Sub-Scientific Mathematics: Observations on a Pre-Modern Phenomenon

981^b14–982^a1 — Quoted from Ross W. D. (ed., transl.), The works of Aristotle, translated into English, viii: Metaphysica, 2nd edn (Oxford, 1928).

Very briefly stated (too briefly!), ‘scientific’ knowledge thus aims at truth, while sub-scientific knowledge is directed toward utility. In the mathematical domain this might make us conclude that ‘scientific’ knowledge is mathematics built on proofs while sub-scientific mathematics builds on recipes and empirical rules. This conclusion is false for several reasons. Firstly, as we know, e.g., from cosmogonies, statements may be considered supremely true because they are old and revered or because they are supposed to stem from sacred revelation. At times, the latter source is encountered even in mathematics: In one of his works, al-Bīrūnī tells that the Indians hold the ratio of circular circumference to diameter to be as “the ratio of 3,927 to 1,250, because it was communicated to them, by divine revelation and angelic disclosure” that such were the proportions of the world —as translated in D. Pingree, “The fragments of the works of al-Fazārī”, Journal of Near Eastern studies, xxix (1970), 103–23, p. 120. Secondly, Nicomachus's ‘science’ is argued solely on the basis of empirical rules. Thirdly, complex mathematics, not least complex mathematics aimed at many-sided application, can be transmitted successfully only if supported by some level of understanding, whence by some sort of argument or proof. Basically, the need for proofs in mathematics comes from teaching; the elevation of the need into a transcendental philosophical principle is historically secondary — cf. my “Influences of institutionalized mathematics teaching on the development and organization of mathematical thought in the pre-modern period”, Materialien und Studien: Institut für Didaktik der Mathematik der Universität Bielefeld, xx (1980), 7–137.

The “Old Babylonian” mathematical texts were produced between c. 1800 b.c. and 1600 b.c. When speaking in the following of “Babylonian mathematics” I shall refer exclusively to this corpus.

This position is asserted in Morris Kline's Mathematical thought from ancient to modern times (New York, 1972), 11.

When, for instance, will you know the area of a trapezoidal field and the portion which broke off from your measuring reed but not the reed itself (cf. below)? The evidence could be multiplied ad libitum.

So, grosso modo, Boyer Carl B. , A history of mathematics (New York, 1968), 45.

See my “Varieties of mathematical discourse in pre-modern socio-cultural contexts: Mesopotamia, Greece, and the Latin Middle Ages”, Science & society, xlix (1985), 4–41, pp. 11–17, and below, Section 2.

Dodge B. (ed., tr.), The Fihrist of al-Nadīm: A tenth-century survey of Muslim culture (2 vols, New York and London, 1970), ii, 634ff. In other parts of the catalogue, al-Nadīm mentions many non-Greek authors; their absence in the chapter on mathematics can hence not be explained as a result of personal prejudice on his part.

This follows, e.g., if one compares Indian and early Islamic algebra; cf. my “The formation of ‘Islamic mathematics’: Sources and conditions”, Science in context, i (1987), 281–329, p. 286.

On the whole, this holds even for the mathematical training of astrologers. Part of the Indian mathematical techniques which they took over was incorporated in books — But as technical chapters in astronomical books, only indirectly derived from the great mathematicians, where, e.g., Āryabhata's value for π would occur as something communicated “by divine revelation and angelic discourse” (cf. above, ref. 2; a general impression of the mathematics of early Islamic astronomy can be gained from Kennedy E. S. , “The lunar visibility theory of Ya^cqûb ibn Târiq”, Journal of Near Eastern studies, xxvii (1968), 126–32; Pingree D. , “The fragments of the works of Ya^cqûb ibn Târiq”, Journal of Near Eastern studies, xxvii (1968), 97–125; and idem, “The fragments of the works of al-Fazārī” (ref. 2)).

This word, like any other modern term, is not completely adequate. We might also speak of ‘crafts’, if only we keep in mind that no guild institution need be involved, and that the groups in question were composed of ‘higher artisans’.

Notable exceptions to this rule are made up by Sumerian and Babylonian accounting tablets and the Mycenean Linear B tablets. Others could be mentioned.

Hermelink H. , “Arabic recreational mathematics as a mirror of age-old cultural relations between eastern and western civilizations”, in al-Hassan A. Y. Karmi Gh. Namnum N. (eds), Proceedings of the First International Symposium for the History of Arabic Science, April 5–12, 1976 (Aleppo, 1978), ii, 44–52, p. 44.

Bhascara II, Lílávati, 53. Quoted from Colebrooke H. T. (ed., tr.), Algebra, with arithmetic and mensuration from the Sanscrit of Brahmagupta and Bhascara (London, 1817; reprinted, Wiesbaden, 1973), 24.

Rudolph Chr., Künstliche Rechnung mit der Ziffer …. 2nd edn (Vienna, 1540), the introduction to the chapter “Schimpfrechnung” (my translation and emphasis).

Liber abaci, my translation from Boncompagm B. (ed.), Scritti di Leonardo Pisano matematico del secolo decimoterzo, i: Il Liber abaci di Leonardo Pisano (Rome, 1857), 228.

Problem 52, version II. My translation from Folkerts M. , “Die älteste mathematische Aufgabensammlung in lateinischer Sprache: Die Alkuin zugeschriebenen Propositiones ad acuendos iuvenes. Überlieferung, Inhalt, Kritische Edition”, Denkschriften der Österreichische Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, cxvi, part 6 (1978), 74.

I shall not go into details concerning the distinction between geometrical and everyday (practical) problems, but only refer to Aristotle's polemics against various sophists' approaches which simply miss that distinction and thus permit trivial solution (Analytica posteriora 75^b40–76^a3; De sophisticis elenchis 171^b16–22, 172^a3–7; Metaphysica 998^a1–4).

See, e.g., Heath Th. L. , A history of Greek mathematics (2 vols, Oxford, 1921), i, 218–70, “Special problems”, which lists these attempts.

Plato , Theaetetus. Sophist, ed. and trans. by Fowler H. N. (Loeb Classical Library, London and Cambridge, Mass., 1921), 24.

There are of course lots of individual exceptions. When mathematics per se has become a profession, people in need of a dissertation subject or another item in the list of publications will easily end up looking out for problems which are likely to be solved with the methods already at their disposal.

VAT 7532, in Neugebauer O. , Mathematische Keilschrift-texte (3 vols, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik. Abteilung A: Quellen, iii, ersterdritter Teil (Berlin, 1935–37)), i, 294f. The translation is mine, and builds on my reinterpretation of the Old Babylonian mathematical terminology. Without going into irrelevant details the text should be comprehensible with the following explanations: (1) Numbers are written in a sexagesimal place-value system (Neugebauer's notation). (2) 1 cubit = nindan; the nindan is the basic length unit and equals approximate 6m. (3) To “detach the igi of n” means finding its reciprocal (). (4) “To raise” means calculation of a concrete entity through multiplication. (5) “To repeat until twice” means (concrete) doubling. (6) “To make a confront itself” means constructing a square with side a; if we do not care about the real (geometric) method of the Babylonians we may translate it “to square”. (7) “a makes b equilateral” means “b is the side of a square with area a“; in numerical interpretation, b = √ a.

6,14,24·z² – 12,0·z = 1,0,0, where z is of the original length of the reed.

According to text IX in Bruins E. M. Rutten M. , Textes mathématiques de Suse (Mémoires de la Mission Archéologique en Iran, xxxiv (Paris, 1961)), 63f. See my “Algebra and naive geometry: An investigation of some basic aspects of Old Babylonian mathematical thought”, Filosofi og videnskabsteori på Roskilde Universitetscenter, 3. Række: Preprints og reprints, 1987, nr 2 (to be published in Altorientalische Forschungen, xvii), 114f.

Quoted from Saidan A. S. (ed., tr.), The Arithmetic of al-Uqlīdisī (Dordrecht, 1978), 337.

This is not the place to go into details about the substance of Babylonian ‘pure’ mathematics. I shall only refer to my “Algebra and naive geometry” (ref. 24), which gives the reasons why its ‘algebra’ must have been built on geometrical (though ‘naive’, not critical) argumentation, and where the overall cognitive orientation of Babylonian mathematics is also discussed (Chapter 9).

Mannheim K. , Ideologie und Utopie , 4. Auflage (Frankfurt a. M., 1965), 71f., 251f., and passim.

The detailed arguments for the following discussion of composite fractions are given in my “On parts of parts and ascending continued fractions: An investigation of the origins and spread of a peculiar system”, Filosofi og videnskabsteori på Roskilde Universitetscenter, 3. Række: Preprints og reprints, 1988, nr 2 (to be published in Centaurus). Here I also discuss why similar usage in different cultures cannot be explained away as a random phenomenon.

Tropfke J. , Geschichte der Elementarmathematik, 4. Auflage, Band 1: Arithmetik und Algebra. Vollständig neu bearbeitet von Kurt Vogel, Karin Reich, Helmuth Gericke (Berlin and New York, 1980), 573–660.

Arithmetica, I, xxiv-xxv are unmistakable stripped versions of the “purchase of a horse” ( Tannery P. (ed., tr.), Diophanti Alexandrini Opera omnia cum graecis commentariis (2 vols, Leipzig, 1893–95), i, 56–61).

Soubeyran D. , “Textes mathématiques de Mari”, Revue d'Assyriologie, lxxviii (1984), 19–48, p. 30. The text is discussed and compared to later versions in my “Al-Khwârizmî, Ibn Turk, and the Liber mensurationum: On the origins of Islamic algebra”, Erdem, ii (Ankara, 1986), 445–84, pp. 477–9.

Connected to a tale about a peasant and his servant, whose wages are determined as successively doubled harvests from one grain of rice; reported in Thompson S. , Motifindex of folk-literature (6 vols, rev. and enl. edition, London, 1975), v, 542, no. Z 21.1.1.

Plato , Laws, ed. and trans. by Bury R. G. (2 vols, Loeb Classical Library, London and Cambridge, Mass., 1926), ii, 104f.

We observe that this oral character of genuine recreational mathematics sets it apart from Old Babylonian sub-scientific mathematics, which was carried by written texts even though the details of didactical explanation have normally been given orally. Genuine recreational mathematics belongs with ‘lay’ traditions; the methodical orchestration of scholasticized mathematics negates its recreational value, even though single problems like the above “broken reed” may betray a recreational origin.

See Thompson S. , The folktale (New York, 1946), 13ff.

My English translation from Suter H. , “Das Buch der Seltenheiten der Rechenkunst von Abū Kāmil al-Miṣrī”, Bibliotheca mathematica, 3. Folge, xi (1910–11), 100–20, p. 100.

Another group of encyclopedias does reflect the sub-scientific traditions, those concerned with the practice of science and not directly with books. A good example is found in Al-Fārābī, Catálogo de las ciencias, edición y traducción Castellana por Palencia A. G. , segunda edición (Madrid, 1953), 39–53, cf. Arabic, 73. Here, seven branches of mathematics are distinguished, the last of which is ^cilm al-ḥiyal, “science of devices/ingenuities”, which according to the description appears to refer to practical applications of ‘scientific’ mathematics — I.e., to ‘applied mathematics' as defined in chapter 1. At the same time, several of the other branches (so arithmetic and geometry) are subdivided into ‘theoretical’ and practical’. No doubt, this division is a reflection of Aristotle's conceptions — But Aristotle's categories are apparently understood exactly as done above, as ‘scientific’ and sub-scientific mathematics, respectively.

In Babylonia this is made fully clear, e.g., in line 27 of the “Examination Text A”, “Kennst du die Multiplikation, die Bildung von reziproken Werten und Koeffizienten, die Buchführung, die Verwaltungsabrechnung, die verschiedensten Geldtransaktionen, (kannst du) Anteile zuweisen, Feldanteile abgrenzen?” (quoted from Sjöberg Å. , “Der Examenstext A”, Zeitschrift für Assyriologie und vorderasiatische Archäologie, lxiv (1975), 137–76, p. 145). In the case of Egypt, it follows from the range of subjects dealt with in the Rhind Mathematical Papyrus, as also from the Papyrus Anastasi I, a ‘satirical letter’ much used in the scribal school to vilify the poor dunce who knows neither how to calculate a ramp, nor to provide rations for troops, nor to find the number of men required to transport an obelisk (etc.); translated in A. H. Gardiner, The Papyrus Anastasi I and the Papyrus Koller, together with parallel texts, Egyptian hieratic texts, Series I: Literary texts from the New Kingdom, Part I (Leipzig, 1911).

See text III, lines 2, 3 and 30, in Bruins Rutten , Textes mathématiques de Suse (ref. 24), 25f., and commentary pp. 31, 33.

See Problems 41ff. in Chace A. B. , The Rhind mathematical papyrus, ii: Photographs, transcription, transliteration, literal translation (Oberlin, Ohio, 1929).

I Kings 7, 23; II Chronicles 4, 2.

See, e.g., Problems nos. 32 and 42 in Parker R. A. , Demotic mathematical papyri (Providence and London, 1972), 40f. and 54ff.

See Cantor M. , Vorlesungen über Geschichte der Mathematik, i: Von den ältesten Zeiten bis zum Jahre 1200 n. Chr, dritte Auflage (Leipzig, 1907), 551; and Gandz S. (ed., tr.), The Mishnat ha Middot, the first Hebrew geometry of about 150 C.E., and the Geometry of Muhammad ibn Musa al-Khowarizmi, the first Arabic geometry [c. 820] (Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung A: Quellen, ii (Berlin, 1932)), 49.

Or even the ratio 3! See Vitruvius , De architectura X, ix, 1 and 5. The text is doubtful, cf. the text with appurtenant notes in Vitruvius, On architecture, ed. and trans. by Granger F. (2 vols, Loeb Classical Library, London and Cambridge, Mass., 1931, 1934), ii, 318 and 322.

Earlier parallels could be pointed out. So, when Middle Kingdom Egyptian scribes began making their accounts in unit fractions instead of metrological sub-units (as had been done in the Old Kingdom), the only reason is that they had learned this system in school — And the reason to introduce it in school will have been that it was easier to argue for (and thus teach unambiguously) the precise calculations in unit fraction notation than to defend the necessarily approximate (and thus ambiguous) solutions of practical problems in metrological units (see my “Influences of institutionalized mathematics teaching …” (ref. 2), 34).

This formulation fits ‘Western’ civilization (the medieval centre of which was the Islamic world). Indian high-level algebra is older, but it did not influence the Western development (cf. above).

Rosen F. (ed., tr.), The Algebra of Muhammad ben Musa (London, 1831). Another book on the subject and probably also the same title (Kitāb al-jabr wa'l-muqābalah) was written by ibn Turk at almost the same time (A. Sayılı, Abdülhamid ibn Türk'ün katıṣık denklemlerde mantıkî zaruretler adlı yazısı ve zamanın cebri (Logical necessities in mixed equations by ^cAbd al Ḥamîd ibn Turk and the algebra of his time) (Publications of the Turkish Historical Society, series VII, 41; Ankara, 1962)). The question of priority can not be settled with certainty, but terminological considerations suggest that ibn Turk was at least independent of al-Khwārizmī (see my “Al-Khwârizmī, Ibn Turk, and the Liber mensurationum …” (ref. 31), 474, note 28).

Luckey P. , “T̠ābit b. Qurra über den geometrischen Richtigkeitsnachweis der Auflösung der quadratischen Gleichungen”, Berichte der Sächsischen Akademie der Wissenschaften zu Leipzig. Mathematisch-physische Klasse, xciii (1941), 93–114.

The detailed arguments for this (which also involve Abū Kāmil's Algebra) are given in my “Al-Khwârizmî, Ibn Turk, and the Liber mensurationum …”. One correction should be given to the exposition in that paper: The distinction between two groups of calculators each with their own specific methods referred on p. 472 to Abū Kāmil is not found in the Arabic text ( Hogendijk J. P. (ed.), Abū Kāmil Shujā^c ibn Aslam, The book of algebra. Kitāb al-jabr wa l-muqābala (Publications of the Institute for the History of Arabic-Islamic Science, series C: Facsimile editions, xxiv (Frankfurt a. M., 1986)). It seems to have been inserted into the Hebrew Renaissance translation and thus to be irrelevant to the early history of the subject.

See Datta B. Singh A. N. , History of Hindu mathematics: A source book (2 vols, Lahore, 1935–38; reprint edn, Bombay, 1962), i, 169. In India, the term is used in connection with the extraction of a square root, and the interpretation is geometrical and thus meaningful. In the al-jabr-tradition, the term is not understood geometrically, which deprives it of metaphorical meaning.

See problems nos. 13 and 17 in Baillet J. , Le Papyrus mathématique d'Akhmīm (Mémoires de la Mission Archéologique Française au Caire, ix, part 1 (Paris, 1892)), 70 and 72. It should be observed that there is nothing strange in the use of the same term for the unknown in problems of type a·x = b and for the second-degree term in problems of type y² + a·y = b, if only we put y² = x, thus transforming the latter problem into x + a·√ x = b. Islamic algebras will normally give the value for both x and for y, thus regarding the māl as an unknown in its own right. The transformation is hence justified by the sources.

See also Datta Singh , History of Hindu mathematics (ref. 50), ii, 9f.

Rosen (ed., tr.), The Algebra of Muhammad ben Musa (ref. 47), 41f.

For other operations al-Khwārizmī did invent geometrical justifications for the rhetorical reductions himself, but then the wording is different (“This is what we intended to elucidate”; “We had, indeed, contrived to construct a figure also for this case, but it was not sufficiently clear”, Rosen (ed., tr.), The Algebra of Muhammad ben Musa (ref. 47), 32 and 34).

Edited critically in Busard H. L. L. , “L'algèbre au moyen âge: Le ‘Liber mensurationum’ d'Abū Bekr”. Journal des savants, Avril-Juin 1968, 65–125.

Detailed analysis in my “Al-Khwârizmî, Ibn Turk, and the Liber mensurationum…” (ref. 31), 456–68. It should be noted that the geometrical character of the argument only follows from indirect arguments — The figures belonging with the solutions are lost in the Latin translation.

Chapter X, xiii, in “Abu-l-Vafa al-Buzdžani, Kniga o torn, čto neobxodimo remeslenniku iz geometričeskix postroenij”, ed. and trans. by Krasnova S. A. in Grigor'jan A. T. Juškevič A. P. (eds), Fiziko-matematičeskie nauki v stranax vostoka (Sbornik statej i publikacij, i; Moscow, 1966), 42–140, p. 115.

This might also be what al-Fārābī tells us when stating in his Catalogue that the science of algebra is “common to arithmetic and geometry” (other interpretations are possible) — My translation from Al-Fārābī, Catálogo de las ciencias, ed. by Palencia (ref. 37), 52.

BM 13901, in Neugebauer , Mathematische Keilschrift-texte (ref. 22), iii, 1–5, cf. translations and discussions in my b (ref. 24), 43–56.

Analysis in my “Algebra and naive geometry”, 48–51.

Rosen (ed., tr.), The Algebra of Muhammad ben Musa (ref. 47), 13–15. The procedure fits the algorithm badly; when it is none the less used (and used first), the reason must be that it was familiar.

Friberg Jöran , “Mathematik”, Reallexikon der Assyriologie, section 5.4.k (forthcoming).

This ‘Islamic miracle’ (a term coined in imitation of the well-known ‘Greek miracle’, the creation of the autonomous ‘scientific’ approach) is dealt with extensively in my “The formation of ‘Islamic mathematics’ …” (ref. 9).