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How to Define a Number? A General Epistemological Account of Simon Stevin’s Art of Defining

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Abstract

This paper explores Simon Stevin’s l’Arithmétique of 1585, where we find a novel understanding of the concept of number. I will discuss the dynamics between his practice and philosophy of mathematics, and put it in the context of his general epistemological attitude. Subsequently, I will take a close look at his justificational concerns, and at how these are reflected in his inductive, a postiori and structuralist approach to investigating the numerical field. I will argue that Stevin’s renewed conceptualisation of the notion of number is a sort of “existential closure” of the numerical domain, founded upon the practice of his predecessors and contemporaries. Accordingly, I want to make clear that l’Aritmetique have to be read not as an ontological analysis or exploration of the numerical field, but as an explication of a mathematical ethos. In this sense, this article also intends to make a specific contribution to the broader issue of the “ethics of geometry.”

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Notes

  1. Stevin (1958, pp. 497–498).

  2. Stevin (1585a, p. 120): “Verhael op de xxvii. definitie der definitien.”

  3. Ibid. p. 121: “De Definitie is een Reden, aenwijsende het wesen der saecken door hare Wesentlickheden; Dese woorden, Door hare Wesentlickheden, legghen sy uyt, Door haer Gheslachte ende Differentie.”

  4. Ibid. p. 121: “…weynich saecken ja beter gheseyt gheen ter werelt, en souden connen Ghedefiniert worden.”

  5. See: Stevin (1605–1608), ‘I bovck des eertclootschrifts’, p. 39: “Om nu te segghen vant woort anschauwen, het is te weten dat wy gheen wesentlicke saeck self en sien, maer alleenlick sijn schaeu,” p. 39.

  6. Clarck (2007, p. 6).

  7. Wilson (1995, p. 218).

  8. Jardine (1988), Reiss (1997, 2000), Hooykaas (1958), Meerhoff (2001).

  9. Ramus and Dassonville (1964, p. 64 and 74).

  10. See Pozzo (2003), Reiss (2000), Nuchelmans (1980), chapter 11. Agricola also makes a strong connection between defining, the operation of thematization and the activity of measurement, see: Agricola (1528), I, 5–8.

  11. Nuchelmans (1979), chapter 12 and 13.

  12. See: Stevin (1955), “Vytspraeck vande weerdicheyt der Duytsche Tael,” pp. 58–93. See also; van der Wal (2004).

  13. Stevin (1585a, p. 53): “…gelijck […]een goedt Schutter is, die niet altijt de pinne en raect, maer daer ontrent schiet.”

  14. Ibid. p. 54: “…een selfde saecke can wel op verscheyden manieren Ghedefiniert worden, die alle goedt sijn”.

  15. Stevin (1605–1609), ‘I bovck des eertclootschrifts’, p. 39 and: Van den Heuvel (2005, pp. 30–31).

  16. See: Malet (2005, pp. 77–79), Verdonk (1969, p. 254), for Stevin’s critique on the authority of Euclid.

  17. Stevin (1605–1609), ‘I bovck des eertclootschrifts’, pp. 17–20.

  18. Ramus eliminates the distinction between art and science of the same object, and the distinction between theoretical and practical sciences. For the same mathesis is utilis ad complendum or ad agendum, that what varies is its use.

  19. Burgersdijk Franco, Institutionum logicarum libri duo, Leiden, 1626. Quoted in Pozzo (2003, p. 10). For Stevin, “formal consequentiality” refers to Aristotelian deduction, “material elimination of our ignorance” refers to induction and reductio ad absurdum.

  20. About usus and usefulness, see: Sellberg (1979) and (2001). In a restricted sense, Stevin still maintained that theory should be developed in its own right. See van den Heuvel (2005, pp. 32–33).

  21. Stevin (1958, pp. 494–495).

  22. Ibid.

  23. Ibid.

  24. Euclid (1956), The Elements, VII, def 1.

  25. For Ramus, see: Verdonk (1966, pp. 129–137). I want also to mention a remark of Lorenzo Valla (1406-1457): “[T]wo women who shared twelve hens and one rooster among them. They agreed that one would have the eggs on day when the number laid was even, but that the other would get them when the number was odd. ‘Say that sometimes single eggs were laid. To which would that egg go; to neither?’ ‘No, to the one who was due the odd number of eggs.’ Therefore, one egg makes a number.” (Valla (1982) (ed.), Laurentii Valle respastinatio dialecticae et philosophiae, G. Zippel, Padua, i. 18–19). Quoted in: Copenhaver and Schmitt (2002, p. 218). On the mathematical influence of Ramus on Stevin, see: Verdonck (1969).

  26. Stevin (1958, pp. 495–496).

  27. Ibid.

  28. Arnauld (1964, p. 316).

  29. Ibid. p. 317.

  30. See: footnote 14 of this text.

  31. Euclid (1956).

  32. Stevin (1955), in: ‘Het eerste bovck vande beghinselen der weegconst,” p. 101.

  33. Stevin (1958, p. 497).

  34. Bovelles (1542), 5 r-v: “le point qui resemble à l’unité en Aritmeticque. Car comme lunité nest pas nombre, mais est le commencement & principe de tous nombres …”.

  35. Stevin (1958, pp. 498–499).

  36. Ibid. pp. 498–500.

  37. Ibid. p. 500.

  38. Nikulin (2008, pp. 157–159), Bos (2001, Chap. 6).

  39. Stevin (1958, p. 500). Stevin (1585a, b, p. 3). See also: Klein: 193–197 and Malet (2005, pp. 76–78).

  40. Stevin (1958, pp. 501–502).

  41. Ibid.

  42. Page (1996, p. 243).

  43. Stevin expresses his attitude towards negative numbers in (1585b, p. 186).

  44. See: Stevin (1585b, p. 309) and (1958, pp. 617–620).

  45. About the difference between syntactically and semantically guided actions: Kaput (1994, pp. 103–104).

  46. Despite similarities between geometric constructions and arithmetic operations, there were important differences between them, which explains why Stevin’s notion of number was not embraced immediately after the publication of his Aritmétique. See: Bos (2001, pp. 128–134).

  47. Reack and Price (2000, p. 345).

  48. An example of the traditional approach was Zabarella. See Vanden Broecke (1998), “Jacob Zabarella specifies that the definition of a circle, triangle and other mathematical things express the essence of an accident.” (my emphasis) pp. 60–61.

  49. Lachterman (1989, p. 165).

  50. Manders (1989, p. 562).

  51. Ibid.

  52. Ibid. p. 554.

  53. See: Hadden (1989), Heeffer (2008b).

  54. Manders (1989, p. 560).: “To accept negative quantities, one had to give up traditionally fundamental properties of quantity, such as the monotonicity of proportions.”

  55. Malet (2005).

  56. Heeffer (2008) makes a similar point about the emergence of symbolic algebra: “The road to symbolic Algebra was paved by several previous stepping stones that have been functional in developing the symbolic mode of reasoning.” (p. 153). About the conceptual changes leading to symbolic algebra, see: Heeffer (2007), Cifoletti (1992, 2006). About the concept of number: Malet (2005).

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Acknowledgments

I would like to thank Leon Horsten for his comments, criticisms and suggestions.

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    Jurgen Naets

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Naets, J. How to Define a Number? A General Epistemological Account of Simon Stevin’s Art of Defining. Topoi 29, 77–86 (2010). https://doi.org/10.1007/s11245-009-9068-1

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