Continuum
CONTINUUM
Broadly speaking, a continuum is a manifold whole whose parts, having continuity, are intrinsically differentiated by their relation to the whole. The various analogical uses of the term continuity, which cover both physical and mathematical domains and encompass extensive and nonextensive manifolds, give breadth to this description. Relative to the primary analogate, which is the extensive continuum, the parts may be understood under two formalities: as analytic or as compositive parts. Thus there are two common definitions of the continuum: (1) a manifold whole that is divisible without end into (analytic) parts, of which there is no smallest; (2) a manifold whole, the extremities of whose (compositive) parts are one. Continuity is logically related to contiguity and consecutiveness, all three referring to extension, but constituting a different ordering of parts. A manifold is consecutive if some element of a different nature intervenes between any part and any other part ordered to the former. The parts are contiguous when nothing of a different nature intervenes, though the parts are bounded; they are continuous when they have a common boundary. A continuum is essentially one, though it has distinguishable parts, whereas both contiguous entities and consecutive manifolds are pluralities, the former with parts distinguished and bounded, the latter with parts separated.
The origin of the notion of continuum is most readily traced to the sensible experience of physical extension, which is the first formal effect of dimensive quantification, manifesting factual material unity but remaining subject to division (see extension). Water provides a good instance, since any sample has parts that are exterior to one another, easily divisible, yet factually not divided: an uninterrupted but divisible extensive whole. Psychological studies show that such continua are grasped by means of a scanning motion, either of the eye or through taction. Though probably secondary, the experience of temporal duration also may be at the basis of man's understanding of the continuum, but this conception differs somewhat from the geometric notion.
Parts Of Extensive Continua
According to the definition given, all continua are divisible into parts that are themselves divisible continua; and since such division does not add anything, the positions of the divisions must be marked by precontained indivisibles. There have been disagreements about the mode of inclusion of the parts in the whole. It would seem that for both aristotle and St. Thomas Aquinas, the parts as well as the intermediate indivisibles are present only in potency (Aristotle, Phys. 233b 33–234b 9; St. Thomas, In 3 meta. 13.502–514). Actual existence of parts as parts is understood to be tantamount to actual division, hence loss of continuity. Nevertheless, john of st. thomas, attempting to represent this position but verbally manifesting a more Platonic tradition, held that both indivisibles and a finite number of extensive parts are actually present, although the boundaries of such parts are shared. This is an interpretation of St. Thomas's remark that an interior point may be the beginning of one and the end of another part of a line (Cursus phil., Phil. nat. 1.20). Yet the further divisibility of these parts indicates further potency, and neither the commentator nor more recent expositors of this view seem able to show how or why there should be some actual and some potential parts. Although there is some difficulty concerning terminology, F. suÁrez seems to hold that parts are distinct but not in act: … illae partes sunt illo modo in potentia, sed non in actu, quamquam melius ac verius dicentur entia in potentia, quam partes in potentia (Disp. meta. 40.4.9). According to the definition, there can be no minimal parts, but in physical continua there will be minimal integral parts, that is, those whose nature would be changed by any further division. An abstract continuum has only aliquot parts, for example, the ten, one-inch segments of a ten-inch line, or proportional parts distinguished by a measure proportional to some constant or varying numerical ratio, for example, ½, ¼, ⅛, 1/16.
DISTINCTION OF PARTS
No one admits actual division in the sense of separation. Actual division of an abstract continuum is accomplished by the mental removal of a portion of the continuum or the establishment of an indivisible boundary, and this is no more than a simple designation. Division produces two continua that are either contiguous or consecutive. But distinction within a continuum does not require actual determinate boundaries of composite parts; material opposition, called situs or position (the differentia of dimensive quantity), suffices [see situation (situs)]. This will be a distinction with a real, that is, nonmental, basis even though the making of the distinction is a mental act. Because mental being is equivalent to being known, it is easy to confuse the real distinction of parts of a geometrical continuum with their actual division, or make the latter a requirement for the former. While it cannot be said that the parts of a continuum have only potential existence—for then the whole composed of such parts would not be actual—the parts qua parts cannot be actually present with their own individuality and unity, for this would mean actual division with actual indivisible boundaries.
INDIVISIBLES
Concerning one point, most scholastics agree: a continuum cannot be composed of indivisibles. An infinity of indivisibles can be no more extended than a finite number. G. W. leibniz (1646–1716) and following him, I. Kant (1724–1804), found a contradiction between a real continuum, which was supposed to be posterior to its simple compositive parts and a mathematical continuum, understood as prior to its analytic parts. Denying composition of the latter, they had to admit it of the former, the composing parts being either real dynamistic points or ideal elements. The continuum, for them, thus became a mere phenomenal unity (G. W. Leibniz, "Letter to Remond," Die philosophichen Schriften …, ed. Gerhardt, 3:622; I. Kant, Metaphysische Anfangsgründe der Naturwissenschaft, 2.4.2). The necessity of composition of a physical continuum is, however, an assumption traceable to the general philosophical position of its proposers, rather than anything proper to the idea of continuum. Arguing on a similar basis, G. berkeley (1685–1753) and D. hume (1711–76) concluded that the continuum is composed of very small extended indivisibles (see indivisible).
Types Of Continua
In addition to extensive magnitudes, motion and time are continua. Except for local motion, whose continuity is reducible in part to that of the magnitude being traversed, it is the temporal flux that usually manifests nonstatic continuity. In the static continuum, all parts coexist and are known immediately, while in flowing continua the parts, successive in existence, are known only through the re-presentations of memory. Both types are understood as wholes divisible into parts, distinguished but not interrupted, which have the same nature as the whole; but a flowing continuum is a becoming, hence its parts are never a being, even when considered abstractly.
Though attained in sensation, the abstracted concept of continuum is primarily mathematical; when used in a physical sense, the term is a secondary analogate. Paradoxically, one must grant notional priority to the mathematical continuum, but experiential priority to the physical continuum. The usual examples are abstractions such as mathematical lines, surfaces or perfectly homogeneous solids. Disagreements about physical continua are often traced to the problem of realization of continuity in matter. It is thought that there can be no continuity where heterogeneity is present. In mathematics this is true because differences would be either qualities proper to quantity (for example, shapes or boundaries), or purely quantitative and hence disruptive. But in the physical order qualitative heterogeneity is consonant with quantitative continuity. The mathematical representation of flowing continua requires greater abstraction than that of static continua, since mathematics abstracts from motion. The mathematical notion is subject to further analogical extensions within that order and has been made to include even discrete quantity.
Mathematical Interpretations
The two mathematical sciences of antiquity, arithmetic (the science of the discrete) and geometry (the science of the continuous), were made to meet in the development of analysis introduced with the conceptions of analytic geometry. The correspondence between points located in a geometric continuum and numbers or sets of numbers suggested an analogous type of continuity in the discrete realm. By a pure convention the integers may be placed in 1 to 1 correspondence with points in a linear continuum marking unit segments; hence, by extension, there should be correspondence between the points within each segment and some numerical value less than one. The genetic development of fractional, irrational and other analogates to natural numbers chronologically followed the requirements for solutions of various algebraic equations of increasing complexity and ultimately involved variables that should assume numerical values continuously according to some functional rule. Thus a function is considered continuous at point t if F (x) has a value at t and approaches F (t) as x approaches t. If the function is continuous at all points in a domain, then it is continuous in the domain. This notion has in turn brought about a new postulational definition of geometric continuity in terms of points rather than parts: a line, for example, is continuous when: (1) between any two points on the line there is a third point and (2) a division of the line always distinguishes an extreme point on one of the parts of the division. The assumption of continuous variability was founded first on the intuitive notions of the calculus and theory of limits, later upon the more rigorous developments in analysis and the theory of functions and series, where transcendental numbers also were explored.
MODERN MATHEMATICS
Until the work of K. Weierstrass (1815–97), R. Dedekind (1831–1916) and G. Cantor (1845–1918), in the latter half of the 19th century, there was no overt attempt to provide a theoretical foundation for a continuum of real numbers. Indeed their theoretical considerations, especially Cantor's, led not only to a definition of a real number continuum, but to a new theory of natural number based upon the concept of "set," which has come to permeate a large part of modern mathematics.
The Dedekind definition of the irrational numbers, which allows a numerical continuum, depends upon the concept of a "cut" in the number field such that two classes are formed: [A ] and [C ] in which every element a of [A ] is numerically less than every element c of [C ]. There exists then but one number b that divides the sets, for: (1) either there is a largest rational number a' or a smallest rational c' that divides the sets, or (2) b must be an irrational number defined by the cut since it can be shown that the difference between a' and c' can be made less than any arbitrarily chosen number, that is, that [A ] and [C ] are arbitrarily close yet divided by a number that is not rational. It is impossible to present the definition rigorously here, but it should be emphasized that there is no proof given but only a rigorous axiomatic definition, which is essentially that of Eudoxus, generalized and freed of its geometric moorings.
Cantor provided an alternate definition of irrational number, which, so to speak, fills the gaps between the elements of the dense set of rational numbers. His theory operates with "nested intervals" that contract toward one element and provides a number system that can be put in one to one correspondence with that of Dedekind. Though the latter accepted an actual infinity of numbers, Cantor first developed an arithmetic of the infinite that operates with infinite sets considered as wholes. If the notion of the continuum is derived in an arithmetic field, as Cantor demands, the geometric continuum will be composed of an infinity of points, a conclusion not universally accepted and philosophically inadmissible.
INSTITUTIONISM AND FORMALISM
H. Weyl (1885–1955) and L. E. J. Brouwer (1881–1966) reject the notion of an actual infinity because it cannot be constructed in intuition, and in their view, the continuum cannot be defined as a system of individual points. The continuum is not a composition for Brouwer, but a matrix in which points can be constructed. The arithmetic continuum does not exist, it "becomes" as it is effectively constructed in intuition, the real numbers being defined in terms of a selective sequence of natural numbers rather than sets. Though more in agreement with the traditional Aristotelian notions, the intuitionist philosophy is largely indebted to Kant, who shared with Aristotle the view that an infinity cannot have the character of a completed whole. The goal of intuitionist mathematics has been to avoid the difficulties inherent in Cantor's set theory.
Early in the 1900s a series of antinomies were discovered that flow from the unguarded use of actual infinities and that are resolved only by postulating certain ad hoc axioms (for example, of choice) concerned largely with infinite sets, the legitimacy of which has raised questions that divide the major philosophical positions. The formalist attempt to escape the implications of the thought of R. Dedekind, G. Frege and B. russell developed parallel difficulties demanding still other assumptions (see antinomy; axiomatic system).
Physical Continua
The history of attempts to understand the physical universe manifests a constant tension between the continuous and the discrete. The earliest attempt to reduce one to the other is found in the paradoxes of zeno of elea, the first two of which, Dichotomy and Achilles, are generated by an unwarranted assimilation of the flowing continua to abstract extension, with the resulting assumption that they are infinitely divided. Mathematical solutions to these paradoxes, based upon convergence theorems of infinite series, make the same baseless assumption. Two others, Arrow and Stadium, assume that continua are composed of an infinity of indivisible parts, another indefensible reduction.
Again, the concepts of plenum and void, found early in natural philosophy, give rise to atomistic versus continuance theories of physical reality. Atoms entail a void—like spatial continuum or arena for local motion, whereas a plenum, or at least a contiguum, may be a continuum heterogeneously and dynamically differentiated. Aristotle countered the atomists, democritus and Leucippus, only by showing that motion could take place in a plenum by "replacement"; in his solution, the obvious discontinuity in the universe would have to be mediated by some sort of interstitial matter or ether. Classical physics, which was thoroughly atomistic, postulated the absolute continua of time and space, and later an ether to serve as a medium for radiation. The failure of the Michelson-Morley experiments to detect such an ether provided the immediate impetus that launched the theories of relativity. A. Einstein's special theory was formulated by H. Minkowski (1864–1909) in terms of a mathematical fusion of space and time, which the general theory interprets as the locale of an energy or force field. Subsequent unsuccessful efforts to construct a unified field theory, had as their goal, the reduction of both gravitational and electromagnetic phenomena to local anomalies and perturbations in an overall curved space-time continuum.
In the 1960s, at least four different fields have to be unified; quantum physics has, moreover, reemphasized the discrete. Classical molecules and atoms have been resolved into subatomic particles that, as models at least, represent the inner construction of matter. The massenergy equivalence relation predicted the conversion of matter into radiant energy and the reverse, thereby providing another avenue of reduction. But efforts to explain the details of various radiation phenomena, especially radiant heat, indicated that what must under one form be considered discrete particles, must under another be regarded as a wave governed by a field. Under the first aspect, an unattended space-time continuum is assumed, not different from that of classical physics; while under the second, a continuous field or set of fields is interpreted as a modulation of either a physico-mathematical space-time continuum or of an ether. The development of the quantum theory has not, therefore, settled the issue. In fact, quantum limitations led some physicists to suggest an atomization of space and time themselves, postulating a smallest interval of space, the hodon, of the order of 10-13 centimeters (more recently the smallest spatial interval has been set at 10-33—Planck's length), and the smallest temporal interval, the chronon, of the order of 10-24 seconds. But such discontinuities only presuppose an underlying continuity, at least as a heuristic device.
The notion of matter as made up of impenetrable bits, though a constantly recurring theme in science, has become unacceptable. In fact, the very notion is considered a gross extrapolation into realms wherein common language is inapplicable. The present classes of subatomic particles, leptons, mesons, baryons and so-called resonance particles, are interrelated in ways that suggest that they are epiphenomenal manifestations of a subject proto-matter or a field. Quantum considerations coupled with general relativity theory have more recently suggested either a compromise "pulsational" universe marked by heterogeneous space-time in which continuity is modified if not completely destroyed, or a rippling universe that manifests foam—like discontinuities in regions of high curvature, identifiable as "particles." It should be borne in mind, however, that all such theories are formally mathematical and explain only dialectically by means of symbolic constructs and models. Thus, for example, the space-time continuum is an abstraction, as Whitehead indicated, and yet it is common to find it considered as an ontological principle similar to primary matter. Though reality cannot be composed of abstractions, it can be explained with the help of constructs that represent abstractions. At present such constructs disclose a modified continuum and hence a plenum as a matrix for the physical universe.
See Also: quantity; mathematics, philosophy of.
Bibliography: p. hoenen, Cosmologia (5th ed. Rome 1956). f. waismann, Introduction to Mathematical Thinking, tr. t. j. benac (New York 1959). r. dedekind, Essays on the Theory of Numbers: I. Continuity of Irrational Numbers, II. The Nature and Meaning of Numbers, tr. w. w. beman (Chicago 1901). g. cantor, Contributions to the Founding of the Theory of Transfinite Numbers, tr. p. e. b. jourdain (New York 1952). m. Čapek, The Philosophical Impact of Contemporary Physics (Princeton 1961). a. grÜnbaum, Philosophical Problems of Space and Time (New York 1963). a. farges, L'Idée de continu dans l'espace et le temps (5th ed. Paris 1908). e. bodewig, "Zahl und Kontinuum in der Philosophie des hl. Thomas," Divus Thomas, 13 (1935) 187–207. v. miano, Enciclopedia filosofica, 4 v. (Venice-Rome 1957) 1:1217–21. r. eisler, Wörterbuch der philosophischen Begriffe, 3v. (4th ed. Berlin 1927–30) 3:154–158.
[c. f. weiher]
continuum
con·tin·u·um / kənˈtinyoōəm/ • n. (pl. -u·a / -yoōə/ ) [usu. in sing.] a continuous sequence in which adjacent elements are not perceptibly different from each other, although the extremes are quite distinct: at the fast end of the fast-slow continuum. ∎ Math. the set of real numbers.