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Exemplarity in Mathematics Education: from a Romanticist Viewpoint to a Modern Hermeneutical One

Interpreting the Teaching of Infinitesimals as an Exemplary Theme

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Abstract

This paper proposes a setting of exemplarity different from the already known one, which is basically a Romanticist philosophical setting. Our general aim is to describe and explore the nature of some exemplary themes and interpretive models in advanced mathematics teaching and learning. In order to do so, we move from Romanticism towards the viewpoint of Modern Hermeneutics, by applying ideas appearing mainly in Gadamer and Ricoeur. We use this new setting as a philosophical framework, to interpret some results from two didactical research studies that have already appeared on infinitesimals.

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Notes

  1. Hans-Georg Gadamer, a leading thinker in modern (philosophical) Hermeneutics, has considered interpretation and understanding of a text as an act of meeting and being confronted with something which is radically different from us but shares with us some common “ground”. This point of view questions objectification of meaning and seems to be particularly important in epistemology and pedagogy.

  2. This is e.g. the general view in the Bourbaki exposition of mathematics, in which interpretations are confined to the Notes Historiques at the end of each book of the Elements de Mathematique.

  3. This second characteristic was first suggested to us by Ricoeur’s own view of the character of Xerxes, in the tragedy Perses (Persians) of Aeschylus, as exemplary: Xerxes is accused as having provoked the Persians’ disaster but, at the same time, he is considered to be a victim of the Gods and therefore he is “an exemplary human person” who deserves the sympathy of the Athenian audience (Ricoeur 1988, p. 363).

  4. The following extract makes things clearer: “Leibniz postulated a system of numbers having the same properties as ordinary numbers but which included infinitesimals. …Yet Leibniz’ position seems absurd on its face. The ordinary real numbers obviously have at least one property not shared by Leibniz’ desired extension. Namely, in the real numbers, there are no infinitesimals. This paradox is avoided by specifying a formal language in the sense of modern logic (mercilessly precise in the same way that programming languages for computers are). Leibniz’ principle is then reinterpreted: there is an extension of the reals that includes infinitesimal elements and has the same properties as the real numbers insofar as those properties can be expressed in the specified formal language. One concludes that the property of being infinitesimal cannot be so expressed, or, as we shall learn to say: the set of infinitesimals is an external set.” (Davis 2005, p.2. Author’s emphasis).

  5. See note 4.

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  1. Department of Mathematics, University of Patras, 26110, Patras, Greece

    Tasos Patronis & Dimitris Spanos

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  1. Tasos Patronis
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  2. Dimitris Spanos
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Correspondence to Dimitris Spanos.

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Patronis, T., Spanos, D. Exemplarity in Mathematics Education: from a Romanticist Viewpoint to a Modern Hermeneutical One. Sci & Educ 22, 1993–2005 (2013). https://doi.org/10.1007/s11191-013-9577-6

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