Quantity is a property that can exist as a multitude or magnitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. 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. Two basic divisions of quantity, magnitude and multitude, imply the principal distinction between continuity (continuum) and discontinuity. Under the name 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 name of magnitude come what is continuous and unified and divisible into divisibles, all cases of non-collective nouns: matter, mass, energy, liquid, material. 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.
In mathematics, the concept of quantity is an ancient one extending
back to the time of
'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,
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.
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).
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 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". A further generalization is given by the theory of
conjoint measurement, independently developed by French economist
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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 tensors,
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
(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.
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 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". OPEC has a few members
Dimensionless quantity Quantification (science) Observable quantity
^ J. Franklin, An Aristotelian Realist Philosophy of Mathematics, Palgrave Macmillan, Basingstoke, 2014, pp. 31-2.
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Aristotle, Logic (Organon): Categories, in Great Books of the Western World, V.1. ed. by Adler, M.J., Encyclopædia Britannica, 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. 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. 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).
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