CONTINUITY. - The notion of continuity is connected with that of extension, which is a primitive concept not susceptible of true definition. Extended is that which consists of distinct parts in space, which is divisible and measurable. Aristotle distinguishes it into three species: consecutive, contiguous (Phys., VI, 1, 231 a 21). Consecutive is that extension whose parts are distinct and separate; contiguous is that extension in which the parts, though distinct, touch one another; continuous is that extension which has no parts distinct in act, but only potentially. The characteristics of continuity are therefore unity and the absence of interruption, whereas the consecutive implies plurality with interruption and the contiguous implies plurality without interruption. Continuity can be static: physical or mathematical continuity; or fluent: movement and time. The essential property of continuity is divisibility into integral parts, that is, parts which have the same nature as the whole from which they derive; for example, a line segment is divisible into smaller line segments, a surface into surfaces, etc. Apart from integral parts, continuity also comprises elements of a different species from the whole, which are indivisible with respect to it, such as a point with respect to a line, a surface, or a volume, or a line with respect to a surface, etc.
I. Philosophy
Important are the consequences which arise from the concept of continuity. If continuity is divisible, even though 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 continuity as distinct and determined entities, because continuity is one, and what is essentially one cannot be essentially many. Continuity is therefore not the aggregate of its parts. The parts in continuity are purely potential realities; that is, although they do not exist in act, they have in continuity the foundation of their being and come to exist as distinct and actual entities as a result of division.The division of continuity into integral parts can proceed indefinitely without reaching indivisible elements. Indeed, from the division of a continuous entity there result integral parts, which are themselves continuous and can in turn be divided into continuous integral parts. For example, a line segment, since it has non-coincident end points, will always have an intermediate point distinct from both end points, which divides it into two smaller segments. This is even clearer if one bears in mind that the point does not exist and that by adding point to point one will always have a point; therefore, between two distinct points of a segment there are always indefinite points. The same reasoning can be repeated for each of the two segments resulting from the division. And so on indefinitely. This indefinite divisibility of continuity can never be exhausted by its very nature. Therefore, from the division of continuity there will never result parts in infinite number. Already Aristotle had noted this: “The infinitely divisible is not so in potentiality that it can also be given divided in act in itself and per se... for division can never cease” (Met., IX, 6, 148 b 15 ff.). From these considerations it follows that continuity cannot be composed of indivisible extended entities, both because the parts of continuity are not actual and because these ultimate, non-existent parts, whose union should give continuity, do not exist, inasmuch as they can never result from the division of continuity. It is therefore not accurate to say that a line is composed of points, a surface of lines, a volume of surfaces. One can indeed determine, in a line, a number of points as large as one wishes, but it is not possible to determine all the points of the line.
The concept of continuity has given rise to significant difficulties: 1) Zeno and Democritus observed that, if the division of continuity implies a potential infinite multitude of parts, since potentiality must be able to be actualized, it follows that the divisibility of continuity leads to admitting an actual infinite multitude of parts; which is impossible, because an actual infinite number cannot be given; therefore continuity is indivisible (cf. Aristotle, De Gen. et Corr., I, 2, 316 a 14 ff.). We have already indicated Aristotle’s response: the divisibility of continuity is actualized in a successive manner and can never be exhausted; the number of parts will grow indefinitely but will never be infinite. 2) Another difficulty arises from a false metaphysical presupposition. Leibniz asserts that, metaphysically, the simple is prior to the composite, which is an aggregate of simples; in real continuity (bodies), the parts (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 continuity one never arrives at indivisible entities; therefore continuity cannot be an aggregate of non-entities. Real continuity thus implies—according to Leibniz—an antinomy between metaphysics and mathematics, and is therefore absurd. Ideal continuity, on the other hand, that which we think, is not contradictory, being prior to the parts according to metaphysics, and in accord 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.” (Letter to Remond, ed. C. I. Gerhardt, Die phil. Schriften von G. W. Leibniz, III, Berlin 1890, p. 622).
Kant, in the precritical period, under the influence of Leibnizian metaphysics, repeats Leibniz’s difficulty: “Cum illa (metaphysics) spatium (and therefore real continuity) in infinitum divisibile esse praefracte neget, haec (mathematics) eadem, qua cetera solet, certitudine asseverat” (Methodologia physica, Bari 1923, p. 2). To resolve this antinomy, Kant proposed realistic dynamism and later arrived at his idealism, according to which space, and therefore continuity, is not real but an a priori form. The difficulty of continuity essentially passed into the Critique of Pure Reason and constitutes the second antinomy. Like Leibniz, Kant also affirms that ideal continuity is not contradictory, because in it the whole is prior to the parts (Metaphysical Foundations of Natural Science, ch. 2, thesis 4, note 2). Similar ideas have recently been repeated by J. Lachelier and C. Renouvier.
The Leibnizian supposition that, according to metaphysics, in real continuity the parts are prior to the whole, is erroneous. As Aristotle already said, the real parts result from the division of real continuity, are potential, are posterior to real continuity, which is, not an aggregate, but one in act and many in potency because of its divisibility. There is therefore no antinomy between metaphysics and mathematics. Real continuity is not contradictory.
II. Mathematics
The development of mathematical analysis is linked with the concept of continuity. Analysis studies continuous numerical magnitudes, in which one passes from one value to another through intermediate values that follow one another without jumps or gaps. To define this continuity, which is the object of analysis, the intuitive geometric continuity was deemed insufficient, and it was desired to create a continuity using only the means that arithmetic can offer. Starting from the integers, relative and rational numbers were defined. To give the set of numbers continuity, it was necessary to define irrational numbers by means of rational numbers. This was accomplished especially by R. Dedekind, with the postulate of continuity. The set of rational and irrational numbers forms the set of real numbers, that is, constructive arithmetic continuity. This, however, if it can be the foundation of analysis, does not have the completeness of geometric continuity: it is in fact not possible to define all real numbers, and there will always remain infinitely many other numbers to be defined. Indeed, there do not exist two consecutive real numbers, because, if they are distinct, it will always be possible to determine between them infinitely many other numbers. For this reason some mathematicians say that continuity cannot be constructed with numbers.A second mathematical theory of the continuum was proposed by G. Cantor, based on his theory of sets. Cantor states that a set is a particular multitude of determinate elements. Sets can be infinite, i.e., contain infinite (transfinite) determinate elements. The continuum is also a set of infinite elements: for example, a line segment is a set of infinite determinate points. In this way, Cantor comes to say that the continuum consists of infinite indivisible actual elements. It has, however, been discovered that set theory, at least in its general form, leads to glaring contradictions. Many mathematicians follow Cantor’s ideas. Others, such as D. Hilbert, believe that actual infinity cannot be realized; it does not exist and cannot serve as a foundation for speculation (D. Hilbert, *Grundlagen der Geometrie*, p. 288). Similarly, 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 L. E. J. Brouwer. Weyl holds that the problem of conceiving intuitive continuum as a totality of discrete elements has made no progress despite R. Dedekind, Cantor, and K. Weierstrass (*Das Kontinuum*, Leipzig 1918, p. 16). Brouwer rejects Cantor’s definition of a set and in particular denies that the continuum as a system of mathematically individuated points exists. The continuum is, as a totality, merely a medium of free becoming (*Medium freien Werdens*). The points of the continuum do not exist as such; rather, they are formed as the endpoints of ever-narrower intervals, which, through a free choice, can be determined endlessly, each of which is a new continuum: “The continuum is given intuitively as a whole. Its construction, the operation that creates all its individuated points through mathematical intuition of numbers, is unthinkable and impossible” (L. E. J. Brouwer, *Over der Grundlagen der Mathematik*, Amsterdam 1907, p. 62). Brouwer therefore rejects Cantor’s theory of the continuum as consisting of infinite distinct points “as an aberration that withers into emptiness and 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 data of geometric intuition. Real numbers can be obtained as ratios between magnitudes resulting from the division of the intuitive geometric continuum.