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Historiography of Mathematics in the 19th and 20th Centuries pp 53–78Cite as

Birkhäuser

Mesopotamian Mathematics, Seen “from the Inside” (by Assyriologists) and “from the Outside” (by Historians of Mathematics)

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Abstract

Since the 1950s, “Babylonian mathematics” has often served to open expositions of the general history of mathematics. Since it is written in a language and a script which only specialists understand, it has always been dealt with differently by the “insiders”, the Assyriologists who approached the texts where it manifests itself as philologists and historians of Mesopotamian culture, and by “outsiders”, historians of mathematics who had to rely on second-hand understanding of the material (actually, of as much of this material as they wanted to take into account), but who saw it as a constituent of the history of mathematics. The article deals with how these different approaches have looked in various periods: pre-decipherment speculations; the early period of deciphering, 1847–1929; the “golden decade”, 1929–1938, where workers with double competence (primarily Neugebauer and Thureau-Dangin ) attacked the corpus and demonstrated the Babylonians to have possessed unexpectedly sophisticated mathematical knowledge; and the ensuing four decades, where some mopping-up without change of perspective was all that was done by a handful of Assyriologists and Assyriologically competent historians of mathematics, while most Assyriologists lost interest completely, and historians of mathematics believed to possess the definitive truth about the topic in Neugebauer ’s popularizations.

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Notes

  1. 1.

    A conceptual clarification: The “Near East” encompasses Egypt, the Palestino-Syrian area, Arabia and Mesopotamia—sometimes other neighbouring areas are included as well. Mesopotamia largely coincides with present-day Iraq. Its northern third is Assyria, and the remainder is Babylonia. Chaldea strictly speaking is the southern third (in the third millennium bce Sumer), but often in the quotations that follow it stands for the whole of Babylonia.

  2. 2.

    My translation, as everywhere below where nothing else is stated. All translated quotations can be found in original language in the preprint version of the article, on http://rudar.ruc.dk//bitstream/1800/10613/1/Hoyrup_2013_c_Mesopotamian_mathematics_from_the_inside_and_from_the_outside_S.pdf. In cases where the titles of publications have been translated, the genuine titles can be found in the bibliography. The notes to the quotation are due to Montucla, my additions are in square brackets. Similarly below for notes within quotations.

  3. 3.

    Geograph.lib.xvii.

  4. 4.

    Cedrenus [an 11th-century Byzantine historian].

  5. 5.

    Diog. Laer. in proemio. [Hicks 1925: I, 12].

  6. 6.

    In Phædro. p. 1240 ed. 1602. [274c].

  7. 7.

    Ibid.

  8. 8.

    [At this point, the astronomer Joseph-Jérôme Lalande (1732–1807) adds the following in the second edition—the earliest reasoned reference to Babylonian mathematics (Montucla 1799: I, 43f): “It is even quite difficult not to affiliate them with the Chaldeans. They, indeed, present us with the first traces of astronomical knowledge, very advanced at that. How would they, without that tool, have been able to discover several astronomical periods, knowledge of which has come down to us!” Apart from that, Montucla ’s passage is unchanged].

  9. 9.

    Ant. Jud. liv. i c. 9. [Actually chapter 8].

  10. 10.

    Abraham is at least absent from Giuseppe Biancani ’s (1566–1624) Clarorum mathematicorum chronologia (1615, 39), and also from Gerardus Vossius’s (1577–1649) De universae mathesios natura et constitutione liber and Chronologia mathematicorum (1650), while Polydorus Vergilius (c. 1470–1555) (1546, 59f) has no chapter reference. Since Montucla does not abstain from identifying Ramus by name when chiding him for following “the inclination of the mob toward everything that seems marvellous” (p. 450), the present reference is most likely at least not to be to Ramus alone.

  11. 11.

    What follows about work done before 1860 is drawn, when no original sources are referred to, from Charles Fossey ’s (1869–1946) very detailed exposition of (good and bad) arguments and results (Fossey 1904, 85–220).

  12. 12.

    Grotefend (1802) was published only in full in (Meyer 1893), for which reason I build on Fossey ’s account (1904, 102–111) of the arguments that circulated.

  13. 13.

    This system is sexagesimal but not positional until 100, after which it is combined with word-signs for 100 and 1000.

  14. 14.

    Archibald Henry Sayce (1845–1933), when returning to the text in (1875, 490; cf. Sayce 1887, 337–340), reinterprets the topic as a table of lunar longitudes. Geometrically, the two interpretations are equivalent, but the final verb of the lines (DU, “to go”) suggested this new understanding.

  15. 15.

    In contrast, the just published Blackwell Encyclopedia of Ancient History planned the same number of pages for Mesopotamian mathematics and Mesopotamian hairstyles. It should be added that those who planned the volume had little idea about Mesopotamia (nor were they very interested in receiving advice, however).

  16. 16.

    We may see this belief as the last scholarly and pseudo-scholarly survivor of the Renaissance faith in ancient prisca sapientia. Paradoxically, Oppert had pointed out already in (1886, 90) that there were “in Assyria and Chaldea, as everywhere else, ceaseless variations in the measures”, which should have warned him against the dangers inherent in the comparative method.

  17. 17.

    See for example p. 427f on the postulated unit “hair”, which leads him to rather far-fetched hypotheses (presented “with all reserve”, it is true).

  18. 18.

    Basing himself on indirect evidence and on Greek writings, Johannes Brandis (1830–1873) had already claimed that the unending sexagesimal fraction system of the Greek astronomers had to be of Chaldean origin “even if we never find direct or indirect testimony ascribing it to them” (Brandis 1866, 18).

  19. 19.

    Hilprecht quotes this passage from Carl Bezold’s (1859–1922) “concise survey of the Babylonian-Assyrian literature” (Bezold 1886, 225): “As far as we know by now, Babylonian-Assyrian mathematics primarily served astronomy, and this on its part a pseudo-science, astrology, which probably arose in Mesopotamia, propagated from there into the Gnostic writings, and was inherited by the Middle Ages, although we are not yet able to reconstruct this whole chain, the links of which are often broken from each other”.

  20. 20.

    Outside Assyriology, in particular among natural scientists taking interest in Antiquity and its mysteries, the comparativist trend is still alive and kicking—see (Berriman 1953; Rottländer 2006; Lelgemann 2004).

  21. 21.

    [A footnote refers to Moritz Cantor ’s Vorlesungen I, on which below.].

  22. 22.

    Now known as BM 85194 and BM 85210.

  23. 23.

    Only the terms for (what can approximately be translated as) square and cube roots were known since Moritz Cantor ’s use of Hilprecht ’s material in (1908). Quite a few of Weidner ’s readings later turned out to be philologically wrong while their technical interpretation was adequate. What was correct, however, was important later on, and some of the philological errors were still taken over in Neugebauer ’s early interpretations without great damage.

  24. 24.

    Already Cantor ’s “mathematical contributions to the cultural life of the nations” had contained a chapter on the Babylonians (Cantor 1863, 22–38). At the time, however, he had only been able to speak about the decipherment; about “Oriental” culture in general; and about the writing system, about integer numerals and about the possible use of some kind of abacus (a hypothesis which he repeats in the Vorlesungen).

  25. 25.

    The exceptions are vols 2, 16, 19–20 and 25, to which I have no access; there is no reason to believe they should change the general picture.

  26. 26.

    Some stylistic features do point to Neugebauer as the writer. As revealed by the quotation in note 41, at least Toeplitz also had an attitude in conflict with the present text.

  27. 27.

    “Moreover, even the readings themselves can be considerably improved, once the substantial contents has been elucidated” (p. 67).

  28. 28.

    [The quotes around the word algebraic indicate that Neugebauer refuses to make hypotheses about which kind of algebraic thought is involved in the texts. The many algebraic formulas in his commentary are not meant to map the thinking of the authors of the texts; they show why the calculations are pertinent (or, rarely, why they are not)].

  29. 29.

    This booklet had no strong impact—it drowned in the fury surrounding the new discoveries of the time. However, a revised English translation (including much about the Babylonian “algebra”) appeared in Osiris in (1939) on George Sarton’s initiative (p. 99).

  30. 30.

    This is what von Soden (1939, 144) tells about the purpose of this parallel edition: “This new work is not meant to replace Neugebauer ’s MKT; indeed, the phototypes and autographs are not repeated, nor are all texts treated anew. Th.-D.s aim was instead, leaving the arithmetical tables completely aside (only the introduction speaks briefly about them) to make those problem texts that are sufficiently well preserved to allow at least a generally satisfying understanding available to as many researchers as possible in a cheaper edition, since the exorbitant price of the MKT unfortunately hampers its wider diffusion.” But further: “While thus the specialist researcher will also in future not be able to give up Neugebauer ’s MKT as the complete source collection, with the just mentioned exception [two small texts from Susa with area calculations published by Vincent Scheil in 1938], then precisely he will also not be able to pass over Th.-D.’s new edition, as nobody will be able to digest in brief the large number of corrected readings and the immensely weighty lexical, grammatical and substantial observations, masterly concise though they are.” In Thureau-Dangin ’s own words (TMB, xl): “The present volume contains no text which has not been published elsewhere in its original form [that is, without a translation of ideograms into syllabic Akkadian]. The main task I have set myself while preparing it has been to make documents accessible to the historians of mathematical thought.”.

  31. 31.

    Curiously enough, (Neugebauer and Sachs 1945) is much less afraid of ascribing modern mathematical concepts to the Babylonians than Neugebauer had been in the 1930s—such as logarithms, p. 35, cf. (Neugebauer (1935–1937), I, 363–365). Whether this is due to Sachs ’s influence or Neugebauer himself had been convinced by what others had read into (Neugebauer 1935–1937) I am unable to say.

  32. 32.

    The edition of one long Seleucid text (BM 34568) in (Neugebauer 1935–1937, III, 14–22) is also, according to Neugebauer , “apart from a few trifles due to Herrn Dr. Waschow ”. This work must be dated between 1935 and 1937.

  33. 33.

    He returns to this link time and again in the numerous angry letters I have from his hand. I suppose he can be believed on this account, his general unreliability notwithstanding. According to the preface (TMS, xi), Rutten made the hand copies and collaborated with Bruins on the translation. However, already the translation of word signs into Akkadian contains so many blunders of a kind no competent Assyriologist would commit that Bruins can be clearly seen to have had the upper hand concerning everything apart from the hand copies.

  34. 34.

    A further text covering three tablets was found on the ground, apparently left behind by illegal diggers as too damaged. It was published by Albrecht Goetze (1897–1971) in (1951).

  35. 35.

    Until then, von Soden had never worked directly on mathematical questions himself; but he had always been interested in the topic, as can be seen from his careful and extensive reviews of Neugebauer 1935–1937 (1937) and TMB (1939). He also made a review of TMS in (1964), an indispensable companion piece to the edition itself.

  36. 36.

    The outcome can be seen as an extension of a division of the corpus into a “northern” and a “southern” group which Neugebauer had suggested in (1932, 6f); but Neugebauer ’s arguments had been of a wholly different nature.

  37. 37.

    In 1978–79, Carlo Zaccagnini thus published at least four papers on the metrologies of peripheral areas.

    I disregard publications in Russian, most noteworthy of which is (Vajman 1961)—my reading of Russian, which reached the level of “rudimentary” 25 years ago, has vanished completely since then for lack of practice. An exhaustive survey, often with discussion, of all at least minimally pertinent publications (also those in Russian) for the period 1945–1980 will be found in (Friberg 1982, 67–130).

  38. 38.

    “Of the creators of Babylonian mathematics we know nothing whatsoever except the result of their work” (p. 6). That the texts are school texts is intimated by photos of presumed schoolrooms from Mari (which are actually store-rooms) and occasional references to a “schoolboy”—but schooling seems to be just as timeless as mathematics. In 1964, we may observe, more was known about the Old Babylonian scribe school than in 1934—cf. (Kramer 1949; Falkenstein 1953; Gadd 1956).

  39. 39.

    However, all of this is described in (Thureau-Dangin 1939), who distinguishes the “abstract” (namely place-value) system “intended only to serve as an instrument of calculation” (p. 117) from the ordinary sexagesimal but non-positional system.

  40. 40.

    “It should be added that an entirely consistent use of the sexagesimal system is to be found only in the mathematical and astronomical texts, and even in astronomical texts one can find year numbers written as, e.g., 1–me 15 (meaning 1 hundred 15) instead of 155. In practical life the Babylonians showed the same profound disregard for rationality in their use of units for weight and measure as does the modern English-speaking world” (p. 20). The year number in question is written in precisely that number system which Hincks had deciphered in 1847, cf. note 13—the very first contribution to the study of Assyro-Babylonian mathematics.

  41. 41.

    “Nothing is farther from me than to teach a history of the infinitesimal calculus; I myself as a student ran away from a lecture of that kind. History is not at stake, but the genesis of problems, acts and demonstrations, and the decisive turning points in this genesis” (Toeplitz 1927, 94).

  42. 42.

    Dirk Struik’s (1894–2000) Concise History of Mathematics from (1948) deals with Mesopotamian mathematics too briefly to allow description in similar depth (pp. 23–32). Struik’s layout, however, is similar: The analysis is embedded in general social history; non-positional as well as place-value system are described; but like Vogel , Struik has no possibility to go beyond Neugebauer .

  43. 43.

    Boyer had written about the “concepts of the calculus” (1949), and Kline ’s title refers to “mathematical thought”. Hofmann had written among other things about Ramon Lull’s squaring of the circle in (1942), and had tried there to penetrate the thinking and motives of Lull (without which he would indeed have been unable to conclude anything of interest).

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    Jens Høyrup

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    Henrik Kragh Sørensen

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Høyrup, J. (2016). Mesopotamian Mathematics, Seen “from the Inside” (by Assyriologists) and “from the Outside” (by Historians of Mathematics). In: Remmert, V., Schneider, M., Kragh Sørensen, H. (eds) Historiography of Mathematics in the 19th and 20th Centuries. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-39649-1_4

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