What, if anything, can Zeno’s paradoxes teach us about motion?
The short answer is “Very little, if anything at all!” However, explaining why so little will take some effort.
A paradox is a logical argument that purports to demonstrate that a contradiction or an absurd consequence follows from apparently reasonable assumptions. A paradox, when reasoned validly, is sufficient evidence that one’s premises are in some way faulty. Zeno’s paradoxes of motion purport to demonstrate that motion is impossible — an obvious absurdity.
Almost everything we know about Zeno is to be found in Plato’s dialogue “Parmenides“. According to Plato, Zeno was a student of Parmenides, and wrote a book of paradoxes defending the philosophy of Parmenides. This book has not survived. The little that we know of his arguments is second hand, principally through Aristotle and Simplicius of Cilicia — both of whom are presumed to have had access to Zeno’s text. Of the over forty “paradoxes of plurality” supposedly in this text, only nine remain in very brief fragments. None of these fragments quote Zeno’s original text in sufficient length to recreate Zeno’s reasoning. All are paraphrases, and leave much to the imagination. Of the nine arguments on which we have any information, only four are directly addressed to motion (“The Dichotomy” or “The Race-Course”, “The Achilles”, “The Arrow”, and “The Stadium”). Three are directly addressed to plurality (generally called “The Argument from Denseness”, “The Argument from Finite Size”, and “The Argument from Complete Divisibility)”, and the remaining two are “The Paradox of Place” and “The Grain of Millet”. In order to understand why these paradoxes teach us so little about motion, I’ll examine briefly the four paradoxes of motion.
The Dichotomy
All that exists of Zeno’s reasoning on this paradox is a very brief description of it in Aristotle(1). His very brief description of Zeno’s original argument, along with his manner of “resolving” the paradox, are all that we now have as the basis of our modern understanding of the Dichotomy.
Suppose a runner running from a starting line to a finish line. Before he reaches the finish line he must run half-way, as Aristotle reports Zeno as saying. But now he must also run half-way from the half-way point — i.e., a 1/4 of the total distance. And after he reaches 1/4 of the way from the finish he must reach 1/2 of 1/4 = 1/8 of the way. And after that a 1/16, and so on without end (bounds). There is obviously no problem at any finite point in this series, but it is assumed that Zeno’s paradox arises when the halving is carried out infinitely many times. The resulting series contains no last distance to run, for any possible last distance could be divided in half, and hence would not be last after all. And now there is presumably a problem, for this description of the run has our runner travelling an infinite number of finite distances. It is suggested by Aristotle that Zeno would have us conclude this could never be completed. But it is not known how Zeno constructed the argument. Depending on how one reconstructs the reasoning, either the runner can never reach the finish line, never start, must cover an infinite distance, or must take an infinite amount of time. Since an argument of this form does not depend on the distance or what the mover is, it follows that no finite distance can ever be traveled. Which is to say that all motion is impossible.
Aristotle, being an empiricist, believed strongly that motion and change are fundamental ingredients of nature. He was thus in elemental disagreement with Zeno and his Parmenidean philosophy of the denial of motion. Following brief descriptions of each of Zeno’s four paradoxes of motion, he “resolves” them by dismissing them altogether as the result of fallacious reasoning. He does not, it should be noted, identify any flaws in Zeno’s application of the rules of logic. Rather, he disputes Zeno’s underlying assumptions. In particular, Aristotle applied a much more advanced notion of infinity to the paradoxes than Zeno would have had available.
The reason that this paradox teaches us nothing about motion, is that the paradox is not actually about motion at all. It is a logical argument entirely dependent on some premises about the nature of infinity. How this paradox gets “resolved,” and what the paradox will therefore end up teaching us, will depend on what additional assumptions a commentator brings to the analysis, and on how the those initial premises are “corrected”. As it happens, the myriad different ways that have been suggested that this paradox can be “resolved” can be loosely grouped into three different approaches.
The first approach is to try to understand the paradox from the perspective that Zeno himself would have brought to the reasoning. In Zeno’s time, he would have had a very limited comprehension of the notion of “infinity”. A better translation of the Greek word he would have used would be “unbounded”. In the context of his paradoxical reasoning, an “unbounded” sequence of “half-runs” is literally “unbounded” — having no last run. Hence, on the basis of this understanding of the paradox, the sequence, even though infinite, has no ending half-run, and the runner can never cross the finish-line. (Or, alternatively, never get started.)
Zeno has given an argument showing why motion is impossible. An adequate “resolution” of the paradox must identify the flaws in his reasoning. If one doesn’t accept Zeno’s reasoning as demonstrating that motion is illusory, as most of us would not, then we need an explanation of what is wrong with his reasoning, not a better understanding of motion. If you accept all of the steps in Zeno’s argument then you must accept his conclusion (assuming that he has reasoned in a logically valid way). It won’t do simply to point out that there are unproblematic ways of thinking about the run (as just two halves, say, or a hundred paces). It is not enough to show an unproblematic way of thinking about the run. You must show why Zeno’s way of thinking about the run is unproblematic.
The resolution of the Dichotomy, understood on the basis of this historical approach, is to challenge (and then suitably alter) the assumptions that Zeno would likely have brought to the notion of infinity, or unboundedness. Aristotle was the first Greek thinker to develop a comprehensive description of infinity. But Aristotle did not have the nineteenth century mathematics of infinity and hence could not argue that the sum of an unbounded sequence of half-runs has a finite answer. So Aristotle’s approach was to argue that, contrary to Zeno’s premise, if a finite distance was infinitely divisible, then so was time. Hence, he argues, a finite distance divisible into an infinity of partial distances could be run in an infinity of partial times. Notice that here we have nothing to be learned about motion, but a lot to be learned about infinity and the notion of unbounded sequences. Although more recent commentators have noticed that Aristotle’s “solution” does not address the problem of the “last” part-run never quite reaching the finish line.
The second approach is to understand the paradox from the perspective of our modern understanding of infinity mathematics. This is taking the scanty hints of what Zeno might have thought, and re-interpreting his “paradox” in terms he would not have had the concepts to consider. On this basis of understanding, the resolution of the paradox is pre-supposed. The sum of fractions specified by the race-track paradox is mathematically defined as one. The infinite series of fractional runs is one unit of distance completed in one unit of time. And the runner crosses the finish line — Hurrah! But again, on this approach, there is nothing to learn about motion. Any lessons to be learned are gleaned from the detailed logic of the anachronistic argument construction, and the details of the mathematics employed.
The third approach, more rarely applied, is to find some subtle aspect of how we currently think about infinity, and re-interpret Zeno’s argument in these terms. One such example is to focus on the premise that even though the mathematical sum of the infinite series is provably one at the limit, there is no “last” fractional part-run that can be completed. Hence, it is suggested, the “finish line” is never part of any of the infinite part-runs. Hence the runner can never reach the finish line — merely come as close as one might wish to it. This is perhaps not a good example, because it trades on a confusion between an infinite sequence, and a “last” member of such a sequence. The paradox is “resolved” by pointing out that, by definition, an infinite sequence does not have a “last” member. Hence there is no “last” member of the sequence to not contain the finish-line. Again, on this approach, there is nothing to learn about motion. As before, any lessons to be learned are gleaned from the detailed logic (or mathematics) of the argument construction.
The Achilles
Zeno’s paradox of Achilles and the Tortoise is very similar in form to the Dichotomy.(2)
Although the details differ somewhat, and some commentators suggest that Zeno’s argument was focussed on a different point than was addressed with the Dichotomy, as with the Dichotomy, we have a logical argument based on a number of premises about the nature of infinite series, and the possibility of completing an “unbounded” series of partial-runs. Again as with the Dichotomy, nothing in this paradox speaks to the reality of motion. Whatever we may learn from analysing a re-creation of the logical argument, either in historical terms or in anachronistic terms, will be about logical arguments and the consequences of premises. It will not be about motion.
The Arrow
With what little information we have on the first two of the motion paradoxes, reconstructing Zeno’s arguments is a very iffy exercise. At least for The Arrow we have a more useful description from Aristotle, brief as it is.(3)
Here, for the first time, we get a lesson on motion. Aristotle’s description makes it seem that Zeno is thinking of the arrow’s flight as an “unbounded” series of instantaneous moments, and then arguing that in any given moment the arrow is necessarily at rest. But of course, motion is not defined in an instantaneous moment. Motion is defined as movement over time. The very concept of motion depends on the prior concept of a duration. It was not until the development of The Calculus (specifically, Isaac Newton’s work on limits, Tractatus de Quadratura Curvarum, in 1704) that we had a mathematical means of defining the concept of an “instantaneous velocity”. So “resolving” this paradox teaches us that motion is dependent on the prior concept of duration, and is not defined in the instant.
The Stadium
This also we have from Aristotle, although in a much more complete form that any of the others.(4)
If Aristotle’s description of Zeno’s argument is correct (not a foregone conclusion, given his sketchy treatment of Zeno’s reasoning in other places), then it is obvious that Zeno was hopelessly confused about the notion of relative motion. It is a shame that Zeno’s original text is not available to us. It would be interesting to determine whether Zeno actually did commit this so very obvious error. The error is so blatant that it is difficult to generate a reconstruction of what Zeno’s logic might have been as the basis for any further analysis. It is so blatant that I am not aware of any commentator who has chosen to address this “paradox” with an anachronistic reconstruction. So if this paradox does teach us anything, it teaches us only the obvious fact that measuring motion is always an act of measuring relative motion.
Conclusion
Many thinkers over the millennia since Zeno have searched for ways to “resolve” the paradoxes. Kant, for example, believed that the contradictions evidenced in Zeno’s paradoxes are intrinsic within our conceptions of space and time. Space and time, he maintained, do not belong to things as they are in themselves, but rather to our way of looking at things. It is our minds which impose space and time upon objects, and not objects which impose space and time upon our minds. Kant also concluded from these contradictions that to comprehend the infinite is beyond the capacity of reason. Hume, as another example, denied the infinite divisibility of space and time, and declared that they are composed of indivisible quanta having magnitude. Hegel regarded Zeno’s paradoxes as examples of the essential contradictory character of reason. With true dialectic style, he believed that any resolution must accommodate both sides of the contradiction.
Zeno’s paradoxes are exercises in logical argument. Analysing them (or rather what little we can reconstruct of them) can teach us about the historical context in which Zeno thought about motion and about infinity (unboundedness). Or they can teach us about the mathematics of infinity and limits if we choose to pursue that route to resolving them. Or they can teach us about premises and logical consequences if we choose an alternative approach such as Kant and Hume. But they can teach us only the very obvious about motion, since they are not actually about motion itself. They are, rather, about reasoning, and about faulty reasoning at that. They are flawed attempts to defend the Parmenidean notion that motion is illusory — necessarily flawed, since motion is not illusory (Kant and Parmenides not withstanding).
Notes & References
(1) “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.”
[Aristotle: Physics, 239b11]
(2) “The [second] argument was called “Achilles,” accordingly, from the fact that Achilles was taken [as a character] in it, and the argument says that it is impossible for him to overtake the tortoise when pursuing it. For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced … . And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount … . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount.”
[Simplicius(b), 1989, On Aristotle’s Physics 6, D. Konstan (trans.), London: Gerald Duckworth & Co. Ltd, as quoted in Huggett, Nick, “Zeno’s Paradoxes”, The Stanford Encyclopedia of Philosophy (Winter 2006 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2006/entries/paradox-zeno/>.]
(3) “The third is … that the flying arrow is at rest, which result follows from the assumption that time is composed of moments … . he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless.”
[Aristotle: Physics, 239b30]
(4) “Fourth is the argument about equal masses moving oppositely along equals masses, some from the end of the stadium, and others from the middle, with equal speed, where he thinks it follows that the half time will equal the double. The fallacy is that the mass moving along one in motion is assumed to move an equal magnitude in an equal time with equal speed as one moving along one at rest. But this is false. For example, let the stationary masses be AA, those starting from the middle be BB, which are equal to the others in size and number, and let GG be those moving from the end, with these too being equal in size and number with those, and let them be equally fast as the B’s. It happens that the first B and the first G will be at the end at the same time, when they are moving alongside one another. It follows that G will traverse all the B’s, while B traverses half (the A’s). Thus, the time will be half. For each is alongside each for an equal time. At the same time the first B will have moved along all the G’s, since the first G and the first B will be at opposite ends [becoming in an equal time alongside each of the B’s as alongside each of the A’s, as he says], since both come to be alongside the A’s in an equal time. And so this is the argument, and it follows according to the mentioned falsehood.”
[Aristotle: Physics, 239b33 — 240a18]
Huggett, Nick, “Zeno’s Paradoxes”, The Stanford Encyclopedia of Philosophy (Winter 2006 Edition), Edward N. Zalta (ed.), June 6, 2007. URL = <http://plato.stanford.edu/archives/win2006/entries/paradox-zeno/>.
Kirk, G.S., Raven, J.E., Schofield, M., The Presocratic Philosophers, 2nd ed., Cambridge University Press, Cambridge, England, 1957.
Sainsbury, R.M. Paradoxes, 2nd Edition. Cambridge University Press, Cambridge, England. 1987. ISBN 0-521-48347-6.
Vlastos, Gregory. Studies in Greek Philosophy: Volume 1 — The Presocratics, Daniel W. Graham Ed., Princeton University Press, Princeton, New Jersey. 1993. ISBN 0-691-01937-1.
“Zeno of Elea”, The Internet Encyclopedia of Philosophy, June 6, 2007. URL= http://www.iep.utm.edu/z/zenoelea.htm