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Several of Zeno's eight surviving paradoxes (preserved in Aristotle's Physics and Simplicius's commentary thereon) are essentially equivalent to one another; and most of them were regarded, even in ancient times, as very easy to refute. Three of the strongest and most famous—that of Achilles and the tortoise, the dichotomy argument, and that of an arrow in flight—are given here. Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Aristotles Physics, frontispice of an 1837 edition Physics (or Physica, or Physicae Auscultationes meaning lessons) is a key text in the philosophy of Aristotle. ... Simplicius, a native of Cilicia, a disciple of Ammonius and of Damascius, was one of the last of the Neoplatonists. ... The Wrath of Achilles, by FranÃ§ois-LÃ©on Benouville (1821â€“1859) (MusÃ©e Fabre) In Greek mythology, Achilles (also Akhilleus or Achilleus) (Ancient Greek: ) was a hero of the Trojan War, the central character and greatest warrior of Homers Iliad, which takes for its theme, not the War... This article or section is in need of attention from an expert on the subject. ... A dichotomy is a division into two non-overlapping or mutually exclusive and jointly exhaustive parts. ...

Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum, also known as proof by contradiction. They are also credited as a source of the dialectic method used by Socrates. This article or section does not cite any references or sources. ... Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ... In classical philosophy, dialectic (Greek: Î´Î¹Î±Î»ÎµÎºÏ„Î¹ÎºÎ®) is an exchange of propositions (theses) and counter-propositions (antitheses) resulting in a synthesis of the opposing assertions, or at least a qualitative transformation in the direction of the dialogue. ... This page is about the ancient Greek philosopher. ...

According to some historians of philosophy, Zeno's paradoxes were a major problem for ancient and medieval philosophers. A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ...

In modern times, calculus has been widely accepted by mathematicians and engineers as at least a practical solution for calculating infinitesimal distances. Other proposed solutions to Zeno's paradoxes from past and present philosophers have included the denial that space and time are themselves infinitely divisible, and the denial that the terms space and time refer to any entity with any innate properties at all. Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ...

Many philosophers still hesitate to say that all paradoxes are completely solved. Some philosophers state that these paradoxes still have modern relevance: attempts to deal with the paradoxes have resulted in intellectual discoveries, and variations on the paradoxes (see Thomson's lamp) continue to produce at least temporary puzzlement in discovering what, if anything, is wrong with the argument. Thomsons lamp is a puzzle that is a variation on Zenos paradoxes. ...

The origins of the paradoxes are somewhat unclear. Diogenes Laertius says that Zeno's teacher, Parmenides, was "the first to use the argument known as 'Achilles and the Tortoise' ", and attributes this assertion to Favorinus. In a later statement, Laertius attributed the paradoxes to Zeno. Diogenes Laërtius, the biographer of the Greek philosophers, is supposed by some to have received his surname from the town of Laerte in Cilicia, and by others from the Roman family of the Laërtii. ... Parmenides of Elea (Greek: , early 5th century BC) was an ancient Greek philosopher born in Elea, a Hellenic city on the southern coast of Italy. ... Favorinus (2nd century AD), was a Greek sophist and philosopher who flourished during the reign of Hadrian. ...

### Achilles and the tortoise

"You can never catch up."

 “ In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead ” —Aristotle, Physics VI:9, 239b15 Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Aristotles Physics, frontispice of an 1837 edition Physics (or Physica, or Physicae Auscultationes meaning lessons) is a key text in the philosophy of Aristotle. ...

In the paradox of Achilles and the Tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is such a fast runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, in which said period the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox. The Wrath of Achilles, by FranÃ§ois-LÃ©on Benouville (1821â€“1859) (MusÃ©e Fabre) In Greek mythology, Achilles (also Akhilleus or Achilleus) (Ancient Greek: ) was a hero of the Trojan War, the central character and greatest warrior of Homers Iliad, which takes for its theme, not the War... This article or section is in need of attention from an expert on the subject. ... A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ...

"You cannot even start."

 “ That which is in locomotion must arrive at the half-way stage before it arrives at the goal. ” —Aristotle, Physics VI:9, 239b10 Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Aristotles Physics, frontispice of an 1837 edition Physics (or Physica, or Physicae Auscultationes meaning lessons) is a key text in the philosophy of Aristotle. ...

Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

$H-frac{B}{8}-frac{B}{4}---frac{B}{2}-------B$

The resulting sequence can be represented as:

$left{ cdots, frac{1}{16}, frac{1}{8}, frac{1}{4}, frac{1}{2}, 1 right}$

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 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 neither be completed nor begun, and so all motion must be an illusion. An illusion is a distortion of a sensory perception, revealing how the brain normally organizes and interprets sensory stimulation. ...

This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness. It is also known as the Race Course paradox. Some, like Aristotle, regard the Dichotomy as really just another version of Achilles and the Tortoise. However, they emphasise different points. In the Achilles and the Tortoise, the focus is that movement by multiple objects is just an illusion whereas in the Dichotomy the focus is that movement is actually impossible. A dichotomy is a division into two non-overlapping or mutually exclusive and jointly exhaustive parts. ...

"You cannot even move."

 “ If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. ” —Aristotle, Physics VI:9, 239b5 Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Aristotles Physics, frontispice of an 1837 edition Physics (or Physica, or Physicae Auscultationes meaning lessons) is a key text in the philosophy of Aristotle. ...

In the arrow paradox, Zeno asks us to imagine an arrow in flight. He then asks us to divide up time into a series of indivisible nows or moments. At any given moment if we look at the arrow it has an exact location so it is not moving. Yet movement has to happen in the present; it can't be that there's no movement in the present yet movement in the past or future. So throughout all time, the arrow is at rest. Thus motion can not happen.

This paradox is also known as the fletcher's paradox—a fletcher being a maker of arrows. This is an article about the projectile; see Arrow (disambiguation) for other meanings. ...

Whereas the first two paradoxes presented divide space into segments, this paradox divides time into points.

## Proposed solutions

### Proposed solutions to the arrow paradox

Aristotle, who recorded Zeno's arguments in his work Physics, disputes Zeno's reasoning. Aristotle denies that time is composed of "nows", as implied by Zeno's argument. If there is just a collection of "nows", then there is no such thing as temporal magnitude. Therefore, if Aristotle is correct in denying that time is composed of indivisible "nows", then Zeno is wrong in saying that the arrow was stationary throughout its flight despite saying that in each "now" the moving arrow is at rest. Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Aristotles Physics, frontispice of an 1837 edition Physics (or Physica, or Physicae Auscultationes meaning lessons) is a key text in the philosophy of Aristotle. ...

According to Zeno, at any instant, the arrow must be at rest. However, this has been disputed, since being at rest is a relative term. One cannot judge, from observing any one instant, that the arrow is at rest. Rather, one requires other, adjacent instants to assert whether, compared to other instants, the arrow at one instant is at rest. Thus, compared to other instants, if the arrow is found to be at a different place than it was and will be at the times before and after, then we have reason to claim the arrow has moved.

A mathematical account would be as follows: in the limit, as the length of a moment approaches zero, the instantaneous rate of change or velocity (which is the quotient of distance over length of the moment) does not have to approach zero. This nonzero limit is the velocity of the arrow at the instant. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

The concepts involved in this paradox are in some ways reminiscent of Heisenberg's Uncertainty principle in which as we become more precise in locating the position of quantum particles, we necessarily become less precise about their momentum — and thus about their velocity. In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ... In classical mechanics, momentum (pl. ...

### Proposed solutions both to Achilles and the tortoise, and to the dichotomy

Both the paradoxes of Achilles and the tortoise and that of the dichotomy depend on dividing distances into a sequence of distances that become progressively smaller, and so are subject to the same counter-arguments. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

Aristotle pointed out that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances.

Before 212 BCE, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Theorems have been developed in more modern calculus to achieve the same result, but with a more rigorous proof of the method. These methods allow construction of solutions stating that (under suitable conditions) if the distances are always decreasing, the time is finite. â€œEra Vulgarisâ€ redirects here. ... Archimedes of Syracuse (Greek: c. ... Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ...

#### Proposed solution using mathematical series notation

Several proposed solutions have at their core geometric series. A general geometric series can be written as In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

$asum_{k=0}^{infty} x^k,$

which is convergent and equal to a/(1−x) provided that |x| < 1 (otherwise the series diverges). Again, if we attempted to solve for a/(1−1) − a/0 we are back to the problem that nothing can be divided by 0. The geometric series that would solve this to convergence reflects our perceived reality, if 0 were that point at which there were no more distance to travel and the destination had been reached. Albert Einstein had written that "what we are sure about in reality is not reflected in mathematics and what we're sure about in mathematics is not reflected in reality." This paradox on the surface is a reflection of the conundrum of math vs reality. In mathematics, a series is the sum of the terms of a sequence of numbers. ... In mathematics, a divergent series is an infinite series that does not converge. ... â€œEinsteinâ€ redirects here. ...

Although these proposed solutions effectively involve dividing up the distance to be travelled into smaller and smaller pieces, it is easier to conceive of the solution as Aristotle did, by considering the time it takes Achilles to catch up to the tortoise.

In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (m/s) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv m/s with x > 1. It takes Achilles time d/xv seconds (s) to travel distance d and reach the point where the tortoise started, at which time the tortoise has travelled d/x m. It then takes further time d/x2v sec for Achilles to travel this new distance d/x m, at which time the tortoise has travelled another d/x2, and so on.

Thus, the time taken for Achilles to catch up is

$frac{d}{v} sum_{k=1}^infty left( frac{1}{x} right)^k = frac{d}{v(x-1)} ,$ seconds.

Since this is a finite quantity, Achilles will eventually catch the tortoise.

Similarly, for the Dichotomy assume that each of Homer's steps takes a time proportional to the distance covered by that step. Suppose that it takes time h seconds for Homer to complete the last half of the distance to the bus; then it will have taken h/2 sec for him to complete the second-last step, traversing the distance between one quarter and half of the way. The third-last step, covering the distance between one eighth and one quarter of the way to the bus, will take h/4 sec, and so on. The total time taken by Homer is, summing from k = 0 for the last step,

$h sum_{k=0}^{infty} left( frac{1}{2} right)^k = 2h ,,$ seconds.

Once again, this is a convergent sum: although Homer must pass through an infinite number of distance segments, most of these are infinitesimally short and the total time required is finite. So (provided it doesn't leave for 2h seconds) Homer will catch his bus.

In both cases, by moving at constant speeds (and in particular not stopping after each segment) Achilles will eventually catch the moving tortoise, and Homer the stationary bus. However, the solutions that employ geometric series have the advantage that they attempt to solve the paradoxes in their own terms, by denying the apparently paradoxical conclusions.

We now give a concrete example. Suppose the tortoise starts 100 meter in advance of Achilles, and moves at 0.1 meter per second, while Achilles moves at 10 meters per second. Then after 10 seconds Achilles will reach the tortoise's earlier location, and the tortoise will be 1 meter ahead of him. After 0.1 seconds Achilles will reach this location, but the tortoise will be 0.01 meter ahead of him. After 0.001 seconds Achilles will reach this location, but the tortoise will be 0.0001 meter ahead of him. It seems as if Achilles will never reach the tortoise. However even though there is infinite numbers of "steps" Achilles will have to go through, it will take him a finite amount of time to do that: The number of seconds is 10+0.1+0.001+0.00001+0.0000001+... = 10 + 10/99 seconds.

#### Proposed solution using calculus notation

Zeno's paradoxes are addressed by calculus, in particular by the mathematical concept of limit, which gives a theoretical framework to deal with the problems brought up by Zeno. Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

• d = distance between runners
• t = time
$lim_{d to 0}f(d) = t$

#### Issues with the proposed calculus-based solution

A suggested problem with using calculus and mathematical series to try to solve Zeno's paradoxes is that these solutions miss the point. To be precise, while these kinds of solutions specify the limit point of infinite series, they do not explain how such a series can actually ever be completed and the limit point be reached. Thus, calculus and mathematical series can be used to predict where and when Achilles will overtake the tortoise, assuming that the infinite sequence of events as laid out in the argument ever comes to an end. But, the problem lies exactly with that assumption, as Zeno's paradox points out that in order for Achilles to catch up with the Tortoise, an infinite number of physical events need to take place, which seems to be impossible in and of itself, independent of how much time such an act would require if it could actually be done.

Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned. To be precise, Zeno *started* with the assumption that a finite interval can be split into infinitely many parts, and then argued that it is impossible to move through such a landscape. For calculus and other series-based solutions to make the point that the sum of infinitely many terms can add up to a finite amount therefore merely confirms Zeno's assumption about the landscape (geometry) of space, but does nothing to answer Zeno's question of how we can actually (dynamically) move through such a space.

An added complication is that many treatments of Zeno's paradox present Zeno's reasoning in such a way that calculus and series-based solutions really do work as objections to that reasoning. To be precise, Zeno's reasoning is often presented as arguing that because there are an infinite number of tasks to be done, it will take an infinite amount of time to complete all these tasks. And, the calculus and mathematical series based solutions are now perfectly correct in objecting that adding an infinite number of intervals can add up to a finite amount of time. However, such a presentation of Zeno's argument makes the argument into a straw man: a weak (and indeed invalid) caricature of the much stronger and much simpler argument that does not at all consider any quantifications of time. This much simpler argument simply states that for Achilles to capture the tortoise an infinite series of physical events need to be completed, which is logically impossible. The calculus and mathematical series based solutions offer no insight into this much simpler, much more stinging, paradox.[2] A straw man argument is a logical fallacy based on misrepresentation of an opponents position. ...

The following thought experiment can be used to illustrate the difference. Imagine that Achilles notes the position occupied by the tortoise, and calls it first; after reaching that position, he once again notes the position the turtle has moved to, calling it second, and so on. If he catches up with the turtle at all, then apparently Achilles must have stopped counting, and we could ask Achilles what the greatest number he counted to was. But of course this is nonsense: there is no greatest number, and Achilles can never stop counting. So, Achilles can't catch up with the Tortoise, whether he has finite time or infinite time to do so.

This mathematical and logical analysis of the physical process of catching up, or of any movement through space at all, would show that it is impossible for Achilles to win or move at all, which is of course the whole paradox. So, in order for the paradox to be resolved, one needs to show why this mathematical analysis cannot be used in our physical world, where things do move. As suggested below, maybe space and time are not so that between any two points one can always find another point, which would indeed prevent this analysis to go through. Possibly our naive conceptions of space and time are mistaken in other ways as well. But whatever it may be, calculus and series-based proposals to resolve the paradox say nothing about the nature of space and time that prevents these infinite sequences to crop up, and as such they do not resolve the paradox.

#### Issues with the issues with the proposed calculus-based solution

If we more closely examine the thought experiment, it is clear that Achilles naming the positions "first", "second", and so forth, is a nonphysical/mathematical act rather than a physical act; as an illustration, try getting your friend to say the word "Bob" on the 1/2 second mark, then the 1/4 second mark, and so on... you just can't do it (you could do this yourself, but it's much more fun to watch somebody else try). Consequently, the "counting process" is a mathematical process, while the "catching up with the turtle" is a physical process. As with most attempts to peddle Zeno's paradoxes, the central element is the conflation of these two processes. But they are simply not to be identified. The mathematical "counting process" goes on to infinity, and this is never something one could complete. However the physical "catching up with the turtle" process is something that can be completed. This is shown by an elementary application of limiting process theory, with time as a parameter.

These considerations (one must divorce the mathematical and physical processes at hand) also apply to the paradox as given in the "much more stinging" form: "for Achilles to capture the tortoise will require him to go beyond, and hence to finish, going through a series that has no finish, which is logically impossible". Here the word finish has been confusingly used for both the physical process and the mathematical process in an effort to conflate the two.

The issue with the statement "Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned." is similar. They (the vast majority) do not assume that one can finish the limiting (mathematical) process, and they do not need to. To finish the physical process it is not required to finish the associated mathematical (limiting) process. The two processes are completely different in nature, and divorcing the two is essential if one is to resolve the paradox.

The mention of time does not make the paradox into a strawman, and telling someone they can't mention time in their solution is extremely unfair, because the problem is posed in the form of the physical, and consideration of time is implicit in any consideration of the physical. Just because someone worded the problem without using the phrase "time" does not make it illegal to use the word "time" in the solution.

### Are space and time infinitely divisible?

Another proposed solution to some of the paradoxes is to consider that space and time are not infinitely divisible. Just because our number system enables us to give a number between any two numbers, it does not necessarily follow that there is a point in space between any two different points in space, and the same goes for time.

If space-time is not infinitely divisible (and thus not perfectly continuous), it is "discrete" (composed of “lumps” and “jumps”, as is experimentally observed in the field of quantum physics e.g. electron orbitals jumping from one level to another). This means that motion is, at the smallest physical level, a series of jumps from one quantum space-time coordinate to the next, each occurring over distance and time intervals that are not divisible into smaller measures. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... Fig. ...

Thus the total number of quantum jumps made while traversing from point A to point B is finite, and therefore there is no paradox.

### Does motion involve a sequence of points?

Augustine of Hippo was the first to posit that time has no precise "moments," in his 4th century C.E. text, Confessions. In Book XI, section XI, paragraph 13, Augustine says, "truly, no time is completely present," and in Book XI, section XV, paragraph 20, Augustine says "the present, however, takes up no space." â€œAugustinusâ€ redirects here. ... Confessions is the name of a series of thirteen autobiographical books by St. ...

Some people, including Peter Lynds, have proposed a solution based on this ancient premise. Lynds posits that the paradoxes arise because people have wrongly assumed that an object in motion has a determined relative position at any instant in time, thus rendering the body's motion static at that instant and enabling the impossible situation of the paradoxes to be derived. Lynds asserts that the correct resolution of the paradox lies in the realisation of the absence of an instant in time underlying a body's motion, and that regardless of however small the time interval, it is still always moving and its position constantly changing, so can never be determined at a time. Consequently, a body cannot be thought of as having a determined position at a particular instant in time while in motion, nor be fractionally dissected as such, as is assumed in the paradoxes (and their historically accepted solutions). Peter Lynds (born May 17, 1975) is a New Zealander who drew sudden attention in 2003 with the publication of an article on the study of time, mechanics and Zenos paradoxes. ...

### Conceptual and semantical approaches

Another approach is to deny that our conceptual account of motion as point-by-point movement through continuous space-time needs to match exactly with anything in the real world altogether. Thus, one could deny that time and space are ontological entities. That is, maybe we should acknowledge our Platonic view of reality, and say that time and space are simply conceptual constructs humans use to measure change, that the terms (space and time), though nouns, do not refer to any entities nor containers for entities, and that no thing is being divided up when one talks about "segments" of space or "points" in time. // In philosophy, ontology (from the Greek , genitive : of being (part. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... In general, a reference is something that refers to or designates something else, or acts as a connection or a link between two things. ...

Similarly, one can say that the number of "acts" involved in anything is merely a matter of human convention and labeling. In the constant-pace scenario, one could consider the whole sequence to be one "act," ten "acts," or an infinite number of "acts." No matter how the events are labeled, the tortoise will follow the same trajectory over time, and all of the acts will be "finished" by the time the tortoise reaches the finish line. Thus, the labeling of acts is arbitrary and has nothing to do with the underlying physical process being described and that it is possible to "finish" an infinite sequence of acts.

### The notion of different orders of infinity

Some people state that the dichotomy paradox merely makes the point that the points on a continuum cannot be counted — that from any point, there is no next point to proceed to. However, it is not clear how this comment resolves the paradox. Indeed, as one variant of Zeno's paradox would state: if there is no next point, how can one even move at all? Also, it is not clear what this comment has to do with different orders of infinity: the rational numbers are countable, i.e. of the same order of infinity as the natural numbers, but on the rational number line, there is for any rational number still no next rational number either.

## Status of the paradoxes today

Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Many philosophers, and certainly engineers, generally agree with the mathematical results. Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... Sir Isaac Newton (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1726][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ...

Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this, he followed Leopold Kronecker, an earlier 19th century mathematician. Some claim that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has not resolved all problems involving infinities, including Zeno's. Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 &#8211; February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 &#8211; May 23, 1857) was a French mathematician. ... Abraham Robinson Abraham Robinson (October 6, 1918 â€“ April 11, 1974) was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics. ...

## Two other paradoxes as given by Aristotle

"… if everything that exists has a place, place too will have a place, and so on ad infinitum". (Aristotle Physics IV:1, 209a25)

Paradox of the Grain of Millet: Look up Ad infinitum in Wiktionary, the free dictionary. ...

"… there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially." (Aristotle Physics VII:5, 250a20)

For an expanded account of Zeno's arguments as presented by Aristotle, see: Simplicius' commentary On Aristotle's Physics.

## The quantum Zeno effect

In recent time, physicists studying quantum mechanics have noticed 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. The quantum Zeno effect is a quantum mechanical phenomenon first described by George Sudarshan and Baidyanaith Misra of the University of Texas in 1977. ...

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Image File history File links This is a lossless scalable vector image. ... Wikiquote is a sister project of Wikipedia, using the same MediaWiki software. ... In philosophy, a supertask is a task occurring within a finite interval of time involving infinitely many steps (subtasks). ... In mathematics and computer science, Zeno machines (abbreviated ZM, and also called Accelerated Turing machine, ACM) are a hypothetical computational model related to Turing machines that allows a countably infinite number of algorithmic steps to be performed in finite time. ... In mathematics, the recurring decimal 0. ... GÃ¶del, Escher, Bach: an Eternal Golden Braid: A metaphorical fugue on minds and machines in the spirit of Lewis Carroll (commonly GEB) is a Pulitzer Prize (1980)-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ... What the Tortoise Said to Achilles is a brief dialog by Lewis Carroll which playfully problematizes the foundations of logic. ...

• Terry Pratchett in Pyramids combines the "Achilles and the tortoise" paradox and the arrow paradox to create the paradox of the arrow chasing the tortoise. [2] The concept of "nows" in the flecher's paradox is also explored in Theif of Time.
• Dilbert has claimed that "No one ever wants to take more than half of what's left of the last doughnut. That's why I call it Xeno's doughnut" (2005-08-13).
• Umberto Eco in his 2004 novel (English language version 2005), The Mysterious Flame of Queen Loana, has the narrator (trying to recover from amnesia by going through old books and possessions) look at a recursive image and remark: "... Chinese boxes or Matrioshka dolls. Infinity, as seen through the eyes of a boy who has yet to study Zeno's paradox. The race towards an unreachable goal; neither the tortoise nor Achilles would ever have reached the last..."
• In Beyond Zork, there is a bridge named "Zeno's Bridge". It is impossible to fully cross this bridge, as you can only go a fraction of the distance to the destination.
• Phillip K. Dick in his 1953 short story The Indefatigable Frog uses Zeno's paradoxes as a basis for an experiment that places a shrinking frog in a tunnel, thus always increasing the length of the tunnel relative to the frog.
• Jorge Luis Borges explores various implications of Zeno's paradoxes in several of his stories and essays. In Avatars of the Tortoise (1932) he discusses how the argument of "Achilles and the tortoise" has manifested itself in the writings of Plato, William James, Lewis Carroll and many others, and goes on to argue that the paradox demonstrates the unreality of the visible world.
• Tom Stoppard in his play Jumpers references Zeno's paradox via the philosopher George; George manages to kill his tortoise Thumper while attempting to disprove Zeno's paradox.
• In Knight Rider Season 1 - Episode 09 Trust Doesn't Rust - 0:24:00 until 0:24:38 - a reference was made by KITT to an upcoming duel between him (KITT) and KARR (his earlier prototype). Kitt says: "...however since KARR is as powerful and as nearly indestructible as myself, Zeno's paradoxes should be affected." Devon Miles then explains who Zeno is. Kitt continues: "Zeno first postulated a question which my twin would most certainly be aware of: To wit; what would happen if an irresistible force met an immovable object?" In a later head-on-collision scene with KITT vs. KARR, Michael says: "..remember Zeno and that immovable object thing? We are about to find out the answer." Scholars do not attribute the irresistible force paradox to Zeno, and its origin is uncertain.
• In Harry Turtledove's novel Wisdom of the Fox, the character Rihwin employs the dichotomy paradox to trick a demon into carelessly halving its distance to him and thus leaving itself vulnerable.
• In the movie I.Q. (1994), Catherine Boyd (played by Meg Ryan), poses Zeno's dichotomy paradox to her love interest, Ed Walters (Tim Robbins), as a flirtingly jocular explanation as to why it is impossible for her to approach and dance with him.

Terence David John Pratchett OBE (born April 28, 1948, in Beaconsfield, Buckinghamshire, England[1]) is an English fantasy author, best known for his Discworld series. ... Dilbert (first published April 16, 1989) is an American comic strip written and drawn by Scott Adams. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... is the 225th day of the year (226th in leap years) in the Gregorian calendar. ... Umberto Eco (born January 5, 1932) is an Italian medievalist, semiotician, philosopher and novelist, best known for his novel The Name of the Rose (Il nome della rosa) and his many essays. ... The Mysterious Flame of Queen Loana (Original Title in Italian: La Misteriosa Fiamma della Regina Loana) is an illustrated novel by Italian writer Umberto Eco. ... See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page â€” a list of pages that otherwise might share the same title. ... This article does not cite any references or sources. ... A Matryoshka doll (Cyrillic &#1084;&#1072;&#1090;&#1088;&#1105;&#1096;&#1082;&#1072; or &#1084;&#1072;&#1090;&#1088;&#1077;&#1096;&#1082;&#1072;) is a Russian nesting doll. ... Zork universe Zork games Zork Anthology Zork trilogy Zork I   Zork II   Zork III Beyond Zork   Zork Zero   Planetfall Enchanter trilogy Enchanter   Sorcerer   Spellbreaker Other games Wishbringer   Return to Zork Zork: Nemesis   Zork Grand Inquisitor Zork: The Undiscovered Underground Topics in Zork Encyclopedia Frobozzica Characters   Kings   Creatures Timeline   Magic   Calendar... Philip Kindred Dick (December 16, 1928 &#8211; March 2, 1982), often known by his initials PKD, or by the pen name Richard Phillips, was an American science fiction writer and novelist who changed the genre profoundly. ... Jorge Luis Borges (August 24, 1899 â€“ June 14, 1986) was an Argentine writer. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... For other people named William James see William James (disambiguation) William James (January 11, 1842 â€“ August 26, 1910) was a pioneering American psychologist and philosopher. ... Charles Lutwidge Dodgson (Lewis Carroll) â€“ believed to be a self-portrait Charles Lutwidge Dodgson (January 27, 1832 â€“ January 14, 1898), better known by the pen name Lewis Carroll, was an English author, mathematician, logician, Anglican clergyman and photographer. ... Sir Tom Stoppard, OM, CBE (born as TomÃ¡Å¡ Straussler on July 3, 1937)[1] is an Academy Award winning British playwright of more than 24 plays. ... Knight Rider was a popular American television series that ran between September 26, 1982, and August 8, 1986. ... KITT on display at Universal Studios. ... KARR from the episode KARR (the Knight Automated Roving Robot), is a fictional character and a villain from the cult television series Knight Rider. ... The Irresistible force paradox is a classic paradox formulated as follows: What happens when an irresistible force meets an immovable object? Common responses to this paradox resort to logic and semantics. ... Harry Norman Turtledove (born June 14, 1949) is an American historian and prolific novelist who has written historical fiction, fantasy, and science fiction works. ... â€œFiendâ€ redirects here. ... I.Q. is a 1994 romantic comedy film directed by Fred Schepisi, starring Tim Robbins, Meg Ryan and Walter Matthau. ... Meg Ryan (born November 19, 1961) is a questionable American actress who specializes in romantic comedies, but has also worked in other film genres. ... Tim Robbins at Cannes, 2001 Height: 6 ft 4 in / 1. ...

Zeno's paradox on a 1990 Croatian election poster
• A political party in Croatia in 1990 used Zeno's paradox of Achilles and the tortoise to encourage voters to choose the path of slow and clever civilian initiatives (the tortoise) rather than militant nationalism (Achilles as an armored warrior). Croatian voters decided that Achilles would win the race.
• Artist Mark Tansey created a painting titled Achilles and the Tortoise which shows the plume from a rising rocket, around which there are observers, which mimics the appearance of a nearby evergreen tree, with the rocket just short of passing the height of the tree. In the foreground are a group of people finishing the activity of planting a new tree. The image, suggesting the rocket as shown will not ever pass the height of the tree, and tantalizing the viewer with the idea that this has something to do with the processes being shown and that the new tree may or may not pass the rocket first--raises questions about how much static representations can have you understand what's being represented. Tansey is alluding to Zeno's own logic as a representation of what he's describing.

Image File history File links Size of this preview: 423 Ã— 599 pixelsFull resolution (701 Ã— 992 pixel, file size: 346 KB, MIME type: image/jpeg) This 1990 poster for the Social Democratic Union of Croatia and Social Democratic Union of Yugoslavia features a picture of Zenos paradox of motion with... Image File history File links Size of this preview: 423 Ã— 599 pixelsFull resolution (701 Ã— 992 pixel, file size: 346 KB, MIME type: image/jpeg) This 1990 poster for the Social Democratic Union of Croatia and Social Democratic Union of Yugoslavia features a picture of Zenos paradox of motion with... Mark Tansey (born 1949- in San Jose, California) is an American postmodern painter best known for monochromatic works which include commentary in their title, for example The Triumph of the New York School from 1984. ...

## Zeno's paradoxes in popular culture

• In the Microsoft/Ensemble Studios game, Age of Mythology: The Titans, the phrase ZENO'S PARADOX in all caps in the chat menu assigns a random selection of god/titan powers.
• On the CD How Can You Be in Two Places at Once When You're Not Anywhere at All by The Firesign Theatre, a track called Zeno's Evil features a series of road signs saying "Antelope Freeway, 1 mile", "Antelope Freeway, 1/2 mile", "Antelope Freeway 1/4 mile", "Antelope Freeway, 1/8 mile", "Antelope Freeway, 1/16 mile" and so on.
• The UK synthpop band Spray's first album contained a track titled "So Close," applies the fletcher's paradox to romance, including the telling line, "How close can I get when Cupid's arrow is always doomed to stall halfway to your heart?"
• Seattle band "Awesome"'s performance piece and album Delaware includes a song entitled Zeno's Rock, with lines such as "I travel half the distance in a single well-placed step" and "I'm always halfway there."

Microsoft Corporation, (NASDAQ: MSFT, HKSE: 4338) is a multinational computer technology corporation with global annual revenue of US\$44. ... Ensemble Studios is a Microsoft-owned company that has developed several computer games, including the famous Age of Empires series. ... Age of Mythology (AoM) is a real-time strategy computer game in the popular Age of Empires series. ... How Can You Be in Two Places at Once When Youre Not Anywhere at All was the second comedy album recorded by The Firesign Theatre. ... Left to right: Phil Proctor, Peter Bergman, Phil Austin, and David Ossman in 2001 The Firesign Theatre is a comedy troupe consisting of Phil Austin, Peter Bergman, David Ossman, and Philip Proctor. ... This article needs additional references or sources for verification. ... Spray is a 21st century synthpop duo from England, starring Jenny McLaren and Ricardo Autobahn. ... City nickname Emerald City City bird Great Blue Heron City flower Dahlia City mottos The City of Flowers The City of Goodwill City song Seattle, the Peerless City Mayor Greg Nickels County King County Area   - Total   - Land   - Water   - % water 369. ...

## Notes

1. ^ Plato, Parmenides, 128c; Kirk, p. 277.
2. ^ Kevin Brown, Reflections on Relativity, [1]

PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... James Kevin (Kevin) Brown (born March 14, 1965) is a former Major League Baseball right-handed starting pitcher. ...

## References

• Plato. Plato: Cratylus. Parmenides. Greater Hippias. Lesser Hippias, H. N. Fowler (Translator), Loeb Classical Library (January 1, 1926). ISBN 0674991850.
• Kirk, G. S., J. E. Raven, M. Schofield. The Presocratic Philosophers: A Critical History with a Selection of Texts. Cambridge University Press; 2 edition (February 24, 1984). ISBN 0521274559.
• Sainsbury, R.M., Paradoxes, Second Ed (Cambridge UP, 2003). ISBN 0521483476.

PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ...

Results from FactBites:

 Zeno's Paradoxes (Stanford Encyclopedia of Philosophy) (0 words) Second, from this Zeno argues that it follows that they do not exist at all; since the result of joining (or removing) a sizeless object to anything is no change at all, he concludes that the thing added (or removed) is literally nothing. Thus we answer Zeno as follows: the argument assumed that the size of the body was a sum of the sizes of the point parts, but that is not the case; according to modern mathematics, a line is an uncountable infinity of points plus a distance function. Zeno abolishes motion, saying "What is in motion moves neither in the place it is nor in one in which it is not".
 Zeno's Paradox, Aporia: Achilles, Tortoise (3104 words) Zeno Of Elea's Aporia: Achilles Can't Outrun The Tortoise (?) Zeno Of Elea's Paradox: Achilles Can't Outrun The Tortoise(?) : : Paradoxes of Zeno The Eleatic: Aporia of Achilles And The Tortoise.
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