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Prolegomena to virtue-theoretic studies in the philosophy of mathematics

  • Virtue Theory of Mathematical Practices
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Abstract

Additional theorizing about mathematical practice is needed in order to ground appeals to truly useful notions of the virtues in mathematics. This paper aims to contribute to this theorizing, first, by characterizing mathematical practice as being epistemic and “objectual” in the sense of Knorr Cetina (in: Schatzki, Knorr Cetina, von Savigny (eds) The practice turn in contemporary theory, Routledge, London, 2001). Then, it elaborates a MacIntyrean framework for extracting conceptions of the virtues related to mathematical practice so understood. Finally, it makes the case that Wittgenstein’s methodology for examining mathematics and its practice is the most appropriate one to use for the actual investigation of mathematical practice within this MacIntyrean framework. At each stage of thinking through mathematical practice by these means, places where new virtue-theoretic questions are opened up for investigation are noted and briefly explored.

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Notes

  1. MacIntyre (1981, p. 186).

  2. Note that my claim isn’t that pursuing, say, a virtue-based epistemology for mathematical knowledge as in Tanswell (2016) needs to wait until we have a perfectly adequate understanding of mathematical practice. However, the more realistic and detailed our picture of the kind of practical and intellectual virtues a mathematical knower can be expected to exhibit is, the more likely a view of this sort is to be compelling. My contention is simply that a clearer view of mathematical practice can help provide this more realistic view of the virtues surrounding the practice.

  3. This type of realistic approach—characterized by “the realistic spirit,” which looks to pay close attention to our ordinary, everyday practices—is largely inspired by the work of Wittgenstein (1953/2009, for example), who in many ways is rightly seen as a philosopher of mathematical practice. Cf., e.g., Shanker (1987), Mühlhölzer (2010), Floyd (2012), and Mühlhölzer (2014). See also Diamond (1996), Laugier (2013, xi–xii) and Methven (2015, Ch. 1) on “ordinary realism” and the realistic spirit.

  4. Floyd (2015, p. 17).

  5. Ferreirós (2016, p. 28, emphasis in the original).

  6. Carter (2019, 24).

  7. It’s not my intention to single out any particular definition of practice as being especially bad of course. On its own, the fact that there are so many attempts to say what a practice is supposed to be in the philosophy of mathematics already suggests that there’s still work to be done.

  8. See Carter (2019, pp. 25–26).

  9. Kitcher (1984, pp. 163–165).

  10. “\(\mathbf {Math[imatical]Pract[ice]} = \langle M, P, F, PM, C, AM, PS, \ldots \rangle \) (as a reminder: \(M = \) community of mathematicians, \(P =\) research program, \(F =\) formal language, \(PM =\) proof methods, \(C =\) concepts, \(AM =\) argumentative methods, \(PS =\) proof strategies)” (Van Bendegem and Van Kerkhove 2004, p. 534).

  11. See, e.g., Rouse (2003, Ch. 5) for discussion.

  12. I am, therefore, basically in agreement with Colin Rittberg that “[t]he philosophy of mathematics needs a body of knowledge which critically assesses our philosophical methods (to engage with mathematical practices and otherwise)” (Rittberg 2019, p. 14).

  13. See Cellucci (2013) for a discussion of “top-down” and “bottom-up” philosophy of mathematics.

  14. This version of the dilemma is taken almost verbatim from Pitt (2001, p. 373).

  15. Burian (2001, p. 388).

  16. Schatzki (1996, p. 89).

  17. Internal goods are also characterized by being less likely to be limited in supply and less likely to be limited to being good just for me than external goods. So, if I get a raise, that means there’s less money in the company available for you, and you’re not particularly benefited by my improved financial standing. But if I invent a new technique in painting—a technique that can perhaps be seen to be the good that it is only by those within the practice—the practice is now no less likely to develop other new techniques and you can also benefit from my advance nearly as much as I can.

  18. Using this terminology, we can say that Bourdieu (1977, p. 183) suggests, contra MacIntyre, that people take part in practices aiming for the external good of “symbolic capital.” Cf. Hicks and Stapleford (2016, p. 463).

  19. See, e.g., Knorr Cetina (1981, p. 152, 2001). Knorr Cetina’s work has been influential in sociology of science studies, but unfortunately seems not to have made its way into the literature of philosophy of mathematics yet.

  20. Knorr Cetina (2001, p. 184).

  21. Heidegger (1927/1962, par. 15:68–70).

  22. Cf. Knorr Cetina (2001, p. 188).

  23. Knorr Cetina (2001, p. 190).

  24. Occasional difficulties in applying tools or understanding how to make use of equipment are significant for Heidegger’s overall story in Being and Time as well, but for different reasons. These sorts of problems—a piece of equipment’s conspicuousness (Auffälligkeit), obtrusiveness (Aufdringlichkeit), or obstinacy (Aufsässigkeit)—can reveal the otherwise hidden “worldliness of the world” to us, but they aren’t themselves motivators of further investigations into particular objects of concern. See Heidegger (1927/1962, par. 16).

  25. Grosholz (2007, p. 47).

  26. See Knorr Cetina (2001, p. 185). In fact, she goes so far as to use the Sartrean language of the epistemic object’s being what it isn’t and not being what it is, like the “for-itself” (Sartre 1943/1993, lxv), at times.

  27. Knorr Cetina (2001, p. 194).

  28. See Knorr Cetina (1999, p. 11) for an account of science in general along these lines.

  29. E.g., one of the motivations for active research into computer-verified proofs, say, using Coq, Mizar, or Isabelle, is both to check long, complicated proofs and to provide an easily accessible store of mathematical results.

  30. Thinking in terms of the virtues seems to be becoming more prevalent in these fields in recent years as well. For example, the most recent edition of Engineering Ethics: Concepts and Cases Harris et al. (2019), one of the most widely-used textbooks on the subject, has now added sections incorporating virtue ethics into the set of tools it hopes to provide its readers.

  31. Cf. Jones (2006) and Su (2020, Ch. 1) for more on the cultivation of virtue through mathematics.

  32. MacIntyre (1981, p. 191).

  33. Cf. MacIntyre (1981, p. 64).

  34. MacIntyre (1988).

  35. Making this case is one of the main goals of MacIntyre (1988).

  36. Moral traditions are also supposed to do some of the work of justifying something that might seem like a virtue: the virtue of understanding yourself and your place within a tradition. See MacIntyre (1981, p. 223).

  37. MacIntyre (1981, p. 222).

  38. MacIntyre (1981, p. 222).

  39. MacIntyre (1981, p. 275, emphasis in the original).

  40. MacIntyre (1988, 402).

  41. See Corfield (2012, pp. 250–255) for an interesting attempt, also within a broadly MacIntyrean setting, to show that “perfected understanding” of mathematical objects is the overarching telos of mathematical research. See also Avigad (2008) on the general aim of understanding in mathematics.

  42. Albers (1994, p. 4).

  43. See, again, Jones (2006) for historical discussion of the question of how mathematics can develop a person’s individual virtues.

  44. See Grayson (2018) for an introduction.

  45. MacIntyre (1988, p. 402).

  46. Cf. MacIntyre (2006). See also Corfield (2012, §5) for rich discussion of how conflicts between mathematical traditions might be settled from a MacIntyrean perspective.

  47. See National Research Council (2013) for more along these lines.

  48. For MacIntyre’s own attempts to justify his version of the claims of tradition-based inquiry, see, e.g., MacIntyre (1988, (1990), and most recently MacIntyre (2016). For more on the approach applied to mathematics, see again Corfield (2012).

  49. MacIntyre (2016, p. 206).

  50. Rittberg (2019, p. 13) provides a long list of possible methodologies for pursuing the study of mathematical practice. The approach advocated here is closest to the one mentioned from Larvor (2010), but it’s not my intention to rule out any of the alternatives. Rather, the methodology to be considered simply suggests ways of thinking about the various objects of study focused on by these other approaches. I should note also that the approach doesn’t fit very naturally into the catalogue of Van Bendegem (2014, p. 221).

  51. See Geuss (2008, p. 10, emphasis in the original). This kind of thinking is also prominent in the work of Max Weber; see, e.g., Weber (1968, Part 2, Ch. X).

  52. The American legal realists can be seen as being realistic in a similar fashion. See, for example, Leiter (2005, pp. 50–53).

  53. Cf. Wittgenstein (1939/1989, p. 55, 103). It remains a matter of controversy, however, whether Wittgenstein really wanted nothing more than for us to look at the workings of mathematics “from close to” Wittgenstein (1953/2009, §51, emphasis in the original).

  54. See, e.g., Dummett (1959, p. 348) for the classic interpretation of this kind.

  55. See in particular the work of Juliet Floyd and Felix Mühlhölzer in the bibliography..

  56. See Wittgenstein (1939/1989, p. 39). It’s interesting to note that Wittgenstein immediately qualifies this claim, calling it an exaggeration and saying that it’s partly true and partly false.

  57. The term ‘proof chauvinism’ comes from D’Alessandro (2018), which argues that not every mathematical explanation is a proof. Lange (2017) addresses this topic as well.

  58. See, e.g., Floyd (2001) and Kienzler and Grève (2016, p. 81).

  59. See, e.g., Friedman (1975) and Simpson (1999), which is the standard reference.

  60. The attempt to minimize philosophical background assumptions also helps to make room for the “pluralism in perspectives” suggested by Michelle Friend, another author that can be seen as attempting to find the best way to be realistic when philosophizing about mathematics. See Friend (2014, p. 25).

  61. This is Felix Mühlhölzer’s translation of the passage more familiarly rendered as “Mathematics is a motley” Wittgenstein (1956/1983, III §46). Mühlhölzer argues that the term ‘motley’ has negative connotations that don’t fit well with the general thrust of Wittgenstein’s remarks about the mixture of proof methods found in mathematics. I use his translation of this remark to signal my agreement on this point. See, however, Hacking (2014, p. 57) for a contrary view.

  62. Cf. Ferreirós (2016, p. 37).

  63. Burgess (2015, p. 60).

  64. Serre et al. (1999, p. 35).

  65. See, e.g., Wittgenstein (1956/1983, I §166): “What, then—does [mathematics] just twist and turn within these rules?—It forms ever new rules: is always building new roads for traffic; by extending the network of the old ones.” See also Wittgenstein (1956/1983, III §31).

  66. E.g., Wittgenstein (1930/1975, §158, 1956/1983, I §168).

  67. Wittgenstein (1935/1958, p. 4, emphasis in the original).

  68. See Frege (1903/1960, §88). This is Frege’s way of restating the views of E. Heine and J. Thomae.

  69. Wittgenstein (1935/1958, p. 4).

  70. Wittgenstein (1953/2009, §432, emphasis in the original).

  71. Wittgenstein (1953/2009, §38).

  72. Wittgenstein (1953/2009, §116).

  73. What exactly ‘metaphysical’ is supposed to mean in this statement is the matter of a debate that needn’t be settled here. For the record, I’m roughly in agreement with Gordon Baker, who suggests that metaphysical uses try to express essences or to pass themselves off as being scientific but are not. Cf. Baker (2009, pp. 96–100).

  74. Thurston (2006, p. 167).

  75. See Shapiro (1991, p. 212) and Putnam (1980).

  76. Inglis and Aberdein (2015, (2016).

  77. Inglis and Aberdein (2016, p. 168).

  78. See Baz (2012).

  79. Baz (2012, p. 105).

  80. The methodological principles advocated in this section are similar to those accepted in ethnomethodology and the sociology of scientific knowledge. (See, e.g., Livingston 1986, p. 1; Lynch 1993, pp. 14–15), and more recently François and Van Kerkhove (2010) for ethnomethodology. Barnes et al. (1996) is a good example of the sociology of knowledge that deals with mathematics in its final chapter.) Many of the authors within these fields also take inspiration from Wittgenstein, so the resemblance isn’t coincidental. The goal of “pure description” for which ethnomethodologists put this kind of methodology to use is, however, likely to be different from the goals of philosophers of mathematics who make use of the methodology outlined here. Being a methodology though, the realistic view on offer doesn’t seek to dictate the uses to which it’s put.

  81. Cf. Toulmin (1972, pp. 505–506).

  82. Cf. Toulmin (1972, pp. 507–508) and Henriksen (1993).

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Martin, J.V. Prolegomena to virtue-theoretic studies in the philosophy of mathematics. Synthese 199, 1409–1434 (2021). https://doi.org/10.1007/s11229-020-02802-0

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