Terence Tao: "Mathematics in the Age of AI" (ICM 2026)

Main ideas and concepts

• AI is forcing a shift in mathematics’ role and priorities

  • Terence Tao argues that mathematics has historically been relatively stable for about a century, but AI is introducing rapid, “existential” change.
  • He frames the situation as not only a technical issue but also a meta-mathematical, ethical, cultural, and economic one.

• Pre-20th-century “semi-formal” mathematics led to a foundational crisis

  • Tao discusses how mathematicians previously relied on informal or semi-formal foundations (e.g., using proofs like Euclid’s without fully axiomatizing logic/set theory/analysis).
  • He highlights unresolved “what is” questions (set, number, function, limit) and the eventual need to more carefully formalize foundations.
  • The key lesson: the community must debate foundational rules, not assume them.

• A community-wide question: what should mathematics/AI communities do about AI?

  • Tao distinguishes:
    • A meta-level question: what to do about AI in terms of goals/values/ethics.
    • The AI community contention (his term): public debate often assumes AI may be able to solve open problems, and that assumption should determine how we respond.
  • He notes the controversy is full of claims and counterclaims, and that “we have a lot of data points.”

• Practical example: AI-assisted progress and evaluation standards

  • Tao describes a recent “batch” of AI-generated results and emphasizes evaluation criteria beyond mere correctness.
  • He suggests researchers tested outputs against benchmark problems and also looked for publication-quality explanations that would plausibly be suitable for journals (even if slightly erroneous).

• Implicit goals must be made explicit

  • Tao argues that communities often state explicit goals (in paper abstracts, grant proposals, press releases) but neglect implicit goals.
  • With AI, this becomes urgent because multiple goals may now collide.

• A structured view of “why we do research”

  • He lists multiple motivations for research, including:
    • solving problems (applied and pure),
    • developing tools,
    • understanding dynamics/world,
    • community-building,
    • educating the next generation,
    • and broader societal aims.

• A major goal under pressure: proof-based problem solving

  • Tao focuses on the goal most directly threatened or “hacked” by AI:
    • not just generating proofs,
    • not just verifying them,
    • but ensuring proofs are explained and communicable so humans can understand them and build a shared mathematical narrative.

• Near-future concern: proofs humans can’t understand

  • He warns of a potential abnormal development where AI produces results that are:
    • formally correct and verified,
    • but unexplainable/ununderstandable by humans, breaking normal mathematical communication.

• Why AI currently falls short in human-facing stages

  • AI may excel at some aspects of writing and formatting (spelling/grammar), but can still produce:
    • frustrating analysis,
    • poor prioritization of what is actually important,
    • overly trivial or irrelevant detail,
    • difficulty capturing the “interesting part” or the hard conceptual core.
  • The emphasis: mathematical writing is not just correctness—it’s curation and pedagogy.

• Peer review, consensus, and narrative coherence remain human bottlenecks

  • Tao argues key human tasks still drive mathematics:
    • editorial referee review,
    • consensus-building (if the community can’t agree on presentation/approach, there’s no meaningful conversation),
    • teaching and iterative feedback,
    • organizing ideas into coherent narratives.
  • He claims these stages are where AI is “completely useless” for now, because they depend on human judgment and communal understanding.

• Abundance problem: education and textbook conversion could lag behind

  • He introduces a scarcity vs. abundance framing:
    • Previously, mathematics often faced scarcity (e.g., lack of methods/results accessible to apply).
    • With AI, there could soon be an “abundance” of published proofs—but not the human-readable, pedagogical material that engineers/physicists/lawyers need (textbooks, digestible explanations, conversions into teachable forms).
  • He implies the community must plan workflows for turning verified results into educational resources.

• A proposed response: set rules for acceptable AI use and create new workflows

  • He argues for:
    • internal initiative in defining what kinds of AI research are acceptable,
    • not letting external pressures dictate norms.
  • Some venues like traditional journals may be unsuitable for certain AI-related workflows, requiring new structures.

• Education/training should not be outsourced early

  • Tao says heavy early reliance on AI can be harmful because learners need to develop foundational skills first.
  • Analogies include:
    • teach arithmetic before calculators,
    • teach walking before giving car keys.
  • He still supports AI use in some places, but with careful boundaries.

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Methodology / structured “process” implied in the talk (problem-solving pipeline)

Tao outlines an implicit multi-stage requirement for mathematical progress (especially when AI is involved), presented as a pipeline:

1. Generate proofs

   • AI can do some proof generation, though he suggests it’s not uniformly strong everywhere.

2. Verify proofs

   • AI can often help with formal verification/correctness checking.

3. Explain proofs so they become human-communicable knowledge

   • Proofs must be:
     • communicated clearly,
     • understood by the mathematical community,
     • placed into narratives that support learning and further development.
   • This stage is crucial; without it, proofs don’t “lead to anything further” in the usual way.

4. Achieve community consensus on exposition and approach

   • Mathematics requires shared standards for understanding and discussion.
   • If the field is split on how to present/interpret results, meaningful collective knowledge formation fails.

5. Convert results into educational resources

   • External users (engineers/physicists/lawyers) typically want textbooks and teachable material, not isolated AI-generated proofs.
   • A key future risk: lots of proofs published, but not enough work turning them into usable learning tools.

6. Maintain peer-review and human editorial/refereeing workflows

   • Verified results still require editorial/referee service and human effort to integrate into journals and the profession.

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Main lessons / takeaways

• AI changes the environment for mathematics, turning it from a mostly stable technical enterprise into a domain requiring explicit value/goal governance.
• The most pressing threat is not just whether AI can prove things, but whether AI can produce understandable, teachable, communicable mathematics.
• The mathematical community must proactively manage AI’s role by:
  • articulating implicit goals,
  • protecting education/training foundations,
  • creating new workflows where traditional systems don’t fit,
  • and ensuring results become shared human knowledge rather than inaccessible formal artifacts.

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Speakers / sources featured

• Speaker: Terence Tao

• Other referenced sources (mentioned, not necessarily quoted verbatim):

  • Euclid (proof tradition)
  • Jean Baudrillard (referenced via “a paper” and “J. Baudrillard”; Tao describes it as educational)
  • Donald Trump (Tao mentions reading Trump’s paper as a learning experience; context suggests an authored text or paper, as described by Tao)