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The emergence of objects from mathematical practices

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Abstract

The nature of mathematical objects, their various types, the way in which they are formed, and how they participate in mathematical activity are all questions of interest for philosophy and mathematics education. Teaching in schools is usually based, implicitly or explicitly, on a descriptive/realist view of mathematics, an approach which is not free from potential conflicts. After analysing why this view is so often taken and pointing out the problems raised by realism in mathematics this paper discusses a number of philosophical alternatives in relation to the nature of mathematical objects. Having briefly described the educational and philosophical problem regarding the origin and nature of these objects we then present the main characteristics of a pragmatic and anthropological semiotic approach to them, one which may serve as the outline of a philosophy of mathematics developed from the point of view of mathematics education. This approach is able to explain from a non-realist position how mathematical objects emerge from mathematical practices.

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Notes

  1. Conceptual metaphors enable metaphorical expressions to be grouped together. A metaphorical expression, on the other hand, is a particular case of a conceptual metaphor. For example, the conceptual metaphor “the graph is a path” appears in classroom discourse through expressions such as “the function passes through the coordinate origin” or “if before point M the function is ascending and after it is descending then we have a maximum.” The teacher is unlikely to say to students that “the graph is a path” but, rather, will use metaphorical expressions that suggest this.

  2. Some authors have argued that some of what are here called primary objects should be considered as objects. For example, Quine proposes referring to propositional objects rather than propositions (Quine, 1969). Another example can be found in some versions of nominalism, which consider that signs are the only mathematical objects that exist.

  3. Depending on the text analysed, some of the elements in this configuration may be missing; this would be the case of epistemic configurations of axiomatic texts.

  4. The notion of institutional cognition is associated with socio-cultural approaches to the construction of knowledge. If mathematical objects do not exist independently of people, but rather are produced intersubjectively through dialogue and convention, then this process by which objects are created constitutes a kind or form of cognition, namely institutional cognition.

  5. The terms ostensive and non-ostensive can be considered as equivalent, respectively, to the terms actual and non-actual.

  6. In phenomenological terms this would be the intentional object.

  7. According to this latter interpretation the point of view being expressed here regarding the emergence of mathematical objects would have certain commonalities with the way in which phenomenology characterizes such objects (Błaszczyk, 2005; Smith 1975).

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Acknowledgment

The research reported in this article was carried out as part of the following projects: EDU2009-08120 and EDU2010-14947 (MICINN, Spain).

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Correspondence to Juan D. Godino.

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Font, V., Godino, J.D. & Gallardo, J. The emergence of objects from mathematical practices. Educ Stud Math 82, 97–124 (2013). https://doi.org/10.1007/s10649-012-9411-0

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