Infinite divisibility

The concept of 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.

In philosophy
This theory is exposed in Plato's dialogue Timaeus and was also supported by Aristotle. Andrew Pyle has one of the most lucid accounts of infinite divisibility in the first few pages of his masterwork 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. 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).

Atomism denies that matter is infinitely divisible. There is no consensus among philosophers as to whether atomism or infinite divisibility is correct, and Peter Simons, author of the classic text Parts, maintains that the issue is undecided. But some philosophers disagree. For example, Jeffrey Grupp of Purdue University has developed a few theories that claim to show that infinite divisibility is incorrect, and therefore atomism is correct. And other philosophers, such as Dean Zimmerman of Rutgers, claim to have developed evidence for the vindication of infinite divisibility.

In physics
Until the discovery of quantum mechanics, no distinction was made between the question of whether matter is infinitely divisible and the question of whether matter can be cut into smaller parts ad infinitum.

As a result, the Greek word átomos (&#7940;&tau;&omicron;&mu;&omicron;&sigmaf;), whose literal meaning is "uncuttable", is usually translated as "indivisible". Whereas the modern atom is indeed divisible, it actually is uncuttable: there is no partition of space such that its parts correspond to material parts of the atom. In other words, the quantum-mechanical description of matter no longer conforms to the cookie cutter paradigm. This casts fresh light on the ancient conundrum of the divisibility of matter. The multiplicity of a material object &mdash; the number of its parts &mdash; depends on the existence, not of delimiting surfaces, but of internal spatial relations (relative positions between parts), and these lack determinate values. According to the Standard Model of particle physics, the particles that make up an atom &mdash; quarks and electrons &mdash; are point particles: they do not take up space. What makes an atom nevertheless take up space is not any spatially extended "stuff" that "occupies space", and that might be cut into smaller and smaller pieces, but the indeterminacy of its internal spatial relations.

Physical space is often regarded as infinitely divisible: it is thought that any region in space, no matter how small, could be further split. Similarly, time is infinitely divisible.

However, the pioneering work of Max Planck (1858-1947) in the field of quantum physics suggests that there is, in fact, a minimum distance (now called the Planck length, 1.616 &times; 10&minus;35 metres) and therefore a minimum time interval (the amount of time which light takes to traverse that distance in a vacuum, 5.391 &times; 10&minus;44 seconds, known as the Planck time) smaller than which meaningful measurement is impossible.

In business
One dollar, or one euro, is divided into 100 cents; one can only pay in increments of a cent. It is quite commonplace for prices of some commodities such as gasoline to be in increments of a tenth of a cent per gallon or per litre (10 x $197.532=$1,975.32). The volume purchased may also be considered divisible, but is measured to some precision, such as hundredth of a liter or gallon, and at some point of division, the car would not run on the added "fuel" (for example, it may take an entire methane molecule or some volume of them to start the necessary chemical reaction). If gasoline costs $197.532 per gallon and one buys 10 gallons, then the "extra" 2/10 of a cent comes to ten times that: an "extra" two cents, so the cent in that case gets paid. If one had bought 9 gallons at that price, one would have rounded to the nearest cent would still be paid. Money is infinitely divisible in the sense that it is based upon the real number system. However, modern day coins are not divisible (in the past some coins were weighed with each transaction, and were considered divisible with no particular limit in mind). There is a point of precision in each transaction that is useless because such small amounts of money are insignificant to humans. The more the price is multiplied the more the precision could matter. For example when buying a million shares of stock, the buyer and seller might be interested in a tenth of a cent price difference, but it's only a choice. Everything else in business measurement and choice is similarly divisible to the degree that the parties are interested. For example, financial reports may be reported annually, quarterly, or monthly. Some business managers run cash-flow reports more than once per day.

Although time may be infinitely divisible, data on securities prices are reported at discrete times. For example, if one looks at records of stock prices in the 1920s, one may find the prices at the end of each day, but perhaps not at three-hundredths of a second after 12:47 PM. A new method, however, theoretically, could report at double the rate, which would not prevent further increases of velocity of reporting. Perhaps paradoxically, technical mathematics applied to financial markets is often simpler if infinitely divisible time is used as an approximation. Even in those cases, a precision is chosen with which to work, and measurements are rounded to that approximation. In terms of human interaction, money and time are divisible, but only to the point where further division is not of value, which point cannot be determined exactly.

In order theory
To say that the field of rational numbers is infinitely divisible (i.e. order theoretically dense) means that between any two rational numbers there is another rational number. By contrast, the ring of integers is not infinitely divisible.

Infinite divisibility does not imply gap-less-ness: the rationals do not enjoy the least upper bound property. That means that one may partition the rationals into two non-empty sets A and B in such a way that every member of A is less than every member of B, and A has no largest member, and B has no smallest member. The field of real numbers, by contrast, is both infinitely divisible and gapless. Any linearly ordered set that is infinitely divisible and gapless, and has more than one member, is uncountably infinite. For a proof, see Cantor's first uncountability proof. Infinite divisibility alone implies infiniteness but not uncountability, as the rational numbers exemplify.

In probability distributions
To say that a probability distribution F on the real line is infinitely divisible means that if X is any random variable whose distribution is F, then for every positive integer n there exist n independent identically distributed random variables X1, ..., Xn whose sum is equal in distribution to X (those n other random variables do not usually have the same probability distribution as X).

The Poisson distribution, the negative binomial distribution, and the Gamma distribution are examples of infinitely divisible distributions; as are the normal distribution, Cauchy distribution and all other members of the stable distribution family. The skew-normal distribution is an example of a non-infinitely divisible distribution (See Domínguez-Molina and Rocha Arteaga (2007))

Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process, i.e., a stochastic process { Xt : t ≥ 0 } with stationary independent increments (stationary means that for s < t, the probability distribution of Xt &minus; Xs depends only on t &minus; s; independent increments means that that difference is independent of the corresponding difference on any interval not overlapping with [s, t], and similarly for any finite number of intervals).

This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.

See also indecomposable distribution.