Dietmar Herrmann - Ancient Mathematics: History of Mathematics in Ancient Greece and Hellenism

From whoever wants to deliver the history of any knowledge, we can rightly demand that he give us news of how the phenomena became known in due course, what has been fantasized about them, what has been meant and thought.

Even though the history of mathematics is not a relevant lecture course at German universities, a supplementary book can be of interest. It offers for all mathematics teachers and -interested people quite a new view on the diverse problems, which were developed over the course of one millennium (Thales 580 BC until Proklos 420 AD) in ancient Greek mathematics. For reasons of space, only facets of the different works can be presented, which however assemble to a kaleidoscope of science. A broad spectrum of tasks, constructions and historical illustrations are put together to a new overall picture, which provides more insight than conventional summary descriptions.

The versatile methods, which the Greek researchers have invented, also demand respect and recognition from todays viewer. These amazing achievements have been created without any aids like calculators and modern communication. It was important to describe the whole range of Greek mathematics, especially by means of literary sources and a well-chosen selection of pictures also to bring in the context of Pythagorean-Platonic philosophy. There are three possibilities of a historical processing: strictly chronological, biographical-person related or subject-related by means of special thematic areas. The present representation chooses a mixture of the latter two.

A first problem in the representation of ancient mathematics is raised by the famous article On the Need to Rewrite the History of Greek Mathematics by Sabetai Unguru. The author expresses the opinion that it is basically inappropriate to present ancient findings with modern formulas. The formula and concept apparatus of modern mathematics included concepts and abstractions, which might veil the authentic historical procedure. As an example the binomial formula is chosen. In modern mathematics, it applies to all abstract elements of a commutative ring; such a concept is completely foreign to Euclid. A product of two numbers or a square is always associated with an area in Euclid and can only be linked with quantities of the same dimension. The Greek word (arithmos) must be seen in the Pythagorean-Platonic environment and cannot be adequately translated with the word number. In order to make the representations readable and compact, the usual formula language is used.

A second goal is the description of the political and cultural environment in which the Greek scientist is located. The cultural flourishing of Athens in a phase of relative peace between the Persian Warsdue to its leading role in the alliance against the Persiansmade it possible to build an academy that attracted educationally motivated people from all over Greece, like Aristotle. Alexander freed Egypt from Persian occupation and caused a shift of power to the southeast. The Egyptian-Syrian province that emerged after his death through the division of the empire became the intellectual and economic center of the Mediterranean region with its capital Alexandria. The schools founded there, the Museion and the Serapeion, survived the collapse of the Ptolemaic Empire and flourished even under Roman occupation. Only the rise of Christianity as the state religion ended the fate of the scientists still clinging to Platonic doctrine, as seen in the fate of Hypatia.

Another concern is the inclusion of new, critical perspectives in comparison to older literature. Stories that the vegetarian Pythagoras sacrificed several bulls when discovering a theorem or that Archimedes set fire to the sails of the Roman fleet with mirrors can be dismissed as fairy tales. A modern interpretation of Diophantus, criticism of the work of Claudius Ptolemy and Heron, as well as new translations of Nicomachus and Theon of Smyrna provide a new view of Greek mathematics. The extensive work of Pappus is completely re-evaluated. The methods used usually require only moderate knowledge.

Geometry is currently taking a back seat in education; but this is not a sufficient reason to completely abolish Euclidean geometry according to the motto of J. Dieudonn (member of the Bourbaki group) Euclid must go!