An Interview with Fernando Zalamea, apropos of Alexander Grothendieck

       Do you recall your first encounter with Alexander Grothendieck’s work or figure? In what context did it take place, and how did it influence your later intellectual interests?

My first encounter with Alexander Grothendieck’s work occurred while I was doing my Master Thesis (1986) with the logician Xavier Caicedo in Bogotá. Xavier introduced me to Lawvere’s elementary toposes, around which we studied some new intuitionistic connectives. From there, I jumped immediately to Grothendieck’s toposes, to try to unveil the history of the subject. It is interesting that I first encountered Grothendieck’s figure in Colombia after having done my Mathematics Licence and Maîtrise in France (Nice and Paris, 1977–1981). This shows how Grothendieck’s work was explicitly ignored by many French mathematicians, even if many others (Dieudonné, Serre, etc.) already considered him a towering figure. Then, while doing my Ph.D. Thesis (1990) in Massachusetts with Ernie Manes, a leading figure in Category Theory, Grothendieck’s place became firmly established in my panorama. After a long decade of studies around Peirce (his pragmaticist maxim can be closely intertwined with Category Theory), I devoted myself, in the period 2004–2019, to read with care Grothendieck’s work, concluding in my monograph Grothendieck. Una guía a la obra matemática y filosófica (2019), perhaps the first complete study of his published work. The figure of Grothendieck spread easily around my other books in philosophy, criticism, literature, and art.

       You encountered Grothendieck’s work at an early stage of your career and have engaged with it over many years since. From your perspective, which of Grothendieck’s personal or intellectual traits drew you toward him, and how did these come to distinguish him, in your experience, from other major figures in mathematics and philosophy with whom you have engaged in your career?

Grothendieck’s traits which most attracted me were (1) his universal practice of concrete mathematics (functional analysis, topology, algebra, complex variables, co/homology, category theory, number theory, etc.), (2) his wide abstract exploration of the main forces involved (schemes, toposes, motives, stacks, derivators, etc.), (3) his in-depth look on mathematical creativity (Récoltes et semailles), (4) his wide-ranging understanding of Cosmos, the Earth, and culture (La Clef des songes), and (5) his humble, generous, altruist personality, driven always to the community. The combination of these five forceful traits makes him stand alone in the recent history of mathematics or philosophy. For example, other giants shared (1), (3) and (4)—Brouwer—, others (1) and (3)—Poincaré—, others (3) and (4)—Peirce—, others (1), (3), (4), (5)—Florensky—, others (3), (4) and (5)—Simone Weil—, etc., but it is difficult to find a single mind which encompassed such diversity and depth in the last two hundred years.

       You described a combination of mathematical, philosophical, and personal traits that sets Grothendieck apart from other major figures. From the standpoint of the historical reception of his work, which aspects of his life or work do you think remain insufficiently understood or insufficiently appreciated within the scientific community and the wider culture today?

It is in fact a paradox that such a major thinker, both as a mathematician and as a philosopher, is still very little understood and appreciated. Even if many scholars have acknowledged his stature, the community of mathematicians at large, and certainly the general public, have not assessed well his figure and his work. This is due to intrinsic reasons—understanding universality and abstractness as a general context of “softness”, where particular obstructions dissolve, with fantastic concrete products as the resolution of the Weil conjectures, or the many advances in the Grothendieck–Teichmüller panorama, contrary to the repeated nonsense of depreciating Category Theory as “abstract nonsense”—and to extrinsic reasons as well—Grothendieck’s revolutionary insights in philosophy, ethics, ecology, linked to a non-compromising person, strongly critical of the egotistic academicism of our epoch. But time will put things in due place: Grothendieck is already seen as a pioneer in caring for our Earth and in offering balanced views of our Community relations, and Category Theory is spreading more and more in a deep comprehension of the foldings and unfoldings of mathematical creativity, and its everyday practice.

       Turning more directly to Grothendieck’s mathematics: his work is often associated with a distinctive use of abstraction and with an unprecedented level of generality in the tools he introduced. From your perspective, what guidance would you offer to younger researchers who approach Grothendieck’s work today? More generally, do you think that what might legitimately be called “Grothendieckian” can be transmitted and shared across generations without necessarily falling into mystification, or does it require a particular historical or intellectual context?

I would recommend arriving to generality based on his very concrete contributions. Particularly, his Résumé de la théorie métrique des produits tensoriels topologiques (1953) is a marvel of tripartite work in topology, abstract algebra, and order (Bourbaki’s three “mother structures”). It is written with utmost care, and the many equivalences between the three structural approaches are just bewildering. From there, one can sense easily his next steps towards Category Theory, in Sur quelques points d’algèbre homologique (1955–56, the “Tôhoku”), a crystal clear gigantic paper, which only needs patience in its reading, in which he works with all kind of examples in abelian groups, modules, sheaves, to attain the vast vision of his abelian categories. In the same vein, his beautiful concrete researches in Riemann surfaces, moduli spaces, and arithmetic questions, in La longue marche à travers la théorie de Galois (1981), constitute another marvel of ingenuity and constant invention. After perusing his many wanderings in concrete mathematics, it is just natural to go ahead, and to arrive at schemes, toposes, motives, derivators, etc. In Récoltes et semailles, Grothendieck described himself as “a mutiple in search of unity”, something which is achieved through abstraction, generality, universality. The “Grothendieckian” spirit has to do with understanding deeply that dialectics between the Many and the One (originating in Plato), not only in mathematics, but also in life. It goes beyond a given historical or intellectual context, but Grothendieck’s very acute, abstract, mathematical constructions have given a gigantic Turn of the Screw to that perennial problematic of civilization. I really consider that “Grothendieckian” could be at the level of “Platonic” for generations to come, now with many new tools to investigate the world of mathematics, and, in extension, the World itself.

       As a former coordinator of the Centre for Grothendieckian Studies (CSG) and a specialist in Grothendieck’s work, what role do you think the CSG can play in shaping the future reception and understanding of his legacy? More broadly, what do you see as the main challenges and opportunities in transmitting his mathematical and intellectual contributions to new generations?

The CSG role in shaping the future and understanding of Grothendieck’s legacy is huge. Olivia Caramello has set the basis for the great role of the CSG. Her initiative will certainly irradiate deeply in the comprehension of Grothendieck. The CSG website, curated by Mateo Carmona, is by far one of the most important resources for accessing Grothendieck’s gigantic output. But much has still to be done, particularly around the edition and study of the unpublished manuscripts. It should be the work of a dozen of specialists along a decade of sustained efforts. In that respect, the continued study, preservation, and eventual wider accessibility of manuscripts like those held at the BnF would constitute an important contribution to the Grothendieckian community. Making these materials more broadly available, in accordance with the applicable rights and institutional frameworks, would open further possibilities for research and contribute to a deeper understanding of Grothendieck’s intellectual legacy, in keeping with the “Grothendieckian” spirit mentioned above.

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