The idea of space is one of the most important in the philosophy of the material world; for centuries it has preoccupied and puzzled philosophers and psychologists, and even today the views as to its nature are far from being harmonious.

It is important first to ascertain the exact meaning of the term. In ordinary language space means empty extension occupied by bodies, and in which local motion takes place. This notion of emptiness is so closely connected with it, that the word is often used to mean the distance between bodies. Space is thus put in contrast with bodies, and we imply, more or less unconsciously, that space by itself contains no body—in a word, that it is empty. Evidently space in this popular sense is the extension of the world. It surpasses in magnitude all that the strongest imagination can picture, and consequently it is assigned no limits. Not indeed that space, in the popular sense, is considered strictly infinite; but rather it is conceived as something "indefinite". Again, space, in the popular mind, is clearly conceived as being tri-dimensional, that is, we can draw in it three straight lines each of which is perpendicular to both of the others, and which exhaust all its dimensional possibilities.

The concept which mathematicians form of space does not correspond in every respect with the popular notion. The geometrician is concerned only accidentally with the space of the world. From it he derives his idea of mathematical space; but he eliminates from it all predicates which are not absolutely necessary to establish his geometrical relations. Mathematical space therefore abstracts from all existence. It is conceived as an extensive, continuous, abstract quantity, in which geometrical points and places can be determined. Mathematical space is said to be infinite—not a metaphysical infinity, which affirms the positive absence of all limits, and with which the mathematician has no concern, but that mathematical infinity, which signifies that the nature of a reality is such that no limit can be assigned to it. The distinction between mathematical and metaphysical infinity is somewhat subtle, but it is real; it prevents much confusion and facilitates the solution of difficult problems. It may be remarked here that mathematical space is not necessarily tri-dimensional or homogeneous, matters to which we shall refer presently.

Philosophers cannot be satisfied with mathematical space, an abstract construction useful for theoretical purposes, for they wish to arrive at the real space of nature. Nor can they restrict themselves to the popular notion, for their task is precisely to purify the data of common sense from all the extraneous factors modifying them and giving rise to latent contradictions. But in their efforts to discover pure and real space, they have sometimes arrived at the most perplexing results; so that many philosophers, while not subscribing to the doctrines of Kantian criticism, consider the idea of space as hopelessly contradictory, as a purely illusory fancy. To recall all the successive explanations of the nature of real space given by the great philosophers it would be necessary to go through the history of philosophy; but, leaving aside the complete negation of extension, all the doctrines, from Hesiod (cf. Aristotle, IV Phys., vi, 213b) to our day, fluctuate between the idea of absolute space, a real substance independent of the bodies it contains, and purely relative space, a mental fiction based on the real extension of material bodies. The most radical expressions of these two conflicting views are those of Newton and Clarke, on the one hand, who consider space as the sensorium of God, and on the other, of Leibniz, who asserts that there is no space independent of extended bodies, and reduces it to "the order of co-existing things".

The traditional philosophy of the Catholic schools rejects absolute space. Newton's idea is incompatible with the concept which the great doctors of the school, following Aristotle, formed of quantity. Suarez declares that space is only "a conceptual entity [ens rationis], not, however, formed at will like chimeras, but extracted form bodies, which by their extension are capable of constituting real spaces" (Met. disp., 51). The expression ens rationis may be equivocal, but it expresses somewhat exaggeratedly the very active part played by the human intellect in the construction of space. Space is not material bodies themselves, since it appears to be rather a receptacle containing them. From this point of view it must be pure extension, an unqualified quantity. In the strict sense of the terms a quantity without quality is contradictory; for quantity is only the multiplicity of the homogeneous parts in the unity of a body; it is the distribution of an essence, simple in its formal determination. Multiplicity implies a thing that is multiplied, and distribution something that is distributed. Every quantity is the quantity of something; all extension is therefore, in itself, the extension of an extended substance. Yet quantity is something more more than a modal accident; it is in truth the absolute accident par excellence ; it confers on a substance a perfection such that, granted the existence of a substance, the corporeal body is measured by its quantity. It is none the less true that quantity postulates a quantitative substance; and, in a sense, entirely different however from the fancies of ancient physics, it may always be said that an empty quantity is a contradiction in terms. From this we must conclude that extension is only a derivative of quantity; a non-qualified extension, pure extension, pure space in the reality of the corporeal world is contradictory. We conceive it, however, and what is, properly speaking, contradictory is inconceivable. The contradiction arises when we add the condition of existence to pure space. Space is not contradictory in the mind, though it would be contradictory in the real world, because space is an abstraction. Extension is always the extension of something; but it is not the thing extended. Mentally we can separate extension from the substances from which we distinguish it; and it is extension thus separated, conceived apart, which constitutes the space of the universe. Space is therefore as real, as objective, as the corporeal world itself, but in itself it exists apart only in the human mind, seeing that in the reality of existing things it is only the extension of bodies themselves.

Space thus conceived avoids many of the difficulties raised against its reality. But there still remain questions that have taxed the ingenuity of philosophers. What is to be thought of the infinity of space, which to many philosophers seems to be an indisputable postulate? Here we must carefully distinguish the two ideas to which we alluded above. Mathematicians do not understand infinity in the same sense as philosophers. The latter consider absolute infinity as the plenitude of being, being itself; spatial infinity for them can signify only plenitude of extension. There are no limits to an infinite space, nowhere can there exist a definite relation to its extremities or even to itself. It is impossible to add even mentally anything to such extension, for it would be an absurdity to conceive anything greater than infinite extension. Mathematical infinity is something quite different. It is not considered solely in relation to the being to which it is attributed, but in relation to this being and to the determinations of limits possible to the intellect. Whatever by its nature surpasses all the limits we can assign it, that is mathematically infinite. It must be carefully noted that these two ideas in no way coincide, since it is possible that the intellect may not grasp the nature of a being fully enough to determine its limits: the possibility that its nature may surpass all assignable limits does not involve the conclusion that the being is in itself unlimited. Mathematical infinity introduces into the problem a factor extrinsic to the nature of the being: the relative perfection, or rather the imperfection, of the human idea; and it is noteworthy that in all problems concerning quantity our intellect is, to a very great extent, dependent on our senses and our imagination. This distinction being established, we may remark that real space evidently surpasses all that experience can teach us. We are forced, consequently, to solve the problem by analysis.

Mathematical space is abstract and mathematically infinite; but we are dealing here with the real universe. The notion of mathematical infinity may be applied to it in a secondary sense. The nature of real space is such as not to demand any definite dimensions. No part of space in itself needs be the last. For all we know, or do not know, about it, space may be greater than any limits whatsoever we might assign. But space cannot be metaphysically infinite. It is impossible to have an actual quantitive infinite being composed of finite parts. To infinite extension nothing can be added, and from it nothing can be taken away, even mentally. For if, by hypothesis, infinite extension is divided in two, neither of the parts is infinite since neither by itself contains the plenitude of extension. Both therefore are finite; by their union they would form the original whole, but it is absurd to imagine that an infinite whole is formed by the union of two finite parts. It is clear that we can mentally take way a portion of space. Hence it is clear that space cannot be metaphysically infinite. An actually infinite quantity is a contradiction in terms. Here of course our imagination cannot follow our intellect. We cannot represent exactly to ourselves what may be the limits of the world; and it is clear that in this case certain physical laws, those of motion for instance, cannot be fully applied. It is useless to discuss the subject further because, owing to the limitations of our experience, we are apt to indulge in mere fantastic and arbitrary speculations.

A still more abstruse subject is reached when we come to deal with the number of dimensions of space and its homogeneity. Our imagination always represents real space as having but three dimensions. We reach this intuitive space (see below) spontaneously; it seems to us so natural, so inevitable, that we have great difficulty in freeing ourselves from the domination of this image, and in conceiving (to imagine it is impossible) a space with more than three dimensions. However, the question has been raised; for geometricians reason frequently about a space of four, of five, or of n dimensions. The problem is not of the experimental order. Our sensory experiences and everything in practical life reveal only three dimensions. But does experience exhaust the possibilities of real space? And can this space have no more than three dimensions? Nothing obliges us to believe that such is the case. The material world requires essentially only quantity, and this is not identical with extension. Quantity confers on substance a multiplicity of parts; extension supposes this multiplicity and gives a relative position to the parts. Quantity implies a distinction of parts, extension adds extraposition, i. e. the placing of part outside of part; hence it will be seen that, in a strict sense, material beings do not necessarily postulate extension. It would then be quite arbitrary to declare a priori that they must have extension according to three mutually perpendicular directions, and that they cannot have any more. The word dimensions is here used, of course, only by analogy with the three dimensions perceived by experience; we can get at pure quantity only through extension. But the intellect in its analysis goes beyond the data offered to it by sense, and it is forced to conclude that space of more than three dimensions implies no contradiction.

By a very similar process we can solve the problem, so perplexing for the average mind, of the homogeneity of space. The essential properties of quantity require no definite number of dimensions. The same may be said of the quality, or rather intensity, of extension: the parts may be more or less extraposed. The parts, remaining the same, may give a greater or a less extension in the ordinary sense of the word. There is nothing contradictory, therefore, in all the parts of space being everywhere equally extraposed, in which case space would be homogeneous. But, on the other hand, there is no reason why space should not be differently extraposed in different parts, and if this be so, space would be heterogeneous; and if the variation be simple and constant, we can formulate the laws of these spaces and determine the properties of the figures formed therein. This explains why geometry, so rigorous in its methods and simple in its postulates, is not necessarily one. The ancient geometry of Euclid takes for granted the homogeneity of space; but it is well known that non-Euclidian geometries have been constructed, notably those of Riemann and of Lobatchewski, differing from Euclid's and yet free form all incoherency.

These speculations on the nature of space cannot, however, do away with the fundamental fact that the human mind is dominated by an image, imposing irresistibly on it a homogeneous tri-dimensional space. One of the central questions of classic psychology concerns the origin of this representation. We dismiss Kant's well-known view, that space is an a priori form of sensory activity. But psychologists fluctuate between two extremes: on the one hand, nativism, represented by Johann Müller, Fichte, Sigwart, Mach, and many others; and on the other hand, empiricism, followed by Locke, Hume, Condillac, Maine de Biran, John Stuart Mill, Bain, Spencer, and others. The former hold that we obtain the image of space from the primordial subjective dispositions of our mentality; and many of them see therein a condition precedent of all experience. The second class, on the contrary, believe that this image is acquired, that it results from visual and tactile impressions and is only a result of association. Many authorities hesitate and try to discover an intermediate position. From the facts adduced and the analysis to which they have been subjected it seems clear that the image of space is in reality acquired like all other images: in very young children we see it, so to say, in process of formation. It is the result of the spontaneous interpretation of all the extensive sensations; and it is because this interpretation takes place in the simplest manner that our intuitive space is homogeneous and tri-dimensional. Evidently this elaboration supposes a special nature in the subject, the faculty of receiving extensive impressions, and that of combining them by synthesis. But this is natural to man, and there is nothing to justify us in speaking of an innate image of space.