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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. Though its use dates back to antiquity, the term first entered into the lexicon of political philosophy w ...
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 multiple of a
unit of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multi ...
.
Mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
,
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, ...
,
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
,
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
, and
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
are among the familiar examples of quantitative properties. Quantity is among the basic classes 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 discipl ...
, substance, 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. 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, 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 and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
and earlier. Aristotle regarded quantity as a fundamental ontological and scientific category. In Aristotle's
ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exi ...
, quantity or quantum was classified into two different types, which he characterized as follows: In his ''Elements'',
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
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 later conceived of ratios of magnitudes as real numbers: 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:

# 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'' 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 that permit tests of hypothesized
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
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 Am ...
, independently developed by French economist Gérard Debreu (1960) and by the American mathematical psychologist R. Duncan Luce and statistician John Tukey (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 when represented by real numbers, or have multiple quantities as do vectors and
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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, arguments of a function, variables in an
expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, o ...
(independent or dependent), or probabilistic as in random and stochastic 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 pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
covers the topics of the discrete quantities as numbers: number systems with their kinds and relations.
Geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
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 In the philosophy of mathematics, Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be immanently realized in the physical world (or in any other world there might be). It contrasts wit ...
, stemming from
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 ph ...
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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
). 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. 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 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. Mathematical ...
and
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, while examples of extensive quantities are
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
, and
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
.

# In natural language

In human languages, including English,
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
is a
syntactic category A syntactic category is a syntactic unit that theories of syntax assume. Word classes, largely corresponding to traditional parts of speech (e.g. noun, verb, preposition, etc.), are syntactic categories. In phrase structure grammars, the ''phrasal c ...
, along with
person A person ( : people) is a being that has certain capacities or attributes such as reason, morality, consciousness or self-consciousness, and being a part of a culturally established form of social relations such as kinship, ownership of prope ...
and
gender Gender is the range of characteristics pertaining to femininity and masculinity and differentiating between them. Depending on the context, this may include sex-based social structures (i.e. gender roles) and gender identity. Most cultures ...
. The quantity is expressed by identifiers, definite and indefinite, and quantifiers, definite and indefinite, as well as by three types of
noun A noun () is a word that generally functions as the name of a specific object or set of objects, such as living creatures, places, actions, qualities, states of existence, or ideas.Example nouns for: * Living creatures (including people, alive, ...
s: 1. count unit nouns or countables; 2.
mass nouns In linguistics, a mass noun, uncountable noun, non-count noun, uncount noun, or just uncountable, is a noun with the syntactic property that any quantity of it is treated as an undifferentiated unit, rather than as something with discrete elemen ...
, uncountables, referring to the indefinite, unidentified amounts; 3. nouns of multitude (
collective noun 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. For example, the collective noun "group" can be applied to people (" ...
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 ( liters) of milk, a continuous quantity * 2''πr'' metres, where ''r'' is the length of a
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
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) * 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".

* Dimensionless quantity *
Quantification (science) In mathematics and empirical science, quantification (or quantitation) is the act of counting and measuring that maps human sense observations and experiences into quantities. Quantification in this sense is fundamental to the scientific method ...
* 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 Encyclopædia") is a general knowledge English-language encyclopaedia. It is published by Encyclopædia Britannica, Inc.; the company has existed since the 18th century, although it has changed ownership various t ...
, 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 affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts (United States). It was established in 1962. History The MIT Press traces its origins back to 1926 when MIT publish ...
. * 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 is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...
. * 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).