There's an interesting philosophical problem about the nature of imperceptiblity, related to the problem of sorites (at some point a baby stops being a baby: when?). I thought I'd post my ideas about this, partly prompted by a recent and somewhat cantankerous essay by the great David Bordwell, with which I agree with in part and disagree in part.
Consider four photos of the hands of a watch, taken at, say, 1/24th of a second apart. Put them in front of someone and ask her which one was taken first:
I didn't manage to crop these quite the same (that's an issue we'll come back to: changes in the perceiver's or recorder's perspective), but just look at the hands of the watch. These shots were taken within about a quarter of a second of each other, so on average they're about 1/12th of a second apart - twice the 1/24th of a second we're hypothesizing. I present them here randomly (I followed the order of the last four digits of my Frequent Flyer number). Can you tell the order they were taken? (Answer: The order is 4,1, 2, 3 - I think you can just make the sequence out if you squint, but if not, highlight all and you'll be able to.)
At some point, though, you'll be able to see that the hands have moved, that it's now (more or less) 9:03 and not 9:01.
So the paradox of imperceptibility is this. Let's say call p the minimal distance that the hand of a normal-sized watch has to move for us to perceive -- by eyeballing two different life-sized photos -- that it has moved. The order of magnitude for p here is probably something like twelve minutes of arc or so (for the big hand that would be about two temporal seconds, for the little hand two temporal minutes). At any rate, the photos above don't show anything like a difference of magnitude p. We're talking about a total of a quarter of a second here, which is why there's no way you can tell the order by looking. (Funes could, I suppose.)
Now consider two photos identical except in the second of them the minute hand is at a distance of 1.6p from the first; i.e. 3.2 seconds have elapsed between the two photos. Just looking you can tell the difference! (Remember that's what p means: the distance greater than which you can see the difference.)
Now imagine interpolating a third photo, which shows the minute hand at a distance of .8p past the first photo, and accordingly at a distance of .8p before the second photo; i.e. this third photo was taken 1.6 seconds after the first, and 1.6 seconds before the second. In other words, let's assume we're looking at a sequence of three photos, taken 1.6 seconds apart.
So here's what exercises the new Zenonians. You can't tell the difference between the first and the second photo (we've stipulated that the minimal difference you can tell is p, the distance it takes the minute hand two seconds to cover), nor can you tell the difference between the second and the third, but you can tell the difference between the first and the third. So you perceive no difference between the first and the second, and no difference between the second and the third, and yet, somehow, somewhere, you must perceive some difference or you wouldn't perceive a difference between the first and the third. (There's a strange failure of the transitivity we expect here.)
To see how this is true, consider ordering the photos to present them to someone else. Since there's no perceptible difference between the first and the second, you should be able to switch them around, and no one will be the wiser. But if you do that there'll be a perceptible difference between the new second and the third. You can't see the difference between one and two, so their order seems not to matter; but switch them and you can now see a difference between two and three, so their order did matter.
That's the problem. People have tried to solve it with vague appeals to threshold differences, but the new Zenonians point out that this is just to rename it, since the whole point is to ask what makes something exceed the threshold of perceptibility. P defines that threshold, but we've already said that. An appeal to a threshold only changes the vocabulary.
Here's my solution. Look at any two indistinguishable photos whatever: even at the same photo twice. There'll always be a perceptible difference between the two. Your head will have moved slightly, the light will have changed, your eyes will have performed some micro-saccades, the beat of your pulse will cause the image to shudder, your breath will inspire it to fitful and inconstant motion. But you'll read the two different retinal images as identical, because part of visual processing consists in abstracting from the incessant flux of experience by fixing on what J.J. Gibson called invariants. The brain uses these invariants (edge ratios, color ratios, etc.) to calculate what's changing in the world vs. what's changing in your perceptual apparatus.
Accordingly, we're always testing the origin of the perceptual changes that occur every moment. This is an argument that Kant was the first to make, as he analyzed the nature of our capacity to distinguish between seeing a boat move downstream and a house standing still. Both visual experiences come to us through the incessant flux of appearance: I see a window, a door, a lintel, an eave, in any order; or I see a prow, a stern, a sail, an oar. But I am proprioceptively aware of the fact that the flux in my view of the house comes from movements I am myself making, that originate in my own will. (William James argues that the difference between proprioceptive awareness of unforced change and the experience of willing is essentially nonexistent.) The flux in my perception of the boat isn't assignable only to my own will.
At some fairly automatic level our brains proprioceptively track our microsaccadic eye-movements and assign the origin of the flux we see in an unmoving object to our own movements. At some point we'll begin to wonder whether, and at a later point decide that, we're seeing more flux than we can explain through our automatic proprioceptive guidance systems.
Vertigo provides an obvious example of this fact. If we mess with the vestibular system (by spinning around, by drinking), we lose a very important proprioceptive clue as to the attitude of our heads. Now we're reeling and seem to stand upon the ceiling: the room is still but we think it's moving. Closing your eyes helps because you stop seeing the motion that they're making, stop projecting it into the world.
So no case of motion may be absolutely assigned to one domain or the other, to the world or to the seeing soul. But usually we start with a very good guess as to where the motion is. This also explains a feature of vertigo: the way we cast our eyes everywhere trying to find something that will stay still to orient ourselves by.
Imperceptibile motion, then, is motion whose perception is swamped by the normal flux of appearance.
Consider the converse idea: that of imperceptible stillness. I feel as though I've been waiting for this class to end forever. Has the clock stopped or am I so bored that every minute seems five? I keep looking at the clock, and after a while, I start realizing that it's broken. It's frozen but I thought it was moving. I decide that it's frozen in several stages: I wonder, I concentrate, I observe closely, I wait a little longer, and after a while I make a judgment: it's not fucking moving.
Likewise imperceptible motion occurs when I think something is still (when my automatic visual processing takes something as still), but then after a while start wondering if it really is still. I am not sure the flux derives only from myself. I start testing the hypothesis that the minute hand itself has moved. That hypothesis is easier to test the more it moves. Nothing makes it certain that the motion comes from the world, and not from me, but I become more and more confident that certain intervals are more likely to be due to a change in the object than to a change in my own perspective.
Body English -- you know, as with Carlton Fisk's stay-fair home run -- provides a vivid example of what I mean: you move your own body to some extreme angle to get a preferred perspective on the ball. Of course you know that this doesn't affect its trajectory, but you can fool or try to fool your perceptual apparatus into thinking that the change you see is a change in what you're seeing, not in yourself.
The fact that perception always involves the will, gyroscopically orienting the telemetry of the senses, means you can dally with false surmise by flooding the proprioceptive stabilizers of perception a little bit. We can pick the locks of perception (by spinning around or crouching and cocking our heads or whatever) and so perceive a little more wishfully than accurately, when we really want to. This shouldn't be surprising: it's the will after all that's being engaged. We move in certain ways because we want certain things to happen, and our motion sometimes affects the world (and so our perception of it), sometimes just our perception of the world. The fact that we engage so much in body English shows that the frontier between what our brains think comes from outside and what they think comes from inside is porous and vague. The two types of willing are continuous, no doubt because all perception requires the same sort of experimental assessment of how (or whether) it's going as action does. A single way of doing that experimental assessment governs both cases, and can turn one into the other.
Film depends on this fact about impercetible motion, in more ways than one. Two frames, 1/24th of a second apart, will often be indistinguishable. (Not to Fred Astaire, apparently, who'd fiddle with single sprockets in his movies, to get the beat exactly right. I believe the technology was eight sprockets per frame when he made his great movies, so that would mean he was able to perceive differences at about 1/200th of a second. But no one had more motor control, and therefore more fineness of proprioception than Astaire.) But of course over eight frames you can see any perceivable motion - that's a third of a second. So when we watch a movie, we're brought to judge motion as imperceptible differences - differences we would ascribe to the noise of the perceptual system - make themselves felt as perceptible.
But that means that we're constantly judging what we're seeing, through the engagement of our will. In well-edited films, the films themselves will, through what we could call cinematic occasionalism, conform to our own acts of willing. Someone looks right. We want to see what they're looking at. The camera or the editor obliges. In classic Hollywood film the audience almost always feels in control of the camera. (The times that we don't, at least in a good movie, are generally the places where we feel the pleasure of being played: think of any Hitchcock film.) Film, and classic Hollywood film especially, exploits our negotiations with our own perceptual flux, and engages our will in ways that minister to narrative desire, to the body English that makes us try to see things as acting the way we want them to act.
I think that what we have here is a natural analogue to Newcomb's Problem. By choosing our perceptions right, we may be able to get what we want. If that's how the psychology of narrative works (through the general phenomenon that I call non-causal bargaining), then on every scale, from 1/24th of a second to two or more hours, our relation to film keeps our occasionalist testing of what we see highly sensitized and engaged. This would make film a particularly apt medium for narrative engagement (probably just about as apt as language, where I think we do a similar sort of testing).
Obviously I wouldn't confine this to Hollywood movies: Chatal Akerman's Jeanne Dielman, 23 quai du Commerce, 1080 Bruxellesis a case in point: we watch Jeanne over three days and start noting the tiny differences in what she's doing. Are they artifacts of film making or are they action? That's the question that animates our perception of the movie.
The perception of all action requires our involvement, and the more our involvement is fictional or factitious, even on the level of the most basic perception, the more it enmeshes our willing and our wishing, our desire and our engagement in non-causal bargaining.