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John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
The supertasks we have considered so far involve an acceleration of the actions performed such that an infinity is only completed once the entire set has been carried out. A reversed supertask reverses this order. In them, in order to get started, an infinity of tasks has to be completed before time has advanced by any amount at all. We have seen one example of this reversal in the reversed dichotomy of Zeno's paradoxes. To advance any distance at all, the runner must pass an infinity of the points of the paradox.
Ordinary supertasks produce problems that can tax
our analytic abilities. Reversed supertasks do so again and present their
own peculiar problems.
This reversed supertask is one of the earliest in the modern tradition. It starts out as a normal supertask, with the infinite complication arising at the end of a motion. In the version developed here, we have two trains approaching each other on a track. A bird flies to and fro between the fronts of the trains until the trains meet. We make the obvious idealizations. The bird flies at constant speed, as do the trains; and the bird flies faster than the trains. The bird can instantly reverse its direction.
It is easy to see that the bird will execute a supertask. The bird will take successively shorter times for each transit between the fronts of the trains; and will execute infinitely many transits to complete the motion. Here is how the motion appears in a spacetime diagram.
To see that an infinity of reversals are needed, imagine otherwise. That is, for purposes of reductio, imagine that some transit--say the tenth from the left to right--is the last one, so that the motion is completed with only finitely many transits. Each transit takes some non-zero time to complete. So, if the tenth is the last transit before the meeting point, then for some finite time prior to meeting, the bird flies from left to right towards the meeting point. The train from the left must also arrive at the meeting point. Since it moves more slowly than the bird, the bird must lag behind the train on its last transit. (This motion is shown in red in the spacetime diagram below.) That contradicts the assumption that the bird always reverses direction when it meets the front of an on oncoming train. It can only lag behind the train if it failed to turn when aligning with the train's front. We have a contradiction and the reductio is complete. Hence the bird's motion cannot be completed in finitely many transits.
We have a familiar supertask problem: in which direction is the bird flying at the meeting point?
•It cannot be to the right, since each
right-directed transit is followed by a left-directed transit.
•It cannot be to the left, since each left-directed transit is followed by a right-directed transit.
The answer, of course, is that the sequence of directions:
right, left, right, left, right, left, ...
is infinite and approaches no single direction as a limit. So that sequence cannot give us any direction as its limit. As far as considerations of this limit are concerned, the bird's direction could be to the left or to the right or nothing at all.
This circumstance is similar to the Thomson lamp. The specification of the infinite sequence of on, off, on, off, ..., is insufficient to specify the state of the lamp once the sequence completes.
We could just stop there and treat the problem as akin to the Thomson lamp. The lamp state or bird direction could be on/off or left/right/neither. However in this case we are going to assume that the bird keeps flying, still following the prescription that it reverses course each time it aligns with the front of the train.
This continued motion is the reversed supertask. A motion that fits this description is shown in the figure below. It is just the reverse of the motion of the bird leading up to the meeting point:
With this extra prescription, this supertask is unlike the Thomson lamp. At the meeting point, the bird cannot have the property of flying to the left or of flying to the right. Its motion can carry no directional property at all. The reason is that a direction property cannot be assigned to a single position of the bird, considered in isolation. Such an assignment necessarily involves considerations of the positions of the bird at neighboring times. These positions are such that no direction assignment can be made.
The reason for this failure is straightforward, but
a little tedious to lay out. Call the time at which the bird and trains
meet "T." For the bird to be moving right at the meeting point, there must
be some sufficiently small time interval
surrounding T such that
• For all times earlier than T in the time interval, the bird's position is to left of the meeting point; and
• For all times after T in the time interval, the bird's position is to the right of the meeting point.
and this condition continues to hold for all smaller intervals of time containing T within this small interval. Such a motion, with the relevant time interval shaded, looks like this:
As the figure shows, this motion is only possible if the bird flies slower than the trains. By stipulation, the bird is always flying faster. At the bird approaches the meeting point, it switches direction ever more rapidly to the right and the left of the meeting point. At no time earlier to that of the meeting does this wild oscillatory motion become confined either to the right or to the left of the meeting point. Hence the condition above cannot be met; no direction property can be assigned to the bird's motion at the meeting point.
There is a further question that arises in this supertask: after the bird passes the meeting point and continues flying, where does the bird end up? If the motion leading up to the meeting point took two minutes, where will the bird be two minutes after it has passed the meeting point?
• Will it be in the same position as when it
• Or will in be in that same position, but reflected to the other side of the meeting point as shown in the figure above?
The answer is that that the bird could be anywhere between the two trains.
This follows from an obvious fact about the possible motions of the bird in the forward supertask, prior to the meeting point. It does not matter where between the trains the bird starts. The bird will always end up at the meeting point. Here are two birds starting at different places between the trains. They end up at the same meeting point, as must all birds whose motions starts between the two trains:
We can take any one of these motions and reverse it in time to produce a motion that recedes from the meeting point. That is, it is a motion that the bird can follow after the meeting point. Since the forward motion can start at any point between the trains, then the reversed motion can end at any point between the trains. The figure shows a second possibility (of the infinitely many possible).
The bird and trains supertask gives us a template for forming more reversed supertasks. We take an existing supertask and form its time reversal. Here we see what results in two cases: Black's transfer machine and the case of the many accelerating spaceships.
In Black's transfer machine, a marble is switched back and forth between two trays until infinitely many switches are completed in finite time. The time reverse of this supertask ends up with the marble located in what was its initial position. When we trace its motion into the past, we find it being switched faster and faster between the two trays, with infinitely many switchings completed at a finite time in the past.
We saw that the location of the marble at the end of the forward supertask was indeterminate. It could be anywhere or nowhere at all. Correspondingly the marble in the reversed supertask can be anywhere prior to the start of the supertask or even nowhere at all.
This case of "nowhere at all" for the forward supertask, the disappearing marble, is a simple choice compatible with all the conditions of the supertask. If the forward supertask has the marble disappearing when the task is complete, the reversed supertask would have the marble appearing--materializing--at the start of the supertask.
Here is a spacetime diagram of the forward supertask:
Here is a spacetime diagram of the reversed supertask:
Can this really be so? It can. Recall that the positions of the marble are governed by a simple dynamics identified in the last chapter:
Continuity dynamics: the position of the marble at any given time is the position to which it was switched at the moment of the LATEST SWITCHING PRIOR to the given time.
This continuity dynamics is respected by this reversed motion:
At time 1/2 past
midnight, the marble is in the right tray, since that left to right
switching was the latest switching.
At time 1/3 past midnight, the marble is in the left tray, since that right to left switching was the latest switching.
At time 1/4 past midnight, the marble is in the right tray, since that left to right switch was the latest switching.
....and so on for all the times 1/5, 1/6, 1/7, ... past midnight.
What about midnight itself? There is by construction no switchings prior to midnight in the motion. So continuity dynamics is unable to specify the position of the marble or even if there is a marble at all. The simplest assumption consistent with all the presumptions made so far is that marble does not exist at midnight and all times prior, but that is just appears and exists at all times after than midnight.
The temptation is to say that there is magic afoot. Somehow we have created matter! That goes too far. There is no magic here. Rather we are dealing with a simplified physics that in unable to provide a definite continuation into the past that includes midnight and earlier times. The forward supertask cannot tell us where the marble is at midnight or even if there is a marble. The reversed supertask says the same thing.
In the forward supertask of the many accelerating spaceships, we start with an infinity of spaceships filling up one dimension of space. They accelerate to arbitrarily high speeds in finite time in such a way that, within a finite time, none remain in space. Their disappearance is sudden. Space is filled with rapidly moving spaceships for all times prior to the moment of the end of the supertask. At the end moment, space is suddenly empty.
The spacetime diagram was:
The reversed supertask ends with the infinity of spaceships at rest, filling the one dimension of space. As we look back into the past, we find the spaceships decelerating from very high speeds to come to a halt at the end of the supertask. As we look to earlier times and approach the first moment of the supertask, the spaceships are moving with ever greater speed. At the first moment, however, space is empty. Looking forward again in time, we find that space suddenly fills with spaceships moving at arbitrarily high speeds.
Here is a spacetime diagram of the reversed supertask.
While an infinity of spaceships pouring in from infinity and filling space seems like some magical apparition, there is nothing magical to it. It is just a consequence of assumptions we have made: that the forward supertask is allowed in our physics and the physics is time reversible. If we time reverse an allowed motion, we have another allowed motion.
While we can describe these reversed supertasks and their curious behaviors, we may wonder about their admissibility. That the forward supertask is admissible may not be enough to ensure that the reversed supertask is also admissible. We can understand how spaceships might fire their highly idealized motors ever more powerfully so that they rocket off to infinity. But can we tell an admissible account of how they do that from infinity? Just how are we to imagine the robot arms of Black's transfer machine is to operate in reverse. The marble has no definite position at the start, so what do they grab?
We shall take up this problem in the next chapter.
A reverse supertask paradox was proposed by Benardete in a book on infinity. It has resulted in considerable muddled thinking about strange force fields (that we will avoid here). He describes it at follows.
"A man decides to walk one mile from A to B. A god waits in readiness to throw up a wall blocking the man's further advance when the man has travelled 1/2 mile. A second god (unknown to the first) waits in readiness to throw up a wall of his own blocking the man's further advance when the man has travelled 1/4 mile. A third god ... &c. ad infinitum. It is clear that this infinite sequence of mere intentions (assuming the contrary-to-fact conditional that each god would succeed in executing his intention if given the opportunity) logically entails the consequence that the man will be arrested at point A; he will not be able to pass beyond it, even though not a single wall will in fact be thrown down in his path. The before-effect here will be described by the man as a strange field of force blocking his passage forward.1"
Benardete recognized how odd it seemed to talk of unrealized intentions of gods creating a "strange field of force" blocking the passage. He proceeded to explain that the intentions of the gods are not really what drives the paradox:
"In regard to the paradox of the gods, the oddity here may be somewhat diminished if we replace each god by a law of nature. It is not, after all, the combined intentions of the gods as such which block the man's progress at A. It is rather the following sum-total of hypothetical facts, namely (I) if the man travels 1/2 mile beyond A, then he will be blocked from further progress, (2) if the man travels 1/4 mile beyond A, then he will be blocked from further progress, (3) ... &c. ad infinitum. It is not surprising that this infinite sequence of contrary-to-fact conditionals should logically entail the categorical fact of the man's being arrested at A. For the cause of this arrest is simply the man's encounter with a field of force, and this field of force is simply the physical equivalent of an omnibus law of nature which is compounded out of an infinite sequence of contrary-to-fact conditionals."
The paradox can be sharpened if we replace the man walking from A to B by a smooth boulder rolling slowing down a slight incline. It precludes the possibility that the man's lack of motion is just a prudent hesitation. He might follow Benardete's reasoning and decide that there is no point in proceeding. Boulders do not think and do not hesitate.
We should also note that no god-like powers are needed to throw up the wall at the right moment. We need only assume that the ball touches a switch just before a wall; and that touching the switch releases a powerful spring that hurls the wall upward.
I will also replace Benardete's sequence of 1/2, 1/4, 1/8,... by 1/2, 1/3, 1/4,.. since it simplifies the presentation and does not alter anything essential.
The paradox results from the following consideration. A boulder is set to roll along an incline from A to B. It is governed by the ordinary physics of bodies that I have labeled "contact mechanics." In it bodies are only deflected from their courses by contact with other bodies.
(a) Contact mechanics: The boulder will continue to roll towards B unless an obstacle obstructs it. It stops at the first obstacle it meets.
The gods will throw up walls to block its progress according to:
(b) Wall physics:
• If the boulder approaches the 1/2 mile point, a god with throw up a wall at the 1/2 mile point.
• If the boulder approaches the 1/3 mile point, a god with throw up a wall at the 1/3 mile point.
• If the boulder approaches the 1/4 mile point, a god with throw up a wall at the 1/4 mile point.
• If the boulder approaches the 1/5 mile point, a god with throw up a wall at the 1/5 mile point.
and so on for an infinity of points.
"Approaches" means "lies in the space between the wall location and location of the preceding wall." "Approaches the 1/2 mile point" means "lies between the 1/3 and 1/2 mile point."
The wall at the 1/2 mile point:
The wall at the 1/3 mile point:
Finally we have:
(c) Initial state: The boulder is initially at A (mile point 0) rolling toward B.
The premises (a), (b) and (c) together produce a contradiction (d). That is the paradox.
(d) (d1) The boulder cannot advance beyond A;
(d2) the boulder cannot stay at A.
The first clause (d1) of (d) follows since there is no consistent account in which the boulder advances past A. For any distance along the road you care to name, its passage has already been blocked by an earlier wall:
It cannot arrive at the 1/2 mile point
since it would have been blocked earlier by the wall at the 1/3 mile
It cannot arrive at the 1/3 mile point since it would have been blocked earlier by the wall at the 1/4 mile point.
It cannot arrive at the 1/4 mile point since it would have been blocked earlier by the wall at the 1/5 mile point.
and so on.
The second clause (d2) of (d) follows since we conclude in the inferences leading to (d1) that no wall is actually thrown up. That means that the boulder's motion is not obstructed, so by contact mechanics (a), it moves from A.
What is strange in this inference is that the
failure to advance is arrived at without ever concluding that a wall is
actually thrown up. The wall at mile point 1/2 is not thrown up, since the
boulder will never approach mile point 1/2. It would already have been
blocked by the wall thrown up at mile point 1/3. And that wall was not
thrown up by analogous arguments. And so on for all the remaining walls.
The boulder cannot advance even though no wall is
actually thrown up to block it.
Benardete halts at this conclusion with a most curious diagnosis. He accepts this halting as a physical effect that would actually happen were we able somehow to realize this system of boulders and gods and walls. The boulder is halted by "a strange field of force blocking his passage forward." My hope is that Benardete was not really serious in this supposition. It amounts to thinking that logical hocus-pocus can somehow materialize a novel force field.
The right way to proceed is to take seriously the second part of the conclusion (d2) and pay attention to the full contradiction. Under premise (a), the motion of the boulder is governed by ordinary contact mechanics. It will continue to roll past A unless something obstructs it. Since we have concluded that no wall is thrown up to obstruct it, in follows from (a) that the boulder continues to roll. That is, (d2) says it cannot stay at A.
We do not have an argument for a mysterious, disembodied force field. We just have a common-garden contradiction, whose origin lies in our acceptance of a contradictory set of assumptions.
(The ball stops moving just in case a wall obstructs it)
and NOT-(The ball stops moving just in case a wall obstructs it)
What is responsible for the contradiction?
It is tempting to blame the rather fanciful wall physics (b). However as a matter of logic that physics by itself does not seem to be to blame. For the wall physics would proceed without problems if the initial position of the boulder were anywhere other than point A at mile point 0. If the boulder were initially at mile point 0.4, then its motion would be blocked by a wall thrown up at mile point 1/2. This is true for any initial point after mile 0.
That is, premises (a) and (b) together almost never produce problems. It is only the selection of the specific initial point A at mile 0 in premise (c) that produces the problem. For, in the collision mechanics, it is supposed that the boulder will halt its rolling with the FIRST obstacle it strikes. By construction, the walls are so configured that there can be no FIRST wall thrown up. The core difficulty is the same as the one we encountered in the reverse dichotomy version of Zeno's paradox. We tacitly suppose a first in a system that has no first; and then arrive at a contradiction.
There is nothing especially troublesome in this result. It is just that a clever specification has lured us into accepting as consistent a collection of assumptions that are not consistent. That is, we have been lured into assigning initial conditions to some physical system that are incompatible with physical laws that we have elsewhere presumed to apply.
There are other occasions in which some physical law is matched with initial conditions incompatible with it; and trouble ensues. A striking example historically occurred in Galileo's magisterial Two New Sciences. He sought to argue against the speed-distance law of fall. According to that law, the speed with which a body falls is proportional to the distance it has fallen. (Galileo replaced this law with the modern law that speed is proportional to the time the body has fallen.) Under the speed-distance law, it turns out that the speed grows exponentially with time. That is, for each equal increment of time, the speed grows by the same multiplicative factor. It follows that, if a body is initially at rest, it must stay at rest for all time. Multiplication of its zero speed by some factor leaves it at zero. Hence, contrary to Galileo's initial supposition, the law cannot treat the most familiar case of a body released initially at rest which then falls. For details see John D. Norton and Bryan Roberts, “Galileo's Refutation of the Speed-Distance Law of Fall Rehabilitated,” Centaurus 54 (2012) pp. 148-164.
What is the time reversal of the Thomson lamp supertask? What is the schedule of on/off switchings? Can you devise a mechanism that implements it?
The chapter leaves open whether the reversal of the
supertasks of Black's transfer machine and the many accelerating
spaceships is admissible Just what separates admissible from inadmissible
June 15, September 17, 2021
Copyright, John D. Norton