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one of the basic mathematical concepts, whose meaning has undergone a number of extensions with the development of mathematics.
I. Even as early as in Euclid’s Elements (third century B.C.) the properties of quantities had been meticulously formulated; in order to distinguish those quantities from subsequent extensions, they are now called positive scalar quantities. This initial concept of quantity is a direct extension of more specific concepts (length, area, volume, mass, and so on).
Each specific kind of quantity is associated with a definite method of comparing physical bodies or other objects—for example, in geometry, segments are compared by means of superposition, and this comparison leads to the concept of length: two segments have the same length if they coincide upon superposition, but if one segment is superposed on part of another and does not cover it completely, then the length of the first is less than the length of the second. More complex methods, which are necessary for the comparison of plane figures by area or three-dimensional bodies by volume, are well known.
In accordance with the foregoing, within the bounds of a system of all similar quantities (that is, within the bounds of a system of all lengths, all areas, or all volumes), a relationship of inequality was established: two quantities a and b of a similar kind either coincide (a = b), or the first is less than the second (a < b), or the second is less than the first (b < a). The manner in which the meaning of the operation of addition is established for each kind of quantity is also well known for cases of lengths, areas, and volumes. Within the bounds of each of the systems of similar quantities that is being examined, the relationship a < b and the operation a + b = c have the following properties:
(1) for any a and b, one and only one of three relationships exists: either a = b, or a < b, or b < a;
(2) if a < b and b < c, then a < c (transitivity of the relationships “less than” and “greater than”);
(3) for any two quantities a and b there exists a uniquely defined quantity c = a + b;
(4) a + b = b + a (commutativity of addition);
(5) a + (b + c) = (a + b) + c (associativity of addition);
(6) a + b > a (monotonicity of addition);
(7) if a > b, then there exists one and only one quantity c for which b + c = a (possibility of subtraction);
(8) for any quantity a and natural number n there exists a quantity b such that nb = a (possibility of division);
(9) for any quantities a and b there exists a natural number n such that a < nb. This property is called Eudoxius’ axiom or Archimedes’ axiom. The theory of measurement of quantities as developed by the ancient Greek mathematicians is based on this property, as are the more elementary properties (l)-(8).
In we take any length l as a standard unit, then the system s′ of all lengths within a rational relationship to / satisfies the requirements of properties (l)-(9). The existence of incommensurable segments (the discovery of which has been ascribed to Pythagoras, sixth century B.C.) indicates that the system s′ still does not encompass the system s of all lengths in general.
In order to obtain an entirely complete theory of quantities, one or another supplementary axiom of continuity must be added to the requirements of (l)-(9)—for example:
(10) if the sequences of quantities a1 < a2 < … < … < b2 < b1 have the property that bn - an < c for any quantity c at a sufficiently large number n, there exists a unique quantity x, which is greater than all an and less than all bn.
Properties (1)-(10) also define the entirely modern concept of the system of positive scalar quantities. If in such a system any quantity l is selected as a standard unit of measurement, then all the remaining quantities in the system are uniquely represented in the form a = αl, where α is a positive, real number.
II. The consideration of directed segments on a straight line, of velocities that may have two opposite directions, and such naturally leads to the extension of the concept of a scalar quantity that is basic in mechanics and physics. A system of scalar quantities in this sense includes zero and negative quantities within itself in addition to positive quantities. Selecting any positive quantity l in such a system as the standard unit of measurement, all the remaining quantities of the system are expressed in the form a = αl, where α is a real number that is positive, negative, or equal to zero. Of course, a system of scalar quantities in this sense may also be characterized axiomatically, without relying on the concept of number. In order to do this, certain changes would have to be made in the requirements of properties (1)-(10), which were used above to characterize the concept of a positive scalar quantity.
III. In a more general sense of the word, vectors, tensors, and other nonscalar quantities may be called quantities. Such quantities may be added, but the inequality relationship (a < b) becomes meaningless for them.
IV. A well-known role in certain more abstract mathematical studies is played by “non-Archimedean” quantities, which have in common with ordinary scalar quantities the general characteristic that for them the usual properties of inequalities are retained, but axiom (9) is not fulfilled. (It is retained for scalar quantities in the sense of point II, with the stipulation that b > 0.)
V. Since the system of real positive numbers satisfies properties (1)—(10) as listed above, and the system of all real numbers has all the properties of scalar quantities, then it is completely justifiable to call the real numbers themselves quantities. This is especially acceptable in examining variable quantities. If any particular quantity—for example, the length l of a heated metal bar—changes in the course of time, then there is also a change in the number that is measuring it: x = l/l0 (with a constant unit of measurement l0). This number x itself—which changes in the course of time—may be called a variable quantity, and it may be said that in any sequential moments of time t1, t2, …, x takes on the “numerical values” x1, x2, … .
In traditional mathematical terminology it is not acceptable to speak of “variable numbers.” However, the point of view that numbers, like lengths, volumes, and so on, are specific instances of quantity and, like all quantities, may be both variable and constant, is more logical. It is equally justifiable to consider variable vectors, tensors, and such in this way.
REFERENCELebesgue, H. Ob izmerenii velichin, 2nd ed. Moscow, 1960. (Translated from French.)
A. N. KOLMOGOROV
(duration), the length of a sound in time. In phonetics, quantity is opposed to the quality of a sound. It is usually measured in milliseconds; the length of speech sounds varies within an extremely broad range, from 20–30 msec to several hundred milliseconds. Absolute quantity, which depends on the speed of the utterance, is not as important for a language as relative quantity (relative duration, or differences in the degree of length of the sound).
In many languages, quantity is used as a distinctive feature of phonemes, particularly vowels—for example, Finnish vapa, “twig” or “switch,” and vapaa, “free.” In most instances two degrees of duration, long and short, are opposed (for example, in German and a number of Turkic languages); sometimes even three degrees are distinguished (as in Estonian). The opposition of long and short consonants is less frequent, although it is possible (for example, in Ukrainian and Dagestani). In many languages, long sounds emerge as the result of fusion of two phonemes at a morphological boundary (compare Russian voobrazhat’, “to imagine,” and vvodit’, “to introduce”). In languages in which quantity has no distinctive function, the duration of sounds depends on location in the word, the adjacent sounds, or the location of stress. Quantity is often one of the markers of the stressed syllable, as for example, in Russian. Along with voice pitch, quantity can be used as a means of intonation.
L. R. ZINDER
the category expressing the external, formal interrelation between objects or their parts, as well as between properties and relations: their magnitude and number and the degree of manifestation of a particular property. The first attempts at a special analysis of the problem of quantity were made by the Pythagoreans, who studied the nature of numbers. Aristotle considered quantity to be a special category: “ ‘Quantity’ means that which is divisible into constituent parts, each or every one of which is by nature some one individual thing. Thus plurality, if it is numerically calculable, is a kind of quantity; and so is magnitude, if it is measurable. ’Plurality’ means that which is potentially divisible into noncontinuous parts; and ’magnitude’ that which is potentially divisible into continuous parts” (Metaphysics, V, 13, 1020a 7–14; Russian translation, Moscow, 1975).
In view of the development of natural science and mathematics, the problem of quantity occupies a special place in modern history. R. Descartes considered quantity to be the real spatial and temporal determination of bodies, which is expressed through number, measure, and magnitude. According to G. Hegel, quantity differs from quality in that while quality characterizes a thing unambiguously in such a way that with a change in its quality the thing becomes something different, a temporary quantitative changes need not transform it into another thing.
In the classics of Marxism-Leninism the category of quantity is considered primarily in connection with the establishment of quantitative (mathematical) lawlike regularities that are linked with qualitative transformations of things. “It is impossible to change the quality of any body without an addition or subtraction of matter or motion, that is, without a quantitative change of this body” (F. Engels, in K. Marx and F. Engels, Soch., 2nd ed., vol. 20, p. 385).
Each aggregate of objects is a multiplicity. If it is finite, it can be counted. All counting consists in the repeated positing of unity. For example, the number 40 is a quantitative characterization of a multiplicity consisting of 40 objects, whether persons or trees. Consequently, numbers and magnitudes are the formal, external, or (in Hegel’s term) “indifferent” aspect of qualitative relations. There are large and small, long and short things, fast and slow motions, high and low levels of development, and so forth, all of which can be measured with the aid of definite standards, such as meters or seconds. In order to establish the quantitative determination of an object, its constituent elements —spatial dimensions, rate of change, degree of development— are compared, using a definite standard as a unit of counting and measurement. The more complex the phenomenon, for example, phenomena in the sphere of morality, politics, and the aesthetic perception of the world, the more difficult it is to study it by quantitative methods. In these cases one has recourse to standards of a different kind. In the process of comprehending the real world, both historically and logically, cognition of quality precedes cognition of quantitative relations. Science moves from qualitative evaluations and descriptions of phenomena to the establishment of quantitative lawlike regularities.
Quantity occurs in unity with the qualitative determinateness of phenomena, things, and processes; this unity constitutes their measure. Up to a certain point a change in the quantitative determination of things does not affect their quality. Beyond this point quantitative changes are accompanied by a change in quality.
REFERENCESEngels, F. Anti-Dühring. Soch., 2nd ed., vol. 20.
Engels, F. “Dialektika prirody.” Soch., vol. 20.
Lenin, V. I. “Filosofskie tetradi.” Poln. sobr. soch., 5th ed., vol. 29.
Matematika, ee soderzhanie, metody i znachenie, vol. 1. Moscow, 1956.
A. G. SPIRKIN