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Mathematical Entities without Objects. On Realism in Mathematics and a Possible Mathematization of (Non)Platonism: Does Platonism Dissolve in Mathematics?

Published online by Cambridge University Press:  18 May 2020

Thierry Paul
Thierry Paul*
Affiliation:
Centre de Mathématiques Laurent Schwartz, CNRS, Ecole polytechnique, 91128, Palaiseau Cedex, France. Email: [email protected]
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Abstract

By looking at three significant examples in analysis, geometry and dynamical systems, I propose the possibility of having two levels of realism in mathematics: the upper one, the one of entities; and a subordinated ground one, the one of objects. The upper level (entities) is more the one of ‘operations’, of mathematics in action, of the dynamics of mathematics, whereas the ground floor (objects) is more dedicated to culturally well-defined objects inherited from our perception of the physical or real world. I will show that the upper level is wider than the ground level, therefore foregrounding the possibility of having in mathematics entities without underlying objects. In the three examples treated in this article, this splitting of levels of reality is created directly by the willingness to preserve different symmetries, which take the form of identities or equivalences. Finally, it is proposed that mathematical Platonism is – in fine – a true branch of mathematics in order for mathematicians to avoid the temptation of falling into the Platonist alternative ‘everything is real’/‘nothing is real’.

Type
Articles
Information
European Review , Volume 29 , Issue 2: Symmetry, Proportion and Seriality: The Semantics of Mirroring and Repetition in Science and the Arts , April 2021 , pp. 253 - 273
Copyright
© 2020 Academia Europaea

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References

Ambrosio, L (2004) Transport equation and Cauchy problem for BV vector fields. Inventiones Mathematicae 158, 227–260, https://doi.org/10.1007/s00222-004-0367-2.CrossRefGoogle Scholar
Benoist, J (2011) Éléments de Philosophie Réaliste: Réflexions sur ce que l’on a. Paris: Vrin.Google Scholar
Benoist, J and Paul, T (2013a) Pour une phénoménologie du formalisme mathématique. In Benoist, J and Paul, T (eds), Le Formalisme en Action: Aspects Mathématiques et Philosophiques. Paris: Hermann, pp. 5–11.Google Scholar
Benoist, J and Paul, T (eds) (2013b) Le Formalisme en Action: Aspects Mathématiques et Philosophiques. Paris: Hermann.Google Scholar
Bouchut, F (2001) Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Archive Rational Mechanics Analysis 157, 75–90, https://doi.org/10.1007/PL00004237.CrossRefGoogle Scholar
Connes, A (1994) Noncommutative Geometry. New York: Academic Press.Google Scholar
DiPerna, RJ and Lions, PL (1989) Ordinary differential equations, transport theory and Sobolev spaces. Inventiones Mathematicae 98, 511–547, https://doi.org/10.1007/BF01393835.CrossRefGoogle Scholar
Gelfand, IM (1941) Normierte Ringe. Recueil Mathématique [Matematicheskii Sbornik] 9, 3–24.Google Scholar
Paul, T (2007) La mécanique quantique vue comme processus dynamique. In Joinet, JB (ed.), Logique, Dynamique et Cognition. Paris: Publications de la Sorbonne, pp. 99–115.CrossRefGoogle Scholar
Paul, T (2009a) Semiclassical analysis and sensitivity to initial conditions. Information and Computation 207, 660–669, https://doi.org/10.1016/j.ic.2008.06.006.CrossRefGoogle Scholar
Paul, T (2009b) A propos du formalisme mathématique de la mécanique quantique. ‘Logique & Interaction: Géométrie de la Cognition’: Actes du Colloque et École Thématique du CNRS ‘Logique, Sciences, Philosophie’ à Cerisy. Paris: Hermann.Google Scholar
Paul, T (2010) Indéterminisme quantique et imprédictibilité classique. Noesis 17, 219–232, http://journals.openedition.org/noesis/1793.Google Scholar
Paul, T (2011) Semiclassical approximation and noncommutative geometry. Comptes Rendus Mathematique 349, 1177–1182, https://doi.org/10.1016/j.crma.2011.10.011.CrossRefGoogle Scholar
Paul, T (2013) Le vierge, le vivace et le bel aujourd’hui: trois mouvements de la structure espace vu des équations différentielles. In Benoist, J and Paul, T (eds), Le Formalisme en Action: Aspects Mathématiques et Philosophiques. Paris: Hermann, pp. 111–133.Google Scholar
Paul, T (2014) Rigueur-contraintes: mathématique-musique. Gazette des Mathématiciens 139, 71–77, https://hal.archives-ouvertes.fr/hal-00867275.Google Scholar
Poincaré, H (1885) Analysis situs. Journal de l’École Polytechnique 1, 1–123.Google Scholar
Teissier, B (2005) Protomathematics, perception and the meaning of mathematical objects. In Grialou, P, Longo, G and Okada, M (eds), Images and Reasoning. Tokyo: Keio University, pp. 135–145.Google Scholar