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Infinite divisibility

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter, space, time, money, or abstract mathematical objects such as the continuum.

This theory is explored in Plato's dialogue Timaeus and was also supported by Aristotle. Andrew Pyle gives a lucid account of infinite divisibility in the first few pages of his Atomism and its Critics. There he shows how infinite divisibility involves the idea that there is some extended item, such as an apple, which can be divided infinitely many times, where one never divides down to point, or to atoms of any sort. Many professional philosophers claim that infinite divisibility involves either a collection of an infinite number of items (since there are infinite divisions, there must be an infinite collection of objects), or (more rarely), point-sized items, or both. Pyle states that the mathematics of infinitely divisible extensions involve neither of these — that there are infinite divisions, but only finite collections of objects and they never are divided down to point extension-less items.

Zeno questioned how an arrow can move if at one moment it is here and motionless and at a later moment be somewhere else and motionless, like a motion picture.

Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.

In reference to Zeno's paradox of the arrow in flight, Alfred North Whitehead writes that "an infinite number of acts of becoming may take place in a finite time if each subsequent act is smaller in a convergent series":

  • Domínguez-Molina, J.A.; Rocha-Arteaga, A. (2007) "On the Infinite Divisibility of some Skewed Symmetric Distributions". Statistics and Probability Letters, 77 (6), 644–648 doi:10.1016/j.spl.2006.09.014


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