CONTINUO. – The notion of continuity is connected with that of extension, which is a primitive concept not susceptible to a true definition. Extended is that which consists of distinct parts in space, and is divisible and measurable. Aristotle distinguishes it into three kinds: successive, contiguous, and continuous (Phys., VI, 1, 231 a 21). Successive is that extended entity whose parts are distinct and separate; contiguous is the extended entity in which the distinct parts touch one another; continuous is the extended entity that has no parts distinct in actuality, but only potentially. The characteristics of continuity are therefore unity and the absence of interruption, whereas the successive implies plurality with interruption, and the contiguous plurality without interruption. Continuity may be static: physical or mathematical; or flowing: motion and time. An essential property of continuity is divisibility into integral parts, that is, into parts having the same nature as the whole from which they derive; for example, a segment is divisible into smaller segments, a surface into surfaces, etc. From continuity, besides integral parts, there also result elements of a kind different from the whole, with respect to which they are indivisible: for example, the point with respect to the line, the surface, and the volume; the line with respect to the surface, etc.
I. PHILOSOPHY
The consequences to which the concept of c. gives rise are important. If c. is divisible, then, although it is 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 c. as distinct and determinate entities, because c. is one, and that which is essentially one cannot be essentially multiple. C. is therefore not the aggregate of its parts. The parts in c. are purely potential realities; that is, although they do not exist in actuality, they have in c. the foundation of their being and come to exist as distinct and actual entities as a result of division.The division of c. into integral parts can proceed indefinitely without ever arriving at indivisible elements. Indeed, from the division of a c. there result integral parts, which are themselves continuous and can in turn be divided into continuous integral parts. For example, since a segment has non-coincident endpoints, it will always have a midpoint distinct from both endpoints, which divides it into two smaller segments. This becomes still 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 any two distinct points of a segment there are always indefinitely many points. The same reasoning can be repeated for each of the two segments resulting from the division. And so indefinitely. This indefinite divisibility of c. can never be exhausted, by its very nature. Hence the division of c. will never yield parts infinite in number. Aristotle had already observed this: « That which is divisible to infinity is not potentially such that it can also be given as divided in actuality in itself and by itself... for division can never come to an end » (Met., IX, 6, 1408 b 13 ff.). From these considerations it follows that c. cannot be composed of extended indivisibles, both because the parts of c. are not actual and because these ultimate unextended parts, whose union would have to produce c., do not exist, since they can never result from the division of c. It is therefore not correct to say that a line is composed of points, a surface of lines, and a volume of surfaces. One may indeed determine, in a line, as many points as one wishes, but it is not possible to determine all the points of the line.
The concept of c. has given rise to significant difficulties:
1) Zeno and Democritus observed that, if the division of c. implies a potentially infinite multitude of parts, then, since potentiality must be capable of being actualized, it follows that the divisibility of c. leads one to admit an actually infinite multitude of parts; this is impossible, because there cannot be an actually infinite number; c. 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 c. is actualized successively and can never be exhausted; the number of parts will increase indefinitely but will never be infinite. 2) Another difficulty originates in a false metaphysical presupposition. Leibniz asserts that, metaphysically, the simple precedes the composite, which is an aggregate of simples; in real c. (bodies), the parts (monads) precede the whole, which is therefore an aggregate of simple, that is, indivisible parts. On the contrary, according to mathematics, division of c. never reaches unextended elements; c. therefore cannot be an aggregate of unextended entities. Real c. consequently implies—according to Leibniz—an antinomy between metaphysics and mathematics, and is therefore absurd. Ideal c., on the other hand, that which we think, is not contradictory, since it is prior to the parts according to metaphysics and in agreement with mathematics: « In the ideal, the whole is prior to the parts... the parts are merely potential; but in the real, the simple is prior to the assemblages, the parts are actual, they precede the whole. » (Lettre à Remond, ed. C. I. Gerhardt, Die phil. Schriften von G. W. Leibniz, III, Berlino 1890, p. 622).
Kant too, in the pre-critical period, under the influence of Leibnizian metaphysics, repeated Leibniz’s difficulty: « Cum illa (la metafisica) spatium (quindi il c. reale) in infinitum divisibile esse praefracte neget, haec (la matematica) 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 c., is not real but is an a priori form. The difficulty of c. essentially passed into the Critique of Pure Reason and constitutes the second antinomy. Kant, like Leibniz, also asserts that ideal c. is not contradictory, because in it the whole precedes the parts (Met. Anfangsgründe der Natur, c. 2, 1hex. 4, note 2). The same ideas have recently been repeated by J. Lachelier and C. Renouvier.
The Leibnizian supposition that, according to metaphysics, in real c. the parts precede the whole is erroneous. As Aristotle already said, real parts result from the division of real c.; they are potential and come after real c., which is not an aggregate but one in actuality and multiple in potentiality because of its divisibility. There is therefore no antinomy between metaphysics and mathematics. Real c. is not contradictory.
II. MATHEMATICS
The development of mathematical analysis is linked to the concept of continuity. Analysis studies continuous numerical magnitudes, that is, magnitudes in which one passes from one value to another through intermediate values, following one another without jumps or gaps. To define this continuity, which is the object of analysis, intuitive geometrical continuity was deemed insufficient, and an attempt was made to create a continuity using only the means arithmetic can offer. Beginning, that is, with whole numbers, the relative and rational numbers were defined. In order to give the set of numbers continuity, it was necessary to define irrational numbers by means of rational ones. This was done especially 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, constructive arithmetical continuity. This, however, although it can serve as the foundation of analysis, does not possess the completeness of geometrical continuity: it is not in fact possible to define all real numbers, and infinitely many other numbers will always remain to be defined. Indeed, no two real numbers are consecutive, 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 continuity cannot be constructed with numbers.A second mathematical theory of continuity was proposed by G. Cantor, in connection with his theory of sets. Cantor says that a set is a particular multitude of determinate elements. Sets may be infinite, that is, contain infinitely many (transfinite) determinate elements. Continuity too is a set of infinitely many elements: for example, a segment is a set of infinitely many determinate points. In this way Cantor comes to say that continuity consists of infinitely many actual indivisible elements. It was discovered, however, that set theory, 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, and that it neither exists nor can serve as the foundation of speculation (D. Hilbert, Grundlagen der Geometrie, p. 288). H. Poincaré likewise held this view (Science et méthode, Paris 1908, p. 41).
A third mathematical theory of continuity is proposed by the mathematical school of intuitionism, especially by H. Weyl and L. E. I. Brouwer. Weyl holds that the problem of conceiving intuitive continuity 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 a set and asserts in particular that continuity, as a system of mathematically individuated points, does not exist. Continuity, as a totality, is merely a medium of free becoming (Medium freien Werdens). The points of continuity are not there; rather, they come into being as the endpoints of ever narrower intervals which, as a consequence of a free choice, can be determined without end (endlos), each of them being a new continuity. “Continuity is given to us intuitively as a whole. Its construction—that is, the operation which creates all its individuated points by means of the mathematical intuition of numbers—is unthinkable and impossible” (L. E. I. Brouwer, Over der Grundlagen der Wissende, Amsterdam 1907, p. 62). Brouwer therefore rejects Cantor’s theory of a continuity consisting of infinitely many distinct points, “as an aberration ending in emptiness and chaos” (A. Fraenkel, Einleitung in die Mengenlehre, Berlin 1928, p. 239), and defines continuity as an indefinitely divisible whole, thus returning to the Aristotelian doctrine (cf. Aristotele, Phys., VI, 2, 232 b 24).
In scholasticism, continuity is defined according to the datum of geometrical intuition. Real numbers can be obtained as ratios between the magnitudes resulting from the division of intuitive geometrical continuity.