25 Fields Medalists Warn: AI Is Turning Math Problems Into Benchmark Scores — When Millennial Problems Become Model Leaderboards
25 Fields Medalists Warn: AI Is Turning Math Problems Into Benchmark Scores — When Millennial Problems Become Model Leaderboards
1. Introduction: A Historic Collective Voice
On September 11, 2026, a declaration titled “A Severe Misalignment of AI in Mathematics” was published on Terence Tao’s blog “What’s New” and made available for collective signing at mathandai.org. The weight of this document can be measured by its signatories: 25 Fields Medalists — roughly one-third of all living recipients of mathematics’ highest honor.
The signatories span nearly half a century of the prize’s history: from Pierre Deligne (1978) to Yu Deng (2026), including Peter Scholze, Maryna Viazovska, Hugo Duminil-Copin, Shigefumi Mori, Ngô Bảo Châu, Cédric Villani, and of course Terence Tao himself. The group wrote that the declaration emerged from a week of urgent discussions, and that they decided to release it immediately rather than follow the slower consultative process typical of such declarations.
This is not an anti-AI manifesto. The letter explicitly acknowledges that “over the last few months, the mathematical capabilities of LLMs have improved dramatically, to the point that they can solve major outstanding problems in many fields of mathematics.” What it opposes is the framing of mathematical problem-solving as a benchmark by AI companies, and the cascade of academic ethics issues that follow.
The immediate trigger was OpenAI’s September 8 announcement: approximately 10,000 concurrent AI agents had tackled the Navier-Stokes existence and smoothness problem — one of the Clay Mathematics Institute’s seven Millennium Prize Problems — in roughly 88 hours, producing a 166-page paper with Lean formal verification. This was accompanied by a priority dispute, allegations that OpenAI may have used unpublished human research, and what many described as the worst academic ethics crisis in the history of mathematics.
This article performs a deep technical analysis of the event, the core arguments of the Fields Medalists’ declaration, and the fundamental question it raises: When AI produces research results faster than humans can absorb them, how do we preserve the value of the scientific process itself?
2. The Navier-Stokes Breakthrough: 88 Hours vs. 90 Years
2.1 What Is the Navier-Stokes Millennium Problem?
The Navier-Stokes equations describe the motion of fluid substances like water and air. Formulated by Claude-Louis Navier and George Gabriel Stokes in the 19th century, they are fundamental to aerodynamics, weather forecasting, and hemodynamics. But a core mathematical question has remained open for nearly 90 years:
In three-dimensional space, given an initially smooth, stationary incompressible fluid under external force, can the velocity become infinite in finite time (a “singularity” or “blow-up”)? Or do smooth solutions exist for all time?
The Clay Mathematics Institute designated this as one of seven Millennium Prize Problems in 2000, each carrying a $1 million prize. As of September 2026, only one — the Poincaré conjecture, proved by Grigori Perelman — has been recognized by the mathematical community.
The mathematical formulation of the problem is deceptively simple:
┌───────────────────────────────────────────────────────────────┐
│ Navier-Stokes Problem — Formal Statement │
├───────────────────────────────────────────────────────────────┤
│ │
│ ∂u │
│ ── + (u·∇)u = -∇p + νΔu + f (Momentum equation) │
│ ∂t │
│ │
│ ∇·u = 0 (Incompressibility condition) │
│ │
│ Where: │
│ u(x,t) : velocity field (3D vector field) │
│ p(x,t) : pressure field (scalar field) │
│ ν : kinematic viscosity coefficient │
│ f(x,t) : external body force │
│ Δ : Laplacian operator │
│ ∇ : Gradient operator │
│ │
│ The core question: │
│ Given u₀∈C^∞(ℝ³) with ∇·u₀=0 and smooth force f, │
│ does a global smooth solution u(x,t) ∈ C^∞(ℝ³×[0,∞)) exist? │
│ Or does there exist finite T<∞ such that ||u(t)||_L^∞ → ∞? │
│ │
└───────────────────────────────────────────────────────────────┘
2.2 OpenAI’s Approach: The Multi-Agent Swarm
OpenAI’s September 8 announcement detailed an unprecedented computational assault on the problem:
Phase 1: Preparation (Aug 28 – Sep 1)
OpenAI began training a new internal model on August 28, described as “significantly more capable than GPT-6 Astra” but not publicly released.
Phase 2: Trigger and Parallel Exploration (Sep 1 – Sep 5)
On September 1, OpenAI researchers heard rumors that “two Millennium Problems had been solved.” They launched a massive multi-agent system, distributing agent clusters across multiple unsolved Millennium Prize Problems. Each agent group was given a different framing — some aimed at constructing a solution, others at disproving the regularity assumption.
A smaller group of nearly 100 agents first spent about 50 hours disproving regularity for the unforced Euler equations (the inviscid limit of Navier-Stokes). This intermediate result was fed back as a crucial hint.
Resources then shifted to the Navier-Stokes problem, with approximately 10,000 concurrent agents working in parallel. Agent groups used Codex to periodically consolidate useful intermediate results and cross-pollinate ideas across groups.
Phase 3: Output (Sep 5 – Sep 6)
On September 5, the agent cluster produced a candidate proof. GPT-6 Astra then spent approximately 17 hours performing Lean formalization and verification.
The final output included a 165-page analytical proof archived on GitHub, with the key conclusion: a smooth, initially stationary three-dimensional incompressible fluid can develop a singularity in finite time — velocity grows unbounded while kinetic energy remains bounded. The solution was described as a spiral-inward vortex that stretches like spaghetti, with the central region contracting and accelerating.
┌───────────────────────────────────────────────────────────────┐
│ OpenAI's 10,000-Agent Parallel Architecture │
├───────────────────────────────────────────────────────────────┤
│ │
│ ┌────────────────────────────────────────────┐ │
│ │ Internal Model (training since Aug 28) │ │
│ └──────────┬─────────────────────────────────┘ │
│ │ │
│ ┌──────────▼─────────────────────────────────┐ │
│ │ Agent Orchestration Layer (Codex) │ │
│ └──┬─────────┬─────────┬─────────┬───────────┘ │
│ │ │ │ │ │
│ ┌────▼──┐ ┌───▼────┐ ┌──▼────┐ ┌──▼────────┐ │
│ │Group 1│ │Group 2 │ │Group 3│ │Group 4... │ ←~10000 Agents│
│ │NS Con.│ │NS Reg. │ │Euler │ │Other Probs│ │
│ └───┬───┘ └───┬────┘ └───┬───┘ └───────────┘ │
│ │ │ │ │
│ └─────────┼──────────┘ │
│ ▼ │
│ ┌────────────────────────────┐ │
│ │ Codex Intermediate Merge │ ← Cross-pollination engine │
│ └──────────┬─────────────────┘ │
│ │ │
│ ┌──────────▼─────────────────┐ │
│ │ Candidate Proof Path │ │
│ └──────────┬─────────────────┘ │
│ │ │
│ ┌──────────▼─────────────────┐ │
│ │ Lean 4 Formalization │ ← GPT-6 Astra, 17 hours │
│ └──────────┬─────────────────┘ │
│ │ │
│ ┌──────────▼─────────────────┐ │
│ │ 166-page Paper + Lean Proof│ │
│ │ Open Source on GitHub │ │
│ │ (Apache 2.0) │ │
│ └────────────────────────────┘ │
│ │
├───────────────────────────────────────────────────────────────┤
│ Key Metrics: ~4.9M agent messages, ~300B output tokens │
│ Compute cost: Millions of USD (confirmed by Mark Chen) │
│ NS portion: ~2.7M messages, ~130B tokens │
└───────────────────────────────────────────────────────────────┘
2.3 The Sheer Scale of Computation
Mark Chen, OpenAI’s Chief Research Officer, confirmed the computing cost reached millions of dollars. Researcher Sébastien Bubeck revealed this was approximately 1,000 times the compute budget of any previous OpenAI mathematical achievement.
Independent analysts calculated that millions of dollars could have funded approximately 600 postdoctoral researchers for a full year.
2.4 The Mathematical Community’s Cautious Response
Despite the Lean formalization — unprecedented for an AI-generated mathematical result — the mathematical community responded with caution. Under Clay Mathematics Institute rules, a Millennium Prize solution must be published in a peer-reviewed journal and undergo at least two years of community acceptance. OpenAI stated it would not claim the $1 million prize, framing the release as a progress report.
Several mathematicians, including Diego Córdoba and Luis Silvestre who have worked extensively on fluid dynamics, stated that the community is still processing what the result actually implies.
3. Tao’s Core Argument: Understanding vs. Answers
3.1 What the Declaration Actually Says
Although the 25 Fields Medalists’ declaration names no company directly, the target is unmistakable — especially given Tao’s supplementary post and The Economist’s coverage titled “Top mathematicians are furious with OpenAI’s approach.”
The declaration’s core thesis can be distilled into a framework of misaligned incentives:
┌───────────────────────────────────────────────────────────────┐
│ AI Company Goals vs. Mathematical Community Goals │
├───────────────────────────────────────────────────────────────┤
│ │
│ AI Company Incentive Chain: │
│ ┌──────────┐ ┌──────────┐ ┌──────────┐ ┌───────────┐ │
│ │ Solve │→ │ PR Event │→ │ Ranking │→ │ Valuation │ │
│ │ Problem │ │ │ │ Score │ │ │ │
│ └──────────┘ └──────────┘ └──────────┘ └───────────┘ │
│ (Misalignment) │
│ ┌──────────┐ ┌──────────┐ ┌──────────┐ ┌───────────┐ │
│ │ Solve │→ │ Understand│→ │ Develop │→ │Transmit │ │
│ │ Problem │ │Structure │ │ Tools │ │Knowledge │ │
│ └──────────┘ └──────────┘ └──────────┘ └───────────┘ │
│ │
│ Mathematical Community Incentive Chain: │
│ Solve → Talks/Discussions/Simplification → │
│ Textbook → Public Understanding │
│ (Timescales of decades or centuries) │
│ │
│ The core misalignment: │
│ "The mass production at faster and faster pace of │
│ 'true/false' statements could destroy fertile ground │
│ instead of breathing life into new ideas." │
│ — Declaration │
│ │
└───────────────────────────────────────────────────────────────┘
The declaration elaborates:
“Research mathematics deals with understanding basic structures of shapes, numbers, and natural phenomena. Over the course of generations, it has built a large corpus of sophisticated ideas, methods, abstractions, and other tools to comprehend the mathematical landscape. Famous problems have often served as landmarks and lighthouses against which one can measure an improved understanding of this landscape. Solving one of these problems has been a certain sign of new insights and interesting methods, which would then be studied by a community of mathematicians, through a long and arduous process of talks, discussions, simplifications.”
This passage reveals the deep distinction between solving a problem and understanding its meaning — between producing an answer and building the intellectual scaffolding that lets others learn, build upon, and eventually transcend that answer.
3.2 Code Demonstration: Understanding vs. Answer
The following code illustrates the chasm between verifying a statement’s truth and understanding why it is true:
"""
From "Answer" to "Understanding" — Fermat's Little Theorem
AI perspective: verify a^(p-1) ≡ 1 (mod p) at scale
Human perspective: understand WHY — it's Lagrange's theorem
applied to the specific group (ℤ/pℤ)^×
"""
import random, math
# ---------- AI: batch verify truth values ----------
def ai_verify_batch(trials: int = 5000) -> tuple:
"""AI verifies thousands of instances at scale — but doesn't know why"""
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
passed, failed = 0, 0
for _ in range(trials):
p = random.choice(primes)
a = random.randint(1, p-1)
if pow(a, p-1, p) == 1:
passed += 1
else:
failed += 1
return passed, failed, trials
p, f, t = ai_verify_batch(5000)
print(f"AI batch verify: {p}/{t} passed, {f} failed")
# ---------- Human: understand structure and generalize ----------
def euler_theorem(a: int, n: int) -> bool:
"""
Euler's theorem — generalization of Fermat:
If gcd(a,n)=1, then a^φ(n) ≡ 1 (mod n)
When n is prime, φ(n)=n-1, recovering Fermat.
AI only verifies instances; humans understand
this as group theory's "order divides group order"
manifesting in different contexts.
"""
def totient(m: int) -> int:
"""Euler's totient function φ(m)"""
result = m; p = 2; temp = m
while p * p <= temp:
if temp % p == 0:
while temp % p == 0: temp //= p
result -= result // p
p += 1 if p == 2 else 2
if temp > 1: result -= result // temp
return result
if math.gcd(a, n) != 1:
return False
return pow(a, totient(n), n) == 1
# Demonstrate several instances of Euler's theorem
for n, a in [(10, 3), (15, 7), (21, 5)]:
result = euler_theorem(a, n)
print(f"Euler: n={n}, a={a}, holds={result}")
This example captures what the Fields Medalists are worried about. An AI can verify Fermat’s Little Theorem across thousands of instances in milliseconds, but it does not — and in its current architecture cannot — see that this theorem is a special case of Lagrange’s theorem applied to a specific group. It cannot recognize the structural pattern that lets mathematicians generalize to Euler’s theorem, and from there to the entire edifice of abstract algebra.
Solving a problem is a tool. Understanding why it works is the goal. When the tool becomes the goal, mathematics loses its soul.
3.3 The Understanding Velocity Gap
Tao’s deeper concern is about what happens when AI output velocity exceeds human absorption velocity. If this trend continues, the mathematical community will face an ever-growing “answer warehouse” — mountains of correct conclusions with too few people who can explain why they are correct or what further insights they might yield.
4. Technical Analysis of the Misalignment
4.1 The Benchmark Design Problem
The declaration’s choice of “misalignment” — a term borrowed from AI safety — is deliberate. The signatories are arguing that the objective function of AI companies is fundamentally misaligned with the value function of the mathematical community:
┌───────────────────────────────────────────────────────────────┐
│ AI Objective Function vs. Math Value Function │
├───────────────────────────────────────────────────────────────┤
│ │
│ AI Company Objective f_AI(x): │
│ ┌───────────────────────────────────────────────────┐ │
│ │ maximize(Model Leaderboard Score) │ │
│ │ = Σ(Problem Difficulty × PR Impact) │ │
│ │ - Σ(Compute Cost × Time) │ │
│ │ + Σ(Exclusivity × News Value) │ │
│ └───────────────────────────────────────────────────┘ │
│ │
│ Math Community Value f_Math(x): │
│ ┌───────────────────────────────────────────────────┐ │
│ │ maximize(Depth of Understanding) │ │
│ │ = Σ(New Concepts/Tools/Methods Born) │ │
│ │ + Σ(Ideas Absorbed into Canon) │ │
│ │ + Σ(Intergenerational Transmission Quality) │ │
│ │ - Σ(Unabsorbed Isolated Conclusions) │ │
│ └───────────────────────────────────────────────────┘ │
│ │
│ Root Conflict: │
│ f_AI maximizes "more, faster, newer" — outcome-driven │
│ f_Math maximizes "deeper, clearer, longer" — process-driven │
│ │
└───────────────────────────────────────────────────────────────┘
4.2 The Three Risks in Detail
The declaration and Tao’s subsequent analysis identify three specific risks:
Risk 1: Proof Velocity Exceeds Human Understanding Velocity
When an AI system “conquers” a 90-year-old problem in 88 hours, but the mathematical community needs years or decades to fully digest the ideas, we face a fundamental bottleneck: the accumulation of answers outpaces the absorption of understanding.
A simulation model illustrates this growing gap:
// Simulating the AI-Human understanding velocity mismatch
package main
import "fmt"
func simulateGap() (int, int) {
aiRate, humanRate := 12.0, 0.5 // AI: 12 proofs/month, Human: 0.5 absorbed
totalAI, totalHuman := 0, 0
// Simulate 24-month trend
for m := 1; m <= 24; m++ {
monthlyAI := int(aiRate * (1 + 0.15*float64(m-1)))
monthlyHuman := int(humanRate * (1 + 0.02*float64(m-1)))
totalAI += monthlyAI
totalHuman += monthlyHuman
if m%6 == 0 {
fmt.Printf("Month %2d: Output=%3d, Absorbed=%2d, Pending=%3d\n",
m, totalAI, totalHuman, totalAI-totalHuman)
}
if m%3 == 0 { aiRate *= 1.1 } // AI output accelerates 10% quarterly
}
return totalAI, totalHuman
}
func main() {
totalAI, totalHuman := simulateGap()
pending := totalAI - totalHuman
fmt.Printf("\nSummary: After 24 months, pending proofs = %d\n", pending)
fmt.Println("Root cause: exponential AI output vs linear human absorption")
fmt.Println("Unabsorbed proofs pile up, risking fragmentation of mathematical knowledge")
}
This simplified model shows exponential vs. linear dynamics. In reality, mathematical absorption involves talks, discussions, verification, teaching — each requiring human time and attention. AI’s accelerating output is breaking this balance.
Risk 2: Hasty Publication — Lack of Rigorous Exposition
The declaration states: “Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others.”
The “slowness” of traditional mathematics is not a bug — it is a feature. The extraction and expression of new ideas is itself a core part of research. A correct proof is only the beginning. What matters is what structural insight it reveals: Why does this approach work? Where else can this method be applied? What intellectual lineage does it belong to?
When OpenAI markets the “speed” of its internal model as the headline feature, it effectively bypasses this entire process.
Risk 3: Attribution and Authorship Disputes
This is the most immediate and sharpest problem. The declaration warns: “As in all creative professions, this raises severe attribution and plagiarism questions.”
What followed was one of the most serious academic ethics controversies in modern mathematics.
5. The Authorship Crisis: Academic Ethics in the AI Era
5.1 The Buckmaster and Alpöge Allegations
NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge had been working on closely related fluid dynamics problems. Their breakthrough came on August 15, with Lean verification completed by August 22.
They planned to refine their paper before publication. But when they suspected their progress had leaked to OpenAI, they were forced to release a statement just before midnight on September 7 — demonstrating finite-time blowup for the forced Euler equations and related problems.
Buckmaster’s account of the timeline was deeply troubling:
- Over the past months, both researchers had extensively used OpenAI’s Codex system for their work, inputting unpublished drafts and reasoning chains
- On September 3, Buckmaster wrote to an OpenAI mathematician expressing concern about leaked progress
- On September 6, OpenAI called to inform him that their internal model had proven Navier-Stokes finite-time blowup
- OpenAI proposed two “cooperation” options:
- Option A: Both mathematicians publish their Euler results first, OpenAI publishes Navier-Stokes the next day
- Option B: Buckmaster writes a solo paper acknowledging OpenAI’s Navier-Stokes result — but Alpöge must be removed from the author list because he works at Anthropic
- Buckmaster rejected both options. He reported being told, “Why would you want to destroy your career?” and “If you don’t want me to be polite, I won’t be.”
OpenAI gave a different account. Sébastien Bubeck stated at a briefing: “Neither our researchers nor our agents saw any of their work before their public release.” He showed text messages claiming he had offered the pair “priority of publication.” Sam Altman defended the team, saying Bubeck and the team behaved with “integrity and generosity” throughout.
However, OpenAI’s official blog contained a crucial caveat: “While unlikely, we cannot rule out that de-identified data from their use of our products may have contributed to improving our model.”
This single sentence was widely interpreted by the mathematical community as a significant concession.
┌───────────────────────────────────────────────────────────────┐
│ Key Questions in the AI Academic Ethics Crisis │
├───────────────────────────────────────────────────────────────┤
│ │
│ Question 1: Did user-inputted unpublished research enter │
│ training data? │
│ ┌──────────────────────────────────────────────────┐ │
│ │ Buckmaster & Alpöge → Codex Prompt │ │
│ │ ↓ │ │
│ │ Did Codex use unpublished user ideas │ │
│ │ to improve the model? │ │
│ │ ↓ │ │
│ │ OpenAI: "cannot rule out de-identified data" │ │
│ └──────────────────────────────────────────────────┘ │
│ │
│ Question 2: Did the inference process access user │
│ conversation history? │
│ ┌──────────────────────────────────────────────────┐ │
│ │ Andreas Thom → ChatGPT (discussing soficity) │ │
│ │ ↓ │ │
│ │ Did OpenAI's inference reference private │ │
│ │ user conversations? │ │
│ │ ↓ │ │
│ │ Thom found OpenAI used the same unconventional │ │
│ │ path as his private research │ │
│ └──────────────────────────────────────────────────┘ │
│ │
│ Question 3: Asymmetric bargaining over authorship │
│ ┌──────────────────────────────────────────────────┐ │
│ │ OpenAI offered: Include Buckmaster → Condition: │ │
│ │ Remove Alpöge (Anthropic employee)│ │
│ │ ↓ │ │
│ │ Authorship used as a competitive bargaining chip │ │
│ │ Academic norms subordinated to corporate rivalry│ │
│ └──────────────────────────────────────────────────┘ │
│ │
└───────────────────────────────────────────────────────────────┘
5.2 The Andreas Thom Case
Perhaps even more troubling was the case of mathematician Andreas Thom. He had spent two decades studying Gromov’s soficity conjecture — one of ten problems OpenAI’s Astra system later claimed to have solved.
In the months before OpenAI’s announcement, Thom had been actively discussing the problem with colleagues using ChatGPT. When OpenAI announced its solution, it employed an unconventional, non-mainstream approach that closely matched the path Thom had been pursuing privately.
Thom wrote to OpenAI researchers directly: Had his private conversations entered the training data or been accessed during inference? OpenAI’s response was evasive: they firmly denied “direct access” but then added, in other contexts, the now-infamous disclaimer about “cannot rule out de-identified data.”
One scholar bitterly commented: “If these allegations are true, we are witnessing one of the most serious academic scandals in the history of science. AI is not discovering new mathematics — it is stealing human discoveries and using compute power to publish first.”
5.3 The Deeper Systemic Risk
Behind these individual disputes lies a systemic problem: When AI companies own closed-source models, user data, and massive compute resources, human researchers face unprecedented asymmetric competition.
One mathematician described the situation vividly: You spend twenty years building a cathedral. An AI company hears the sound of your foundation being laid, uses ten thousand bulldozers to erect the walls in 88 hours, and then announces, “We built this cathedral.”
6. The Millennium Problem Benchmark Race
6.1 A New Leaderboard
Following the Navier-Stokes announcement, a chart titled the “Millennium Problem Benchmark” circulated widely on social media:
| Entity | Score | Notes |
|---|---|---|
| Meta/DeepMind/xAI (public models) | 0 | No Millennium Problems solved |
| Human baseline (all top mathematicians) | 1 | Poincaré conjecture (Perelman) |
| Anthropic internal unreleased model | 1 | Reportedly close on Hodge conjecture |
| OpenAI internal unreleased system | 2 | Navier-Stokes + Hodge progress |
More alarmingly, OpenAI has reportedly begun applying its internal model to the Riemann Hypothesis and P vs NP. A proof of P = NP would upend the foundations of computer science entirely.
The Hodge conjecture — a second Millennium Problem — was reported on September 11 to be nearing resolution by Anthropic. The O/A rivalry appears to be accelerating toward the Birch and Swinnerton-Dyer conjecture as well.
6.2 What Mathematics Loses When Reduced to Benchmark Scores
The “Millennium Problem KPI race” reveals a deep pattern in how AI companies think: reducing humanity’s highest intellectual achievements to scoreable, rankable games.
This is precisely what infuriated the 25 Fields Medalists. As the declaration states:
“Solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal.”
Hugo Duminil-Copin used a vivid analogy: Air-dropping someone to the summit of Everest is fundamentally different from climbing there yourself. Both arrive at the same point, but the experiences they carry are entirely different.
┌───────────────────────────────────────────────────────────────┐
│ The Irreplaceable Value of the Mathematical Process │
├───────────────────────────────────────────────────────────────┤
│ │
│ Traditional Research Path: │
│ ┌──────┐ ┌────────┐ ┌──────┐ ┌──────┐ ┌──────────┐ │
│ │Conjec│→ │Explore │→ │Build │→ │Strict│→ │Extract & │ │
│ │ture │ │Failures│ │Intuit│ │Proof │ │Generalize│ │
│ └──────┘ └────────┘ └──────┘ └──────┘ └──────────┘ │
│ │ │
│ ┌──────▼──────┐ │
│ │Talks/Review │ │
│ └──────┬──────┘ │
│ │ │
│ ┌──────▼──────┐ │
│ │Textbook │ │
│ │Transmission │ │
│ └─────────────┘ │
│ │
│ AI Shortcut (skipping the process): │
│ ┌──────┐ ┌──────┐ │
│ │Problem│ 88 hours later │Proof │ │
│ │ │───────────────────────→│Output│ │
│ └──────┘ └──────┘ │
│ │
│ What is Lost: │
│ ✗ Lessons from exploring dead ends │
│ ✗ The process of building mathematical intuition │
│ ✗ Naming and framing new concepts │
│ ✗ Connecting results to the existing body of knowledge │
│ ✗ Creating teachable, textbook-ready presentations │
│ ✗ The human-to-human transmission network │
│ │
└───────────────────────────────────────────────────────────────┘
7. Paths Forward: Coexistence of Mathematics and AI
7.1 AI’s Positive Potential
The 25 Fields Medalists are not Luddites. The declaration explicitly acknowledges: “AI offers the potential of enhancing and accelerating genuine mathematical study and understanding.”
The question is not whether to use AI, but how to establish the boundaries and norms for its use.
7.2 The Right Way to Use AI in Mathematics
"""
A framework for healthy AI-assisted mathematical research
Principle 1: AI explores, humans decide
Principle 2: Formal verification aids understanding
"""
from dataclasses import dataclass
from typing import List
@dataclass
class MathInsight:
"""Mathematical insight — cognitive output AI cannot autonomously produce"""
name: str
category: str # 'concept' | 'method' | 'connection'
significance: float # 1-10
class HealthyAIMath:
def __init__(self):
self.insights: List[MathInsight] = []
def ai_explore(self, conjecture: str) -> list:
"""AI exploration: suggest strategies, identify difficulties"""
return ["Strategy A: constructive proof", "Strategy B: contradiction", "Strategy C: analogy"]
def human_develop(self, strategy: str) -> str:
"""Human-led development: core concepts, proof path, theory connections.
AI provides Lean/Coq formalization support,
but the architect's role belongs to humans."""
return "Human-led proof architecture"
def extract_insights(self, proof: str) -> List[MathInsight]:
"""Insight extraction — uniquely human work"""
self.insights = [
MathInsight("Deep connection between structures X and Y", "connection", 9.0),
MathInsight("New tool YY extracted from the proof", "method", 8.5),
MathInsight("Previously overlooked symmetry revealed", "concept", 7.5),
]
return self.insights
def pedagogify(self, insight: MathInsight) -> str:
"""Transform frontier results into teachable knowledge"""
return f"Transformed '{insight.name}' into textbook-ready material"
7.3 Changes Needed
Drawing from the declaration and community responses:
For AI companies:
- Stop treating major mathematical problems as benchmark competitions
- Establish academic ethics protocols with the mathematical community
- Maintain transparency about how user data flows into model training
- Allow sufficient time for peer review when publishing AI-generated mathematical results
For the mathematical community:
- Proactively embrace AI as a research assistant tool
- Establish academic norms for AI-assisted research
- Rethink core elements of mathematics education
For society at large: As the declaration concludes: “These issues must be addressed urgently, in the mathematical community, by the companies developing these technologies and, more broadly, by a society that will confront similar problems in many other forms of intellectual work.”
8. Looking Forward: Knowledge Transmission and Discovery
8.1 The Paradox of Tao’s Position
Tao’s position in this controversy is particularly interesting. He has repeatedly demonstrated using ChatGPT with the Lean proof assistant for formalization, describing the AI as a “junior collaborator” that “circles back on itself and cites the wrong theorems until nudged.” Yet he now stands firmly against how AI companies are approaching mathematics.
This apparent contradiction reveals a crucial insight: Using AI to do mathematics and allowing AI companies to define what mathematics is worth are completely different things.
The latter carries risks — when “problem-solving speed” replaces “depth of understanding” as the measure of value, when compute scale rather than intellectual insight becomes the competitive advantage — that could fundamentally alter the direction and meaning of mathematical research.
8.2 The Question That Remains
As one widely circulated commentary asked:
“When every intellectually starry sea that once demanded the full force of human thought can be reduced by GPUs in 88 hours to a cold ‘True’; when every Millennium Problem is lit up on a benchmark leaderboard one by one — how, then, shall humanity prove that it once thought, and felt, and wondered in this universe?”
The 25 Fields Medalists’ answer seems to be this: The true value of mathematics lies not in producing cold answers, but in the process of nurturing and transmitting ideas itself. These processes inevitably take time and are built on human interaction. If we discard them in pursuit of efficiency, “efficiency” itself loses its meaning.
As the declaration’s closing words remind us:
“Whether these changes ultimately benefit the field or have a destructive effect will in large part be determined by the decisions of the humans in control of this new technology.”
Further Reading:
- Full declaration and signing: https://mathandai.org/
- Tao’s blog post: https://terrytao.wordpress.com/2026/09/11/a-severe-misalignment-of-ai-in-mathematics/
- OpenAI Navier-Stokes paper: https://cdn.openai.com/navier-stokes-paper.pdf
- Economist coverage (linked from Tao’s blog)