§2 From Zeno’s Arrow to Time’s Arrow

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§2 From Zeno’s Arrow to Time’s Arrow

Zeno’s paradox of the arrow may not be quite so familiar as the “dichotomy” or “Achilles”, but it certainly repays serious scrutiny; it also has some more general implications that are not always recognized.

To start with, let’s suppose, in line with the orthodox view of contemporary physics (and one of the main views in Greek times), that time is infinitely divisible, and that any temporal interval is composed of an infinite number of durationless instants. Now imagine an arrow in flight. The arrow’s speed will not be constant as it traces out its trajectory—if it follows an arcing path, it will slow down as it approaches the highest point of its path—but it is nonetheless in constant motion: at no point does it stop, it is continually moving from the moment it is fired to the moment it arrives at its destination, or so common sense suggests. But now consider. At each instant during its flight the arrow is at some specific location, exactly filling a particular volume of space, a volume which exactly matches the arrow’s own dimensions. Given that motion takes time, and instants lack any duration, the arrow isn’t in motion at any one of these instants: it is entirely stationary. But the arrow’s entire journey, from start to finish, is entirely composed of instants. If it isn’t in motion at any one of these instants, then doesn’t it follow that the arrow is never in motion, and hence that it hasn’t undergone any motion at all? This reasoning seems impeccable, yet the arrow does reach its destination (as many ancient warriors discovered to their cost).

There is a standard response to this Zenonian paradox: the arrow does move, but motion isn’t what we normally suppose it to be. Whether or not an object O is moving at a time t isn’t a matter of how the object itself is at t, rather it depends entirely on what is happening at the instants neighbouring t. If O is at different spatial locations at these neighbouring instants, then O is in motion; if it is occupying the same spatial location as O occupies at t, then it is motionless. Or as Russell, with his habitual concision, puts it: “Motion consists merely in the fact that bodies are sometimes in one place and sometimes in another, and that they are at intermediate places at intermediate times (1917: 84). For readily comprehensible reasons, this view of motion is often called the “at-at” theory.

Since motion plainly does involve an object’s varying its spatial location over time, its proponents maintain that the at-at view captures everything that motion invariably and essentially involves. It may well be that, from the perspective of physical theory, they are right about this. But the at-at view can also seem incomplete and inadequate: the claim that motion is in no way, shape or form an intrinsic feature of objects can easily seem entirely wrong-headed, if not entirely impossible to believe. The full story about why this is the case is no doubt a complex one, but one important element of this story is not difficult to discern: motion is something we can directly perceive, and the appearance of a moving object is (typically) quite different—intrinsically different—from that of its static counterpart.3

If this is not immediately obvious, think of what it is like to see a succession of still images depicting a dog trotting along the street (e.g. digital photographs taken at one second intervals, say, displayed for a second each) with what it is like to see the dog in propria persona trotting down the street—or for that matter, the playback of a video recording of the same event. Self-evidently, the two courses of experience are vastly different: in the latter case we see a dog moving in the former, a sequence of entirely motion-free images. Moreover, we will perceive the animal’s motion in two different ways. Unless it is running very quickly indeed, although the dog’s body looks clearly to be moving, we can also see it clearly, without any trace of blurring or smudging; in contrast, the dog’s legs—provided it is trotting at a reasonable speed—will appear as nothing but a blur: there is something there, something that is in rapid motion, but we are unable to discriminate precisely what. What goes for dogs goes for other objects. Move your hand back and forth in front of your eyes quite slowly and you will see it moving in a clear and clean way; if you increase the speed, your hand disappears into a blur. In a quite general way, our perceiving of moving objects—everyday objects, at everyday speeds—are associated with distinctively dynamic sensible appearances, sui generis forms of experience which are not reducible to (or composed of) sequences of static appearances. But if the at-at theory is correct, motion as it in itself—motion as an unobserved purely physical phenomenon—does not involve anything resembling these sensible appearances: it consists of nothing more (or less) than an object being located at a different locations at different instants. Since this is not how motion appears to us, it is reasonable to conclude that our perceptual systems are responsible for creating the dynamic qualities that we perceive moving objects as possessing.

There are some long-established, if still somewhat contentious, empirical results which are also very relevant in this context. Suppose you are sitting in front of a video screen which is displaying nothing but a white background, and two black spots, which flash on and off at regular intervals, but always at the same two locations on the screen; each spot is shown for quite a short time, e.g. for around 10 msec. How you perceive these spots will depend on the temporal interval separating them. If this interval is quite small, less than 30 msec or so, you won’t see an alternation or succession at all: it will seem to you that spots are being shown simultaneously. If the intervening interval is a fair bit longer, above around 200 msec, then you will see the spots appearing in succession rather than simultaneously, and as quite separate occurrences—in short, you are now seeing them as they in fact are. Things get more interesting when the interval is between 30 and 200 msec: so-called “apparent motion” effects now come into play. If the gap is between around 30 and 60 msec, you will have the impression there is just one spot that is moving to and fro, but it is moving so quickly that you don’t actually see the spot cross the intervening space, you merely have the blurred impression of that something is doing so; this effect goes by the name of “pure motion”. When the gap is around 60 msec you will also see a single spot moving back and forth, but now you won’t merely have the impression of motion: as it smoothly slides from side to side you see the spot itself moving across the screen. For obvious reasons this effect is often known as “optimal motion”.4

The pure and optimal motion effects studied by psychologists may well coincide with the “clean” and “blurry” forms of motion that we find in our everyday experience, and since the various forms of apparent motion play a crucial role in cinema, TV and video displays, they are of more than purely academic interest. For present purposes they provide further confirmation that the perceiving of a sequence of still images can lead to highly dynamic visual experience, which is grist to the at-at theorist’s mill: if a succession of still images shown at a mere 15 or 20 frames per second can give rise to perfectly smooth-seeming experiences of motion, it is hardly surprising that the vastly greater number of stills presented to us by an object that is really moving can do likewise. There is, however, an important respect in which the psychological terminology of “apparent” motion is potentially misleading. It is of course true that when we perceive the pair of spots are flashing on and off at a rate which generates the optimal motion effect, we are seeing something which is not really there: in reality there is no spot moving back and forth across the screen, just two stationary spots flashing on and off. However, it would be a mistake also to conclude that the motion which features in our visual experience is also merely “apparent”, for it is nothing of the kind. As noted earlier, the contents of our visual experience are often very dynamic in character, and these sui generis dynamic characteristics are generally vivid and robust: think again of the enormous (phenomenological) difference between seeing a sequence of still images of a dog running across a field, and seeing a dog running across a field.