Jens Høyrup

ON A COLLECTION OF GEOMETRICAL RIDDLES AND THEIR ROLE IN THE SHAPING OF FOUR TO SIX "ALGEBRAS"

For more than a century, there has been some discussion about whether medieval Arabic al-jabr (and hence also later European algebra) has its roots in Indian or Greek mathematics. Since the 1930s, the possibility of Babylonian ultimate roots has entered the debate. This article presents a new approach to the problem, pointing to a set of quasi-algebraic riddles that appear to have circulated among Near Eastern practical geometers since c. 2000 BCE, and which inspired first the so-called “algebra” of the Old Babylonian scribal school and later the geometry of Elements II (where the techniques are submitted to theoretical investigation). The riddles also turn up in ancient Greek practical geometry and Jaina mathematics. Eventually they reached European (Latin and abbaco ) mathematics via the Islamic world. However, no evidence supports a derivation of medieval Indian algebra or the original core of al-jabr from the riddles.

Investigations of an early Sumerian division problem, c. 2500 B.C.

Abstract Two Sumerian school tablets from c. 2500 B.C. —containing respectively a correct and an erroneous solution of the same problem of dividing a very large amount of grain into rations of 7 sila each—are analyzed. It is suggested that the error committed cannot reasonably be explained by two earlier conjectures on the method of solution (normal “long division,” and multiplication by the reciprocal), but only by conversion to an intermediary unit and calculation in two steps analogous to the principle of “long division.” Also discussed are some possible implications of this result for contemporary Sumerian arithmetical abilities and general techniques.

Double Regge analysis and the extended Dolen-Horn-Schmid duality applied to the processes pp → Δ++ π−p and pp → ppω

Abstract A simplified double Regge model analysis is applied down to threshold to the processes pp → Δ++π−p and pp → pωp in order to test the extension of Dolen-Horn-Schmid duality proposed by Chew and Pignotti. This gives rather good agreement with experiment, and further leads to quantitative conclusions concerning relative couplings of different reggeons, and concerning the vertex functions. The meson-exchange coupling is found to be stronger than the Pomeron exchange in multi-Regge diagrams even at the highest accelerator energies available at present.

The Babylonian Cellar Text BM 85200 + VAT 6599 Retranslation and Analysis

In a number of earlier publications, I have proposed a new understanding of the Old Babylonian mathematical technique known as “algebra„, concentrating on problems of the second and to some extent of the first degree.1 As a background to the following investigation of a particular text dealing in part with problems of the third degree I shall need a summary of my methods and my main results.

Mathematics Education in the European Middle Ages

This chapter is subdivided chronologically into periods that correspond to changes in educational system, aims, and ideals: c. 500–750, c. 750–1100, c. 1100–1200, and c. 1200–1500. To the extent it is possible, Latin and lay education (in its geographical and professional diversity) are distinguished, but the scarcity of sources for lay teaching causes the Latin (first quadrivial, later broader universitarian) type to preponderate in the exposition.

Mahāvīra’s geometrical Problems: Traces of unknown Links between Jaina and Mediterranean Mathematics in the classical Ages

In various publications [Hoy95, Hoy96, Hoy01] I have argued for the existence in (what Western Europe sees as) the Near East of a long-lived community of practical geometers - first of all surveyors - which was not or only marginally linked to the scribe school traditions, and which (with branchings) carried a stock of methods and problems from the late third millennium BCE at least into the early second millennium CE. The arguments for this conclusion constitute an intricate web, and I shall only repeat those of them which are of immediate importance for my present concern: the links between the geometrical section of Mahāvīra’s Gaṇita-sāra-saṅgraha and the practical mathematics of the Mediterranean region in the classical ages.

Fibonacci - Protagonist or Witness? Who Taught Catholic Christian Europe about Mediterranean Commercial Arithmetic?

Abstract Leonardo Fibonacci (ca. 1170 - after 1240) 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. Hindu- Arabic numerals, and Arabic mathematics, was thus transmitted through a narrow and unique gate. This piece of conventional wisdom 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, at least until ca. 1350 within an open space, apparently concentrated around the Iberian region.

Conceptual divergence¿canons and taboos¿and critique: reflections on explanatory categories

Abstract Since the late 19th century it has been regularly discussed whether, e.g., the ancient Egyptian way to deal with fractions or the Greek exclusion of fractions and unity from the realm of numbers was a mere matter of imperfect notations or due to genuine “conceptual divergence,” that is, to a mathematical mode of thought that differed from ours. After a discussion of how the notion of a “mode of thought” can be made operational through the linking of concepts to mathematical operations and practices it is argued (1) that cases of conceptual divergence exist, but (2) that the discussion of notational imperfection versus conceptual divergence is none the less too simplistic, since differences may also be due to deliberate choices and exclusions on the part of the authors of the ancient texts—for instance because such a choice helps to fence off a profession, because it expresses appurtenance to a real or imagined tradition, or as a result of a critique in the Kantian sense, an elimination of expressions and forms of reasoning that are found theoretically incoherent. The argument is based throughout on historical examples.

Mathematics and war: An invitation to revisit

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in-chief, Chandler Davis.

As the Outsider Walked in the Historiography of Mesopotamian Mathematics Until Neugebauer

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 to 1937 and Thureau-Dangin’s Textes mathematiques 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 their best will, had been blocked. One may see it as the anatomy of a particular “Kuhnian revolution”.