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Poincaré’s geometric worldview and philosophy. (English) Zbl 07265533
Dani, S. G. (ed.) et al., Geometry in history. Cham: Springer (ISBN 978-3-030-13608-6/hbk; 978-3-030-13611-6/pbk; 978-3-030-13609-3/ebook). 435-450 (2019).
Henri Poincaré is one of the pioneers in non-Euclidean geometry and topology. The author gives first short insights into Poincaré’s work on hyperbolic geometry and its connections to his research in function theory and differential equations. He then goes briefly into Poincaré’s topological papers referring to more detailed analysis by K. S. Sarkaria [in: History of topology. Amsterdam: Elsevier. 123–167 (1999; Zbl 0959.54002)], and being very short on Poincaré’s famous conjecture which was recently resolved. The shorter second part of the paper is devoted to Poincaré’s philosophical work for which the author mainly refers to Poincaré’s semi-popular books of the first decade of the 20th century. The author builds on work by the philosopher L. Althusser (1974) on “spontaneous philosophies of scientists”. While Althusser looks at these philosophies rather critical, the author interprets Poincaré’s “spontaneous philosophy” in a rather positive sense. He underlines that Poincaré’s “geometric conventionalism” which “depends on his research in geometry and topology” was way beyond the level of understanding of contemporary philosophers such as Husserl, Lenin, and Russell. Several topics discussed in this article are also included in biographies of Poincaré such as the one by J. Gray [Henri Poincaré. A scientific biography. Princeton, NJ: Princeton University Press (2013; Zbl 1263.01002)], which is not cited.
For the entire collection see [Zbl 1426.01005].
MSC:
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
01A70 Biographies, obituaries, personalia, bibliographies
30-03 History of functions of a complex variable
51-03 History of geometry
54-03 History of general topology
57-03 History of manifolds and cell complexes
00A30 Philosophy of mathematics
Biographic References:
Poincaré, Henri
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