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Philosophy, Rigor, and Axiomatics in Mathematics: Imposed or Intimately Related?

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Ongoing Advancements in Philosophy of Mathematics Education

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Abstract

Two main tendencies of mathematical development in the nineteenth and twentieth centuries seem to have been rigor and formalization. Rigor and formalization took place on an axiomatic basis leading to more abstraction. A Euclidean type of axiomatic model became a model of mathematics even for constructively developed analysis. Although rigor and axiomatic method differ, and rigor does not need to be based on axiomatic method, in practice, that basis has been required for mathematical validity. Among different interpretations made about it, some have explained it as a mathematical necessity, while others have attributed it to the philosophical underpinnings of formalism and foundationism. The debate has motivated me to examine how philosophy, rigor, and axiomatics are related. It seems that philosophy has a distant but decisive impression on the nature of mathematical knowledge, whereas rigor and axiomatic seem to be relatively internal to mathematics. However, because such trends have mostly been associated with European tradition, they need to be examined in the light of non-European traditions, including Hindu mathematical traditions, which have made significant contributions to mathematics without any axiomatic proof or philosophical presumption of absolute certainty.

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Author information

Authors and Affiliations

  1. Tribhuvan University, Kathmandu, Nepal

    Min Bahadur Shrestha

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  1. Min Bahadur Shrestha
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Editors and Affiliations

  1. Mathematics Department, São Paulo State University Campus of Rio Claro, Rio Claro, Brazil

    Maria Aparecida Viggiani Bicudo

  2. Mathematics Department Hostos CC, City University of New York, New York, NY, USA

    Bronislaw Czarnocha

  3. Department of Teaching and Curriculum, Federal University of Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil

    Maurício Rosa

  4. LaGuardia Community College, Long Island, NY, USA

    Małgorzata Marciniak

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Shrestha, M.B. (2023). Philosophy, Rigor, and Axiomatics in Mathematics: Imposed or Intimately Related?. In: Bicudo, M.A.V., Czarnocha, B., Rosa, M., Marciniak, M. (eds) Ongoing Advancements in Philosophy of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-35209-6_20

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  • DOI: https://doi.org/10.1007/978-3-031-35209-6_20

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  • Online ISBN: 978-3-031-35209-6

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