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[Math Lair] Supertasks

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A supertask (or super-task) is a countably infinite number of operations that are performed in a finite period of time. What constitutes a supertask, and whether supertasks are possible or not, are interesting problems in the field of philosophy. Supertasks can also raise various paradoxes.

Perhaps the first philosopher to investigate supertasks, although he didn't refer to them as such at the time, was the ancient Greek philosopher Zeno, who formulated Zeno's paradox somewhere around 450 B.C. Zeno's paradox consists of several arguments suggesting that motion is impossible. In essence, Zeno argued that motion from one point to another is a supertask, because space can be infinitely divided; in order to travel a distance, you need to travel half the distance, a quarter of the distance, an eighth of the distance, and so on ad infinitum. Zeno implicitly assumes that a supertask is impossible and thus that motion is impossible. On the Zeno's paradox page is discussed one way of resolving the paradox, by noting that, even though there are an infinite number of terms in the sum of 1 + ½ + ¼ + 18 + ... , the sum is a finite number, namely 2.

Following Zeno, we'll skip ahead around 2,400 years to the 1950s. In 1953, J. E. Littlewood described what is now called the Ross-Littlewood paradox. The paradox can be described as follows: There is an empty but infinitely large vase and a infinite number of balls, numbered sequentially from 1. At 1 minute to noon, balls 1–10 are put into the vase, and ball number 1 is taken out. At ½ minutes to noon, balls 11–20 are put into the vase, and number 2 is removed. At ¼ minutes to noon, balls 21–30 are put into the vase, and number 3 is removed. This process continues indefinitely, with the time increment being halved each time. How many balls are in the vase at noon? The number of balls appears to grow without limit; however, what balls are left in there? Ball 1 was removed at 11:59:00, ball 2 at 11:59:30, ball 3 at 11:59:45, ball 4 at 11:59:52.5, and this sequence is infinite. Given a ball with any number, you can find a time when it was removed from the vase. Therefore, there are no balls in the vase. But this makes no sense because each operation increases the number of balls in the vase by 9.

Philosopher James F. Thomson invented the word "supertask" in 1954. He argued that motion is not a supertask and that things that actually are supertasks are impossible, because they result in logical contradictions. He originated the Thomson lamp paradox, first described in his paper "Tasks and Super-Tasks," to support his argument that supertasks are impossible. In this paradox, assume that we have a perfect machine for turning a lamp off and on. First, we have the light on for one minute. Then, we have the light off for ½ minutes. Then, it is off for ¼ minutes. This continues, with the light being turned off and on after one-half of the previous time period. At the end of two minutes, an infinite sequence of offs and ons will have taken place. At this time, will the light be off or on? Would the answer be any different if the light had started being off instead of on?

However, in the 1962 paper "Tasks, Super-Tasks, and the Modern Eleatics," American mathematician Paul Benacerraf noted that the conditions described above don't logically determine the state of the lamp at exactly 2 minutes. For any time before the two-minute mark, the state of the lamp is determined, but at exactly two minutes no value can be determined, as an infinite series has no last term. As the state of the lamp is not determined after 2 minutes, the contradictions Thomson describes do not apply. Thomson acknowledges that Benacerraf's points mostly invalidate his original argument, and the question of whether supertasks are logically contradictory or not is still unresolved.

Sources used (see bibliography page for titles corresponding to numbers): 5, 59. Note that the papers mentioned in this article are reprinted in source #59.