Editing Zeno of Elea (section)

== Legacy ==
=== Antiquity ===
[[File:Zeno of Elea Tibaldi or Carducci Escorial.jpg|thumb|upright=2|Zeno shows the Doors to Truth and Falsity (''Veritas et Falsitas''). Fresco in the Library of [[El Escorial]], Madrid.]]
Zeno's greatest influence was within the thought of the Eleatic school, as his arguments built on the ideas of Parmenides,{{Sfn|Vlastos|1995|p=259}} though his paradoxes were also of interest to [[Greek mathematics|Ancient Greek mathematicians]].{{Sfn|Vlastos|1995|p=258}} Zeno is regarded as the first philosopher who dealt with attestable accounts of mathematical [[infinity]].<ref>{{cite book |last1=Boyer |first1=Carl B. |title=A History of Mathematics |last2=Merzbach |first2=Uta C. |date=2011 |publisher=John Wiley & Sons |isbn=978-0-470-52548-7 |edition=Third |location=Hoboken, New Jersey |page=538 |quote=Ever since the days of Zeno, men had been talking about infinity,... |author1-link=Carl Benjamin Boyer |author2-link=Uta Merzbach}}</ref> Zeno was succeeded by the [[Atomism#Greek atomism|Greek Atomists]], who argued against the infinite division of objects by proposing an eventual stopping point: the atom. Though [[Epicurus]] does not name Zeno directly, he attempts to refute some of Zeno's arguments.{{Sfn|Vlastos|1995|p=259}}

Zeno appeared in Plato's dialogue ''[[Parmenides (dialogue)|Parmenides]]'', and his paradoxes are mentioned in ''[[Phaedo]]''.{{Sfn|Vamvacas|2009|p=151}} Aristotle also wrote about Zeno's paradoxes.{{Sfn|Vamvacas|2009|p=153}} Plato looked down on Zeno's approach of making arguments through contradictions.{{Sfn|Sherwood|2000}} He believed that even Zeno himself did not take the arguments seriously.{{Sfn|Sanday|2009|p=209}} Aristotle disagreed, believing them to be worthy of consideration.{{Sfn|Sherwood|2000}} He challenged Zeno's dichotomy paradox through his conception of infinity, arguing that there are two infinities: an actual infinity that takes place at once and a potential infinity that is spread over time. He contended that Zeno attempted to prove actual infinities using potential infinities.{{Sfn|Vamvacas|2009|p=153}}{{Sfn|Palmer|2021}} He also challenged Zeno's paradox of the stadium, observing that it is fallacious to assume a stationary object and an object in motion require the same amount of time to pass.{{Sfn|Vamvacas|2009|p=155}} The paradox of Achilles and the tortoise may have influenced Aristotle's belief that actual infinity cannot exist, as this non-existence presents a solution to Zeno's arguments.{{Sfn|Vlastos|1995|p=259}}

=== Modern era ===
Zeno's paradoxes are still debated, and they remain one of the archetypal examples of arguments to challenge commonly held perceptions.{{Sfn|Vlastos|1995|p=260}}{{Sfn|Vamvacas|2009|p=156}} The paradoxes saw renewed attention in 19th century philosophy that has persisted to the present.{{Sfn|Palmer|2021}} Zeno's philosophy shows a contrast between what one knows logically and what one observes with the senses with the goal of proving that the world is an illusion; this practice was later adopted by the modern philosophic schools of thought, [[empiricism]] and [[post-structuralism]]. [[Bertrand Russell]] praised Zeno's paradoxes, crediting them for allowing the work of mathematician [[Karl Weierstrass]].{{Sfn|Sherwood|2000}}

Scientific phenomena have been named after Zeno. The hindrance of a quantum system by observing it is usually called the [[Quantum Zeno effect]] as it is strongly reminiscent of Zeno's arrow paradox.<ref>{{Cite book |last=Anastopoulos |first=Charis |url=https://books.google.com/books?id=iwXXEAAAQBAJ&pg=PA213 |title=Quantum Theory: A Foundational Approach |publisher=[[Cambridge University Press]] |year=2023 |isbn=978-1-009-00840-2 |edition=1st |pages=213 |language=en}}</ref><ref name="u0">{{cite journal |author1=W.M.Itano |author2=D.J. Heinsen |author3=J.J. Bokkinger |author4=D.J. Wineland |year=1990 |title=Quantum Zeno effect |url=http://www.boulder.nist.gov/timefreq/general/pdf/858.pdf |url-status=dead |journal=[[Physical Review A]] |volume=41 |issue=5 |pages=2295–2300 |bibcode=1990PhRvA..41.2295I |doi=10.1103/PhysRevA.41.2295 |pmid=9903355 |archive-url=https://web.archive.org/web/20040720153510/http://www.boulder.nist.gov/timefreq/general/pdf/858.pdf |archive-date=2004-07-20 |access-date=2004-07-23}}</ref> In the field of verification and design of [[Timed event system|timed]] and [[Hybrid system|hybrid systems]], the system behavior is called ''Zeno'' if it includes an infinite number of discrete steps in a finite amount of time.<ref name="Fishwick2007">{{cite book |title=Handbook of dynamic system modeling |date=1 June 2007 |publisher=CRC Press |isbn=978-1-58488-565-8 |editor=Paul A. Fishwick |edition=hardcover |series=Chapman & Hall/CRC Computer and Information Science |location=Boca Raton, Florida, USA |pages=15–22 to 15–23 |chapter=15.6 "Pathological Behavior Classes" in chapter 15 "Hybrid Dynamic Systems: Modeling and Execution" by Pieter J. Mosterman, The Mathworks, Inc. |access-date=2010-03-05 |chapter-url=https://books.google.com/books?id=cM-eFv1m3BoC&pg=SA15-PA22}}</ref>

Zeno's arguments against plurality have been challenged by modern [[atomic theory]]. Rather than plurality requiring both a finite and infinite amount of objects, atomic theory shows that objects are made from a specific number of [[atom]]s that form specific elements.{{Sfn|Vamvacas|2009|p=152}} Likewise, Zeno's arguments against motion have been challenged by modern mathematics and physics.{{Sfn|Vamvacas|2009|p=154}} Mathematicians and philosophers continued studying infinitesimals until they came to be better understood through [[calculus]] and [[Limit (mathematics)|limit theory]]. Ideas relating to Zeno's plurality arguments are similarly affected by [[set theory]] and [[transfinite number]]s.{{Sfn|Vamvacas|2009|p=156}} Modern physics has yet to determine whether space and time can be represented on a mathematical continuum or if it is made up of discrete units.{{Sfn|Palmer|2021}}

Zeno's argument of Achilles and the tortoise can be addressed mathematically, as the distance is defined by a specific number. His argument of the flying arrow has been challenged by modern physics, which allows the smallest instants of time to still have a minuscule non-zero duration.{{Sfn|Vamvacas|2009|p=154}} Other mathematical ideas, such as [[internal set theory]] and [[nonstandard analysis]], may also resolve Zeno's paradoxes.{{Sfn|Vamvacas|2009|p=157}} However, there is no definitive agreement on whether solutions to Zeno's arguments have been found.{{Sfn|Vamvacas|2009|p=156}}