Executive Summary

On June 2, 2026, sixteen mathematicians from fifteen universities released the Leiden Declaration. On the first day, 130 people signed; within 24 hours the count passed 1,000. Fields Medalist Peter Scholze added his name, as did the International Mathematical Union (IMU). It is rare for a discipline to converge so quickly and so clearly on a stance toward AI.

Start with the point that is easy to get wrong. This declaration is not a call to stop using AI in mathematics. What it opposed was three practices: taking papers as training data without consent, breaking results to the press before peer review instead of going through it, and failing to say whose work the result was indebted to. The problem lay not in the tool but in the procedure.

For anyone who works with data, the declaration translates into a familiar question. Is the provenance traceable, was there consent for its use, and can the result be independently verified? The mathematicians wrote that these three are not administrative formalities but the essence of trust. This article looks at what the declaration opposed, what it sought to protect, and how that bears on the problem of AI training data.

Key Figures

The shape of the episode fits into a few numbers. Within a single day of the release, more than a thousand mathematicians had signed; meanwhile the disproof of the 80-year-old conjecture that sparked it all came with not a single piece of public verification material. And two months later, the demand to disclose training data becomes a legal obligation in Europe.

Sources: Scientific American, Leiden Declaration

1,000+

Signatures within 24 hours

Mathematicians who signed on within a day of release

80 years

Age of the conjecture AI claimed to disprove

A result released by press release, with no peer review

Zero

Pieces of public verification material

A proprietary model, so method, training data, and compute stayed hidden

Aug 2, 2026

EU AI Act Article 53 enforcement

The day disclosing training data and honoring opt-out becomes a legal duty

1

An 80-Year Conjecture and the Spark

In May 2026, OpenAI announced that an AI had disproved an 80-year-old mathematical conjecture. The target was an old Erdős conjecture concerning unit distances among points in the plane. The news that a machine had cracked a problem mathematicians had wrestled with for decades was striking in itself. The trouble was how to make sense of it.

Paul Erdős portrait — his 80-year-old unit-distance conjecture became the spark for the Leiden Declaration
▲ Paul Erdős (1913–1996), Hungarian mathematician. His planar unit-distance conjecture faced an AI challenge 80 years later. | Source: Wikimedia Commons (CC BY 3.0)

The announcement came as a press release, not a journal paper. The model used was proprietary and never disclosed; there was no way to check the method, the training data, or the compute. In mathematics, a proof does not end with a correct conclusion. It is only a proof if anyone can follow the reasoning and confirm why the conclusion holds. Here the path to follow was blocked from the start. The community's reaction was natural: how are we supposed to trust this result?

In fact, the announcement was only the fuse. The fire had been smoldering for nine months. In September 2025, more than sixty mathematicians from ten countries gathered at the Lorentz Center of Leiden University in the Netherlands to debate the problems arising as AI enters mathematics. Those discussions led sixteen of them to draft a statement, released on June 2, 2026, as an eleven-page declaration. The 130 signatories of the first day passed 1,000 within a day, and the International Mathematical Union voiced official support. Fields Medalist Peter Scholze, Kevin Buzzard (known for his work on formal proof tools), and popular mathematics author Steven Strogatz signed side by side.

The core: The moment an unverifiable result reached the public through a press release, mathematicians took issue with something more fundamental than whether the result was true. If the procedure of verification itself collapses, what can be believed?

2

What the Declaration Opposed Was Not AI

The declaration plainly acknowledges that AI can help mathematics. Ilka Agricola, chair of the IMU's publishing committee, says AI can be "extremely useful." Yet from the same mouth comes the warning that the current approach is causing "enormous confusion." What the declaration took aim at was not the tool but three specific practices.

Practice opposed → Condition required ✕ Training Without Consent Exploiting licenses · Copyright violation → ✓ Consent Author consent before training · Opt-out guaranteed ✕ Bypassing Verification Press release first · Peer review skipped → ✓ Peer Review Journal review · Human explains central argument ✕ Failure to Attribute No citations · Method and training data hidden → ✓ Attribution Cite prior work actively · Disclose AI tools used
▲ Three practices the Leiden Declaration opposed and three conditions it required | Pebblous original diagram

2.1Training Without Consent

The first is using papers as training data without consent. Preprints posted to arXiv and published papers flow into AI training without their authors' consent. Rodrigo Ochigame, an anthropologist at Leiden, puts it this way: the output of mathematicians who never imagined their work would be used to develop AI is being used for exactly that purpose, without consent. The declaration describes this as collection that "exploits licenses designed before AI existed or violates copyright protections."

2.2Bypassing Verification

The second is skipping peer review. Results are put out through press releases and blogs before they pass through journal publication or conference presentation. Corporate market timelines outrun the academy's verification procedures. As a result, journal editors are swamped by a flood of unverified, AI-generated proofs. The Erdős conjecture announcement was the very template of this practice.

2.3Failure to Attribute

The third is failing to say whom the work is indebted to. The results AI produces do not properly cite the human work that made them possible. And because proprietary models disclose neither method nor training data, there is no way to trace which prior research is dissolved inside them. Jim Portegies of Eindhoven University of Technology says commercial interests are pushing mathematics behind a "closed door."

Correcting the misreading: The declaration is not an AI ban. Training without consent, bypassing verification, failing to attribute — it rejected three practices that blur provenance and responsibility. It is closer to a statement that the standards long demanded of people will now be demanded of machines too.

3

Beyond Opposition: What the Declaration Asks For

The declaration does not only object. It puts five values it means to protect on record, and specifies who must do what. The five values are these: that a proof confers the highest level of certainty; that a result must be attributed to a specific, accountable human; that an argument must be transparent enough to be independently verified; that there must be shared standards for measuring the depth and importance of work; and that the direction of research must be led by expert judgment.

So as not to remain an abstract statement, the declaration assigns actions by actor. It asks individual mathematicians to add a new "tools and computational resources" section disclosing the AI tools used in a paper. Responsibility for accuracy and sufficiency rests entirely with the human author, and AI is not listed as an author. The prior research that led to the conclusion must be actively sought out and cited.

More concrete demands are directed at institutions and journals: to maintain peer-review standards even for AI-generated proofs, and to require that a human explain the central argument. Above all, they should put in place licensing agreements that prevent training data from being used without consent, and guarantee authors the right to opt their work out of AI training. Of AI companies, the declaration asks that they hold to the very standards the academy expects of a peer.

The core: The single line running through all five values is provenance and responsibility. Whose work the result is indebted to, who is accountable for its accuracy, and whether a third party can follow the process and confirm it. Mathematics drew a line: a result that cannot answer these three questions will not be recognized as a proof.

4

Verifiable Provenance Is the Foundation

The three conditions the mathematicians drew are not unfamiliar to anyone who works with data. Consent is the question of whether training data was collected legitimately; attribution is the question of whether you can trace where that data came from; and peer review is the question of whether the result can be independently confirmed. Provenance, consent, verifiability. In an age when AI can churn out plausible but wrong answers, these three are the minimum conditions that hold trust up.

This principle is also being translated into the language of law. From August 2, 2026, Article 53 of the EU AI Act will require AI providers to publish a summary of their training data and to respect opt-outs under EU copyright law. Italy already, in a 2025 statute, pulled text and data mining for AI training inside existing copyright protection. The provenance of and consent for training data are beginning to be treated not as an incidental use but as a separately negotiable right. What the Leiden Declaration said in the language of scholarship, regulation is writing down in the language of enforcement.

From declaration to law — the same principle gains enforcement Sep 2025 Lorentz Center 60+ mathematicians, 10 countries Jun 2, 2026 Leiden Declaration 1,000+ signatures in 24 hours Aug 2, 2026 EU AI Act Art. 53 Training data disclosure mandatory
▲ From the Leiden Declaration to EU AI Act Article 53 — scholarly principle becomes legal obligation | Pebblous original diagram

So the question shifts to this. Is the data we feed AI traceable to its source, was there consent for its use, can the result be verified again? The three things one discipline nailed down as the essence of trust are also the questions of every site that grapples with data quality. Verifiable provenance is a foundation not only in mathematics.

Editor's Note: The heart of what Pebblous means by AI-Ready Data sits in the same place. Traceable provenance, legitimate consent, verifiable quality. The Leiden Declaration is the first time mathematics has formalized these three as conditions of trust, and the same question is being put, just as squarely, to everyone who works with data.

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References

Official Document & Declaration

News Coverage

Legislation & Analysis