Abstract
Historians of mathematics have long exalted the achievements of the ancient Greeks as symbolized by a single name, Euclid of Alexandria. The thirteen books that comprise his Elements hold a place within Greek mathematics comparable to the Parthenon in its architectural tradition. Appreciation for Greek classicism was long reinforced by the formal ideal of Euclidean geometry, a style that persisted until well into the nineteenth century. Not until the early decades of the twentieth did a new picture of ancient mathematics emerge, advanced by the pioneering researches of Otto Neugebauer on Egyptian and especially Mesopotamian mathematics. Although grounded in detailed analysis of primary sources, Neugebauer ’s work was guided by a broad vision of the exact sciences in ancient cultures that predated the Greeks. He thereby broke with the traditional Greco-centric understanding of European science. Neugebauer ’s historical views and methodological approach, which elevated mathematical techniques while diminishing the importance of philosophical commentary, came under strong attack after he immigrated to the United States in 1939.
The common belief that we gain “historical perspective” with increasing distance seems to me utterly to misrepresent the actual situation. What we gain is merely confidence in generalizations which we would never dare make if we had access to the real wealth of contemporary evidence.
— Otto Neugebauer , The Exact Sciences in Antiquity (Neugebauer 1969, viii)
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- 1.
The sharpest attack against Neugebauer ’s methodological approach came from Sabetai Unguru in (Unguru 1975); for this text and reactions to it, see (Christianidis 2004). I discussed this within the larger context of the historiographical debates from that period in (Rowe 1996). See also the contribution by Schneider to the present volume “Contextualizing Unguru’s 1975 Attack on the Historiography of Ancient Greek Mathematics”.
- 2.
It was later translated into English by Eva Brann (1929) (Klein 1968).
- 3.
Ancient sources only hint at the circumstances surrounding this discovery, which probably took place during the latter half of the fifth century. Before this time, it was presumed that magnitudes of the same kind, for example two lengths, could always be measured by a third, hence commensurable. This is equivalent to saying that their ratio will be equal to the ratio of two natural numbers. This theory had to be discarded when it was realized that even simple magnitudes, like the diagonal and side of a square, have an irrational ratio because their lengths are incommensurable lengths. The discovery of such irrational objects in geometry had profound consequences for the practice of Greek geometry in the fourth century, see (Fowler 1999).
- 4.
Die Antwort auf diese Frage, d. h. auf die Frage nach der geschichtlichen Ursache der Uraufgabe der gesamten geometrischen Algebra, kann man heute vollständig geben: sie liegt einerseits in der aus der Entdeckung irrationaler Größen folgenden Forderung der Griechen, der Mathematik ihre Allgemeingültigkeit zu sichern durch Übergang vom Bereich der rationalen Zahlen zum Bereich der allgemeinen Größenverhältnisse, andererseits in der daraus resultierenden Notwendigkeit, auch die Ergebnisse der vorgriechischen “algebraische” Algebra in eine “geometrische” Algebra zu übersetzen. Hat man das Problem einmal in dieser Weise formuliert, so ist alles Weitere vollständig trivial und liefert den glatten Anschluß der babylonischen Algebra an die Formulierungen bei Euklid.
- 5.
Jeder Versuch, Griechisches an Vorgriechisches anzuschließen begegnet einem intensiven Widerstand. Die Möglichkeit, das gewohnte Bild der Griechen modifizieren zu müssen, ist immer wieder unerwünscht, trotz aller Wandlungen … stehen also die Griechen in der Mitte und nicht mehr am Anfang.
- 6.
See, for example, the essays by Knorr in (Christianidis 2004).
- 7.
See (Christianidis 2004) for a recent account of older as well as the newer historiography on Greek mathematics.
- 8.
Neugebauer here alludes to the so-called “foundations crisis” that supposedly ensued with the discovery of incommensurable magnitudes. This interpretation became popular during the 1920s, but later fell out of favour (Christianidis 2004).
- 9.
This refers to “The Unicorn in Captivity,” one of seven tapestries dating from ca. 1500 located in The Cloisters in New York. In the pagan tradition, the unicorn was a one-horned creature that could only be tamed by a virgin; whereas Christians made this into an allegory for Christ's relationship with the Virgin Mary.
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Rowe, D.E. (2016). Otto Neugebauer’s Vision for Rewriting the History of Ancient 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_7
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