Editing Zeno's paradoxes (section)

== Proposed solutions ==

=== In classical antiquity ===
According to [[Simplicius of Cilicia|Simplicius]], [[Diogenes the Cynic]] said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions.<ref name=":2" /><ref name=":1" /> To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Throughout history several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.

[[Aristotle]] (384 BC–322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.<ref>Aristotle. Physics 6.9
</ref>{{failed verification|reason=In the section cited, Aristotle says nothing about the distance decreasing |date=October 2019}}<ref>
Aristotle's observation that the fractional times also get shorter does not guarantee, in every case, that the task can be completed. One case in which it does not hold is that in which the fractional times decrease in a [[Harmonic series (mathematics)|harmonic series]], while the distances decrease geometrically, such as: 1/2 s for 1/2 m gain, 1/3 s for next 1/4 m gain, 1/4 s for next 1/8 m gain, 1/5 s for next 1/16 m gain, 1/6 s for next 1/32 m gain, etc. In this case, the distances form a convergent series, but the times form a [[divergent series]], the sum of which has no limit. {{Original research inline|date=October 2020}} Archimedes developed a more explicitly mathematical approach than Aristotle.</ref>
Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities").<ref>Aristotle. Physics 6.9; 6.2, 233a21-31</ref>
Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."<ref>{{cite book |author=Aristotle |title=Physics |url=http://classics.mit.edu/Aristotle/physics.6.vi.html |volume=VI |at=Part 9 verse: 239b5 |isbn=0-585-09205-2 |access-date=2008-08-11 |archive-date=2008-05-15 |archive-url=https://web.archive.org/web/20080515224131/http://classics.mit.edu//Aristotle/physics.6.vi.html |url-status=live }}</ref> [[Thomas Aquinas]], commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."<ref>Aquinas. Commentary on Aristotle's Physics, Book 6.861</ref><ref>{{Cite book |last=Kiritsis |first=Paul |title=A Critical Investigation into Precognitive Dreams |date=2020-04-01 |publisher=Cambridge Scholars Publishing |isbn=978-1527546332 |edition=1 |pages=19 |language=en}}</ref><ref>{{Cite web |last=Aquinas |first=Thomas |author-link=Thomas Aquinas |title=Commentary on Aristotle's Physics |url=https://aquinas.cc/la/en/~Phys.Bk6.L11 |access-date=2024-03-25 |website=aquinas.cc}}</ref>

=== 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>

=== Applications ===
==== Quantum Zeno effect ====
{{Main article|Quantum Zeno effect}}
In 1977,<ref>{{Cite journal |bibcode=1977JMP....18..756M |last1=Sudarshan |first1=E. C. G. |author-link=E. C. G. Sudarshan |last2=Misra |first2=B. |title=The Zeno's paradox in quantum theory |journal=Journal of Mathematical Physics |volume=18 |issue=4 |pages=756–763 |year=1977 |doi=10.1063/1.523304 |osti=7342282 |url=http://repository.ias.ac.in/51139/1/211-pub.pdf |access-date=2018-04-20 |archive-date=2013-05-14 |archive-url=https://web.archive.org/web/20130514062722/http://repository.ias.ac.in/51139/1/211-pub.pdf |url-status=live }}</ref> physicists [[E. C. George Sudarshan]] and B. Misra discovered that the dynamical evolution ([[motion]]) of a [[quantum system]] can be hindered (or even inhibited) through [[observation]] of the [[system]].<ref name="u0">{{cite journal |url=http://www.boulder.nist.gov/timefreq/general/pdf/858.pdf |author1=W.M.Itano |author2=D.J. Heinsen |author3=J.J. Bokkinger |author4=D.J. Wineland |title=Quantum Zeno effect |journal=[[Physical Review A]] |volume=41 |issue=5 |pages=2295–2300 |year=1990 |doi=10.1103/PhysRevA.41.2295 |pmid=9903355 |bibcode=1990PhRvA..41.2295I |access-date=2004-07-23 |archive-url=https://web.archive.org/web/20040720153510/http://www.boulder.nist.gov/timefreq/general/pdf/858.pdf |archive-date=2004-07-20 |url-status=dead }}
</ref> This effect is usually called the "[[Quantum Zeno effect]]" as it is strongly reminiscent of Zeno's arrow paradox. This effect was first theorized in 1958.<ref>{{Cite journal  |last=Khalfin |first=L.A. |journal=Soviet Phys. JETP |volume=6 |page=1053 |year=1958 |bibcode = 1958JETP....6.1053K  |title=Contribution to the Decay Theory of a Quasi-Stationary State }}</ref>

==== Zeno behaviour ====
In the field of verification and design of [[timed event system|timed]] and [[hybrid system]]s, the system behaviour is called ''Zeno'' if it includes an infinite number of discrete steps in a finite amount of time.<ref name="Fishwick2007">{{cite book | editor=Paul A. Fishwick | title=Handbook of dynamic system modeling | chapter-url=https://books.google.com/books?id=cM-eFv1m3BoC&pg=SA15-PA22 | access-date=2010-03-05 | edition=hardcover | series=Chapman & Hall/CRC Computer and Information Science | date=1 June 2007 | publisher=CRC Press | location=Boca Raton, Florida, USA | isbn=978-1-58488-565-8 | 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. | archive-date=2023-12-29 | archive-url=https://web.archive.org/web/20231229215249/https://books.google.com/books?id=cM-eFv1m3BoC&pg=SA15-PA22#v=onepage&q&f=false | url-status=live }}</ref> Some [[formal verification]] techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.<ref>{{cite journal |last=Lamport |first=Leslie |author-link=Leslie Lamport |year=2002 |title=Specifying Systems |journal=Microsoft Research |publisher=Addison-Wesley |isbn=0-321-14306-X |url=http://research.microsoft.com/en-us/um/people/lamport/tla/book-02-08-08.pdf |page=128 |access-date=2010-03-06 |archive-date=2010-11-16 |archive-url=https://web.archive.org/web/20101116164613/http://research.microsoft.com/en-us/um/people/lamport/tla/book-02-08-08.pdf |url-status=live }}</ref><ref>{{cite journal  |last1=Zhang |first1=Jun |last2=Johansson| first2=Karl | first3=John  |last3=Lygeros |first4=Shankar |last4=Sastry   |title=Zeno hybrid systems | journal=International Journal for Robust and Nonlinear Control |year=2001 |access-date=2010-02-28  |doi=10.1002/rnc.592  |volume=11  |issue=5  |page=435 |s2cid=2057416 |url=http://aphrodite.s3.kth.se/~kallej/papers/zeno_ijnrc01.pdf |url-status=dead |archive-url=https://web.archive.org/web/20110811144122/http://aphrodite.s3.kth.se/~kallej/papers/zeno_ijnrc01.pdf |archive-date=August 11, 2011}}</ref> In [[systems design]] these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.<ref>{{cite book|last2=Henzinger |first2=Thomas |last1=Franck |first1=Cassez |first3=Jean-Francois |last3=Raskin |url=http://mtc.epfl.ch/~tah/Publications/a_comparison_of_control_problems_for_timed_and_hybrid_systems.html |title=A Comparison of Control Problems for Timed and Hybrid Systems |year=2002 |access-date=2010-03-02 |url-status=dead |archive-url=https://web.archive.org/web/20080528193234/http://mtc.epfl.ch/~tah/Publications/a_comparison_of_control_problems_for_timed_and_hybrid_systems.html |archive-date=May 28, 2008 }}</ref>