Change and intervals

Suppose a Newtonian universe where an elastic and perfectly round ball is dropped. At some point in time, the surface of the ball will no longer be spherical. If an object is F at one time and not F at another, while existing all the while, at least normally the object changes in respect of being F. I am not claiming that that is what change in respect of F is (as I said recently in a comment, I think there is more to change than that), but only that normally this is a necessary and sufficient condition for it. So the ball changes with respect to sphericity, and specifically changes from being spherical to being non-spherical.

When does the ball change from spherical to non-spherical? There are two kinds of times: times when the ball is still spherical and times when the ball is no longer spherical. At any time t at which the ball is no longer spherical it is already true that for some time the ball wasn’t spherical. Why? Well, whenever the ball isn’t spherical, it differs from sphericity by some non-zero amount, and it takes some time for the ball to deform by that amount. But if at a time t the ball had not been spherical for a while, then it’s not changing from being spherical to being non-spherical—rather, it had already changed.

What about times at which the ball is still spherical? These can be further subdivided into the pre-impact times and the time of impact. It’s clear that at the pre-impact times, the ball isn’t changing from being non-spherical to being spherical.

That leaves exactly one possible answer to the question of when the ball changes from being non-spherical to being spherical: at the time of impact. Now, at the time of impact, the ball is still spherical. We now have two interesting issues. The first is that if the future is open, there need be no fact of the matter at the time of impact that the ball will ever be anything but spherical (a powerful being could, for instance, make the ball penetrate the ground without changing shape). So if the future is open, it is not true at the time of impact that the ball is changing from spherical to non-spherical, since change with respect to sphericity requires being spherical and being non-spherical at different times. The second is that even if the future is closed, it seems awkward to say that at the time of impact the ball is changing with respect to sphericity. After all, the ball still is spherical then, and has been spherical for a while, and so it doesn’t seem right to say that something that is in the same state as it’s been for a while is changing with respect to that state.

So it seems that at no time is the ball changing from spherical to non-spherical. At any given time it either has already changed or it is going to change, but it never is changing.

What if time is necessarily discrete? That doesn’t change the arguments that the ball isn’t changing pre-impact or at the time of impact. But it allows for one more option: perhaps the ball counts as changing at the instant right after impact. On a discrete-time view, that is the first moment at which the ball is non-spherical. I am inclined to say: “No, the ball isn’t changing any more. It already has changed.”

Here’s a super-quick way of putting the above, neutrally between discrete and continuous time:

• When the ball is spherical, it will change but isn’t changing yet.

• When the ball is non-spherical, it has already changed but isn’t changing any more.

Since obviously we don’t want to deny that change happens, what should we say? I see two options. The first is to say that change is something that only makes sense from a four-dimensional perspective. To say that change happens is not to say anything about how the world is at a time, but how the world is at two or more times, just as to say that the road narrows at the 10 mile point isn’t really to say just what the road is like at the 10 mile point, but what it’s like before the 10 mile point and what it’s like after the 10 mile point.

But I think there is another option. Suppose that time is discrete, but that in addition to having instants it also has intervals between the instants. Then if t₁ is the instant of impact and t₂ is the next instant, there will be an interval I from t₁ to t₂. This interval is not like the intervals of mathematics—it isn’t a set of points of time between t₁ and t₂ inclusive, because on the theory in question there are only two points of time between t₁ and t₂ inclusive. Rather it is at least as fundamental as the instants themselves (and perhaps grounds the instants—but we don’t need that right now). Then we can say that the ball is changing from spherical to non-spherical at I.

On this story, we can say that change always happens at some time. But times include both instants and intervals. And change is something that doesn’t happen at an instant—that seems obvious when put that way—but something that happens at an interval.

But here is an interesting problem. It seems that for every time t at which the ball exists, either it is spherical at t or it’s not spherical at t. But what if t is the interval I? Then the ball is spherical at the beginning of the interval and non-spherical at its end. It seems it’s neither spherical nor non-spherical at I.

But that doesn’t follow. I think we can simply say that the ball is not spherical at I, because it’s not the case that it’s spherical throughout I. (A pipe that is square at some point in its length is not round.)

So we have come back to the idea that the ball changes from being non-spherical to being spherical at a time when it is already non-spherical. But that’s OK, because that time is an interval, and we cannot say that it is wholly non-spherical at that interval. It is non-spherical because it is partly non-spherical and partly spherical on that interval, because it is changing from spherical to non-spherical.

So, change happens at intervals. Or at least first-order change does. Second-order change, however, can be taken to occur at instants. Thus, if t₁ is the instant of impact and t₂ is the next instant and I₀ is the minimal interval just preceding t₁ while I₁ is the interval from t₁ to t₂ (which I previously just called I), then at I₀ the ball isn’t changing in sphericity, while at I₁ it is. And we can say that at t₁ it is changing from not changing in sphericity to changing in sphericity. Third-order change, then, will take place at intervals, fourth-order change at instants, and so on. There is no vicious regress: we just need two kinds of things, instants and intervals.

This is pretty complicated, more complicated than the simple story that change doesn’t happen at a time but at a pair (or more) of times. But it also gives me a nice story about what’s lacking in the at-at theory of change. It may be necessarily the case that an object changes if and only if it is one way at one time and another way at another time. But that isn’t what change is. What change is is having an interval of time such that the object is one way at one endpoint and another way at the other endpoint. But an interval is something over and beyond its endpoints. If, perhaps per impossibile, God were to annihilate the interval I between t₁ and t₂, the ball would be first spherical and then non-spherical, but it wouldn’t have changed from spherical to non-spherical.