SECTION II. THE CONCEPT OF SPACE.
The idea of space has given rise to so many theories that it is difficult to restrain ourselves within the strict limits of psychology, and of our particular subject. Whether or no this concept be innate, given à priori or derived from our cerebral constitution, we have here—setting aside all question of origin—only to inquire by what ways and means we attain full consciousness of it and determine it to be a fundamental concept.
In order to follow its development we must necessarily set out from experience; since space, like number or time, is perceived before it is conceived. For the sake of clearness and precision, let us designate the primitive concrete data, the result of perception, as extension, and the concept, the result of abstraction, as space—properly so called.
I. At the outset what is given us by intuition is extension under a concrete form. What first becomes known to us is not space but a limited and determined extension—what the child can hold in its hand, reach by a movement of its arms, later on the room which it crosses with uncertain steps; it is a street, a square traversed, a journey made by carriage or by train, the horizon which the eye embraces, the nebulæ vaguely seen in the nocturnal sky, etc. All this is concrete and measurable, and can be reduced to a measure, i. e., to a concrete extension such as the metre and its fractions.
These different extensions, although given by the senses, and therefore concrete, are already abstract; since they co-exist with other qualities (resistance, color, cold, heat, etc.) from which a spontaneous analysis separates them, in order to consider them individually. This analysis is translated by the common terms, long, short, high, deep, near, far, to the right, to the left, in front, behind, etc.
By a simplification which occurred much later (for it implies the foundation of geometry) this somewhat confused and incoherent list is replaced by a more rational analysis: height, breadth, depth, distance, position. It marks the transition from the concrete-abstract to the abstract period. It is in fact certain that before constituting itself as a science founded upon reasoning, geometry traversed a semi-empirical stage, it was born of practical needs—the necessity of measuring fields, building houses, and the rest. Moreover certain great mathematicians have by no means disdained to admit its relations with experience: Gauss called it the “science of the eye,” and Sylvester declared “that most if not all the chief ideas of modern mathematics originated in observation.”
Let us, without insisting further, recollect that extension is given us by touch and sight. Touch is par excellence the sense of extension: thus geometry reduces the problems of equality or inequality to superpositions, and all measure of extension is finally reducible to tactile and muscular sensations. The terms touch and vision ought in fact to be completely co-extensive, representing not merely a passive impression upon the cutaneous surface, or the retina, but an active reaction of the motor elements proper to the sensorial organs.
The term acoustic space has recently come into use. Much work has been done on the semi-circular canals, leaving no doubt as to the part they play in the sense of bodily equilibrium;[98] some authors have even localised a “space-sense” in them. Münsterberg relates from his personal experience that while the vestibule and the cochlea receive excitations whence result the purely qualitative sensations of sound (height, intensity, etc.), the semi-circular canals receive others which depend upon the position of the source of the sound: these excitations would produce reflexes, probably in the cerebellum, the purpose of which would be to bring the head into the position best adapted for clear audition. The synthesis of sounds, of the modifications perceived in the canals, and of the aforesaid movements (or images of movement) would constitute the elements of an acoustic space. Wundt, who opposes this theory, sees nothing more in the semi-circular canals than internal tactile organs, auxiliary to external touch.[99]
Leaving this hypothesis of acoustic space (which is by no means well-established), we know from numerous observations that the different modalities of tactile and visual extension, notably that of distance, are only known with precision after much groping and long apprenticeship.[100]
Extension under all its aspects, whether perceived or imagined, presents according to constitution, age, or circumstances, a character of variability which is in complete contrast with the stability and fixity of the concept of space. The conditions of this relativity have been exposed at length by Herbert Spencer. “A creature without eyes cannot have the same conception of space as one that has eyes; and it is the same with the congenitally blind as compared with those who are in full possession of their eyesight; and for the creature whose locomotion is rapid and powerful as compared with the creature which moves slowly and painfully.” Our bodily bulk and organic dimensions also affect conceptions of space; distances which seemed great to the boy seem moderate to the man, and buildings once thought imposing in height and mass dwindle into insignificance. Without speaking of nervous subjects, who illusively imagine their bodies enormously large or infinitely small, there are also transient and momentary states of the organism which considerably modify the consciousness of space; thus, De Quincy, describing some of his opium dreams, says that “buildings and landscapes were exhibited in proportions so vast as the bodily eye is not fitted to receive. Space swelled, and was amplified to an extent of unutterable infinity.”[101] “Deliberate analysis of their movements,” says Lotze, “is so little practised by women that it can be asserted without fear of error that such expressions as ‘to the right,’ ‘to the left,’ ‘forwards,’ ‘backwards,’ etc., express in their language no mathematical relations whatever, but merely certain particular feelings which they experience when during work they perform movements in these directions.”[102] In fine, the consciousness of concrete extension varies in quantity and quality with the structure, position, age, and momentary condition of the feeling subject.
II. Starting from these concrete data—extension as included in our perceptions—how does the intellect arrive at the abstract notion of space?
The immense majority of men left to their own resources do not rise above a confused notion, half-concrete, half-abstract, of the properties of extension, and what Lotze says (supra) applies even better to their total idea of space-relations. The fundamental conception in such minds is simply the possibility of going very far in all directions, or of placing a succession of bodies one behind the other. As to limit, this operation remains vaguely undetermined. It is however translated into current parlance, e. g., “bodies are in space,” and other analogues. Space is conceived, or rather imagined, as an immense sphere which encloses everything, as the receptacle of all extension. It contains bodies, as a barrel holds wine. The primitive cosmologies which yet demand a certain development of reflexion and of abstraction reveal the nature of this conception to us when they speak of the circle of the horizon, the vault of heaven, the firmament (a kind of firm enclosure), and other expressions which denote belief in an insurmountable limit. This conception, which is wholly imaginative at bottom, is a fine example of abstraction elevated into an entity, and the phantom thus created becomes in its turn the source of idle or badly-stated problems such as the following.
‘We have never,’ contends J. S. Mill, ‘perceived an object, or a portion of space, without there being space beyond it, and from the moment of our birth we have always perceived objects or parts of space. It follows from this that the idea of an object or part of space must be inseparably associated with the notion of a further space beyond it. Each moment of our life tends to rivet this association, no experience has ever interrupted it. Under the actual conditions of existence, this association is indissoluble.... Yet we can conceive that, given other conditions of existence, it might be possible to transport ourselves to the limits of space, and that after there receiving impressions of a kind totally unknown to our present state, we might instantly become capable of conceiving the fact and of stating the truth of it. After some experience of the new conception, the fact would seem as natural to us as the revelations of sight to the blind man whose cure is of long standing.’
This argument is founded upon an equivocation. Mill appears to admit as the basis of his discussion the semi-concrete, semi-abstract notion of space, described above; namely, that of common sense. Consequently he confounds and mixes up two perfectly distinct questions; that of Extension, the concrete fact perceived or imagined, and Space, abstract and conceived. In the case of the former, the problem is cosmological and objective, and we are not concerned with it; it is, under another form, the repetition of the discussion on infinite number,—are we, or are we not, to admit a continuous, real magnitude? In the latter, the problem is psychological, subjective, relative to abstraction alone, and will be answered later on.
Up to this point, in fact, the concept of Space corresponds to the state of evolution that we have so frequently denoted as concrete-abstract. The true concept of space—of purely abstract space—was only constituted when the geometricians (Greek and otherwise) disengaged from different extensions those essential features which they termed dimensions, showing by their science that elements thus abstracted and considered separately can be substituted for all the rest. Stallo justly observes that the geometrical elements are neither real, nor ideal, nor hypothetical; they are conceptual, the result of abstraction. “In the processes of discursive thought the intellect never has before it either sensible objects or the whole complement of relations which make up their mental images or representations, but only some single relation or class of relations. It operates along lines of abstraction, the final synthesis of whose result never yields anything more than outlines of the objects represented. During all its operations the intellect is fully aware that neither any one link in the chain of abstraction nor the group of abstract results which we call a concept (in the narrower sense of a collection of attributes representing an object of intuition or sensation) is a copy or an exact counterpart of the object represented. It is always conscious that to bring about true conformity between concepts or any of their constituent attributes with forms of objective reality, the group of relations embodied in these concepts would have to be supplemented with an indeterminate number of other relations which have not been apprehended and possibly are insusceptible of apprehension....”[103]
No one imagines that there are in Nature points, surfaces, lines, volumes, such as geometry proposes them, nor that its concepts are copies of these, but it is not therefore necessary to take refuge in the à priori: abstraction suffices, the act, i. e., by which fundamental qualities are abstracted, to be subsequently fixed by definition. It is strange that Stuart Mill in his long and untoward discussion of this subject should content himself with saying that “we have a power, when a perception is present to our senses, or a conception to our intellect, of attending to a part only of that perception or conception.”[104] In this remark upon attention he is very near recognising the rôle of abstraction (which for the rest he fails to name), but instead of insisting upon it he returns to his thesis, that “the foundation of all sciences, even deductive and demonstrative sciences, is induction....”
The concept of space such as the geometers have made it, namely at its highest degree of abstraction, is thus the result of association. It is extension emptied of all its constitutive qualities, save the necessary conventions which determine it. This schema (apart from all transcendental considerations) appears to us as the total of the conditions of bodily existence in so far as they are endowed with extension. Thus constituted, with the marks which are proper to it, and distinguish it from all others, this concept, like that of number, is susceptible of multiple application, while moreover it has no assignable limits in any direction (i. e., according to the time-honored expression, it is infinite).
Just as concrete number represents real unities or collections, while abstract number detached from discontinuous realities admits of infinite numeration; so concrete space (extension) corresponds to the intuition of certain bodies, while abstract space, by an unrepresentable concept, if not by words, implies an unlimited extension.
If, hypothetically, it were possible to count all the leaves of all the trees in the world, this prodigious number corresponding to concrete unities would be as nothing to the mind that can count for ever beyond that. So for the extension determined by the movement of our arms or legs, by days of railway travelling or sailing, by balloon ascensions, and finally by the most powerful telescopes that can scrutinise the infinity of the heavens,—in all these concrete, fixed, measured extensions we can always imagine a beyond, because the end of one extension is the beginning of the next. All that, however, is but the work of the imagination manipulating abstraction. The law of construction for infinite space is the same as for infinite number: this infinity is only in the operation of our mind, it is a pure psychological process; we believe we are dealing with real magnitudes, and we are only acting upon our own judgment: we are but adding states of consciousness one upon another. Space is only potentially infinite, and this potentiality is in us, and in nothing but ourselves; it is a virtuality which can neither be exhausted nor achieved. To erect it into an entity is to reify an abstraction, to attribute an undue objective value to an entirely subjective concept. The journey to the end of space as suggested by Stuart Mill in the passage above cited (if by space he means the simple possibility of containing extended bodies) would really be a journey to the end of our minds: if he means a journey to the end of the real world, i. e., determinable and measurable extension—which has no limits beyond the imperfection of our instruments—then he admits implicitly that the universe has bounds, he takes sides in a debate in which experimental psychology (we repeat) sees nothing, and which it is even totally incompetent to decide.
During this century certain illustrious mathematicians,—Gauss, 1792, in an unpublished work, Lobachévski in 1829, Riemann, Beltrami, Helmholtz and many others after them, constructed a new geometry known under various names: astral, imaginary, pangeometric, metageometric, and lastly non-Euclidean geometry. Its fundamental principle is that our Euclidean space is only one particular case among several possible cases, and our Euclidean geometry one species of which pangeometry is the genus—that the sole determining reason in its favor is that Euclidean geometry alone is practically applicable to, and verified by, experience. These essays, beyond their direct interest to mathematicians, have already given rise to a considerable number of philosophical considerations. While they have only very distant relations with psychology, they deserve notice, because they enable us the better to understand the genesis of the concepts of space, and are moreover a striking proof of the constructive power of the mind, emancipated from experimental data, and subject only to the rules of logic.
Our space being of three dimensions, the neo-geometers speculated in the first place as to the hypothesis of a space of 4, 5, or n-dimensions; later on they chose as their base of operations a space of three dimensions, considered no longer as plane (Euclidean space) but as spherical or pseudo-spherical, having, i. e., instead of a zero curvature, either a positive (spherical space) or a negative curvature (pseudo-spherical space). Their point of departure is the rejection of Euclid’s postulate—they do not admit that it is impossible to draw through a point more than one parallel to a given straight line. In spherical space there is nothing analogous to the Euclidean axiom of parallels; in pseudo-spherical space two parallels to a line can be drawn through any point. In the first hypothesis, the sum of the three angles of a triangle is greater than two right angles; in the second it is smaller. Thus by deduction after deduction, the neo-geometers constructed an edifice very different from ordinary geometry, subject to no other conditions than that of being free from internal contradiction.
In our connexion, the sole utility of the invention of imaginary geometries is to have reinforced, as if by a magnifying process, the distinction between space perceived and conceived; this assumes various forms according to the process of abstraction employed and fixed in definitions. “Euclidean” space has only one advantage, that it is the simplest, the most practical, the best adapted to facts: in short, that which involves the least disparity between the ideal and our experience, and consequently the most useful. “Certain neo-geometers have in fact maintained that it is uncertain whether space can, or cannot, have the same properties throughout the whole universe ... and that it is possible that in the rapid march of the solar system across space we might gradually pass into regions in which space has not the same properties as those we know”; yet this thesis, which, fundamentally, reifies an entity, does not seem to have gained many partisans. Stallo criticises it at length (op. cit., Chap. XIII).
There is no agreement as to the measure in which the new concepts agree or disagree with the theory of space, “the à priori form of sensibility.” Some hold them to be indifferent, others to be unfavorable to Kantism: this discussion which, for the rest, does not concern us is still in progress.
In conclusion, extension is a primary datum of perception and cannot be further reduced: it is multiple, full, heterogeneous, continuous (at least in appearance), variable, perhaps finite; while space (concept) is void, unified, homogeneous, continuous, and without limits.
Many men and races never get beyond this stage of concrete representation, which corresponds with the first moment of evolution in the individual and in the species. The first step towards the concept of space (concrete-abstract period) consists in representing it to oneself as the place, the receptacle of all bodies. This is the direct result of primitive reflexion: image rather than concept, to which the mind attributes an illusive reality.
The true concept, resultant of abstraction, has been the elaboration of geometricians. It is actually constituted by a synthesis of abstracts or extracts which are, according to Riemann, size, continuity, dimension, simplicity, distance, measure. This synthesis or association of abstracts has nothing necessary about it; its elements may be combined in several ways; hence the possibility of different concepts of space (Euclidean, non-Euclidean). Space conceived as infinity reduces itself to the power that the human mind has of forming sequences, and it forms them thanks to abstraction, which admits of its seizing the law of their formation.
Intuition is the common basis of all concepts of space. Euclidean space rests directly upon this, and upon definitions. Non-Euclidean space rests directly upon it, but more particularly upon definitions.
Although inapplicable to the real world, these last—which are constructions in which the mind is submitted to no other laws than agreement with itself—are brilliant examples of the power of abstraction, when it attains its highest degree of development.