Terence Tao on Mathematics in the Age of AI

Terence Tao on Mathematics in the Age of AI

The Shift from Proof Scarcity to Proof Abundance

Terence Tao posits that the mathematical community is entering a turbulent period comparable to the early 20th-century crisis in foundations. The primary challenge is no longer just whether AI can solve mathematical problems, but how the community should respond when AI tools begin performing a significant fraction of research-level tasks.

Tao introduces the Working Hypothesis: AI tools will soon be capable of performing a reasonable fraction of research-level mathematical tasks with reasonable success, quality, and cost. Under this hypothesis, mathematics transitions from an era of "proof scarcity" to an era of "proof abundance," creating a critical bottleneck in how mathematical knowledge is verified and integrated.

The AI Capability Conjecture and First Proof

To analyze the impact of AI, Tao frames the possibility of AI success as the AI Capability Conjecture. This conjecture suggests that AI tools, with some human supervision, will be able to accomplish research-level mathematical tasks in various fields with a non-trivial success rate.

Evidence for this is emerging through initiatives like First Proof, an independent assessment of frontier AI models. In a May 2026 test batch of ten novel research-level problems, at least one AI team solved seven of the ten problems at publication-level quality, with compute costs ranging from $10 to $1,000 per problem.

The Bottleneck: Proof Digestion and Canonicalization

Tao argues that solving a problem is only the first step in a long chain of value creation. He identifies a sequence of stages that a mathematical result must pass through to truly contribute to the field:

  1. Proof Generation: Creating a potential solution.
  2. Proof Verification: Confirming the solution is correct (accelerated by autoformalization in languages like Lean or Rocq).
  3. Proof Exposition: Communicating the result clearly so others can understand it.
  4. Proof Publication: Having the result peer-reviewed and accepted by the community.
  5. Proof Digestion: The process where other mathematicians incorporate the result into their own work.
  6. Proof Canonicalization: The final stage where a result becomes part of definitive textbooks and reference material.

While AI can drastically accelerate generation and verification, it struggles with exposition and cannot replace the human-centric processes of digestion and canonicalization. Tao warns of "proof indigestion," where AI-generated proofs accumulate faster than humans can verify, understand, or integrate them.

The Risks of Over-Optimization

Applying Goodhart's Law—which states that when a measure becomes a target, it ceases to be a good measure—Tao warns that optimizing solely for the number of solved problems may degrade the quality of mathematics.

He notes that AI-generated exposition often lacks "natural friction." In human-written proofs, the difficulty of a specific step is often reflected in the writing, signaling the reader to slow down. AI-polished proofs may remove this friction, making difficult and routine parts appear equally easy, which can paradoxically hinder actual learning and understanding.

Recommendations for the Mathematical Community

To mitigate the risks of proof abundance, Tao suggests several cultural and policy shifts:

  • Normalize Responsible Disclosure: Authors should openly disclose the use of AI tools to avoid covert usage and peer criticism.
  • Shift Value Metrics: The community should decrease the emphasis on being the "first" to solve a problem (generation) and increase the value placed on exposition, publication, and canonicalization (digestion).
  • Verify Human Understanding: Tao proposes a rule of thumb: if authors cannot give a clear, expert-level talk on their results, the result should not be published, regardless of whether an AI verified the proof's correctness.

Community Perspectives and Counterpoints

Discussion surrounding Tao's presentation highlights several critical tensions:

  • Brute Force vs. Insight: Some observers argue that current AI success is largely a result of "large model + harness" executing broad brute-force searches rather than the invention of new mathematical machinery. One commenter noted, "AI cracking a problem and giving a solution, inspires a human to invent new techniques."
  • The Nature of Discovery: There is a debate over whether AI can produce "genius" level insights (akin to Nash or Einstein) or if it is limited to "PhD grunt work."
  • The Risk of Secrecy: Some predict that as AI capabilities grow, mathematics may become less open, leading to the "balkanization and hoarding of 'secret maths'" by entities that no longer require the validation of traditional publication.
  • The Burden of Review: Similar to open-source software, the mathematical community may face a flood of trivial or slightly incorrect AI submissions that waste the productive time of human reviewers.

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