Zeno's paradoxes are a set of
philosophical
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
problems generally thought to have been devised by
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
philosopher
Zeno of Elea
Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known fo ...
(c. 490–430 BC) to support
Parmenides
Parmenides of Elea (; grc-gre, Παρμενίδης ὁ Ἐλεάτης; ) was a pre-Socratic Greek philosopher from Elea in Magna Graecia.
Parmenides was born in the Greek colony of Elea, from a wealthy and illustrious family. His dates a ...
' doctrine that contrary to the evidence of one's senses, the belief in
plurality and change is mistaken, and in particular that
motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and mea ...
is nothing but an
illusion
An illusion is a distortion of the senses, which can reveal how the mind normally organizes and interprets sensory stimulation. Although illusions distort the human perception of reality, they are generally shared by most people.
Illusions may o ...
. It is usually assumed, based on
Plato's ''Parmenides'' (128a–d), that Zeno took on the project of creating these
paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
es because other philosophers had created paradoxes against Parmenides' view. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one." Plato has
Socrates
Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no te ...
claim that Zeno and Parmenides were essentially arguing exactly the same point. Some of Zeno's nine surviving paradoxes (preserved in
Aristotle's ''Physics''[Aristotle's ''Physics'']
"Physics" by Aristotle translated by R. P. Hardie and R. K. Gaye
and
Simplicius's commentary thereon) are essentially equivalent to one another. Aristotle offered a refutation of some of them.
[ Three of the strongest and most famous—that of Achilles and the tortoise, the ]Dichotomy
A dichotomy is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be
* jointly exhaustive: everything must belong to one part or the other, and
* mutually exclusive: nothing can belong simulta ...
argument, and that of an arrow in flight—are presented in detail below.
Zeno's arguments are perhaps the first examples of a method of proof called ''reductio ad absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
'', also known as proof by contradiction
In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as ...
. They are also credited as a source of the dialectic
Dialectic ( grc-gre, διαλεκτική, ''dialektikḗ''; related to dialogue; german: Dialektik), also known as the dialectical method, is a discourse between two or more people holding different points of view about a subject but wishing ...
method used by Socrates. Some mathematicians and historians, such as Carl Boyer
Carl Benjamin Boyer (November 3, 1906 – April 26, 1976) was an American historian of sciences, and especially mathematics. Novelist David Foster Wallace called him the "Gibbon of math history". It has been written that he was one of few histori ...
, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
provides a mathematical solution.[
] Some philosopher
A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
s, however, say that Zeno's paradoxes and their variations (see Thomson's lamp
Thomson's lamp is a philosophical puzzle based on infinites. It was devised in 1954 by British philosopher James F. Thomson, who used it to analyze the possibility of a supertask, which is the completion of an infinite number of tasks.
Conside ...
) remain relevant metaphysical
Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
problems.[
][
] The origins of the paradoxes are somewhat unclear. Diogenes Laërtius
Diogenes Laërtius ( ; grc-gre, Διογένης Λαέρτιος, ; ) was a biographer of the Ancient Greece, Greek philosophers. Nothing is definitively known about his life, but his surviving ''Lives and Opinions of Eminent Philosophers'' is a ...
, a fourth source for information about Zeno and his teachings, citing Favorinus
Favorinus (c. 80 – c. 160 AD) was a Roman sophist and academic skeptic philosopher who flourished during the reign of Hadrian and the Second Sophistic.
Early life
He was of Gaulish ancestry, born in Arelate (Arles). He received a refin ...
, says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise. But in a later passage, Laërtius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.
Paradoxes of motion
Dichotomy paradox
Suppose Atalanta
Atalanta (; grc-gre, Ἀταλάντη, Atalantē) meaning "equal in weight", is a heroine in Greek mythology.
There are two versions of the huntress Atalanta: one from Arcadia, whose parents were Iasus and Clymene and who is primarily known ...
wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.
ImageSize= width:800 height:100
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at:0.09765625 mark:(line,black)
The resulting sequence can be represented as:
:
This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.
This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can be neither completed nor begun, and so all motion must be an illusion
An illusion is a distortion of the senses, which can reveal how the mind normally organizes and interprets sensory stimulation. Although illusions distort the human perception of reality, they are generally shared by most people.
Illusions may o ...
.
This argument is called the "Dichotomy
A dichotomy is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be
* jointly exhaustive: everything must belong to one part or the other, and
* mutually exclusive: nothing can belong simulta ...
" because it involves repeatedly splitting a distance into two parts. An example with the original sense can be found in an asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
. It is also known as the Race Course paradox.
Achilles and the tortoise
In the paradox of Achilles
In Greek mythology, Achilles ( ) or Achilleus ( grc-gre, Ἀχιλλεύς) was a hero of the Trojan War, the greatest of all the Greek warriors, and the central character of Homer's ''Iliad''. He was the son of the Nereid Thetis and Peleus, k ...
and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. As Aristotle noted, this argument is similar to the Dichotomy. It lacks, however, the apparent conclusion of motionlessness.
Arrow paradox
In the arrow paradox, Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that at any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not.
It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.
Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.
Three other paradoxes as given by Aristotle
Paradox of place
From Aristotle:
Paradox of the grain of millet
Description of the paradox from the ''Routledge Dictionary of Philosophy'':
Aristotle's refutation:
Description from Nick Huggett:
The moving rows (or stadium)
From Aristotle:
For an expanded account of Zeno's arguments as presented by Aristotle, see Simplicius's commentary ''On Aristotle's Physics''.
Proposed solutions
Diogenes the Cynic
According to Simplicius, Diogenes the Cynic
Diogenes ( ; grc, Διογένης, Diogénēs ), also known as Diogenes the Cynic (, ) or Diogenes of Sinope, was a Greek philosopher and one of the founders of Cynicism (philosophy). He was born in Sinope, an Ionian colony on the Black Sea ...
said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions (see ''solvitur ambulando
''Solvitur ambulando'' is a Latin phrase which means "it is solved by walking" and is used to refer to a problem which is solved by a practical experiment. It is often attributed to Saint Augustine.
Citations
The argument is retold by Pushkin i ...
''). To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.
Aristotle
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
(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.
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").
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."
Archimedes
Before 212 BC, Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. (See: Geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each succ ...
, 1/4 + 1/16 + 1/64 + 1/256 + · · ·, ''The Quadrature of the Parabola
''Quadrature of the Parabola'' ( el, Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regar ...
''.) His argument, applying the method of exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area bet ...
to prove that the infinite sum in question is equal to the area of a particular square, is largely geometric but quite rigorous. Today's analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
achieves the same result, using limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
(see convergent series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted
:S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k.
The th partial sum ...
). These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.
Thomas Aquinas
Thomas Aquinas
Thomas Aquinas, OP (; it, Tommaso d'Aquino, lit=Thomas of Aquino; 1225 – 7 March 1274) was an Italian Dominican friar and priest who was an influential philosopher, theologian and jurist in the tradition of scholasticism; he is known wi ...
, 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."
Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
offered 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.[
]
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
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
, the assumption that space is made of finite and discrete units is subject to a further problem, given by the " tile argument" or "distance function problem".[
]
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
Jean Paul Van Bendegem (born 28 March 1953 in Ghent) is a mathematician, a philosopher of science, and a professor at the Vrije Universiteit Brussel in Brussels.
Career
Van Bendegem received his master's degree in mathematics in 1976. Afterwards ...
has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.[
]
Henri Bergson
An alternative conclusion, proposed by Henri Bergson
Henri-Louis Bergson (; 18 October 1859 – 4 January 1941) was a French philosopherHenri Bergson. 2014. Encyclopædia Britannica Online. Retrieved 13 August 2014, from https://www.britannica.com/EBchecked/topic/61856/Henri-Bergson in his 1896 book ''Matter and Memory
''Matter and Memory'' (French: ''Matière et mémoire'', 1896) is a book by the French philosopher Henri Bergson. Its subtitle is ''Essay on the relation of body and spirit'' (''Essai sur la relation du corps à l’esprit''), and the work prese ...
'', is that, while the path is divisible, the motion is not. In this argument, instants in time and instantaneous magnitudes do not physically exist. An object in relative motion cannot have an instantaneous or determined relative position, and so cannot have its motion fractionally dissected.
Peter Lynds
In 2003, Peter Lynds put forth a very similar argument. All of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.[Time’s Up, Einstein]
Josh McHugh, Wired Magazine
''Wired'' (stylized as ''WIRED'') is a monthly American magazine, published in print and online magazine, online editions, that focuses on how emerging technologies affect culture, the economy, and politics. Owned by Condé Nast, it is headquar ...
, June 2005
[S E Robbins (2004) ''On time, memory and dynamic form''. Consciousness and Cognition 13(4), 762-788: "Lynds, his reviewers and consultants (e.g., J.J.C. Smart) are apparently unaware of his total precedence by Bergson"]
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. For more about the inability to know both speed and location, see Heisenberg uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
.
Nick Huggett
Nick Huggett argues that Zeno is assuming the conclusion when he says that objects that occupy the same space as they do at rest must be at rest.[
]
Paradoxes in modern times
Infinite processes remained theoretically troublesome in mathematics until the late 19th century. With the epsilon-delta
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
definition of limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
, Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
and Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.[ B Russell (1956) ''Mathematics and the metaphysicians'' in "The World of Mathematics" (ed. J R Newman), pp 1576-1590.]
While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Kevin Brown[
] and Francis Moorcroft[
claim 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.
Popular literature often misrepresents Zeno's arguments. For example, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite. However, none of the original ancient sources has Zeno discussing the sum of any infinite series. Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding the ''sum'', but rather with ''finishing'' a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?]
A humorous take is offered by Tom Stoppard
Sir Tom Stoppard (born , 3 July 1937) is a Czech born British playwright and screenwriter. He has written for film, radio, stage, and television, finding prominence with plays. His work covers the themes of human rights, censorship, and politi ...
in his 1972 play ''Jumpers'', in which the principal protagonist, the philosophy professor George Moore, suggests that according to Zeno's paradox, Saint Sebastian
Saint Sebastian (in Latin: ''Sebastianus''; Narbo, Gallia Narbonensis, Roman Empire c. AD 255 – Rome, Italia, Roman Empire c. AD 288) was an early Christian saint and martyr. According to traditional belief, he was killed during the Dioclet ...
, a 3rd Century Christian saint martyred by being shot with arrows, died of fright.
Debate continues on the question of whether or not Zeno's paradoxes have been resolved. In ''The History of Mathematics: An Introduction'' (2010) Burton writes, "Although Zeno's argument confounded his contemporaries, a satisfactory explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.'"
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
offered a "solution" to the paradoxes based on the work of Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
, but 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 image' onto which people can project their most fundamental phenomenological concerns (if they have any)."[
]
Similar Chinese paradoxes
Roughly contemporaneously during the Warring States period
The Warring States period () was an era in History of China#Ancient China, ancient Chinese history characterized by warfare, as well as bureaucratic and military reforms and consolidation. It followed the Spring and Autumn period and concluded ...
(475–221 BCE), ancient Chinese philosophers from the School of Names
The School of Names (), sometimes called the School of Forms and Names (), was a school of Chinese philosophy that grew out of Mohism during the Warring States period in 479–221 BCE. The followers of the School of Names were sometimes called the ...
, a school of thought similarly concerned with logic and dialectics, developed paradoxes similar to those of Zeno. The works of the School of Names have largely been lost, with the exception of portions of the ''Gongsun Longzi''. The second of the Ten Theses of Hui Shi
Hui Shi (; 370–310 BCE), or Huizi (; "Master Hui"), was a Chinese philosopher during the Warring States period. He was a representative of the School of Names (Logicians), and is famous for ten paradoxes about the relativity of time and space, ...
suggests knowledge of infinitesimals:
''That which has no thickness cannot be piled up; yet it is a thousand li in dimension.''
Among the many puzzles recorded in his ''Zhuangzi'' is one very similar to Zeno's Dichotomy:
The Mohist canon appears to propose a solution to this paradox by arguing that in moving across a measured length, the distance is not covered in successive fractions of the length, but in one stage. Due to the lack of surviving works from the School of Names, most of the other paradoxes listed are difficult to interpret.
Quantum Zeno effect
In 1977, physicists E. C. George Sudarshan
Ennackal Chandy George Sudarshan (also known as E. C. G. Sudarshan; 16 September 1931 – 13 May 2018) was an Indian American theoretical physicist and a professor at the University of Texas. Sudarshan has been credited with numerous contrib ...
and B. Misra discovered that the dynamical evolution (motion) of a quantum system can be hindered (or even inhibited) through observation of the system.[
] 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.
Zeno behaviour
In the field of verification and design of timed
timed (time daemon) is an operating system program that maintains the system time in synchronization with time servers using the Time Synchronization Protocol (TSP) developed by Riccardo Gusella and Stefano Zatti. Gusella and Zatti had done ea ...
and hybrid system A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a differential equation) and ''jump'' (described by a state machine or automaton). Often, the te ...
s, the system behaviour is called ''Zeno'' if it includes an infinite number of discrete steps in a finite amount of time.[
] Some formal verification
In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal metho ...
techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour. In systems design
Systems design interfaces, and data for an electronic control system to satisfy specified requirements. System design could be seen as the application of system theory to product development. There is some overlap with the disciplines of system a ...
these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.
Lewis Carroll and Douglas Hofstadter
'' What the Tortoise Said to Achilles,'' written in 1895 by Lewis Carroll
Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ...
, was an attempt to reveal an analogous paradox in the realm of pure logic. If Carroll's argument is valid, the implication is that Zeno's paradoxes of motion are not essentially problems of space and time, but go right to the heart of reasoning itself. Douglas Hofstadter
Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, an ...
made Carroll's article a centrepiece of his book '' Gödel, Escher, Bach: An Eternal Golden Braid,'' writing many more dialogues between Achilles and the Tortoise to elucidate his arguments. Hofstadter connects Zeno's paradoxes to Gödel's incompleteness theorem in an attempt to demonstrate that the problems raised by Zeno are pervasive and manifest in formal systems theory, computing and the philosophy of mind.
See also
* Incommensurable magnitudes
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
* Infinite regress
An infinite regress is an infinite series of entities governed by a recursive principle that determines how each entity in the series depends on or is produced by its predecessor. In the epistemic regress, for example, a belief is justified beca ...
* Philosophy of space and time
Philosophy of space and time is the branch of philosophy concerned with the issues surrounding the ontology and epistemology of space and time. While such ideas have been central to philosophy from its inception, the philosophy of space and time wa ...
* Renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
* Ross–Littlewood paradox
The Ross–Littlewood paradox (also known as the balls and vase problem or the ping pong ball problem) is a hypothetical problem in abstract mathematics and Mathematical logic, logic designed to illustrate the paradoxical, or at least Intuition ( ...
* School of Names
The School of Names (), sometimes called the School of Forms and Names (), was a school of Chinese philosophy that grew out of Mohism during the Warring States period in 479–221 BCE. The followers of the School of Names were sometimes called the ...
* Supertask
In philosophy, a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time. Supertasks are called hypertasks when the number of operations becomes uncountably infinite. A hypertask that inc ...
* " What the Tortoise Said to Achilles", an allegorical dialogue on the foundations of logic by Lewis Carroll
Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ...
(1895).
* Zeno machine
In mathematics and computer science, Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that are capable of carrying out computations involving a co ...
* List of Paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their ...
Notes
References
* Kirk, G. S., J. E. Raven, M. Schofield (1984) ''The Presocratic Philosophers: A Critical History with a Selection of Texts, 2nd ed.'' Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
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* Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
(1926) ''Plato: Cratylus. Parmenides. Greater Hippias. Lesser Hippias'', H. N. Fowler (Translator), Loeb Classical Library
The Loeb Classical Library (LCL; named after James Loeb; , ) is a series of books originally published by Heinemann in London, but is currently published by Harvard University Press. The library contains important works of ancient Greek and L ...
. .
* Sainsbury, R.M. (2003) ''Paradoxes'', 2nd ed. Cambridge University Press. .
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External links
* Dowden, Bradley.
Zeno’s Paradoxes
" Entry in the Internet Encyclopedia of Philosophy
The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. The IEP combines open access publication with peer reviewed publication of original pape ...
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Introduction to Mathematical Philosophy
Ludwig-Maximilians-Universität München
* Silagadze, Z. K.
Zeno meets modern science
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Zeno's Paradox: Achilles and the Tortoise
' by Jon McLoone, Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
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Kevin Brown on Zeno and the Paradox of Motion
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Philosophical paradoxes
Supertasks
Mathematical paradoxes
Paradoxes of infinity
Physical paradoxes