«david link Scrambling T-R-U-T-H Rotating Letters as a Material Form of Thought “Know that the secrets of God and the objects of His science, the ...»
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Rotating Letters as a Material Form of Thought
“Know that the secrets of God and the objects of His science, the subtle
realities and the dense realities, the things of above and the things from
below, belong to two categories: there are numbers and there are letters.
The secrets of the letters are in the numbers, and the epiphanies of the
numbers are in the letters. The numbers are the realities of above, belonging to the spiritual entities. The letters belong to the circle of the material realities and to the becoming.” A mad al-Būnī (d. 1225)1 The works of the Majorcan philosopher and missionary Ramon Llull (1232–1316) are commonly and rightly regarded as foundational for the development of West- ern combinatorics and logic. Circular disks inscribed with the letters from B to K, which can be rotated in relation to each other, play a central role in his Fig. 1: Alphabetic disk in Ars Generalis Ultima, 1308.
1–Quoted by Henri Corbin, Histoire de la Philosophie Islamique (Paris, 1964), p. 205: “Sache que les secrets de Dieu et les objects de Sa science, les réalités subtiles et les réalités denses, les choses d’en haut et les choses d’en bas, sont de deux catégories: il y a les nombres et il y a les lettres. Les secrets des lettres sont dans les nombres, et les épiphanies des nombres sont dans les lettres. Les nombres sont les réalités d’en haut, appartenant aux entités spirituelles. Les lettres appartiennent au cercle des réalités matérielles et du devenir.” Translation from the French, D.L.
215 kwb.45-variantology_inhalt_neu:kwb.31-variantology_end_5.0 19.02.10 18:04 Seite 216 Ars Inveniendi Veritatem. A working model of the paper machine was included in several of his publications; a thread through the middle held the disks together.
Llull’s motives are not easily understood today, but it seems safe to state that in his quest to convert the “inﬁdels” to Christianity, the disk construction served theoretical functions as an encyclopaedia of religious thought, a tool to inspire meditation about its main topics, a way to generate new propositions, or even to abolish language altogether: “The idea is to present in a single book everything that can
of the procedure executed on the “Zā’irajah of the world” (as one of the translators into English, Franz Rosenthal, spells it).5 The device is attributed to the Maghribi Suﬁ Mu ammad b. Mas-ūd Abū l—cAbbās as-Sabtī, who lived in Marrakech at the end of the twelfth century.6 A footnote in the ﬁrst printed edition of the work in Arabic from Bulaq near Cairo relates that Ibn Khaldūn derived the knowledge about the artefact “from people who work with the zā’irajah and whom [he had] met”.7 An untitled manuscript by Shaikh Jamāl ad-dīn Abd alMalik b. Abd Allāh al-Marjānī in the library of Rabat, Morocco, establishes it was he who introduced Ibn Khaldūn to the procedure in Biskra, now Algeria, in 1370/1371 (the year 772 A. H.).8 Al-Marjānī had heard from the qādī (judge) of Constantine, today also Algeria (who in turn had allegedly been informed by one of the companions of the prophet, Hudhaifa b. Yamān), that the zā’irjah was a traditional and ancient science.9 When Ibn Khaldūn questioned the correctness
of the proposition, the shaikh suggested simply to ask the device itself:
Üè‚Î æ_ o¦ Â íq†è]ˆÖ] “Az-zā’irja ‘ilm muhdath au qadīm” – “[Is the] zā’irja [a] recent or [an] ancient science?”10 When the self-referential question was posed, the ascendant (the sign of the zodiac cycle rising at the eastern horizon) stood in the ﬁrst degree of Sagittarius.
Al-Marjānī performed the complicated procedure and explained it to Ibn Khaldūn. It yielded the answer “The Holy Spirit will depart, its secret having been brought forth / To Idrīs, and through it, he ascended the highest summit” 5–“Muqaddima” signiﬁes “Prolegomenon, introduction” in Arabic. The main printed editions of this work are, in chronological order: Ibn Khaldūn, Kitāb al-‘Ibar wa-dīwān al-mūbtadā’ wa-l- abar etc., ed.
Nas}r al-Hūrīnī (Bulaq near Cairo, 1857, 1 vol.; reprint Bulaq, 1867/8, 7 vols.; in Arabic); Prolégomènes d’Ebn-Khaldoun, ed. Étienne M. Quatremère, 3 vols. (Paris, 1858; in Arabic); Prolégomènes Historiques d’Ibn Khaldoun, trans. William MacGuckin de Slane, 3 vols. (Paris, 1863–1868); Ibn Khaldūn, al-Muqaddimah, ed. ‘Alī A. Wāfī, 4 vols. (Cairo, 1957–1962; in Arabic); Ibn Khaldūn, The Muqaddimah. An Introduction to History, trans. Franz Rosenthal, 3 vols. (New York, 1958); Ibn Khaldūn, Discours sur l’Histoire Universelle (al-Muqaddima), trans. Vincent Monteil, 3 vols. (Paris – Beyrouth, 1967/8); Ibn Khaldūn, Le Livre des Exemples, vol. 1: Autobiographie, Muqaddima, trans. Abdesselam Cheddadi (Paris, 2002).
6–Muqaddimah, trans. Rosenthal, vol. 1, p. 239 [I, 213] and fn. 365. Cf. Carl Brockelmann, Geschichte der Arabischen Literatur, Suppl. vol. 1 (Leiden, 1937), p. 909.
7–Muqaddimah, trans. Rosenthal, vol. 3, p. 196, fn. 880.
8–Dates in this chapter generally follow the Christian system.
9–Henri P. J. Renaud, Divination et histoire nord-africaine au temps d’Ibn Khaldun. Hespéris 30 (1943): 213– 221, esp. 213– 215. How the latter learned the fact from the former is unclear, since more than 600 years separate them.
10–The single letters of the question are: “’ l z ’ y r j t / ‘ l m / m h} d th / ’ w / q d y m”.
“s‘afas}”, “qurshat”, “thakhudh” and “d}az}ugh”, which were later thought to represent archaic kings, demons, or the days of the week.14
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 Fig. 2: Arabic letters, Western numerical interpretation.15 The zimām symbols follow the same system, but only designate numerals and do not contain a character for thousand (Figure 3). They translate easily into Arabic letters of the same value. In both sets of signs, composite numbers are formed by joining together the respective symbols for units, tens, hundreds, and so on; in the ﬁrst decreasing in value from right to left (the “natural” writing direction), and increasing in the second.16 Hence 333 would be represented as Â‰ in abjad, or F3 in zimām.17
Fig. 3: Zimām numerals, according to Rosenthal (1958).
14–Georges Ifrah, The Universal History of Numbers. From Prehistory to the Invention of the Computer (London, 1998), p. 243f.
15–The Arabic Orient and Occident match letters and numbers slightly differently; cf. Muqaddimah, trans. Rosenthal, vol. 3, p. 173, fn. 809. Throughout this article, alif will be transliterated as ’ and ‘ayn as ‘.
16–Azzedine Lazrek, personal communication, 14 May 2008.
17–The symbols employed in this paper, slightly different from the ones below, which Rosenthal copied from one of the manuscripts of the Muqaddima, were published by Azzedine Lazrek, Cadi Ayyad University, Marrakech, Morocco, as part of a proposal to include them in the Unicode standard; cf. A. Lazrek, Unicode–Proposals symbols for Unicode Consortium, 17 March 2008.
http://www.ucam. ac.ma/fssm/rydarab/unicode.htm. The proposal was accepted on 25 April 2008, and the characters will probably be allocated in the range 10E60..10E7E; cf. Unicode Consortium, Proposed New Characters—Pipeline Table, 1 May 2008. http://unicode.org/unicode/alloc/Pipeline.html.
18–Cf. Florian Cajori, A History of Mathematical Notation, vol. 1: Notations in Elementary Mathematics  (New York, 1993), pp. 21–30; Georges S. Colin, L’origine grecque des “chiffres de Fès” et de nos “chiffres Arabes”. Journal Asiatique 222 (1933): 193–215, esp. 193–203; cf. p. 202f., fn. 3: “It is odd to ﬁnd that still at present, the majority of the functionaries of the Egyptian ministry of ﬁnances are Copts.” 19–A. Lazrek, Rumi Numeral System Symbols, 15 July 2006. http://www.ucam. ac.ma/fssm/rydarab/ doc/unicode/amos5l.pdf, p. 6.
20–Cf. Mahdi Abdeljaouad, Le manuscrit mathématique de Jerba. Une pratique des symboles algébriques Maghrébins en pleine maturité, in: Actes du 7ième Colloque Maghrébin sur l’Histoire des Mathématiques Arabes, ed. Abdallah El Idrissi and Ezzaim Laabid (Marrakech, 2005), pp. 9 – 98. Online http://math.unipa.it/~grim/MahdiAbdjQuad11.pdf, p. 21.
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One of the main procedures on the reckoning-board is the scaling of values, where one bead is assumed to represent ﬁves, tens, and so on. When a number needs to be carried over to the next counter, a second operation is employed, the calculation of the remainder after division through a certain quantity, 5 in the case of this device. It evolves naturally when a group of units needs to be compressed into a sign with a higher value, and may derive from the cyclical character of time and the practice of clock-building. The modern term is modulo. In
Fig. 4: Ancient Greek abacus found on Salamis, ﬁfth to fourth century B.C.22
21–Most basic calculations are executed faster on an abacus than with an electric calculator, as proven by a contest held in Japan shortly after the Second World War, in which the soroban (Japanese abacus) champion won 4 to 1 against his North American counterpart; cf. Ifrah, Universal History, p. 289f.
22–The tablet measures approximately 149 ∑ 75 ∑ 4.5 cm. The right side of the device represents the number 9,823, 1 (∑ 5000) + 4 (∑ 1000) + 1 (∑ 500) + 3 (∑ 100) + 0 (∑ 50) + 2 (∑ 10) + 0 (∑ 5) + 3 (∑ 1); cf.
Cajori, Mathematical Notation, p. 22f., and Ifrah, Universal History, pp. 201– 203, for a more exact treatment. A similar table is already mentioned in the Athenian Constitution (written between 332 and 322 B.C.): “And when all have voted, the attendants take the vessel that is to count and empty it out on to a reckoning-board [“ábaka”] with as many holes in it as there are pebbles, in order that they may be set out visibly and be easy to count, and that the perforated and the whole ones may be clearly seen by the litigants. And those assigned by lot to count the voting-pebbles count them out on to the reckoningboard [“ábakos”], in two sets, one the whole ones and the other those perforated.” Aristotle in 23 Volumes, vol. 23: Athenian Constitution, trans. H. Rackham (Cambridge, MA, 1952), sect. 69. 1.
23–Cf. Ifrah, Universal History, pp. 291– 294.
24–The origins of the system in India is supported by many documents, but is by no means undisputed; cf. Ifrah, Universal History, pp. 356 – 439. For a controversial position, see Solomon Gandz, The origin of the Ghubār numerals, or The Arabian abacus and the articuli. Isis 16 (1931): 393– 424.
25–Muqaddimah, trans. Rosenthal, vol. 3, p. 197, fn. 882. For the various forms of the numerals in the course of their history, see Ifrah, Universal History, pp. 534 –539. According to Ifrah, “the oldest
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modern positional system by quoting his “friend and teacher” Sylvestre de Sacy, who had discovered them in a manuscript in the library of the old Abbey of St. Germain-des-Prés: “The gobar has a strong relation with the Indian number, but it does not possess a zero.”26 Humboldt, however, believed zero was expressed by marks placed above the signs: tens were indicated by one, hundreds by two, and thousands by three dots in superscript, constituting a “hybrid” counting system.27 He speculated that it was these marks that had given the numbers their “strange name gobar or dust writing”.28 Literally, “ghubār” (…^fÆ) does indeed signify “dust”, not in the sense of the miraculously appearing house dirt of our days, but rather the ﬁnest possible kind of sand.29 This led many scientists in the nineteenth century to believe that the name derived from an ancient means of calculation. Since antiquity, a board of wood covered with dust (called “lawha” in Maghreb) had been used for this purpose, writing numbers with a stick on it and deleting others in a performative kind of arithmetic.30 It complemented the wax tablets of Aristotelian fame. This setup allowed relatively ﬂexible processing of data, unlike working with clay or stone, and could be described as volatile “random access memory”, since its ﬂuidity permitted the modiﬁcation of any grain on the surface at any time.31 known documents which refer to Ghubar numerals and calculation date back to 874 and 888 CE” and were apparently found in India (p. 536).
26–Translation D. L. De Sacy also translated parts of the Muqaddima; cf. Baron Antoine Isaac Silvestre de Sacy, Extraits de Prolégomènes d’Ebn Khaldoun, in: Relation de l’Égypte, par Abd-Allatif, Médecin Arabe de Bagdad, trans. S. de Sacy (Paris, 1810; reprint Frankfurt am Main, 1992), pp. 509 –524 (translation vol. 5, chap. 4 and vol. 4, chaps. 3 and 4), pp. 558 – 64 (Arabic text), and ‘Abd ar-Rahmān alJāmī, Vie des Souﬁs ou Les Haleines de la Familiarité, trans. S. de Sacy (Paris, 1831; reprint Paris, 1977), pp. 16 – 20 (Arabic text), pp. 20 – 27 (translation “Du Soﬁsme”).
27–In this representation, the number is decomposed into units, tens, hundreds, and so on, and then each is written down in two parts in a multiplicative manner. 7,659 would be expressed as 7000 600 50 9, 7 ∑ 1000 + 6 ∑ 100 + 5 ∑ 10 + 9 ∑ 1. The abbreviation of such systems led to positional notation very early on; cf. Ifrah, Universal History, pp. 330 –340.
28–Alexander von Humboldt, Über die bei den verschiedenen Völkern üblichen Systeme von Zahlzeichen und über den Ursprung des Stellenwerthes in den indischen Zahlen. Crelle’s Journal für die reine und angewandte Mathematik 4  (1829): 205–231, esp. 213, 222–224. In the manuscript that Rosenthal took the ghubār numerals from, zero is placed to the right side of the sign; cf. the ten in Figure 6.
29–Lane, Arabic–English Lexicon, vol. 6, p. 2224.