The ArgumentThe development of autonomous theoretical science is often considered a “Greek miracle.” It is argued in the present paper that another “miracle,” necessary for the creation of modern science, took place for the first time in the Islamic Middle Ages, viz. the integration of theory and practice.The discussion focuses on the mathematical disciplines. It starts by investigating the plurality of traditions which were integrated into Islamic mathematics during its formation, emphasizing practitioners' “sub-scientific” traditions, and shows how these were synthesized (...) in a way virtually unknown in earlier cultures. A discussion of the sociocultural roots of this specific synthesis concludes that a major role was played in the earlier period by the combination of fundamentalist convictions characteristic of Islam – that the most humble daily activity is directly responsible to the highest ontological level, while conversely this highest level is concerned with the humblest ranks of daily existence – with the absence of an institutionalized “Church” able to monopolize the interpretation of the mutual bond of the divine and the everyday levels.As the institutions of learning crystallized around the turn of the millennium, the integrative attitude to theory and practice was fixated institutionally; the latter process is discussed, first with the example of the madrasah institution as the carrier of an arithmetical textbook tradition, and second with that of the bond between astronomy and theoretical geometry. (shrink)

For more than a century, there has been some discussion about whether medieval Arabic al-jabr 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. The riddles also turn up in ancient Greek practical geometry and Jaina mathematics. Eventually they reached European 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. (shrink)

Eine Reihe von mittelalterlichen Schriften zur Landmessung (vom 9. islamischen Jahrhundert bis zu Fibonacci und Pacioli) enthält eine besondere Art von „algebraischen” Aufgaben. Darin werden z.B. die Summe der Fläche und einer oder alle vier Seiten eines Quadrates beschrieben und nach der Seite gefragt. Es zeigt sich erstens, daß dieser Aufgabentyp mindestens seit dem frühesten 2. vorchristlichen Jahrtausend von geometrischen Praktikern tradiert wurde, und zweitens, daß er die Entwicklung einer „Algebra” in der altbabylonischen Schreiberschule inspirierte. Der Aufsatz untersucht, in welcher (...) Weise die Überführung der „sub-wissenschaftlichen” Praktikertradition in einen systematischen Schulunterricht den mathematischen Inhalt und den Denkstil künftiger Praktiker präge. Im letzten Kapitel wird diskutiert, inwieweit es sinnvoll ist, innerhalb dieser Schulalgebra von „Schulen” zu sprechen. (shrink)

In French, haute vulgarisation is a recognized genre. Not so in English, but even without the term A New History of Greek Mathematics is still an excellent representative. Regularly—and especially...

pt. I. Institutions, professions and ideas -- pt. II. Human science and human nature -- pt. III. The art of knowing.

A theme like “town and science” invites to comparative analysis, and suggests questions like these: Is the urban context a particularly fertile soil for the development of scientific thinking? Or rather the contrary? Is it fertile or barren under specific circumstances? Or does it favour a particular kind of scientific activity?General answers to such questions can hardly be found; still, they may provide case studies with a guiding perspective. Case studies, on the other hand, may lead to better understanding of (...) the implications of the general questions if not to answers. In the following I therefore try to read the emergence of Mesopotamian mathematics into the context of the “urban revolution” and the rise of states (c. 3300 to c. 2300 B.C.).Mesopotamian city states appeared in the late fourth millennium B.C., and were eventually absorbed in larger territorial states a thousand years later. Contemporary with the incipient state formation was the creation of a script and of genuine, coherent mathematics. Both were based on earlier accounting techniques, both were used by Temple officials for administrative purposes, and both were marked decisively by this application. At the same time, both presuppose modes of thought which are not produced directly by their application but only as mediated by a school system.In the mid‐third millennium, an autonomous scribal profession appears for the first time. At the same time emerge the first literary texts and the first “pure” mathematical problems (i.e., problems not directed at or following the pattern of genuine application). It seems as if the scribes, as soon as they were emancipated as a craft, tried out the range of their professional tools.Neither the use of writing for literary purposes nor the possibility to use numbers for “higher” computation were forgotten when the city states were absorbed in larger social systems. Both, however, were subsumed by the interest of the new territorial state, the former as a means to propagate royal ideology, the latter in the royal administration.The final section of the article compares this early Mesopotamian development with other instances of the “urban revolution” and with the development of scientific thought in the Ancient Greek city state and in the context of Medieval city communes. (shrink)

The “unknown heritage” is the name usually given to a problem type in whose archetype a father leaves to his first son 1 monetary unit and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{n}}$$\end{document} (n usually being 7 or 10) of what remains, to the second 2 units and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{n}}$$\end{document} of what remains, and so on. In the end, all sons get the same, and nothing remains. The earliest known (...) occurrence is in Fibonacci’s Liber abbaci, which also contains a number of much more sophisticated versions, together with a partial algebraic solution for one of these and rules for all which do not follow from his algebraic calculation. The next time the problem turns up is in Planudes’s late thirteenth century Calculus according to the Indians, Called the Great. After that the simple problem type turns up regularly in Provençal, Italian and Byzantine sources. It seems never to appear in Arabic or Indian writings, although two Arabic texts (one from c. 1190) contain more regular problems where the number of shares is given; they are clearly derived from the type known from European and Byzantine works, not its source. The sophisticated versions turn up again in Barthélemy de Romans’ Compendy de la praticque des nombres (c. 1467) and, apparently inspired from there, in the appendix to Nicolas Chuquet’s Triparty (1484). Apart from a single trace in Cardano’s Practica arithmetice et mensurandi singularis, the sophisticated versions never surface again, but the simple version spreads for a while to German practical arithmetic and, more persistently, to French polite recreational mathematics. Close examination of the texts shows that Barthélemy cannot have drawn his familiarity with the sophisticated rules from Fibonacci. It also suggests that the simple version is originally either a classical, strictly Greek or Hellenistic, or a medieval Byzantine invention; and that the sophisticated versions must have been developed before Fibonacci within an environment (located in Byzantium, Provence, or possibly in Sicily?) of which all direct traces has been lost, but whose mathematical level must have been quite advanced. (shrink)

One of the ways in which the artificial languages of mathematics are “generous”, that is, in which they assists the advance of thought, is through its establishment of advanced operatory structures that permit an even further advance of intuition. However, this generosity may be delusive, suggest ideas which in the longer run turn out to be untenable. The paper analyses two cases of “honest generosity”, namely a “proof” of the sign rule “less times less makes plus” from the 1340s and (...) a result in partition theory obtained by Euler by means of rash manipulations of infinite series and products, case-Cantor’s introduction of transfinite numbers from 1895-and (in modern terms) a failed attempt to extend the semi-group of algebraic powers into a complete group, also from c. 1340. Gewöhnlich glaubt der Mensch, wenn er nur Worte hört es müsse sich dabei wohl auch was denken lassen Goethe, Faust I, 2565-2566He gives the kids free samples because he knows full well that today’s young innocent faces will be tomorrow’s clientele Tom Lehrer, “The Old Dope Peddler”. (shrink)