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.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.The argument, so far as it is valid, elicits a contradiction from the two premises: (i) that in a becoming something (res vera) becomes, and (ii) that every act of becoming is divisible into earlier and later sections which are themselves acts of becoming. Consider, for example, an act of becoming during one second. The act is divisible into two acts, one during the earlier half of the second, the other during the later half of the second. Thus that which becomes during the whole second presupposes that which becomes during the first half-second. Analogously, that which becomes during the first half-second presupposes that which becomes during the first quarter-second, and so on indefinitely. Thus if we consider the process of becoming up to the beginning of the second in question, and ask what then becomes, no answer can be given. For, whatever creature we indicate presupposes an earlier creature which became after the beginning of the second and antecedently to the indicated creature. Therefore there is nothing which becomes, so as to effect a transition into the second in question. 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. The idea of divisibility of matter was first proposed by Indian philosopher Kanada (philosopher) around 500 BCEThis 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. 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': 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 (ἄτομος), which literally means '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—the number of its parts—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—quarks and electrons—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. Time is similarly considered as 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, 1.616229(38)×10−35 metres) and therefore a minimum time interval (the amount of time which light takes to traverse that distance in a vacuum, 5.39116(13) × 10−44 seconds, known as the Planck time) smaller than which meaningful measurement is impossible. 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. If gasoline costs $3.979 per gallon and one buys 10 gallons, then the 'extra' 9/10 of a cent comes to ten times that: an 'extra' 9 cents, so the cent in that case gets 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.