Continuous

CONTINUOUS. - The notion of the continuum is connected with that of extension, which is a primitive concept not susceptible of a true definition. Extended is that which consists of parts distinct in space, which is divisible and measurable. Aristotle distinguishes it into three species: consecutive, contiguous, and continuous (Phys., VI, 1, 231 a 21). Consecutive is that extended thing whose parts are distinct and separate; contiguous the extended thing in which the parts, though distinct, touch one another; continuous the extended thing that has no parts distinct in act, but only potentially. The characteristics of the continuum are therefore unity and the absence of interruption, whereas the consecutive supposes on the other hand plurality with interruption and the contiguous plurality without interruption. The continuum can be static: the physical or mathematical one; or flowing: movement and time. An essential property of the continuum is divisibility into integral parts, that is, into parts that have the same nature as the whole from which they derive; e.g., a segment is divisible into smaller segments, a surface into surfaces, etc. From the continuum, besides integral parts, there result elements of a species different from the whole, with respect to which they are indivisible, e.g., the point with respect to the line, the surface, and the volume, the line with respect to the surface, etc.

I. PHILOSOPHY

Important are the consequences to which the concept of the continuum gives rise. If the continuum is divisible, although it is an ontologically one entity, it must in some way contain the parts into which it is divisible. On the other hand, these parts cannot exist in the continuum as distinct and determinate entities, because the continuum is one, and what is essentially one cannot be essentially multiple. The continuum is therefore not the aggregate of its parts. The parts in the continuum are purely potential realities; that is, although they do not exist in act, they have in the continuum the foundation of their being and come to exist as distinct and actual entities as a result of division.

The division of the continuum into integral parts can proceed indefinitely without ever arriving at indivisible elements. In fact, from the division of a continuum there result integral parts, which are themselves also continuous and can in turn be divided into continuous integral parts. For example, a segment, since it has non-coincident endpoints, will always have an intermediate point distinct from both endpoints, which divides it into two smaller segments. This is all the clearer if one bears in mind that the point is unextended and that by adding point to point one will always have a point; therefore between two distinct points of a segment there always exist indefinitely many points. For each of the two segments resulting from the division the same reasoning can be repeated. And so on indefinitely. This indefinite divisibility of the continuum can never be exhausted by its very nature. Therefore from the division of the continuum there will never result parts infinite in number. Aristotle had already noted this: "The divisible to infinity is not so in potency that it can also be given divided in act in itself and by itself... for the division can never fail" (Met., IX, 6, 1408 b 13 ff.). From these considerations it follows that the continuum cannot be composed of extended indivisibles, both because the parts of the continuum are not actual, and because these ultimate unextended parts, whose union should give the continuum, do not exist, inasmuch as they can never result from the division of the continuum. It is therefore not exact to say that the line is composed of points, the surface of lines, the volume of surfaces. One can indeed determine, in a line, as great a number of points as one wishes, but it is not possible to determine all the points of the line.

The concept of the continuum has given rise to considerable difficulties: 1) Zeno and Democritus observed that, if the division of the continuum implies a potentially infinite multitude of parts, since potency must be able to be actualized, it follows that the divisibility of the continuum leads to admitting an actually infinite multitude of parts; which is impossible, because there cannot be an actually infinite number; the continuum is therefore indivisible (cf. Aristotle, De Gen. et Corr., I, 2, 316 a 14 ff.). Aristotle's answer has already been indicated: the divisibility of the continuum is actualized successively and can never be exhausted; the number of parts will grow indefinitely but will never be infinite. 2) Another difficulty originates from a false metaphysical presupposition. Leibniz affirms that, metaphysically, the simple is prior to the composite, which is an aggregate of simples; in the real continuum (bodies), the parts (the monads) are prior to the whole, which is therefore an aggregate of simple, that is, indivisible, parts. On the contrary, according to mathematics, in the division of the continuum one never arrives at unextended elements, therefore the continuum cannot be an aggregate of unextended things. The real continuum therefore implies - according to Leibniz - an antinomy between metaphysics and mathematics, and is therefore absurd. The ideal continuum, on the other hand, the one thought by us, is not contradictory, being prior to the parts, according to metaphysics, and in agreement with mathematics: "Dans l'idéal le tout est antérieur aux parties... les parties ne sont que potentielles; mais dans le réel, le simple est antérieur aux assemblages, les parties sont actuelles, sont avant le tout." (Lettre à Remond, ed. C. I. Gerhardt, Die phil. Schriften von G. W. Leibniz, III, Berlin 1890, p. 622).

Kant too, in the pre-critical period, under the influence of Leibnizian metaphysics, repeats Leibniz's difficulty: "Cum illa (metaphysics) spatium (therefore the real continuum) in infinitum divisibile esse praefracte neget, haec (mathematics) eadem, qua cetera solet, certitudine asseverat" (Monadologia physica, Bari 1923, p. 2). To resolve this antinomy, Kant proposed realistic dynamism and subsequently arrived at his idealism, according to which space, and therefore the continuum, is not real but is an a priori form. The difficulty of the continuum has substantially passed into the Critique of Pure Reason and constitutes the second antinomy. Kant too, like Leibniz, affirms that the ideal continuum is not contradictory, because in it the whole is prior to the parts (Met. Anfangsgründe der Natur, c. 2, 1hex. 4, not. 2). The same ideas have recently been repeated by J. Lachelier and C. Renouvier.

The Leibnizian supposition that, according to metaphysics, in the real continuum the parts are prior to the whole, is erroneous. As Aristotle already said, the real parts result from the division of the real continuum, are potential, are after the real continuum, which is, not an aggregate, but one in act and multiple in potency because of its divisibility. There exists therefore no antinomy between metaphysics and mathematics. The real continuum is not contradictory.

II. MATHEMATICS

The development of mathematical analysis is bound up with the concept of continuity. Analysis studies continuous numerical magnitudes, that is, magnitudes in which one passes from one value to another through intermediate values, which follow one another without jumps or gaps. To define this continuum, which is the object of analysis, the intuitive geometrical continuum was deemed insufficient, and it was sought to create a continuum with only the means that arithmetic can offer. That is, starting from the integers, relative numbers and rational numbers were defined. To give the set of numbers continuity, it was necessary to define irrational numbers by means of rational ones. This was done chiefly through the work of R. Dedekind, with the postulate of continuity. The set of rational and irrational numbers forms the set of real numbers, that is, the constructive arithmetical continuum. This, however, even if it can be the foundation of analysis, does not have the completeness of the geometrical continuum: it is in fact not possible to define all real numbers, and infinitely many other numbers will always remain to be defined. Indeed, there are no two consecutive real numbers, because, if they are distinct, it will always be possible to determine infinitely many other numbers between them. For this reason some mathematicians say that the continuum cannot be constructed with numbers.

A second mathematical theory of the continuum was proposed by G. Cantor, as a consequence of his theory of sets. Cantor says that a set is a particular multitude of determinate elements. Sets can be infinite, that is, contain infinitely many (transfinite) determinate elements. The continuum too is a set of infinite elements: for example, a segment is a set of infinitely many determinate points. In this way Cantor comes to say that the continuum is constituted by infinitely many actual indivisible elements. It has been discovered, however, that the theory of sets, at least in its general form, leads to glaring contradictions. Cantor's ideas are followed by many mathematicians. Others, for example D. Hilbert, hold that the actual infinite cannot be realized, does not exist and cannot be the foundation of speculation (D. Hilbert, Grundlagen der Geometrie, p. 288). So too H. Poincaré (Science et méthode, Paris 1908, p. 41).

A third mathematical theory of the continuum is proposed by the mathematical school of intuitionism, especially by H. Weyl and by L. E. J. Brouwer. Weyl holds that the problem of thinking the intuitive continuum as a totality of discrete elements, despite R. Dedekind, Cantor and K. Weierstrass, has made no progress (Das Kontinuum, Leipzig 1918, p. 16). Brouwer rejects Cantor's definition of set and affirms in particular that the continuum, as a system of mathematically individuated points, does not exist. The continuum as a totality is only a means of free becoming (Medium freien Werdens). The points of the continuum are not there, but rather are formed as the endpoints of ever narrower intervals, which, as a consequence of a free choice, can be determined without end (endlos), and each of which is a new continuum. "The continuum is given to us intuitively as a whole. Its construction, that is, the operation that creates all its individuated points by means of the mathematical intuition of numbers, is unthinkable and impossible" (L. E. J. Brouwer, Over der Grundlagen der Wissende, Amsterdam 1907, p. 62). Therefore Brouwer rejects Cantor's theory of a continuum constituted by infinitely many distinct points, "as an aberration issuing into the void and into chaos" (A. Fraenkel, Einleitung in die Mengenlehre, Berlin 1928, p. 239), and defines the continuum as a whole indefinitely divisible, thus returning to the Aristotelian doctrine (cf. Aristotle, Phys., VI, 2, 232 b 24).

In scholasticism the continuum is defined according to the datum of geometrical intuition. Real numbers can be obtained as relations between the magnitudes resulting from the division of the intuitive geometrical continuum.

III. PHYSICS

In science the conception of a discontinuous matter was introduced when G. Dalton, basing himself on experimental chemical laws, proposed the hypothesis that bodies are composed of particles, called atoms, which, combining in various ways, give rise to the different substances. This theory was developed by atomic physics, which conceived the atom as composed of a nucleus and electrons rotating around it, and by nuclear physics, which has demonstrated that the nucleus too is composed of particles. This scientific result is not against hylomorphism (v.). Note that, in order for real discontinuity to be possible, real c. must exist. If indeed real c. did not exist, real space would not exist and it would make no sense to admit the existence of indivisibles distant from one another; distance presupposes real space and this presupposes real c. As regards the discontinuity of energy, v. QUANTI, teoria dei.

Roberto Mass