The Arrow Paradox
An arrow is at rest whenever it is in a space equal to itself. A launched arrow goes through its flight one instant at a time. Since the arrow is in a space equal to itself each instant of the flight (just as it is when it is at rest), then the arrow must be at rest at each such instant as well. Since it is at rest at any one instant, it must be at rest for the entire duration of the flight. Hence, the flight is apparent, not real; the arrow does not move.
This is a paradox since its conclusion is based on a logical argument that contradicts the apparent reality of sense perception according to which a flying arrow changes positions each moment of its flight and thus apparently moves. Again, is Zeno’s logic flawed or are our senses?
I believe the formulation of the arrow paradox must have been triggered by a simple observation, that an object at rest occupies a space equal to its own size. A book, for example, resting on a desk occupies a space exactly equal to its own size. That said, I am not implying that such observation validates Zeno’s conclusion that an arrow in apparent flight does not move. But could he be right? Could it be true that an apparently flying arrow is really motionless? Using quantum theory, I will argue that, at best, it is not verifiable whether the arrow is moving or not. Motion, in general, is an ambiguous concept.
Motion Is Ambiguous
While motion is part of apparent reality and is also the very premise of important theories of physics, on a fundamental level (i.e., concerning the motion of microscopic particles, to say the least) motion is not an experimentally verifiable phenomenon, ever! Therefore, motion is essentially a postulate inferred from sense-perceived experiences, but its truth is actually ambiguous. This is so because inherent in the Heisenberg uncertainty principle observations are disconnected, discrete events; consecutive observations have time and space gaps—we can observe only discontinuously (as seen in chapter 7). The concept of continuity in observation must be dismissed. It is a false habit of the mind created by the observations of daily phenomena—as of an arrow in flight (although, as explained in the section titled “Observations Are Disconnected Events” in chapter 7 and as will be reemphasized later, the arrow’s apparent continuity of motion is an illusion due to its large mass that makes the time and space gaps between consecutive observations undetectably small). Now without the ability to observe continuously, motion not only is observed to be discontinuous but the very notion of motion itself becomes ambiguous. How so?
Motion occurs when during a time interval a particle (e.g., an electron) changes positions with regard to some observer; a particle should be now here and later there in order to say that it moved. But since nature does not allow us to keep a particle under continuous observation and follow it in a path, and also since a particle is identical to all other particles of the same family (for example, all electrons are identical), it is impossible to determine whether, say, an electron observed in one position has moved there from another, or whether it is really one and the same electron with that observed in the previous position, regardless of their proximity. Since observations are disconnected, discrete events—with time (and space) gaps in between, during (and within) which we don’t know what a particle might be doing—subsequent observations of identical particles might in fact be observations of two different particles belonging in the same family and not observations of one and the same particle that might have moved from one position (that of the first observation) to the next (that of a subsequent observation). Without the ability to determine experimentally whether a particle has changed position, its motion—and motion in general—is a questionable concept.
In summary, (1) without the ability to keep a particle under continuous observation, (2) it is impossible to establish experimentally its identity, and therefore (3) it is also impossible to experimentally verify that it has moved.
Reinforcing this conclusion is the fact that, when we observe a microscopic particle, all we see through a microscope is just a flash of light, and somewhere within it is the particle. But where exactly within it the particle is each moment of time, and whether it is at rest or in motion, are all indeterminable; while we do detect a particle, we detect it neither at rest nor in motion. Hence, indeed, neither immobility nor motion is experimentally verifiable. Motion is ambiguous.
But if we accept it does occur, motion is quantum! It occurs through “quantum jumps,” a foundational notion in modern quantum mechanics: an elementary particle has moved from here to there but without moving through any of the points in between! In a sketchy analogy, you see a pawn on a square of a chessboard. Then you close your eyes for a bit. Upon opening your eyes, you see a pawn on another square. Although you don’t know if your observations are of the same pawn, and although you haven’t seen a pawn moving, you may still say, the pawn has moved to a new square without moving. Epicurus (see chapter 13) taught this very quantum nature of motion! He arrived at it by countering Aristotle’s arguments which were against quantum motion. Paraphrased by classicist David J. Furley, Aristotle (in his Physics Z) thought if quantum motion were true, then “a thing may have moved without ever moving; a thing may be simultaneously at rest and moving; and motion may be composed of non-motions.”6 Amazingly, trying to capture the peculiar consequences of the Heisenberg uncertainty principle concerning motion, physicist J. Robert Oppenheimer (1904–1967) said something similar: “If we ask, for instance, whether the position of the electron remains the same, we must say ‘no’; if we ask whether the electron’s position changes with time, we must say ‘no’; if we ask whether the electron is at rest, we must say ‘no’; if we ask whether it is in motion, we must say ‘no.’ ”7
Now, since motion is ambiguous for microscopic particles, then in a stringent sense it must be ambiguous for arrows, too, for arrows are made of microscopic particles. Observing an arrow in flight moving continuously does not prove (in the strictest sense of the word) that one and the same arrow has endured and moved, simply because there is no proof that any of its component microscopic particles have endured and moved. Besides, quantum theory (hence the uncertainty principle) is true for both the world of the large and the small. It is only for practical purposes that the world of the large is assumed to behave according to classical physics—for which objects appear to endure and move in definite traceable paths—because the consequences of the uncertainty principle for large objects are undetectably small, although not zero.
Lastly, in Einstein’s block universe, all arrow events—where the arrow is each moment of time, that is, all its space-time points—exist in the continuum always, without the arrow to have to change position each moment.
Well, then, how do we explain everyday apparent motion and in fact the apparent continuity of apparent motion for any object, such as an arrow in flight, Achilles chasing a tortoise, or anyone taking a trip?
Cinematography and Apparent Motion
We can explain them with cinematography. In an analogy, consider a series of identical and disconnected red lightbulbs, closely spaced along the arched outline of George Washington Bridge. Now imagine it is nighttime and that the first lightbulb in the series is turned on briefly for a few seconds and then off forever; after a brief time gap, lasting a minuscule fraction of a second, the second lightbulb is turned on and off the same way, then the third, and so on, until each lightbulb is turned on and off in this sequential manner. The events, the on-off turnings of each lightbulb, are (1) identical (in the sense that an observer sees the same red light) and (2) disconnected: the space gaps are the distances between the bulbs; the time gaps are the minuscule fractions of a second. Furthermore, (3) assuming that the space and time gaps of these events are small enough, to a distant observer, this phenomenon will appear as if one red object (the first lightbulb) has moved and has moved continuously along the outline of the bridge, when in fact no object has. Motion in this case is an illusion of the senses created by observing a series of identical and appropriately disconnected events. In particular, the
first two facts, (1) and (2), create the illusion of motion, and requirement (3) creates the illusion of continuity of motion.
The way the red light appears to be moving is similar to the way an arrow in flight appears to be moving. In each case, apparent motion and apparent continuity of motion are in reality the result of (1) a chain of identical observations (of an apparently identical red light or arrow), which (2) are also disconnected (for the arrow, this is due to the uncertainty principle) with (3) undetectably small space and time gaps in between (for the arrow, this is due to its large mass). Specifically, (1) and (2) create the illusion of motion, of one and the same object, the red light or the arrow, and (3) creates the illusion of continuity of motion.
But to refine the analogy, we must add this: unlike the case of the bridge for which several identical lightbulbs are assumed to preexist along its outline, for an arrow in flight we cannot assume several identical arrows to preexist along its apparent path; only that, each one of our observations is of an apparently identical arrow; though it is uncertain whether our observations are absolutely of one and the same arrow, for, as we learned in chapter 7, it is impossible to establish experimentally the identity of microscopic particles, and since arrows are made up of such particles, it is also impossible to prove unambiguously whether our observations are actually of one and the same arrow (that is, of an arrow composed of the same microscopic particles at each location of its apparent flight)—a fact that makes motion an ambiguous notion for any object, microscopic or macroscopic. The most we can say is that, at subsequent observations, the observed arrows have similar bulk properties, similar general form or organization (just as the Heraclitean river); it is this general organization that seems to endure, at least during some interval of time and within some region of space, creating the illusion of a permanent thing (e.g., an arrow) in motion.
While on the one hand, all around us certain things appear to endure (appear as permanent things at least for some time and within some region of space), and whenever they appear to move, there appears to be continuity in their motion, on the other hand, neither permanency in things nor motion is an experimentally demonstrable idea. Thus, motion is, to say the least, an ambiguous concept, a result to be expected because of the very definition of motion, which requires that permanent things exist so they can move: motion occurs when during a time interval a particle (a thing in general) changes positions relatively to an observer; but to refer to a particle and define its motion, the particle must remain the same for the duration of its motion; motion cannot be defined if a particle does not remain the same for a period of time. Now for Heraclitus and modern physics there are no identifiable particles, no permanent things, only events. And without an enduring thing, without the ability to establish the sameness of a thing at two different moments, motion remains an ambiguous concept. While ambiguous, can it nonetheless be a practically useful way to explain the phenomena?
Adequacy Versus Truth
The answer to the previous questions is sometimes. Causality in classical physics is deterministic: a single cause produces a single effect, and both cause and effect are precisely determinable at least in principle. In quantum theory, causality is probabilistic: causes and effects are expressed in terms of a probability; this is actually the reason it is impossible to determine whether the observations of two identical particles might be observations of one and the same particle, for we cannot causally connect these observations with deterministic (absolute) accuracy. Still, to make sense of the phenomena from the point of view of quantum theory, we often assume certain causal chains of events. For example, an electron here collides with a photon and recoils there (as if the electron endures). Thus, while neither a cause nor an effect is certain, and motions are untraceable, still the assumption of a certain chain of events and of motion is often an adequate way to model a practical explanation. Supposing that particles endure, we have previously argued that they are constantly moving.
But while motion may be an adequate and useful concept in devising a certain practical explanation of nature—especially so for macroscopic objects such as arrows, cars, and planes—as a true property of nature, it is, to say the least, an ambiguous concept, for it lacks the support of experimental confirmation from the microscopic constituents of matter that make up all macroscopic objects. Therefore, the merit of modeling an object (an electron or an arrow) as moving is a practical necessity of everyday life, not a confirmable truth. It should also be pointed out that even practicality cannot be applied consistently (especially so in the microscopic world).
Before quantum theory, and within the context of classical physics, concepts such as position, velocity, and motion (in general) were intuitive, self-evident, and could be used in a definitive way to characterize an object; an object moves with a specific velocity, it is now passing from here and will later go there. However, after quantum theory all these concepts became counterintuitive and could not be used in the same definitive way to describe the behavior of microscopic particles; such particles have neither definite position nor velocity nor a path of motion—as seen, this ambiguity of motion first came up in the natural philosophy of Zeno. A better way to describe the behavior of a particle (and in general the phenomena), then, is not through motion but through the probabilities calculated from the wave functions of quantum theory. It is the concept of probability that is the fundamental (intrinsic) property of matter and not properties such as position, velocity, and motion. Within this context, as initially argued in chapter 6, a particle is truly a mathematical form. Could this mean something concerning change and motion in nature?
A Compromise: Yes to Change, No to Motion
In the view of Zeno, the arrow itself exists but does not move, and there is no change. On the other hand, in the view of Heraclitus, the arrow as a permanent thing does not exist (only events exist), but constant change and constant motion do; only the organization of the arrow exists and endures at least for some time. Could these antithetical views be reconciled by modern physics? Well, we can observe an electron (or an arrow) here now and an electron (or an arrow) there later. So obviously we can experimentally confirm that there is a certain change of events (at least in what we observe and where and when we observe it; that is, our observations are of different phenomena, electrons or arrows, at different places and times); but we cannot experimentally confirm that anything has moved. So the compromise might just be this: that constant changes do exist in nature (as Heraclitus posited), but motion does not (as Zeno theorized). But do these changes occur in a passive, playground-like space, or are space, time, and matter somehow related?
The Space Paradox
“If everything that exists is in some space, then that space, too, will exist in some other space, ad infinitum” (Aristotle’s account of Zeno).8 We may reconstruct this quote as follows.
(1) Things that exist do so in some space.
(2) Space exists (for if it didn’t [1] wouldn’t hold and things could not exist).
(3) Since space exists and everything that exists is in some space, then space, too, must exist in some other space, ad infinitum.
With this paradox Zeno seems to be arguing that requiring space, that is, void (as the atomists do, by treating it as a sort of a playground to put things in), is as problematic as denying space (as Parmenides does). And that, if we shouldn’t completely deny space, we also shouldn’t treat space as a playground—as if space is supposed to exist independently of the objects merely for the objects’ sake, namely, for them to exist in it.
As discussed in chapters 6 and 7, the special and general theories of relativity replaced the immutable playground-like space of Newtonian physics (for which space and time are absolute) with the malleable space-time continuum (for which space and time are relative). For relativity, things do not just exist in a passive space with time flowing steadily in the background. Instead, space, time, and matter are complexly intertwined—with astonishing effects such as length contracti
on, time dilation, relativity of simultaneity, and space distortions (the latter being true only in the theory of general relativity).
So Zeno’s space paradox is a paradox because while, on the one hand, his argument against a passive playground-like space is logical, on the other hand, it contradicts sense perception of exactly that kind of space. With the theory of relativity in mind, the space paradox may be considered resolved.
The peculiarity of a playground-like space implied by the space paradox is appreciated further when we try to construct a similar-type time paradox (though this is not one of Zeno’s paradoxes). For example, if everything that exists does so for some time, then that time, too, will exist for some other time, ad infinitum. This time paradox argues against an absolute (Newtonian) time, flowing the same way for everyone while things happen.
Last, in his effort to show that a nature made up by many things is as problematic and contradictory as the Parmenidean oneness, Zeno devised several other paradoxes. Based on them, he concluded that if in nature there are many things, they must simultaneously be (a) infinitely small and (b) infinitely huge and (c) finitely many and (d) infinitely many.9 We will not cover these paradoxes here, however, briefly, while the simultaneous existence of opposite qualities (such as of a thing being both large and small) appears as a contradiction, still don’t dismiss it easily; recall the Copenhagen interpretation and also read chapter 11.
Conclusion
Zeno’s paradoxes challenge our views of space, time, and matter. Are these notions somehow connected? Einstein thought that, yes, space-time is a continuum in constant and intricate interaction with matter. Is the nature of matter continuous—spread everywhere and also infinitely divisible for which matter can be cut into ever smaller pieces—as Empedocles and Anaxagoras thought? Or is the nature of matter atomic—and so finitely divisible, for which matter cannot be cut beyond some fundamental pieces that are spread unconnectedly because they are surrounded by void—as Democritus taught? Are there also atoms of space and of time, as Epicurus taught? Is there just one primary substance of matter, or are there many? Although all (Empedocles, Anaxagoras, Democritus, and Epicurus) had speculated plurality in the number of primary materials, each had a unique take on it.
In Search of a Theory of Everything Page 14