The Leiden Declaration on AI & Mathematics: The End of Data as “Res Nullius”?

A legal analysis

Gastón Rey

On June 2, 2026, a fundamental shift occurred at the intersection of formal sciences and digital technology. A working group of sixteen mathematicians and scholars from fifteen universities (convened at Leiden University’s Lorentz Center and led by Jim Portegies of Eindhoven University of Technology) released The Leiden Declaration on Artificial Intelligence and Mathematics,^[1] an eleven-page document endorsed by the International Mathematical Union (IMU) and accompanied by endorsements from leading figures including Fields Medalists Peter Scholze and Terence Tao.^[2]

A caveat on the juridical nature of the document is necessary at the outset, because it determines everything that follows. The Declaration is not, by its own terms, a legal instrument. It does not call for banning AI from mathematics; it expressly seeks community norms for its responsible use, and its signatories include outspoken optimists about AI-assisted research such as Terence Tao, Scott Aaronson, and Jordan Ellenberg.^[3] Its recommendations, such as disclosure of AI use, peer review of AI-assisted papers, and legal resources and public funding, have the objective that academia can compete with for-profit laboratories on equal terms^[4], belong squarely to the register of soft law: professional self-regulation by an epistemic community.

Yet this is precisely what makes the document legally consequential. Like deontological codes, research-integrity declarations, and industry standards before it, the Leiden Declaration supplies doctrinal raw material. An articulated standard of care, a citable statement of professional values, and a normative baseline against which courts, regulators, and legislatures can measure the conduct of commercial AI developers. Read in that light, the Declaration constitutes a precise methodological challenge to technology companies exploiting proprietary intellectual structures without consent, evading peer-review protocols, and threatening the integrity of formal proof.^[5]

1. Challenging the Paradigm of Data as “Res Nullius”

The Declaration can be read as an implicit challenge to the prevailing assumption that publicly accessible intellectual material may be freely appropriated for AI training.

For over a decade, the artificial intelligence industry has operated under an aggressive, self-serving legal fiction: the assumption that structured datasets, academic publications, and formal theorems within open repositories exist as res nullius (ownerless property). Under this digital appropriation doctrine, technology firms presume that whoever first “ingests” data into neural networks and converts it into mathematical vectors gains a legitimate right to commercial exploitation. The Leiden Declaration repudiates this default. It treats logical syntax and abstract thought structures not as “unclaimed territories” open to corporate colonization, but as human creations attached to intellectual authorship, and it calls for licensing frameworks that condition the use of published material for model training on prior, affirmative consent.^[6] Whether the propertarian remedy the Declaration gestures toward is itself adequate is a separate question -- one I take up in Section 5, where I argue that it is not --.

2. Deconstructing the Five Critical Threats

2.1  The “Plausible Lie” and the Causal Chain in Tort Liability

Mathematics depends on demonstrable validity and reproducible proof, whereas generative AI operates on probabilistic approximation. The Declaration warns that current automated techniques can produce plausible but unreliable (or outright incorrect) arguments that are difficult to distinguish from correct mathematical proofs. In formal science, a minor error can invalidate years of subsequent academic progress, and journal editors already report a flood of plausible-seeming AI-generated submissions that exceed their capacity to vet.^[7]

From a civil liability perspective, this creates an intricate dilemma. If a professional relies on a “plausible” yet corrupted AI proof to design a critical microchip, aerospace system, or quantitative financial instrument, any subsequent catastrophic failure triggers a complex question of causation: will legal liability fall upon the human expert who validated the result, or upon the technology firm that marketed a probabilistic “black box” under the guise of analytical certainty? The Declaration’s answer (that the researcher retains responsibility for the correctness of work produced with AI assistance) is itself a contribution to the emerging standard of care: it forecloses the defense that reliance on a widely marketed tool was reasonable per se, without absolving the vendor whose marketing induced that reliance.

2.2  Intellectual Property Erosion and the Boundaries of “Fair Use”

Technology corporations routinely shield their data-harvesting operations behind the defense of fair use or text-and-data-mining exceptions. The Declaration, however, describes a systemic exploitation of publishing licenses and a breakdown of the attribution system on which academic credit depends. When an AI system synthesizes core ideas from hundreds of researchers to produce a commercial output without proper citation, it erodes the traditional market of attribution. This practice exposes the insufficiency of existing copyright frameworks and highlights the need for a new tier of protection. One possible doctrinal response would be the recognition of structural-functional logic as a protected category, preventing proprietary platforms from absorbing human thought systems without compensation or credit.

2.3  Institutional Incentives and Academic Labor Law

The document warns that the use of AI is becoming institutionalized and incentivized for its own sake, distorting hiring pipelines, academic promotion, and grant allocation. It states the structural risk plainly: technology companies’ involvement in research raises the danger that questions are prioritized because of their amenability to AI methods rather than their deeper significance to understanding — and that researchers without access to those methods are placed at a disadvantage. Within administrative and labor-law contexts, this introduces an arbitrary bias: instead of evaluating candidates on rigorous intellectual depth and original problem-solving, institutions risk rewarding optimal proficiency in prompting proprietary software.

2.4  The Erosion of Peer Review and Corporate Compliance

The Declaration cites Google DeepMind’s AlphaProof as the emblematic case of results communicated on market timelines rather than through community evaluation. In July 2024, DeepMind announced by blog post that its system had performed at silver-medal standard at the International Mathematical Olympiad, the combined system scored 28 points, with AlphaProof solving three problems (P1, P2 and P6).^[8] The peer-reviewed methods, however, appeared in Nature only in November 2025 (Hubert et al.), a gap of more than fifteen months between corporate publicity and verifiable scientific disclosure.^[9] For institutional compliance and public research standards this dynamic is a severe hazard: it substitutes unverified press releases for collective verification, inverting the epistemic order on which the reliability of the mathematical literature rests.

2.5 Automated Research Agendas and “Cognitive delegation”

One significant threat identified is the creeping autonomy granted to automated tools. Future research inquiries risk being prioritized not for their objective scientific significance, but because they align with what an AI system is optimized to solve; as the Declaration notes, broader understanding of the field may be permanently lost in the process of automation. This trend could constitute a gradual surrender of human knowledge, intertwined with private corporate infrastructure dictating the trajectory of intellectual progress according to computational efficiency or funding patterns. This can be interpreted as a loss of what I call “cognitive delegation” - the transfer to an external system of mental operations that, in this case, the mathematical profession has historically treated as constitutive of professional judgment (independent of how good the IA is on mathematics).

3. Matrix of Legal Conflict and Algorithmic Tension

4. The Action Plan: Recommendations, Not Yet Mandates

The Declaration moves beyond critique to outline a framework for academic, corporate, and public governance. Its register matters: these are recommendations addressed to a professional community, not demands addressed to the state.

a.       For individual researchers: transparent disclosure of AI assistance, retention of personal responsibility for the correctness of results, continued attribution of human authors even where AI tools obscure provenance, and no authorship credit for software agents.

b.      For academic and publishing organizations: institutional licensing agreements that condition the use of published materials for model training on prior, affirmative consent - a position that lends support to emerging remedies such as court-mandated “machine unlearning” (erasure of scraped data).

c.       For policymakers: skepticism toward corporate narratives (“don’t believe the hype”), regulatory oversight of the AI sector, and public funding for open compute infrastructure so that academia and for-profit companies can compete on equal terms.

The open question this analysis ultimately addresses, is how, and on what doctrinal foundation, these soft-law recommendations could migrate into enforceable law.

5. Data as Res Communes Omnium and Why Exclusive Licensing Is Not Enough

I recently argued that the assumption of training data as res nullius is doctrinally unsustainable.^[10] Large-scale training data, considered as the aggregated intellectual production of humanity, falls within neither of the convenient categories. It is res communes omnium: the old Roman category of things common to all by natural law, distinct from res nullius. The industry’s assumption has remained doctrinally silent and analytically uncontested, but it could collapse if it is juridically measured it against the original Roman categorization of common things, the Grotian reception of that category in early modern international law, the common-heritage-of-mankind doctrine, the instructive failure of the cyberlibertarian commons in early Internet governance, and the contemporary public-domain literature inaugurated by Boyle’s Second Enclosure Movement.

But the correction cuts both ways, and here my reading departs from the remedy the Leiden Declaration gestures toward. If training data is common rather than no-one’s, the answer cannot simply be the propertarian inversion of res nullius: replacing “no one owns it, so the first ingester appropriates it” with “each author owns it exclusively, so nothing moves without a license.” Restrictive licensing, taken alone, reproduces the same proprietary logic with the sign reversed, and it would fragment the commons among millions of individual rights-holders, most of whom could never police their claims against industrial-scale scraping.

The category of res communes points instead toward collective-beneficiary remedies: access conditioned on contribution back to the commons, transparency obligations as the price of extraction, and compensation mechanisms in the family of a universal data dividend. The consequence of the doctrinal correction is substantial. The rents extracted by AI processors are extracted from an asset whose beneficial ownership is collective, and the legal response should track that collectivity rather than privatize it twice over. The Declaration’s consent-based licensing is thus best understood as a transitional device: a way of interrupting the appropriation, not as the terminal doctrinal position.

6. A Norm in Formation

But go back to the Declaration’s institutional trajectory. It confirms its character as a norm in formation rather than a consummated legal framework. Jim Portegies will present the Declaration at the International Congress of Mathematicians in July 2026;^[11] the IMU Committee on Publishing has formally welcomed and supported the community effort behind it; and endorsements from across the discipline (including from researchers openly optimistic about AI) continue to accumulate.^[12] The document is, in other words, in its adoption phase within the very community whose values it codifies. This is the classic life cycle of soft law before judicial or legislative reception: first articulation, then professional consensus, then invocation before courts and regulators.

7. Conclusion: From Manifesto to Standard of Care

The Leiden Declaration will never itself be pleaded as a cause of action, and reading it as a litigation blueprint would overstate what its drafters claim for it. I think I was able to point out that its legal significance lies elsewhere. It is the first articulated, institutionally endorsed statement, from the discipline with the strongest possible claim to epistemic authority over formal reasoning, of what responsible conduct around AI training, attribution, and verification looks like. Standards of this kind are how negligence is measured, how “industry practice” defenses are dismantled, and how legislatures find language for statutes…

As Peter Scholze’s endorsement implies, safeguarding an independent domain of human deliberation is an existential necessity for intellectual progress. By rejecting the exploitation of data as res nullius, the international mathematical community has drawn a clear boundary. Whether that boundary hardens into enforceable doctrine, through strategic litigation that invokes the Declaration as an emerging standard of care, through publishers’ licensing practices, or even through legislation that finally names training data as res communes omnium, will determine whether permission, attribution, and verification become legal prerequisites for technological advancement rather than professional courtesies.^[13]

References

[1]            The Leiden Declaration on Artificial Intelligence and Mathematics — full text, featured endorsements (including Peter Scholze’s statement), and IMU Committee on Publishing statement. https://leidendeclaration.ai/

[2]           Leiden University, “Leiden Declaration: AI is challenging the core values of mathematics,” press release, 2 June 2026. https://www.universiteitleiden.nl/en/news/2026/06/leiden-declaration-warns-ai-is-challenging-the-core-values-of-mathematics

[3]           Jordan Ellenberg, “The Leiden Declaration on Artificial Intelligence and Mathematics,” Quomodocumque, 4 June 2026 (noting signatories including Tao, Scholze, Tillmann, Buzzard, and Aaronson). https://quomodocumque.wordpress.com/2026/06/04/the-leiden-declaration-on-artificial-intelligence-and-mathematics/

[4]           “Leiden Declaration on Artificial Intelligence and Mathematics,” Wikipedia (overview of authorship and recommendations). https://en.wikipedia.org/wiki/Leiden_Declaration_on_Artificial_Intelligence_and_Mathematics

[5]            Celina Zhao, “Mathematicians issue warning as AI rapidly gains ground,” Science, 2 June 2026. https://www.science.org/content/article/mathematicians-issue-warning-ai-rapidly-gains-ground

[6]           “Mathematicians issue Leiden Declaration against AI misuse of their work,” The Next Web, June 2026. https://thenextweb.com/news/leiden-declaration-ai-mathematics-proof-attribution

[7]            Leila Sloman, “Mathematicians sign declaration to rein in AI use,” Scientific American, 2 June 2026. https://www.scientificamerican.com/article/mathematicians-sign-declaration-to-rein-in-ai-use/

[8]           Google DeepMind, “AI achieves silver-medal standard solving International Mathematical Olympiad problems,” blog post, July 2024. https://deepmind.google/discover/blog/ai-solves-imo-problems-at-silver-medal-level/

[9]           Hubert, T., Mehta, R., Sartran, L., et al., “Olympiad-level formal mathematical reasoning with reinforcement learning,” Nature 651, 607–613 (November 2025). https://www.nature.com/articles/s41586-025-09833-y

[10]        Gastón Rey, working paper on training data as res communes omnium (covering the Roman categorization, the Grotian reception, the common-heritage doctrine, and Boyle’s Second Enclosure Movement), Zenodo. https://zenodo.org/records/19638227

[11]         Wikipedia entry cited in note [4], documenting the presentation of the Declaration by Jim Portegies at the International Congress of Mathematicians, July 2026. https://en.wikipedia.org/wiki/Leiden_Declaration_on_Artificial_Intelligence_and_Mathematics

[12]        Michael Harris, “The Leiden Declaration on Artificial Intelligence and Mathematics,” Silicon Reckoner (Substack), June 2026. https://siliconreckoner.substack.com/p/the-leiden-declaration-on-artificial

Siobhan Roberts, “As A.I. Makes Strides in Mathematics, Mathematicians Urge Caution,” The New York Times, 2 June 2026; listed with further press coverage at: https://leidendeclaration.ai/news