Pierrot T. Seban

This paper has a modest ambition : to argue for the existence of a particular logical form, the form of the “indefinite iterative”, and to argue that this form is typical of the Ancient Greek corpus in some of its early accomplishments both literary, philosophical and mathematical. We hope to describe and reveal something like an invention of the infinite in Ancient Greece, by tracing a specific form of thought from its presentation in some Homeric myths up to its mature and accomplished philosophical analysis in Aristotle’s Physics, and through its use in the paradoxes of Zeno of Elea and the mathematical formulas of incommensurability.

This paper considers some aspects of the early conception and use of the infinite in ancient Greece, in the spirit of recent results in the history of ancient mathematics. It follows aspects of the practice of reasoning ad infinitum from the extant corpus of and about Zeno of Elea up to early Hellenistic examples in Aristotle and Euclid. Starting with the idea of ‘reasoning from indefinite iteration’, based on the metalogical recognition of the unachievability of an inference process, it identifies several different classes of more or less sophisticated arguments that make use of this idea, and examines the logical devices and notions required for their acceptance in the philosophical practice. Those include ‘Infinite regress’ properly speaking, where Non-Contradiction is used in the formation of indirect infinitary arguments.