1. The Current Mathematical Ecosystem

• Cultural Context: Mathematics is not just formal symbols—it thrives within a human ecosystem composed of:
  • Journals (publishing),
  • Referees (peer validation),
  • Conferences (community discussion),
  • Informal networks (idea exchange).
• Nature of Understanding: We often claim to "understand" a theorem when:
  • We grasp examples and applications,
  • We know how to prove it,
  • We can explain it informally to peers.
• Caveat: This “understanding” is not formally defined—it’s socially and culturally negotiated.

2. The Issue of Trust in Modern Mathematics

• Growing Complexity: Advanced math, like Scholze’s Liquid Tensor Experiment, is:
  • Beyond the grasp of most humans,
  • Trusted due to the reputation of the author, not mass verification.
• Paradigm Shift: We are moving:
  • From trusting proofs (that are verifiable),
  • To trusting people (who we believe are correct).
• Buzzard’s Concern: This is philosophically shaky—should truth depend on a mathematician's prestige?

3. Formal Proof Systems

• Enter Lean (proof assistant):
  • Designed for formal verification—no step omitted, no ambiguity.
  • Has verified nontrivial theorems like:
    • Feit–Thompson (Odd Order Theorem),
    • Four Color Theorem,
    • Perfectoid Spaces (via Liquid Tensor project).
• Key Comparison:
  • Traditional proof: Can be incomplete, intuitive, informal.
  • Formal proof: Complete, precise, lossless, but hard to write.
• Buzzard’s Hope: With better tools, mathematicians can use computers not just to compute, but to think rigorously.

4. Mathematical AI: Where Are We Now?

• Current State:
  • AI can autocomplete proofs (like Copilot for code),
  • Generate full proof scripts in Lean,
  • Suggest lemmas or tactics.
• But Limitations Remain:
  • AI doesn’t "understand" in human terms.
  • It cannot yet decide which lemmas are central or insightful.
• Open Questions:
  • Can AI discern mathematical beauty, depth, or significance?