Editing Zeno's paradoxes (section)

=== In modern mathematics ===
Some mathematicians and historians, such as [[Carl Boyer]], hold that Zeno's paradoxes are simply mathematical problems, for which modern [[calculus]] provides a mathematical solution.<ref name=boyer>{{cite book |last=Boyer |first=Carl |title=The History of the Calculus and Its Conceptual Development |url=https://archive.org/details/historyofcalculu0000boye |url-access=registration |year=2012|orig-date=1959 |publisher=Dover Publications  |access-date=2010-02-26 |page=[https://archive.org/details/historyofcalculu0000boye/page/295 295] | quote=If the paradoxes are thus stated in the precise mathematical terminology of continuous variables (...) the seeming contradictions resolve themselves. |isbn=978-0-486-60509-8 }}</ref> Infinite processes remained theoretically troublesome in mathematics until the late 19th century. With the [[(ε, δ)-definition of limit|epsilon-delta]] definition of [[Limit (mathematics)|limit]], [[Karl Weierstrass|Weierstrass]] and [[Augustin Louis Cauchy|Cauchy]] developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.<ref name=Lee>{{cite journal |last=Lee |first=Harold | title=Are Zeno's Paradoxes Based on a Mistake? |jstor=2251675 |year=1965 |journal= [[Mind (journal)|Mind]] |volume=74 |issue=296 |publisher=Oxford University Press |pages= 563–570 |doi=10.1093/mind/LXXIV.296.563}}</ref><ref name=russell>[[Bertrand Russell|B Russell]] (1956) ''Mathematics and the metaphysicians'' in "The World of Mathematics" (ed. [[James R. Newman|J R Newman]]), pp 1576-1590.</ref>

Some [[philosopher]]s, however, say that Zeno's paradoxes and their variations (see [[Thomson's lamp]]) remain relevant [[Metaphysics|metaphysical]] problems.<ref name=KBrown/><ref name=FMoorcroft>{{cite web |first=Francis |last=Moorcroft |title=Zeno's Paradox |url=http://www.philosophers.co.uk/cafe/paradox5.htm |archive-url=https://web.archive.org/web/20100418141459/http://www.philosophers.co.uk/cafe/paradox5.htm |archive-date=2010-04-18 }}</ref><ref name=Papa-G>{{cite journal |url=http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf |first=Alba |last=Papa-Grimaldi |title=Why Mathematical Solutions of Zeno's Paradoxes Miss the Point: Zeno's One and Many Relation and Parmenides' Prohibition |journal=The Review of Metaphysics |volume=50 |year=1996 |pages=299–314 |access-date=2012-03-06 |archive-date=2012-06-09 |archive-url=https://web.archive.org/web/20120609113959/http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf |url-status=live }}</ref> While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Kevin Brown<ref name=KBrown>{{cite web|first = Kevin |last = Brown |title = Zeno and the Paradox of Motion |work = Reflections on Relativity |url = http://www.mathpages.com/rr/s3-07/3-07.htm |access-date = 2010-06-06 |url-status = dead |archive-url  = https://archive.today/20121205030717/http://www.mathpages.com/rr/s3-07/3-07.htm |archive-date = 2012-12-05}}</ref> and Francis Moorcroft<ref name=FMoorcroft/> hold that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise. Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of '[[Rorschach test|Rorschach image]]' onto which people can project their most fundamental phenomenological concerns (if they have any)."<ref name=KBrown/>

==== Henri Bergson ====
An alternative conclusion, proposed by [[Henri Bergson]] in his 1896 book ''[[Matter and Memory]]'', is that, while the path is divisible, the motion is not.<ref>{{cite book|last=Bergson|first=Henri|title=Matière et Mémoire|trans-title=Matter and Memory|url=https://antilogicalism.com/wp-content/uploads/2017/07/matter-and-memory.pdf|author-link=Henri Bergson|date=1896|pages=77–78 of the PDF|publisher=Translation 1911 by Nancy Margaret Paul & W. Scott Palmer. George Allen and Unwin|access-date=2019-10-15|archive-date=2019-10-15|archive-url=https://web.archive.org/web/20191015184719/https://antilogicalism.com/wp-content/uploads/2017/07/matter-and-memory.pdf|url-status=live}}</ref><ref>{{Cite book |last=Massumi |first=Brian |title=Parables for the Virtual: Movement, Affect, Sensation |publisher=Duke University Press Books |year=2002 |isbn=978-0822328971 |edition=1st |location=Durham, NC |pages=5–6 |language=English}}</ref> 

==== Peter Lynds ====

In 2003, Peter Lynds argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.<ref>{{cite web|url=http://philsci-archive.pitt.edu/1197/|title=Zeno's Paradoxes: A Timely Solution|date=January 2003|access-date=2012-07-02|archive-date=2012-08-13|archive-url=https://web.archive.org/web/20120813040121/http://philsci-archive.pitt.edu/1197/|url-status=live}}</ref><ref> Lynds, Peter. Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity. Foundations of Physics Letter s (Vol. 16, Issue 4, 2003). doi:10.1023/A:1025361725408</ref><ref name="Time’s Up Einstein">[https://www.wired.com/wired/archive/13.06/physics.html Time’s Up, Einstein] {{Webarchive|url=https://web.archive.org/web/20121230100640/http://www.wired.com/wired/archive/13.06/physics.html |date=2012-12-30 }}, Josh McHugh, [[Wired Magazine]], June 2005</ref> Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes.  Nick Huggett argues that Zeno is [[begging the question|assuming the conclusion]] when he says that objects that occupy the same space as they do at rest must be at rest.<ref name=HuggettArrow/>

==== Bertrand Russell ====
Based on the work of [[Georg Cantor]],<ref>{{cite book |last=Russell |first=Bertrand |date=2002 |title=Our Knowledge of the External World: As a Field for Scientific Method in Philosophy |chapter=Lecture 6. The Problem of Infinity Considered Historically |publisher=Routledge |page=169 |orig-year=First published in 1914 by The Open Court Publishing Company |isbn=0-415-09605-7}}</ref> [[Bertrand Russell]] offered a solution to the paradoxes, what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is just change in position over time.<ref name=HuggettBook>{{ cite book |title=Space From Zeno to Einstein |first=Nick |last=Huggett |year=1999 |publisher=MIT Press |isbn=0-262-08271-3}}</ref><ref>{{cite book |url=https://books.google.com/books?id=uPRbOOv1YxUC&q=at+at+theory+of+motion+russell&pg=PA198 |title=Causality and Explanation |first=Wesley C. |last=Salmon |author-link=Wesley C. Salmon |page=198 |isbn=978-0-19-510864-4 |year=1998 |publisher=Oxford University Press |access-date=2020-11-21 |archive-date=2023-12-29 |archive-url=https://web.archive.org/web/20231229215244/https://books.google.com/books?id=uPRbOOv1YxUC&q=at+at+theory+of+motion+russell&pg=PA198#v=snippet&q=at%20at%20theory%20of%20motion%20russell&f=false |url-status=live }}</ref>

==== Hermann Weyl ====
Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. According to [[Hermann Weyl]], the assumption that space is made of finite and discrete units is subject to a further problem, given by the "[[Weyl's tile argument|tile argument]]" or "distance function problem".<ref>{{cite encyclopedia| last=Van Bendegem| first=Jean Paul| title=Finitism in Geometry| url=http://plato.stanford.edu/entries/geometry-finitism/#SomParSolProDea| encyclopedia=Stanford Encyclopedia of Philosophy| access-date=2012-01-03| date=17 March 2010| archive-date=2008-05-12| archive-url=https://web.archive.org/web/20080512012132/http://plato.stanford.edu/entries/geometry-finitism/#SomParSolProDea| url-status=live}}</ref><ref name="atomism uni of washington">{{cite web| last=Cohen| first=Marc| title=ATOMISM| url=https://www.aarweb.org/syllabus/syllabi/c/cohen/phil320/atomism.htm|work=History of Ancient Philosophy, University of Washington| access-date=2012-01-03|date=11 December 2000  |url-status=dead |archive-url=https://web.archive.org/web/20100712095732/https://www.aarweb.org/syllabus/syllabi/c/cohen/phil320/atomism.htm |archive-date=July 12, 2010}}</ref> According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry. [[Jean Paul Van Bendegem]] has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.<ref name=boyer/><ref>{{cite journal |jstor=187807 |title=Discussion:Zeno's Paradoxes and the Tile Argument |first=Jean Paul |last=van Bendegem |location= Belgium |year=1987 |journal=Philosophy of Science |volume=54 |issue=2 |pages=295–302|doi=10.1086/289379|s2cid=224840314 }}</ref>