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definitions - quantity

quantity (n.)

1.how much there is or how many there are of something that you can quantify

2.an adequate or large amount"he had a quantity of ammunition"

3.(math)the concept that something has a magnitude and can be represented in mathematical expressions by a constant or a variable

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Merriam Webster

QuantityQuan"ti*ty (?), n.; pl. Quantities (#). [F. quantite, L. quantitas, fr. quantus bow great, how much, akin to quam bow, E. how, who. See Who.]


1. The attribute of being so much, and not more or less; the property of being measurable, or capable of increase and decrease, multiplication and division; greatness; and more concretely, that which answers the question “How much?”; measure in regard to bulk or amount; determinate or comparative dimensions; measure; amount; bulk; extent; size. Hence, in specific uses: (a) (Logic) The extent or extension of a general conception, that is, the number of species or individuals to which it may be applied; also, its content or comprehension, that is, the number of its constituent qualities, attributes, or relations. (b) (Gram.) The measure of a syllable; that which determines the time in which it is pronounced; as, the long or short quantity of a vowel or syllable. (c) (Mus.) The relative duration of a tone.

2. That which can be increased, diminished, or measured; especially (Math.), anything to which mathematical processes are applicable.

☞ Quantity is discrete when it is applied to separate objects, as in number; continuous, when the parts are connected, either in succession, as in time, motion, etc., or in extension, as by the dimensions of space, viz., length, breadth, and thickness.

3. A determinate or estimated amount; a sum or bulk; a certain portion or part; sometimes, a considerable amount; a large portion, bulk, or sum; as, a medicine taken in quantities, that is, in large quantities.

The quantity of extensive and curious information which he had picked up during many months of desultory, but not unprofitable, study. Macaulay.

Quantity of estate (Law), its time of continuance, or degree of interest, as in fee, for life, or for years. Wharton (Law Dict. ) -- Quantity of matter, in a body, its mass, as determined by its weight, or by its momentum under a given velocity. -- Quantity of motion (Mech.), in a body, the relative amount of its motion, as measured by its momentum, varying as the product of mass and velocity. -- Known quantities (Math.), quantities whose values are given. -- Unknown quantities (Math.), quantities whose values are sought.

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Wikipedia

Quantity

                   

Quantity is a property that can exist as a magnitude or multitude. 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. 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, 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.

Contents

  Background

In Mathematics the concept of quantity is an ancient one extending 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 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".

  Quantity in mathematics

Being of two types, magnitude and multitude (or number), quantities are further divided as mathematical and physical. In formal terms, 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 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 physical quantities, 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.

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

  References

  • Aristotle, Logic (Organon): Categories, in Great Books of the Western World, V.1. ed. by Adler, M.J., Encyclopaedia Britannica, Inc., Chicago (1990)
  • Aristotle, Physical Treatises: Physics, in Great Books of the Western World, V.1, ed. by Adler, M.J., Encyclopaedia Britannica, Inc., Chicago (1990)
  • Aristotle, Metaphysics, in Great Books of the Western World, V.1, ed. by Adler, M.J., Encyclopaedia Britannica, Inc., Chicago (1990)
  • 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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