Zeno's paradoxes in the context of "Series (mathematics)"

Zeno of Elea (/ˈziːnoʊ ... ˈɛliə/; Ancient Greek: Ζήνων ὁ Ἐλεᾱ́της; c. 490 – c. 430 BC) was a pre-Socratic Greek philosopher from Elea, in Southern Italy (Magna Graecia). He was a student of Parmenides and one of the Eleatics. Zeno defended his instructor's belief in monism, the idea that only one single entity exists that makes up all of reality. He rejected the existence of space, time, and motion. To disprove these concepts, he developed a series of paradoxes to demonstrate why they are impossible. Though his original writings are lost, subsequent descriptions by Plato, Aristotle, Diogenes Laertius, and Simplicius of Cilicia have allowed study of his ideas.

Zeno's arguments are divided into two different types: his arguments against plurality, or the existence of multiple objects, and his arguments against motion. Those against plurality suggest that for anything to exist, it must be divisible infinitely, meaning it would necessarily have both infinite mass and no mass simultaneously. Those against motion invoke the idea that distance must be divisible infinitely, meaning infinite steps would be required to cross any distance.

In physics and the philosophy of science, instant refers to an infinitesimal interval in time, whose passage is instantaneous. In ordinary speech, an instant has been defined as "a point or very short space of time," a notion deriving from its etymological source, the Latin verb instare, from in- + stare ('to stand'), meaning 'to stand upon or near.'

The continuous nature of time and its infinite divisibility was addressed by Aristotle in his Physics, where he wrote on Zeno's paradoxes. The philosopher and mathematician Bertrand Russell was still seeking to define the exact nature of an instant thousands of years later.

In philosophy, an antinomy (/ænˈtɪnəmi/; Ancient Greek: antí 'against' + nómos 'law') is a real or apparent contradiction between two conclusions, both of which seem justified. It is a term used in logic and epistemology, particularly in the philosophy of Immanuel Kant.

Antinomy is a common form of argument in the dialogues of Plato. Kant credited Zeno of Elea (see Zeno's paradoxes) as the inventor of the antinomic mode of argumentation, which he described as a "skeptical method" of "watching, or rather provoking, a conflict of assertions, not for the purpose of deciding in favor of one or the other side, but of investigating whether the object of the controversy is not perhaps a deceptive appearance which each vainly tries to grasp, and in regard to which, even if there were no opposition to overcome, neither can arrive at any result".

In philosophy and theology, infinity is explored in articles under headings such as the Absolute, God, and Zeno's paradoxes.

In Greek philosophy, for example in Anaximander, 'the Boundless' is the origin of all that is. He took the beginning or first principle to be an endless, unlimited primordial mass (ἄπειρον, apeiron). The Jain metaphysics and mathematics were the first to define and delineate different "types" of infinities. The work of the mathematician Georg Cantor first placed infinity into a coherent mathematical framework. Keenly aware of his departure from traditional wisdom, Cantor also presented a comprehensive historical and philosophical discussion of infinity. In Christian theology, for example in the work of Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity.