Infinite divisibility arises in different ways in

^{−35} metres, named after one of the fathers of Quantum Theory, ^{−44} seconds, known as the Planck time) at which the Standard Model is expected to break down – effectively making this the smallest physical scale about which meaningful statements can be currently made. To predict the physical behaviour of space-time and fundamental particles at smaller distances requires a new theory of

_{1}, ..., ''X''_{''n''} whose sum is equal in distribution to ''X'' (those ''n'' other random variables do not usually have the same probability distribution as ''X'').
The _{''t''} − ''X''_{''s''} depends only on ''t'' − ''s''; ''independent increments'' means that that difference is

* 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

philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. S ...

, physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...

, economics, order theory
Order theory is a branch of mathematics that investigates the 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 intro ...

(a branch of mathematics), and probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

(also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic parti ...

, space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...

, time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...

, money
Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, such as taxes, in a particular country or socio-economic context. The primary functions which distinguish money are ...

, or abstract mathematical objects such as the continuum.
In philosophy

The origin of the idea in the Western tradition can be traced to the 5th century BCE starting with the Ancient Greek pre-Socratic philosopherDemocritus
Democritus (; el, Δημόκριτος, ''Dēmókritos'', meaning "chosen of the people"; – ) was an Ancient Greek pre-Socratic philosopher from Abdera, primarily remembered today for his formulation of an atomic theory of the universe. N ...

and his teacher Leucippus
Leucippus (; el, Λεύκιππος, ''Leúkippos''; fl. 5th century BCE) is a pre-Socratic Greek philosopher who has been credited as the first philosopher to develop a theory of atomism.
Leucippus' reputation, even in antiquity, was obscured ...

, who theorized matter's divisibility beyond what can be perceived by the senses until ultimately ending at an indivisible atom. The Indian philosopher, Maharshi Kanada also proposed an atomistic theory, however there is ambiguity around when this philosopher lived, ranging from sometime between the 6th century to 2nd century BCE. Around 500 BC, he postulated that if we go on dividing matter ('' padarth''), we shall get smaller and smaller particles. Ultimately, a time will come when we shall come across the smallest particles beyond which further division will not be possible. He named these particles ''Parmanu''.Another Indian philosopher, Pakudha Katyayama, elaborated this doctrine and said that these particles normally exist in a combined form which gives us various forms of matter.
Atomism
Atomism (from Greek , ''atomon'', i.e. "uncuttable, indivisible") is a natural philosophy proposing that the physical universe is composed of fundamental indivisible components known as atoms.
References to the concept of atomism and its atoms ...

is explored in Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...

's dialogue Timaeus. Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...

proves that both length and time are infinitely divisible, refuting atomism. 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
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ex ...

, 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.
In reference to Zeno's paradox of the arrow in flight, Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applic ...

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":
In quantum physics

Until the discovery ofquantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

, 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'' is a Latin phrase meaning "to infinity" or "forevermore".
Description
In context, it usually means "continue forever, without limit" and this can be used to describe a non-terminating process, a non-terminating ''repeating'' ...

.
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
The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...

of particle physics, the particles that make up an atom—quark
A quark () is a type of 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. All commonly o ...

s and electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...

s—are point particle
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take ...

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. Time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...

is similarly considered as infinitely divisible.
However according to the best currently accepted theory in physics, The Standard Model, there is a distance (called the Planck length, 1.616229(38)×10Max Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical ...

) and therefore a time interval (the amount of time which light takes to traverse that distance in a vacuum, 5.39116(13) × 10Quantum Gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...

, which unifies the hitherto incompatible theories of Quantum Mechanics and General Relativity.
In economics

Onedollar
Dollar is the name of more than 20 currencies. They include the Australian dollar, Brunei dollar, Canadian dollar, Hong Kong dollar, Jamaican dollar, Liberian dollar, Namibian dollar, New Taiwan dollar, New Zealand dollar, Singapore dollar, ...

, or one euro
The euro ( symbol: €; code: EUR) is the official currency of 19 out of the member states of the European Union
The European Union (EU) is a supranational political and economic union of member states that are located prim ...

, 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.
Although time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...

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 thefield
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...

of rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...

s is infinitely divisible (i.e. order theoretically dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

) means that between any two rational numbers there is another rational number. By contrast, the ring of integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s is not infinitely divisible.
Infinite divisibility does not imply gaplessness: the rationals do not enjoy the least upper bound property
In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if eve ...

. 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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...

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
In mathematics, an uncountable set (or uncountably infinite 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 num ...

. For a proof, see Cantor's first uncountability proof
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is unco ...

. Infinite divisibility alone implies infiniteness but not uncountability, as the rational numbers exemplify.
In probability distributions

To say that aprobability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

''F'' on the real line is infinitely divisible means that if ''X'' is any random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

whose distribution is ''F'', then for every positive integer ''n'' there exist ''n'' independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...

identically distributed random variables ''X''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 or space if these events occur with a known c ...

, the stuttering Poisson distribution, the negative binomial distribution
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non- ...

, and the Gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distr ...

are examples of infinitely divisible distributions — as are the normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...

, Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) func ...

and all other members of the stable distribution
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be sta ...

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
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...

, i.e., a stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

with stationary independent increments (''stationary'' means that for ''s'' < ''t'', the probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

of ''X''independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...

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 (13 June 1906 – 20 July 1985) was an Italian probabilist statistician and actuary, noted for the "operational subjective" conception of probability. The classic exposition of his distinctive theory is the 1937 "La prévision: ...

.
See also

*Divisible group In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive i ...

, a mathematical group in which every element is an arbitrary multiple of some other element
*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''. ...

*Salami slicing
Salami slicing tactics, also known as salami slicing, salami tactics, the salami-slice strategy, or salami attacks, is the practice of using a series of many small actions to produce a much larger action or result that would be difficult or unlawf ...

*Zeno's paradoxes
Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plura ...

References

External links

* Infinite Hierarchical Nesting of Matter (translation of Russian Wikipedia page) {{DEFAULTSORT:Infinite Divisibility Order theory Metaphysics Quantum mechanics de:Unendliche Teilbarkeit