After this general study of the nature of the most elevated forms of abstraction, we must take the principal concepts one by one, and retrace their evolution in outline. Let us once more note that we are concerned with pure psychology, and are to eliminate all that depends on the theory of knowledge and other transcendental speculations. As regards the first origin of our notions of time, space, cause, etc., each may adopt the opinion that pleases him. Whether we admit the hypothesis of à priori forms of the mind (Kant), or that of an innate sense acquired by repetition of experience in the species, and fixed by heredity in the course of centuries (Herbert Spencer), or any other whatsoever—it is clear that the appearance of these concepts, and the data of their evolution, depend on experimental conditions, and consequently fall within our province. Accordingly it is with their empirical genesis, and development through experience, that we are concerned—and with that alone.
SECTION I. CONCEPT OF NUMBER.
The lower phases of this concept are already known to us. We have traversed them in considering numeration, in brutes, children, and aborigines. And here we return to it finally under its higher aspects.
At the outset, counting was, as we found, merely the perception of a plurality; abstraction being practically at zero. Later on a rudiment of numeration appeared, under a practical concrete form: we have perception plus the word—a poor auxiliary, whose part is so insignificant as to be mostly negligible. We noted the different stages of this concrete abstract period, marked by the increasing importance of the word. Finally we arrived at the point at which it is the prime and almost the only factor. Number under its abstract form, as it results, from a long elaboration, consists in a collection of unities that are, or are reputed, similar. We have therefore first to examine how the idea of unity is formed. Next by what mental operation the sequence of numbers is constituted, lastly what is the part played by the sign.
I. To common sense nothing appears more easy than to explain how the idea of unity is formed. I see a man, a tree, a house; I hear a sound; I touch an object; I smell an odor, and so on: and I distinguish this single state from a plurality of sensations. John Stuart Mill seems to admit that number (at least in its simplest forms) is a quality of things that we perceive, as white, black, roundness, hardness: there is a distinct and special state of consciousness corresponding to one, two, three, etc.
Even if we admit this very doubtful thesis, we should arrive at last only at perceived numbers, with which consistent and extended numeration is impossible. It can only be carried on with homogeneous terms, i. e., such as are given by abstraction.
The notion of unity must however find its point of departure in experience, at first under a concrete form; Although it may enter consciousness by several doors, some psychologists, with no legitimate reason, have attributed its origin to one definite mode of external, or even internal, perception which they have chosen to the exclusion of all others.
For some, it is the primordial sense, the sense par excellence; touch. The child regards as a unity the object which it can hold in its hand (a ball, a glass), or follow uninterruptedly in all its boundaries. Wherever his operations are interrupted, where there are breaks of surface continuity, he perceives plurality. In other terms unity is the continuous, plurality is the discontinuous. Numerous observations prove that children actually have a far more exact and precocious notion of continuous quantity (extension), than of discontinuous, discrete quantity (number).[91]
For others, it is sight, for which all that was said above may obviously be repeated. The retina replaces the cutaneous surface: an image clearly perceived without discontinuity is the unity; the perception of simultaneous images leaving intermediate lacunæ in the field of vision, gives plurality.
The same may be said of the acoustic sensations. Preyer, in a work on “Arithmogenesis,” claims that “hearing takes first rank in the acquisition of the concept of number.” Number must be felt before it is thought. Ideas of number and of addition have to be acquired, and this, according to him, takes place in the child when it hears and compares sounds. Subsequently, touch and sight complete this first outline. It is known that Leibnitz assimilated music to an unconscious arithmetic. Preyer reverses the proposition and says: Arithmetica est exercitium musicum occultum nescientis se sonos comparare animi.[92]
As against those who seek the origin of the idea of unity in external events, others attribute it to internal experience.
Thus it has been maintained that consciousness of the ego as a monad which knows itself, is the prototype of arithmetical unity. Obviously this assertion is open to numerous objections. To wit, the late formation of the notion of the ego, the fruit of reflexion; its instability,—still more, this unity, like all the preceding, is concrete, complex; it is a composite unity.
The thesis of W. James is very superior: “Number seems to signify primarily the strokes of our attention in discriminating things. These strokes remain in the memory in groups, large or small, and the groups can be compared. The discrimination is, as we know, psychologically facilitated by the mobility of the thing as a total.... A globe is one if undivided; two, if composed of hemispheres. A sand heap is one thing, or twenty thousand things, as we may choose to count it.”[93] This reduction to acts of attention brings us back definitely to the essential and fundamental conditions of abstraction.
Save this last, the hypotheses enumerated (and internal sensation might also have been invoked; e. g., a localised pain as compared with several scattered pains) give only percepts or images, i. e., the raw material of abstract unity. This is itself a subjective notion. We said above (Chapter II) that the question whether consciousness starts from the general or the particular is a misstatement, because it applies to the mind which is in process of formation, categories valid only for the adult intelligence. So here. At the outset there is no clear perception of primary unity and subsequent plurality, or vice versa: neither observation nor reasoning justifies an affirmation. There is a confused, indefinite state, whence issues the antithesis of continuous and discontinuous, the primitive equivalents of unity and plurality. It took centuries to arrive at the precise notion of abstract unity as it exists in the minds of the first mathematicians, and this notion is the result of a decomposition, not of any direct and immediate act of postulation. It was necessary to decompose an object or group into its constituent parts, which are or appear to be irreducible. Then a new synthesis of these parts was required to reconstitute the whole, in order that the notion of relation between unity and plurality should be perceived clearly. It cannot be doubted that for the lesser numbers two, three, four, the successive perception of each separate object, and then of the objects apprehended together at a single glance, has aided the work of the mind in the conception of this relation. We have seen that many human races never passed beyond this phase. The abstract notion of unity is that of the indivisible (provisory). It is this abstract quality of the indivisible, fixed by a word, that gives us the scientific idea of unity as opposed to the vulgar notion. Perceived unity is a concrete, conceived unity is a quality, an abstract; and in one sense it may be said that unity, and consequently all abstract number, is a creation of the mind. It results like all abstraction from analysis—dissociation. Like all abstraction, it has an ideal existence; yet this in no way prevents it from being an instrument of marvellous utility.
II. It is owing to this that the sequence of numbers, homogeneous in material, can be constituted; for the identity of unities is the sole condition in virtue of which they can be counted, and the scant numerations of the concrete-abstract period transcended. The sequence is constituted by an invariable process of construction, which may be reduced to addition or subtraction. “Thus the number 2, simplest of all numbers, is a construction in virtue of which unity is added to itself; the number 3 is a construction in virtue of which unity is added to the number 2, and so on in order. If numbers are composed by successively adding unity to itself, or to other numbers already formed by the same process, they are decomposed by withdrawing unity from the previously constructed sums; and thus, to decompose is again to compose other numbers. For example, if 3 is 2 + 1, it is also 4 - 1. Addition and subtraction are two inverse operations whose results are mutually exclusive: they are the sole primitive numerical functions.”[94]
The simplicity and solidity of this process result from its being always identical with itself, and although the series of numbers is unlimited, some one term of the sequence is rigorously determined, because it can always be brought back to its point of departure, unity. In this labor of construction by continuous repetition, two psychological facts are to be noted:
1. No sooner is unity passed, in the elaboration of numbers, than intuition fails altogether. Directly we reach 5, 6, 7, etc, (the limit varies with the individual), objects can no longer be perceived or represented together: there is now no more in consciousness than the sign, the substitute for the absent intuition: each number becomes a sum of unities fixed by a name.
2. For our unity-type we substitute higher unities, which admit of simplification. Thus in the predominating decimal system, ten and a hundred are unities ten and a hundred times larger than unity, properly so called. They may be of any given magnitude: the Hindus, whose exuberant imagination is well known, invented the koti, equivalent to four billions three hundred and twenty-eight millions of years, for calculating the life of their gods; each koti represents a single day of the divine life.[95]
Inversely, we may consider the unity-type as a sum of identical parts, and represent 1 = 10/10 or 100/100, etc. A tenth, a hundredth, are unities ten times, a hundred times, smaller than unity properly so called, but they obey the same laws in the formation of fractional numbers.
It is well for the psychologist to note the privileged position of what we term the unity-type, or simply 1. It originates in experience, because unity, even when concrete, and apprehended by gross perception, appears as a primitive element, special and irreducible. So long as the mind confines itself to perceiving or imagining, there is in the passage from one object to two, three, or four objects, or inversely in the passage from four objects to three, two, or only one, an augmentation or diminution. But below unity in the first case, and above unity in the second, there is no longer any mental representation; unity seems to border on nonentity and to be an absolute beginning.
From this privileged point the mind can follow two opposite directions, by an identical movement: the one towards the infinitely great, with constant augmentation; the other towards the infinitely small, with constant diminution—but in one sense or the other, infinity is a never exhausted possibility. Here we reach the much disputed question of infinite number: psychology is not concerned with this. For some, infinite number has an actual existence. For others, it only exists potentially, i. e., as an intellectual operation which may, as was said above, add or subtract, without end or intermission.[96]
III. The importance of signs, as the instruments of abstraction and generalisation, is nowhere so well shown as in their multiple applications to discrete or continuous quantity. The history of the mathematical sciences is in part that of the invention, and use of symbols of increasing complexity, whose efficacy is clearly manifested in their theoretical or practical results. In the first place, words were substituted for the things that were held to be numerable; next, particular signs, or figures; later still, with the invention of algebra, letters took the place of figures, or at any rate assumed their function and part in the problem to be solved; later still, the consideration of geometrical figures was replaced by that of their equations; finally, the use of new symbols corresponded with calculations for infinitesimal quantities, negative quantities, and imaginary numbers.
These symbols are such a powerful auxiliary to the labor of the mathematicians that those among them who affect philosophy have gladly discoursed upon their nature and intrinsic value. They seem to be divided into two camps.
One faction attribute reality to the symbols, or at least incline that way. It is the introduction of the nomina numina into mathematics. They maintain that these pretended conventions are only the expression of necessary relations which the mind is obliged, on account of their ideal nature, to represent by arbitrary signs, but which are not invented by caprice, or by the necessity of the individual mind—since this contents itself with laying hold of that which is offered by the nature of the things. Do we not see moreover that the labor accomplished by their aid is, with necessary modifications, applicable to reality?
To the other, symbols are but means, instruments, stratagems. They mock at those who “look upon relations once symbolised as things which have in themselves an à priori scientific content, as idols, which we supplicate to reveal themselves” (Renouvier). Signs, whatever they may be, are nothing more than conventions: negative quantities represent a change in the direction of thought. Imaginary numbers “represent important relations under a simple and abridged form.” Symbols are an aid in surmounting difficulties, as, empirically, the lever and its developments serve for the lifting of weights. “It is not calculation,” said Poinsot, “that is the secret of this art which teaches us to discover; but the attentive consideration of things, wherein the intellect seeks above all to form an idea of them, endeavoring by analysis properly so called to decompose them into other more simple ideas, and to review them again subsequently as if they had been formed by the union of those simpler things of which it had full knowledge.”[97]
In sum: numbers consist in a series of acts of intellectual apprehension, susceptible of different directions, and of almost unlimited applications. They serve for comparison, for measurement, for putting order into a variety of things. If we compare now the two extremes,—viz., the first attempt at infantine numeration and the highest numerical inventions of the mathematician,—we must recognise the notion of number to be a fine example of the complete evolution of the faculty of abstraction, as applied to a particular case, the principal stages of which we have been able to note in bringing out the ever-increasing importance of signs.
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.
SECTION III. THE CONCEPT OF TIME.
In evolving the idea of time, as in that of space, we must first examine the concrete fact which is its starting-point, i. e., real duration; next the concept which is extracted from it, time in abstracto—and this must be followed in the successive stages of its development.
I.
Real, concrete duration is a quality known by itself, included among internal and external sensations, as later on in the representations which psychology, in what concerns it, must accept as an ultimate datum.
What is this concrete duration, furnished by experience? It might strictly be said to be the present; yet this answer is somewhat theoretical, for it must be admitted that what we term “the present” has vague and fluctuating limits. Moreover, its clear and precise distinction from what has preceded and what is to follow it—the past and the future—is a somewhat late production. Of this we have objective witness (see Ch. II.) in primitive languages, during the period in which the value of the verbs was undetermined. Take again the fact, as frequently observed, that children even at the age of three, or older, having vague notions of past and future, make a general confusion, and do not distinguish between “yesterday” and last week, between “to-morrow” and next week (Sully).
Still, we must admit that the present has the privilege of appearing in consciousness as the typical duration, the standard, the measure to which everything must be referred. Nor can this be otherwise, seeing that (as is too often forgotten) we live only in the present; that the past and the future have no existence for us, are only known to us under the condition of becoming present, of occupying actual consciousness. The present is the only psychical element, which, consciously or unconsciously, gives a content and reality to duration.
It is essential to rid ourselves of the opinion accredited by many authors, that the present is only an elusive moment, a transition, a passage, a flash, a mathematical point, a zero, a nullity; on the contrary—it alone has duration, is now long, now short. It is even possible, to some extent, to determine its limits, and to transcend this vague description. Here we are aided by the labors of the psycho-physicists. We may say that this study, long restricted to metaphysical dissertations, entered upon a positive phase with Czermak, who (in 1857) opened out a new line, in which he has been followed by many others. The numerous researches and experiments made upon the “sense of time” may be omitted without prejudice to our subject, while the discussion of them would deter us from our principal aims. We must, however, give a rapid summary of those which relate either to the actual perception of duration, or to the reproduction in memory of past duration.[105]
1. This present, declared to be inapprehensible, has however been determined as regards its minimum duration. For the discrimination period between two different sensations (taken as the type of the briefest and simplest psychical act), Kries and Auerbach found durations varying between 0·01 and 0·07 second with an average of 0·03 second. At a later period, Exner, experimenting with Savart’s wheel, stated that the interval at which two successive taps can be perceived separately may be reduced to 0·05 second: as also for the sound produced by two electric sparks. For the eye, the minimum perceptible interval between two impressions falling on the yellow spot, is 0·044 second. Below this, one of the conditions necessary to consciousness—an adequate duration—is wanting.
Certain experiments contributed by Wundt and his pupils throw light also upon the maximum duration that can be apprehended by consciousness. They made use, almost exclusively, of auditory sensations, which are more closely allied than any others to the sense of time. Wundt finds that twelve impressions equivalent to a duration varying from 3·6 to 6 seconds can be clearly perceived to form a group. Dietze admits the continuous perception of 40 beats of the metronome, provided the mind arranges them spontaneously in 5 sub groups of 8, or 8 sub-groups of 5. Total duration: 12 seconds. Others vary in their conclusions from 6 to 12 seconds and even more.[106] James is inclined to think that the actual present may extend to a minute.
Of course these figures, of which we can only give a few, vary with the subjects, quality of the impressions received, conditions of experience, exercise, etc. Nor must we forget that these laboratory researches are somewhat artificial, and concerned with the perception of “the present” under studied conditions of simplicity which are not precisely those of spontaneous consciousness. Still it is plain that “the present” is by no means an abstraction, a nothing, and we may conclude, in the words of James, “by saying that we are constantly conscious of a certain duration—the specious, present—varying in length from a few seconds to probably not more than a minute, and that this duration (with its content perceived as having one part earlier and the other part later) is the original intuition of time. Longer times are conceived by adding, shorter ones by dividing, portions of this vaguely broached unit, and are habitually thought by us symbolically.”[107]
2. Experiments relating, not to consciousness of actual duration, but to the reproduction of durations, and determination of the errors involved, are numerous, and contradictory. I refer to them in passing only because they are eminently suited to show the very relative and precarious character of our concrete notions of duration.
Through all divergencies, the formula enunciated by Vierordt, the principal initiator of these researches, remains stable; our consciousness of time comes, not from a sensation, but from a judgment, and in our retrospective appreciation of duration, we diminish intervals that are long, and increase those that are short. The debates and disagreements of the experimenters relate above all to the determination of the “indifferent point.” Vierordt denoted by this term the interval of time which we appreciate the most exactly, which we have no tendency to lengthen or abridge, so that if we are required to repeat it experimentally, the error is nil, or very rare. This duration, reproduced according to reality, is 0·35 sec. (according to Vierordt and Mach); 0·4 sec. (Buccola); 0·72 sec. (Wundt); 0·75 sec. (Kollert); 0·71 sec.; etc. According to another author, Glass, there is a series of points at which we find maximum relative accuracy; 1·5 sec., 2·5 sec., 3·75 sec., 5 sec., 6·25 sec., etc. Münsterberg again criticises the entire series of figures and experiments, for reasons that will be given below.
Independent of these experiments, which are restricted to very simple events, the facts of our daily life show to superabundance that our memory of duration is almost always inexact. Thus it has often been remarked that the years seem to be shorter with advancing age: which is again an instance of abbreviation of the longer intervals.[108] It is hardly necessary to say that our appreciation of duration (concrete), like that of extension (concrete), depends upon multiple conditions, and varies with these. Such are pre-eminently constitution and temperament: compare a phlegmatic with a nervous individual; an Oriental for whom time is not, with an Occidental, agitated by a feverish existence. Add to these, age, number, and vivacity of impressions received, certain pathological states, etc., and we find here, as for space, that the variability of concrete knowledge is opposed to the fixity of the concept.
This consciousness of duration, fluctuating, variable, and unstable as it may be, is nevertheless the source whence our abstract notion of time is derived: but whence comes it, itself? Where does it originate? “Time has been called an act of mind, of reason, of perception, of intuition, of sense, of memory, of will, of all possible compounds and compositions to be made up from all of them. It has been deemed a General Sense accompanying all mental content in a manner similar to that conceived of pain and pleasure.”[109]
Here are many answers. We may add that among these supposed origins some authors admit only one, to the exclusion of the rest, though nothing justifies them in such arbitrary selection.
Some prefer external sensations, inasmuch as they give us the consciousness of a sequence. Hearing has been termed the sense of time par excellence. This thesis has notably been sustained by Mach:[110] since in a melody we can separate the rhythm from the sounds which compose it, he concludes that rhythm forms a distinct sequence, and that there must be in the ear, as in the eye, a mechanism of accommodation which is perhaps the organ of the “time-sense.” Others decide in favor of the general sense, touch, capable in all animals of receiving a succession of impressions at once distinct and forming a series. Sight, with its fine and rapid perception of movements and changes, is an organ admirably adapted to the formation of relations of sequence, the constitutive elements of time. Were not, moreover, the first essays at determining time (succession of days and nights, seasons, etc.) founded upon visual perceptions?
The majority of contemporary psychologists are, however, inclined with reason to seek the principal origin of the notion of duration in internal sensations; and these derive their prerogative from the primordial and rhythmical nature which pertains to the principal functions of life.
“A stationary creature,” says Herbert Spencer, “without eyes, receiving distinct sensations from external objects only by contacts which happen at long and irregular intervals, cannot have in its consciousness any compound relations of sequence save those arising from the slow rhythm of its functions. Even in ourselves, the respiratory intervals, joined sometimes with the intervals between the heart’s pulses, furnish part of the materials from which our consciousness of duration is derived; and had we no continuous perceptions of external changes, and consequently no ideas of them, these rhythmical organic actions would obviously yield important data for our consciousness of Time: indeed, in the absence of locomotive rhythms, our sole data.”[111]
“Rhythm,” to quote Horwicz, “is the measure, and the sole measure, of time; a being incapable of regular periodic intervals could not attain to any conception of time. All the rhythmic functions of the body subserve this end: respiration, pulse, locomotor movements, hunger, sleep, work, habits and needs of whatever kind.”—Guyau maintains essentially the same thesis, under a more metaphysical aspect: “Time is embryonically in primitive consciousness; under the form of force and effort; succession is an abstraction of motor effort, exerted in space. The past is the active become passive.”[112]
More recently, Münsterberg[113] has attributed a preponderant and almost exclusive part to respiration. Although he affirms that the origin of our notions of duration must be sought in our consciousness of muscular effort in general, and that its primitive measure lies in the rhythm of the bodily processes; yet the gradual rise and fall of the sense of effort which accompanies the two phases of the respiratory function (inspiration, expiration) seem to him the principal source of our appreciation of duration. After a rather sharp criticism of the attempts of his predecessors (which we have already reviewed) to determine the “indifferent point,” he maintains that their disagreements were caused by incomplete comprehension of the psychical events produced in the course of experience. In the perception of the successive beats of a metronome, or taps of Wundt’s electric hammer, only the auditory impressions are attended to; this is a mistake. It is supposed that the sensation-limits form the entire content of consciousness, and that the intervals between them are empty. On the contrary, they are filled by an act of attention. We are conscious of a process of variable tension which, from the initial moment, goes decreasingly towards zero, and then rises again, to adapt itself to the sonorous impression that should follow it. In other words, there are, in the perception of three successive taps, not three, but five states of consciousness: three external and two internal sensations. We must reckon thus if we are rigorously to determine the psychological conditions of experience. As evidence, Münsterberg brings forward the following results, which are from his own experiments.
The “normal time” is first determined, i. e., the standard of duration that should be reproduced by the experimenter as exactly as possible (“time of comparison”).
In one case, different durations were given, such as 15, 7, 22, 18 secs., etc., without attending to the respiration (expiration or inspiration) of the subject, who reacted independently of it. In the reproduction of normal time, the mean error was 10·7 per cent.
In the second case, the same numbers were given again, but care was taken that the subject began his estimation at precisely that respiratory period which coincided with the beginning of the normal time. The mean error did not now exceed 2·9 per cent.
In the two cases cited, there was no interruption between the determination of the normal time and its reproduction; the two operations succeeded each other immediately. If, on the contrary, a short pause, or arrest, was introduced between the two, varying from 1 to 60 seconds, the results are—proceeding at random as in the first case—a mean error of 24 per cent.; as in the second, a mean error of 5·3 per cent.
Münsterberg has been not unreasonably reproached for attributing to respiration, among all the other internal sensations, the exclusive privilege of measuring time. A less justifiable criticism asserts that his thesis is devoid of value because we can appreciate the variations in duration in the beats of a clock more readily than the changes in the rhythm of respiration. This is confounding two distinct factors in the genesis of the idea of duration: its period of formation and its period of constitution; that which occurs at the commencement, and that which takes place in the adult. Our measure is at first subjective, variable; progress consists in the substitution of a fixed, objective measure. Doubtless, the latter is superior in clearness and in precision; yet this is no proof, not even presumption, that it is first in order: we shall return to this point later on.
In short, our consciousness of duration is a complex state, more exactly, a process—since it is less a state than a becoming. The rhythmical visceral sensations are its core; it is an internal chronometer, fixed in the depths of our organisation. Around this subjective element, other objective elements are added and co-ordinated—the regular sequences which are caused by external sensations. They form the sheath of the core, and constitute the sensible portion of our consciousness of duration, not, however, its totality.
II.
Until now we have considered time under its concrete form alone, whether given as an actual event in consciousness, or revived as a past event in memory. We have now to follow the complete development of this idea to its extreme limit. In this study we may conveniently distinguish two stages:
The first, which depends on memory and imagination, consists in thinking a certain extension of duration, that may be more or less vaguely represented: a day, a week, a year, etc.
The second, which depends on abstraction alone, gives time in general, the pure concept, which cannot be represented, and is determined by signs alone.
First Stage.—Certain minds never get beyond this first stage. In respect of time, this corresponds to the lower forms of abstraction which we have so often designated by the terms, generic image and, at a higher degree, concrete-abstract notions (intermediate abstracts).
The lowest form, which is just higher than the recognition of concrete duration, results like the generic image from the repetition of a sequence of events that recur constantly, and are approximately uniform. They are series of which the terms are variable, but which begin and end always in the same manner. Such are the appearance and disappearance of the sun, lying down to sleep and waking up again, and similar facts of common life. The points of departure, the start and finish, are always the same, whatever the variations in the intermediate states. These generic images are met with among the higher brutes, children, and primitive races.
To what extent are the higher animals capable of having a certain representation of time, constructed from their experience of real duration? This is an obscure problem which has been little studied. We are not of course referring to time in abstracto, to the concept, but to the recognition of certain often repeated cycles. Many animals are known to have an approximate appreciation of sufficiently protracted periods, supplied by the periodicity of their needs (hours at which they get food, are taken out, etc., etc.). Prejudice apart, we know of others which, in addition to this subjective physiological knowledge, possess a fairly exact notion of certain regular and objectively caused periods, determined by the progress of natural phenomena, especially by the path of the sun.[114]
In all these instances we may assign as cause, the incontestable preponderance (in animal life) of automatism and of routine: which is equivalent to saying that the notion of these durations is formed by a passive assimilation, and this—as we have seen—is the creative process of generic images.
According to some authors, there are instances of exact appreciation of much less simple periods. Brehm says that during a long passage an ourang-outang visited the sailors every Wednesday and Friday at 8 o’clock, because on those days sago, sugar, and cinnamon were served out, of which he got his share. An anecdote has often been cited after Romanes, of “the geese who came regularly every fortnight, from afar, the morning after the market, in a small English town, to pick up the corn scattered on the marketplace. One year the market was postponed for a day of national humiliation, but the geese came as usual.”[115]
These and other analogous facts seem scarcely sufficient in number, nor strictly enough observed, to warrant any scientific conclusion.
We have previously remarked that, up to the age of three years or more, children who already have an approximate knowledge of space relations, (distance, proximity, within, without, upper, lower, etc.) have a very confused notion of periods as short as three to four days, a week, etc. It has been hypothetically suggested that the extension of the notion of duration must for them arise in expectation rather than in memory, in an orientation towards the future rather than the past.
The concrete-abstract period with its different degrees, limited on the one extreme by generic images, on the other by the pure concept, is met with among savage races, and in rising civilisations. It is a stage that has to be traversed by every human race; many now existing have not got beyond it. Days (solar revolution), months (lunar revolution), and seasons, the round of the changing aspects of nature, give the primitive and simplest notions of time in extension. No tribe is so low in the scale as not to have reached this level. The determination of the (solar) year, even when only approximate, marks a decisive progress.
The peculiar feature of this period, in its lowest degrees, is that the notion of time cannot as yet be separated, or extracted, from the sequence of events. We have already given many examples of this state of intelligence. It is not poetical feeling that makes the savage reckon the age of his children by the flowering of certain plants (and other analogous locutions abound among primitive races,)—nor any innate taste for metaphor: it is merely that he requires concrete marks to determine duration. He cannot think the longer periods in abstracto; they must be imagined, represented in virtue of a more or less arbitrary choice, imprisoned in a concrete mould. Moreover, in the absence of any extended, coherent, systematic numeration, the mind loses itself after the first step. It lacks the necessary vehicle for movement in front and behind, knowing whither it is tending. The natural phenomena which it takes as its starting-point are poor substitutes for the absent sign, and moreover rivet it invincibly to the concrete.
In my opinion, the culminating point of this period is arrived at in the popular conception of time—considered as a vague entity which unrolls itself, as it gives birth to events. This is the notion that is general among most men of medium culture, who are ignorant of philosophical speculation on the subject. It is the final term of common, spontaneous reflexion, left to its own resources. Thus it is said of time that it brings the unexpected, consoles sorrow, extinguishes passion, changes tastes, solves difficulties, and the like; it seems to be an active power, a thing in itself. In fact, no other abstraction has perhaps been so often reified. We may further remark that time has often been personified and even deified in several religions. Such an honor has never been conceded to space. The cause of this difference is that time has an internal, human character: above all, that it is opposed to space as dynamic to static. It is an entity manifested in movement and change, and thereby essentially acting and living. While, in the popular conception, space is the passive receptacle of bodies, time is the active spring by which the whole is set in motion.
Second Stage.—The generic images of duration, and later, the semi-concrete, semi-schematic representation of more prolonged lapses of time, provide the material whence we obtain the purely abstract concept of time. It was stated above (p. 153) that the true concept of space was constituted on the day when the ancient geometers disengaged from the different extensions, the essential features which they termed dimensions. So must the first astronomers, without knowing or seeking for what they did, have laboriously disengaged the essential characteristics of time conceived in abstracto. First, they purified the notion of duration from all anthropomorphic features, studying it objectively, in the course of the regular phenomena of nature. Moreover, they introduced measure. The Chaldæans of Alexander’s time, who possessed a series of astronomical observations embracing a period of 1,900 years, who made an error of only two minutes in their computation of the sidereal year, who determined a cycle of 6,585 days by which they were able to calculate eclipses;[116] who were later on the inventors of the clepsydra, hour-glass, and other more or less imperfect instruments for measuring the subdivisions of the day; all these counted for more than metaphysical speculation in ridding our subject of popular conceptions—or at least to a large extent prepared the way. Accustomed as we are in civilised life to a convenient and exact knowledge of the flow of time, measuring it off at any moment by clocks and watches, we forget how widely different must be the state of mind in the man whose only guides are approximations: such, e. g., as the varying height of the sun in different seasons, with other natural changes apt to be misinforming. The one life is precise, the other vague, or at least mysterious. That our measure of time (as of aught else) is relative, matters little, and the vexed problems of this subject do not concern us. By measure, the notion of time acquires a quantitative mark; it no longer appears as an entity, but as a possibility of successive events, as a divisible and subdivisible process; as an extract or abstract, set apart from the events, dissociated from them by an intellectual operation: in short—time is a thing no longer real or imaginary, but conceptual.
It is wasted labor to repeat for time what has already been said for space, and is applicable to both concepts. Time, like space and number, can be conceived as illimitable; but here again the infinity is only in our mental operation. We can add century to century, million upon million of years. This infinite time is potential only—constituted by a two-fold process: either as a sequence of numbers, which is the ordinary, simplest, and most abstract proceeding; or by filling it with fictitious events, with arbitrary constructions, for the future; by evoking the image of vanished states, when we go back to the first geological ages of our globe, to the nebulous period, and so on. This conception of infinite time is however quite subjective, and in itself reveals nothing as to the nature of things: we do but add one state of consciousness to another; it is an inexhaustible possibility of progression and retrogression; and it is nothing more.
It is a common illusion to transform this conceived infinity into a real infinity; we forget that the mind is only working upon the abstract, i. e., upon a fiction, useful no doubt, but created by ourselves alone, according to our intellectual nature.
Let us suppose that, in consequence of gradual cooling, the disappearance of the sea, or any other cause, man and all animals capable of appreciating duration were to disappear from the surface of the earth; time would disappear with them. Doubtless the earth would continue to turn round its axis, the moon round our planet, the sun to take its course; yet nothing would exist beyond the movements. Just as—if every eye were to disappear—there would be neither light nor color; if every ear failed, there would be neither sounds nor noises, but only the bare potentiality of luminous and auditory sensations if the appropriate organs were to appear again: so, on our hypothesis, there could only be a potentiality of time.
Consciousness is the necessary condition of any notion whatever of the time which appears and disappears with it.
It is no part of our subject to discuss the various theories that have been advanced as to the nature of the psychological process by which the primitive notion of time is constituted in consciousness. This question is, on the one hand, distinct from the history of its development as an abstract idea, which we have been endeavoring to follow, and, on the other, from all hypotheses as to its ultimate origin (Kant’s à priori form, Renouvier’s law of the mind, Spencer’s cerebral innateness) which explains neither its appearance as a fact, nor its genesis in experience. We may, however, complete our account by summarising the latest psychological opinions.[117]
It is clear that a simple sequence of impressions will not suffice to constitute the idea of time; the series must be cognised as such, felt or thought as a sequence. How is it to be cognised? Contemporary opinion upon this point appears to be capable of reduction into two principal types.
1. Some admit, as adequate conditions, sensations and their consecutive images, strong states and weak states; provided, however, that the latter arise before the former have disappeared from consciousness.
Wundt supposes that similar beats of a clock succeed each other at regular intervals in a vacant consciousness. When the first has disappeared its image remains until the second succeeds it. This reproduces the first, in virtue of the law of association by similarity, but at the same time encounters the still persisting image. Hence the simple repetition of the sound contains all the elements of time-perception. The first sound (recalled by association), gives the commencement, the second the end, and the persistent image represents the length of the interval. At the moment of the second impression, the entire perception of time exists simultaneously, since all the elements are presented together: the second sound and the image directly, and the first impression by reproduction.
“The phenomena of ‘summation of stimuli’ in the nervous system prove,” says James, “that each stimulus leaves some latent activity behind it which only gradually passes away. Psychological proof of the same fact is afforded by those ‘after-images’ which we perceive when the sensorial stimulus is gone.... With the feeling of the present thing there must at all times mingle the fading echo of all those other things which the previous few seconds have supplied. Or, to state it in neural terms, there is at every moment a cumulation of brain-processes overlapping each other, of which the fainter ones are the dying phases of processes which but shortly previous were active in a maximal degree. The amount of the overlapping determines the feeling of the duration occupied.... Why such an intuition should result from such a combination of brain-processes, I do not pretend to say. All I aim at is to state the most elemental form of the psycho-physical conjunction.” James is careful to repeat in several places that he makes no attempt at explanation.
2. Others admit sensations and intervals; sensations that are no longer images, but internal sensations of tension, of effort; more properly a subconscious element, which consciousness is able to apprehend by observation or induction. This theory has a more active character than that first discussed.
The cleanest and most complete form of this interpretation is that of Münsterberg,—as set forth above.
Fouillée supports the same thesis as a particular case of his general theory of idées-forces. The apparent present is a synthesis of real presents. Our primitive perception is of change, not of stability; we are conscious of transition. The static point of view must be completed by the dynamic.
The complete separation of present and past is a mathematical fiction. The sum of transition which is a factor in appetite aids in forming the series. Time is a form of appetite; beneath the floating image there is a tendency to movement. A non-volitional being would have no representation of time: time is a form of appetition.[118]
“It is probable,” says Mach, “that time sensation is connected with the organic consumption necessarily associated with consciousness,—that we feel the work of attention as time.... The fatiguing of the organ of consciousness goes on continually in waking hours, and the labor of attention increases just as continually. These sensations connected with greater expenditure of attention appear to us to happen later.”[119]
Others again (Waitz, Guyau, and more particularly Ward) admit temporal signs in imitation of Lotze’s “local signs.” Our successive acts of attention leave a series of residua, variable in intensity and precision; these “temporal signs” permit the conception of representations as successive, and no longer as simultaneous. “What is this distance that separates A from B, B from C, and so on?... It is probably that the residuum of which I have called a temporal sign; or, in other words, it is the movement of attention from A to B.”[120]
These extracts will suffice to show the character of the second theory, which seems to me the more acceptable. It is the more complete, inasmuch as it takes into consideration, not only the clear states, existing in consciousness, but the subconscious states also; it is not confined to intellectual elements alone (sensations and images), but recognises the necessary rôle of the active, motor elements.