Different approaches to past mathematical texts are reviewed. The question addressed is: should we stress the continuity of past mathematics with the mathematics practiced today, or should we emphasize its difference, namely what makes it a product of a distant mathematical culture?

Scholars tend to assume that the mathematical sciences and philosophy were distinct disciplines in antiquity, as they are today. From the fourth century B.C.E. onward, mathematicians and philosophers did distinguish themselves. They criticized each other’s work and, in some areas of the Greek world, strong rivalries developed between philosophers and mathematicians. I argue, however, that the distinction between philosophers and mathematicians did not entail that their fields of inquiry were distinct. This chapter examines the relationship between the mathematical sciences and philosophy from the perspective of the practitioners of the mathematical sciences, in particular, Archytas of Tarentum, Hero of Alexandria, and Claudius Ptolemy. I argue that these practitioners viewed the relationship between the mathematical sciences and philosophy as more complex, where the mathematical sciences are not only in relation to philosophy but, even stronger, forms of philosophy.  Moreover, the mathematical sciences answer some of the most fundamental questions of philosophy, e.g., how to obtain knowledge, how to form a just society, and how to attain the good life.

This chapter examines the historical roots of deduction in Ancient Greek philosophy and mathematics. It relies extensively on the work of G.E.R. Lloyd and Reviel Netz to argue that dialogical debating practices in a democratic city-state like Athens were causally instrumental for the emergence of the axiomatic-deductive method in mathematics. The same sociocultural political background was decisive for the emergence of practices of dialectic, the kinds of dialogical interactions famously portrayed in Plato’s dialogues. In turn, dialectic provided the background for the emergence of the first fully-fledged theory of deduction in history, namely Aristotle’s syllogistic.