Smash Notes

Back to #503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory, with Ankur Moitra & Steven Brunton.

How Long Is Fermat's Last Theorem in Lean?

Alex Kontorovich says the Lean formalization of Fermat's Last Theorem is 13 million lines. He was sure that formalization was at least three years away.

He had just told a colloquium in the Princeton mathematics department that current systems build one or two layers on Mathlib and are weak at canonization, the cleanup that puts a lemma where it belongs in the library. A manuscript that pulls many areas together sat outside that halo. The statement of Fermat's Last Theorem is elementary, so there is no hidden meaning to check, and Lean reports the proof sorry-free. Whole textbooks formalize when they sit one or two steps from Mathlib. Three or four steps out, definitions made for different purposes clash and will not stitch back together.

Transcript

#503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory ·

Lex Fridman Podcast

More from this episode

  1. Would Alex Kontorovich Call AI in Mathematics a Crisis?Alex Kontorovich does not know whether he would call the moment in AI and mathematics a crisis. A lot of opportunity sits in the crisis.
  2. Does Ankur Moitra Feel More Intellectually Lonely?Ankur Moitra says he might feel more intellectually lonely than before. He feels very excited.
  3. Why Does Steven Brunton Stay Hopeful About AI?Steven Brunton says most of his feelings are hopeful.
  4. Did AI Solve the Riemann Hypothesis?No. Alex Kontorovich says the Riemann hypothesis is not solved.
  5. Why Is the Doorman Fallacy Useful for Math and AI?Ankur Moitra treats the doorman fallacy as a good way to think about mathematics in the age of AI.
  6. Are Machines Already Finding Physical Laws?Steven Brunton says that search is already happening for physical laws.
  7. Why Is AI Math More Than a Convex Hull?Alex Kontorovich says AI math is more than a convex hull because known moves, chained together, reach much further than that label allows.
  8. Is the quasi-Riemann Hypothesis Proof a New Idea?Alex Kontorovich says the quasi Riemann Hypothesis case is not a genuinely new idea. It is a beautiful combination of things that were already very well known.
  9. How Are Research Papers Changing Because of AI?Steven Brunton says an academic contribution in engineering and applied science is already shifting, and with AI a small reusable piece is worth more than one…
  10. Why Do Journals Need New Models for AI Proofs?Ankur Moitra wants new models that ease the load on journals and conferences, including a shared effort to verify AI proofs, while the older publishing models…
  11. What Is the Riemann Hypothesis?Alex Kontorovich says the Riemann hypothesis predicts that every wave behind the prime numbers has exactly the same amplitude.
  12. Can AI Formalize a Proof a Human Already Wrote?Alex Kontorovich says autoformalization is basically solved, and not in the way he wants.
  13. What Did AI Prove About the quasi-Riemann Hypothesis?Alex Kontorovich says the proof gives an infinite vertical strip with no zeros whose real part lies between seven-eighths and one.
  14. Why Is a Fluid So Hard to Solve?Steven Brunton says force equals mass times acceleration is simple to write down, yet a fluid becomes very complicated to solve in any practical sense.
  15. What Is the Basic Building Block of a Fluid?Steven Brunton says the basic building block of a fluid is a velocity field, together with the pressures that force it to move.
  16. What Did AI Prove About Navier-Stokes?Steven Brunton says the proven result is the forced version on the Clay website, one of four versions, not every Navier-Stokes problem.
  17. Why Is Turbulence So Hard?Steven Brunton says turbulence is a complex fluid roiling, mixing, and moving, and it is hard because it is multi-scale.
  18. What Did Kamal Jain Say About Laptops and Markets?Ankur Moitra cites Kamal Jain for the line he likes: if my laptop can't find it, then neither can the market.
  19. Did AI Prove the Unique Games Conjecture?Ankur Moitra says a headlining result in the OpenAI drop is that the unique games conjecture is true.
  20. How Many Authors Does a Math Paper Have?Alex Kontorovich says almost every math paper written until the 20th century had one author. He thinks the median in mathematics is now two.
  21. How Do AI Swarms Act Like the Game of Life?Lex Fridman says a single large LLM that does a lot of reasoning and can interact with tools is already a dynamical process, and multi-agent swarms built from…
  22. What Should Pure Math PhD Students Learn?Alex Kontorovich says PhD students in pure mathematics should do what mathematicians have always done: learn the most difficult, most abstract mathematics…
  23. Why Learn Linear Algebra If a Computer Does It Better?Steven Brunton says you should still learn linear algebra even if a computer can do linear algebra better than you, because the point is to build intuition…
  24. Why Does Ankur Moitra Say to Pretend You Already Won?Ankur Moitra says the best way to live a rich and fulfilling life is to pretend you have already won, then figure out what you would do.
  25. What Matters When AI Commodifies Intelligence?Lex Fridman believes that as AI commodifies intelligence, people will treat intelligence and the puzzles they solve as a cherry on top, or a cherry to the…

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