A Critique of Russell’s “At-At” Theory of Motion as a Solution to Zeno’s Arrow Paradox

Zeno’s paradoxes are aimed at supporting the Parmenidean thesis, which argues for the impossibility of change. Movement is a kind of change, so if the Parmenidean thesis is correct, it should be impossible. Zeno outlines paradoxes designed to show why motion is impossible. We might be inclined to reject the idea that change and motion are impossible on the basis that we all move and observe movement around us all the time. In this sense, it might appear that Zeno’s paradoxes are disproved by simple observation and that any question about the reality of motion is nonsensical. To this, Zeno would respond that his paradoxes demonstrate the logical impossibility of motion. From Zeno’s perspective, if he can show motion to be logically impossible, then any perceptual notion we have of the existence of motion is something like a mere illusion and not an accurate representation of reality. The disconnect between Zeno’s logical conclusions and observed reality is what makes these paradoxes. Therefore, in order to resolve these paradoxes, one must show why Zeno is wrong in logically concluding from his paradoxes that motion does not exist. In other words, if there is a way to logically explain the motion we observe in Zeno’s examples, Zeno has failed to cast doubt, via his examples, on the reality of motion. Bertrand Russell claims to resolve several of Zeno’s paradoxes by providing such a logical explanation of motion through what has come to be known as the At-At Theory of Motion. In this paper, I will be focusing specifically on Russell’s response to Zeno’s Arrow Paradox. While I do not propose to resolve the paradox myself, I argue that Russell’s solution, while it is in one sense successful in disproving Zeno’s conclusion that motion is logically impossible, fails to fully account for the fundamental discrepancy between how we know ourselves to move through the world, namely through discrete time composed of specious presents, and what must be logically true about time for Russell’s solution to work: that space and time are continua composed of instants without duration.

Zeno’s first two paradoxes of motion, the dichotomy paradox and Achilles and the tortoise, attack the logical possibility of motion by dividing space into segments. Zeno’s arrow paradox is unique in that it attacks motion by instead dividing time into instants. Zeno’s arrow paradox argues that a flying arrow is always at rest. Zeno’s arrow paradox begins with the idea that at any instant, the arrow in flight occupies a space equal to itself where an instant is understood as an indivisible and minimal part of time. The arrow cannot move during an instant because this would require the instant to have parts which it, by definition, cannot have: “throughout an instant, it is said, a moving body is where it is: it cannot move during the instant, for that would require that the instant should have parts” (Russell, 140). This is because in order to be moving during the instant, the arrow must be at one place during one part of the instant and at another place during another part of the instant. Furthermore, for the arrow to be moving during the instant would mean for the arrow to occupy a space larger than itself during the instant because it would be moving through space which would violate the very first premise that the arrow in flight occupies a space equal to itself at any given instant. 

“If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion.” ( Aristotle, Physics VI:9, 239b5)

So, at every instant of the arrow’s path, the arrow cannot be in motion and must be still. If time is composed of these instants and at each instant the arrow is still, the arrow is never in motion and therefore cannot move.

“Suppose we consider a period consisting of a thousand instants, and suppose the arrow is in flight throughout this period. At each of the thousand instants, the arrow is where it is, though at the next instant it is somewhere else. It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever.” (Russell, 140)

Russell claims to resolve this arrow paradox by developing what has come to be known as the “At-At” Theory of Motion. He begins his solution by attacking an assumption Zeno has made about the fundamental nature of time: “the view that a finite part of time consists of a finite series of successive instants seems to be assumed; at any rate the plausibility of the argument seems to depend upon supposing that there are consecutive instants” (Russell, 140). Therefore, in order to dispute Zeno, Russell does away with the assumption that space and time are composed of finite series of successive instants and instead conceives of space and time as composed of continuous series: “We find it hard to avoid supposing that, when the arrow is in flight, there is a next position occupied at the next moment; but in fact there is no next position and no next moment, and when once this is imaginatively realized, the difficulty is seen to disappear” (Russell, 140).

Russell’s At-At theory of motion argues that the arrow’s motion consists in being at a particular place at a particular time. On this view, there is no difference between an arrow being in motion at a specific point or an arrow being at rest at a specific point, if we consider each particular position of the arrow at each particular moment. If we consider the position of the arrow at many different moments, we can say that the arrow is moving as long as it is at different places at different moments. Russell argues that there is no additional process to moving other than simply being at the right place at the corresponding time. Therefore, Russell argues that it is not illogical to conceive of motion as being composed of immobilities: 

"Weierstrass, by strictly banishing all infinitesimals, has at last shown that we live in an unchanging world, and that the arrow, at every moment of its flight, is truly at rest. The only point where Zeno probably erred was in inferring (if he did infer) that, because there is no change, therefore the world must be in the same state at one time as at another. This consequence by no means follows.” 

A difficulty in understanding Russell’s solution to the paradox might arise from a psychological demand to regard space and time as discrete. Russell’s solution demands that we regard space and time as continua composed of instants with no duration, but we psychologically experience time as composed of specious moments, discrete segments of time with duration.

We might demand that Russell explain how the arrow gets from one place at one moment to another place at a different moment, but Russell is freed from this burden because the At-At Theory of Motion regards space and time as continuous and therefore composed of densely ordered points. If A and B represent two different points on an arrow’s path at which the arrow must be at two different moments, asking how the arrow gets from A to B or what is the next point after A between A and B, presupposes that there is such a next point. Zeno would argue that the infinite regress of his dichotomy paradox demonstrates how there cannot be such a next point because between any two points there are an infinite number of intermediate points so the arrow cannot move. Russell agrees with Zeno’s conclusion that there is no “next point” or “first interval” that the arrow must cross because Russell considers the arrow’s path as a continuum. However, Russell disagrees with Zeno’s conclusion that because there is no “next” or “first” point, the arrow cannot move. Instead, Russell argues that while there is no next or first point in space for the arrow to be, there is also no next or first point in time in which the arrow must occupy this space. On Russell’s view, identifying the next or first interval which the arrow must cross is not important as long as for each moment in time there is a corresponding point in space and for each point in space there is a corresponding moment in time.

In one sense, Russell has resolved Zeno’s paradox in that he has successfully found a mathematical and logical way to allow motion to exist. As difficult as it might be to understand a moving world made of immobilities, Russell has shown that, contrary to Zeno, it is nevertheless logically sound. That said, there is another sense in which Russell has not fully resolved the very core of Zeno’s argument: the logical reality of motion differs from how we take motion to work in our experience. While Zeno concluded that motion must not exist and our error is in believing it does, Russell concludes that motion does exist but that it is still different in fundamental nature from our experience of it. If Zeno accounts for the discrepancy between his mathematical conclusions and our experience of reality by asking us to conclude that our experience is something like an illusion, we must ask how Russell accounts for the discrepancy between the logic and experience of time in his own solution. If one argues that only a logical account of motion is necessary to resolve the paradox, then Russell has succeeded and there is nothing more to be done. However, such a conception of the burden of Zeno’s paradox misses the deeper point that the paradoxes, particularly the arrow paradox, are drawing out. While, as a Parmenidean, it is reasonable to attribute to Zeno a will to disprove the existence of motion via his paradoxes, the paradoxes are pointing to a more profound issue, whose resolution would require an account which goes beyond a simple logical account of motion. Such an account would require explaining how the reality of motion could differ so fundamentally from our experience of it and what this difference means. For Zeno, there were consequences to a logic which did not align with how we go through reality, namely that reality resides in the logic and anything which departs is an illusion. Where does reality sit in Russell’s picture? If in Russell’s account, the logic of the structure of time still departs from how we know ourselves to experience time, how could Russell’s answer count as a proper resolution when the paradoxical discrepancy between logical reality and experienced reality has not been entirely accounted for and resolved? 

It is nevertheless indubitable that a description of time as continuous, like Russell’s, has been crucial to modeling motion. One might argue that there is no reason to impose the discrete structure of psychological time on our mathematical understanding of time other than an inclination towards anthropomorphism. To this, I would argue that Russell himself describes “the change of position [that] has to occur between the instants” as “miraculous” (Russell, 140). Furthermore, in order to understand his solution, Russell asks us to “imaginatively realize” the idea that “there is no next position and no next moment” (Russell, 140). I argue that in this regard, Zeno’s paradox is very much alive in Russell’s solution: Zeno wanted us to take issue with the discord between our anthropomorphic experience and logical demands. As Russell says “we live in an unchanging world, and…the arrow, at every moment of its flight, is truly at rest,” but we nevertheless “find it hard to avoid supposing that, when the arrow is in flight, there is a next position occupied at the next moment” (Russell, 140). Russell is in a very real sense agreeing with Zeno that the arrow is never truly in motion. There is no time when the arrow moves, only different times at which the arrow can be at rest in different places. As mathematically or logically sound as this explanation might be, Russell himself agrees with Zeno that we do not experience motion as being composed of instants without duration and it is in this sense that we have not accounted for how we could possibly move through the world the way we do. I can accept Russell’s mathematical explanation while still acknowledging that from my perception, the arrow is constantly moving on its path. If Russell proposes to resolve this discrepancy by, like Zeno, proposing that our perception and intuitions of motion as well as space and time are illusory, then Russell has failed to resolve Zeno’s paradox in such a way that fully eliminates the gap between lived experience and logical reality. While Russell may have shortened the gap between logical possibility and lived reality, he has not eliminated it and has thus failed to fully resolve the paradox. Still, Russell has made a stride in at least allowing for motion to, in some sense, logically exist. While Russell has allowed for things to be in different places at different times, and therefore allows for the state of things to change from one time to another, he has not allowed for things to be actively in motion the way we experience them to be.

A proper solution to Zeno’s paradox would be one where our perceptual and intuitive understanding of motion through space and time is accounted for. Either we are provided with a logical account of motion which aligns with our understanding of things in active motion or we are provided with an account of the discrepancy between our experience of active motion and the reality of an unchanging world which is truly at rest. Russell, while moving beyond Zeno in mathematically conceiving of motion, fails to provide either of these two accounts and thus has not fully resolved Zeno’s arrow paradox.

Works Cited

McKeon, Richard. “Physics.” Basic Works of Aristotle, Modern Library, 2001. 

Russell, Bertrand. “On the Problem of Infinity Considered Historically.” Our Knowledge of the 

External World, 1st ed., Routledge, London, 2009.