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Quantity or amount is a property that can exist as a
multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact of existence. The term has a history of use reaching back to antiquity, but took on a strictly political concept when it ...
or
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement.
Mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
,
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

time
,
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

distance
,
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

heat
, and
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

angle
are among the familiar examples of quantitative properties. Quantity is among the basic
classes Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
of things along with
quality Quality may refer to: Concepts *Quality (business), the ''non-inferiority'' or ''superiority'' of something *Quality (philosophy), an attribute or a property *Quality (physics), in response theory *Energy quality, used in various science disciplin ...
,
substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composition * Matter, anything that has mass and takes up space * Substance th ...
, change, and relation. Some quantities are such by their inner nature (as number), while others function 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. Under the name of multitude comes what is discontinuous and discrete and divisible ultimately into indivisibles, such as: ''army, fleet, flock, government, company, party, people, mess (military), chorus, crowd'', and ''number''; all which are cases of
collective nouns In linguistics, a collective noun is a word referring to a collection of things taken as a whole. Most collective nouns in everyday speech are not specific to one kind of thing, such as the word "group", which can be applied to people ("a group of ...
. Under the name of magnitude comes what is continuous and unified and divisible only into smaller divisibles, such as: ''matter, mass, energy, liquid, material''—all cases of non-collective nouns. Along with analyzing its nature and
classification Classification is a process related to categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such as Object (philosophy), objects, eve ...
, the issues of quantity involve such closely related topics as 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.


Background

In mathematics, the concept of quantity is an ancient one extending back to the time of
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

Aristotle
and earlier. Aristotle regarded quantity as a fundamental ontological and scientific category. In Aristotle's
ontology Ontology is the branch of philosophy that studies concepts such as existence, being, Becoming (philosophy), becoming, and reality. It includes the questions of how entities are grouped into Category of being, basic categories and which of these ...

ontology
, quantity or quantum was classified into two different types, which he characterized as follows: In his ''Elements'',
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
developed the theory of ratios of magnitudes without studying the nature of magnitudes, as Archimedes, but giving the following significant definitions: For Aristotle and Euclid, relations were conceived as whole numbers (Michell, 1993).
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

John Wallis
later conceived of ratios of magnitudes as
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

real numbers
: That is, the ratio of magnitudes of any quantity, whether volume, mass, heat and so on, is a number. Following this,
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...

Newton
then defined number, and the relationship between quantity and number, in the following terms:


Structure

Continuous quantities possess a particular structure that was first explicitly characterized by Hölder (1901) as a set of axioms that 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 ''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...
'' for any given property. The linear
continuum Continuum may refer to: * Continuum (measurement) Continuum theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, categorical theories or models explain variatio ...
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 that permit tests of hypothesized
observable In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
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 = ''r''a". A further generalization is given by the
theory of conjoint measurement The theory of conjoint measurement (also known as conjoint measurement or additive conjoint measurement) is a general, formal theory of continuous quantity. It was independently discovered by the French economist Gérard Debreu (1960) and by the A ...
, independently developed by French economist
Gérard Debreu Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist An economist is a practitioner in the social sciences, social science discipline of economics. The individual may also study, develop, and apply theories and concep ...
(1960) and by the American mathematical psychologist R. Duncan Luce and statistician
John Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the Cooley–Tukey FFT algorithm, Fast Fourier Transform (FFT) algorithm and box plot. The Tukey's range test, ...
(1964).


In mathematics

Magnitude (how much) and multitude (how many), the two principal types of quantities, are further divided as mathematical and physical. In formal terms, quantities—their ratios, proportions, order and formal relationships of equality and inequality—are studied by mathematics. The essential part of mathematical quantities consists of having a collection of variables, each assuming a set of values. These can be a set of a single quantity, referred to as a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
when represented by real numbers, or have multiple quantities as do
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
and
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

tensor
s, two kinds of geometric objects. The mathematical usage of a quantity can then be varied and so is situationally dependent. Quantities can be used as being
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
, arguments of a function, variables in an expression (independent or dependent), or probabilistic as in random and
stochastic Stochastic () refers to the property of being well described by a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wi ...
quantities. In mathematics, magnitudes and multitudes are also not only two distinct kinds of quantity but furthermore relatable to each other.
Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

Number theory
covers the topics of the discrete quantities as numbers: number systems with their kinds and relations.
Geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

Geometry
studies the issues of spatial magnitudes: straight lines, curved lines, surfaces and solids, all with their respective measurements and relationships. A traditional Aristotelian realist philosophy of mathematics, stemming from
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

Aristotle
and remaining popular until the eighteenth century, held that mathematics is the "science of quantity". Quantity was considered to be divided into the discrete (studied by arithmetic) and the continuous (studied by geometry and later
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
). The theory fits reasonably well elementary or school mathematics but less well the abstract topological and algebraic structures of modern mathematics.


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 physical quantities, by postulating that all material bodies marked by quantitative properties or physical dimensions 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
quanta
quanta
. A distinction has also been made between
intensive quantityIn grammar, an intensive word form is one which denotes stronger, more forceful, or more concentrated action relative to the root on which the intensive is built. Intensives are usually lexical formations, but there may be a regular process for formi ...
and
extensive quantity Extensive may refer to: * Extensive property * Extensive function * Extensional See also * Extension (disambiguation) {{Dab ...
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 The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

density
and
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

pressure
, while examples of extensive quantities are
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

energy
,
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
, and
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
.


In natural language

In human languages, including
English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading lan ...

English
,
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...
is a
syntactic category A syntactic category is a syntactic unit that theories of syntax In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics enco ...
, along with
person A person (plural people or persons) is a being that has certain capacities or attributes such as reason Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is ...

person
and
gender Gender is the range of characteristics pertaining to, and differentiating between femininity Femininity (also called womanliness or girlishness) is a set of attributes, behaviors, and roles generally associated with women A woman is ...

gender
. The quantity is expressed by identifiers, definite and indefinite, and
quantifiers Quantifier may refer to: * Quantifier (linguistics), an indicator of quantity * Quantifier (logic) * Quantification (science) See also

*Quantification (disambiguation) {{disambiguation ...
, definite and indefinite, as well as by three types of
noun A noun () is a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), meaning. In many l ...

noun
s: 1. count unit nouns or countables; 2.
mass nouns In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis includ ...
, uncountables, referring to the indefinite, unidentified amounts; 3. nouns of multitude (
collective noun In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most languag ...
s). 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 (
liter The litre (British English spelling) or liter (American English spelling) (SI symbols L and l, other symbol used: ℓ) is a metric units, metric unit of volume. It is equal to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cub ...
s) of milk, a continuous quantity * 2''πr'' metres, where ''r'' is the length of a
radius In classical geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

radius
of a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

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 type of
count data In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wit ...
) * a ''couple'' conventionally refers to two objects. * ''a few'' usually refers to an indefinite, but usually small number, greater than one. * ''quite a few'' also refers to an indefinite, but surprisingly (in relation to the context) large number. * ''several'' refers to an indefinite, but usually small, number – usually indefinitely greater than "a few".


See also

*
Dimensionless quantity In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric cur ...
*
Quantification (science) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
* Observable quantity * Numerical value equation


References

* Aristotle, Logic (Organon): Categories, in Great Books of the Western World, V.1. ed. by Adler, M.J.,
Encyclopædia Britannica The (Latin for "British Encyclopaedia") is a general knowledge English-language encyclopaedia which is now published exclusively as an online encyclopedia, online encyclopaedia. It was formerly published by Encyclopædia Britannica, Inc., ...
, Inc., Chicago (1990) * Aristotle, Physical Treatises: Physics, in Great Books of the Western World, V.1, ed. by Adler, M.J., Encyclopædia Britannica, Inc., Chicago (1990) * Aristotle, Metaphysics, in Great Books of the Western World, V.1, ed. by Adler, M.J., Encyclopædia Britannica, Inc., Chicago (1990) * Franklin, J. (2014)
Quantity and number
in ''Neo-Aristotelian Perspectives in Metaphysics'', ed. D.D. Novotny and L. Novak, New York: Routledge, 221-44. * Hölder, O. (1901). Die Axiome der Quantität und die Lehre vom Mass. ''Berichte über die Verhandlungen der Königlich Sachsischen Gesellschaft der Wissenschaften zu Leipzig'', Mathematische-Physicke Klasse, 53, 1-64. * Klein, J. (1968). ''Greek Mathematical Thought and the Origin of Algebra. Cambridge''. Mass:
MIT Press The MIT Press is a university press A university press is an academic publishing Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Traditionally, the ...
. * Laycock, H. (2006). Words without Objects: Oxford, Clarendon Press
Oxfordscholarship.com
* Michell, J. (1993). The origins of the representational theory of measurement: Helmholtz, Hölder, and Russell. ''Studies in History and Philosophy of Science'', 24, 185-206. * Michell, J. (1999). ''Measurement in Psychology''. Cambridge:
Cambridge University Press Cambridge University Press (CUP) is the publishing business of the University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowled ...
. * Michell, J. & Ernst, C. (1996). The axioms of quantity and the theory of measurement: translated from Part I of Otto Hölder's German text "Die Axiome der Quantität und die Lehre vom Mass". ''Journal of Mathematical Psychology'', 40, 235-252. * Newton, I. (1728/1967). Universal Arithmetic: Or, a Treatise of Arithmetical Composition and Resolution. In D.T. Whiteside (Ed.), ''The mathematical Works of Isaac Newton'', Vol. 2 (pp. 3–134). New York: Johnson Reprint Corp. * Wallis, J. ''Mathesis universalis'' (as quoted in Klein, 1968).


External links

{{Authority control Concepts in metaphysics Measurement Ontology