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Everything tagged mathematics, across every stream.

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On Terence Tao's blog, mathematicians debate what their work is for after OpenAI's 700-paper dump

Two guest essays in two days, one practical and one philosophical, from working mathematicians trying to make sense of AI-produced proofs.

Terence Tao's blog ran two guest essays this week on how mathematics should respond to AI. In "Mathematics has changed. Let's act like it" (Nov. 1), mathematician Frank Calegari calls OpenAI's Oct. 6 release of more than 700 research papers a watershed, while saying the papers are "atrociously written," violate the field's norms on attribution, and that some are wrong. He compares the moment to the classification of finite simple groups, which he says cooled interest in that subject, and argues mathematicians should reward people who choose worthwhile problems and understand AI-found proofs, and reform hiring and journals rather than try to change AI companies.

In "Egoless Mathematics in the AI Era" (Nov. 2), Karthik C. S., Amit Sahai and Jie Xue argue that anxiety about being replaced comes from measuring mathematicians by output, and that a truth "has become accessible; its worth is not diminished by the absence of our authorship."

Tao notes both posts were converted to the blog's format using AI.

Cascadic Analysis 2
UndertowRole underspecification / the Doorman Problem

What hidden risk could pull against this, even if the news is good?

If a mathematician's job were producing proofs, 700 papers at once would look like the job being done. Frank Calegari's complaints point to what that leaves out: he calls the papers "atrociously written," says they break the field's norms on attribution and that some are wrong. He argues the field should reward choosing worthwhile problems and understanding AI-found proofs, which are the parts of the role a paper count never captured.

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UndertowDelayed Consequence / False Success Problem

What hidden risk could pull against this, even if the news is good?

Calegari says some of the papers are wrong, and finding out which takes careful reading by people, at the slow pace mathematics checks proofs. A flood of results that look finished can be cited and built on before the errors surface. The volume itself makes the review lag longer.

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Rabbit Holes 1
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OpenAI says its models disproved the Exponential Time Hypothesis, among three new results it handed to outside experts

If the computer-science claim holds, hundreds of papers built on that assumption are affected; Engadget explains the three results and how OpenAI is releasing them.

OpenAI announced three results it says its models produced, in harmonic analysis, complexity theory and quantum field theory, and released them differently, Engadget reports. Two manuscripts are on GitHub: one proves Sogge's local smoothing conjecture for waves in every spatial dimension, and one claims the Exponential Time Hypothesis (ETH) is false. The preprint describes a deterministic algorithm that counts the satisfying assignments of any k-CNF formula in subexponential time.

ETH, a widely believed assumption that some problems can't be solved quickly, underpins hundreds of papers; if it is false, better algorithms could exist in fields from databases to computational biology. OpenAI asked outside experts, including the scientists who proposed ETH, to examine and explain the computer-science result; the physics result (on interacting quantum field theories in four dimensions) has been left to a group of physicists that has not yet published. OpenAI says it won't seek academic credit. TechCrunch reports that Anthropic released its own refutation of ETH hours later, after Anthropic mathematician Levent Alpöge suggested OpenAI's solution drew on pre-existing work. The claims have not been peer reviewed; MIT's Ryan Williams, organizing the field's response with ETH's proposers, told TechCrunch parts of the proof look "alien."

Hacker News commenters said the headline "buries the lede," asked whether this shifts the odds on P versus NP, and one questioned the size of OpenAI's donation to academia.

Cascadic Analysis 1
UndertowWe don't completely understand why frontier models behave as they do

What hidden risk could pull against this, even if the news is good?

MIT's Ryan Williams says parts of OpenAI's proof are routine but "two or three alien-looking parts" need the whole community to work through them, and Anthropic's Levent Alpöge suggested the solution drew on pre-existing work. A model's output doesn't come with an account of where its ideas came from, so people have to reconstruct both whether a proof is right and how new it is. Until that checking is done, these are claims, not results.

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Rabbit Holes 2
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OpenAI publishes math results from an unreleased model, with Lean proofs

The release notes explain how the results were checked and what OpenAI disclosed about how they were produced.

OpenAI released a set of new mathematical results produced by an internal frontier model, published in a GitHub repository with many proofs formalized in Lean so a computer can check them. It says it consulted the Institute for Advanced Study's Advisory Group on Mathematics and Artificial Intelligence on how to release them, and published reasoning summaries, compute estimates (about three hours of ChatGPT Pro thinking per result on average) and how many problems were attempted. OpenAI says it is working to release the model.

Cascadic Analysis 2
UndertowWe don't completely understand why frontier models behave as they do

What hidden risk could pull against this, even if the news is good?

OpenAI shared summaries of the model's reasoning for 10 results, not the reasoning itself, from a model the public can't use yet. Lean can confirm a proof is valid; it can't show how the model found it or how often it fails.

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UndertowBenchmark scores can overstate real-world reliability

What hidden risk could pull against this, even if the news is good?

OpenAI chose which results to publish and reported how many problems were attempted. Success on selected research problems, at about three hours of top-tier compute each, says little about how the model does on an ordinary user's math.

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Rabbit Holes 1
  • Lean (proof assistant)

    After exploring Sharing AI progress in mathematics, the tool that checks these proofs is Lean (proof assistant).

    Lean is a proof assistant and functional programming language, free and open source, now supported by the nonprofit Lean Focused Research Organization.

    Wikipedia · Swim · 30 min

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