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On Mathematical Validity and Its Human Origins

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Ongoing Advancements in Philosophy of Mathematics Education

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Abstract

This chapter suggests the desirability of a fully integrated philosophy of mathematics education that builds on several distinct, mutually compatible foundational pillars. These pillars have their intellectual bases in different philosophical school of thought. The discussion focuses on one aspect of those epistemological foundations – the interplay between the human origins and uses of mathematics and its objective truth and validity. Various philosophical trends in education have centralized just one of these aspects, often to the extent of denying or dismissing the other. I argue here for their compatibility, maintaining that objective mathematical truth and the fact of culturally situated human mathematical invention should both be guiding teaching and learning, with neither diminished in importance. Several meanings given to the “why” of mathematics are discussed – logical and empirical reasons that underlie mathematical truths and relationships, sociocultural and contextual reasons for develo** and teaching mathematics, and the in-the-moment experiences afforded students to motivate their study. Some sources of cultural relativism, historical change, and “fallibility” in mathematics are identified, and the value of “mistakes” in powerful mathematical problem solving is highlighted. My goal is to argue that an intellectually sound philosophy of mathematics education must incorporate all of the aforementioned features of mathematics and its practice, dismissing none.

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Notes

  1. 1.

    However, there have been historical periods of disagreement, followed by radical shifts in belief among mathematicians, over some metamathematical issues, for example: whether Euclid’s postulates are “self-evident truths,”, whether negative numbers “really exist,”, or whether mathematics can be formalized in a complete and consistent way (e.g., Kline, 1980).

  2. 2.

    Of course, chess is a finite game and thus such a question has a definite answer—though we may not know now what the answer is. In contrast, mathematics involves infinite sets, where the work of Gödel demonstrates that with a given set of rules of inference, not every proposition can be resolved as either true or false. The notion of “truth” extends beyond what is provable with the specified rules of inference.

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Authors and Affiliations

  1. Departments of Mathematics and Physics, Rutgers University, New Brunswick, NJ, USA

    Gerald A. Goldin

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  1. Gerald A. Goldin
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Correspondence to Gerald A. Goldin .

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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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Goldin, G.A. (2023). On Mathematical Validity and Its Human Origins. 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_7

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  • DOI: https://doi.org/10.1007/978-3-031-35209-6_7

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  • Online ISBN: 978-3-031-35209-6

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