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Zeno's Paradox: Unraveling the Ancient Mystery Behind the Science of Space and Time
The fascinating story of an ancient riddle and what it reveals about the nature of time and space
Three millennia ago, the Greek philosopher Zeno constructed a series of logical paradoxes to prove that motion is impossible. Today, these paradoxes remain on the cutting edge of our investigations into the fabric of space and time. Zeno's Paradox uses the motion paradox as a jumping-off point for an exploration of the twenty-five-hundred-year quest to uncover the true nature of the universe. From Galileo to Einstein to Stephen Hawking, some of the greatest minds in history have tackled the problem and made spectacular breakthroughs, but through it all, the paradox of motion remains.
Three millennia ago, the Greek philosopher Zeno constructed a series of logical paradoxes to prove that motion is impossible. Today, these paradoxes remain on the cutting edge of our investigations into the fabric of space and time. Zeno's Paradox uses the motion paradox as a jumping-off point for an exploration of the twenty-five-hundred-year quest to uncover the true nature of the universe. From Galileo to Einstein to Stephen Hawking, some of the greatest minds in history have tackled the problem and made spectacular breakthroughs, but through it all, the paradox of motion remains.
274 pages, Kindle Edition
First published March 25, 2008
About the author
Joseph Mazur
13 books21 followersJoseph C. Mazur is Professor Emeritus of Mathematics at Marlboro College in Vermont. He earned his Ph.D. in algebraic geometry from MIT and has held visiting positions at MIT and the University of Warwick. A recipient of Guggenheim, Bellagio, and Bogliasco Fellowships, he has written widely on the history and philosophy of mathematics, with books translated into over a dozen languages.
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Displaying 1 - 8 of 8 reviews
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December 9, 2024"I don't know who Zeno was, but he sounds like a bit of a dick to me."
- Australian citizen who wishes to remain anonymous
- Australian citizen who wishes to remain anonymous
July 18, 2021
Bel saggio sempre interessante e accurato, fruibile a piu' livelli in relazione alla propria preparazione e interesse.
February 18, 2019
C'è un concetto molto bello che apre questo libro. Generalmente si afferma che i paradossi di Zenone siano banalmente stati risolti quando i matematici hanno formalizzato i concetti del calcolo infinitesimale. Peccato che questo non sia affatto vero: quello che abbiamo è un formalismo matematico che è in accordo con la nostra esperienza macroscopica, ma che non può dirci nulla su cosa succede a livello ultramicroscopico. Insomma, saper sommare una serie infinita non ci dice nulla se spazio e tempo siano continui o discreti, né cosa succede effettivamente. Mazur, ben tradotto da Claudio Piga, racconta come i paradossi di Zenone siano stati trattati nei secoli, con una serie di divagazioni che solo a posteriori mostrano la relazione con il tema del libro. Insomma, un lato diverso dal solito per la trattazione dei paradossi, che ci ricorda come la matematica, pur essendo indispensabile, non è totalizzante.
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May 23, 2014This book starts out 2500 years ago and marches us thru mathematical history to arrive at the door step of modern physics. It is fairly easy to follow is you have read other books on math and physics. I wpuod reccommend it to anyone who is not scared my mathematic formulas or physics.
January 1, 2023
Zeno’s Paradox, Resolved! Maybe, Well, Actually, No
It is important to understand that the longevity of Zeno's paradoxes is owing to the longevity of the problems he tackled. That is, many of his paradox puzzles address infinity and since Zeno's time, there has always been more to say on the subject of infinity. Also, Zeno was a student of Parmenides and his paradoxes were designed to defend the views of Parmenides by challenging the common sense notions that come from experience, perception and sense reality. Interestingly, the Greek politician and orator Pericles was a student of Zeno and Aristotle credited Zeno with inventing the Socratic method of relentless dialectical challenge and dispute in seeking the truth which was greatly refined by Socrates himself.
For the remainder of this review, I would like to share my thoughts from the book on the most well-known of Zeno’s paradoxes, that of Achilles and the tortoise. I think this is enough to illustrate the fundamental challenges of the remaining paradoxes and the deep problem that remains with us.
Zeno’s paradox forces us to confront a scenario wherein both sides of a contradiction appear to be true, simultaneously. Any paradox, including Zeno’s most well-known paradox, becomes a tool for advancing knowledge and confirming or rejecting observations about the world. The result of Zeno’s Achilles paradox is that we are forced to question whether mathematics adequately describes reality, and to face the proposition that neither reason nor perception of the physical world yields the truth about what is real.
The conflicting information is found in two seemingly incontrovertible separate datum, both of which cannot be true. The first datum, to be found in the empirical observation, is that Achilles, the fleet-footed hero of the Trojan War, will of course pass the lowly tortoise, generous head start notwithstanding. The second datum is found through reason. (A precursor of the rationalism versus empiricism debate to come many centuries later). The question posed by reason is, what must Achilles do to catch-up with the tortoise? Zeno tells us that he must cross an infinite number of points. Zeno explains that there is an infinite number of points contained within a finite space by asking us to continuously divide the given space by half. We can keep dividing, well, forever. Each resulting interval of space gets small by half, ad infinitum. This is the paradox, we observe that Achilles can overtake the tortoise but doing so requires him to do something infinite, which should be impossible, cross over an infinite number points. Both data seem to be true, so something is wrong, either in our reasoning, or with the data, or with reality itself.
Aristotle’s Answer:
Aristotle claimed that the paradox results from a misunderstanding of infinity. Aristotle answered this paradox by hypothesizing that there are two different types of infinities, potential infinities, (which is the case in Zeno’s paradox) and actual infinities. The potential infinity does not present Achilles with any insurmountable challenges and the paradox is not resolved, but dissolved. However, this seems a bit contrived. What is this new thing called a ‘potential infinity’? Can we bifurcate, or even further subdivide infinities in this manner to find the one that meets our needs? Is this a newly invented metaphysics made-up on the spot to resolve the paradox? Is this perhaps just a linguistic trick? Aristotle can be said to be simply defining the problem away rather than resolving the paradox. Alternatively, perhaps Aristotle was on the correct course in anticipating the modem development in mathematics of multiple infinities. Still, our problem remains in trying to resolve infinities with finite mathematics.
Mathematical Answers:
It can be shown mathematically that the sum of an infinite division by two equals one, or unity, or the whole, and that is what Achilles is actually doing, crossing the whole space, one finite space, between him the tortoise. This is just a geometric series that converges absolutely. Paradox resolved. However, this is not true if the subdivision of the finite space is not a sequence of halves and there is no reason to think that the finite space must be divided by halves. Why not a combination of halves, thirds, ninths and sixteenths? Such a serious will not converge absolutely.
A more sophisticated mathematical resolution is possible with the use of calculus if we concede that space and time are infinitely divisible. The author explores this approach but ends by questioning whether any mathematical (calculus) based answer tells us what is happening in the physical world. This, to me, is the entire point of the paradox.
Atomic Answer:
The Atomists resolved this paradox by denying the premise. That is, a finite space is not divisible into an infinite number of points. There is a stopping place at the level of a fundamental particle at which no further division is possible, e.g., the atom. In this case, there is a very large number of very small points in the finite space, but not an infinite number. So, there is no paradox after all. This eliminates the dubious notion of a potential infinity, it does not open the possibility of a nonconverging series and has the added benefit of conforming to our physical observation that Achilles does indeed overcome the tortoise with great ease. This notion of a fundamental particle is also consistent with our most modern particle physics. Paradox resolved, finally, well maybe not. There is no theoretical reason that prevents an infinite regress and perhaps time and space are infinitely divisible in which case the use of calculus offers the best resolution to the paradox, but this solution does comport to our lived experience in the physical world. Hence, the paradox remains that our mathematics, beneficial though it is, does not adequately describe our experience. In this sense, at least, Zeno’s paradox is still with us.
The Deeper Problem:
Add to this that mathematics is a human invention. Godel's Incompleteness Theorems showing that mathematics does not have a complete and consistent set of axioms seems to bear this out. However, mathematics works well in explaining so much about how the universe works; a universe discovered and not invented by human beings. We have a human invention of the intellect that very accurately describes the operation of physical reality, at least that reality within our ontological level, or range of common experience. This is the even deeper problem still with us that Zeno’s paradox uncovers.
It is important to understand that the longevity of Zeno's paradoxes is owing to the longevity of the problems he tackled. That is, many of his paradox puzzles address infinity and since Zeno's time, there has always been more to say on the subject of infinity. Also, Zeno was a student of Parmenides and his paradoxes were designed to defend the views of Parmenides by challenging the common sense notions that come from experience, perception and sense reality. Interestingly, the Greek politician and orator Pericles was a student of Zeno and Aristotle credited Zeno with inventing the Socratic method of relentless dialectical challenge and dispute in seeking the truth which was greatly refined by Socrates himself.
For the remainder of this review, I would like to share my thoughts from the book on the most well-known of Zeno’s paradoxes, that of Achilles and the tortoise. I think this is enough to illustrate the fundamental challenges of the remaining paradoxes and the deep problem that remains with us.
Zeno’s paradox forces us to confront a scenario wherein both sides of a contradiction appear to be true, simultaneously. Any paradox, including Zeno’s most well-known paradox, becomes a tool for advancing knowledge and confirming or rejecting observations about the world. The result of Zeno’s Achilles paradox is that we are forced to question whether mathematics adequately describes reality, and to face the proposition that neither reason nor perception of the physical world yields the truth about what is real.
The conflicting information is found in two seemingly incontrovertible separate datum, both of which cannot be true. The first datum, to be found in the empirical observation, is that Achilles, the fleet-footed hero of the Trojan War, will of course pass the lowly tortoise, generous head start notwithstanding. The second datum is found through reason. (A precursor of the rationalism versus empiricism debate to come many centuries later). The question posed by reason is, what must Achilles do to catch-up with the tortoise? Zeno tells us that he must cross an infinite number of points. Zeno explains that there is an infinite number of points contained within a finite space by asking us to continuously divide the given space by half. We can keep dividing, well, forever. Each resulting interval of space gets small by half, ad infinitum. This is the paradox, we observe that Achilles can overtake the tortoise but doing so requires him to do something infinite, which should be impossible, cross over an infinite number points. Both data seem to be true, so something is wrong, either in our reasoning, or with the data, or with reality itself.
Aristotle’s Answer:
Aristotle claimed that the paradox results from a misunderstanding of infinity. Aristotle answered this paradox by hypothesizing that there are two different types of infinities, potential infinities, (which is the case in Zeno’s paradox) and actual infinities. The potential infinity does not present Achilles with any insurmountable challenges and the paradox is not resolved, but dissolved. However, this seems a bit contrived. What is this new thing called a ‘potential infinity’? Can we bifurcate, or even further subdivide infinities in this manner to find the one that meets our needs? Is this a newly invented metaphysics made-up on the spot to resolve the paradox? Is this perhaps just a linguistic trick? Aristotle can be said to be simply defining the problem away rather than resolving the paradox. Alternatively, perhaps Aristotle was on the correct course in anticipating the modem development in mathematics of multiple infinities. Still, our problem remains in trying to resolve infinities with finite mathematics.
Mathematical Answers:
It can be shown mathematically that the sum of an infinite division by two equals one, or unity, or the whole, and that is what Achilles is actually doing, crossing the whole space, one finite space, between him the tortoise. This is just a geometric series that converges absolutely. Paradox resolved. However, this is not true if the subdivision of the finite space is not a sequence of halves and there is no reason to think that the finite space must be divided by halves. Why not a combination of halves, thirds, ninths and sixteenths? Such a serious will not converge absolutely.
A more sophisticated mathematical resolution is possible with the use of calculus if we concede that space and time are infinitely divisible. The author explores this approach but ends by questioning whether any mathematical (calculus) based answer tells us what is happening in the physical world. This, to me, is the entire point of the paradox.
Atomic Answer:
The Atomists resolved this paradox by denying the premise. That is, a finite space is not divisible into an infinite number of points. There is a stopping place at the level of a fundamental particle at which no further division is possible, e.g., the atom. In this case, there is a very large number of very small points in the finite space, but not an infinite number. So, there is no paradox after all. This eliminates the dubious notion of a potential infinity, it does not open the possibility of a nonconverging series and has the added benefit of conforming to our physical observation that Achilles does indeed overcome the tortoise with great ease. This notion of a fundamental particle is also consistent with our most modern particle physics. Paradox resolved, finally, well maybe not. There is no theoretical reason that prevents an infinite regress and perhaps time and space are infinitely divisible in which case the use of calculus offers the best resolution to the paradox, but this solution does comport to our lived experience in the physical world. Hence, the paradox remains that our mathematics, beneficial though it is, does not adequately describe our experience. In this sense, at least, Zeno’s paradox is still with us.
The Deeper Problem:
Add to this that mathematics is a human invention. Godel's Incompleteness Theorems showing that mathematics does not have a complete and consistent set of axioms seems to bear this out. However, mathematics works well in explaining so much about how the universe works; a universe discovered and not invented by human beings. We have a human invention of the intellect that very accurately describes the operation of physical reality, at least that reality within our ontological level, or range of common experience. This is the even deeper problem still with us that Zeno’s paradox uncovers.
August 22, 2020
This book was a lot more mathematical and technical than I was expecting. There were quite a few parts that I just glazed over and didn’t understand. Still super interesting though.
December 31, 2021
Una lettura paradossale in tutti i sensi.
September 23, 2008
A well-written history of the idea of motion, starting from Zeno's famous paradoxes (e.g., if before each step I take I first must take half that step, and so on, how is any motion possible?). Mazur, a mathematician by trade, develops the useful idea that calculus and other such tools don't really resolve the motion paradox, they just give us tools to deal with motion. His background into the development of calculus is an enjoyable read. He then ventures into relativity, quantum mechanics and string theory, although it seems to add little to his central argument that perhaps Zeno was right and motion is merely an illusion.
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