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The Ontological Problems of Mathematics and Mathematics Education

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Abstract

This chapter addresses the ontological problems of mathematics and mathematics education. The ontological problem of mathematics is that of accounting for the nature of mathematical objects. The ontological problem of mathematics education concerns the chief entities in the domain, namely persons. What is the nature of persons, restricting the inquiry to the nature of their mathematical identities and their associated powers? Conversation theory is adopted as the fundamental mechanism through which both mathematical objects and persons as mathematicians are socially constructed. In consequence, the two ontological problems converge. It is claimed that the rules and conventions of mathematical culture help build up and constitute both of these types of entity. The objects of mathematics are abstracted actions encapsulating these rules. Mathematical identities are constituted, shaped and constrained through the internalization and appropriation of these rules. It is claimed that the necessity that is so characteristic of mathematics is deontic. Mathematical tasks and mathematical texts primarily employ the imperative mode, and mathematical necessity is thus based on the rules, customs and norms of the institution of mathematics.

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Notes

  1. 1.

    I use the term nature without presuming essentialism or assuming ‘natural’ states of being. I shall answer the question of how the properties and characteristics of mathematical objects and human beings as mathematical subjects are inscribed within them as a process of becoming without the presuppositions of essentialism.

  2. 2.

    Various scholars in mathematics education research make this point (Sfard, 1994; Tall, 2013; Dubinsky, 1991).

  3. 3.

    This has been used as an explicit strategy within mathematics. Henkin (1949) defines the reference of each sign within the system to be itself, in his classic proof of the completeness of the first-order functional calculus.

  4. 4.

    This is supported both within philosophy (for example, Machover, 1983) and empirically by research into the psychology of learning mathematics (Ernest, 2006; Tall, 2013).

  5. 5.

    In writing that signs create their own meanings, it is taken for granted and unwritten here that it is signs-in-use by persons that perform actions, for it is persons that use signs and create and comprehend meanings.

  6. 6.

    This answers the criticism of Cole (2008) that names cannot be constituent of mathematical objects because there are too many objects to be named.

  7. 7.

    V is the von Neumann class of hereditary well-founded sets.

  8. 8.

    Lakatos (1976) points out that the ‘logical time’ of justification often subverts the ‘chronological time’ of discovery, when the presentation of a completed proof inverts the order in which it was created.

  9. 9.

    I reject a possible fourth aspect of time. This is the consideration of mathematics as having universal validity across time and space, as this contradicts the sequential emergence of mathematics, as well as the social constructionist assumptions of this chapter.

  10. 10.

    If P is a subset of N, and 0 belongs to P, and if n belonging to P implies S(n) also belongs to P, then P=N.

  11. 11.

    Note that some of the axioms and assumptions that underpin mathematics can be contingent, as they may follow from mathematicians’ choices, albeit constrained choices. The same holds for mathematical logic.

  12. 12.

    In this chapter I use dialogue and conversation interchangeably. Some authors use ‘dialogue’ to mean a democratic and ethically more valuable type of conversation. Here I am using these terms descriptively without prescriptive attribution of greater ethical value to one over the other.

  13. 13.

    This is not to deny that persons working together in a shared practice can have different goals, such as reluctant student not fully participating in the classroom practice that the teacher is directing.

  14. 14.

    Conversation can also be used to further separate the interlocutors by asserting and reinforcing power and status differences, such as a teacher imposing order on an unruly class.

  15. 15.

    Gerofsky (1996) adds that tasks, especially ‘word problems’, also bring with them a set of assumptions about what to attend to and what to ignore among the available meanings.

  16. 16.

    There are also more open mathematical tasks such as problem solving (choose your own methods) and investigational work (pose your own questions) in school but these are not frequently encountered.

  17. 17.

    Much of the mathematics education literature concerns optimal teaching approaches intended to enhance cognitive, affective, critical reasoning or social justice gains (prescriptive). Here my concern is just with teaching as a process that enables students to learn mathematics, without problematizing the teaching itself, that is, purely descriptive.

  18. 18.

    I use the word text broadly to include whatever multimodal representations are required in the task including writing, symbolism, diagrams and even 3-D models.

  19. 19.

    Normally learners of school mathematics are not expected to specify the transformations used. Rather they are implicitly evidenced in the difference between the antecedent and the subsequent text in any adjacent (i.e., transformed) pair of texts in the sequence. In some forms of proof, including some versions of Euclidean geometry not generally included in modern school curricula, a proof requires a double sequence. The first is a standard deductive proof and the second a parallel sequence providing justifications for each step, that is specifications for each deductive rule application. Only in cases like this are the transformations specified explicitly.

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Authors and Affiliations

  1. School of Education, University of Exeter, Exeter, Devon, UK, EX1 2LU

    Paul Ernest

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  1. Paul Ernest
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Correspondence to Paul Ernest .

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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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Ernest, P. (2023). The Ontological Problems of Mathematics and Mathematics Education. 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_1

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