Supertask

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

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Zeno

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.

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Zeno's argument takes the following form:

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• Motion is a supertask
• Supertasks are impossible
• Therefore motion is impossible

Most subsequent philosophers reject Zeno's bold conclusion in favour of common sense. Instead they turn his argument on its head (assuming its 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.

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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 it is off, t=7/8 on etc etc... The natural question arises: at t=1 is the lamp on or off? There doesn't 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.

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Benacerraf

Paul Benacerraf is of the latter category. Benacerraf believes that supertasks are at least logically possible despite Thomsons 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. For all logical implication has to say about this the lamp could be on, off or vanish 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, a bit like the way nothing can be found in Shakespeares play to determine whether Hamlet was right or left handed.

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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. Why does it not apply to 1? Because 1 does not appear in the sequence 0, 1/2, 3/4, 7/8... whereas Thomsons experiment only specified the state of the lamp for times in this sequence.

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Modern Literature

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

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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 delineate between things which are not humanly possible to prove (because they are too long or complicated - see Boolos "A Curious Inference") but nonetheless are considered "provable" and those which are provable by infinite brute force in the above sense.

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Spontaneous Self-Excitement

Some ingeniously devised supertasks can involve spontaneous self excitement. Earman and Norton, Alper and Briger, and Laraudogoitia have explored these examples in depth.

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Physical Possibility

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

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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")

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Philosophy: Philosophy is a discipline or field of study involving the investigation, analysis, and development of ideas at a general, abstract, or fundamental level. It is the discipline in search for a general understanding of values and reality by chiefly speculative rather than observational means. The term...

Infinitely: REDIRECT Infinity...

Hypertask: REDIRECT Supertask...