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Quantity

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Quantity is a kind of property which exists as magnitude or multitude.
   It is among the basic classes of things along with quality, substance,
   change, more than one and relation. Quantity was first introduced as
   quantum, an entity having quantity. Being a fundamental term, quantity
   is used to refer to any type of quantitative properties or attributes
   of things. Some quantities are such by their inner nature (as number),
   while others are functioning as states (properties, dimensions,
   attributes) of things such as heavy and light, long and short, broad
   and narrow, small and great, or much and little. A small quantity is
   sometimes referred to as a quantulum.

   Two basic divisions of quantity, magnitude and multitude (or number),
   imply the principal distinction between continuity ( continuum) and
   discontinuity.

   Under the names of multitude come what is discontinuous and discrete
   and divisible into indivisibles, all cases of collective nouns: army,
   fleet, flock, government, company, party, people, chorus, crowd, mess,
   and number. Under the names of magnitude come what is continuous and
   unified and divisible into divisibles, all cases of non-collective
   nouns: the universe, matter, mass, energy, liquid, material, animal,
   plant, tree.

   Along with analyzing its nature and classification, the issues of
   quantity involve such closely related topics as the relation of
   magnitudes and multitudes, dimensionality, equality, proportion, the
   measurements of quantities, the units of measurements, number and
   numbering systems, the types of numbers and their relations to each
   other as numerical ratios.

   Thus quantity is a property that exists in a range of magnitudes or
   multitudes. Mass, time, distance, heat, and angular separation are
   among the familiar examples of quantitative properties. Two magnitudes
   of a continuous quantity stand in relation to one another as a ratio,
   which is a real number.

Background

   The concept of quantity is an ancient one which extends back to the
   time of Aristotle and earlier. Aristotle regarded quantity as a
   fundamental ontological and scientific category. In Aristotle's
   ontology, quantity or quantum was classified into two different types,
   which he characterized as follows:

          'Quantum' means that which is divisible into two or more
          constituent parts of which each is by nature a 'one' and a
          'this'. A quantum is a plurality if it is numerable, a magnitude
          if it is measurable. 'Plurality' means that which is divisible
          potentially into non-continuous parts, 'magnitude' that which is
          divisible into continuous parts; of magnitude, that which is
          continuous in one dimension is length; in two breadth, in three
          depth. Of these, limited plurality is number, limited length is
          a line, breadth a surface, depth a solid. (Aristotle, book v,
          chapters 11-14, Metaphysics).

   In his Elements, Euclid developed the theory of ratios of magnitudes
   without studying the nature of magnitudes, as Archimedes, but giving
   the following significant definitions:

          A magnitude is a part of a magnitude, the less of the greater,
          when it measures the greater; A ratio is a sort of relation in
          respect of size between two magnitudes of the same kind.

   For Aristotle and Euclid, relations were conceived as whole numbers
   (Michell, 1993). John Wallis later conceived of ratios of magnitudes as
   real numbers as reflected in the following:

          When a comparison in terms of ratio is made, the resultant ratio
          often [namely with the exception of the 'numerical genus'
          itself] leaves the genus of quantities compared, and passes into
          the numerical genus, whatever the genus of quantities compared
          may have been. (John Wallis, Mathesis Universalis)

   That is, the ratio of magnitudes of any quantity, whether volume, mass,
   heat and so on, is a number. Following this, Newton then defined
   number, and the relationship between quantity and number, in the
   following terms: "By number we understand not so much a multitude of
   unities, as the abstracted ratio of any quantity to another quantity of
   the same kind, which we take for unity" (Newton, 1728).

Quantitative structure

   Continuous quantities possess a particular structure which was first
   explicitly characterized by Hölder (1901) as a set of axioms which
   define such features as identities and relations between magnitudes. In
   science, quantitative structure is the subject of empirical
   investigation and cannot be assumed to exist a priori for any given
   property. The linear continuum represents the prototype of continuous
   quantitative structure as characterized by Hölder (1901) (translated in
   Michell & Ernst, 1996). A fundamental feature of any type of quantity
   is that the relationships of equality or inequality can in principle be
   stated in comparisons between particular magnitudes, unlike quality
   which is marked by likeness, similarity and difference, diversity.
   Another fundamental feature is additivity. Additivity may involve
   concatenation, such as adding two lengths A and B to obtain a third A +
   B. Additivity is not, however, restricted to extensive quantities but
   may also entail relations between magnitudes that can be established
   through experiments which permit tests of hypothesized observable
   manifestations of the additive relations of magnitudes. Another feature
   is continuity, on which Michell (1999, p. 51) says of length, as a type
   of quantitative attribute, "what continuity means is that if any
   arbitrary length, a, is selected as a unit, then for every positive
   real number, r, there is a length b such that b = ra".

Quantity in mathematics

   Being of two types, magnitude and multitude (or number), quantities are
   further divided as mathematical and physical. Formally, quantities
   (numbers and magnitudes), their ratios, proportions, order and formal
   relationships of equality and inequality, are studied by mathematics.
   The essential part of mathematical quantities is made up with a
   collection variables each assuming a set of values and coming as
   scalar, vectors, or tensors, and functioning as infinitesimal,
   arguments, independent or dependent variables, or random and stochastic
   quantities. In mathematics, magnitudes and multitudes are not only two
   kinds of quantity but they are also commensurable with each other. The
   topics of the discrete quantities as numbers, number systems, with
   their kinds and relations, fall into the number theory. Geometry
   studies the issues of spatial magnitudes: straight lines (their length,
   and relationships as parallels, perpendiculars, angles) and curved
   lines (kinds and number and degree) with their relationships (tangents,
   secants, and asymptotes). Also it encompasses surfaces and solids,
   their transformations, measurements and relationships.

Quantity in physical science

   Establishing quantitative structure and relationships between different
   quantities is the cornerstone of modern physical sciences. Physics is
   fundamentally a quantitative science. Its progress is chiefly achieved
   due to rendering the abstract qualities of material entities into the
   primary quantities of physical things, by postulating that all material
   bodies marked by quantitative properties or physical dimensions, which
   are subject to some measurements and observations. Setting the units of
   measurement, physics covers such fundamental quantities as space
   (length, breadth, and depth) and time, mass and force, temperature,
   energy and quantum.

   Traditionally, a distinction has also been made between intensive
   quantity and extensive quantity as two types of quantitative property,
   state or relation. The magnitude of an intensive quantity does not
   depend on the size, or extent, of the object or system of which the
   quantity is a property whereas magnitudes of an extensive quantity are
   additive for parts of an entity or subsystems. Thus, magnitude does
   depend on the extent of the entity or system in the case of extensive
   quantity. Examples of intensive quantities are density and pressure,
   while examples of extensive quantities are energy, volume and mass.

Quantity in logic and semantics

   In respect to quantity, propositions are grouped as universal and
   particular, applying to the whole subject or a part of the subject to
   be predicated. Accordingly, there are existential and universal
   quantifiers. In relation to the meaning of a construct, quantity
   involves two semantic dimensions: 1. extension or extent (determining
   the specific classes or individual instances indicated by the
   construct) 2. intension (content or comprehension or definition)
   measuring all the implications (relationships and associations involved
   in a construct, its intrinsic, inherent, internal, built-in, and
   constitutional implicit meanings and relations).

Quantity in natural language

   In human languages, including English, number is a syntactic category,
   along with person and gender. The quantity is expressed by identifiers,
   definite and indefinite, and quantifiers, definite and indefinite, as
   well as by three types of nouns: 1. count unit nouns or countables; 2.
   mass nouns, uncountables, referring to the indefinite, unidentified
   amounts; 3. nouns of multitude (collective nouns). The word ‘number’
   belongs to a noun of multitude standing either for a single entity or
   for the individuals making the whole. An amount in general is expressed
   by a special class of words called identifiers, indefinite and definite
   and quantifiers, definite and indefinite. The amount may be expressed
   by: singular form and plural from, ordinal numbers before a count noun
   singular (first, second, third…), the demonstratives; definite and
   indefinite numbers and measurements (hundred/hundreds,
   million/millions), or cardinal numbers before count nouns. The set of
   language quantifiers covers ‘’a few, a great number, many, several (for
   count names); a bit of, a little, less, a great deal (amount) of, much
   (for mass names); all, plenty of, a lot of, enough, more, most, some,
   any, both, each, either, neither, every, no’’. For the complex case of
   unidentified amounts, the parts and examples of a mass are indicated
   with respect to the following: a measure of a mass (two kilos of rice
   and twenty bottles of milk or ten pieces of paper); a piece or part of
   a mass (part, element, atom, item, article, drop); or a shape of a
   container (a basket, box, case, cup, bottle, vessel, jar).

Further examples

   Some further examples of quantities are:
     * 1.76 litres (liters) of milk, a continuous quantity
     * 2πr metres, where r is the length of a radius of a circle expressed
       in metres (or meters), also a continuous quantity
     * one apple, two apples, three apples, where the number is an integer
       representing the count of a denumerable collection of objects
       (apples)
     * 500 people (also a count)
     * a couple conventionally refers to two objects

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