Quantity
Quantity or amount is a property that can exist as a
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 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
Along with analyzing its nature and
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, Metaphysics, Book V, Ch. 11-14
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.
— Euclid, Elements
For Aristotle and Euclid, relations were conceived as
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,
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
Structure
Continuous quantities possess a particular structure that was first explicitly characterized by
In mathematics
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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.
A traditional Aristotelian realist philosophy of mathematics, stemming from 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). The theory fits reasonably well elementary or school mathematics but less well the abstract topological and algebraic structures of modern mathematics.[1]
In science
Establishing quantitative structure and relationships between different quantities is the cornerstone of modern science, especially but not restricted to physical sciences. Physics is fundamentally a quantitative science; chemistry, biology and others are increasingly so. Their 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
In natural language
This section may be confusing or unclear to readers. (May 2021) |
In human languages, including
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 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
The number one is recognized as a dimensionless base quantity.[5] Radians serve as dimensionless units for angular measurements, derived from the universal ratio of 2π times the radius of a circle being equal to its circumference.[6]
Dimensionless quantities play a crucial role serving as parameters in differential equations in various technical disciplines. In calculus, concepts like the unitless ratios in limits or derivatives often involve dimensionless quantities. In differential geometry, the use of dimensionless parameters is evident in geometric relationships and transformations. Physics relies on dimensionless numbers like the Reynolds number in fluid dynamics,[7] the fine-structure constant in quantum mechanics,[8] and the Lorentz factor in relativity.[9] In chemistry, state properties and ratios such as mole fractions concentration ratios are dimensionless.[10]See also
- Quantification (science)
- Observable quantity
- Numerical value equation
References
- ISBN 9781137400734.
- ^ "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. Retrieved 2011-03-22.
- ^ "SI Brochure: The International System of Units, 9th Edition". BIPM. ISBN 978-92-822-2272-0.
- ^ Mohr, Peter J.; Phillips, William Daniel (2015-06-01). "Dimensionless units in the SI". Metrologia. 52.
- ISSN 0026-1394.
- ISBN 978-0-8135-2898-4.
- ISBN 978-0-07-717359-3.
- .
- .
- ISSN 0743-7463.
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (July 2010) |
- 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).