Temporal purism

Say that a fact is temporally pure about an instantaneous time t provided that it holds solely in virtue of how things are at t. (The term is due to Richard Gale, but I am not sure he would have wanted the “instantaneous” restriction.) Thus, that Alice is swallowed the fatal poison at noon is not temporally pure because a part of why it holds is that she died after noon. The concept of a temporally pure fact is intuitively related to the Ockhamist notion of a hard fact: any fact that’s temporally pure about the past or present is a hard fact.

I will allow two ways of filling out “instantaneous time t” in the definition of temporal purity: the time t can be a B-theoretic time like “12:08 GMT on January 2, 2084 AD” and it can be an A-theoretic time like “exactly three hours ago” or “now”.

We can now define a theory:

• Temporal purism: Necessarily, all temporal facts are grounded in the temporally pure facts and/or facts about the existence (including past and future existence) of instantaneous times and of their temporal relationships.

Presentists, open futurists and eternalists can all embrace temporal purism.

Probably the best way to deny temporal purism is to hold that there are fundamental truths about temporal reality that irreducibly hold over an interval of times—this is the temporal equivalent of holding that there are fundamental distributional properties.

I think there are reasons to deny temporal purism. First, it is plausible that (a) some states of our consciousness are fundamental features of reality and (b) they irreducibly occur over an interval of time of some positive length. Claim (a) is pretty standard among dualists. Claim (b) seems to follow from the plausibility that no state of consciousness shorter than, say, a nanosecond can be felt by us, but of course there are no unfelt states of consciousness.

Second, temporal purism pushes one pretty hard to an at-at analysis of change, and many people don’t like that.

Eternalists can deny temporal purism. This is pretty clear: eternalists have no difficulty with temporally distributional properties.

I think it is difficult for open futurists to deny temporal purism. For suppose that some fundamental feature F of our temporal reality occurs over an interval from t₁ to t₂, and cannot occur over a much shorter interval. Then at some time very shortly after t₁, but well before t₂, the feature is already present. But its being present seems to depend on a future that is open. So open futurism plus temporal impurism pushes one to a view on which the present and even the near past is open, because it depends on what will happen in the future.

Closed-future presentists can deny temporal purism. However, this feels uncomfortable to me. There is something odd on presentism about the idea that a present reality depends on the near past and/or the near future. At the same time, many odd things are actually true.

I think the denial of temporal purism pushes one somewhat towards eternalism.

Murder by slowdown?

Zeno wants Alice dead and he has the following plan. He slows down Alice’s functioning—say, by cooling her or by sending her around the earth on a spaceship so fast that relativistic time dilation does the job—so much that each second of Alice’s internal time takes a billion years of external time. In six seconds of Alice’s internal time, she’s dead, because the sun runs out of hydrogen and turns into a red giant.

Did Zeno kill Alice or did the sun kill Alice? Both: Zeno kills Alice by shifting her future life into a spatiotemporal position where that life would be destroyed by the sun. This is akin to sending Alice now into the sun on a speeding rocket.

(I am not a lawyer, but I expect Zeno could only be convicted of attempted murder, since a conviction for murder requires the victim to be dead; similarly, I assume that an 80-year-old person who gives someone a poison that takes forty years to work can only be convicted of attempted murder, because by the time the poison does its work, the murderer will be dead.)

But now imagine that Zeno lives in a universe where the earth will be habitable forever. He sets up an automated system that slows down Alice’s internal time to such a degree that in the first year of external time, Alice’s internal time moves ahead only 3 seconds; in the next external year, it moves ahead by 1.5 seconds; in the next year, it moves ahead by 0.75 seconds; and so on. What happens? Well, Alice still cannot have more than six seconds of life ahead of him. In n years of external time, she will have had 6 − 6/2ⁿ seconds of internal time.

So just as in the first scenario, Zeno has ensured that Alice has less than six seconds of internal time left. It sure sounds like murder. But wait! In the second scenario, it seems that Alice never dies: she is alive this year, just sluggish; she will be alive next year, though even more sluggish; and so on.

But Alice will be dead in exactly six seconds of internal time. So what will be the cause of death? The unfortunate misalignment between Alice’s internal time and the external time of the universe, together with the universe running out of time “once year ω rolls around”? Maybe. I am not sure. This is paradoxical.

There is a way of getting out of this paradox. Suppose internal time must be discrete. Then to slow down Alice’s time means to space out the discrete ticks of her time. Suppose for simplicity that Alice has a hundred ticks per internal second. Then in the next year, she will have 300 ticks. Some time in year ten, the 599th tick of Alice’s future life happens. And the 600th tick will never happen. So, the gradual slowdown story is is impossible. The speed hits zero after the tenth year. The best (or worst?) Zeno can do is ensure that the 599th tick of Alice’s life is the last one. But if that’s what he does, then he causes her death by ensuring that the 600th tick never happens. But if that’s what he does, there is no gradual slowdown paradox.

Movement

The following seem quite plausible:

1. It is possible for an object both (a) to have both a first and a last moment of its existence and (b) to be moving at every time during its existence.
2. It is not possible for an object (a) to exist at only one time and yet (b) be moving.

By (2), movement is not an instantaneous property: it is not a property an object has solely in virtue of how it is at one moment. By (1), however, movement is not a property defined in terms of the past and present states of an object (say, "an object moves at a time provided that it is a different location from where it was in the past"), since it can move at the first moment of its existence; nor is it a property defined in terms of the present and future states of the object since it can move at the last moment of its existence.

So what is movement? We could say that an object is moving at time t provided that there are arbitrarily close moments t* at which the object is in a different location. This would make sense of both (1) and (2). But this account falsifies the following intuition:

3. If a ball is thrown vertically into the air, then at the high point of its flight it is not moving.

(If it were moving, would it be moving upward or downward?) For at moments arbitrarily close to that top-point time, the ball is at different locations.

We could try to define movement in terms of there being a well-defined non-zero derivative of the position with respect to time, with the derivative being one-sided at the beginning and end of the object's existence. But then, given continuous time (which we need anyway to have time-derivatives), an object could continuously change location without ever moving, since there are continuous nowhere differentiable functions.

So what should we say? I think it is that the concept of "moving at time t" is underspecified, and specifications of it simply aren't going to cut nature at the joints. Being at different places at different times (at least relative to a reference frame) makes good and fairly precise sense. But moving (or changing) at a time does not. Zeno was right about that much.

A funny discrete view of time

A number of my posts are exercises in philosophical imagination rather than serious philosophical theories. These exercises can have several benefits, including: (a) they're fun, (b) they expand the range of possibilities to think about and thus might contribute to a new and actually promising approach, and (c) they potentially contribute to philosophical humility by making us question whether the views that we take more seriously are actually better supported than these. This is one of those posts.

Suppose that time is discrete and made up of instants. However instead of saying that always some instant is present, we now allow for two possibilities. Sometimes an instant is present. But sometimes presently we are between instants. When an instant is present, there is a present moment. When an instant is not present, when we are between instants, there is a present interval, bounded by the last past instant and the first future instant.

Why posit that sometimes we are between instants? Because this lets us get out of Zeno's paradox of the arrow. Zeno notes that at no instant is the arrow moving, because at no instant does it occupy two places, and so the arrow never moves. But now that we have two possibilities, that of an instant being present and of an interval being present, we see that Zeno's inference from

1. At no instant is the arrow moving

to

2. The arrow never moves

uses the implicit assumption that we are always at an instant. But if sometimes instead of being at an instant we are between them, we are at an interval, then the inference fails. And indeed when instead of a present moment we have a present interval, we can say that the arrow really is moving in the present—it is in two places in the present, in one place in the last past instant and in another in the first future instant.

So we have positions when an instant is present and velocities when an interval is present.

Of course there are other ways out of the Zeno paradox of the arrow, the best of which is to adopt the at-at theory of motion. But it's nice to have other solutions besides the usual ones.

Antipresentism

Presentists think that the past and future are unreal but the present is real. I was going to do a tongue-in-cheek post about an opposed view where we have the past and future but no present. But as I thought about it, the position grew a little on me philosophically, at some expense of the tongueincheekness. Still, please take all I say below in good fun. If you get a plausible philosophical view out of it, that's great, but it's really just an exercise in philosophical imagination.

One way to think about antipresentism is to imagine the eternalist's four-dimensional universe, but then to remove one slice from it. Thus, we might have 1:59 pm and 2:01 pm, but no 2:00 pm. Put that way, the view isn't particularly attractive. Still, I do wonder why it would be more unattractive to remove just one time slice than to remove everything but that one time slice as the presentist does. It would, of course, be weird for the antipresentist to say that events first exist in the future, then pop out of existence just as one would have thought that they would come to be present, and then pop back into existence in the past. But perhaps no weirder than events coming out of nothing and going back into nothing, as on presentism. This way to think about antipresentism makes it a species of the A-theory.

But the antipresentisms I want to think about are ones that might be compatible with the B-theory. Start with the famous puzzles of Zeno and Augustine about the now. Augustine worried about the infinite thinness of the now. Zeno on the other hand worried about the fact that there are no processes in the now; there is no change in the now since within a single moment all is still.

One way of taking these ideas seriously is to see the present as an imaginary dividing line between the past and the future. There is in fact no dividing line: there is just the past and the future. (I think Joseph Diekemper's work inspired this thought.)

We might, for instance, instead of thinking of times as instants think of the basic entities as temporally extended events or time intervals, not made out of instantaneous events or moments. An event or interval might be past, or it might be future, or—like the writing of this post—it might be both past and future. (Thus, "past" and "future" is taken weakly: "at least partly past" and "at least partly future".) Some events or time intervals have the special property of being both past and future. We can stipulate that those events or time intervals are present. But they aren't real because they are present. They're just lucky enough to have two holds on reality: they are past and they are present. (In this framework, the presentist's claim that only present events are real sounds very strange. For why should reality require both pastness and futurity—why wouldn't one be enough?) There are no events or time intervals that are solely present.

There is a natural weakly-earlier-than relation e on events. If we had instants of time, we would say that EeF if and only if some time at which E happens is earlier than some time at which F happens. But that's just to aid intuition. Because there are no instantaneous events, every event is weakly earlier than itself: e is reflexive. It is not transitive, however. The antipresentist theory I am sketching takes e to be primitive. There is also a symmetric temporal overlap relation o that can be defined in terms of e: EoF if and only if EeF and FeE.

If we like, we can now introduce abstract times. Maybe we can say that an abstract time is a maximal pairwise overlapping set of time intervals (or of events, if we prefer). We can say that t₁ is earlier than t₂ provided that some element of t₁ is strictly earlier than some element of t₂ (where E is strictly earlier than F provided EeF but not FeE). I haven't checked what formal properties this satisfies—I need to get ready for class now (!).

A weakness of eliminative materialism

I was telling my kids about eliminative materialism, the view that there are only material objects, and that there are no minds, persons, beliefs, perceptions, etc. My kids are used to hearing about nutty philosophical views, such as those of Zeno, but they noticed that the standard tool in defending wacky philosophical views is unavailable here. For while Zeno can say that motion is an illusion, eliminative materialists can't say that thought is an illusion. For illusions are among the things the eliminative materialist eliminates. I hadn't noticed this.

Zeno's arrow, Newtonian mechanics and velocity

Start with Zeno's paradox of the arrow. Zeno notes that over every instant of time t₀, an arrow occupies one and the same spatial location. But an object that occupies one and the same spatial location over a time is not moving at that time. (One might want to refine this to handle a spinning sphere, but that's an exercise to the reader.) So the arrow is not moving at t₀. But the same argument applies to every time, so the arrow is not moving, indeed cannot move.

Here's a way to, ahem, sharpen The Arrow. Suppose in our world we have an arrow moving at t₀. Imagine a world w* where the arrow comes into existence at time t₀, in exactly the same state as it actually has at t₀, and ceases to exist right after t₀. At w* the arrow only ever occupies one position—the one it has at t₀. Something that only ever occupies one position never moves (subject to refinements about spinning spheres and the like). So at w* the arrow never moves, and in particular doesn't move at t₀. But in the actual world, the arrow is in the same state at t₀ as it is at w* at that time. So in the actual world, the arrow doesn't move at t₀.

A pretty standard response to The Arrow is that movement is not a function of how an object is at any particular time, it is a function of how, and more precisely where, an object is at multiple times. The velocity of an object at t₀ is the limit of (x(t₀+h)−x(t))/h as h goes to zero, where x(t) is the position at t, and hence the velocity at t₀ depends on both x(t₀) and on x(t₀+h) for small h.

Now consider a problem involving Newtonian mechanics. Suppose, contrary to fact, that Newtonian physics is correct.

Then how an object will behave at times t>t₀ depends on both the object's position at t₀ and on the object's velocity at t₀. This is basically because of inertia. The forces give rise to a change in velocity, i.e., the acceleration, rather than directly to a change in position: F(t)=dv(t)/dt.

Now here is the puzzle. Start with this plausible thought about how the past affects the future: it does so by means of the present as an intermediary. The Cold War continues to affect geopolitics tomorrow. How? Not by reaching out from the past across a temporal gap, but simply by means of our present memories of the Cold War and the present effects of it. This is a version of the Markov property: how a process will behave in the future depends solely on how it is now. Thus, it seems:

1. What happens at times after t₀ depends on what happens at time t₀, and only depends on what happens at times prior to t₀ by the mediation of what happens at time t₀.

But on Newtonian mechanics, how an object will move after time t₀ depends on its velocity at t₀. This velocity is defined in terms of where the object is at t₀ and where it is at times close to t₀. An initial problem is that it also depends on where the object is at times later than t₀. This problem can be removed. We can define the velocity here solely in terms of times less than t₀, as lim_h→0−(x(t+h)−x(t))/h, i.e., where we take the limit only over negative values of h.[note 1] But it still remains the case that the velocity at t₀ is defined in terms of where the object is at times prior to t₀, and so how the obejct wil behave at times after t₀ depends on what happens at times prior t₀ and not just on what happens at t₀, contrary to (1).

Here's another way to put the puzzle. Imagine that God creates a Newtonian world that starts at t₀. Then in order that the mechanics of the world get off the ground, the objects in the world must have a velocity at t₀. But any velocity they have at t₀ could only depend on how the world is after t₀, and that just won't do.

Here is a potential move. Take both position and velocity to be fundamental quantities. Then how an object behaves after time t₀ depends on the object's fundamental properties at t₀, including its velocity then. The fact that v(t₀)=lim_h→0(x(t₀+h)−x(t₀))/h, at least at times t₀ not on the boundary of the time sequence, now becomes a law of nature rather than definitional.

But this reneges on our solution to The Arrow. The point of that solution was that velocity is not just a matter of how an object is at one time. Here's one way to make the problematic nature of the present suggestion vivid, along the lines of my Sharpened Arrow. Suppose that the arrow is moving at t₀ with non-zero velocity. Imagine a world w* just like ours at t₀ but does not have any times other than t₀.[note 2] Then the arrow has a non-zero velocity at t₀ at w*, even though it is always at exactly the same position. And that sure seems absurd.

The more physically informed reader may have been tempted to scoff a bit as I talked of velocity as fundamental. Of course, there is a standard move in the close vicinity of the one I made, and that is not to take velocity as fundamental, but to take momentum as fundamental. If we make that move, then we can take it to be a matter of physical law that mlim_h→0(x(t₀+h)−x(t₀))/h=p(t₀), where p(t) is the momentum at t.

We still need to embrace the conclusion that an object could fail to ever move and yet at have a momentum (the conclusion comes from arguments like the Sharpened Arrow). But perhaps this conclusion only seems absurd to us non-physicists because we were early on in our education told that momentum is mass times velocity as if that were a definition. But that is definitely not a definition in quantum mechanics. On the suggestion that in Newtonian mechanics we take momentum as fundamental, a suggestion that some formalisms accept, we really should take the fact that momentum is the product of mass and velocity (where velocity is defined in terms of position) to be a law of nature, or a consequence of a law of nature, rather than a definitional truth.

Still, the down-side of this way of proceeding is that we had to multiply fundamental quantities—instead of just position being fundamental, now position and momentum are—and add a new law of nature, namely that momentum is the product of mass and velocity (i.e., of mass and the rate of change of position).

I think something is to be said for a different solution, and that is to reject (1). Then momentum can be a defined quantity—the product of mass and velocity. Granted, the dynamics now has non-Markovian cross-time dependencies. But that's fine. (I have a feeling that this move is a little more friendly to eternalism than to presentism.) If we take this route, then we have another reason to embrace Norton's conclusion that Newtonian mechanics is not always deterministic. For if a Newtonian world had a beginning time t₀, as in the example involving God creating a Newtonian world, then how the world is at and prior to t₀ will not determine how the world will behave at later times. God would have to bring about the initial movements of the objects, and not just the initial state as such.

Of course, this may all kind of seem to be a silly exercise, since Newtonian physics is false. But it is interesting to think what it would be like if Newtonian physics were true. Moreover, if there are possible worlds where Newtonian physics is true, the above line of thought might be thought to give one some reason to think that (1) is not a necessary truth, and hence give one some reason to think that there could be causation across temporal gaps, which is an interesting and substantive conclusion. Furthermore, the above line of thought also shows how even without thinking about formalisms like Hamiltonian mechanics one might be motivated to take momentum to be a fundamental quantity.

And so Zeno's Arrow continues to be interesting.

The Grim Reaper Paradox

Here is a version of the Grim Reaper paradox. Say that a Grim Reaper is a being that has the following properties: It wakes up at a time between 8 and 9 am, both exclusive, and if you're alive, it instantaneously kills you, and if you're not alive, it doesn't do anything.[note 1] Suppose there are countably infinitely many Grim Reapers, and before they go to bed for the night, each sets his alarm for a time (not necessarily the same time as the other Reapers) strictly between 8 and 9 am. Suppose, also, that no other kind of death is available for you, and that you're not going to be resurrected that day.

Then, you're going to be dead at 9 am, since as long as at least one Grim Reaper wakes up during that time period, you're guaranteed to be dead. Now whether there is a paradox here depends on how the Grim Reapers individually set their alarm clocks. Suppose now that they set them in such a way that the following proposition p is true:

(p) for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t.

Here's a useful Theorem: If the Grim Reapers choose their alarm clock times independently and uniformly over the 8-9 am interval, then P(p)=1.

Now, if p is true, then no Grim Reaper kills you. For suppose that a Grim Reaper who wakes up at some time t₁, later than 8 am, kills you. If p is true, there is a Grim Reaper who woke up strictly between 8 am and t₁, say at t₀. But if so, then you're going to be dead right after t₀, and hence the Grim Reaper who woke up at t₁ is not going to do anything, since you're dead then. Hence, if p is true, no Grim Reaper kills you. On the other hand, I've shown that it is certain that a Grim Reaper kills you. Hence, if p is true, then no Grim Reaper kills you and a Grim Reaper kills you, which is absurd.

The above argument shows that some arrangements of Grim Reaper alarm clock times, namely the ones that make p be true, are impossible, because they result in your being dead and not dead at the same time. But no such objection can be made to other arrangements of Grim Reaper alarm clock times. For instance, if Grim Reaper 177 wakes up at 8:05 am, and all the other Grim Reapers happen to wake up later, there is no difficulty--Number 177 kills you, and you're dead at 9 am.

Now we have a trilemma. Either all mathematical combinations of Grim Reaper alarm clock times strictly between 8 and 9 am are possible in the above story, or some but not all, or none (in the last case, the story above is impossible whatever the times are). The hypothesis that some but not all are possible seems unlikely. Look: it's midnight, say, and we have all of these Grim Reapers setting their alarm clocks. It would be really, really odd if they were somehow compelled by the metaphysics of the situation to set their times in one of the privileged ways, unless it turns out that there are only finitely many moments of time between 8 and 9 am, so that p cannot be true. (Indeed, by the Theorem given above, these privileged ways of setting times are very unlikely if the Reapers are choosing independently, assuming that all real-numbered times between 8 and 9 am exist, which the Theorem assumes.) That leaves two hypotheses: That all the combinations are possible or none. If all the combinations are possible, so will be the ones that make p true (e.g., Reaper 1 waking up at 8:30:00, Reaper 2 at 8:15:30, Reaper 3 at 8:07:30, Reaper 4 at 8:03:45, and so on). And that's not possible.

So either there are only finitely moments of time between 8 and 9 am, or no combination of Grim Reaper alarm clock settings is possible. In the latter case, it basically follows that it's just impossible to have infinitely many Grim Reapers, whether their wakeup times are arranged so as to result in a paradox or not. So why can't there be infinitely many Grim Reapers? It seems that the only reason to suppose there can't be infinitely many Grim Reapers, even in cases where no paradox is generated, is if one thinks there can't be an actual infinity of objects in existence. And if there can't be an actual infinity of objects in existence, then there can't be an actual infinity of times in the past, since if there were an actual infinity of times, surely a new object could come into existence at each of those times.

So either there are only finitely moments of time between 8 and 9 am, or there are only finitely moments of time in the past. But if there are only finitely many moments of time in the past, there were only finitely many moments of time yesterday between 8 and 9 am, and today is no different. So in either case, a bounded interval of times contains only finitely many moments.

I am not fully convinced by this argument, but I don't have a very good response.

[This post is revised. I am grateful to Bill Craig for pointing out some sloppiness in the original.]