Space

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

I. PHILOSOPHY

1. Cenni storici

Un luogo o s.
affermarono i Pitagorici (Aristotele, Phys., IV, 6, 213 b
22-27). Parmenide identifica il vuoto, cioè lo s., con il
non-being: "Being (the full) is, non-being (the void)
non è» (H. Diels, Die Pragm. d. Vors., Berlino 1934-36,
fr., 6, v. 1-2). Zenone formula la prima aporia dello s.
(place, τόπος): if everything is in space, then space too is in another
s. e così via (Aristotele, Phys., IV, 1, 209 a 23); Leucoppo
e Democrito sostengono che il non-essere, cioè il vuoto
(κενόν) esiste e rende possibile la molteplicità e il moto degli
atomi (Aristotele, Met., I, 4, 985 b 8-9). Aristotele parla
del luogo (τόπος), definito come la parte dello s. occupata
from body (Phys., IV, 1). Epicurus, like Democritus, admits space as empty and infinite, in which there are infinite atoms
(Epist. a Erod., 41-42, ed. E. Bignone, Bari 1920; cf.
Lucrezio, De rerum natura, I, 951 sgg.). Secondo Proclo,
lo s. è costituito di luce finissima (Simplicio, Phys.,
142 a, 143 b; ed. H. Diels, Berlino 1882-95). S. Tommaso, che sostiene la teoria aristotelica del luogo, afferma
«non fuisse locum aut spatium ante mundum» (Sum.
Theol., 9, 46, a. 1, ad 4), cioè lo s. dipende dalla realtà
corporea. La stessa dottrina in Sudres: «Quatenus hoc
spatium apprehenditur per modum entis positivi, distincti
a corporibus, mihi videtur esse ens rationis... sumpto
fundamento in ipsis corporibus, quatenus sua estensione
apta sunt constituere spatia realia» (Disp. Met., d. 51,
s. 1, n. 12).

Nella filosofia moderna Descartes identifica lo s.
(estensione) con la materia: «Revera enim extensio in
longum latum et profundum, quae spatium constituit,
eadem plane est cum illa quae constituit corpus» (Princ.
philos., II, 10); differiscono solamente «in nostro modo
concipiendi» (ibid., II, 12). Questo s. è indefinito, perché
non si possono immaginare i suoi limiti (ibid., II, 21).
Secondo Spinoza, lo s. (estensione) è un attributo della
sostanza divina (Ethica, parte 1ª, prop. 15, schol.). P. Gassendi, riprendendo Epicuro, afferma lo s. «non pendere
a corporibus corporeumque adeo accidens non esse»
(Phys., sez. 1ª, I, 1; ed. Lione 1668, t. I, p. 182); ma
lo s., infinito, improdotto, indipendente da Dio, immobile,
incorporeo, non è sostanza: «vox incorporei nihil aliud
sonet, quam negationem corporis, corporearumve di-mensionum; non autem positivam ullam naturam» (ibid.,
p. 183). I. Newton, dall'esperienza meccanica delle forze
centrifughe del moto rotatorio, deduce l'esistenza dello s.
assoluto, indipendente dai corpi: «Spatium absolutum,
natura sua absque relatione ad externum quodvis, semper
manet similare et immobile» (Philos. naturalis principia
mathematica, Amsterdam 1723, p. 6; cf. p. 9). Newton
suggerisce che Dio «non est aeternitas vel infinitas, sed
aeternus et infinitus; non est duratio vel spatium, sed
durat et adest... essistendo semper et ubique durationem
et spatium, aeternitatem et infinitatem constituit» (ibid.,
p. 483). Clarke, discepolo di Newton, da tali premesse
deduce che «lo s. e la durata non sono fuori di Dio:
sono nelle conseguenze immediate e necessarie della sua
esistenza; senza le quali egli non sarebbe eterno e presente dappertutto» (Quatrième replique de M. Clarke,
n. 10; ed. Dutens, Ginevra 1768, t. 1ª, parte 2ª, p. 136).
G. Leibniz oppone che «se lo s. infinito è l'immensità
di Dio... bisognerà dunque dire che ciò che è nello s.
è nell'immensità di Dio e per conseguenza nella sua
essenza» (Cinquième écrit de M. Leibniz, n. 4, ed. cit.,
p. 151). «Come sarebbe accettabile l'opinione che i corpi
passeggiano nelle parti dell'essenza divina?» (ibid., n. 50; ed. cit., p. 154). According to Leibniz, space is "quelque chose de purement relatif... un ordre de coexistences" (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). Nevertheless, because of the antinomy of the continuum (v.), Leibniz affirms that space is a phenomenon; but a phoenomenon bene fundatum, phénomén réel. For G. Locke, space is a simple idea, given to us by the perception of the distance between two objects (space properly so called) or between parts of one and the same body (extension properly so called). Against R. Descartes, Locke favors space in the Newtonian sense, independent of bodies (An essay concerning human understanding, l. 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, n. 110 ff.), and therefore, like them, it is an idea. In Hume, the idea of space or of extension is nothing other than "l'idea di punti visibili e tangibili distribuiti con un certo ordine; ne segue che non possiamo avere nessuna idea del vuoto, ossia di uno s. in cui non sia niente di visibile o tangibile" (Treatise on human nature, l. I, part 2, sec. 5, n. 1; Italian trans. by A. Carlini, Bari 1926, pp. 77-78).

E. Kant was first interested in the Leibnizian antinomy of the continuum (cf. Metaphysicae cum geometria iunctae usus 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 purely ideal" (It. 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 would be repeated in the Kritik der reinen Vernunft (1781). Kant affirms that "space is not an empirical concept derived from external experiences...; it is a necessary representation a priori, which lies at the foundation of all external intuitions...; it is not a discursive or, as it is said, universal concept of the relations of things in general, but a pure intuition" (Critique of Pure Reason, It. 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 point of view. But if we step outside the subjective condition... the idea of space would no longer mean anything" (ibid.).

For G. Hegel, nature is the first externalization of the idea extending itself in space and time. Space is the wholly abstract exteriority, without determinate difference; it is being outside of itself: "The first or immediate determination of nature is abstract universality in its exteriority, whose indifference devoid of mediation is space" (Encyclopedia, Ital. trans. by B. Croce, Bari 1923, p. 204). According to G. Gentile, "nature, the realm of the existent... has been represented precisely as the totality of individuals coexisting in space and successive in time" (General Theory of the Spirit as Pure Act, Florence 1938, p. 112). Therefore space is the very multiplicity of positive things mutually excluding one another: "A pure multiplicity immediately given is space" (ibid., p. 116); and this multiplicity is invented in the dialectical unity of the spirit (ibid., p. 127).

2. Conclusions

The notion of absolute s. in the sense of Gassendi and Newton, existing independently of bodies and as their necessary place, originates from our imagination, but proves inconsistent. Illogically, Newton deduces absolute s. from the centrifugal forces of a rotating body, as a real term of reference for rotary motion; which can instead be referred to the whole universe, as G. Berkeley (De motu, n. 64, ed. Fraser, I, London 1871, p. 524) and E. Mach (I principi della meccanica, Milan-Rome 1929, pp. 242-46) noted. Others posit absolute s. as the condition of the possibility of position and movement of an isolated body; thus L. Euler, C. Flammarion, C. G. Neumann, etc. (cf. D. Nys, La notion d'espace, Louvain 1930, pp. 28-29). But it is not demonstrated that position and movement are necessary conditions of material reality. Moreover, absolute s., as Leibniz (loc. cit.) and Berkeley (Treatise on the principles of human knowledge, no. 117) observed, implies divine attributes and therefore involves the denial of God's transcendence.

Subjectivistic conceptions of space are either proposed in order to overcome particular difficulties intrinsic to the notion of space, or are the consequence of more general metaphysical concepts. With phenomenal space Leibniz attempts to avoid the antinomy of the continuum (v.). Kant does not take care to distinguish real space from imaginary space (cf. Critique of Pure Reason, ed. cit., p. 66 ff.). One understands, then, how, by attributing to real space the properties of imaginary space, the author of criticism judges real space absurd and has recourse to the a priori form. Kant's assertion that geometry as a necessary science is possible only by admitting the a priori form of space is unfounded (op. cit., p. 52, 69): the value of universality and necessity of geometrical propositions is in fact 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 real universe itself, abstracting from every physico-chemical property and considering only its dimensional reality. Real space is therefore not determined by the real extension of bodies, of which it is a particular aspect. Understanding by dimension the direction according to which, through a displacement, a geometrical entity comes 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, nos. 2, 3, 5, 6, ed. F. Enriques, Rome 1925, p. 1 ff.), real space has three dimensions; there does not in fact exist a fourth displacement generating a geometrical entity different from volume (cf. F. Enriques, s.v. Dimensioni, in Enc. Ital., XII, pp. 849-50).

Real space is not dilatable or condensable, nor heterogeneous (that is, different at different points), nor curved: these are physical properties, from which real space prescinds; and which are not conceivable except by supposing them in a non-condensable and rectilinear space.

II. MATHEMATICS

Geometrical space is space as considered by geometry, the locus of geometrical figures. It came to be commonly identified with imaginary space, which possesses the properties of real space. Thus Euclid and all ancient geometry understood it (Euclidean space). In the critical study of Euclid's principles, especially of the fifth postulate (through a point there passes only one parallel to a given line), chiefly through the work of G. Saccheri, S.J. (Euclides ab omni naevo vindicatus, Milan 1733), of G. F. Gauss, N. I. Lobachevsky, G. Bolyai (cf. C. Fano, Geometria non euclidea, Bologna 1935, p. 1 ff.), a perfectly coherent geometry was eventually constructed 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 again altered: 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 turns out to have zero curvature. Hence again the notion of 4-dimensional space, some of which are curved (non-Euclidean spaces) and some not (pseudo-Euclidean space) with respect to a fourth: e.g., in an analogous way the spherical surface is 2-dimensional and curved with respect to a third (finite radius), the plane is 2-dimensional and not curved with respect to a third (infinite radius). But one can also conceive of a 4-dimensional space curved differently among themselves, with curvature varying from point to point; analogously, e.g., 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 respectively in space of 1 (line), 2 (plane), 3 (ordinary Euclidean space) dimensions, so functions of 4, 5, ... n variables can be represented in space of 4, 5, ... n dimensions. These can be curved differently among themselves. In this way the theories of hyperspaces arose.

For the study of the calculus of functionals, thinkers were led to the creation of abstract spaces, in which the variables are lines, surfaces, functions, etc. Thus one obtains metric, linear, topological spaces, etc. Among them are spaces of infinite dimensions, e.g., Hilbert space, which is Euclidean and of infinite dimensions.

The fourth dimension and spatial curvature are unimaginable and therefore, since « in mathematicis oportet cognitionem secundum iudicium terminari ad imaginationem » (St. Thomas, In Boetium de Trin., q. 6, a. 2), they have no meaning of reality (cf. J. Maritain, Les degrés du savoir, Paris 1949, p. 329 ff.). Hyperspaces are conceivable only analogically to Euclidean space; they count as ideal entities with a foundation in reality, without internal contradictions, and are therefore an object of study for geometry. All the more does this hold for spaces of 5, 6, ... n dimensions and for abstract spaces.

III. PHYSICS

Physical space is space as considered by physics, in which bodies with their chemico-physical properties are located. Until the beginning of the present century, physical space had been assumed to be Euclidean, independent of motion and of the properties of matter. The theory of relativity (v.) of A. Einstein introduced matter and its properties into the structure of space. From special relativity, by virtue of a new concept of simultaneity (v. TEMPO), founded on the postulate that light travels at a constant speed (ca. 3.10¹⁰ cm/sec) for all reference systems in inertial motion, it follows that space is relative to the reference system: therefore a rod of length K, in motion relative to an observer, becomes of length K √{1 - frac{v²}{c²}}, that is it shortens, not for physical causes, but because of the very contraction of space in the direction of relative motion, remaining uncontracted in the other directions. Moreover, because of a certain similarity between them, space and time were linked together by H. Minkowski (1908) in a 4-dimensional universe (Wols, chronotope), of which three are spatial, x, y, z, and the fourth temporal, c √{-1} (v. UNIVERSO). In the general theory of relativity, by the principle of equivalence, according to which an accelerated system may be considered equivalent to one at rest in a gravitational field, space-time curves. In a vacuum space-time is 4-dimensional Euclidean; where there is matter and energy it is 4-dimensional Riemannian; for example, a ray of light curves when passing near a star. Extending these considerations to the whole of material reality, one deduces that all of space-time is curved upon itself according to Riemannian geometry; this resolves, with the conception of a finite but unbounded space (just as a spherical surface is finite but without limits), the question of the extension of the universe. Interpreting the law of gravitation of general relativity G_{μν} = λ g_{μν}, in which G_{μν} indicates Newtonian attraction, and λ g_{μν} the cosmic dispersion proportional to distance, various models of physical space (universe) were proposed: that of A. Einstein, according to which the universe has the properties of the surface of a 4-dimensional hypersphere, with fixed radius, and contains uniformly distributed matter; that of W. De Sitter, according to which the universe has the properties of the surface of a 5-dimensional hypersphere with imaginary radius and is empty of matter: in it two material points continually move away from each other. Besides these two static solutions of the gravitational equation, there are intermediate dynamic solutions. That proposed by A. Friedmann (1922) and by G. Lemaître (1927) admits that the radius of the universe grows with time, and this gives rise to an increase of all magnitudes, like the figures drawn on the surface of a rubber balloon that is being inflated. This would explain (Doppler-Fizeau effect) the redshift observed in the spectral lines of extra-galactic nebulae as the effect of a recession indicated by the equations of general relativity (cf. A. Eddington, The expanding universe, Cambridge 1933; G. Lemaître, L'univers, Louvain 1950). It is understandable how it is possible, according to these theories, to calculate the extension of the universe and the quantity of matter contained in it.

In order to construct a more general geometry of the universe, it was thought to introduce into the structure of space, besides the gravitational field, also electromagnetic fields in a unified theory; thus the concept of space was further broadened. Attempts of this kind were made by H. Weyl, A. Eddington and, on various occasions, by A. Einstein (The meaning of relativity, 2ⁿᵈ ed. including the generalized theory of gravitation, Princeton 1950), with varying success.

The space of relativity cannot have the value of objective reality. In fact the contraction of space derives from the postulate of the constancy of the speed of light, which cannot be said to be experimentally demonstrated, since the experiments of Michelson and others, however precise, had a negative result. The homologation of space with time in Minkowski's universe is evidently only formal, because of the irrepressible diversity between space and time. Equally unproven is the objective value of the principle of equivalence, which is moreover obvious from the mathematical point of view, and the experimental proofs of general relativity are uncertain. It is clear furthermore that the unified field theories are attempts at synthesis. Finally, condensation and curvature are properties of matter, which should not be attributed to space: in fact the one or the other cannot be understood and cannot exist except by presupposing metric immutability and spatial straightness (cf. P. Landucci, Lo s. 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 physico-mathematical space, a logical entity with a foundation in reality, a symbol that summarizes, in a gentle geometrization of physics, the observable properties of matter. It synthesizes, in the experiences that confirm it, the reading of measuring instruments, without claiming to express the nature of things, in accordance with the direction of modern experimental science, directed toward the observable as such.

Roberto Masi