AI Summary
5 min readIn 2024, OpenAI’s reasoning model, Astra, solved ten longstanding open problems in mathematics, including a 70-year-old question about sphere packing in high dimensions and a foundational conjecture about the nature of infinite groups. The researchers behind these results, mathematicians Metav Swani and Mark Selki, joined a16z partner Lisha Lee to explain how the model works, why its reasoning traces look surprisingly human, and what this means for the future of mathematical practice.
What makes AI different from a human mathematician
The most striking quality of Astra’s reasoning is not raw speed but a kind of doggedness that humans struggle to replicate. When a human mathematician has an idea and tries to execute it, they typically spend hours or weeks working through details. If the approach fails, the failed attempt pollutes their thinking — the context window, as Selki puts it, becomes “polluted” with dead ends, making it hard to restart with a fresh perspective. Astra does not suffer from this. It can backtrack cleanly, discard a failing approach, and try another without carrying the emotional or cognitive weight of the previous failure.
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What you'll learn
- 1 (00:00) **The Renaissance of Reachable Results** - A mathematician describes how GPT-5 solved a problem he and his friends had spent hours on, marking a turning point.
- 2 (05:13) **AI's Strengths: Execution, Search, and Parallel Thinking** - The model excels at executing ideas, searching for connections, and avoiding the mental pollution that hinders human backtracking.
- 3 (10:56) **The Problem with Training on Math Papers** - Math papers are a poor training set because they hide the struggle and motivation behind definitions, unlike code which has more structure.
- 4 (14:40) **Why Reasoning Traces Matter** - The models' reasoning traces, released as "summarized chains of thought," show the model is not just guessing but thinking in a way that is recognizable to mathematicians.
- 5 (15:53) **Case Study 1: Sphere Packing in High Dimensions** - The model solved a long-standing problem about the densest possible sphere packing in high dimensions, improving on a 50-year-old bound.
- 6 (27:50) **A Personal Connection to the Problem** - One of the mathematicians had worked on this sphere packing problem for six months as a graduate student, making the model's solution deeply satisfying.
- 7 (28:30) **Case Study 2: Spherical and Binary Codes** - The model solved a related problem about error-correcting codes, using representation theory to find better bounds.
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Show Notes
a16z Infra Partner Lisha Li sits down with OpenAI mathematicians Mehtaab Sawhney and Mark Sellke to discuss how quickly AI’s mathematical capabilities are advancing, what recent results reveal about model reasoning, and what happens when AI begins making progress on problems mathematicians have struggled with for decades.
Mehtaab and Mark unpack several recent results from OpenAI’s models, including advances in sphere packing and the construction of a non-sofic group. They explain why the surprising part isn’t simply that models can search more possibilities or work longer than humans: in many cases, the reasoning traces look remarkably similar to the work of an expert mathematician, including choosing promising approaches, backtracking when they fail, and combining ideas from across the literature.
They also explore what this means for mathematics itself: how the role of human taste and judgment may change, whether AI could produce far more mathematics than humans can absorb, and why models that accelerate discovery may also make sophisticated results easier to understand.
Resources:
Follow Lisha Li on X: https://x.com/lishali88
Follow Mehtaab Sawhney on X: https://x.com/mehtaab_sawhney
Follow Mark Sellke on X: https://x.com/MarkSellke
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