SPAZIO

SPACE. – In common usage, it is the place in which bodies and movements are imagined. It involves some fundamental distinctions. Real space is space in its objective reality; space considered by physics is the place of bodies with their physical-chemical properties; ideal space is the abstract idea of extension; imaginary space is the image of an extension subsisting without bodies; geometric space is that studied by geometry, the place of geometric bodies.

I. PHILOSOPHY

I. Historical outline

The Pythagoreans affirmed a place or space (Aristotle, Phys., IV, 6, 213 b 22-27). Parmenides identifies void, that is, space, with non-being: “Being (the full), non-being (the void) is not” (H. Diels, Die Fragm. d. Vors., Berlin 1934-36, fr., 6, V. 1-2). Zeno formulates the first antinomy of space (place, τόπος): if everything is in space, space itself is in another space, and so on (Aristotle, Phys., IV, 1, 209 a 23); Leucippus and Democritus maintain that non-being, that is, void (χενόν), exists and makes possible the multiplicity and motion of atoms (Aristotle, Met., I, 4, 985 b 8-9). Aristotle speaks of place (τόπος), defined as the part of space occupied by the body (Phys., IV, 1). Epicurus, like Democritus, admits empty and infinite space, in which there are infinite atoms (Epist. ad Herod., 41-42, ed. E. Bignone, Bari 1920; cf. Lucretius, De rerum natura, I, 951 ff.). According to Proclus, space consists of the finest light (Simplicius, Phys., 142 a, 143 b; ed. H. Diels, Berlin 1882-95). St. Thomas, who upholds the Aristotelian theory of place, states: “non fuisse locum aut spatium ante mundum” (Sum. Theol., q. 46, a. 1, ad 4), that is, space depends on corporeal reality. The same doctrine is found in Suárez: “Quatenus hoc spatium apprehenditur per modum entis positivi, distincti a corporibus, mihi videtur esse ens rationis... sumpto fundamento in ipsis corporibus, quatenus sua extensione apta sunt constituere spatia realia” (Dist. Met., d. 51, s. 1, n. 12).

In modern philosophy, Descartes identifies space (extension) with matter: “Revera enim extensio in longum latum et profundum, quae spatium constituit, eadem plane est cum illa quae constituit corpus” (Princ. philos., II, 10); they differ only “in nostro modo concipiendis” (ibid., II, 12). This space is indefinite, because its limits cannot be imagined (ibid., II, 21). According to Spinoza, space (extension) is an attribute of the divine substance (Ethica, part 1, prop. 15, schol.). P. Gassendi, following Epicurus, maintains that space “non pendere a corporibus corporeumque adeo accidens non esse” (Phys., sect. 1, I, 1; ed. Lyon 1658, vol. I, p. 182); but space, infinite, uncreated, independent of God, immobile, incorporeal, is not substance: “voi incorporei nihil aliud sonet, quam negationem corporis, corporearumve dimensionum; non autem positivam ullam naturam” (ibid., p. 183). I. Newton, from the mechanical experience of centrifugal forces in rotational motion, deduces the existence of absolute space, independent of bodies: “Spatium absolutum, natura sua absque relatione ad externum quodvis, semper manet similare et immobile” (Philosophiae naturalis principia mathematica, Amsterdam 1723, p. 6; cf. p. 9). Newton suggests that God “non est aeternitas vel infinitas, sed aeternus et infinitus; non est duratio vel spatium, sed durat et adest... exsistendo semper et ubique durationem et spatium, aeternitatem et infinitatem constituit” (ibid., p. 483). Clarke, a disciple of Newton, from such premises deduces that “space and duration are not outside of God: they are in the immediate and necessary consequences of His existence; without which He would not be eternal and omnipresent” (Quatrième réplique de M. Clarke, n. 10; ed. Dutens, Geneva 1768, vol. I, part 2, p. 136). G. Leibniz objects that “if infinite space is the immensity of God... it will be necessary to say that what is in space is in the immensity of God and consequently in His essence” (Cinquième écrit de M. Leibniz, n. 4, ed. cit., p. 151). “How could the opinion be acceptable that bodies wander in the parts of the divine essence?” (ibid., n. 50; ed. cit., p. 154). According to Leibniz, space is “quelque chose de purement relatif... un ordre de coéxistences” (Troisième écrit de M. Leibniz, n. 4; ed. cit., p. 121). Space is the complex of the reciprocal positions of bodies (cf. Cinquième écrit de M. Leibniz, n. 47; ed. cit., pp. 151-53). However, because of the antinomy of the continuum (v.), Leibniz affirms that space is a phenomenon; but a well-founded phenomenon, a real phenomenon. For G. Locke, space is a simple idea given by the perception of the distance between two objects (space properly so called) or between parts of the same body (extension properly so called). Contrary to R. Descartes, Locke favours a Newtonian sense of space, independent of bodies (An essay concerning human understanding, bk. II, ch. 13; cf. nn. 16, 17). For Berkeley, on the other hand, space cannot be absolute, but only relative to bodies (Treatise on the principles of human knowledge, nn. 110 ff.), and thus, like them, is an idea. In Hume, the idea of space or extension is nothing but “the idea of visible and tangible points distributed in a certain order; it follows that we cannot have any idea of void, that is, of space in which there is nothing visible or tangible” (Treatise on human nature, bk. I, part 2, sect. 5, n. 1; Italian trans. by A. Carlini, Bari 1926, pp. 77-78).

E. Kant first concerned himself with the Leibnizian antinomy of the continuum (cf. Metaphysica cum geometria iunctae suis in philosophia naturali, monadologia physica, 1756). In Von dem ersten Grunde des Unterschiedes der Gegenden im Raume (1768), he speaks of an absolute space “not merely ideal” (Italian trans. by P. Carabellese, Bari 1923, p. 208). The De mundi sensibilis atque intelligibilis forma et principiis (1770) contains the definitive doctrine of space, which will be repeated in the Kritik der reinen Vernunft (1781). Kant states that “space is not an empirical concept, derived from external experiences...; it is a necessary a priori representation, which underlies all outer intuitions...; it is not a discursive or, as it is called, universal concept of the relations of things in general, but a pure intuition” (Critique of Pure Reason, Italian trans. by G. Gentile and G. Lombardo-Radice, Bari 1924, pp. 66-67). “We can therefore speak of space, extended beings, etc., only from the human standpoint. But if we go beyond the subjective condition... the idea of space would signify nothing more” (ibid.).

For G. Hegel, nature is the first externalization of the idea extending itself in space and time. Space is the most abstract externality, without any determinate difference; it is being outside itself: “The first or immediate determination of nature is the abstract universality in its externality; its indifference without mediation is space” (Encycl., Italian trans. by B. Croce, Bari 1923, p. 204). According to G. Gentile, “nature, the realm of the existent... is represented precisely as the totality of individuals coexisting in space and succeeding one another in time” (Teoria generale dello spirito come atto puro, Florence 1938, p. 112). Hence space is the very multiplicity of positive entities mutually excluding one another: “A pure multiplicity immediately given is space” (ibid., p. 116); this multiplicity is realized in the dialectical unity of spirit (ibid., p. 127).

2. Conclusions

The notion of absolute space in the sense of Gassendi and Newton, existing independently of bodies and their necessary place, originates from our imagination, but proves inconsistent. Newton, illogically, deduces absolute space from the centrifugal forces of a rotating body as a real term of reference for rotational motion; yet this motion can instead be referred to the entire universe, as noted by G. Berkeley (De motu, n. 64, ed. Fraser, I, London 1871, p. 524) and E. Mach (Die Mechanik in ihrer Entwicklung, Leipzig 1883, pp. 242-46). Others posit absolute space as a condition for the possibility of position and motion of an isolated body; thus L. Euler, C. Flammarion, C. G. Neumann, etc. (cf. D. Nys, La notion d'espèce, Louvain 1930, pp. 28-29). But it is not demonstrated that position and motion are necessary conditions of material reality. Moreover, absolute space, as observed by Leibniz (loc. cit.) and Berkeley (Treatise on the Principles of Human Knowledge, n. 117), implies divine attributes and thus involves the negation of God’s transcendence.

Subjectivist conceptions of space are either proposed to overcome particular intrinsic difficulties in the notion of space, or are consequences of more general metaphysical concepts. With phenomenal space, Leibniz attempts to avoid the antinomy of the continuum (v.). Kant does not trouble to distinguish real space from imaginary space (cf. Critique of Pure Reason, ed. cit., p. 66 ff.). It is thus understandable how, by attributing to real space the properties of imaginary space, the author of critical philosophy deems real space absurd and resorts to the a priori form. Kant’s assertion that geometry as a necessary science is possible only if the a priori form of space is admitted (op. cit., p. 52, 60) is fortunate: the universal and necessary value of geometric propositions is sufficiently grounded in the abstract character of the representation of space. Therefore, real space must be understood as the real extension of the universe, or, in other words, the universe itself in its real dimensionality, abstracting from all physical-chemical properties and considering only its dimensional reality. Real space is not determined by the real extension of bodies, of which it is a particular aspect.

If we define dimension as the direction in which, through displacement, a geometric entity passes entirely out of itself and gives rise to another of a different kind (point-line, line-surface, surface-volume; cf. Aristotle, De An., I, 4, 409 a 4; Euclid, Elements, Definitions, nn. 2, 3, 5, 6, ed. F. Enriques, Rome 1925, p. 1 ff.), then real space has three dimensions; for there does not exist a fourth displacement generating a geometric entity different from volume (cf. F. Enriques, s.V. Dimensioni, in Enc. Ital., XII, pp. 849-50).

Real space is neither dilatable nor condensable, nor heterogeneous (i.e., different at different points), nor curved: these are physical properties from which real space abstracts; and they are inconceivable unless one supposes the union of non-condensable and rectilinear space.

II. Mathematics

Geometric space is the space considered by geometry, the locus of geometric figures. It has commonly been identified with imaginary space, which possesses the properties of real space. This is how Euclid and all ancient geometry understood it (Euclidean space). In the critical study of the principles of Euclid, especially the Fifth Postulate (through a point there passes only one parallel to a given line), particularly through the work of G. Saccheri, S.J. (Euclides ab omni naevo vindicatus, Milan 1733), G.F. Gauss, N.I. Lobachevsky, G. Bolyai (cf. C. Fano, Geometria non euclidea, Bologna 1935, p. 1 ff.), there arose geometries perfectly coherent in which the Fifth Postulate was replaced by another: through a point there pass infinitely many parallels to a given line. B. Riemann constructed a third geometry in which the Fifth Postulate was altered once more: through a point there passes no parallel to a given line. These non-Euclidean geometries lead to the notion of curved space: Lobachevsky’s space has negative curvature, Riemann’s has positive curvature. Euclidean space proves to have zero curvature. From this also comes the notion of four-dimensional space, some of which are curved (non-Euclidean spaces) and others not (pseudo-Euclidean spaces) with respect to a fourth dimension: for example, in a similar way a spherical surface is two-dimensional and curved with respect to a third dimension (finite radius), while a plane is two-dimensional and not curved with respect to a third dimension (infinite radius). But one can also conceive of a four-dimensional space diversely curved among itself with variable curvature from point to point; similarly, for example, a surface can have curvature varying from point to point.

In analytic geometry, functions can have a spatial representation; just as functions of 1, 2, 3 variables are represented in spaces of 1 (line), 2 (plane), 3 (common Euclidean space) dimensions, so functions of 4, 5, … n variables can be represented in spaces of 4, 5, … n dimensions. These can be diversely curved among themselves. In this way theories of hyperspaces arose.

For the study of functional calculus, the creation of abstract spaces was necessitated, in which the variables are lines, surfaces, functions, etc. Thus are obtained metric, linear, topological, etc. spaces. Among these are spaces of infinite dimensions, e.g., Hilbert space, which is Euclidean in infinite dimensions.

The fourth dimension and spatial curvature are unimaginable and therefore, since “in mathematics cognition must be guided by judgment to the imagination” (St. Thomas, In Boetium de Trin., q. 6, a. 2), they have no meaning in reality (cf. J. Maritain, Les degrés du savoir, Paris 1949, p. 329 ff.). Hyperspaces are conceivable only analogically to Euclidean space; they function as ideal entities with a foundation in reality, without internal contradictions, and are thus the object of study in geometry. All the more does this apply to spaces of 5, 6, … n dimensions and to abstract spaces.

### III. PHYSICS

The physical space is the space considered by physics, in which bodies exist with their chemical-physical properties. Until the beginning of the present century, physical space was assumed to be Euclidean, independent of motion and the properties of matter. The theory of relativity (v.) by A. Einstein introduced matter and its properties into the structure of space. From special relativity, through a new concept of simultaneity (v. TIME), based on the postulate that light travels at a constant speed (ca. 3·10¹⁰ cm/sec) for all inertial reference systems, it follows that space is relative to the reference system: thus a ruler of length K, in motion relative to an observer, becomes K√(1−v²/c²) in length, i.e., it contracts, not due to physical causes, but because of the contraction of space itself in the direction of relative motion, remaining unaltered in other directions. Moreover, due to some similarities, space and time were united by H. Minkowski (1908) into a four-dimensional universe (Welt, chronotope), of which three dimensions are spatial, x, y, z, and the fourth is temporal c√(1−t) (v. UNIVERSE).

In the theory of general relativity, due to the principle of equivalence, according to which an accelerated system can be considered equivalent to one at rest in a gravitational field, space-time becomes curved. In a vacuum, space-time is four-dimensional Euclidean; where there is matter and energy, it remains four-dimensional; e.g., a ray of light bends when passing near a star. Extending these considerations to all material reality, it is deduced that all space-time is curved upon itself according to Riemannian geometry, which resolves, with the conception of a finite but boundless space (as a spherical surface is finite but without limits), the question of the extent of the universe.

Interpreting the law of gravitation in general relativity G_{μν} = λG_{μν}, in which G_{μν} indicates Newtonian attraction, and λG_{μν} the cosmic dispersion proportional to distance, two models of physical space (universe) are proposed: that of A. Einstein, according to which the universe has the properties of the surface of a four-dimensional hypersphere, with a fixed radius and containing uniformly distributed matter; that of W. De Sitter, according to which the universe has the properties of the surface of a five-dimensional hypersphere with an imaginary radius and is devoid of matter: in it, two material points continuously move away from each other. In addition to these two static solutions of the gravitational equation, there are intermediate dynamic solutions. The one proposed by A. Friedmann (1922) and G. Lemaître (1927) admits that the radius of the universe increases with time, giving rise to an increase in all magnitudes, as the figures drawn on the surface of an inflating balloon. This would explain (Doppler-Fizeau effect) the redshift observed in the spectral lines of extragalactic nebulae as an effect of recession indicated by the equations of general relativity (cf. A. Eddington, *The Expanding Universe*, Cambridge 1933; G. Lemaître, *L’univers, L’avenir*, 1950). It is understood how, according to these theories, it is possible to calculate the extent of the universe and the quantity of matter it contains.

To construct a more general geometry of the universe, it was thought to introduce into the structure of space, in addition to the gravitational field, also the electromagnetic fields in a unified theory; thus the concept of space was further expanded. Attempts of this kind were made by H. Weyl, A. Eddington, and, at various times, by A. Einstein (*The Meaning of Relativity*, 3rd ed. including the generalized theory of gravitation, Princeton 1950), with varying degrees of success.

The space of relativity cannot have the value of objective reality. Indeed, the contraction of space derives from the postulate of the constancy of the speed of light, which cannot be said to be experimentally demonstrated, given that the experiments of Michelson and others, however precise, yielded negative results. The homologation of space with time in Minkowski’s universe is evidently only formal due to the insuppressible difference between space and time. The objective value of the principle of equivalence is also destroyed, besides being obvious from a mathematical point of view, and the experimental proofs of general relativity are uncertain. It is clear moreover that the unified field theories are attempts at synthesis. Finally, condensation and curvature are properties of matter, which should not be attributed to space: indeed, neither can be understood nor can exist unless presupposing metric immutability and spatial rectilinearity (cf. P. Landucci, *Lo spazio e la fisica moderna*, Rome 1935, pp. 160-62).

Moreover, if experience confirms the conclusions of relativity, it follows that this is a good physical theory, not necessarily that it has the meaning of objective truth. Therefore, the space of relativity is a physical-mathematical space, a logical entity with a foundation in reality, a symbol that summarizes, in a brilliant geometrization of physics, the observable aspects of matter. It synthesizes, in the experiences that confirm it, the readings of measuring instruments, without claiming to express the nature of things, according to the direction of modern experimental science, aimed at the observable as such.