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🧵 My 3rd book is now out! It is on the fascinating history of math.
How did people throughout history think about math?What problems did they care about? How did they express them? How were they solved?
This book includes a lot of historical math.
a.co/d/gsoNKxP
How did people throughout history think about math?What problems did they care about? How did they express them? How were they solved?
This book includes a lot of historical math.
a.co/d/gsoNKxP
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I’m still amazed that there aren’t more mistakes being found.
100 mathematics wrote 500 journal articles totaling 15,000 pages of high level research which when combined proves a theorem. And they were correct!
Humans are incredible!
100 mathematics wrote 500 journal articles totaling 15,000 pages of high level research which when combined proves a theorem. And they were correct!
Humans are incredible!
The largest math proof in human history is now computer-verified.
Today, Axiom announces a complete Lean auto-formalization of the Classification of Finite Simple Groups, the "periodic table of symmetry."
About 15,000 pages, built by over 100 mathematicians across generations.
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Math 1 was crazy
Andrew Wiles worked on Fermat's Last Theorem alone in his attic for 7 years and told nobody except his wife.
The problem was 358 years old. Fermat scribbled in a margin in 1637 that he had a proof and no room to write it, and every mathematician since had failed.
Wiles saw the book at 10 years old. He never let it go.
He was a Princeton professor when he started in 1986. He stopped publishing new work and released old results in pieces to hide the silence.
He announced it in Cambridge in June 1993, over 3 lectures, revealing the target only at the end. Newspapers ran it worldwide.
Then a referee found a gap. One step in the proof did not hold, and the error was fundamental.
He spent 14 months trying to close it in public view, with the whole field watching. He was ready to quit.
The fix arrived in September 1994. He has said he stared at it in disbelief for 20 minutes.
The BBC filmed him describing that moment. He gets a few words out, then turns away from the camera in tears.
He kept the greatest result of the century secret so nobody could take it.
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My son asked me what infinity plus ten equals.
I’ve been training for this moment my whole life.
I’ve been training for this moment my whole life.
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“The reports of my death have been greatly exaggerated”
—human math researchers
—human math researchers
Some fascinating developments following OpenAI's shock breakthrough proof of the Quasi-Riemann Hypothesis, which was released publicly on October 6:
1. Building on OpenAI's proof, a recent Hexagon preprint (released on October 12) by Cureton and Thorner establishes that ALL nctions are zero-free to the right of 1 - 1/6000.
2. And now, today, an arXiv preprint by Frei, Somos and Widmer uses this result to prove unconditionally the 2007 Ellenberg–Venkatesh bounds on ℓ-torsion in class groups of number fields.
I was amused to see the AI declaration in the second paper: "No artificial intelligence tools were used in this work."
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Jay Cummings retweeted
We want to support the mathematics community in turning progress in AI into deeper human understanding and new discoveries. To support that goal, we are contributing $5 million to a new Simons Foundation initiative to help mathematicians examine, understand, and build on results produced by AI, including the results we recently published.
The funding, managed and awarded by the @SimonsFdn will support workshops, conferences, and special programs. We will also provide access to our models and computing resources to Simons Foundation-supported researchers.
The funding, managed and awarded by the @SimonsFdn will support workshops, conferences, and special programs. We will also provide access to our models and computing resources to Simons Foundation-supported researchers.
How can the math community forge the best path forward in the AI age? We're committing $15M to workshops and initiatives — plus free AI computing resources for foundation-supported researchers — through new partnerships with @AnthropicAI and @OpenAI: bit.ly/4zpfIbW
#math #science #AI
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Jay Cummings retweeted
you could be living in sacramento
you could be sacramentomaxxing
we have the the kings, old sac, the waterfront
you will build your life in sacramento, california
you could be sacramentomaxxing
we have the the kings, old sac, the waterfront
you will build your life in sacramento, california
America’s most livable cities, at local median salary and local tax rate.
New York: this is probably the reason why Mamdani wonCommunity noteThe chart's "most livable cities" rankings and quality-of-life data rely on Numbeo, a crowdsourced site where anyone can submit or alter entries; it carries explicit disclaimers on accuracy and has been widely criticized by experts and media as unreliable.
numbeo.com/common/terms_o…
en.wikipedia.org/wiki/Numbeo
tf1info.fr/international/…
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Jay Cummings retweeted
Just getting started on @LongFormMath’s Proofs, already much more lively than how proofs were taught in high school. Looking forward to the rest of it!
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Jay Cummings retweeted
10 remarks on AI and math that can be true all at once:
1. The recent progress is stunning and exciting.
2. The Quasi-Riemann Hypothesis is arguably Fields-medal level.
3. Mathematics is being transformed and there is no way back.
4. OpenAI doesn't give a fuck about the future of mathematics.
5. Half-baked unreadable preprints claiming major results can effectively stall mathematical progress.
6. Some mathematicians are responding with naive, overly defensive comments.
7. The social-media wankers framing this as "OpenAI tramples gatekeeping" have no experience of mathematical research.
8. The product of mathematics is clarity and understanding. Not theorems, by themselves.
9. Mathematics is a critical infrastructure of human cognition. The human need for understanding isn't obsolete and it can't be offloaded to machines.
10. We haven't "solved" mathematics and we won't get there before the Sun burns out.
1. The recent progress is stunning and exciting.
2. The Quasi-Riemann Hypothesis is arguably Fields-medal level.
3. Mathematics is being transformed and there is no way back.
4. OpenAI doesn't give a fuck about the future of mathematics.
5. Half-baked unreadable preprints claiming major results can effectively stall mathematical progress.
6. Some mathematicians are responding with naive, overly defensive comments.
7. The social-media wankers framing this as "OpenAI tramples gatekeeping" have no experience of mathematical research.
8. The product of mathematics is clarity and understanding. Not theorems, by themselves.
9. Mathematics is a critical infrastructure of human cognition. The human need for understanding isn't obsolete and it can't be offloaded to machines.
10. We haven't "solved" mathematics and we won't get there before the Sun burns out.
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During K-12 school, I didn’t particularly like math. It was only when I took college-level math courses that I fell in love with the subject.
Now that AI is making the fun and fascinating parts of math much more accessible, I hope far more people will also come to love math.
Now that AI is making the fun and fascinating parts of math much more accessible, I hope far more people will also come to love math.
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For examples of how AI might do this:
@theonlymolnarould have been very interested in learning about advanced topics in math, but nobody around me knew those topics or had time to share them with me. If I had AI to chat with, I very likely would have asked it to teach me things.
Similarly, for many people in the world, even buying math books and other resources is challenging.
Also, people who love to teach math now have new resources to do a better job at it.
Lastly, AI will be discovering new and beautiful things. Which both increases the quantity of fun, fascinating things to study, and also produces a lot of new things which in some ways is more exciting than studying very old math
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Btw it seems like this one is not completely verified yet with Lean. And many others also seem to not be verified with Lean.
Seems strange that they would release so many where a lean check is doable but not done.
Seems strange that they would release so many where a lean check is doable but not done.
I love this theorem. And I hope many people work to improve AI’s proof of it. The Book Proof is still out there! :-)
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I love this theorem. And I hope many people work to improve AI’s proof of it. The Book Proof is still out there! :-)
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Very exciting prospects for applied math! And AI holds the promise of enabling scientists to more capably work with these deeper areas of math.
I hope OpenAI leans much more in this direction, where the benefits extend to humanity more broadly.
I hope OpenAI leans much more in this direction, where the benefits extend to humanity more broadly.
“okay, so what? it’s just math.”
no, you don't get it.
here’s what OpenAI’s math breakthroughs could eventually help make possible in the real world:
Vlasov–Maxwell → stronger foundations for fusion research, whose ultimate prize is virtually limitless clean energy to power civilization
Calderón’s Problem → portable body scanners using electrical signals, making medical imaging cheaper and easier to access
Maximum-Cardinality Matching → faster searches for compatible kidney swaps across large donor pools, helping hospitals coordinate lifesaving transplants
Inverse Elasticity Problem → scans that map tissue stiffness, helping doctors locate suspicious growths inside the body
Mumford–Shah Conjecture → better tools for spotting tissue changes in brain scans, helping doctors identify signs of disease
Bose–Einstein Condensation → stronger foundations for quantum sensors that could help vehicles navigate without GPS
Simple Stochastic Games → better safety checks for self-driving cars and robots before dangerous mistakes reach the real world
Matrix Multiplication → cheaper AI and bigger scientific simulations using the same computers
Edit Distance → faster DNA comparisons, helping researchers study genetic changes linked to disease
The k-Server Problem → robots that waste less movement and energy, making warehouses more efficient and goods cheaper to move
a reminder that math is the foundation of science.
so accelerating mathematical discovery could compress centuries of scientific progress into years.
insane timeline to be alive for!
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