Frédéric Jaëck

This book offers an original combination of historical, philosophical, and mathematical perspectives on the life and work of Albert Lautman (1908–1944), a philosopher of mathematics and member of the French Resistance. Lautman’s work demonstrates the mutual illumination that philosophy and mathematics can provide. It describes the central place of mathematics in our relationship to reality and advances a distinctive conception of mathematical unity—not as a closed system, but as an evolving field. Structured by productive dualities—local and global, continuous and discontinuous, intrinsic and induced properties, symmetry and asymmetry, among others—mathematics contributes to the effective resolution of problems first raised by philosophy, including those of existence, the absolute, and finitude. Conversely, philosophy—in the form of a dynamic conception of Platonic Ideas—makes it possible to understand the structure of physical and mathematical theories and their shared participation in an ideal reality. The volume is divided into three parts. The first retraces Lautman’s path as a Resistance fighter and author during the war years. The second situates his work within the long history of philosophy, tracing its roots back to the beginnings of the discipline and emphasizing its continuing capacity to inspire contemporary philosophy of mathematics, particularly certain forms of structuralism. The third shows how fruitful Lautman’s work remains for understanding the most recent developments in mathematics. The result of a collective undertaking by historians, philosophers, and mathematicians, this volume introduces contemporary readers to a man committed to carrying the rigor of ideas into concrete action. ### The Editors Christophe ECKES is Associate Professor of the Contemporary History of Mathematics at the University of Lorraine and a member of the Henri Poincaré Archives (UMR 7117, Nancy). His dissertation, defended in 2011, examined Hermann Weyl’s scientific and philosophical work from the early 1910s to the early 1930s. Since 2016, his research has focused on the social history of the mathematical sciences during the German occupation of France, with particular attention to the teaching of mathematics in universities established in prisoner-of-war camps. He currently coordinates a project, supported by the French National Research Agency, devoted to preserving and promoting the archives of the Nicolas Bourbaki group of mathematicians. His recent work has led him to reconstruct the mathematical sources from which Albert Lautman’s philosophy of mathematics developed. Frédéric JAËCK is a philosopher of mathematics at Aix-Marseille University and a member of the Centre Gilles-Gaston-Granger. His research lies at the intersection of a tradition of critical philosophy and a descriptive philosophy adapted to mathematics. He recently coedited, with Gabriella Crocco, a double special issue of the journal *Annals of Mathematics and Philosophy* devoted to a tradition of thought initiated by the work of Brunschvicg, Lautman, and Cavaillès: *La philosophie mathématique: Mathematical and Philosophical Inspirations from Brunschvicg to Granger* (2025). Baptiste MÉLÈS, a graduate of the École Normale Supérieure on rue d’Ulm and holder of the agrégation in philosophy, is a CNRS Research Fellow affiliated with the Henri Poincaré Archives (UMR 7117, Nancy). His work examines the relationships among philosophical rationality, the formal sciences—logic, mathematics, and computer science—and language. He co-organizes the “Codes sources” seminar and is the author of *Classifications des systèmes philosophiques* (Vrin, 2016). Jean-Jacques SZCZECINIARZ, a graduate of the École Normale Supérieure on rue d’Ulm, is Professor Emeritus at Paris Diderot University. From 2005 to 2019, he directed the university’s Department of History and Philosophy of Science. He holds the agrégation in philosophy, advanced postgraduate degrees in pure mathematics and logic, and a State Doctorate on cosmology and its history. He has worked in the Complex Geometry Laboratory at Paris Diderot University and in mathematical physics. In philosophy, he studied under Louis Althusser and Pierre Macherey. His work has focused particularly on historical epistemology, including Duhem, Cavaillès, Lautman, and Lecourt. His current project consists primarily of a philosophical study of Grothendieck’s work. He has received a Grand Prize from the French Academy of Sciences and the Romieu Prize. With contributions by Alya AGLAN (Paris 1 Panthéon-Sorbonne University), Hourya BENIS-SINACEUR (CNRS), Sorin DUMITRESCU (Université Côte d’Azur), Christophe ECKES (University of Lorraine), Gerhard HEINZMANN (University of Lorraine), Frédéric JAËCK (Aix-Marseille University), Jacques LAUTMAN (CNRS), Olivier LAUTMAN, Sébastien MARONNE (Toulouse Mathematics Institute), Baptiste MÉLÈS (CNRS), Frédéric PATRAS (CNRS), Jean PETITOT (International Academy for Philosophy of Science), Damian RÖSSLER (University of Oxford), Jean-Michel SALANSKIS (Paris Nanterre University), Jean-Jacques SZCZECINIARZ (Paris Diderot University), and Bernard TEISSIER (CNRS). *Foreword by Frédéric WORMS, Director of the École Normale Supérieure.*

This article examines Stefan Banach’s contributions to the field of functional analysis based on the concept of structure and the multiply-flavored expression of generality that arises in his work on linear operations. More specifically, it discusses the two stages in the process by which Banach elaborated a new framework for functional analysis where structures were bound to play an essential role. It considers whether Banach spaces, or complete normed vector spaces, were born in Banach’s first paper, the 1922 doctoral dissertation On operations on abstract spaces and their application to integral equations. It also analyzes what appears to be the core of Banach’s 1922 article and the transformation into a general setting that it represents. The main achievements of Banach’s dissertation, as well as all the essential features that bear witness to the birth of a new theory, are concentrated in the study of linear operations.

*Lectures grothendieckiennes* brings together the papers that grew out of a seminar held in the Department of Mathematics at the École Normale Supérieure from 2017 to 2018. The volume presents a complex body of thought in action: that of Alexander Grothendieck, one of the most influential and enigmatic mathematicians of the twentieth century. The contributors—Pierre Cartier, Olivia Caramello, Alain Connes, Laurent Lafforgue, Colin McLarty, Gilles Pisier, Jean-Jacques Szczeciniarz, and Fernando Zalamea—each explore, in their own way, the mathematical and philosophical implications of a monumental body of work that transformed the mathematical landscape of the twentieth century and may well have inaugurated a new mathematical era that we have only begun to explore. Foreword by Peter Scholze. Video recordings of all the lectures are available online.

Dans cet article nous étudions trois notes de M. Fréchet sur les opérations linéaires et leur rôle dans l’émergence de l’analyse fonctionnelle au début du XXème siècle. Dans un premier temps nous mettons en évidence les processus de sélection, d’extraction et de réinterprétation mis en oeuvre par Fréchet à partir de matériaux publiés antérieurement. A partir de cette analyse nous nous attachons à montrer comment la progression vers une vision plus générale met en jeu des éléments structurels fondamentaux dans la constitution d’un champ nouveau des mathématiques. Notre étude permet finalement de préciser le concept de structure dans son rôle central dans l’émergence de l’analyse fonctionnelle.

This paper provides an analysis of the use of axioms in Banach’s Ph.D. and their role in the progression of Banach’s mathematical thought. In order to give a precise account of the role of Banach’s axioms, we distinguish two levels of activity. The first one is devoted to the overall process of creating a new theory able to answer some prescribed problems in functional analysis. The second one concentrates on the epistemological role of axioms. In particular, the notion of norm completeness, as it appears in Banach’s text, can be interpreted as an epistemic linchpin between several a priori inhomogeneous domains of mathematical thought.