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Quantity.
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, 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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