After three deep posts on Parmenides, Zeno of Elea will be a refreshing change of pace where we can rest our minds a bit and have some fun. Zeno is known almost exclusively for his intriguing paradoxes.1 For example, according to Zeno, did you know that if you set off to reach a destination, you will never make it? It is impossible. In other words, you can’t get there from here. And his argumentation is pretty convincing. In fact, people have been trying for years to find solutions to his perplexing paradoxes. But before we get into all of that, let’s learn a little bit about his life.
Zeno of Elea’s Life
We do not know much about the man who created these intriguing paradoxes for there are various accounts of his life that are contradictory. Nevertheless, there are things about him that we do know for sure. He was born around 495 B.C. and died around 430 B.C.2 This made him 25 years younger than his mentor and teacher, Parmenides.
But Zeno was more than a student of Parmenides. Diogenes Laertius states that Parmenides adopted him as his son.3 This was a common practice in certain parts of the Greek world where even adults could be adopted.4 This put Zeno’s relationship with Parmenides on an entirely different level. Rather than being just a student, this implies that he was to be Parmenides’ successor in the very important role of Priest of Apollo. Please see post 43 for more information on this.
Zeno and the Tyrant
The Greek philosophers of old actually tried to make practical differences in their world. This is refreshing to see since nowadays we tend to view philosophers as impractical and irrelevant academics whose primary aim is to keep publishing lest they perish.
Not so for Zeno. Diogenes stated that
“Zeno was a man of great nobility, both in philosophy and politics, his extant books at any rate, are full of intelligence.”5
As the story goes, a tyrant named Nearchus arose in Elea. Zeno was caught taking weapons to rebels who were opposing the tyrant and he was arrested.6 According to Diogenes, Zeno was brought before the tyrant and interrogated about the plot and who his co-conspirators were. He named all of the tyrant’s friends, hoping to separate Nearchus from his supporters.
When he was asked if there were any other supporters, he stated that he wanted to whisper that in Nearchus’ ear. When he got close to his ear, he said, “Yes, you, the scourge of the city,” and then he did a Mike Tyson and bit down on his ear.
Nearchus screamed and his guards pulled the two men apart. After the two men were separated, Zeno spit out the piece of ear from his bloody mouth, looked around at the crowd and said, “I am surprised at your cowardice, if in fear of what I am now enduring you could be the tyrant’s slaves.”
In all of this, he did not betray his friends. He was then ordered thrown into a giant mortar where he was beaten and crushed to death. Needless to say, this is not the usual way that a philosopher’s academic career typically ends.
But in reality, many philosophers of old have had such experiences for ideas can be dangerous things, especially to a corrupt establishment.7 Think about Socrates who was executed for simply asking questions. We once again face tyranny in the West and it would seem that some philosopher out there should be able to develop a philosophy, expanding upon the traditional philosophies, that would speak to the errors of modern times. If such a philosophy does emerge that is true and sound, we should also expect a fierce backlash – if not death – for those who hold such views. This also holds true for pastors, artists, poets, and writers who seek to present the truth.
Diogenes said of Zeno:
“You desired, Zeno, and your desire was noble, to slay the tyrant and deliver Elea from slavery. But you were defeated; for the tyrant caught you and beat you in a mortar. But what am I saying? It was your body that he beat and not you.”8
Zeno and Parmenides
Plato’s Parmenides centers around a dialogue between a young Socrates and the distinguished philosophers Parmenides and Zeno, among others. Parmenides and Zeno traveled to Athens from Elea.9 Parmenides is the main speaker throughout this fictional dialogue, but the dialogue opens with Zeno reading a book he had written to a group that had eagerly gathered to hear it.
What was all of the excitement about concerning Zeno’s new book? Why did they travel all the way from Elea to read it? Well, in post 44 we learned about Parmenides’ main idea that being is the unifying principle of all things and thus is the only thing that exists. The corollary of this is that all reality is one. There is no diversity or multiplicity. Change and plurality are thus illusions.
As you can imagine, this must have caused great reverberations throughout the philosophical world that reached all the way to Athens. So Parmenides came to Athens, the philosophical capital of the world, in order to defend his doctoral dissertation, so to speak. It’s as if Plato is emphasizing that his Academy is the central philosophical authority in the world as opposed to Miletus or Elea.
In actuality, Parmenides does not defend his dissertation but instead leaves that to Zeno who wrote a book defending his ideas. Zeno said that he wrote the book to defend Parmenides, who was being made fun of because of his ideas. In his book, Zeno does a very clever thing: He turns the tables and goes on the offense. The best defense is a good offense, as the saying goes.
Rather than defend oneness, Zeno attacks plurality as absurd. The central thesis of Zeno’s book is that everything has to be one because it is utterly impossible for plurality to exist. He lays out a simple reason, which young Socrates spends the majority of the dialogue trying to disprove, and to no avail! This is the amazing genius of Zeno that we will also see when studying his paradoxes. He comes up with the most absurd propositions that defy what our senses tell us about reality, and yet are impossible to disprove logically or mathematically. His genius lies in the simplicity of his assertions.
Why, according to Zeno, is it impossible for plurality to exist? Zeno’s central thesis is as follows:
“If things are many, they must be both like and unlike, but that is impossible, because unlike things can’t be like or like things unlike?10
Now this is interesting to say the least. What he is saying is that it is impossible for things to be many. For if we had a plurality of dogs, for example, then each dog would be like the other dogs since they are dogs, but each dog would be unlike all of the other dogs because no two dogs are the same. Since it is impossible, then, for an object to be like and unlike at the same time, it is impossible for things to be many. This sounds like a simple matter, but it could not be refuted in the 50 pages of Plato’s Parmenides.
Zeno’s challenge is thus summarized:
“Accordingly, my book speaks against those who assert the many and pays them back in kind with something for good measure, since it aims to make clear that their hypothesis, the many exists, would, if someone examined the matter thoroughly, suffer consequences even more absurd than those suffered by the hypothesis of its being one.”11
Young Socrates posits the theory of Forms (really Plato’s idea put in the mouth of Socrates) in order to refute Zeno. According to Plato, the theory of Forms gives a solution to the problem of how the one and the many can both exist. This is the problem of universals, which is a recurring theme in my blog since, as I have stated before, it is the fundamental philosophical problem to be solved.
In the rest of the dialogue, Parmenides goes on the offense and grills Socrates, making him defend the theory of Forms. Parmenides succeeds in poking holes in all of Socrates’ arguments until Plato’s theory collapses in upon itself in a heap of nonsensical propositions.
The interesting question is this: Why would Plato, since he wrote the dialogue, show the weakness of his own theory of Forms which he otherwise defends as being true? There are several answers to this question which I hope to discuss in a future post dedicated to the Parmenides dialogue and Plato’s theory of Forms.
Zeno developed around 40 paradoxes, but unfortunately only 10 have survived.12 Zeno’s paradoxes on motion are known primarily from Aristotle’s Physics. In this post, I will discuss the most popular and interesting ones. In the next post, I will dive a little deeper into some of his lesser-known paradoxes on plurality including one of my favorites, infinite divisibility.
Zeno’s three most popular paradoxes have to do with motion: The Dichotomy (The Racetrack), Achilles and the Tortoise, and The Arrow. These are the most challenging and difficult to resolve. Aristotle seemed to imply in Physics that there were other Zenonian arguments of motion easier to resolve.13
Paradox One: The Dichotomy or Racetrack
According to this paradox, a runner is preparing to run a race. Let’s say it is a 100-meter dash. The gun fires and he starts to run. He makes it to the 50-meter mark. Now he only has 50 meters to go. He then runs half of that distance to the 75-meter mark. Now he has only 25 meters to go. He then runs half of that distance to the 87.5-meter mark. Now he has only 12.5 meters to go and so he runs half of that distance and so on. You get the picture. We have an infinite number of half distances that can never be traversed because they are infinite.
Mathematically, the distance toward the goal that the runner must cover is (1/2, 1/4, 1/8, 1/16, 1/32…∞). So according to Zeno, the runner must cover an infinite amount of finite distances, which is impossible since it would take an infinite amount of time.14
This can also be looked at in reverse, from a regressive versus a progressive perspective. In order for the runner to run 1/2 the distance to the goal, he would have to run half of that distance or 1/4. But in order for him to run a fourth of the distance, he would have to run 1/8 the distance, and so on. In the regressive scenario, the runner would never really be able to begin, thus showing that motion is impossible.
Paradox Two: Achilles and the Tortoise
Achilles and the Tortoise is probably Zeno’s most famous paradox, and it’s really a variation of The Racetrack. Aristotle introduces this paradox immediately after the Racetrack paradox. Achilles was the fastest runner in antiquity. In Zeno’s scenario, Achilles is racing against a tortoise that has a head start. And contrary to what intuition would tell us, Achilles will never overcome the tortoise. Why?
According to this paradox, in order to overcome the tortoise, Achilles must first run to where the tortoise is, who has a head start. Once Achilles gets to that point, the tortoise will have moved on ahead. Now Achilles has to move to that place. Once he gets to that place, then the tortoise will have moved ahead further by a little bit. This is best illustrated by the diagram below.15
So in the time that it takes Achilles to go from a2(=t1) to a3(=t2), the tortoise would have moved on a further distance d3 beyond t2 to t3 (t2 + d3).16 Because of this, the fastest runner in the world Achilles will never be able to overcome the tortoise, proving again that motion is impossible.
Paradox Three: The Arrow
Of the motion paradoxes, this is my personal favorite. In this paradox, Zeno takes a slightly different approach in demonstrating that motion is impossible. Think about an arrow flying through the air. We can really think of it in two ways. Let’s say that you record the arrow using the camera on your smart phone. You can replay the video and watch the arrow travel from the bow to the target.
In addition to the video, you can also take individual “slices” of that video and make a still photograph out if it. You can now view the arrow in motion or as a snapshot in time. What is the difference between the two?
Let’s look at the still photo. With the still photo, you have captured the arrow taking up a specific space equal to itself in an indivisible amount of time. According to Aristotle, Zeno makes the assumption that time is composed of moments, and so the arrow can only take up an amount of space equal to itself at a particular place in a specific moment in time.17
Now here is where the brain twister comes in. At any moment in time, the arrow is occupying a space equal to itself at a particular place. Places do not move. So at that specific moment in time, the arrow is not moving since the particular place that the arrow is in does not move. Motion would require even a small unit of time and we are already at the smallest unit of time possible.18 The same argument can be made for any other specific moment of time in the flight in which the arrow is also not moving.
Therefore, according to Zeno, the arrow never moves in its “flight” and therefore he argues from that conclusion that nothing else moves, that motion is impossible. According to the image below, the arrow occupies the specific space shown at a particular time and is at rest.
Solutions to Zeno’s Motion Paradoxes
I have read many possible solutions to Zeno’s motion paradoxes that are readily available on the Internet. So rather than just copying and pasting these to this post, I thought that I would discuss some of the more interesting solutions and then, more importantly, take a peek behind these paradoxes in order to see what Zeno was really trying to teach us. When we deal with philosophers such as Zeno and Parmenides, it is easy to get immersed in the details and lose sight of the overall picture. Just like the magician who distracts us with the shiny object so we miss the sleight of hand, so too we must be careful, lest we miss the true points of their teaching.
When talking about solutions to these paradoxes, I found that they fall roughly into mathematical and non-mathematical solutions. For many years, various thinkers have tried to solve these paradoxes mathematically, but more recently, some have turned to non-mathematical solutions such as physics.
Aristotle Weighs In
Before proceeding to the mathematical solutions, we should let Aristotle speak first since, after all, without his writings, we would not even be aware of these paradoxes, not to mention that he was the greatest philosopher who had ever lived.
In regard to The Racetrack, Aristotle, in Physics Z9, said that “it is possible for a runner to come in contact with a potentially infinite number of things in a finite time provided the time intervals becomes shorter and shorter.”19 Aristotle said that Zeno assumed that this was impossible, but it was not.
Aristotle makes a similar argument in regard to Achilles and the Tortoise. He claimed that Achilles did not have to cover an infinite distance, but only potential infinity since the sum of potential infinity is a finite number at any given time.20 Achilles could achieve an infinity of tasks since he is just covering a subset of that infinity, which is a finite distance at a finite duration.
Aristotle is saying that there is an infinite number of the finite, whereas Zeno is saying that the infinite is contained in the finite. It would appear that Aristotle wins that battle on two counts. First of all, it makes mathematical sense and secondly, it comports with what our senses tell us. And I would concede that that is true.
But the debate is far from over, for I would say that, from a suprarational perspective, Zeno is also correct and that he had actually stumbled upon the intersection between the eternal and temporal. From a materialistic point of view, the finite is a subset of the infinite, but from an eternal perspective, the infinite is contained within the finite. (More on that in the next post.)
Aristotle’s rebuttal to The Arrow is as follows:
“Zeno reasons fallaciously; for he says that if every thing always is resting whenever it is against what is equal, and what moves is always in the now, then the moving arrow is motionless” (Ph. 6.9, 239b5-7)21
Aristotle is saying that Zeno relies on a false assumption that time is composed of individual instances or “nows.” I think that is a very keen insight on Aristotle’s part.
Regardless, Aristotle did the best he could at refuting these paradoxes considering what was not known about mathematics at the time, such as calculus and the clear notion of a continuous space. In other words, Aristotle just made intuitive assertions that he could not support mathematically. These would not come for many centuries.
So is it calculus to the rescue? Maybe.
Since Aristotle’s time, great strides have been made in mathematics, especially with the discovery of calculus. Because of this, more sophisticated mathematical solutions have been developed in order to solve Zeno’s paradoxes. I will cover this section generally, but if you want to get into the tall weeds, please check out the link below entitled “A Contemporary Look at Zeno’s Paradoxes” by Wesley C. Salmon.22
In trying to solve Zeno’s paradoxes, many mathematicians have turned to calculus for after all, calculus deals with the concepts of instantaneous change, continuum, and the summation of infinitesimal differences. Let’s see how these concepts apply.
The Racetrack and Achilles can be solved the same way since they are really two versions of the same thing. According to Bolzano, Dedekind, and Cantor, the set of those points (1/2, 1/4, 1/8…) moving toward the goal does not have to be considered potentially infinite, but is actually an infinite set of points abstracted from a continuum of points.23
You do not have to understand this, but simply know that they are saying that there are different types of infinities and this particular type does not really lead to infinity, but rather converges to 1. Because of these mathematical manipulations, the runner can reach his goal. Please see the links below if you wish to take a deeper dive into this topic.24
People have used the concept of instantaneous change that originated with Newton and Leibniz in the 16th century to explain The Arrow paradox.25 This is a little easier to understand than the above idea of different types of infinities.
The concept of instantaneous change is also called the “is-at” theory of instantaneous motion. This theory says that instantaneous motion can be defined at an instant without defining motion during an instant. So because of this concept, even though the arrow is frozen in time, motion can still be defined.
Another concept used to solve The Arrow paradox is the concept of the infinitesimal, which refers to an infinitely small quantity approaching zero but that is not zero. Rather than exploring this, I thought that I would just put links in the footnotes below on two interesting articles on this topic.26 What is noteworthy is that the two authors disagree.
In my opinion, Zeno’s Racetrack and Achilles paradoxes can be illustrated by the mathematic function y(x) = 1/x so as x goes to zero, then y goes to infinity. This is called an asymptote – a concept yet to be discovered in Aristotle’s day. An asymptote is a curve that continually approaches a line, but never touches it. The graph of this function is as follows:
So imagine in The Racetrack paradox that the runner is running from the right side of the graph and is following the red line toward zero. The closer the runner gets to the goal of zero, the interval he has to cover increasingly diminishes. If we plug those increasingly smaller intervals into the formula 1/x, the more y goes to infinity. It moves ever closer to the y axis, but never touches it, i.e., the runner never crosses the goal. This graph does not solve the paradox; it simply describes the paradox mathematically.
The point of this section is not to understand the math necessarily, but to show the depths to which Zeno’s seemingly simple paradoxes have prompted mathematicians, physicists, and philosophers to go in understanding the secrets of the universe.
With all of the different approaches and disagreements among the mathematicians, others have taken a different approach eschewing the mathematical approach altogether.
I found an interesting article in Forbes entitled “This Is How Physics, Not Math, Finally Resolves Zeno’s Famous Paradox” where the author states that physics can solve The Racetrack paradox by taking into consideration the element of time.27 The author’s argument is based upon the following physics formula:
Distance = Velocity x Time
According to the author:
“If you halve the distance you’re traveling, it takes you only half the time to traverse it. To travel (½ + ¼ + ⅛ + …) the total distance you’re trying to cover, it takes you (½ + ¼ + ⅛ + …) the total amount of time to do so. And this works for any distance, no matter how arbitrarily tiny, you seek to cover.”28
This is illustrated by the diagram below:
This assumes Newton’s first law of motion that states that a body at rest remains at rest and a body in motion remains in motion unless acted on by an outside force. This case assumes constant motion. So, time is the variable in the equation that brings the runner across the goal line.
A Metaphysical Solution
By far, the most interesting paper that I have read on the subject is one by Alba Papa-Grimaldi entitled “Why Mathematical Solutions of Zeno’s Paradoxes Miss the Point: Zeno’s One and Many Relation and Parmenides’ Prohibition.”29 In my mind, he gets first prize for best interpretation of Zeno’s paradoxes.
In his paper he states the following:
“…mathematical ‘solutions’ miss, and always will miss, the point of Zeno’s arguments. I do not think that any mathematical solution can provide the much sought after answers to any of the paradoxes of Zeno.”
He claims, and I agree, that Zeno’s paradoxes need to be approached from a metaphysical perspective. Zeno seeks to communicate something larger than just solving a math puzzle about a runner or an arrow. The key to interacting with these philosophers is to realize that they are pointing beyond the physical to the more mysterious metaphysical realm.
The One and the Many
Papa-Grimaldi has discovered what nearly everyone else has missed – that Zeno is really pointing us to the relation between the One and the Many, the problem of universals. This is the exact thing that I wrote about in my previous post that Parmenides’ discussion of Being actually leads us to the problem of universals. The solution is to look up and out rather than down into the weeds. Papa-Grimaldi points out that even if the math is correct, they still miss the point.
In one way or another, all of the Presocratics struggled with the one-many relation for our senses tells us the world is composed of many things, yet our intuition tells us that there has to be a unity to everything. These ancient philosophers tended to fall within two camps – the Monists and the Pluralists. Following his teacher Parmenides, Zeno claimed that all is One and unchanging, whereas the pluralists claimed that reality consisted of the Many and of change or becoming.
According to Papa-Grimaldi, The Racetrack and Achilles paradoxes are concerned with the passage from the one to the many (infinite divisibility of a quantity) and The Arrow paradox has to do with the passage from the many to the one (from identity to complete concrete multiplicity).
In other words, the point of Zeno’s paradoxes is to show the lack of relation between the One and the Many. It is impossible to transition between the One and the Many and vice versa. Therefore, the only option, which is also illustrated by his paradoxes, is the One, since motion or change is impossible.
Diogenes said that Aristotle called Zeno the inventor of the dialectic as Empedocles was of rhetoric.30 He is also considered to be the inventor of the reductio ad absurdum argument by which one proves a proposition true by showing the absurdity of the opposite.
Metaphysically speaking, Zeno has left an incredible legacy on many levels. As mentioned above, he has forced mathematicians, physicists, and philosophers to be more rigorous and precise in formulating possible solutions to paradoxes such as Zeno’s.
His greatest legacy is probably captured best by a quote from Bertrand Russell in 1914:
“Zeno’s arguments, in some form, have afforded grounds for almost all theories of space and time and infinity which have been constructed from his time to our own.”31
Now that’s a legacy!
Taking a cue from Bertrand Russell, a group of philosophers, most famously Adolf Grünbaum, set out to show how modern mathematics could solve all of Zeno’s paradoxes. They failed.32 They surmised that if Zeno’s paradoxes could not be solved by pure mathematics, then his paradoxes question not only abstract mathematics, but also the nature of reality itself! In other words, if mathematics could not refute Zeno’s paradoxes, then one would have to assume that mathematics could not adequately describe space, time, and motion.
On the other hand, nearly all mathematical laws and physical theories of infinity describe reality as a space-time continuum, three dimensions of space fused with the one dimension of time to form a single, continuous entity. This is also supported by our experiences. The problem is that while most physical theories support the space-time continuum, not all of them do. For example, many quantum theories seem to contradict traditional theories.33
Zeno and Quantum Physics
The philosopher Zeno who gave us paradoxes about arrows and runners and tortoises finally ends up making an appearance in quantum physics. He even has a principle named after him called the Quantum Zeno Effect, also known as “a watched arrow never moves.”
Experiments have shown that observing a quantum system such as the movement of a beam of electrons or radioactive decay actually changes the reality of the system. In other words, the mere act of observing changes the way the quantum world behaves.34
Scientists have found that taking “glimpses” of radioactive decay, i.e., measurements, actually halts the process of radioactive decay.35 The more glimpses that the observer makes, the more the process of decay slows. The analogy to the arrow is obvious. The more slices of time we take to observe the arrow, the more it becomes “frozen” in space, just like the decay of the atomic particle becomes frozen in time. The flight of the arrow left to itself is a continuous action, but if we seek to take a video of the arrow and analyze it at various segments of time, then it becomes a series of arrows frozen in space and time. Did Zeno predict the findings of quantum physics with his simple analogy of The Arrow?
Below is another example of the Quantum Zeno Effect. It is called the two-slit experiment. If a beam of electrons emanates from a source and is left unobserved, then when examined after the fact, it appears that the electrons behaved as a wave and as a result, went through both slits. But when the beam of electrons was observed, it behaved as a series of particles instead and only went through one slit.36 Just like the arrow and radioactive decay, the act of observing converted these processes from a continuous process to an intermittent one. This is remarkable indeed.
These experiments would seem to tell us that the mere act of observing the physical world actually changes it. If this is the case, then we have deeper connections to the world around us than we could have ever imagined.
The point is that there is a deeper reality and mystery to the universe than we realize, and the door to these realities was opened over 2000 years ago by a Greek philosopher in Southern Italy talking about runners and tortoises and arrows.
I end with a quote by Zeno:
“By silence I hear other men’s imperfections, and conceal my own.”
Which of Zeno’s paradoxes is your favorite and why? Please leave a comment below. If you enjoyed this post, please share, subscribe, and press the follow button. Thank you!
Footnotes and Endnotes
- Palmer, John, “Zeno of Elea”, The Stanford Encyclopedia of Philosophy (Summer 2021 Edition), Edward N. Zalta (ed.)
- Britannica, The Editors of Encyclopaedia. “Zeno of Elea”. Encyclopedia Britannica, 23 Apr. 2020, https://www.britannica.com/biography/Zeno-of-Elea. Accessed 19 September 2021.
- Diogenes Laertius, Lives of the Eminent Philosophers, Translated by Pamela Mensch, Edited by James Miller, pp. 326-327, Oxford University Press, Oxford. 2018
- Kingslsey, Peter, In the Dark Places of Wisdom, pp. 150-154, Golden Sufi Center, Point Reyes, California, 1999
- Diogenes Laertius, Lives of the Eminent Philosophers, Translated by Pamela Mensch, Edited by James Miller, p. 326
- Dowden, Bradley, “Zeno’s Paradoxes,” Internet Encyclopedia of Philosophy; There are several versions of this story so I just combined them into a composite.
- Grayling, A.C., The History of Philosophy, p. 36, Penguin Press, New York, 2019
- Diogenes Laertius, Lives of the Eminent Philosophers, Translated by Pamela Mensch, Edited by James Miller, p. 327
- Plato, Parmenides, Translated by Mary Louise Gill and Paul Ryan with an introduction by Mary Louise Gill, p. 3, Hackett Publishing Company, Inc., Indianapolis, 1996
- Plato, Parmenides, Translated by Mary Louise Gill and Paul Ryan with an introduction by Mary Louise Gill, p. 126
- Plato, Parmenides, pp. 127-128
- Grayling, A.C., The History of Philosophy, p. 36
- Palmer, John, “Zeno of Elea”, The Stanford Encyclopedia of Philosophy (Summer 2021 Edition), Edward N. Zalta (ed.), 2.2 The Paradoxes of Motion
- Palmer, John, “Zeno of Elea”, The Stanford Encyclopedia of Philosophy, 3.1 The Dichotomy
- Palmer, John, “Zeno of Elea”, The Stanford Encyclopedia of Philosophy, 2.2.2 The Achilles
- Dowden, Bradley, “Zeno’s Paradoxes,” Internet Encyclopedia of Philosophy, iii. The Arrow
- Dowden, Bradley, “Zeno’s Paradoxes,” Internet Encyclopedia of Philosophy, Paradoxes of Motion
- Palmer, John, “Zeno of Elea”, The Stanford Encyclopedia of Philosophy, 2.2.3 The Arrow
- Salmon, Wesley C., “A Contemporary Look at Zeno’s Paradoxes,” University of Minnesota Press, Minneapolis, 1970
- Dowden, Bradley, “Zeno’s Paradoxes,” Internet Encyclopedia of Philosophy, ii. The Dichotomy (The Racetrack).
- DeMichele, Thomas, “There are Different Types of Infinity.”; Otis, Waldo, “Understanding Cantor’s Mathematical Infinity,” Jan. 18, 2020, Davies, Paul, “Some infinities are bigger than others,” February 18. 2018
- Dowden, Bradley, “Zeno’s Paradoxes,” Internet Encyclopedia of Philosophy, iii. The Arrow
- Reeder Patrick, Zeno’s arrow and the infinitesimal calculus, Synthese Vol. 192, No. 5 (May 2015), pp. 1315-1335, Published by Springer; White, Michael J., “Zeno’s Arrow, Divisible Infinitesimals, and Chrysippus,” Phronesis Vol. 27, No. 3 (1982), pp. 239-254, Published by: Brill
- Siegel, Ethan, “This Is How Physics, Not Math, Finally Resolves Zeno’s Famous Paradox,” Editor’s Pick, May 5, 2020.
- Papa-Grimaldi, Papa, “Why Mathematical Solutions of Zeno’s Paradoxes Miss the Point: Zeno’s One and Many Relation and Parmenides’ Prohibition,” http://philsciarchive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf
- Diogenes Laertius, Lives of the Eminent Philosophers, p. 326
- Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, Inc., 1959. Previously published under the title, The Concepts of the Calculus as cited in Salmon, Wesley C., “A Contemporary Look at Zeno’s Paradoxes,”
- Palmer, John, “Zeno of Elea”, The Stanford Encyclopedia of Philosophy, Zeno’s Influence on Philosophy
- “Confined quantum Zeno dynamics of a watched atomic arrow,” Nature Physics, Volume 10, Issue 10, pp. 715-719 (2014)
- Ojha, Piyush, “Science: A watched atom never decays,” New Scientist, March 10, 1990
- Weizmann Institute Of Science. “Quantum Theory Demonstrated: Observation Affects Reality.” ScienceDaily. ScienceDaily, 27 February 1998
- Ananthaswamy, Anil, Through Two Doors at Once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality, Dutton Publishers, Boston, (August 7, 2018)
- Aristotle, Physics, Translated by Robin Waterfield with introduction and notes by David Bostock, Oxford University Press, Oxford, 1996
- Diogenes Laertius, Lives of the Eminent Philosophers, Translated by Pamela Mensch, Edited by James Miller, pp., Oxford University Press, Oxford. 2018
- Grayling, A.C., The History of Philosophy, p. 36, Penguin Press, New York, 2019
- Halpern, Paul, Einstein’s Dice and Schrödinger’s Cat: How Two Great Minds Battled Quantum Randomness to Create a Unified Theory of Physics, Basic Books, New York, First Edition (April 14, 2015)
- Kingslsey, Peter, In the Dark Places of Wisdom, Golden Sufi Center, Point Reyes, California, 1999
- Plato, Parmenides, Translated by Mary Louise Gill and Paul Ryan with an introduction by Mary Louise Gill, Hackett Publishing Company, Inc., Indianapolis, 1996
- Smith, Wolfgang, Physics and Vertical Causation: The End of Quantum Reality, Angelico Press, Brooklyn, N.Y., (January 21, 2019)