Infinite divisibility
Encyclopedia
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 such as the continuum
.
'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).
Atomism
holds that matter is composed ultimately of indivisible parts, or 'atoms'. Perhaps counter-intuitively, atomism is compatible with infinite divisibility. For example, a line segment
composed of indivisible points can be infinitely divided into ever smaller line segments. There is no consensus among philosophers as to whether atomism is correct, and Peter Simons
, author of the classic text Parts, maintains that the issue is undecided. But some philosophers disagree. Dean Zimmerman
of Rutgers claims to have developed evidence for the rejection of atomism .
, 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 — quark
s and electron
s — are point particle
s: 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 × 10−35 metres) and therefore a minimum time interval (the amount of time which light takes to traverse that distance in a vacuum, 5.391 × 10−44 seconds, known as the Planck time
) smaller than which meaningful measurement is impossible.
, 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 × $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.
of rational number
s 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 integer
s 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 if one were to partition
the rationals into two non-empty sets A and B where A contains all rationals less than some irrational number (π
, say) and B all rationals greater than it, then A has no largest member and B has no smallest member. The field of real number
s, 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.
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 − Xs depends only on t − 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
.
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, order theory
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
(a branch of mathematics), and probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
(also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
, space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
, time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, money
Money
Money is any object or record that is generally accepted as payment for goods and services and repayment of debts in a given country or socio-economic context. The main functions of money are distinguished as: a medium of exchange; a unit of account; a store of value; and, occasionally in the past,...
, or abstract mathematical objects such as the continuum
Continuum (theory)
Continuum theories or models explain variation as involving a gradual quantitative transition without abrupt changes or discontinuities. It can be contrasted with 'categorical' models which propose qualitatively different states.-In physics:...
.
In philosophy
This theory is exposed in PlatoPlato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
's dialogue Timaeus
Timaeus (dialogue)
Timaeus is one of Plato's dialogues, mostly in the form of a long monologue given by the title character, written circa 360 BC. The work puts forward speculation on the nature of the physical world and human beings. It is followed by the dialogue Critias.Speakers of the dialogue are Socrates,...
and was also supported by Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
. 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).
Atomism
Atomism
Atomism is a natural philosophy that developed in several ancient traditions. The atomists theorized that the natural world consists of two fundamental parts: indivisible atoms and empty void.According to Aristotle, atoms are indestructible and immutable and there are an infinite variety of shapes...
holds that matter is composed ultimately of indivisible parts, or 'atoms'. Perhaps counter-intuitively, atomism is compatible with infinite divisibility. For example, a line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
composed of indivisible points can be infinitely divided into ever smaller line segments. There is no consensus among philosophers as to whether atomism is correct, and Peter Simons
Peter Simons
Peter Simons, FBA, is a professor of philosophy at Trinity College Dublin.He studied at the University of Manchester, and has held teaching posts at the University of Bolton, the University of Salzburg, where he is Honorary Professor of Philosophy, and the University of Leeds...
, author of the classic text Parts, maintains that the issue is undecided. But some philosophers disagree. Dean Zimmerman
Dean Zimmerman
Dean Zimmerman is an American professor of philosophy at Rutgers University. He specializes in metaphysics and the philosophy of religion. He earned his PhD in philosophy from Brown University, and has taught at the University of Notre Dame and Syracuse University...
of Rutgers claims to have developed evidence for the rejection of atomism .
In physics
Until the discovery of quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, 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
Ad infinitum
Ad infinitum is a Latin phrase meaning "to infinity."In context, it usually means "continue forever, without limit" and thus can be used to describe a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever," among other uses...
.
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
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
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
Conundrum
Conundrum may refer to:* A riddle whose answer is or involves a pun or unexpected twist* A logical postulation that evades resolution, an intricate and difficult problem- Literature :...
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
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of particle physics, the particles that make up an atom — quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
s and electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s — are point particle
Point particle
A point particle is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension: being zero-dimensional, it does not take up space...
s: 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
Quantum indeterminacy
Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics...
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
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
is infinitely divisible.
However, the pioneering work of Max Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...
(1858–1947) in the field of quantum physics suggests that there is, in fact, a minimum distance (now called the Planck length, 1.616 × 10−35 metres) and therefore a minimum time interval (the amount of time which light takes to traverse that distance in a vacuum, 5.391 × 10−44 seconds, known as the Planck time
Planck time
In physics, the Planck time, , is the unit of time in the system of natural units known as Planck units. It is the time required for light to travel, in a vacuum, a distance of 1 Planck length...
) smaller than which meaningful measurement is impossible.
In economics
One dollarDollar
The dollar is the name of the official currency of many countries, including Australia, Belize, Canada, Ecuador, El Salvador, Hong Kong, New Zealand, Singapore, Taiwan, and the United States.-Etymology:...
, or one euro
Euro
The euro is the official currency of the eurozone: 17 of the 27 member states of the European Union. It is also the currency used by the Institutions of the European Union. The eurozone consists of Austria, Belgium, Cyprus, Estonia, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg,...
, 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 × $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
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
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 fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s is infinitely divisible (i.e. order theoretically dense
Density (disambiguation)
Density and dense usually refer to a measure of how much of some entity is within a fixed amount of space. Types of density include:In physics, density of mass:* Density, mass per volume...
) means that between any two rational numbers there is another rational number. By contrast, the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s is not infinitely divisible.
Infinite divisibility does not imply gap-less-ness: the rationals do not enjoy the least upper bound property
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
. That means that if one were to partition
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
the rationals into two non-empty sets A and B where A contains all rationals less than some irrational number (π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
, say) and B all rationals greater than it, then A has no largest member and B has no smallest member. The field of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, by contrast, is both infinitely divisible and gapless. Any linearly ordered set
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
that is infinitely divisible and gapless, and has more than one member, is uncountably infinite
Uncountable set
In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
. For a proof, see Cantor's first uncountability proof
Cantor's first uncountability proof
Georg Cantor's first uncountability proof demonstrates that the set of all real numbers is uncountable. This proof differs from the more familiar proof that uses his diagonal argument...
. Infinite divisibility alone implies infiniteness but not uncountability, as the rational numbers exemplify.
In probability distributions
To say that a probability distributionProbability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
F on the real line is infinitely divisible means that if X is any random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
whose distribution is F, then for every positive integer n there exist n independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
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
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
, the negative binomial distribution
Negative binomial distribution
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number of failures occur...
, and the Gamma distribution are examples of infinitely divisible distributions; as are the normal distribution, Cauchy distribution
Cauchy distribution
The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...
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
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...
, i.e., a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
{ Xt : t ≥ 0 } with stationary independent increments (stationary means that for s < t, the probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
of Xt − Xs depends only on t − s; independent increments means that that difference is independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
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
Bruno de Finetti
Bruno de Finetti was an Italian probabilist, statistician and actuary, noted for the "operational subjective" conception of probability...
.
See also indecomposable distribution
Indecomposable distribution
In probability theory, an indecomposable distribution is a probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables: Z ≠ X + Y. If it can be so expressed, it is decomposable:...
.