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The Theoretical Framework and the Subject of Study

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How We Understand Mathematics

Part of the book series: Mathematics in Mind ((MATHMIN))

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Abstract

The following sections will introduce the tools of study and the subject to be studied—mental operations of story and conceptual blending and modern algebra.

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Notes

  1. 1.

    The full title is The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities.

  2. 2.

    cf. Lakoff and Johnson (1980), Lakoff and Turner (1989), Lakoff (1993), Gibbs and Steen (1999), Lakoff and Johnson (1999)

  3. 3.

    cf. Fauconnier (1994), Fauconnier (1997), Fauconnier and Sweetser (1996)

  4. 4.

    cf. Fauconnier and Turner (1998), Coulson and Oakley (2000), Fauconnier and Turner (2002)

  5. 5.

    Conceptual integration theory (aka conceptual blending theory). Stadelmann (ibid.) uses the abbreviation MSCI (Mental Spaces & Conceptual Integration)

  6. 6.

    Mental spaces and conceptual integration

  7. 7.

    Unless we finally find an answer to “the ultimate question of life, the universe and everything” (Douglass Adams)

  8. 8.

    The capitalization may seem excessive here, but I am following a convention adopted by George Lakoff and Mark Johnson in their famous Metaphors We Live By (1980).

  9. 9.

    To learn more about image schemas see, for example, Johnson (1987), Talmy (1988), Brugman (1998), Sweetser (1990), Mandler (1992), Turner (1996).

  10. 10.

    To learn more about conceptual blending theory, see, for example, Fauconnier and Turner (2002) and Turner (2014).

  11. 11.

    More precisely, complex numbers are a vector space over the field of real (or complex) numbers (see Chap. 6 for more details).

  12. 12.

    cf., for example, the Chicago undergraduate mathematics bibliography, where we can read, “[...] classic text by one of the masters [...] wonderful exposition—clean, chatty but not longwinded, informal—and a very efficient coverage of just the most important topics of undergraduate algebra.” ( https://www.ocf.berkeley.edu/~abhishek/chicmath.htm, accessed 2017-10-06)

  13. 13.

    According to Nicolas Bourbaki (a collective pseudonym for a famous group of mathematicians), “The axiomatization of algebra was begun by Dedekind and Hilbert, and then vigorously pursued by Steinitz (1910). It was then completed in the years following 1920 by Artin, Nöther and their colleagues at Göttingen (Hasse, Krull, Schreier, van der Waerden). It was presented to the world in complete form by van der Waerden’s book (1930).” (http://www.math.hawaii.edu/~lee/algebra/history.html, accessed 2017–10-06)

  14. 14.

    Herstein (1975) does not define ordered pairs and neither did Frege (1879). Other mathematicians suggested various definitions. For example, Hausdorff (1914: 32) gave the definition of the ordered pair (a, b) as {{a,1}, {b, 2}}, but, as we argue below, this does resolve the problem implicit circularity of the static definition of a mapping.

  15. 15.

    “Mathematics may be defined as the subject in which we never know what we are talking about.” (https://en.wikisource.org/wiki/Mysticism_and_Logic_and_Other_Essays, accessed 2017–10-06)

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Authors and Affiliations

  1. Institute of English Studies, University of Wrocław, Otmuchów, Poland

    Jacek Woźny

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Woźny, J. (2018). The Theoretical Framework and the Subject of Study. In: How We Understand Mathematics. Mathematics in Mind. Springer, Cham. https://doi.org/10.1007/978-3-319-77688-0_2

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