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What Are Mathematical Cultures?

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Cultures of Mathematics and Logic

Part of the book series: Trends in the History of Science ((TRENDSHISTORYSCIENCE))

Abstract

In this paper, I will argue for two claims. First, there is no commonly agreed, unproblematic conception of culture for students of mathematical practices to use. Rather, there are many imperfect candidates. One reason for this diversity is there is a tension between the material and ideal aspects of culture that different conceptions manage in different ways. Second, normativity is unavoidable, even in those studies that attempt to use resolutely descriptive, value-neutral conceptions of culture. This is because our interest as researchers into mathematical practices is in the study of successful mathematical practices (or, in the case of mathematical education, practices that ought to be successful).

I first distinguish normative conceptions of culture from descriptive or scientific conceptions. Having suggested that this distinction is in general unstable, I then consider the special case of mathematics. I take a cursory overview of the field of study of mathematical cultures, and suggest that it is less well developed than the number of books and conferences with the word ‘culture’ in their titles might suggest. Finally, I turn to two theorists of culture whose models have gained some traction in mathematics education: Gert Hofstede and Alan Bishop. Analysis of these two models corroborates (in so far as two instances can) the general claims of this paper that there is no escaping normativity in this field, and that there is no unproblematic conception of culture available for students of mathematical practices to use.

I am grateful to Paul Ernest, as well as to two anonymous referees and the editor of this collection for comments on drafts of this paper. I am also grateful to the organisers of Cultures of Mathematics and Logic (9–12 November 2012, Institute for Logic and Cognition, Sun Yat-Sen University, Guangzhou, China)

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Notes

  1. 1.

    Culture and Anarchy, preface (Cambridge University Press 1978 reprint of the second (1875) edition, p. 6). Also of interest is Schiller (1794), because he attempted to explain in detail how exposure to high art can do its ennobling work.

  2. 2.

    There is, for example, more than a trace of Matthew Arnold in Paul Lockhart’s Mathematician’s Lament (Bellevue Literary Press, 2009). Paul Ernest makes an Arnold-like argument in ‘Why Teach Mathematics?’ (in White and Bramall (2000) Why Learn Maths? London: Institute of Education, 2000), including a distinction between capability and appreciation that emphasises the value of mathematics as cultural achievement. In the closing remarks of his presentation at the third London conference on mathematical cultures (2014), Ernest advocated inspiring pupils with “the poetry of mathematics”.

  3. 3.

    See Skovsmose’s principal programmatic work Towards a Philosophy of Critical Mathematics Education (1994) Dordrecht: Kluwer. For his more recent reflections, see the interview in Alrø, Ravn and Valero (eds) Critical Mathematics Education: Past, Present and Future pp. 1–9 (2010), in which he relates his thinking to philosophers associated with critical theory such as Habermas, Adorno and Foucault. Naturally, his relation to these figures is not uncritical. From such perspectives, the present argument (that our interests in mathematics and mathematics education must inevitably erode the is/ought distinction) will appear as a naïve statement of the obvious.

  4. 4.

    Harvard University Peabody Museum of American Archeology and Ethnology Papers 47.

  5. 5.

    On the is/ought relation, see Hegel (1821) Philosophy of Right.

  6. 6.

    Kroeber & Kluckhohn 1952 p. 181.

  7. 7.

    Kroeber & Kluckhohn 1952 p. 340.

  8. 8.

    Parsons, Talcott (1949) Essays in Sociological Theory. Glencoe, IL, p. 8. Similarly, Useem & Useem define culture as, “…the learned and shared behavior of a community of interacting human beings” (Useem, J., & Useem, R. 1963 p. 169). That is, culture is patterned behaviour rather than ideas and values, and is reproduced non-biologically.

  9. 9.

    Banks, J.A., Banks, & McGee, C. A. (1989). Multicultural education. Needham Heights, MA: Allyn & Bacon.

  10. 10.

    See Larvor’s review of The Philosophy of Mathematical Practice Paolo Mancosu (ed.) OUP 2008 in Philosophia Mathematica (2010) 18(3): 350–360 for a representative list.

  11. 11.

    http://www.philmathpractice.org/.

  12. 12.

    Oxford University Press, 2008.

  13. 13.

    Wilder offered this definition of culture: “We use [the term ‘culture’] in the general anthropological sense… In this sense, a culture is the collection of customs, beliefs, rituals, tools, traditions, etc., of a group of people… It is not the use of the term as in “a cultured person” that we have in mind.” Introduction to the Foundations of Mathematics (John Wiley; second ed. 1965 (first published 1952)) p. 282.

  14. 14.

    Pergamon Press, 1981.

  15. 15.

    For a description of the series, see “The Mathematical Cultures Network Project” 2012 Journal of Humanistic Mathematics 2 (2): 157–160.

  16. 16.

    This claim is argued many times over in the literature; something like a locus classicus is Rav’s ‘Why do we prove theorems?’ Philosophia Mathematica (1999) 7 (1): 5–41.

  17. 17.

    In Mancosu 2008

  18. 18.

    Cambridge University Press, 1999.

  19. 19.

    Netz makes this observation, “…mathematics survived even Christianity, and in the totally dissimilar culture of Islam the very same mathematics went on.” (p. 237, emphasis added).

  20. 20.

    1999, p. 3.

  21. 21.

    Mancosu 2008, pp. 80–133.

  22. 22.

    Mancosu 2008, p. 67.

  23. 23.

    https://youtu.be/umuKvJFR_7U see also François, Karen & Stathopoulou, Charoula (2012). ‘In-Between Critical Mathematics Education and Ethnomathematics. A Philosophical Reflection and an Empirical Case of a Romany Students’ group Mathematics Education.’ Journal for Critical Education Policy Studies, 10(1), 234–247 ISSN 1740-2743.

  24. 24.

    Paul Andrews made extensive reference to Hofstede’s scheme in his international comparison of school mathematics teaching; Albrecht Heeffer used part of Bishop’s framework in his presentation of the Abbaco mathematical culture. Slides and video recordings of their talks and Bishop’s keynote address are available on the Mathematical Cultures project website https://sites.google.com/site/mathematicalcultures/.

  25. 25.

    International Journal of Intercultural Relations Vol 10 pp. 301–320, 1986. In an earlier paper, Hofstede defines culture as “…the collective programming of the mind which distinguishes the members of one category of people from another.” (p. 51). Hofstede, G. (1984). ‘National cultures and corporate cultures’. In L.A. Samovar & R.E. Porter (Eds.), Communication Between Cultures. Belmont, CA: Wadsworth. Note that this definition is about categorising people rather than reproducing ideas (though the ideas are there, under the computer metaphor).

  26. 26.

    He is not inconsistent in this; he prepared for this eventuality when writing the 4-dimensional model, “There is nothing magic about the number of four dimensions…” (p. 306).

  27. 27.

    Critique of Pure Reason (1787) A80/B106ff for the table of categories; A70/B95ff for the ‘clue’ found in logical theory; A95-130/B129-169 for his detailed account.

  28. 28.

    See Arnold (1875) Culture and Value chapter iv ‘Hebraism and Hellenism’.

  29. 29.

    See Nietzsche, principally the Genealogy of Morals (1887).

  30. 30.

    See Max Weber (1930) The Protestant Ethic and the Spirit of Capitalism.

  31. 31.

    This essay focuses on Bishop’s third chapter, ‘The Values of Mathematical Culture’. Heeffer’s discussion (alluded to above) draws on other parts of Bishop’s book. White’s scheme included technological values as a separate category; Bishop does not. White’s magnum opus is (1959) The evolution of culture: the development of civilization to the fall of Rome. New York: McGraw Hill.

  32. 32.

    Hardy, G.H. A Mathematician’s Apology. (2004) [1940]. Cambridge: University Press. Note the date! Hardy published his thoughts on the uselessness of mathematics at the outbreak of war.

  33. 33.

    Peirce, C.S. ‘The logic of relatives’ The Monist Vol. VII. January, 1897. No. 2. p. 192. I learned of this passage from Hacking, I. Why is there philosophy of mathematics at all? Cambridge: Cambridge University Press, 2014, p. 255.

  34. 34.

    See, for example, Amir Alexander’s Duel at Dawn. Harvard University Press, 2010.

  35. 35.

    https://youtu.be/FE6m-z61VTs, accessible from https://sites.google.com/site/mathematicalcultures/.

  36. 36.

    Why Learn Maths? London: Institute of Education, 2000, pp. 6–7.

  37. 37.

    ‘Mathematics and Mysticism, Name Worshipping, Then and Now’ Jean-Michel Kantor Theology and Science Vol. 9, Issue 1, 2011.

  38. 38.

    Plato Republic 525 ff. For a useful discussion of this in a contemporary context, see Peter Huckstep ‘Mathematics as a Vehicle for Mental Training’ in Why Learn Maths? London: Institute of Education, 2000, pp. 88–91.

  39. 39.

    Republic 511a.

  40. 40.

    Though see Brian Rotman’s claim that contemporary Platonism is a “theological obfuscation of number” (Ad Infinitum: the ghost in Turing’s machine. Stanford: Stanford University Press, 1993, p. xii.

  41. 41.

    Hacking, I. Why is there philosophy of mathematics at all? Cambridge: Cambridge University Press, 2014, chapter five.

  42. 42.

    Oswald Spengler The Decline of the West Volume I: Form and Actuality. Tr. C. F. Atkinson. London: George Allen & Unwin Ltd., 1926, p. 59.

  43. 43.

    See, for example, Greiffenhagen, C. (2014), ‘The materiality of mathematics: Presenting mathematics at the blackboard’. The British Journal of Sociology. The abstract begins “Sociology has been accused of neglecting the importance of material things in human life and the material aspects of social practices.” and goes on to describe a ‘material turn’ in sociology.

  44. 44.

    The Novembertagung series of meetings is intended for young historians and philosophers of mathematics.

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Author information

Authors and Affiliations

  1. University of Hertfordshire, College Lane, Hatfield, AL10 9AB, UK

    Brendan Larvor

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  1. Brendan Larvor
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Correspondence to Brendan Larvor .

Editor information

Editors and Affiliations

  1. Institute of Logic and Cognition, Sun Yat-Sen University, Guangzhou, China

    Shier Ju

  2. Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany

    Benedikt Löwe

  3. Fachbereich Philosophie, Universität Konstanz, Konstanz, Germany

    Thomas Müller

  4. Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China

    Yun Xie

Appendix: Recent Conferences on Mathematical Cultures

Appendix: Recent Conferences on Mathematical Cultures

24–26 October 2002. Perspectives on Mathematical Practices, Brussels, Belgium.

2006–2010. The PhiMSAMP network funded by the Deutsche Forschungsgemeinschaft with events and nodes in seven countries:

  1. 1.

    PhiMSAMP-0: Bonn, Germany, 8 May 2005.

  2. 2.

    GAP.6 Workshop: Towards a new epistemology of mathematics (= PhiMSAMP-1). Berlin, Germany, 14–16 September 2006.

  3. 3.

    PMP 2007. Perspectives on Mathematical Practices II, Brussels, 26–28 March 2007.

  4. 4.

    PhiMSAMP-2. Utrecht, The Netherlands, 19–21 October 2007.

  5. 5.

    18th Novembertagung.Footnote 44 Bonn, Germany, 1–4 November 2007.

  6. 6.

    PhiMSAMP-3. Vienna, Austria, 16–18 May 2008.

  7. 7.

    Foundations of the Formal Sciences VII . Brussels, Belgium, 21–24 October 2008.

  8. 8.

    PhiMSAMP-4. Brussels, Belgium, 24–25 October 2008.

  9. 9.

    PhiMSAMP-5. Hatfield, UK, 29–30 June 2009.

  10. 10.

    Two Streams in the Philosophy of Mathematics: Rival Conceptions of Mathematical Proof. Hatfield, UK, 1–3 July 2009.

  11. 11.

    PhiMSAMP-6 . Utrecht, The Netherlands, 22–23 April 2010.

27–29 May 2010. Mathematics as Culture and Practice, Bielefeld, Germany.

9–11 December 2010. First International Meeting of the Association for the Philosophy of Mathematical Practice, Brussels, Belgium

2–3 December 2011. Mathematics as Culture and Practice II, Greifswald, Germany.

2010–2012. Symposia on mathematical practice and cognition at the conventions of the Society for the Study of Artificial Intelligence and Simulation of Behaviour.

  1. 1.

    Symposium on Mathematical Practice and Cognition, Leicester, UK, 29–30 March 2010.

  2. 2.

    Symposium on Mathematical Practice and Cognition II, Birmingham, UK, 2–4 July 2012.

2012–2014. Mathematical Cultures: workshop series funded by the UK Arts and Humanities Research Council and the London Mathematical Society:

  1. 1.

    Mathematical Cultures 1: Contemporary mathematical cultures, London, UK, 10–12 September 2012.

  2. 2.

    Mathematical Cultures 2: Values in mathematics, London, UK, 17–19 September 2013.

  3. 3.

    Mathematical Cultures 3: Mathematics in public culture, London, UK, 10–12 April 2014.

9–12 November 2012. Cultures of Mathematics and Logic, Guangzhou, China (the origin of this book).

20–23 September 2013. Foundations of the Formal Sciences VIII: History and Philosophy of Infinity, Cambridge, UK.

3–4 October 2013. Second International Meeting of the Association for the Philosophy of Mathematical Practice Urbana-Champaign, USA.

22–25 March 2015. Cultures of Mathematics IV New Delhi, India.

2–4 November 2015. Third International Meeting of the Association for the Philosophy of Mathematical Practice, Paris, France.

Most of these meetings have associated books of proceedings.

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Larvor, B. (2016). What Are Mathematical Cultures?. In: Ju, S., Löwe, B., Müller, T., Xie, Y. (eds) Cultures of Mathematics and Logic. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-31502-7_1

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