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Zeno's Paradoxes. Edited by W.C. Salmon. Hackett. 2001.
A reprint of the Bobbs-Merrill edition of 1970.These essays lead the reader through the land of the wonderful shrinking genie to the warehouse where the infinity machines are kept. By careful examination of a lamp that is switched on and off infinitely many times, or the workings of a machine that prints out an infinite decimal expansion of pi, we begin to understand how it is possible for Achilles to overtake the tortoise. The concepts that form the basis of modern science—-space, time, motion, change, infinity-—are examined and explored in this edition. Includes an updated bibliography.
- GenresPhilosophyScience
320 pages, Hardcover
First published March 1, 2001
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January 25, 2016
An excerpt from my blog post on time here:
What little we know of Zeno comes primarily from the writings of Aristotle and Plato. The Zeno that we are talking about here is the Zeno who lived in Elea and who was, according to Plato’s Parmenides dialog, a contemporary of Socrates. Elea was a colony of Greeks who had settled in Western Italy. A small ruin remains of the ancient city near to the contemporary town of Velia, south of Salerno. The train from Salerno travels through a tunnel directly beneath the acropolis of the ancient city. It is visited today by a very few tourists (including the author) and nearby school students on field trip. The Porta Rosa above separates the main town from the acropolis.
Diogenes Laertius has a very short note on Zeno of Elea as compared with close to the 150 pages of Greek / English in his Loeb volume devoted to Zeno of Citium, the Stoic. Plato tells us that Zeno was a student, or at least a fellow-traveler with, Parmenides, the great philosopher and law giver of Elea and that the two discoursed with Socrates on their trip to Athens for the Great Panathenaea. Zeno is known for his paradoxes of motion which are bound up with the interplay of motion and time:
1) The Dichotomy
2) Achilles and the Tortoise
3) The Arrow
4) The Stadium
These arguments were apparently aimed at supporting the Parmenidean position that reality is one (although there is some doubt (See here.) As Plato says in the Parmenides,
“In reality, this writing is a sort of reinforcement for the argument of Parmenides against those who try to turn it into ridicule on the ground that, if reality is one, the argument becomes involved in many absurdities and contradictions. This writing argues against these who uphold a Many, and give them back as good as they gave; its aim is to show that their assumption of multiplicity will be involved in still more absurdities than the assumption of unity, if it is sufficiently worked out.” (Burnet’s translation in Early Greek Philosophy)
The only bits of writing that we have directly attributed to Zeno by Diels amount to one page in the Freeman’s English translation; three bits treat of the One versus the Many and one tiny, but interesting, fragment deals with motion, “That which moves, moves neither in the place in which it is, nor in that in which it is not.” (Ancilla to the Pre-Socratic Philosophers)
Aristotle relates the arguments on motion (and time) as follows:
“Zeno’s arguments about motion, which cause so much trouble to those who try to answer them, are four in number. The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. . . .
The second is the so-called Achilles, and amounts to this, that 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. . . .
The third is that . . . . to the effect that the flying arrow is at rest, which result follows from the assumption that time is composed of moments: if this assumption is not granted, the conclusion will not follow.
The fourth argument is that concerning equal bodies which move alongside equal bodies in the stadium from opposite direction – the ones from the end of the stadium, the others from the middle – at equal speeds, in which he thinks it follows that half the time is equal to its double.” (Physics, 239b10-240, revised Oxford translation by Hardie and Gaye.)
Burnet summarizes the goal of these arguments as follows: “This argument (the fourth, the Stadium), like the others, is intended to bring out the absurd conclusions which follow from the assumption that all quantity is discrete, and what Zeno has really done is to establish the conception of continuous quantity by a reductio ad absurdum of the other hypotheses.” In other words, Zeno was arguing not that moments are all that we have, but that to think so is absurd. It was apparently an attempt to support the idea of a block universe, a concept that was reborn 2,500 years later in Einstein’s general relativity theory, of which more anon.
The twentieth century saw an explosion of interest in Zeno by philosophers. Most of the concern with Zeno’s paradoxes focused how they interact with the concept of infinity and infinite series. Bertrand Russell’s lecture on “The Problem of Infinity considered historically” appeared in his 1914 book, Our Knowledge of the External World. Towards the end of this lecture, he writes,
“Zeno’s arguments, in some form, have afforded grounds for almost all the theories of space and time and infinity which have been constructed from his day to our own. We have seen that all of his arguments are valid (with certain reasonable hypotheses) on the assumption that finite spaces and times consist of a finite number of points and instants, and that the third and fourth almost certainly in fact proceeded on this assumption, while the first and the second, which were perhaps intended to refute the opposite assumption, were in that case fallacious. We may therefore escape from his paradoxes either by maintaining that, (1) though space and time do consist of points and instants, the number of them in any finite interval is infinite; or (2) by denying that space and time consist of points and instants at all; or (3) lastly, by denying the reality of space and time altogether.”
Russell goes on to say that it seems Zeno was arguing for the third possibility, that Bergson argued for the second, but that Russell himself sees that the first is justified by the mathematical theories of Cantor in the nineteenth century.
The last half of Russell’s lecture is reprinted as an introductory essay in the 1970 book edited by Wesley Salmon, Zeno’s Paradoxes. This book, which has been used in many philosophy courses in the US since, includes classic essays on Zeno by Henri Bergson, J.O Wisdom, Max Black, G.E.L. Thompson and concluding articles by Adolf Grunbaum, to whom Salmon gives the credit for coming closest to “solving” the paradoxes, following the suggestions of Russell and based on the mathematics of Cantor: “Thus Zeno’s mathematical paradoxes are avoided in the formal part of a geometry built on Cantorian foundations”. (Grunbaum, “Zeno’s Metrical Paradox of Extension,” in Salmon, Zeno’s Paradoxes)
Many other philosophers have taken a crack at Zeno, including very stimulating essays by Gregory Vlastos, reprinted in the book Studies in Greek Philosophy, Volume I, The Presocratics, Gilbert Ryle in his book, Dilemmas and even David Foster Wallace in his book Everything and More, A complete History of ∞. No one really doubts that we can’t walk across the street, as would be implied by The Dichotomy, for example, but Zeno’s paradoxes have been heroically difficult to unravel. DFW concludes his infinity book with the words,
“Gödel’s own personal view was that the Continuum Hypothesis (that there exists no set whose power is greater than that of the naturals and less than that of the reals) is false, that there are actually a whole ∞ of Zeno-type ∞s nested between 0 and c, and that sooner or later a principle would be found that proved this. As of now, no such principle’s ever been found. Gödel and Cantor both died in confinement bequeathing a world with no finite circumference. One that spins, now, in a new kind of all-formal Void. Mathematics continues to get out of bed.”
The important issue for the current project is Zeno’s concept of time: essentially that time is an illusion. But note Russell’s conclusion that Zeno’s argument for the particularity of time, in spite of his goal to prove the opposite, was impregnable, at least up to the nineteenth century. We will see that Zeno’s arguments really set the table for the strife that we have seen in contemporary cosmology.
What little we know of Zeno comes primarily from the writings of Aristotle and Plato. The Zeno that we are talking about here is the Zeno who lived in Elea and who was, according to Plato’s Parmenides dialog, a contemporary of Socrates. Elea was a colony of Greeks who had settled in Western Italy. A small ruin remains of the ancient city near to the contemporary town of Velia, south of Salerno. The train from Salerno travels through a tunnel directly beneath the acropolis of the ancient city. It is visited today by a very few tourists (including the author) and nearby school students on field trip. The Porta Rosa above separates the main town from the acropolis.
Diogenes Laertius has a very short note on Zeno of Elea as compared with close to the 150 pages of Greek / English in his Loeb volume devoted to Zeno of Citium, the Stoic. Plato tells us that Zeno was a student, or at least a fellow-traveler with, Parmenides, the great philosopher and law giver of Elea and that the two discoursed with Socrates on their trip to Athens for the Great Panathenaea. Zeno is known for his paradoxes of motion which are bound up with the interplay of motion and time:
1) The Dichotomy
2) Achilles and the Tortoise
3) The Arrow
4) The Stadium
These arguments were apparently aimed at supporting the Parmenidean position that reality is one (although there is some doubt (See here.) As Plato says in the Parmenides,
“In reality, this writing is a sort of reinforcement for the argument of Parmenides against those who try to turn it into ridicule on the ground that, if reality is one, the argument becomes involved in many absurdities and contradictions. This writing argues against these who uphold a Many, and give them back as good as they gave; its aim is to show that their assumption of multiplicity will be involved in still more absurdities than the assumption of unity, if it is sufficiently worked out.” (Burnet’s translation in Early Greek Philosophy)
The only bits of writing that we have directly attributed to Zeno by Diels amount to one page in the Freeman’s English translation; three bits treat of the One versus the Many and one tiny, but interesting, fragment deals with motion, “That which moves, moves neither in the place in which it is, nor in that in which it is not.” (Ancilla to the Pre-Socratic Philosophers)
Aristotle relates the arguments on motion (and time) as follows:
“Zeno’s arguments about motion, which cause so much trouble to those who try to answer them, are four in number. The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. . . .
The second is the so-called Achilles, and amounts to this, that 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. . . .
The third is that . . . . to the effect that the flying arrow is at rest, which result follows from the assumption that time is composed of moments: if this assumption is not granted, the conclusion will not follow.
The fourth argument is that concerning equal bodies which move alongside equal bodies in the stadium from opposite direction – the ones from the end of the stadium, the others from the middle – at equal speeds, in which he thinks it follows that half the time is equal to its double.” (Physics, 239b10-240, revised Oxford translation by Hardie and Gaye.)
Burnet summarizes the goal of these arguments as follows: “This argument (the fourth, the Stadium), like the others, is intended to bring out the absurd conclusions which follow from the assumption that all quantity is discrete, and what Zeno has really done is to establish the conception of continuous quantity by a reductio ad absurdum of the other hypotheses.” In other words, Zeno was arguing not that moments are all that we have, but that to think so is absurd. It was apparently an attempt to support the idea of a block universe, a concept that was reborn 2,500 years later in Einstein’s general relativity theory, of which more anon.
The twentieth century saw an explosion of interest in Zeno by philosophers. Most of the concern with Zeno’s paradoxes focused how they interact with the concept of infinity and infinite series. Bertrand Russell’s lecture on “The Problem of Infinity considered historically” appeared in his 1914 book, Our Knowledge of the External World. Towards the end of this lecture, he writes,
“Zeno’s arguments, in some form, have afforded grounds for almost all the theories of space and time and infinity which have been constructed from his day to our own. We have seen that all of his arguments are valid (with certain reasonable hypotheses) on the assumption that finite spaces and times consist of a finite number of points and instants, and that the third and fourth almost certainly in fact proceeded on this assumption, while the first and the second, which were perhaps intended to refute the opposite assumption, were in that case fallacious. We may therefore escape from his paradoxes either by maintaining that, (1) though space and time do consist of points and instants, the number of them in any finite interval is infinite; or (2) by denying that space and time consist of points and instants at all; or (3) lastly, by denying the reality of space and time altogether.”
Russell goes on to say that it seems Zeno was arguing for the third possibility, that Bergson argued for the second, but that Russell himself sees that the first is justified by the mathematical theories of Cantor in the nineteenth century.
The last half of Russell’s lecture is reprinted as an introductory essay in the 1970 book edited by Wesley Salmon, Zeno’s Paradoxes. This book, which has been used in many philosophy courses in the US since, includes classic essays on Zeno by Henri Bergson, J.O Wisdom, Max Black, G.E.L. Thompson and concluding articles by Adolf Grunbaum, to whom Salmon gives the credit for coming closest to “solving” the paradoxes, following the suggestions of Russell and based on the mathematics of Cantor: “Thus Zeno’s mathematical paradoxes are avoided in the formal part of a geometry built on Cantorian foundations”. (Grunbaum, “Zeno’s Metrical Paradox of Extension,” in Salmon, Zeno’s Paradoxes)
Many other philosophers have taken a crack at Zeno, including very stimulating essays by Gregory Vlastos, reprinted in the book Studies in Greek Philosophy, Volume I, The Presocratics, Gilbert Ryle in his book, Dilemmas and even David Foster Wallace in his book Everything and More, A complete History of ∞. No one really doubts that we can’t walk across the street, as would be implied by The Dichotomy, for example, but Zeno’s paradoxes have been heroically difficult to unravel. DFW concludes his infinity book with the words,
“Gödel’s own personal view was that the Continuum Hypothesis (that there exists no set whose power is greater than that of the naturals and less than that of the reals) is false, that there are actually a whole ∞ of Zeno-type ∞s nested between 0 and c, and that sooner or later a principle would be found that proved this. As of now, no such principle’s ever been found. Gödel and Cantor both died in confinement bequeathing a world with no finite circumference. One that spins, now, in a new kind of all-formal Void. Mathematics continues to get out of bed.”
The important issue for the current project is Zeno’s concept of time: essentially that time is an illusion. But note Russell’s conclusion that Zeno’s argument for the particularity of time, in spite of his goal to prove the opposite, was impregnable, at least up to the nineteenth century. We will see that Zeno’s arguments really set the table for the strife that we have seen in contemporary cosmology.
December 3, 2024
I read ~40% of the book, but calling it done at this point, since I'm not a philosophy student or professional. I think I've gotten great exposure to what Zeno's paradoxes are, and an excellent general overview of the modern philosophical arguments, from the essays I did read. Giving this 3 stars...hard to rate a compilation of essays from different authors. Bertrand Russell's essay was my favorite, well written--the remnants of an era when some philosophers wrote for a general audience as well as for their contemporaries. That essay by itself deserves 5 stars. The other essays I read were dry (albeit, yes, intellectually interesting....but dry), hence the overall 3-star rating.
February 18, 2024
best book I have ever read. I took philosophy and my teacher did a lesson about Zeno and it made me want to buy it
100% worth it
100% worth it
July 27, 2025
Much to think about but I feel you must be really in the mood to hear these short paradoxes reexplained many times and it is especially grating if you are not as into this mathematical philosophy
Displaying 1 - 4 of 4 reviews