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Space and Time in the Foundations of Mathematics, or Some Challenges in the Interactions with Other Sciences

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Handbook of the Mathematics of the Arts and Sciences

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Abstract

Our relation to phenomenal space has been largely disregarded, and with good motivations, in the prevailing foundational analysis of Mathematics. The collapse of Euclidean certitudes, more than a century ago, excluded “geometric judgments” from certainty and contributed, by this, to isolate the foundation of Mathematics from other disciplines. After the success of the logical approach, it is time to broaden our foundational tools and reconstruct, also in that respect, the interactions with other sciences. The way space (and time) organize knowledge is a cross-disciplinary issue that will be briefly examined in Mathematical Physics, Computer Science, and Biology. This programmatic paper focuses on an epistemological approach to foundations, at the core of which is the analysis of the “knowledge process,” as a constitutive path from cognitive experiences to mathematical concepts and structures. When first presented, in 2001, it opened to way to the idea that phylogenetic trajectories, in biology, co-construct the space of possibilities, in contrast to physical theories which assume a pregiven phase space of all possible dynamics.

Revised version of an Invited Lecture, first American Mathematical Society/SMF Conference, Lyon, July, 2001.

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Acknowledgments

(2001) In the last few years, the activity in two working groups (CeSEF, on the epistemology of Quantum Physics) and “Géométrie et Cognition” has been a fantastic occasion to meet and discuss with several colleagues in Physics, Biology and Philosophy. I am particularly indebted to the joint work and uncountably many discussions with Francis Bailly, Jean Petitot, Bernard Teissier, Catherine Vidal, Paul Bourgine, RossanaTazzioli, MioaraMugur-Schachter.

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Authors and Affiliations

  1. Centre Cavaillès, CNRS and Ecole Normale Supérieure, Paris, France

    Giuseppe Longo

  2. School of Medicine, Tufts University, Boston, MA, USA

    Giuseppe Longo

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  1. Giuseppe Longo
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Correspondence to Giuseppe Longo .

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  1. Department of Mathematical Sciences, University of Montana, Missoula, MT, USA

    Bharath Sriraman

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  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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Longo, G. (2021). Space and Time in the Foundations of Mathematics, or Some Challenges in the Interactions with Other Sciences. In: Sriraman, B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-57072-3_116

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