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Discourse Transformed: Changing Modes of Argumentation

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Abstract

Moving from the linguistic to the discursive level, this chapter examines the changing modes of argumentation and analyses the ways in which the authority of these modes of argumentation was built upon both Chinese philosophical and foreign elements. Whether it is inductive proof, visual tools, or arguments by analogy in number-theoretical contexts, new standards seem to emerge in nineteenth century due to the social pressure of claiming the validity of the obtained results. From a comparative and transcultural perspective, the chapter investigates in particular the Comparable Categories of Discrete Accumulations (Duoji bilei 垛積比類) by Li Shanlan (1811–1882), a Chinese mathematician who was familiar with foreign mathematical discourse through his translation work, and who systematically proceeded in his number-theoretical writings by “analogical extension” (leitui 類推), to infer from the particular to the universal. To conclude, the historiographic problem of claims about a specifically Western scientific rigour of inductive proofs in Blaise Pascal’s Traité du Triangle Arithmétique (1665) is questioned.

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Notes

  1. 1.

    Also: “Li Renshu Identity.” Renshu was the style name of Li Shanlan. See Horng (1991) p. 206.

  2. 2.

    See Martzloff (2006) p. 341–342.

  3. 3.

    See Zhang (1939).

  4. 4.

    For an extensive discussion of the strands of the Chinese tradition of considering piles of discrete objects in different geometric shapes as figured numbers, see Bréard (1999a). On figured numbers (and in extension, polygonal numbers) in the Greek tradition, see Diofanto (2011) p. 39–40.

  5. 5.

    These correspond to the coefficients \((C^{n+k}_k)^2\) for the values k = 1 and n = 0,  1,  2.

  6. 6.

    The squared triangular numbers correspond to the binomial coefficients \((C^{n+k}_k)^2\) for the values k = 2 and n = 0,  1,  2,  3.

  7. 7.

    Pólya (1954).

  8. 8.

    The treatise was composed towards the end of 1654 and published posthumously, together with others connected to it, in 1665 under the complete title Traité du Triangle arithmétique, avec quelques autres petits traités sur la même matière.

  9. 9.

    See, for example Martzloff (2006) p. 20, 158 and 342 and Martzloff (1993) p. 165.

  10. 10.

    Translation from Legge (1885) Section III. Part II. 9. Another example can be found in the chap. “Record of Music” (Yue ji 樂記). See Legge (1885) sec. II.14.

  11. 11.

    See Wang (2012) p. 85–93, here p. 92.

  12. 12.

    For the earlier significance of “analogy” in mathematics in China as a form of reasoning based on examples, see Volkov (1992).

  13. 13.

    Schironi (2007) shows that the originally mathematical concept of analogy was later taken over by Greek grammarians.

  14. 14.

    On the genesis of the Shuli jingyun, which extends from 1690 to 1723, see Jami (1998) p. 118–119: Between 1690 and appr. 1695 Jesuits—in particular by Antoine Thomas (1644–1709) and Jean-François Gerbillon (1654–1707)—wrote manuals for the emperor, essentially in three fields: arithmetic, algebra and geometry. From 1713 on, these texts were revised and rewritten by Chinese scholars working in the Academy of Mathematics (Suanxue guan 算學館), an institution created especially at the Mengyang zhai 蒙養齋—a Bureau where scholarly books were compiled under imperial patronage—for the compilation of the Origins of musical harmony and the calendar (Lüli yuanyuan 律曆淵源), of which the Essence of Numbers and their Principles were the mathematical part.

  15. 15.

    Translated from Yunzhi 允祉 (1723) vol. 20, p. 18A and already quoted on page 38. I provide the quote again here because my focus is on argumentation rather than on the mathematical aspects of the ellipse.

  16. 16.

    Translated from Yang (1261) p. 78A–78B.

  17. 17.

    Measure of length, where 1 zhang=10 chi.

  18. 18.

    The procedure here corresponds to the following calculation (for the various dimensions, see Fig. 5.7 to the left):

    $$\displaystyle \begin{aligned}A=[(2d+b)\cdot c+(2b+d)\cdot a]\cdot h \div 6.\end{aligned}$$
  19. 19.

    Text framed in red in Fig. 5.6.

  20. 20.

    The procedure of “comparable category” here corresponds to the following calculation (for the various dimensions, see Fig. 5.7 to the right):

    $$\displaystyle \begin{aligned}{}[A=(2d+b)\cdot c+(2b+d)\cdot a+(b-d)]\cdot h \div 6. \end{aligned}$$
  21. 21.

    For an extensive discussion of Yang Hui’s contributions see Bréard (1999a) chap. 3.2, p. 119–158.

  22. 22.

    The interested reader might consult Bréard (1999a) chap. 4 and 5 or Tian (2003).

  23. 23.

    Martzloff translates the title as Accumulated “Heaps” Studied from an Analogical Point of View. See Selin (2008) p. 1225.

  24. 24.

    By relying on diagrams with separate explanations, Wang Lai uses (incomplete) induction to prove rhetorically the correctness of a recursive algorithm that he represents graphically up to a certain finite step. For details on the inductive pattern and the visual tools in Wang Lai’s text, see Bréard (2019).

  25. 25.

    I use the terms “algorithm” and “procedure” interchangeably, the latter being the literal translation of shu 術, which in the Chinese mathematical tradition designates the list of operations to follow for calculating the numerical solution of a mathematical problem.

  26. 26.

    See Mueller (1981).

  27. 27.

    Li (1867) p. 1a. Li qualifies identically Dong Youcheng’s 董佑誠 (style name Fangli, 1791–1823) work on quadrangular piles, but since Li does not reproduce discursive elements from Dong’s text, I do not analyse it here with respect to Li Shanlan’s schemes of justification.

  28. 28.

    Depicted in the Traité du Triangle Arithmétique “apparently printed in 1654 (though circulated in 1665).” See Daston (1988) p. 9.

  29. 29.

    Todhunter (1865) chap. 2 as well as many other historians of probability theory considers the Problem of Points, which prompted the seminal correspondence between Pascal and Pierre Fermat in 1654, as the beginning of the theory of probability. See also Todhunter (1865) chap. 9 on the history of the arithmetic triangle.

  30. 30.

    See Bréard (1999a) chap. 4.2 and Bréard (1999b).

  31. 31.

    On the meaning of the procedures and diagrams in Wang’s text, see Bréard (2015).

  32. 32.

    Recursive algorithms calculate the solution to a problem depending on solutions to previously calculated instances of the same problem. It is thus possible to define a mathematical object for any instance by a finite statement.

  33. 33.

    Expressed formally, the recursive character of Wang’s construction becomes even more apparent: \(C_2^{10}=\sum _{i=1}^{n-1}C_1^i=C_1^1+C_1^2+\cdots +C_1^9\).

  34. 34.

    Idem: \(C_3^{10}=\sum _{i=2}^{n-1}C_2^i=C_2^2+C_2^3+\cdots +C_2^9\).

  35. 35.

    It can be grouped and ordered in two ways. A horizontal reading from top to bottom gives the terms:

    $$\displaystyle \begin{aligned}C^{10}_5= 6\cdot 1 + 5\cdot 4 + 4 \cdot 10 + 3\cdot 20+2\cdot 35 + 1\cdot 56\end{aligned}$$

    whereas a diagonal reading produces different terms, yielding the same sum:

    $$\displaystyle \begin{aligned}C^{10}_5=1 + (1+4) + (1+4+10) + (1+4+10+20)+\end{aligned}$$
    $$\displaystyle \begin{aligned}+ (1+4+10+20+35) + (1+4+10+20+35+56).\end{aligned}$$
  36. 36.

    Li (1867) vol. 1, p. 3B.

  37. 37.

    Translated from Li (1867) scroll 2, p. 2B-3A. Punctuation in the Chinese quote is mine.

  38. 38.

    For more details, see my introduction to Li (2019).

  39. 39.

    Translated from Li (1867) scroll 2, p. 3A. Punctuation in the Chinese quote is mine.

  40. 40.

    For example, the number 57 in the second cell from the left in the green line in Fig. 5.17 is the sum of 5 ⋅ 1 and 2 ⋅ 26, since the 1 on its upper left is in the fifth position of its line that goes diagonally down to the left and 26 on its upper right is in the second position in the line going diagonally down to the right.

  41. 41.

    Translated from Li (1867) vol. 2, p. 3B–4A.

  42. 42.

    See Li (1867) vol. 3, p. 3A–3B.

  43. 43.

    A familiar formulation is in terms of the sum of squares of natural numbers:

    $$\displaystyle \begin{aligned}\sum^n_{k=1}k^2=\frac{n(n+1)(2n+1)}{6}.\end{aligned}$$
  44. 44.

    See scroll 1, p. 5B and scroll 2, p. 7B: “By the Celestial Element [method] induce this accordingly.” (Yi tianyuan fang ci tui zhi 以天元仿此推之).

  45. 45.

    See Kim (2004) p. 41.

  46. 46.

    Quote from Zhu Xi’s Classified Dialogues of Master Zhu (Zhuzi yulei 朱子語類, compiled in 1270), translated in Kim (2004) p. 57.

  47. 47.

    See Li and Wylie (1859) scroll 8, p. 8A and Loomis (1868) p. 105.

  48. 48.

    Li (1867) scroll 1, p. 1A–1B.

  49. 49.

    Chen (17th) p. 685.

  50. 50.

    Jia 甲, Yi 乙, Bing 丙 and Ding 丁 are the first four of the Ten Heavenly or Celestial Stems (tiangan 天干) which, together with the Twelve Earthly Branches (dizhi 地支), beginning with Zi 子, Chou 丑, Yin 寅 and Mao 卯, form China’s traditional calendrical system. They were also used since the first translation of the first six books of Euclid’s Elements (Ricci and Xu 1607) for representing the alphabetic letters in the geometric diagrams.

  51. 51.

    Translated from Chen (1904) p. 1A–1B.

  52. 52.

    See Tian (2003) p. 58–69.

  53. 53.

    By “proof” I mean an argument very much in the style of Greek proofs, considered rigorous by contemporary authors and later commentators.

  54. 54.

    As an application of the recursive algorithm, he only gives one example drawn from divination with hexagrams. In this case, the algorithm allows for the calculation of the total number of possible mutations of a diagram of six lines, with two possibilities for each line. In Bréard (2012) I have shown that in late imperial China, among cultural practices like gaming or divination, hexagrams became the paradigmatic model for observing and analysing stable combinatorial patterns. Bréard (2019) contains a complete translation of Wang Lai’s text.

  55. 55.

    For examples from the Babylonian and Greek mathematical traditions, see Høyrup (2012) and Mueller (2012).

  56. 56.

    At 149A-C. See Acerbi (2000). The text “displays a series of phrases, adverbs, and syntactical constructs which enable him to word in a very refined way the explicitly iterative character of the proof.” Acerbi (2003) p. 477. More generally, see idem p. 476–481 for an overview of recursive or quasi-inductive proofs in Greek mathematical writings. Outlining “the composition of the combinatorial humus in which calculations must have grown out” (p. 466), Acerbi shows, for example through a passage of Plutarch’s De Stoicorum repugnantiis, that, even without an explicitly formulated proof scheme, an astonishing result related to Schröder numbers could be obtained.

  57. 57.

    Pascal (1665) p. 5:

    En tout Triangle Arithmetique, la somme des cellules de chaque base, est double de celles de la base precedente.

  58. 58.

    Pascal (1665) p. 5:

    En tout Triangle Arithmetique, la somme des cellules de chaque base, est un nombre de la progression double, qui commence par l’unité, dont l’exposant est le mesme que celuy de la base.

  59. 59.

    Idem. Final phrases to corollary 7 and 8 respectively: “La mesme chose se demonstre de mesme de toutes les autres” and “Et ainsi à l’infiny.”

  60. 60.

    Pascal expresses this relation proportionally. For more details, see Edwards (1987) p. 63–65.

  61. 61.

    Idem p. 91 in the Latin version: “Quamvis infiniti sint hujus propositionis casus, […], breviter tamen demonstrabo, positis duobus assumptis.”

  62. 62.

    Pascal (1665) p. 7:

    Le 1. qui est évident de soy-mesme, que cette proportion se rencontre dans la seconde base; car il est bien visible, que f est às comme 1 est à 1.

  63. 63.

    Quoted from Conséquence Douziesme in Pascal (1665) p. 7.

  64. 64.

    Translated from Conséquence Douziesme in Pascal (1665) p. 8.

  65. 65.

    See Kim (2014) p. 35–52, previously published as Kim (2004).

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Bréard, A. (2019). Discourse Transformed: Changing Modes of Argumentation. In: Nine Chapters on Mathematical Modernity. Transcultural Research – Heidelberg Studies on Asia and Europe in a Global Context. Springer, Cham. https://doi.org/10.1007/978-3-319-93695-6_5

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