Podcast Summary: The Last Invention is AI

Episode: AI: Humanity's Last Math Tool?
Date: January 15, 2026

Main Theme and Purpose

In this episode, the host explores the rapidly evolving capabilities of AI in mathematics—not just as a tool for solving problems the traditional way, but as an entity now inventing new mathematical methods. The discussion focuses on how AI is revolutionizing the field by solving longstanding problems, enhancing human research, and potentially becoming a generational force multiplier for all technical innovation.

Key Discussion Points & Insights

1. AI Correcting and Inventing Mathematics

• The episode opens with an anecdote about Mo Gadda (former Google X executive), who highlights a striking breakthrough:

  • AI moved beyond simply optimizing human-created software to inventing an entirely new method for matrix multiplication.
  • This change resulted in a 26% performance boost and dramatically reduced costs and energy for Google.
  • The host reflects:

    "AI is... creating new ways to solve math and coming up with completely new methods when it thinks that our methods are flawed." (02:00)

• Implication: AI is not just "getting better at math," but discovering fundamentally new methods, changing the paradigms mathematicians have long accepted.

2. AI Solving Previously Unsolved Problems

• The host discusses recent news:

  • GPT-5 Pro solved Yu Semrutsu's 554th problem — a major first for an AI model (05:20).
  • AI scored a perfect 120/120 in the 'Putnam' math competition via Axiom’s Prover in Lear platform.

    "AI just achieved a perfect perfect score on the hardest math competition in the world." (06:00)

• Beyond competitions, AI models are now routinely tackling unsolved problems.

  • Neil Simani, a software engineer, inputted a long-standing unsolved problem into ChatGPT. After 15 minutes, the model returned a solution, which independent verification tool Harmonic confirmed as correct (08:50).

  • The AI's chain-of-thought process involved:

    • Researching and connecting multiple mathematical theories (Legendre's formula, Bertrand's postulate, Star of David theorem).
    • Pulling related work from a 2013 MathOverflow post by Harvard mathematician Noam Elkies.
    • Synthesizing and creating a unique, more complete answer incorporating Paul Erdős' work.

3. How Is AI Achieving This?

• The host emphasizes the importance of "tooling"—the additional functions like calculators and algorithmic helpers now available to AI models (12:10).

• AI isn’t just copying from human sources; it's proving capable of self-directed research, synthesizing across sources, and innovating beyond its training data.

  "Its reasoning was so good, like it was going and doing research, finding old posts, finding, you know, solutions to similar problems, adapting them to this problem, and writing more complete versions of it, which was incredible." (13:20)

4. Human-AI Collaboration & the New Role for Mathematicians

• The current generation of AIs isn't independently seeking out problems—it’s still human experts who “point it in a direction.” (15:40)

• The Erdős Problem List: Since Christmas, 15 previously open problems have been solved; 11 of those solutions credited AI tools, either fully solving or supporting human researchers (16:30).

  "In my mind this is just no doubt that this is really pushing the field forward and in really novel, interesting new ways." (17:10)

• Terence Tao (renowned mathematician) observes:

  • AI sometimes generates genuinely "new ideas" or retrieves relevant prior research for humans to build upon.
  • Tao notes fully independent "AI mathematicians" are distant, but the impact of current AI tools is already huge (19:40).

  "Of course it's using human research. Like, where do you think it's training data came from?... But it is coming up with new, novel things." (20:20)

5. Verification and Scaling the Math Frontier

• New tools help translate traditional natural-language math proofs into machine-verifiable arguments (22:00).
• Tudor Achimic (founder of Harmonic, a proof-checker for math):

  "What matters is that serious math and computer science professors are actually using them. These are people whose careers depend on being careful and credible. So when they say they rely on AI tools, that says a lot." (24:30)

6. Broader Impact – Beyond Math

• The host projects these advancements will extend to fields like engineering, medicine, science, and economics, where complexity and verification bottlenecks slow discovery (25:20).

  "As the AI systems get better at actually exploring ideas and checking work and connecting past knowledge, they are going to dramatically speed up research and innovation... The reasoning capability of these models improving means that so many different areas are going to improve." (26:00)

Notable Quotes & Memorable Moments

• On AI’s creative leap:

  "Instead [of optimizing existing software] it invented a completely new way of doing math... a 26% performance boost and the removal of hundreds of millions of dollars in cost and energy use for Google." (01:50, summarizing Mo Gadda’s example)

• On validation and trust:

  "What matters is that serious math and computer science professors are actually using them... When they say they rely on AI tools, that says a lot."
  — Tudor Achimic (24:30)

• On changing research culture:

  "Fully independent mathematicians are still a long way off... But those tools are already making a huge difference."
  — Summarizing Terence Tao (21:00)

Timestamps for Important Segments

• 00:00–02:30 — Mo Gadda's keynote: AI invents new math, matrix multiplication breakthrough
• 05:00–07:00 — News: AI solves major problems (Yu Semrutsu’s 554th, perfect Putnam score)
• 08:30–14:00 — AI’s chain of reasoning and novel synthesis in problem solving
• 15:00–18:00 — Collaboration between humans and AI; impact on Erdős problem set
• 19:30–21:30 — Terence Tao’s views on AI-generated ideas and research assistance
• 22:00–24:45 — Machine-verifiable proofs; significance of expert adoption (Harmonic, Tudor Achimic)
• 25:00–26:20 — Ripple effect: how AI’s math reasoning will accelerate all scientific innovation

Conclusion

This episode makes a compelling case that AI is not simply automating human mathematical work, but is actively broadening and deepening the boundaries of mathematical discovery. With AI inventing new mathematical methods, solving renowned open problems, and enabling researchers to verify and build on each other's work more efficiently than ever, the nature of mathematical research—and by extension, scientific progress—may be on the cusp of dramatic transformation.