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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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