Metaphysicians have defined two distinct conceptions of how objects persist through time. Under one conception, objects are extended in time as they are extended in space. Just as a single point in space can contain only part of an extended object, a spatial part; so a single point in time can contain only part of a persisting object, a temporal part. The object fills time by having one temporal part after another, just as it fills space by having one spatial part next to another. An object that persists through time in this way is said to perdure.
Under the alternative conception, an object’s extension in time is different from its extension in space. Whereas only part of an object can be present at a single point in space, the object can be wholly present at a single point [in] time. An object that persists [through] time in this way is said to endure.
As Velleman goes on to point out, this latter conception seems transparently incoherent. To exist wholly in a moment is precisely to not exist at any other moment, so such "endurance" is impossible. The endurantist doesn't want to say that all of the object's temporal parts are present. So they must deny that it has any temporal parts at all. Yet such claims are hopeless, for "what is to prevent us from considering the object as it is at a single moment, and then denominating that aspect of it as a temporal part?"
Velleman goes on to explain the illusion of endurance as arising from the structure of first-personal experience and memory. We think of ourselves as momentary objects, but there is a conflation of the remembered and remembering selves in memory. This contrasts with other forms of imaginative representation, as when "[you say] “I’ve imagined ‘I am the birthday boy’,” where the the first occurrence of ‘I’ refers to you but the second refers to him." When Velleman himself remembers being the birthday boy, he automatically conflates the two 'I's, leading to the incoherent notion that these two momentary selves, existing at different moments, could somehow be one and the same momentary self. Or, as he describes the conflicting intuitions:
I am tempted to say that all of my temporal parts are present at a single point in time because I tend to think of myself as my present self — a momentary subject whose existence is indeed complete in the here-and-now. I am tempted to say that I nevertheless persist through time because I tend to think of this self, complete in the moment, as nevertheless existing at other moments.
So that's the illusion of endurance. Now let's consider the incoherence of denying the eternalist picture of an Unchanging Time. As Velleman explains the basic problem:
An event’s changing from future to present to past must unfold in time: the event must be first in the future, then in the present, and then again in the past... The event is timelessly later than the one time, simultaneous with the second, and earlier than the third; and so its transit from future to past appears to be no more than a set of temporal relations that it occupies statically. In order to complete our description of how time passes, we have been forced to describe it once again in terms that seem to make it stand still.
[W]hen we tried to identify something toward which a future event draws nearer or from which a past event recedes, we focused our attention on other events. Yet each event depends for its identity on when it occurs: it could not be closer to a future event, or further from a past event, without occupying a different temporal position and hence being a different event. This conception of the problem suggests the solution. Whatever the future draws nearer to, or the past recedes from, must be something that can exist at different positions in time with its identity intact. And we have already found such a thing — or the illusion of one, at least — in the form of the enduring self.
Suppose that I endure in the admittedly incoherent sense that is suggested by experiential memory and anticipation. In that case, I exist in my entirety at successive moments in time, thereby moving in my entirety with respect to events. As I move through time, future events draw nearer to me and past events recede. Time truly passes, in the sense that it passes me.
This contrasts with what is actually going on:
If I merely perdure, however, then I do not move with respect to time. I extend through time with newer and newer temporal parts, but all of my parts remain stationary. A perduring self can be compared to a process, such as the performance of a symphony. The performance doesn’t move with respect to time; it merely extends newer and newer temporal parts to fill each successive moment. The last note of the performance is of course closer to midnight than the first, but we wouldn’t say that midnight and the performance move closer together.
Which brings us to the crucial conclusion:
If this enduring ‘me’ is an illusion, however, then so is the passage of time. And ceasing to think of myself as an enduring subject should result in my ceasing to experience the passage of time. Coming to think of myself as perduring should result in my coming to experience different temporal parts of myself at different moments, but no enduring self past which those moments can flow.
Fascinating stuff, no?
This is a painfully quote-heavy post already, but I can't resist finishing up with a quote Velleman offers from Vonnegut's Slaughterhouse Five:
“[W]hen a person dies he only appears to die. He is still very much alive in the past, so it is very silly for people to cry at his funeral. All moments, past, present, and future, always have existed, always will exist. ... It is just an illusion ... that one moment follows another one, like beads on a string, and that once a moment is gone is it gone forever.”