Definitions

In philosophy, a supertask is a task occurring within a finite interval of time involving infinitely many steps (subtasks). A hypertask is a supertask with an uncountable number of subtasks. The term supertask was coined by the philosopher James F. Thomson, and the term hypertask by Clarke and Read in their paper of that name.

## History

### Zeno

#### Motion

The origin of the interest in supertasks is normally attributed to Zeno of Elea. Zeno claimed that motion was impossible. He argued as follows: suppose our burgeoning "mover", Achilles say, wishes to move from A to B. To achieve this he must traverse half the distance from A to B. To get from the midpoint of AB to B Achilles must traverse half this distance, and so on and so forth. However many times he performs one of these "traversing" tasks there is another one left for him to do before he arrives at B. Thus it follows, according to Zeno, that motion (travelling a non-zero distance in finite time) is a supertask. Zeno further argues that supertasks are not possible (how can this sequence be completed if for each traversing there is another one to come?). It follows that motion is impossible.

Zeno's argument takes the following form:

1. Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
3. Therefore motion is impossible

Most subsequent philosophers reject Zeno's bold conclusion in favor of common sense. Instead they turn his argument on its head (assuming it's valid) and take it as a proof by contradiction where the possibility of motion is taken for granted. They accept the possibility of motion and apply modus tollens (contrapositive) to Zeno's argument to reach the conclusion that either motion is not a supertask or supertasks are in fact possible.

#### Achilles and the tortoise

Zeno himself also discusses the notion of what he calls "Achilles and the tortoise". Suppose that Achilles is the fastest runner, and moves at a speed of 1 m/s. Achilles chases a tortoise, an animal renowned for being slow, that moves at 0.1 m/s. However, the tortoise starts 0.9 metres ahead. Commonsense seems to decree that Achilles will catch up with the tortoise after exactly 1 second, but Zeno argues that this is not the case. He instead suggests that Achilles must inevitably come up to the point where the tortoise has started from, but by the time he has accomplished this, the tortoise will already have moved on to another point. This continues, and every time Achilles reaches the mark where the tortoise was, the tortoise will create a new point that Achilles will have to catch up with; while it begins with 0.9 metres, it becomes an additional 0.09 metres, then 0.009 metres, and so on, infinitely. While these distances will grow very small, they will remain finite, while Achilles' chasing of the tortoise will become an unending supertask. Much commentary has been made on this particular paradox; many assert that it finds a loophole in common sense, while others say it reveals an inadequacy in mathematics to resolve the problem.

### Thomson

James F. Thomson was of the former category. He believed that motion was not a supertask, and he emphatically denied that supertasks are possible. The proof Thomson offered to the latter claim involves what has probably become the most famous example of a supertask since Zeno. Thomson's lamp may either be on or off. At time t = 0 the lamp is off, at time t = 1/2 it is on, at time t = 3/4 (= 1/2 + 1/4) it is off, t = 7/8 (= 1/2 + 1/4 + 1/8) it is on, etc. The natural question arises: at t = 1 is the lamp on or off? There does not seem to be any non-arbitrary way to decide this question. Thomson goes further and claims this is a contradiction. He says that the lamp cannot be on for there was never a point when it was on where it was not immediately switched off again. And similarly he claims it cannot be off for there was never a point when it was off where it was not immediately switched on again. By Thomson's reasoning the lamp is neither on nor off, yet by stipulation it must be either on or off — this is a contradiction. Thomson thus believes that supertasks are impossible.

### Benacerraf

Paul Benacerraf is of the latter category. Benacerraf believes that supertasks are at least logically possible despite Thomson's apparent contradiction. Benacerraf agrees with Thomson insofar as that the experiment he outlined does not determine the state of the lamp at t = 1. However he disagrees with Thomson that he can derive a contradiction from this, since the state of the lamp at t = 1 need not be logically determined by the preceding states. Logical implication does not bar the lamp from being on, off, or vanishing completely to be replaced by a horse-drawn pumpkin. There are possible worlds in which Thomson's lamp finishes on, and worlds in which it finishes off not to mention countless others where weird and wonderful things happen at t = 1. The seeming arbitrariness arises from the fact that Thomson's experiment does not contain enough information to determine the state of the lamp at t = 1, rather like the way nothing can be found in Shakespeare's play to determine whether Hamlet was right- or left-handed. So what about the contradiction? Benacerraf showed that Thomson had committed a mistake. When he claimed that the lamp could not be on because it was never on without being turned off again — this applied only to instants of time strictly less than 1. It does not apply to 1 because 1 does not appear in the sequence {0, 1/2, 3/4, 7/8, …} whereas Thomson's experiment only specified the state of the lamp for times in this sequence.

### Modern literature

Most of the modern literature comes from the descendants of Benacerraf, those who accept the possibility of supertasks. Philosophers who reject their possibility tend not to reject them on grounds such as Thomson's but because they have qualms with the notion of infinity itself (of course there are exceptions; for example, McLaughlin claims that Thomson's lamp is inconsistent if it is analyzed with internal set theory, a variant of real analysis).

#### Philosophy of mathematics

If supertasks are logically possible, then the truth or falsehood of unknown propositions of number theory, such as Goldbach's conjecture, or even undecidable propositions could be determined in a finite amount of time by a brute-force search of the set of all natural numbers. This would, however, be in contradiction with the Church-Turing thesis. Some have argued this poses a problem for intuitionism, since the intuitionist must distinguish between things that it is not humanly possible to prove (because they are too long or complicated; see Boolos, "A Curious Inference") but nonetheless are considered "provable", and those that are provable by infinite brute force in the above sense.

#### Physical possibility

Some have claimed Thomson's lamp is physically impossible since it must have parts moving at speeds faster than the speed of light (e.g., the lamp switch). Adolf Grünbaum suggests that the lamp could have a strip of wire which, when lifted, disrupts the circuit and turns off the lamp; this strip could then be lifted by a smaller distance each time the lamp is to be turned off, maintaining a constant velocity. However, such a design would ultimately fail, as eventually the distance between the contacts would be so small as to allow electrons to jump the gap, preventing the circuit from being broken at all.

Other physically possible supertasks have been suggested. In one proposal, one person (or entity) counts upward from 1, taking an infinite amount of time, while another person observes this from a frame of reference where this occurs in a finite space of time. For the counter, this is not a supertask, but for the observer, it is. (This could theoretically occur due to time dilation, for example if the observer were falling into a black hole while observing a counter whose position is fixed relative to the singularity.)

Davies in his paper "Building Infinite Machines" concocted a device which he claims is physically possible up to infinite divisibility. It involves a machine which creates an exact replica of itself but has half its size and twice its speed.

#### Super Turing machines

The impact of supertasks on theoretical computer science has triggered some new and interesting work (see Hamkins and Lewis — "Infinite Time Turing Machine").

### The diary of Tristram Shandy

Tristram Shandy, the hero of a novel by Laurence Sterne, writes his autobiography so conscientiously that it takes him one year to lay down the events of one day. If he is mortal he can never terminate; but if he lived forever then no part of his diary would remain unwritten, for to each day of his life a year devoted to that day's description would correspond.

### Kafka's 'Great Wall of China'

Many of Franz Kafka's characters seem to be struggling with a supertask of some kind - but one of the clearest examples is from his story 'The Great Wall of China':

Suppose there is a jar capable of containing infinitely many marbles and an infinite collection of marbles labelled 1, 2, 3, and so on. At time t = 0, marbles 1 through 10 are placed in the jar and marble 1 is taken out. At t = 0.5, marbles 11 through 20 are placed in the jar and marble 2 is taken out; at t = 0.75, marbles 21 through 30 are put in the jar and marble 3 is taken out; and in general at time t = 1 − 0.5n, marbles 10n + 1 through 10n + 10 are placed in the jar and marble n + 1 is taken out. How many marbles are in the jar at time t = 1?

One argument states that there should be infinitely many marbles in the jar, because at each step before t = 1 the number of marbles increases from the previous step and does so unboundedly. A second argument, however, shows that the jar is empty. Consider the following argument: if the jar is non-empty, then there must be a marble in the jar. Let us say that that marble is labeled with the number n. But at time t = 1 − 0.5n - 1, the nth marble has been taken out, so marble n cannot be in the jar. This is a contradiction, so the jar must be empty. The Ross-Littlewood paradox is that here we have two seemingly perfectly good arguments with completely opposite conclusions.

Further complications are introduced by the following variant. Suppose that we follow the exact same process as above, but instead of taking out marble 1 at t = 0, one takes out marble 2. And, at t = 0.5 one takes out marble 3, at t = 0.75 marble 4, etc. Then, one can use the same logic from above to show that while at t = 1, marble 1 is still in the jar, no other marbles can be left in the jar. Similarly, one can construct scenarios where in the end, 2 marbles are left, or 17 or, of course, infinitely many. But again this is paradoxical: given that in all these variations the same number of marbles are added or taken out at each step of the way, how can the end result differ?

Some people decide to simply bite the bullet and say that apparently, the end result does depend on which marbles are taken out at each instant. However, one immediate problem with that view is that one can think of the thought experiment as one where none of the marbles are actually labeled, and thus all the above variations are simply different ways of describing the exact same process, and it seems unreasonable to say that the end result of the one actual process depends on the way we describe what happens.

Moreover, Allis and Koetsier offer the following variation on this thought experiment: at t = 0, marbles 1 to 9 are placed in the jar, but instead of taking a marble out they scribble a 0 after the 1 on the label of the first marble so that it is now labeled "10". At t = 0.5, marbles 11 to 19 are placed in the jar, and instead of taking out marble 2, a 0 is written on it, marking it as 20. The process is repeated ad infinitum. Now, notice that the end result at each step along the way of this process is the same as in the original experiment, and indeed the paradox remains: Since at every step along the way, more marbles were added, there must be infinitely marbles left at the end, yet at the same time, since every marble with number n was taken out at t = 1 − 0.5n - 1, no marbles can be left at the end. However, in this experiment, no marbles are ever taken out, and so any talk about the end result 'depending' on which marbles are taken out along the way is made impossible.

A bare-naked variation that really goes straight to the heart of all of this goes as follows: at t = 0, there is one marble in the jar with the number 0 scribbled on it. At t = 0.5, the number 0 on the marble gets replaced with the number 1, at t = 0.75, the number gets changed to 2, etc. Now, no marbles are ever added to or removed from the jar, so at t = 1, there should still be exactly that one marble in the jar. However, since we always replaced the number on that marble with some other number, it should have some number n on it, and that is impossible because we know exactly when that number was replaced, and never repeated again later. In other words, we can also reason that no marble can be left at the end of this process, which is quite a paradox.

Of course, it would be wise to heed Benacerraf’s words that the states of the jars before t = 1 do not logically determine the state at t = 1. Thus, neither Ross’s or Allis’s and Koetsier’s argument for the state of the jar at t = 1 proceeds by logical means only. Therefore, some extra premise must be introduced in order to say anything about the state of the jar at t = 1. Allis and Koetsier believe such an extra premise can be provided by the physical law that the marbles have continuous space-time paths, and therefore from the fact that for each n, marble n is out of the jar for t < 1, it must follow that it must still be outside the jar at t = 1 by continuity. Thus, the contradiction, and the paradox, remains.

One obvious solution to all these conundrums and paradoxes is to say that supertasks are impossible. If supertasks are impossible, then the very assumption that all of these scenarios had some kind of 'end result' to them is mistaken, preventing all of the further reasoning (leading to the contradictions) to go through.

Similar paradoxes involving a man trying to walk a mile from A to B but the demons building a wall in front of him if he reaches 1/2, 1/4, 1/8, … of a mile past A, so he hits some sort of invisible barrier when he leaves A even though no wall has been built.

This supertask is an example of indeterminism in Newtonian mechanics. The supertask consists of an infinite collection of point masses all of which are stationary and will spontaneously self-excite (start moving for no apparent reason). The point masses are all of mass m and are placed along a line AB that is a meters in length at positions B, AB / 2, AB / 4, AB / 8, and so on. The first particle at B is accelerated to a velocity of one meter per second towards A. According to the laws of Newtonian mechanics, when the first particle collides with the second, it will come to rest and the second particle will inherit its velocity of 1 m/s. This process will continue as an infinite amount of collisions, and after 1 second, all the collisions will have finished since all the particles were moving at 1 meter per second. However no particle will emerge from A, since there is no last particle in the sequence. It follows that all the particles are now at rest, contradicting conservation of energy. Now the laws of Newtonian mechanics are time-reversal-invariant; that is, if we reverse the direction of time, all the laws will remain the same. If time is reversed in this supertask, we have a system of stationary point masses along A to AB / 2 that will, at random, spontaneously start colliding with each other, resulting in a particle moving away from B at a velocity of 1 m/s. Alper and Bridger have questioned the reasoning in this supertask invoking the distinction between actual and potential infinity.

### Davies' super-machine

This is a machine that can, in the space of half an hour, create an exact replica of itself that is half its size and capable of twice its replication speed. This replica will in turn create an even faster version of itself with the same specifications, resulting in a supertask that finishes after an hour. If, additionally, the machines create a communication link between parent and child machine that yields successively faster bandwidth and the machines are also capable of simple arithmetic, the supertask can be used to perform brute-force proofs of unknown conjectures. For example, for Goldbach's Conjecture, the first machine will spend the first half-hour checking if 4 can be expressed as the sum of two primes (albeit an inefficient use of the whole half hour), the next quarter of an hour checking 6, and so on. This can even be extended to undecidable number-theoretical problems, which some claim can lead to problems for intuitionism in the philosophy of mathematics. See also technological singularity.