How did Plato and the Pythagoreans influence the development of theoretical geometry?

The Pythagoreans and Plato were instrumental in elevating geometry from a practical, empirical craft to a field of supreme intellectual and philosophical importance.

The Pythagorean Influence

The Pythagoreans fundamentally shifted the perception of geometry by asserting that all things are governed by numbers. While they may not have developed proofs in the modern Euclidean sense, they established the foundational idea that geometry is a key to interpreting the universe. Their contributions include:12
* Numerical Relationships: They explored the properties of geometric figures through numerical symbolism, such as associating points, lines, and triangles with specific numbers.23
* Foundational Concepts: They are credited with early insights into geometric proportions and the discovery that the sum of the angles in a triangle equals two right angles.2
* Advanced Speculation: Later Pythagoreans, such as Archytas, contributed to three-dimensional geometry, including a solution for doubling the cube.2

The Platonic Influence

Plato furthered this development by championing geometry as a rigorous, deductive science essential for understanding reality. His influence is most notable in:14
* The Platonic Solids: Plato systematically developed a cosmological theory based on the five regular polyhedra, which now bear his name.5
* Philosophical Status: By integrating geometry into his philosophical framework, Plato helped solidify its reputation as an exemplar of precise reasoning, influencing the transition toward the deductive methods later perfected in Euclid’s Elements.14