Kuhnian view of Early Greek Mathematics

Applying Kuhn (Chapters 1–3) to Early Greek Mathematics

Pre-Paradigm Phase

Egyptian and Babylonian mathematics were largely practical, focused on arithmetic and geometry for land measurement, construction, and astronomy. - No proof, little self-correction mechanism, and no shared theoretical framework. - They did use 3-4-5 rope circles to lay out right angles, but this was a practical tool rather than a theoretical insight. Let’s try it!

Paradigm of Proof

We begin with compass & straightedge constructions.

Normal Science

• Hippocrates of Chios (c. 470-410 BCE) showed how to square a lune.
• First we square a rectangle
• Hippocrates’ Lune
• Not every lune!
• Hexagon, then the lune on it
• Pentagon, golden ratio

Anomalies

• Doubling the cube
• Trisecting an angle
• Squaring the circle
• Parallel postulate & Poincare Disk model

We will discuss Chapter 1-2 of Dunham’s Journey Through Genius (1990) and how it relates to Kuhn’s ideas about paradigms, normal science and anomalies.