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Indeterminacy, coincidence, and “Sourcing Newness” in mathematical research

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Abstract

Far from being unwelcome or impossible in a mathematical setting, indeterminacy in various forms can be seen as playing an important role in driving mathematical research forward by providing “sources of newness” in the sense of Hutter and Farías (J Cult Econ 10(5):434–449, 2017). I argue here that mathematical coincidences, phenomena recently under discussion in the philosophy of mathematics, are usefully seen as inducers of indeterminacy and as put to work in guiding mathematical research. I suggest that to call a pair of mathematical facts (merely) a coincidence is roughly to suggest that the investigation of connections between these facts isn’t worthwhile. To say of this pair, “That’s no coincidence!” is to suggest just the opposite. I further argue that this perspective on mathematical coincidence, which pays special attention to what mathematical coincidences do, may provide us with a better view of what mathematical coincidences are than extant accounts. I close by reflecting on how understanding mathematical coincidences as generating indeterminacy accords with a conception of mathematical research as ultimately aiming to reduce indeterminacy and complexity to triviality as proposed in Rota (in: Palombi (ed) Indiscrete thoughts, Birkhäuser, 1997).

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Notes

  1. Hilbert (1926/1983, p. 191).

  2. [Dante, Paradiso, Canto XXXII, 52–56].

  3. On mathematical coincidence, see, e.g., Baker (2009), Lange (2010, 2017, Ch. 8). See also Davis (1981).

  4. [Homer, Odyssey 11.487–503].

  5. The idea of “sourcing newness” is drawn from Hutter and Farías (2017).

  6. Dewey (1938, pp. 104–105, emphasis in the original).

  7. See Dewey (1938, pp. 105–106). Dewey’s use of ‘indeterminate situation’ in this semi-technical sense helps block Russell’s “counterexample” that, according to Dewey, a bricklayer’s dealings with a pile of bricks is a form of inquiry (Russell, 1946/1961, p. 823). See Gale (1959) for more on Russell on Dewey on inquiry.

  8. Hutter and Farías (2017).

  9. Something like this induced indeterminacy may also be familiar as what the character Paul aims to produce in his rented apartment in Last Tango in Paris.

  10. Hutter and Farías (2017, pp. 438–440, pp. 441–442, p. 444).

  11. See Wittgenstein (1953/2009, §189) for discussion of this sense of ‘determinacy’.

  12. See, e.g., Bartle and Sherbert (2011, Section 6.3).

  13. See, e.g., Mac Lane and Birkhoff (1999, Ch. III.6).

  14. See Saccheri (1733/2014, Book I).

  15. See Corfield (2003, p. 152) for more on the role of axioms in experimentation and creativity within mathematical research.

  16. Cf. Hatcher (2002, p. 21): “Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces.” See also the notion of “transport of structure” from Bourbaki (1968, Ch. IV, §1).

  17. This is one way to view the resurgence of interest in type-theory and constructive logic initiated by the so-called univalent foundations program. See Univalent Foundations Program (2013).

  18. See, e.g., Lakatos (1976).

  19. See Shapiro and Roberts (2021) for discussion of open texture and mathematics more generally.

  20. See Tanswell (2018) for a compelling case that the project of conceptual engineering has much to learn from mathematical practice. See, e.g., [Burgess et al.(2020)] for a general introduction to the conceptual engineering project.

  21. Of course, none of this indeterminacy quite suggests the kind of “ontological indeterminacy” investigated in more metaphysically-focused literature: see, e.g., Rosen and Smith (2004) or Barnes and Williams (2011) for more on this purported type of indeterminacy. (See, e.g., Lewis (1986, p. 212) for the ‘purported’ qualification.).

  22. In addition to indeterminacy, Jones suggests that this kind of situation also “induces obsessive and anti-social behaviour”.

  23. See Aberdein (2010) for an attempt to mine the mistaken half of this kind of argument pair for a deeper understanding of mathematical error, mathematical fallacies, and the role of and justification for informal reasoning in mathematics.

  24. Another way talk of coincidence can arise in mathematics is when a bad argument “coincidentally” reaches a true conclusion. As Aberdein (2010) shows, this kind of coincidence can be usefully investigated, but it is not of the type central to recent discussions of mathematical coincidence, so I’ll set it aside in what follows.

  25. So-called “monstrous moonshine” Conway and Norton (1979) is another more famous and much more complicated source of examples of surprising (non)coincidences relating facts about finite groups and modular functions. See, e.g., Gannon (2006).

  26. Davis (1981, p. 312).

  27. Guy (1990, p. 10).

  28. Guy (1988, p. 699).

  29. This operation is named after Dattathreya Kaprekar, who studied it and discovered a number of facts about it. See, e.g., Kaprekar (1955). Kaprekar also gave the name ‘harshad’ to the harshad numbers in Example 2.

  30. For four-digit numbers, the operation always ends with 6174. Neither two-digit nor five-digit numbers have a repeating value (or “kernel”). See Nishiyama (2012, p. 370).

  31. A similar procedure can be used to show that 6174 is the fixed point of the Kaprekar operation applied to normal four-digit numbers. See Nishiyama (2012, pp. 364–365).

  32. This may also suggest, as Fine (1994) has argued regarding essence, that necessity is too “coarse-grained” of a notion to capture the phenomena and so an account of mathematical coincidences as not stemming from the “essences” of the objects involved is needed instead. Taking this line would produce a rather different view than the one to be investigated here, and will have to wait for another opportunity to be explored in any case.

  33. See Davis (1981, pp. 312–313).

  34. See Augustine (1961, I.xvii). Cf. the fairy tale discussed by Wittgenstein and cited in Säätelä (2011, p. 173), where the prince asks a smith to bring him a “hubbub”.

  35. See Muntersbjorn (2003) for an argument that this is a false dilemma anyway. Muntersbjorn argues that mathematics is best thought of as being “cultivated” rather than invented or discovered.

  36. Cf. Gowers (2011) and the commentary in Rosen (2011).

  37. See, e.g., Wittgenstein (1930/1975, §158, 1956/1983, I §168).

  38. See, e.g., Martin (2020, §5.1).

  39. “[L]es saints [\(\ldots \)] disent en parlant des choses divines qu’il faut les aimer pour les connaître”.

  40. Note that Guy (1988, p. 698), e.g., disagrees with this kind of view. He suggests that early coincidences like the ones seen in Example 2 and 3 above are actually “the enemy of mathematical discovery” since they tend to send us on wild goose chases for proofs of theorems that are simply false. Of course, I’m not suggesting that there is some kind of foolproof path to successful mathematical discovery. In any given case, the mathematician will have to rely on background knowledge, experience, intuition, and so on to determine whether a direction of research suggested by a mathematical coincidence is worth the time and effort. My point is that these coincidences are useful ways of inducing indeterminacy that can spur research, not that they’re the only guide available.

  41. Seeing the ways in which “[m]athematics is for human flourishing” (Su, 2020, p. 10, emphasis added) is another route to this same end.

  42. Although he wouldn’t agree with the details of the Pascalian view presented here, Marc Lange, e.g., agrees that coincidence talk can sometimes make it easier to recognize interesting issues. See Lange (2010, p. 331).

  43. Lange very briefly discusses and dismisses a view like this which holds that a coincidence “does not repay further study, it is not fruitful, it leads to no further interesting mathematics” (Lange, 2017, p. 286). (He also suggests that Roy Sorensen makes a proposal like this in an unpublished manuscript, “Mathematical Coincidences.”) I’ll discuss Lange’s view further in Sect. 5 and comment on how the proposal under consideration differs from this dismissed one there as well.

  44. Nishiyama (2012, p. 372).

  45. \(123456789=3^2\cdot 3607\cdot 3803=10821\cdot 11409\) and \(123456784=2^4\cdot 11^2\cdot 43\cdot 1483=10406\cdot 11864\).

  46. See, for example, Buchbinder and Zaslavsk (2011) and Diaconis and Mosteller (1989, p. 859).

  47. For examples of this in a few popular textbooks across a variety of disciplines consider, e.g., Artin (1991, p. 233), Axler (1997, p. 177), Dummit and Foote (2004, p. 55), and Spivak (1994, p. 199, p. 528), where this kind of language only turns up in “no coincidnce” (or “no accident”) contexts.

  48. <https://mathoverflow.net/tour> Accessed 25 March 2021. See [Martin and Pease(2013)] for some reflections on MathOverflow as a resource and the light it sheds on the production of mathematics.

  49. <https://mathoverflow.net/questions/15444/examples-of-eventual-counterexamples> Accessed 29 August 2021.

  50. <https://mathoverflow.net/a/15506> Accessed 29 August 2021.

  51. See, e.g., Brookfield (2016, p. 186).

  52. <https://mathoverflow.net/questions/109149/cyclotomic-polynomials-with-coefficients-0-pm1> Accessed 29 August 2021.

  53. Cf. Sylvain Bromberger’s advice to someone seeking an answer to a why-question: “My guess is that the rational thing for him to do is to forget about the why-question and to turn to other questions instead, remembering that answers to why-questions usually emerge from work on questions with more reliable credentials” (Bromberger, 1992, p. 169).

  54. See Martin and Pease (2013) for more on this “fact-gathering”-role played by MathOverflow.

  55. Note that this kind of question can be settled without coming to the conclusion that it’s true or false that X is a coincidence. One way of settling the question would be to come to the opinion that the fact isn’t interesting and so just a coincidence or that it is interesting and so is no mere coincidence.

  56. Cf. Dannenberg (2008, p. 155): “It would probably be difficult to discover a novelist more consummate in the art of coincidence than Dickens”.

  57. See Wittgenstein (1956/1983, III §46, emphasis in the original): “[M]athematics is a multicoloured mixture of techniques of proof”.

  58. See, e.g., Williams (2010) for a general framework for offering an explanation of meaning in terms of use along these lines. See Pérez Carballo (2016) on a non-representational view of mathematics as a whole.

  59. A full theory might proceed by providing more explicit rules for “introducing” and “eliminating” coincidence-talk; explaining how coincidence-claims embed in non-asserted contexts; etc. There seem to be ample tools and methods available for filling in some of these details if one were so inclined. Cf. Thomasson (2020, Ch. 3) for a similar approach to handling talk of necessity and possibility especially in the area of metaphysics.

  60. Cf. Floyd (2012, p. 232) for a similar claim about surprises in mathematics from a Wittgenstein-inspired perspective.

  61. X may be worthy of attention for other reasons, of course.

  62. See Lange (2017), Part III and especially Chapter 8.

  63. Cf. Lange (2017, p, 304): That some theorem is no coincidence “is a fact about mathematics no less than that the theorem holds.” Whether or not this thought can only be captured by an account like Lange’s depends on how robustly we want to understand the concept of a “fact.” See, e.g., Price (2011, §4).

  64. “coincidence, n.” OED Online. Oxford University Press. Accessed 25 March 2021.

  65. Cf. Lange (2017, p. 277).

  66. See Lange (2010, pp. 316–322) and Baker (2009, p. 141).

  67. Given an account like this, an alternative to the line I’m pursuing here, but one that would be congenial to the general outlook, might be to follow Locke (2020) in his account of “metaphysical explanation for modal normativists” and use Lange’s definition of ‘coincidence’ along with a non-descriptive story about non-causal explanation.

  68. The qualification, “single, unified,” aims to prevent one from claiming that by putting together an explanatory proof of A and an explanatory proof of B one has given an explanation for A and B.

  69. For more detailed accounts, see, e.g., Steiner (1978), Kitcher (1989), or Lange (2017). For worries about having mathematical explanation focus only on explanatory proofs, see Lange (2018) and D’Alessandro (2020a).

  70. See also Inglis and Aberdein (2015) for a similar kind of study.

  71. It’s also possible, as already suggested in n.67, that substituting a different account of explanation or modifying Lange’s could do the required work just as well.

  72. See Davis (1981, p. 320) and Moore (1909), which is cited in Krieger (2003, p. 214). Lange makes note of this part of the Davis article in a footnote, but thinks that it would be better interpreted in a less extreme form. See Lange (2017, 445n.8).

  73. There remain questions about how literally to take and how heavily to weigh the opinions of mathematicians on these sorts of issues though of course. Cf. Martin (2020, §5.2).

  74. This is one important way in which the Pascalian account isn’t just the account discussed in Lange (2017, p. 286). Lange commits the view there to the claim that non-coincidences are non-coincidental because they suggest further interesting mathematics. On a Pascalian view, there isn’t any ‘because’ playing a significant role. Further, the view in Lange is primarily a story about what mathematical coincidences are—according to the view, they’re the facts that don’t repay further study. The Pascalian viewpoint is, instead, primarily a story about what we do with coincidence talk that then takes mathematical coincidences themselves to be roughly the shadows of this talk.

  75. Note that this potentially commits the view to what might be called “explanation chauvinism”: Is it really the case that explanations are all that are sought when a coincidence is investigated in mathematics? Compare with “proof chauvinism” as discussed in D’Alessandro (2020a) and Lange (2018).

  76. See Smith (1994, pp. 71–76) for an influential argument for this claim from someone not moved to expressivism by this “motivation problem”. Shafer-Landau (2000) and Railton (1986), on the other hand, object to this sort of motivational internalism.

  77. See, e.g., Schroeder (2010, Ch. 1.4).

  78. For the desirability of such a feature, see Baker (2009, p. 148).

  79. See Davis (1981, p. 320): “To some extent, [the mathematician] even brings [mathematical coincidence] about”.

  80. See Lange (2010, p. 316). According to an alternative account of explanation for which explanations are intimately related to the answering of why-questions, this claim of Lange’s would likely be judged to be mistaken though. E.g., Fraassen (1980, p. 130) suggests that an omniscient being wouldn’t be in the business of explanation at all, so the distinction between coincidence and non-coincidence in Lange’s terms would disappear.

  81. See, e.g., MacIntyre (1981, Ch. 14) on this role of practice. See Martin (2020) for a general MacIntyrean pespective on mathematical practice.

  82. See Field (2001) for a similar view in relation to his “evaluativist” account of apriority. See also Rosen (1994) on the general difficulty of saying what exactly objectivity comes to in the first place.

  83. A simpler way out of the worry may be by adopting a deflationary account of what it is for there to be “facts of the matter” in this domain. See, e.g., Thomasson (2020, Ch. 6.1).

  84. See, e.g., Harman (1977, Ch. 1.3) and Wiggins (1987, p. 147, pp. 149–151).

  85. Wright (1992, pp. 92–93).

  86. This fact about arbitrary collections of mathematical facts is also noted by Lange and accounted for by his view. See, e.g., Lange (2017, p. 280).

  87. Cf. Rota (1997, p. 93): “The quest for ultimate triviality is characteristic of the mathematical enterprise”.

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Martin, J.V. Indeterminacy, coincidence, and “Sourcing Newness” in mathematical research. Synthese 200, 28 (2022). https://doi.org/10.1007/s11229-022-03493-5

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