In the second version, particle 0 moves off the plane and coincides in location with particle 1. Particle 0 stays there, and particle 1 moves to the plane and then off the plane, ending up in the starting-position of particle 2. Particle 2 then moves to and from the plane, and so on. In this version, every particle n ends up in the starting-position of particle n+1, and in the end there is no particle on the plane. Yet, the two versions of the super-task are exactly the same except for the question of which of two indistinguishable particles moves away from their common position at each phase of the task. Intuitively, this kind of difference cannot make a substantive difference to the end-state, that is, whether or not there is any particle at all on the plane at the end of the process.
The most plausible explanation of the impossibility of the Super-Urn task is Temporal Finitism: it is impossible to carry out any super-task in a finite period of time, because no finite period of time can have infinitely many actual temporal parts.
PRUSS'S GRIM REAPER PARADOX Alexander Pruss (2009) has posed the following version of the Grim Reaper paradox (Benardete 1964, Hawthorne 2000) as an argument for the discrete character of time. We are to suppose that there are an infinite number of Grim Reaper mechanisms, each of which is engineered to do two things. First, each checks whether the victim, Fred, is still alive at the Grim Reaper's appointed time. Second, if he is still alive, each Reaper kills him instantaneously. The last Grim Reaper, Reaper 1, performs this dual task at exactly one minute after noon. The next-to-last Reaper, Reaper 2, is appointed to perform the task at exactly one-half minute after noon. In general, each Reaper number n is assigned the moment 1/n minute after noon. There is no first Reaper: for each Reaper n, there are infinitely many Reapers who are assigned moments of time earlier than Reaper n's appointment.
It is certain that Fred does not survive the ordeal. In order to survive, he must still be alive at one minute after 12 p.m., but we have stipulated that, if he survives until 12:01 p.m., then Reaper 1 will kill him. We can also prove that Fred will not survive until 12:01, since in order to do so, he must be alive at 30 seconds after 12, in which case Reaper 2 will have killed him. In the same way, we can prove that Fred cannot survive until 1/n minutes after 12, for every n. Thus, no Grim Reaper can have the opportunity to kill Fred. Thus, it is impossible that Fred survive, and also impossible that any Reaper kill him! However, it seems also to be impossible for Fred to die with certainty and yet without any cause.
If one worries that this paradox depends somehow on the vagueness of the life/death distinction, consider the following variant: the Grim Mover. In this case, we have a particle that is located exactly on a plane. At 12:01 p.m., Mover 1 will move the particle off the plane exactly one meter, if it hasn't already been moved. If it has already been moved then, he does nothing. Mover 2 is primed to perform the task of moving the particle one-half meter away from the plane at one-half minute after noon, if it hasn't already been moved. And so on. We can now prove that the particle is moved off the plane after noon, even though none of the Movers has moved it. Even worse, it must have been moved off the plane, even though there is no finite distance that it has been moved, since every particular distance corresponds to exactly one Mover!
The whole set-up must be metaphysically impossible. We can use Infinite Patchwork (PMeta 5.2) to turn this impossibility into a positive argument for Temporal Finitism (Koons 2014a). We can build a reductio ad absurdum of the hypothesis that any finite interval is divisible into an infinite number of sub-intervals. We will use an even simpler version of the paradox: the paradox of the Grim Signaler. Each Signaler has a unique natural number n, and each is capable of sending a signal representing that number n to its successor. Each Signaler #n is built in such a way that (i) if it receives an appropriate signal (of some number m > n) at its appointed time, then it simply transmits that signal to its successor, Signaler #(n+1), and (ii) if it does not receive an appropriate signal at its appointed time, then it sends a signal representing the number n to its successor. Here is the reductio:
Assume for contradiction that it is possible for there to exist a finite temporal interval that is divided into an infinite number of sub-intervals, with a last sub-interval but no first sub-interval.
It is possible to build a Grim Signaler with the disposition to respond to a signal from its predecessor and to send a signal to its successor in the specified manner.
The specification of each Signaler is intrinsic to its situation (that is, the passive and active powers that are attributed to it are intrinsic to it during its interval of activity).
Assume Infinite Patchwork (PMeta 5.2).
It is possible for there to exist an infinite series of Signalers, with a final Signaler but no first one.
At least one number n is such that Signaler #n initiated a signal representing n. If we assume that no Signaler sent such a signal, then no Signaler #m with m > 1 would have done so. On this assumption, Signaler #1 would have sent a signal representing 1 to its successor, a contradiction.
If Signaler #n sent a signal representing n, then there is no m > n such that Signaler #m sent a signal representing m.
So, there is no m> n+1 such that Signaler #(n+1) sent a signal representing n+1.
So, Signaler #(n+1) would have sent a signal representing n+1. But n+1> n, contradicting (7).
Hence, there cannot be a beginningless infinite series of sub-intervals.
In fact, the Grim Signaler paradox suggests not only that no finite time period can be divided into infinitely many sub-periods but also that it is impossible that there should exist infinitely many time periods at all. It seems to provide grounds for thinking that time must be bounded at the beginning: there must be a first period of time.
Can we also show that time is bounded in the future, that there will be a last period of time? Apparently not. The only way to construct the Grim Signaler paradox in reverse would be to stipulate that each Signaler is able to check whether or not any future Signaler has initiated its characteristic signal. The apparent connections between time, knowledge, and action all seem to rule out the possibility of such a paradox, without requiring any limitations concerning the end of time.
One important proviso. The Grim Reaper/Signaler arguments depend on the assumption that the active and passive powers attributed to each Reaper or Signaler is intrinsic to that Reaper or Signaler during its assigned period of activity. This intrinsicality assumption is needed in order to apply the Infinite Patchwork (PMeta 6.2). That principle states only that an intrinsically described possible situation can be infinitely duplicated within a possible temporal structure. Situations that are not intrinsically described may be incapable of duplication. For example, consider a situation that is described in the following way:
(1) The First Reaper. Reaper n is the first Reaper to swing his scythe.
It seems possible that a situation of type (1) could occur: we can imagine a world in which Reaper n is the first one to swing his scythe. However, it would be fallacious to try to apply a patchwork principle to (1) in order to conclude that there could be an infinite series of Reapers each of which is the first one to swing his scythe. Why is this fallacious? Because to describe a Reaper as the first one to swing his scythe is not to describe the Reaper intrinsically. Instead, it is to describe the relation between Reaper n and a large number of extrinsic situations.
Thus, to apply Infinite Patchwork (PMeta 5.2) to the Grim Signaler story, we have to assume that the description we gave of the Signalers' dispositions to receive and send signals are intrinsic to each of them. This involves the Intrinsicality of Powers (PMeta 2), the claim that having a power is an intrinsic property of a thing. As we saw in Chapters 4 and 5, Powerists (4.4A.3) have good reason to accept this principle, while Neo-Humeists (4.4T) will reject it. For Neo-Humeists, whether one of the Grim Signalers has a certain power or not depends on the actual history of the whole world, including the actions and inactions of all of the other Grim Signalers. Neo-Humeist
s will conclude, not that an infinite regress of times is impossible, but merely that it is impossible for an infinite series of Signalers, arranged in the way described, to have the powers we ascribed to them. Neo-Humeists feel no pressure from Infinite Patchwork (PMeta 5.2) to think otherwise, since for them powers are not intrinsic.
19.2 Instants as Dependent Entities
In this section, we will assume for the sake of argument that Intervalism is true, that is, that some temporally extended intervals and processes are among the world's fundamental entities. Given that assumption, we can then ask if instants or instantaneous events are among the fundamental entities of the world, or if all fundamental entities are finite in duration. Strong Intervalism denies that instants are fundamental, while Moderate Intervalism, or Interval-Boundary Dualism, embraces the fundamentality of both instants and intervals.
19.1A.1T Strong Intervalism. Instants either don't exist at all or are derived entities—mere logical constructions from finite intervals.
19.1A.1A Interval-Instant Dualism (Moderate Intervalism). Both instants and intervals are fundamental entities.
The burden of proof is on Moderate Intervalists, since Strong Intervalism is a simpler, more economical theory.
Is the present, the “now”, an instant or an interval? William James (1890) introduced into philosophy the psychological notion of the specious present: the present experienced as encompassing a short interval of time, some fraction of a second. Do we really have no experience of the present moment as an instant? What about the leading edge of the specious present? Do we have an experience of that edge as a dimensionless surface? Or, do we experience a present instant sweeping through the specious present? If we didn't, how would we be able to distinguish between earlier and later parts of that specious present?
Aristotle argued that we must suppose the present to be a dimensionless instant, since it is the boundary between the future and the past. It is impossible for a whole interval to be that boundary, since the earlier parts of such an interval would have to be already in the past. To resist Aristotle's argument, Strong Intervalists have to maintain that the specious-present interval has no temporal parts at all, no earlier or later. How, then, can it include motion and change? If a ball rolls down a ramp during the specious present, won't there have to be an earlier part of that interval during which it is on the top half of the ramp? This certainly seems to be the case. Denying the veridicality of such experiences is a theoretical cost to Strong Intervalists who appeal to the specious present.
A better tack for Strong Intervalists might be to locate the fundamental intervals at a very small scale, a matter of nanoseconds.
The principal advantage of Strong Intervalism is that it enables us to give answers to Zeno's paradox of the arrow and to the paradox of death. The paradox of the arrow points out that the arrow in flight does not move anywhere during each instant of flight. During each instant, it has exactly one precise location, and so is (in a sense) completely at “rest” “during” the instant itself. However, if the flight of the arrow merely consists in the sum of the arrow's state at each instant during the flight, then the arrow is motionless throughout its flight. How can the arrow move during this period if it is at rest during every instant of the period? The paradox of death starts with the simple question: is one alive or dead at the very instant t of one's death? If we say that one is dead at t, then it seems that t cannot be the instant of death, since one is already dead at t, and someone who is already dead cannot die. Alternatively, if we say that one is alive at t, then once again it seems that t cannot be the moment of death, since one is still alive at t, and so one has not yet died at t.
Strong Intervalists can respond to both paradoxes. The flight of the arrow is not made up of momentary (and stationary) instants: rather, each part of the arrow's flight occupies a finite interval of time, during which the arrow covers some distance. Similarly, there is no instant of death, and so no issue about whether one is then alive or dead. The “instant” of death for Strong Intervalists is merely a logical construction built up from finite intervals. For example, we could define the instant of death as the set containing all final intervals of the person's life, that is, as the set containing all intervals throughout which the person is alive, and which are not succeeded by a still later period of life.
However, there is a clear analogy between instants and spatial boundaries like surfaces, curves, and points. Just as it seems plausible that extended things have superficial (zero-, one-, and two-dimensional) boundaries, so it seems plausible that temporally extended events have instants as initial and final boundaries. Moderate Intervalists can embrace this view, Instants as Dependent Entities:
19.1A.1A.1 Instants as Dependent Entities. Instants exist only when they are the actual boundaries of extended processes or events.
What about Zeno's paradoxes? Can Moderate Intervalists dissolve them? Yes, because they do not claim that all intervals are ultimately composed of instants in a strong, metaphysical sense of composition, that is, one according to which the instants are supposed to be actual and independent parts. Instead, instants are dependent parts in that they are actual and potential boundaries of extended processes. Processes, like motion, do not derive their properties from their instantaneous parts. If anything, the dependency goes the other way around, from parts to wholes. Thus, Moderate Intervalists are not bothered by the fact that the arrow is not moving “within” any single instant. Instants aren't the fundamental units of motion; intervals are.
In the case of the paradox of death, the most plausible response for Moderate Intervalists may be to suppose that there can be more than one simultaneous instant, just as there can be more than one spatially coincident surface. In this case, there would be two, simultaneous instants of death. One is the final boundary of one's life, and the second is the initial boundary of one's death.
19.3 Does Time have a Beginning?
The Grim Reaper or Grim Signaler paradox that we considered in Section 19.1 gives us reason to believe that time must have a first period or interval. Does it follow then that time must have a beginning?
19.5T Beginning of Time. Time necessarily has a beginning.
The Beginning of Time could be understood in one of two ways:
19.5T.1 Existence of a First Temporal Part. Time necessarily has a single part that is earliest.
19.5T.2 Metrical Finitude of the Past. The past is necessarily finite in duration or measure.
The Grim Reaper paradox gives us a direct argument for the Existence of a First Temporal Part, but not for the Finitude of the Past. There is a gap between the two. Imagine a universe in which time begins with a single, undivided but infinitely long period of time. Let's call such a period an ‘infinite simple past’. To get from the Existence of a First Temporal Part to the Metrical Finitude of the Past, we would need to assume that an infinite simple past is impossible:
No Simple Infinite Past. It is impossible for a temporal interval to extend infinitely far (in duration) into the past unless (i) it actually has infinitely many proper parts or (ii) it overlaps with an infinite number of disjoint intervals or events.
It seems reasonable to assume that for a simple region to have measurable temporal duration without parts, the simple region would have to either contain a process with a natural beginning and end or temporally overlap with one or more such processes. Thus, we can reasonably embrace the possibility of simple regions with finite duration, a duration corresponding to the natural distance between the two endpoints in processes of this kind. However, a simple region with an infinite duration in the past would have to contain only processes without a natural beginning, and we might well ask how any such process could have a temporal measure, without having actual proper parts or overlapping in time with other regions. Time is the measure of change, which seems to require both a terminus a quo and a terminus ad quem, a starting point and an end point for the change. This assumes, of course, that time has no intrinsic met
ric of its own.
Here's another argument for the principle No Simple Infinite Past. A simple region can have a temporal measure only if it is at least potentially divisible into parts. A region is divisible into temporal parts only if it contains one or more processes that can potentially be stopped or interrupted. A process P is potentially stoppable only under certain conditions:
P itself has a natural, finite measure, based on the normal distance in time between its terminus a quo and terminus ad quem, a measure that can be shortened by accelerating P, or
There is another process P2 that, when it reaches its terminus ad quem, has the power of terminating P, and P2 is stoppable before the termination of P.
However, if the early history of the world consisted entirely of processes without finite measures, then none of those processes would be potentially stoppable, and hence none of the temporal intervals containing them would be even potentially divisible. Intervals that contain no temporal parts at all, whether actual or potential, and that temporally overlap only other regions without temporal parts cannot have a temporal measure. Hence, a simple past must be a quantitatively finite past.
Here's a third argument against a simple infinite past. Either there is an intrinsic metric to the pure passage of time or not. If there is, then the infinite past is actually divided, in and of itself, into an infinite number of actual periods. This contradicts the conclusion of the Grim Reaper paradox. If there is no intrinsic measure of time, then a period of time with no actual parts cannot have any measure unless it contains both the beginning and end of a process with an intrinsic duration.
The Atlas of Reality Page 69
