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One of the supreme categories of being, and as such incapable of strict definition. According to Aristotle, quantity answers the question "how much?" (Gr. ποσόν; Lat. quantum ). It may be described as that by which a thing is said to be large or small, or to have part outside of part, or to be divisible into parts. Most philosophers in the Western tradition admit that something like quantity exists, although they are not all agreed as to precisely what it is. Those who may be identified as materialists or as mechanists, such as lucretius and Thomas hobbes, regard quantity as a primary attribute of bodies, even though they conceive matter as bereft of all qualities. Others, such as John locke and Sir Isaac Newton, list the quantitative attributes of material entities as primary or universal qualities, evidently understanding quality in the broad sense of attribute. Scholastic philosophers, following St. thomas aquinas, distinguish between physical and mathematical quantities, i.e., quantities that inhere in bodies and those that are abstracted from them. René descartes, in considering extension the one primary attribute of material substance, also insisted on the reality of quantity.

Those who have denied the reality of quantity have generally done so because of an idealistic theory of knowledge or because of a dynamist theory of physical reality. Absolute idealists, denying the real existence of bodies, and therefore of quantity and extension, hold that mind or spirit is the only reality, of which bodies are only a representation. Immanuel Kant, though not denying the existence of quantity, held that it was completely unknowable in itself and made it a pure category of the mind. G. W. von leibniz also subscribed to a subjectivist view of quantity, insofar as he taught that reality is composed of simple and unextended entities, or monads, which accounted for the appearance of extension (see dynamism; monad).

Many of these difficulties arise from the fact that quantity can be studied under various formalities. For example, it is considered differently by the logician, who sees it as a measure of substance and is concerned with the properties that distinguish it from the other categories; by the metaphysician, who sees it as a mode of being and considers its ontological properties; and by the mathematician, who considers quantity and the relations that follow from it as the proper subject of his science. To clarify these different formalities, this article presents an analysis of quantity as studied in logic, in metaphysics, and in mathematics, with particular reference to the Aristotelian origins and elaboration of the doctrine.

In Logic. The logician, following Aristotle, considers quantity as the first category after substance (Cat. 4b 206a 36). For him, quantity is the measure of substance in its material aspect. The logician divides quantity into the continuous and the discrete, although he is not concerned with the actual existence of either, considering only what these terms imply in their signification. Gross examples of both continuous and discrete quantity are available to the senses. A bookshelf appears to be continuous, even though the books resting on it are discrete; whether the bookshelf, upon further analysis, actually turns out to be discrete is not a logical question but one pertaining to physical science.

Parts. As a somewhat parallel division, Aristotle lists quantities whose parts have a position relative to other parts and quantities whose parts have no such relative position. The position to which he refers is the order of the parts in the whole, considering these parts in themselves, which is to be distinguished from position as a separate category, considering the parts of a whole with respect to place (see location [ubi]). The former position requires permanence, situation, and continuity. Permanence of parts is necessary for there to be order, since a relation of order among parts requires that they coexist. Situation further clarifies this relation: each part must be somewherei.e., above or below, before or after, etc.with respect to the others. This does not mean that as parts they are in place in any proper sense, since only bodies have place. The last requirement, continuity, explains how position properly distinguishes continuous quantity from discrete. By continuity is meant that all parts are joined by some common term. Thus the parts of a line are united by a point; those of the surface, by a line; and those of the body, by a surface.

Instances. As a logician Aristotle gives number and speech as instances of discrete quantity, and lines, surfaces, and bodies, together with time and place, as instances of continuous quantity. By speech the logician refers to sounds made by the voice and grouped into syllables that may be long or short; in pronunciation such sounds are separated, thus discrete. Among the examples of continuous quantity, the logician regards time and place as extrinsic measures of material substance, as opposed to line, surface, and body, which are intrinsic measures.

In addition to the above quantities, which the logician considers as proper, or per se, there are also things that are quantified by reason of something else, or per accidens. White, for example, is greater or smaller because of the surface in which it is found. An action is longer or shorter by reason of its duration or time. These and others like them are not considered to be quantified as such, but merely in reference to something else.

Properties. Finally, as is proper to his science, the logician gives three properties by which the quantified can be distinguished from what is not quantified. The first is that quantity has no contrary, such as is found in quality. Second, quantity does not permit of more or less; e.g., one line is not more so than another, nor is one pentad more five than another. Third, quantities can be said to be equal and unequal.

In Metaphysics. The investigation by the metaphysician is quite different from the foregoing (Meta. 1020 a734). He treats quantity not as the measure of substance but as it depends upon being. Thus he studies the various modes of being that are found in the quantified. This method of dealing with the quantified leaves out certain kinds of quantity that are mentioned in the Categories. Thus place, as an extrinsic measure, does not indicate a different mode of being from that of surface. Also time, which is quantified by motion, is not considered in metaphysics as a per se quantity, nor is speech, which is quantified by both motion and time.

Definition. The metaphysician provides a proper definition also of the quantified, namely, that which is divisible into constituent parts each of which is naturally apt to be a unit and a particular being. Thus plurality is a quantity if it is numberable, and magnitude if it is measurable. Plurality differs from magnitude in that it is divisible into noncontinuous parts, whereas magnitude is divisible into continuous parts.

Number. Since number by definition is a plurality, the question arises how it can be one and thus a true species of quantity. The answer to this question, involving the notion of homogeneity, also presupposes that number arises from the division of the continuum. That number, as a plurality measured by the unit, arises from such division can be seen from the example of the line divided into two or more parts. The number of parts is given by the application of one of them to the multitude that results from the division. This implies that the unit or measure be homogeneous with the measured, and so a line must be measured by a line, and a surface by a surface.

In this understanding, the unity of number is possible despite its discontinuity, multiplicity, or aggregation. Yet its unity is different from that of substance, being a unity of order. The unity of number is an ordering of all the parts of the whole under one part or unit. Only a material or quantitative multitude is capable of such orderings; thus the numbering of nonmaterial things is an analogical use of the term "numbering" (see multitude). The last unit of the plurality is what gives the order, or number, to the other units; it determines that the particular multitude be three, seven, or some other number. In this way a heterogeneous whole with homogeneous parts can be expressed as a numbered plurality.

Not every order, however, gives a per se unity to that which is ordered. The unity of an army or of a city is one of order; yet the being that results is an accidental being. This is so because the ordering of these is one of relation only, and relation is not sufficient to constitute an essential unity. The unity of quantitative order, on the other hand, derives from the homogeneity of its parts and thus constitutes an essential unity.

Magnitude. Some question whether line and surface are true species of quantity because of what they call the imperfections contained in these notions. They maintain that just as point is an indivisible and is quantitative only reductively as a principle of the line, so line and surface are indivisibles and thus imperfect quantities. They argue further that these exist only in the body and have extension and measure only because of the body. Thus line and surface cannot be true species of quantity.

The answer to this objection is to be found in the description of quantity given above: that which is divisible into constituent parts each of which is naturally apt to be a unit and a particular being. Since this description is verified of both line and surface, they are proper species of quantity. If it be said that these have this divisibility by reason of body, and are thus quantified per accidens, it should be pointed out that the reverse is also true. Body is divisible by reason of surface and line.

Properties. Objections such as these serve to make the properties of quantity, as set forth by the metaphysician, more precise. The metaphysician sees order as the most important aspect of quantity. It is by reason of order that quantity effects a distinction of the parts of material substance. Without quantity such substance would have parts only in a confused way. The ordering of parts in a material subject, in fact, makes quantity a primary factor in the individuation of material substance.

The order of homogeneous parts is the basis of other properties of quantity: divisibility, extension, measurability, and impenetrability. If such parts are to be distinct, they must be separated, and this requires extension. If such extension results in an actual plurality, the quantity is discrete and the unity that of number. If the distinction produces only potential parts, the quantity is continuous and the extension that of line, surface, or body. The latter extension, in turn, is the basis of divisibility and impenetrability.

Although these properties are real and are founded in the nature of material things, they can nevertheless be separated from their subject. For example, in the Sacrament of the Eucharist, the effects of quantity as well as of other accidents are found separated from the substances of bread and wine, which are changed into the Body and Blood of Christ while retaining their former appearances or accidents (see transubstantiation). Certain other miracles performed by Christ, as when He appeared through closed doors, would indicate that impenetrability also can be separated from material substance (see glorified body).

In Mathematics. The treatment of quantity by the mathematician is different from that of the logician or of the metaphysician. The science of mathematics considers its subject not only as abstracted from singulars, as does every science, but also as abstracted from sensible matter (see abstraction; sciences, classification of). Such a consideration is possible because quantity is the first accident of substance and, by reason of its role in determining the parts of a material thing, serves as the foundation of all other accidents. Since it is thus prior to quality, quantity can be considered without the qualities that render substance sensible.

Procedure. In his proper treatment of quantity the mathematician begins by defining different species of quantity, e.g., unit, number, line, surface, and circle. With these definitions he demonstrates properties that differ from those of the logician and the metaphysician. In fact, his method of abstracting renders many properties of quantity more evident than those discerned through the more material consideration of the metaphysician. This was one of the reasons why the mathematical arts were called disciplinales by the scholastics; they are the easiest for the student to grasp, since they require little experience and their proofs depend on constructions that are controlled by the imagination. Yet the mathematician is not concerned with quantity exclusively. As the development of modern mathematics has shown, he can be concerned also with relations and qualities that are only remotely connected with quantity. His science, however, considers these entities in abstraction from the data of sense perception and precisely as they can be visualized in the imagination or through some form of symbolic construction.

Infinity. This explains why the mathematician, for example, can speak of lines being divisible to infinity. In the division of the mathematical line it is obvious that every division results in line segments that are further divisible. This is so because the only relevant consideration is that set by requirements of quantity itself. From the viewpoint of quantitative extension, there is no reason division should stop at any particular place. In the case of natural things, however, some point exists beyond which division cannot continue. In the division of water, for example, a point is reached when further division does not give water but some other thing; thus the division of water is terminated. The same is true for all natural substances, whose forms require a minimum of matter for their existence. This is not true of the quantities studied in mathematics, where one is not concerned with existence in the extramental sense.

See Also: continuum; extension; indivisible; mathematics, philosophy of.

Bibliography: m. j. adler, ed., The Great Ideas: A Syntopicon of Great Books of the Western World, 2 v. (Chicago 1952); v. 2, 3 of Great Books of the Western World 2:527545. s. caramella, Enciclopedia filosofica, 4 v. (Venice-Rome 1957) 3:179297. f. selvaggi, Cosmologia (Rome 1959). h. g. apostle, Aristotle's Philosophy of Mathematics (Chicago 1952). v. e. smith, The General Science of Nature (Milwaukee 1958). john of st. thomas, Cursus philosophicus thomisticus, ed. b. reiser, 3 v. (new ed. Turin 193037). t. de vio cajetan, Scripta philosophica: Commentaria in praedicamenta Aristotelis, ed. m. h. laurent (Rome 1939).

[r. a. kocourek]

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