Further questions to the historiography of Arabic (but not only Arabic) mathematics from the perspective of Romance abbacus mathematics

Contribution to the

9ième Colloque Maghrébin sur l'Histoire des Mathématiques Arabes

Tipaza, 12–13–14 mai 2007


    For some years I have been engaged in a close reading of early Italian abbacus books and related material from the Ibero-Provençal orbit and in comparison of this material with Arabic mathematical writings. At the 7th North African Meeting on the History of Arab Mathemat­ics in Marrakesh in 2002 I presented the first outcome of this investigation: namely that early Italian abbacus algebra was not influenced by the Latin algebraic writings of the 12th-13th centuries (neither the translations of al-Khwārizmī nor the works of Fibonacci); instead, it received indirect inspiration from a so far unknown link to the Arabic world, viz to a level of Arabic algebra (probably linked to mu‛āmalāt mathematics) of which very little is known. At the 8th Meeting in Tunis in 2004 I presented a list of linguistic clues which, if applied to Arabic material, might enable us to say more about the links between the Romance abbacus tradition and Arabic mu‛āmalāt teaching.

 Here I investigate a number of problem types and techniques which turn up in some but not necessarily in all of the following source types:

– Romance abbacus writings,

– Byzantine writings of abbacus type,

– Arabic mathematical writings of various kinds,

– Sanskrit mathematical writings,

in order to display the intricacies of the links between these – intricacies which force us to become aware of the shortcomings of our current knowledge, and hence formulate questions that go beyond the answers I shall be able to present.




“As the Outsider Walked In: The Historiography of Mesopotamian Mathematics Until Neugebauer”

Paper prepared for the conference

Otto Neugebauer between History and Practice of the Exact Sciences

Institute for the Study of the Ancient World, New York University, November 12–13, 2010



    Those who nowadays work on the history of advanced-level Babylonian mathematics do so as if everything had begun with the publication of Neugebauer’s Mathematische Keilschrift-Texte from 1935–37 and Thureau-Dangin’s Textes mathématiques babyloniens from 1938, or at most with the articles published by Neugebauer and Thureau-Dangin during the few preceding years. Of course they/we know better, but often that is only in principle. The present paper is a sketch of how knowledge of Babylonian mathematics developed from the beginnings of Assyriology until the 1930s, and raises the question why an outsider was able to create a breakthrough where Assyriologists, in spite of the best will, had been blocked. One may see it as the anatomy of a particular “Kuhnian revolution”.




“Fibonacci – Protagonist or Witness? Who Taught Christian Europe about Mediterranean Commercial Arithmetic?”

Paper prepared for the workshop

Borders and Gates or Open Spaces? Knowledge Cultures in the Mediterranean During the 14th and 15th Centuries

Departamento de Filosofia y Lógica, Universidad de Sevilla, 17–20 December 2010



    Fibonacci during his boyhood went to Bejaïa, learned about the Hindu-Arabic numerals there, and continued to collect information about their use during travels to the Arabic world. He then wrote the Liber abbaci, which with half a century’s delay inspired the creation of Italian abbacus mathematics, later adopted in Catalonia, Provence, Germany etc.
    This story is well known – too well known to be true, indeed. There is no
doubt, of course, that Fibonacci learned about Arabic (and Byzantine) commercial arithmetic, and that he presented it in his book. He is thus a witness (with a degree of reliability which has to be determined) of the commercial mathematics thriving in the commercially developed parts of the Mediterranean world. However, much evidence – presented both in his own book, in later Italian abbacus books and in similar writings from the Iberian and the Provençal regions – shows that the Liber abbaci did not play a central role in the later adoption. Romance abbacus culture came about in a broad process of interaction with Arabic non-scholarly traditions, interaction at first apparently concentrated in the Iberian region.




“The fifteenth-seventeenth century transformation of abbacus algebra: Perhaps - though not thought of by Edgar Zilsel - the
best illustration of the 'Zilsel-Needham thesis'”

Paper prepared for

Summer School on the History of Algebra

Institute for the History of the Natural Sciences, Chinese Academy of Science, 1-2 September 2011◄



    In 1942, Edgar Zilsel proposed that the sixteenth-seventeenth-century emergence
of Modern science was produced neither by the university tradition, nor by the
Humanist current of Renaissance culture, nor by craftsmen or other practitioners
but through an interaction between all three in which all were indispensable
for the outcome. He only included mathematics via its relation to the 'quantitative spirit'. The present study tried to apply Zilsel's perspective to the emergence
of the Modern algebra of Viète and Descartes (etc.), by tracing the reception of
algebra within the Latin-Universitarian tradition, the Italian abbacus tradition
and Humanism, and the exchanges between them, from the twelfth through the
late sixteenth and early seventeenth century.



Mesopotamian Calculation: Background and Contrast to Greek Mathematics

Contribution to

IX Congresso della Società Italiana di Storia della Matematica

Genova, 17–19 novembre 2011



    The fourth-millennium state formation process in Mesopotamia was intimately linked to accounting and to a writing system created exclusively as support for accounting. This triple link between the state, mathematics and the scribal craft lasted until the end of the third millennium, whereas the connection between learned scribehood and accounting mathematics lasted another four hundred years. Though practical mathematics was certainly not unknown in the Greco-Hellenistic-Roman world, a similar integration was never realized.
    Social prestige usually goes together with utility for the power structure (not to be confounded with that mere utility for those in power which characterizes a working and tax/tribute-paying population), and until the 1600 BCE scribes appears to have enjoyed high social prestige.
    From the moment writing and accounting was no longer one activity among others of the ruling elite (c. 2600 BCE) but the task of a separate profession, this profession started exploring the capacity of the two professional tools, writing and calculation.Within the field of mathematics, this resulted in the appearance of “supra-utilitarian mathematics”: mathematics which to a superficial inspection appears to deal with practical situations but which, without having theoretical pretensions, goes beyond anything which could ever be encountered in real practice. After a setback in the late third millennium, supra-utilitarian mathematics reached a high point – in particular in the so-called “algebra” during the second half (1800–1600) of the “Old Babylonian” period.
    Analysis of the character and scope of this “algebraic” discipline not only highlights the difference between theoretical and high-level supra-utilitarian mathematics, it also makes some features of Greek theoretical mathematics stand out more clearly. Babylonian “algebra” was believed by Neugebauer (and by many after him on his authority) have inspired Greek so-called “geometric algebra”. This story, though not wholly mistaken, is today in need of reformulation; this reformulation throws light on one of the processes that resulted in the creation of Greek theoretical mathematics.




Written Mathematical Traditions in Ancient Mesopotamia: knowledge, ignorance, and reasonable guesses

Contribution to the conference

Traditions of Written Knowledge in Ancient Egypt and Mesopotamia

Frankfurt am Main, 3.–4. December 2011



    Writing, as well as various mathematical techniques, were created in proto-literate Uruk in order to serve accounting, and Mesopotamian mathematics as we know it was always expressed in writing. In so far, mathematics generically regarded  was always part of the generic written tradition.

   However, once we move away from the generic perspective, things become much less easy. If we look at elementary numeracy from Uruk IV until Ur III, it is possible to point to continuity and thus to a “tradition”, and also if we look at place-value practical computation from Ur III onward – but already the relation of the latter tradition to type of writing after the Old Babylonian period is not well elucidated by the sources.

    Much worse, however, is the situation if we consider the sophisticated mathematics created during the Old Babylonian period. Its connection to the school institution and the new literate style of the period is indubitable; but we find no continuation similar to that descending from Old Babylonian beginnings in fields like medicine and extispicy. Still worse, if we look closer at the Old Babylonian material, we seem to be confronted with a small swarm of attempts to create traditions, but all rather short-lived. The few mathematical texts from the Late Babylonian (including the Seleucid) period also seem to illustrate attempts to create traditions rather than to be witnesses of a survival for sufficiently long to deserve the label “traditions”.