely understand them.
5. Mathematics proofs are hard. Today proofs are hundreds of pages long. And because of high specialization, there is routinely less than 10 people on Earth able to barely understand them. ↠ Grigori Perelman proof of Poincaré conjecture in 2002: after several years of review - it seems ok
6. Mathematics proofs are hard. Today proofs are hundreds of pages long. And because of high specialization, there is routinely less than 10 people on Earth able to barely understand them. ↠ Grigori Perelman proof of Poincaré conjecture in 2002: after several years of review - it seems ok ↠ Shinichi Mochizuki proof of ABC conjecture in 2012: 500 pages of terse proofs and totally new concepts - nobody knows if ok
7. “The world of mathematics is becoming very large, the complexity of mathematics is becoming very high, and there is a danger of an accumulation of mistakes. Proofs rely on other proofs; if one contains a flaw, all others that rely on it will share the error.” Vladimir Voevodsky
8. Vladimir Voevodsky • In 1990, enters Harvard without applying after publishing a major theorem proof. • Proves Milnor’s conjecture in 1996. • Fields Medal in 2002 at only 38 years old for Milnor’s conjecture proof. • After 10 years of work, proves Bloch-Katos conjectures in 2009.
9. Building castle on sand Sometimes, major proofs are wrong.
10. Building castle on sand Sometimes, major proofs are wrong. ↠ Kenji Fukaya foundational theorem on symplectic geometry proof was mostly wrong (and it takes almost 15 years to get noticed and corrected)
11. Building castle on sand Sometimes, major proofs are wrong. ↠ In 1998, Carlos Simpson find an error in Voevodsky’s 1990’s work. It was an epiphany for Voevodsky. ↠ Kenji Fukaya foundational theorem on symplectic geometry proof was mostly wrong (and it takes almost 15 years to get noticed and corrected)
12. “I got worried. I stopped doing curiosity driven research. I might make a mistake. And, as I’ve just learned, no one was likely to be checking up with any diligence.” Vladimir Voevodsky
13. “I got worried. I stopped doing curiosity driven research. I might make a mistake. And, as I’ve just learned, no one was likely to be checking up with any diligence.” Vladimir Voevodsky Machine checked proof ?
14. In software world - proof assistants Goal: make interactive, formal, machine verified proof of theorems They are like a game: • you have some basic rules (higher order logic) • you defined goals (theorems), and steps to reach them (lemma) • and then you spend hours arguing with the computer (the interactive proof part) • in the end, you get a reusable machine checked proof
15. In software world - proof assistants Goal: make interactive, formal, machine verified proof of theorems They are like a game: • you have some basic rules (higher order logic) • you defined goals (theorems), and steps to reach them (lemma) • and then you spend hours arguing with the computer (the interactive proof part) • in the end, you get a reusable machine checked proof • Martin-Löf type theory ◦ 1972 • Isabelle/HOL ◦ 1986 ◦ Higher Order Logic ◦ University of Cambridge & Munich • Coq ◦ 1989 ◦ Calculus of Inductive Constructions (Coquant, Bertot, Leroy...) ◦ INRIA 󰏃 • Lean ◦ 2013 ◦ Calculus of Inductive Constructions ◦ Microsoft
16. Making proof assistants suitable for Maths Proof assistant equivalence language is alien for mathematicians. (type theory) (set theory)
17. Making proof assistants suitable for Maths Voevodsky spent 7 years rebuilding Mathematics foundation to make them easy to use in proof assistant: Homotopy Type Theory (HoTT) Proof assistant equivalence language is alien for mathematicians. (type theory) (set theory)
18. Making proof assistants suitable for Maths And then, magic happened... Then, he tries to convince people to use that foundation to build up things
19. HoTT book - Doing science in our Millennium In 2012-2013, a special team is built. Achievement: “collaborative open science” - in less than 6 months - more than 30 Mathematicians, - from different backgrounds, - created a new branch of Mathematics, - with machine-checked proofs in Coq, - reported in a 600 pages book, - in github, under a Creative Common license !
20. HoTT book - Doing science in our Millennium In 2012-2013, a special team is built. Achievement: “collaborative open science” - in less than 6 months - more than 30 Mathematicians, - from different backgrounds, - created a new branch of Mathematics, - with machine-checked proofs in Coq, - reported in a 600 pages book, - in github, under a Creative Common license ! “Truly open research habitats cannot be obstructed by copyright, profit-grabbing publishers, patents, commercial secrets, and funding schemes that are based on faulty achievement metrics.”
21. End of story? Proof assistants don’t have it easy Most mathematicians don’t like nor use proof assistants: • "they remove the human creativity and insights" • "they force to tediously prove each and all little aspects of a theory, obscuring the bigs ideas behind a heap of irrelevant technical details" * Plus: • UnivMath effort stalled, its legacy is not clear yet • Voevodsky died in septembre 2017 after difficult years • HoTT needs some time to be more broadly assimilated * If you are a developer, it may sound familiar (yes, that holy war between statically and dynamically typed language :)
22. Peter Scholze • maths prodigy (but don't call it like that) ◦ 3 gold / 1 silver medal in International Mathematical Olympiad, ◦ license in 3 semesters, master in 2 semesters, PhD done at 24y old... • youngest Fields medal en 2018 at 30 for “the revolution that he launched in arithmetic geometry.”
23. • Scholze main work: Condensed Sets ◦ new foundations for topology • July 2019: works on a central theorem ◦ (a complicated equivalence: "real functional analysis still works if you replace topological spaces with condensed sets") ◦ make the proof in his head with few notes for 4 days Proving theorem in his head
24. • Scholze main work: Condensed Sets ◦ new foundations for topology • July 2019: works on a central theorem ◦ (a complicated equivalence: "real functional analysis still works if you replace topological spaces with condensed sets") ◦ make the proof in his head with few notes for 4 days Proving theorem in his head
25. • Scholze main work: Condensed Sets ◦ new foundations for topology • July 2019: works on a central theorem ◦ (a complicated equivalence: "real functional analysis still works if you replace topological spaces with condensed sets") ◦ make the proof in his head with few notes for 4 days ◦ not sure about one central part of the proof ◦ even after writing it down 4 months latter Proving theorem in his head
26. • Scholze main work: Condensed Sets ◦ new foundations for topology • July 2019: works on a central theorem ◦ (a complicated equivalence: "real functional analysis still works if you replace topological spaces with condensed sets") ◦ make the proof in his head with few notes for 4 days ◦ not sure about one central part of the proof ◦ even after writing it down 4 months latter • Asked Kevin Buzzard and Johan Commelin, from Lean community, for help ↠ Liquid Tensor Experiment: "help me prove that hard theorem in Lean" Proving theorem in his head
27. The Liquid Tensor Experiment - success In december 2020, the Liquid Tensor experiment is started. Achievement: “collaborative open science” - in less than 6 months - more than 12 Mathematicians, - from different backgrounds, - proved a hard theorem in a new branch of Mathematics, - with machine-checked proofs in Lean, - in github !
28. Maths as Collaborative open science Formal, machine-checked proof allowed to build on sound ground, to scale-up the team and push forward quickly. Distributed collaborative work improved efficiency: multiple backgrounds, different expertise and insights, more reviewers and helps, more docs and async tasks. Open processes created commons: helped to spread the theory, get feedbacks, removed virtual toll and IP risk. Massive gains for Human knowledge 3 1 2
29. Collaborative open science Machine-checked automation allows to build on sound ground, scale-up the team and push forward quickly. Distributed collaborative work improves efficiency: multiple backgrounds, different expertise and insights, more reviewers and helps, more docs and async tasks. Open processes creates commons: help to spread the theory, get feedbacks, remove virtual toll and IP risk. Massive gains for Human knowledge 3 2 1
30. … for massive gains for Human knowledge ! Let's bring it everywhere ... Collaborative open science
31. References (Amazing pictures and quotes also mainly taken from following links *) • On Shirishi’s ABC proof: ◦ http://projectwordsworth.com/the-paradox-of-the-proof/ ◦ https://www.quantamagazine.org/hope-rekindled-for-abc-proof-20151221/ • On Perelman’s proof: ◦ https://www.newyorker.com/magazine/2006/08/28/manifold-destiny • On Kenji Fukaya foundational theorem problems: ◦ https://www.quantamagazine.org/the-fight-to-fix-symplectic-geometry-20170209/ • On Voevodsky error revealed by Carlos Simpson: ◦ https://nautil.us/issue/24/error/in-mathematics-mistakes-arent-what-they-used-to-be • Actually, the HoTT book (and author’s feedback on open science): ◦ https://homotopytypetheory.org/ ◦ http://math.andrej.com/2013/06/20/the-hott-book/ • A superbe, must read article about Voevodsky, univalent foundation, set theory, and Maths roots: ◦ https://www.quantamagazine.org/univalent-foundations-redefines-mathematics-20150519/ • On Peter Scholze Liquid Tensor Experiment: ◦ https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-experiment/ ◦ https://www.quantamagazine.org/lean-computer-program-confirms-peter-scholze-proof-20210728/ ◦ https://github.com/leanprover-community/lean-liquid * please let me know if it’s not ok! • The Coq Proof assistant: ◦ https://coq.inria.fr/ • Lean theorem prover: ◦ https://leanprover.github.io/about/ • Stay Up - Journey of a Free Software Company ◦ https://medium.com/@fanf42/stay-up-5 b780511109d