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Encyclopedia > 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. The philosopher Socrates about to take poison hemlock as ordered by the court. ... This article needs additional references or sources for verification. ... This article needs additional references or sources for verification. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... Matter is the substance of which physical objects are composed. ... Space has been an interest for philosophers and scientists for much of human history. ... A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ... Various denominations of currency, one form of money Money is any good or token that functions as a medium of exchange that is socially and legally accepted in payment for goods and services and in settlement of debts. ...

Contents

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). PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... Timaeus is a theoretical treatise of Plato in the form of a Socratic dialogue, written circa 360 BC. The work puts forward speculation on the nature of the physical world. ... Aristotle (Greek: Aristotélēs) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Andrew Pyle (born 17 March 1955) is a British philosopher who is an expert on the history of philosophical atomism. ...


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 [1] has developed a few theories[2] that claim to show that infinite divisibility is incorrect, and therefore atomism is correct. And other philosophers, such as Dean Zimmerman of Rutgers[3], claim to have developed evidence for the vindication of infinite divisibility. In natural philosophy, atomism is the theory that all the objects in the universe are composed of very small, indestructible elements - atoms. ... Peter Simons is a professor of philosophy at the University of Leeds and a director of the Franz Brentano Foundation. ... This page may meet Wikipedias criteria for speedy deletion. ...


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. Fig. ... Look up Ad infinitum in Wiktionary, the free dictionary. ...


As a result, the Greek word átomos (ἄτομος), 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 — 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. A partition of U into 6 blocks: an Euler diagram representation. ... Cookie cutter paradigm (CCP) refers to the following set of notions: The parts of a material object are defined by the parts of the space it occupies. ... A conundrum is a puzzling question. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... The six flavours of quarks and their most likely decay modes. ... e- redirects here. ... A point particle is an idealized particle heavily used in physics. ... 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. ...


Physical space has often is 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. A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ...


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. Max Karl Ernst Ludwig Planck (April 23, 1858 in Kiel, Germany – October 4, 1947 in Göttingen, Germany) was a German physicist. ... 1858 (MDCCCLVIII) is a common year starting on Friday of the Gregorian calendar (or a common year starting on Sunday of the 12-day-slower Julian calendar). ... Year 1947 (MCMXLVII) was a common year starting on Wednesday (link will display full 1947 calendar) of the Gregorian calendar. ... The Planck length, denoted by , is the unit of length approximately 1. ... In physics, the Planck time (tP), is the unit of time in the system of natural units known as Planck units. ...


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. United States one-dollar bill Canadian one-dollar coin (Loonie) One New Taiwan dollar Australian one-dollar coin 500 old Zimbabwean dollars The dollar (often represented by the dollar sign: $) is the name of the official currency in several countries, dependencies and other regions. ... “EUR” redirects here. ...


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. A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ... The 1920s is a decade that is sometimes referred to as the Jazz Age or the Roaring Twenties, usually applied to America. ...


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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... The word density or dense has a variety of senses in the physical, mathematical, and quantitative sciences. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... The integers are commonly denoted by the above symbol. ...


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 mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... A partition of U into 6 blocks: an Euler diagram representation. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ... Contrary to what most mathematicians believe, Georg Cantors first proof that the set of all real numbers is uncountable was not his famous diagonal argument, and did not mention decimal expansions or any other numeral system. ...


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). In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...


The Poisson distribution, 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. In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of exponentially distributed random variables, each of which has mean . ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is actually a family of probability distributions which are characterized by four parameters: α, β, μ and c , as well as the distributed value, x . The μ and c  are shift and scale parameters which...


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 XtXs depends only on ts; 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). In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that has stationary independent increments -- this phrase will be explained below. ... In the mathematics of probability, a stochastic process is a random function. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...


This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. Year 1929 (MCMXXIX) was a common year starting on Tuesday (link will display the full calendar) of the Gregorian calendar. ... Bruno de Finetti (Innsbruck, June 13, 1906 - Rome, July 20, 1985) was an Italian probabilist and statistician, noted for the operational subjective conception of probability. ...


See also indecomposable distribution. In probability theory, an indecomposable distribution is any probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables. ...


See also

Image:Bvn-small.png Probability distributionsview  talk  edit ]
Univariate Multivariate
Discrete: BenfordBernoullibinomialBoltzmanncategoricalcompound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniform • Yule-Simon • zetaZipf • Zipf-Mandelbrot Ewensmultinomialmultivariate Polya
Continuous: BetaBeta primeCauchychi-squareDirac delta function • Coxian • Erlangexponentialexponential powerFfading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gammageneralized extreme valuegeneralized hyperbolicgeneralized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussianinverse gamma (scaled inverse gamma) • KumaraswamyLandauLaplaceLévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speedNakagaminormal (Gaussian) • normal-gamma • normal inverse Gaussian • ParetoPearson • phase-type • polarraised cosineRayleigh • relativistic Breit-Wigner • Riceshifted Gompertz • Student's t • triangulartruncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigtvon MisesWeibullWigner semicircleWilks' lambda Dirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kentmatrix normalmultivariate normalmultivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantorconditionalequilibriumexponential familyinfinitely divisible • location-scale family • marginalmaximum entropyposteriorprior • quasi • samplingsingular

  Results from FactBites:
 
Infinite divisibility - Wikipedia, the free encyclopedia (1274 words)
By contrast, the ring of integers is not infinitely divisible.
The Poisson distributions, the normal distributions, and the gamma distributions are infinitely divisible probability distributions.
This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.
The Internet Classics Archive | Physics by Aristotle (7134 words)
Moreover, it is plain that everything continuous is divisible into divisibles that are infinitely divisible: for if it were divisible into indivisibles, we should have an indivisible in contact with an indivisible, since the extremities of things that are continuous with one another are one and are in contact.
That infinite time will not be occupied in passing over BE is evident if the time be taken as limited in one direction: for as the part will be passed over in less time than the whole, the time occupied in traversing this part must be finite, the limit in one direction being given.
And we now see that in most cases the fact that all the terms are divisible or infinite is a direct consequence of the fact that the thing that changes is divisible or infinite: for the attributes 'divisible' and 'infinite' belong in the first instance to the thing that changes.
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