Geometry, Classical

Geometry has a special role in mathematics because of its connection with physical space. Today this might not be the only force, or the driving force, behind the development of new geometries, but in the ancient world it was an important aspect. Geometry was considered an access point to deeper insights about how the world is constructed and how it connected to our reasoning.

Geometry can be thought about in many different ways. We know this from today’s geometries, which can be quite challenging for our experiences, which are derived from a normal three-dimensional perception of the universe. There were also different ways to develop geometry in, for example, ancient Greece, but one of them came to play a dominant role—namely Euclidean geometry. This geometry has played a paradigmatic role in mathematical thinking, and this entry traces its development and its sources as well as its impact on modern conceptions of nature.

The first section discusses pre-Euclidean geometry and the manifold roots of geometry. Next we present the Pythagorean ideas on geometry, and in particular the idea that the world consists of numbers (or of number units). This is followed by a discussion of Plato’s idea that geometric concepts belong to an eternal world of ideas. The entry continues with a presentation of Euclid’s Elements, with reference to his definitions, axioms, and theorems. We then explain how Euclid’s conception of geometry differs from Archimedes’ conception, and how Euclid’s conception turned into an epistemic paradigm, stating that knowledge has to be organized in an axiomatic framework. This epistemic paradigm dominated for centuries, and also influenced how Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, and Isaac Newton developed their approaches toward a modern science. The entry concludes with a summary of the ontological, epistemological, sociological, and [Page 317]ethical issues related to the development of classical geometry.