The transition from praxis to theoria—from mathematics as tekhne (techniquesfor dealing with practical activities) to mathematics as episteme and gnosis (a form of pure knowledge)—occurred only once in human history, namely, among the classical Greeks. No earlier mathematical tradition gives evidence of such a theoretical dimension,and where one encounters mathematical theory among later traditions, it is in the context of some manner of borrowing from the ancient Greek precedent. The goal of this paper is to examine the above “Greek precedent,” their process of developing mathematics as well as other scientific theory. We will first examine the rise of rational thought and beginnings of deduction in the Archaic Era, using Thales as representative of the time period. Then we will see how Aristotle added the syllogism and emphasis on observation during the Classical Era in order to enhance and codify deductive reasoning. Finally, Archimedes enters the picture and introduces an inductive method of discovery that crosses the disciplines of mathematics and mechanics.Throughout the paper are examples of how the Greeks applied their mathematical and scientific knowledge in order to demonstrate what new developments in reasoning added to the times.
Foss, Michael, "Archimedes's Impact on the Discovery Process in Ancient Greece" (2006). 2006 AHS Capstone Projects. 12.