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Mathematizing Power, Formalization, and the Diagrammatical Mind or: What Does “Computation” Mean?

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Abstract

Computation and formalization are not modalities of pure abstractive operations. The essay tries to revise the assumption of the constitutive nonsensuality of the formal. The argument is that formalization is a kind of linear spatialization, which has significant visual dimensions. Thus, a connection can be discovered between visualization by figurative graphism and formalization by symbolic calculations: Both use spatial relations not only to represent but also to operate on epistemic, nonspatial, nonvisual entities. Descartes was one of the pioneers of using this kind of two-dimensional spatiality as a cognitive instrument.

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Notes

  1. An earlier version of the paper was presented at the conference “History and Philosophy of Computing 11”, Ghent, Belgium November 2011. I am grateful for comments received of three anonymous reviewers.

  2. Husserl 1970 (1954), §2, §9.

  3. The idea of the exteriority of the human mind does not come into being with the debate on the ‘extended mind’. Philosophers as Leibniz, Peirce, Wittgenstein and Cassirer contributed to the idea that all rational cognition is made up by the use of perceptible (visual, acoustic, tactile) signs.

  4. On the concept and culture of diagrams: Bender and Marrinan (2010), Pombo and Gerner (2010), and Stjernfelt (2007).

  5. On zero: Kaplan (2001), Reid (1992), Rotman (1987).

  6. Datta and Singh (1962, 118); Bose (1971, 177).

  7. Abu Dscha’far Muhammed ibn Musa al-Chwarizmi (ca. 780–850 AD) contributed mostly to the introduction of Indian numerals in Europe: al-Hwarizmi 1857. See al-Daffa 1977; Hill 1915.

  8. Krämer (2008), pp. 464.

  9. Viète 1970 (1646), vol. 1 pp. 1–12, 42–81, 305–327.

  10. On Indian numerals: Datta and Singh (1962).

  11. Leibniz (1979), pp. 42.

  12. Krämer (1991a), pp. 88.

  13. Jonas (1997), p. 248.

  14. As an exception from bottom to top: the writing of Tagbanuwa on Palawan, Philippines (http://de.wikipedia.org/wiki/Tagbanuwa-Schrift). Barbara Tversky has done inspiring research on directionality in graphics; for example: Tversky (2004).

  15. Kant (1977) (1768).

  16. The internal arithmetic order—from right to left—is the opposite to the Western order of writing and reading from left to right.

  17. Giaquinto (2007), pp. 127.

  18. See Krämer (1991a), pp. 143.

  19. Leibniz (1961), p. 531.

  20. Krämer (1988).

  21. See Krämer (2011).

  22. Wittgenstein (1967, 1984), vol. I , Philosophische Untersuchungen § 122, p. 302.; see Hacker (2004), Puhl (2001, 2006); Krämer (2011), p. 285.

  23. Wittgenstein (1984), vol. VII, Bemerkungen über die Philosophie der Psychologie, §889, p. 163.

  24. On the connection between spatiality, diagrammatics, and mind: Châtelet (2000), pp. 38; on the connection between flat spaces and arts: Summers (2003), pp. 43.

  25. Leibniz (1846); see Krämer (1991b).

  26. Leibniz (1961) pp. 326; Leibniz (1960/1961), vol. VII (1961), p. 206.

  27. The term “coordinate system” only characterizes a tendency: in handling the Pappus-problem within his Géométrie Descartes selected two non right-angled lines as referential system.

  28. Descartes (1973); German: Descartes (1981); English (1954); see Krämer (1989).

  29. Descartes (1981), p. 2.

  30. Zittel constructs a gap between the diagrammatic and iconic visualization in Descartes’ natural science and philosophy on the one side and his methodical and mathematical orientation on the other. Our paper starts from the assumption, that there is an interesting connection between Descartes’ visualization praxis as mathematician and as natural scientist and philosopher, although this is not the subject of this essay.

  31. Descartes (1966), vol. 10, Regulae ad directionem ingenii; German (1997).

  32. Descartes (1966), vol. 10, Rule 3, p. 367.

  33. Descartes (1966), vol 10, Rule 12, p. 418.

  34. Cogitationes Privatae (Descartes 2011), 196: “Ut imaginatio utitur figuris ad corpora concipienda, ita intellectus utitur quisbusdam corporibus sensibilibus ad spiritualia figuranda, ut vento, lumine.” One anonymous reviewer mentioned, that the bodily dimension of Descartes’ concept of imagination could be sustained by Descartes (1966), Vol. X, Rule 14, pp 438, 440–441.

  35. Plato (1971); see Krämer (2010).

  36. Descartes (1966), vol. 10, Rule 14, pp. 441.

  37. Descartes (1966), vol. 10, Rule 12, pp. 411.

  38. Descartes (1966), vol. 10, Rule 8, pp. 393.

  39. See Schuster (1980); Krämer (1991a), pp. 159.

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  1. Department of Philosophy, Freie Universität Berlin, Habelschwerdter Allee 30, 14195, Berlin, Germany

    Sybille Krämer

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Krämer, S. Mathematizing Power, Formalization, and the Diagrammatical Mind or: What Does “Computation” Mean?. Philos. Technol. 27, 345–357 (2014). https://doi.org/10.1007/s13347-012-0094-3

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