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On Certainty, Change, and “Mathematical Hinges”

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Abstract

Annalisa Coliva (Int J Study Skept 10(3–4):346–366, 2020) asks, “Are there mathematical hinges?” I argue here, against Coliva’s own conclusion, that there are. I further claim that this affirmative answer allows a case to be made for taking the concept of a hinge to be a useful and general-purpose tool for studying mathematical practice in its real complexity. Seeing how Wittgenstein can, and why he would, countenance mathematical hinges additionally gives us a deeper understanding of some of his latest thoughts on mathematics. For example, a view of how mathematical hinges relate to Wittgenstein’s well-known river-bed analogy enables us to see how his way of thinking about mathematics can account nicely for a “dynamics of change” within mathematical research—something his philosophy of mathematics has been accused of missing (e.g., by Robert Ackermann (Wittgenstein’s city, The University of Massachusetts Press, Amherst, 1988) and Mark Wilson (Wandering significance: an essay on conceptual behavior, Oxford University Press, Oxford, 2006). Finally, the perspective on mathematical hinges ultimately arrived at will be seen to provide us with illuminating examples of how our conceptual choices and theories can be ungrounded but nevertheless the right ones (in a sense to be explained).

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Notes

  1. Given his understanding of mathematics as “twisting and turning about within” and creating rules (Wittgenstein 1956/1983, I.§§165–168), discussion of mathematical propositions is sometimes deemed to be inappropriate in relation to Wittgenstein’s thought. See especially (Moyal-Sharrock 2004, Ch. 2) for this judgment in the present context. As a number of commentators have pointed out however [see, e.g., (Baker and 2009) Ch. VII. 4–6) and (Coliva 2010, pp. 80–82)], by the time of the Investigations Wittgenstein seems to view proposition itself as a family-resemblance concept [cf. Wittgenstein (1953/2009, §136) and Wittgenstein (1969, §320)]. His adoption of this perspective makes the objections to proposition-talk raised by Moyal-Sharrock et al. less pressing, at least for present purposes. See (Travis 2012) for a helpful discussion of the different understandings of ‘proposition’ Wittgenstein endorsed over the course of his career.

  2. See Moyal-Sharrock (2004), Coliva (2015), and Pritchard (2016) for examples of the hinge approach to epistemological problems in action. Coliva and Moyal-Sharrock (2017) is a recent collection of essays on the subject.

  3. This phrase comes from (Coliva 2020, p. 365), which will be discussed in some detail presently.

  4. For the purposes of this paper, I’ll be consistently using the now-firmly-established language of “hinges” and “hinge propositions.” Whether or not the general hinge approach to On Certainty is the most effective way to get a handle of this swath of Wittgenstein’s thinking is a larger question not taken up here.

  5. Wilson (2006, pp. 279, 566) also raises some worries about Wittgenstein’s ability to understand the complexities of conceptual change in real-world mathematical cases. Wilson (2020) is more circumspect about this judgment.

  6. See Eriksen (2020) for another recent discussion of Wittgenstein on the “dynamics of change.” Understanding the dynamics of change is of course also central to the large, on-going project of making sense of mathematical progress more generally. For a recent discussion of the state and goals of this project see Weisgerber (2022).

  7. See, e.g., Wittgenstein (1969, §§110, 166) for the grounding metaphor.

  8. I add this qualification because Coliva considers how hinges might find a place in Wittgenstein’s thought about mathematics both “on the vulgata,” where he takes mathematical statements to behave like rules, and according to a view where he gives them a more traditional understanding (Coliva 2020, p. 347). I take Shanker (1987) and Baker and (2009, Ch. VII), among others, to establish that a rule-like/normative role for mathematical statements must be taken on board for a view to count as being Wittgensteinian, so aside from parts of Sect. 3 I’ll only be speaking with the vulgar in what follows.

  9. See Moyal-Sharrock (2004, pp. 94–97) and Kusch (2016, Sect. 6) for discussion of this question.

  10. For a more detailed account of hinges as understood by advocates of the so-called framework view of On Certainty, see Coliva (2015). Coliva (2010), McGinn (1989), Moyal-Sharrock (2004), Schönbaumsfeld (2016), and Wright (1985) all advocate versions of the framework reading of the text.

  11. Given that the project of the paper is simply to consider the prospects for a hinge-based philosophy of mathematics deriving from On Certainty, I’ll not be trying to justify Wittgenstein’s general approach to certainty or our special commitment to hinges in what follows.

  12. See, for example, Wittgenstein (1969, §§57, 268, 308). Cf. Moyal-Sharrock (2004, pp. 85–87). Coliva (2020, p. 439n6) suggests that Kusch doesn’t accept this normative role as being essential to hinges, but I take the following passage to put that conclusion in question: “McGinn and Moyal-Sharrock are right to stress this grammatical/linguistic role of certainties. But they pay too little attention to the various epistemic roles that certainties do also play” (Kusch 2016, p. 138).

  13. See, e.g., Wittgenstein (1969, §§56, 231, 359, 454). Cf. Moyal-Sharrock (2004, p. 75–80). Kusch (2016, Sect. 9) argues that this characteristic isn’t shared by all hinges. One of the reasons he makes this claim, however, is that he holds mathematical propositions to be hinges and, citing (Wittgenstein 1969, §563), counts proofs as evidence for them. Since the status of mathematical propositions as hinges is in question here, it seems worth keeping this item as part of our characterization of hinges at least for now. Cf. Coliva (2020, p. 439n6).

  14. See, for example, Wittgenstein (1969, §§4, 87, 117). Cf. Moyal-Sharrock (2004, pp. 72–74). Kusch (2016) doesn’t raise any special objections to this characteristic, but once one puts hinges in the realm of the justifiable, it seems unlikely that they can be spared from being doubtable as well.

  15. See, for example, Wittgenstein (1969, §§93–97, 138, 162, 233). Cf. Moyal-Sharrock (2004, pp. 80–85). Kusch (2016, p. 135) shows Kusch to be in agreement that this is a characteristic of hinges at least some of the time.

  16. Cf. Kusch (2016, pp. 121–122). See Pritchard (2016, Chs. 3 and 4) for a thorough consideration of the merits of each of the two readings.

  17. Cf. Moyal-Sharrock (2004, p. 119).

  18. There are subtle questions about what exactly justification is supposed to come to here raised by Wittgenstein’s repeated characterization of mathematics as “akin both to what is arbitrary and to what is non-arbitrary” (Wittgenstein 1967, §358). See, e.g., (Baker and 2009, Ch. VII.11–12) for discussion. The main question under discussion in what follows is addressable without fully engaging these other difficult questions though, so I’ll set them aside wherever possible.

  19. Wittgenstein (1969, §563).

  20. See Coliva (2020, p. 351) and Kusch (2016, p. 128): “[A]ll mathematical sentences are certainties.” (‘Certainties’ is Kusch’s way of referring to hinges.) On the difference between ‘subjective’ and ‘objective’ certainty, see, e.g., Wittgenstein (1969, §194).

  21. See Coliva (2020, pp. 347–348) for the later worry.

  22. See Coliva (2020, p. 352).

  23. Another important part of Coliva’s case is her explanation of why the following passage doesn’t immediately settle the question: “The mathematical proposition has, as it were officially, been given the stamp of incontestability. i.e.: “Dispute about other things; this is immovable—it is a hinge on which your dispute can turn” Wittgenstein (1969, §655). See Coliva (2020, pp. 359–360). I think she’s correct that this passage on its own is not enough to put the debate to rest, so I don’t rely on it here.

  24. Wittgenstein (1956/1983, III.§18)

  25. That is, if the system’s consistency isn’t called into question in the process.

  26. Given that our system of naming numbers is set up appropriately. See (Kim 2021) for a discussion of what an appropriate setup might look like.

  27. Whitehead and Russell (1927\(*\)110.643).

  28. Cf. Mühlhölzer (2020, pp. 193–195).

  29. Church (1956, p. 76).

  30. See Coliva (2020, p. 352).

  31. See Coliva (2020, pp. 352–354).

  32. “In some sense” because it’s not clear how the \(1057\times 216=228,312\) learned prior to, say, \(2\times 2=4\) would relate to the equation \(1057\times 216=228,312\) as we know and relate to it now.

  33. See, e.g., Wittgenstein (1969, §§106, 264).

  34. This feature of the method to be discussed is noted, but not commented on at Coliva (2020).

  35. See, e.g., https://www.popularmechanics.com/science/a32131826/ancient-multiplication-method/ (accessed 13 September 2022).

  36. This sort of check is especially valuable because all the counting involved makes the intersection-counting technique more liable to error than our normal method of multiplying.

  37. A practice that used Roman numerals instead of Arabic ones for calculation would similarly have a different collection of equations with a special logical status supporting justification within the practice. E.g., to multiply MLVII by CCXVI one would need to know that V times V is XXV, that L times V is CCL, and that five Cs add up to one D, and so on. These considerations are perhaps mundane, but they are part of a realistic investigation of this mathematical practice nonetheless. See Detlefsen et al. (1976) for more on techniques of Roman numeral arithmetic.

  38. Something like this worry seems to lie behind the general thrust of Coliva’s arguments. See, e.g., Coliva (2020, pp. 347–348).

  39. See, e.g., Baker (2009), Davis and Hersh (1981), Lange (2017, Ch. 8), and Martin (2022) for discussion of the phenomenon.

  40. See van Gennip (2003) for discussion of the timeline of the manuscripts mined for the content of On Certainty.

  41. See, e.g, Wittgenstein (1953/2009, Part II.xi,§165) and Wittgenstein (1977, §10).

  42. Frege (1893/1964, p. xvii).

  43. Pritchard (2021) provides a recent overview of this account.

  44. This is likely the line Frege, for example, would take given the rest of his discussion in the Grundgesetze.

  45. I’m thinking especially of Juliet Floyd’s recent work on Wittgenstein’s philosophy of mathematics that appears to suggest that Wittgenstein may have once had some sympathy for Frege’s line of thought in at least some form. See Floyd (2021, Sec. 3.4) and Wittgenstein (1930/1975, §163).

  46. See, e.g., Wittgenstein (1953/2009, §216), where Wittgenstein suggests that there is “no finer example of a useless proposition” that the law of identity.

  47. Cf. The discussion in Wittgenstein (1939/1989, pp. 41–42) considering the ways that even \(25\times 25=625\) might be deemed mathematical or not depending on usage.

  48. See, e.g., Wittgenstein (1930/1975, §163): “‘Every symbol is what it is and not another symbol.’”

  49. See Poincaré (1914/2001, pp. 466–467). Cf. Hilbert (1899/1980, pp. 39–40) and Russell (1903, Sec. 1). Wilson (2020, p. 39) suggests, however, that this “if-thenist” picture of mathematical research is “deeply insincere.” For more on that line of thought, see Sect. 7.

  50. Sometimes logicians are interested in studying systems that do assert their own consistency though. E.g., Peano Arithmetic (PA) + “PA is consistent” is a so-called consistency extension of Peano Arithmetic that has been investigated. See, e.g., Franzén (2004, Ch. 13). The consistency of this larger system must still be asserted somehow too though if Poincaré’s idea is on the right track.

  51. Pedersen (2021) argues that this type of consistency claim is in fact a hinge (or a “cornerstone proposition” as he puts it).

  52. In order to make this commitment into a more evidently mathematical claim, it could be reformulated as a commitment to the claim that the axioms have a model.

  53. See, e.g., Floyd (2021, Sec. 3.6) for discussion.

  54. See, e.g., Wittgenstein (1939/1989, p. 210).

  55. Again, see Pedersen (2021) here.

  56. This is where Coliva thinks it’s wisest for us to look for mathematical hinges as well. See Coliva (2020, Sec. 6). The main differences between the account on offer here and Coliva’s own suggestions are that the present account is able to incorporate Wittgenstein’s insight that mathematical statements seem to behave uniformly in a rule-like fashion (see Friederich (2011) for persuasive way of making this case); it posits more hinges than just axioms; and it attempts to give a more detailed explanation of the difference between a proposition’s being listed as an axiom when stating a theory and its playing a truly axiomatic role within the practice. (Coliva briefly touches on this distinction at Coliva (2020, p. 363).)

  57. Cf. Hilbert (1899/1980, pp. 40–41). See also Resnik (1974).

  58. Cf. Coliva (2020, p. 363).

  59. Wittgenstein (1956/1983, VII.§73). This passage from Remarks on the Foundations of Mathematics is from the latest manuscript year to be represented in the work, 1944: it appears at MS 124, 197.

  60. Wittgenstein (1956/1983, III.§46, emphasis in the original) (in Felix Mühlhölzer’s amended translation).

  61. Wittgenstein (1956/1983, VII.§73).

  62. cf. van Gennip (2003) for the relation between “Cause and Effect” and the On Certainty manuscripts.

  63. Wittgenstein (1993, p. 397, emphasis in the original).

  64. See Bocompagni (2012, Sec. 2). The metaphor of a stream of thought is from James (1890, Ch. IX).

  65. See MS 165, 24–25 and Bocompagni (2012, p. 3).

  66. Wittgenstein (1993, p. 397, emphases added).

  67. Wittgenstein (1969, §96).

  68. Wittgenstein (1969, §97).

  69. Wittgenstein (1969, §99). See also Wittgenstein (1969, §63).

  70. See, e.g., Quine (1951, p. 41).

  71. See, e.g., the essays in Arrington and Glock (1996).

  72. I essentially agree with Hacker (1996) when he suggest that there is “proximity at great distance” between the two. See also Moyal-Sharrock (2000) especially on the danger of bringing a Quinian perspective to bear on the river-bed analogy.

  73. See Wilson (2020, p. 54) for a similar point about all “the mathematics science needs.”

  74. See also Friedman (2001) and its notion of a “relativized a priori” for a better object of comparison to Wittgenstein’s late views.

  75. “I can enumerate various typical cases, but not give any common characteristic” (Wittgenstein 1969, §674).

  76. Arnold (2000, p. 404), cited in Wilson (2020, p. 57) where Wilson also connects this idea to Wittgenstein’s river-bed analogy.

  77. Cf. Travis (2011, p. 52) in relation to the question of when a system of concepts would be recognized by us as color-concepts. The idea that “the parochial” must do this kind of work for us in general plays a central role in Travis’s reading(s) of Wittgenstein. See also Thomasson (2020a, pp. 73–76) and (2020b) for discussions of how these judgments about sameness play a role in other Wittgenstein-inspired theorizing.

  78. See Herstein (1964, p. 83) and Lang (2002, p. 83).

  79. See Kleiner (1996) for more on the history of the abstract ring concept.

  80. See Gross et al. (2013). For Conway’s problem, see https://oeis.org/A248380/a248380.pdf (accessed 13 September 2022).

  81. See, e.g., Corfield (2012), MacIntyre (1988), and Martin (2021) for more on the role of “traditions” in guiding inquiry.

  82. See Burgess et al. (2020) for an overview of the conceptual engineering project.

  83. Haslanger (2000, p. 35).

  84. Hodges (1993, p. 10) suggests one reason that we might want to make this choice.

  85. Although, see Mortensen (1995).

  86. Cf. Frege (1893/1964, p. xvii). We are here again considering the hinge concept in general and not only as Wittgenstein would want to apply it.

  87. See, e.g., Wittgenstein (1969, §92). See Floyd (2021, Sec. 1.2) for more discussion of Wittgenstein on coming to see aspects in mathematics.

  88. See Wittgenstein (1969, §92). It’s worth noting that this comment comes just before the river-bed analogy. I’ll be suggesting below that it’s the realization, “Oh, that’s how things must be,” that leads to the shifting of the river-bed in this sort of case.

  89. Wittgenstein (1939/1989, p. 24).

  90. Wittgenstein (1969, §256). He doesn’t always seem to be settled on his views of this matter though. See, e.g., Wittgenstein (1977, III.§124).

  91. Cf. “How must we look at this problem in order for it to become solvable?” (Wittgenstein 1977, II.§11).

  92. Wittgenstein (1953/2009, §1)

  93. Weyl (1955/2009, p. vii). Mark Wilson often appeals to this example as well.

  94. Wittgenstein (1969, §206).

  95. See (Manders 1989) and more recently (Bellomo 2021) for further discussion of the advantages of domain extension.

  96. This example has been discussed by philosophers in numerous places: see, e.g., Lange (2017, pp. 290–292), Leng (2011, p. 68), Steiner (1978, pp. 18–19), Waismann (1954/1982, pp. 29–30), and Wilson (2006, pp. 313–314). See (Shanker 1987, p. 338) for a very different take on the significance of the example.

  97. Gamelin (2001, p. 146).

  98. Ponnusamy and Silverman (2006, p. 188).

  99. “Maclaurin series” is just another name for the Taylor series expansion about zero.

  100. Needham (1997, pp. 64–67).

  101. Waismann (1954/1982, p. 30).

  102. See, e.g., Nahin (1998).

  103. Cf. Wittgenstein (1969, §206).

  104. Wilson (2020, p. 4).

  105. See (Wilson 2020, p. 5).

  106. Wilson himself thinks that the profession of the “if-thenist” doctrine is primarily a way for mathematicians to avoid pestering questions from philosophers anyway. See Wilson (2020, p. 39).

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  1. Department of Philosophy, Western Michigan University, Kalamazoo, MI, 49008, USA

    James V. Martin

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Martin, J.V. On Certainty, Change, and “Mathematical Hinges”. Topoi 41, 987–1002 (2022). https://doi.org/10.1007/s11245-022-09834-w

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