Abstract
In this essay we give a general picture about the evolution of Grothendieck’s ideas regarding the notion of space. Starting with his fundamental work in algebraic geometry, where he introduces schemes and toposes as generalizations of classical notions of spaces, passing through tame topology and ending with the formulation of a geometry of forms, we show how the ideas of Grothendieck evolved from pure mathematical considerations to physical and philosophical questions about the nature and structure of space and its mathematical models.
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Acknowledgments
The author wants to thank Fernando Zalamea for many illuminating discussions on Grothendieck’s work over the years, his constant and warm support and having read a draft version of this essay. Of course, any mistake that remains is completely responsibility of the author. He also thanks Winfried Scharlau for providing him with Grothendieck’s correspondence with Yamashita and Tsuji and for conversations on Grothendieck’s life, and Yuri Manin for supporting the idea of writing this text. Finally, he thanks Pierre Deligne for many interesting emails and his patience in explaining his ideas.
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Cruz Morales, J.A. (2020). The Notion of Space in Grothendieck. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_5-1
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