Francesco Preta: Who Owns the Vector? A Critical Genealogy of Vector Space Theory

The rise of Large Language Models (LLMs) and other technologies of artificial intelligence (AI) has solidified the idea that any and all entities can be represented as embeddings in high-dimensional “vector spaces”. While recent scholarship (Pasquinelli, 2023; Brunila, 2025; Weatherby, 2026) has explored different facets of the controversial history that this notion entails, its mathematical foundations have largely remained unexamined. Developed jointly with Mikael Brunila (Sciences Po), this talk aims to fill this gap by proposing a genealogy of the concept of the vector through a comparison of the perspectives of pure and applied mathematics, understood here as two distinct “styles of reasoning” (Hacking, 1990).

Drawing on Hacking’s framework, the talk examines the different objects, forms of evidence, laws, and discourses validated within each field through their respective approaches to the themes of foundations, infinity, and geometry. By considering the divergent evolution of these two epistemological approaches as a subject of critical evaluation, it argues that both are underpinned by a shared Platonic thread that sustains the idea of modern science and underwrites the authority of those who produce and privately control mathematical knowledge. Against this backdrop, Francesco Preta proposes an alternative that situates both epistemic practices within the historical context of their development while avoiding the relativization of their validity.

Following the tradition of historical epistemology (Hacking, 2002; Renn, 2020), Preta argues that the material conditions and institutional structures enabling each practice are fundamental in determining both epistemic horizons (what is researched) and the principles motivating them (why it is researched). Through a computational analysis of citation graphs and linguistic drift, the talk further corroborates the argument—already advanced in the existing literature (e.g., Abdalla et al., 2023)—that private institutions increasingly dominate scientific research and that the pure sciences are progressively disappearing. It concludes with an open discussion of how and why scientists might pursue pure research in the age of AI, and what role such inquiry can still play within the current societal structure.

The talk will be commented by Julian A. Morgan.

The event is organized by Sebastian Berg, Research Group “Technology, Power and Domination”.

About the speaker

Francesco Preta holds a PhD in Mathematics from New York University (2021) and has collaborated with academic and private institutions on a freelance basis since 2022. His research interests include the history and philosophy of mathematics, critical AI theory, historical epistemology, and the political economy of mathematical knowledge production. He is currently a guest researcher in the research group “Technology, Power and Domination” at the Weizenbaum Institute.