Kitabı oku: «Fundamental Philosophy, Vol. 2 (of 2)», sayfa 15

CHAPTER VII.
ANALYSIS OF THE IDEA OF NUMBER IN ITSELF AND IN ITS RELATIONS WITH SIGNS

58. In order clearly to conceive the idea of number, and the way it is engendered in our mind, let us study its formation in a deaf and dumb person.

We have no better way of giving such a one an idea of unity than by presenting an object to him. Now, if we would convey to him the idea of two, we show him two fingers, then two oranges, then two books, and in each of these operations make a sign which must be always the same. If we repeat this operation a number of times, the deaf and dumb person will associate the idea of two with that of the sign, and one will suggest the others; and he will endeavor to show us that he has seen two objects of some kind, by uniting the expression of the object with the sign of two. The same will take place with three, or four. When we reach higher numbers, the sign becomes more indispensable; since the less easily the idea of number is represented, the more necessary is the sign to secure it. But what we do to convey an idea of number to the deaf and dumb person, what he himself must do to express the number which he conceives, we must all do if we would obtain the idea.

59. Numeration is a repetition of operations; and the art of facilitating it consists in instituting signs which recall to our memory what we have done. It is an exceedingly complicated labyrinth, and we cannot trust ourselves to its windings with any expectation of finding our way out again, if we do not take care to mark the path we have followed.

It is to the admirable simplicity of the decimal system, united to its inexhaustible variety, that the facility and fecundity of our arithmetic are due. Algebra, going a step beyond, expresses without determining numbers, and presents the results of its operations without effacing its footsteps on the road travelled, is far superior to arithmetic, and has made the human mind take gigantic strides. But how? Solely by aiding the memory. Thus, the very principle that enables the child to say four and one, five, instead of adding unity five times to unity, the dumb man to express five by a hand, a hundred by a grain, enables the algebraist to express the result of his longest operations by a formula easy of retention by the memory. Both attain their object simply by aiding the memory. A grain of wheat denotes to the dumb man the idea of hundred, and this he applies to all similar collections; a few letters combined in a simple manner designate to the mathematician a property of certain quantities, and this he applies to all which are found in the same case.

60. Numeration is only an aggregation of formulas; and the more easy these are of mutual transformation with a slight modification, the more perfect will be the numeration. The better one knows the relations of these formulas and the manner of transforming them, the better will he know how to count. The greater a person's intellectual power of fixing simultaneously the attention upon many formulas, and of composing them, the more perfect arithmetician will he be, because the simultaneous comparison of many, leads to the perception of new relations.

61. What is our idea of hundred? The union of the units composing it, a union which we have made more or less frequently when learning to count. But how do we know that it is the same union? Because we have a formula called a hundred, expressed by a sign 100. This formula is so easily recollected that we have no difficulty in recollecting the idea of hundred and all the properties connected with it. We may be asked if a hundred is more than ninety. Were we under the necessity counting one and one and one, we should be bewildered, and never succeed in distinguishing the greater; but knowing as we do that to reach the formula hundred, we must pass by another formula ninety, and that this was in ascending, we know, once for all, that hundred expresses ninety and something more, that is, a hundred is more than ninety. And if it be further inquired what is the excess, we shall not undertake to ascertain this by adding units, but by the two formulas ninety and ten which compose the formula hundred.

62. By generalization we unite many similar things in one idea. The general idea is a kind of formula. Numeration unites in one sign many things contained in a general idea, but this sign has, at the same time, its own distinctive character. Thus the general idea belongs as a predicate to each of its particular objects; number belongs to no one in particular, but to all joined. We perceive in abstraction a common property, and lay aside all the particular objects which it presents; in numeration, we perceive similarity, but always with distinction. Abstraction is the result of comparison, but not comparison. Numeration implies a permanent comparison, or the recollection of it.

63. The idea of number is not conventional; a hundred is always a hundred with all its properties and relations, and this, too, prior to all convention and even to all human perception. The sign, and the sign only, is conventional. Were there no intellectual creature, and a hundred beings distinct among themselves were to exist, there would really be this number. The number three exists in the august mystery of the Trinity, from all eternity, and of absolute necessity. Number requires only the existence of distinct things; since, however unlike they may be, they always have something in common being, which may be included in a general idea, and consequently they fulfil the two conditions necessary to number.

64. The perception of being and of distinction, that is, of substantive being and of relative not-being, is the perception of number. The science of the relations of every collection, with its measure, which is unity, is the science of numbers.

BOOK SEVENTH.
ON TIME

CHAPTER I.
IMPORTANCE AND DIFFICULTY OF THE SUBJECT

1. The explanation of the idea of time is not a matter of mere curiosity, but of the highest importance. To convince ourselves of this we have only to consider that the explanation of the whole edifice of human cognitions is based upon it. The most fundamental and indispensable principle which supports all others, includes the idea of time. A thing cannot be and not be at the same time: "impossibile est idem simul esse et non esse." The impossibility of being and of not-being regards only the simul, the same time. Therefore, the idea of time necessarily enters into the very principle of contradiction.

2. The idea of time is involved in all our perceptions; it extends to many more objects than does the idea of space. We estimate not only the movements of bodies by time, but also the operations of the mind. We know that a series of thoughts may be measured by time the same as a series of corporal movements.

3. The idea of succession necessarily enters into that of time, and vice versa, the idea of time into that of succession. We may conceive that one thing succeeds another; but this would be impossible without succession, without a before and after, that is, without time. This reasoning, apparently vicious, shows, perhaps, that we must not explain the ideas of time and succession, the one by the other, since they are identical.

4. Time does not seem to be distinct from things; for who can imagine duration without that which lasts, or a succession without that which succeeds? Is it a substance? Is it a modification inherent in things, or distinct from them? Whatever is something exists; and yet we nowhere meet time existing. Its nature is composed of instants divisible to infinity, essentially successive, and consequently incapable of simultaneousness. Imagine the minutest instant you can, and it does not exist, for it is composed of others infinitely minute, which cannot exist united. To conceive an existing time, we must conceive it as actual, and in order to do this, we must surprise it in an indivisible instant; but even this is not time; it involves no succession; it is not duration, containing a before and an after.

5. Nothing is easier than to calculate time, and nothing more difficult than to conceive it in its essence. As to the former the learned and the ignorant are on the same footing; both have equally clear ideas; the latter is excessively difficult even to the most eminent men. The passage in the Confessiones of St. Augustine, in which the Holy Doctor endeavors to penetrate this mystery is well known.

CHAPTER II.
IS TIME THE MEASURE OF MOVEMENT?

6. Time is said by many philosophers to be the measure of movement. This idea is fruitful, but it needs to be illustrated.

When we measure movement we refer to something fixed. Thus we measure the rapidity with which we have traversed a certain space by noticing the time denoted by a watch. But how do we measure time by a watch? By the space passed over by the hand on the dial. If we reflect carefully, we shall see that this is purely conventional, or rather, that it depends upon an arbitrary condition. For if we suppose the time marked to be an hour, the space passed over by the minute hand, that is, the circumference of the dial, has no relation with the hour except what the artificer gave it by so constructing the watch that the minute hand would make one revolution every hour. If the watchmaker had constructed it differently, as he did the hour hand, the time would be the same, but the space passed over is very different.

7. The time, therefore, indicated by the watch is no measure, save as itself is subject to another measure; consequently it is not the primitive measure. The same can evidently be said of all other watches which must have been regulated one after another, until we come to the first of all watches. There was no other watch to regulate this; it follows, therefore, that no one of the measures furnished by art is the primitive measure.

8. Not finding this measure in the works of man, we must seek it in nature; and here we discover fixed measures. If we regard the course of the sun, and take for unity the time it requires from the time it leaves the meridian until it returns, we shall have the day; this divided into twenty-four parts gives us the hours. Here we have a great watch which will serve to regulate all others.

9. Nevertheless, however lightly we reflect upon this, we cannot help seeing that the solution is not so satisfactory as it seems at first sight.

Solar time and sidereal time do not agree. Thus, if we note the moment when a star is in the meridian conjointly with the sun, we shall the next day see that the star reaches the meridian a little before the sun. Which is right? Has the star taken just twenty-four hours, or the sun? If time be a fixed thing independently of movement, neither of these measures corresponds exactly to time.

10. This argument, which may be called practical, is corroborated by another purely theoretical. If we take celestial movement for the measure of time, will it be true that whenever the movement, which serves as the rule, shall be verified, that there has passed a fixed and determinate time? If we be answered in the affirmative, we must infer, that even were this movement to be accelerated or retarded, as, for instance, if a solar revolution were to be made with a half, or with twice its ordinary velocity, it would continue to mark the same time, which, however, is absurd. If it be said that the movement is supposed to be uniform, we reply, that this is a begging of the question. Uniformity of movement consists in equal times recurring after equal intervals. Did time, then, in its nature depend upon the movement of the sun, or of any star, as primitive measure, neither uniformity nor variety would have any meaning. If the space of twenty-four hours depended upon a revolution's being made, no matter in what manner whether at a snail's pace, or with the velocity of light, we should never have more or less than twenty-four hours. But if these depend upon another measure, if prior to them, there was a time which measured the velocity of movement, and determined whether it had been accelerated or retarded, then the movement of the stars is not the primitive measure; they are in the same category as our watches, they marked the time passed, but time has not passed because they mark it. Time is the measure of their movement, not their movement the measure of time. Movement is in time, not time in movement.

11. To appeal to the movement of the superior heavens, is evidently no solution of this difficulty, for what has been said of the sun, may also be said of the remotest star in the firmament. Whether we appeal to annual, solar, or sidereal movements, the same difficulty remains. Would sidereal years be the same, if the movement be made with greater or less velocity. If they would, an absurdity would follow; if not, this is not the primitive measure.

12. Moreover, we perceive, when considering movement, that we seem to conceive of greater and less velocity; and thus the idea of time, of necessity, enters into that of velocity, since velocity is the relation of space passed over in a given time. The idea of time is therefore prior to, consequently independent of, every particular measure.

13. We measure time by movement, and in order to measure the velocity of movement we need that of time. Here then, perhaps, is a vicious circle; but possibly this only shows that these are correlative ideas, the one explanatory of the other; or, rather, they are different aspects of one and the same idea. The difficulty of separating them, and the intimate union which unites them on the one hand as much as it divides them on the other, confirms this conjecture. To show this, we ask, what time has passed? Two hours. How do we know this? By our time-piece. But what if it be too fast or too slow? The measure fails. This time is thus to us as a fixed measure, prior to that of the watch by which we undertake to measure it. But what are these two hours, if we abstract the measure of the watch, that also of the stars, and every other measure? Two hours, in the abstract, can be found in no category of real or possible beings; and we cannot, without a measure, give any idea of them, nor form one for ourselves. The idea of hour refers to a determinate movement of known bodies; and this in its turn refers to others; and finally, we come to one in which we can discover no reason why it should be exempted from the general law to which the others are subject. No farther reference being possible, all measure fails; and this failing, time, by the force of analysis, vanishes.

14. Therefore, the referring of time to movement, explains nothing; it only expresses a thing known, and that is, the mutual relation between time and movement, a relation known to the unlearned, and of constant and common use; but the philosophic idea stands intact; the same difficulty remains; what is time?

CHAPTER III.
SIMILARITIES AND DIFFERENCES BETWEEN TIME AND SPACE

15. Time seems to us to be something fixed. An hour is neither more nor less than an hour, no matter how our time-pieces go, or the world itself; just as a cubic foot of space is always a cubic foot, neither more nor less, whether occupied or not occupied by bodies.

16. Time exists independent of all movement, of all succession; if it is something absolute, has a determinate value of its own, is applicable to all that changes without itself changing, the measure of all succession without itself being measured, what is it? That it is something accidental cannot be reconciled with its immutability and universality. Every thing lives in it, but it lives in nothing; every thing dies in it, but death has no power over it. When the substance perishes, the accident perishes; but time continues the same although no substance exist. Before all created beings, we conceive ages and ages, that is, time; and after the destruction, the annihilation of all beings, we still conceive a successive although unending succession, which is time. The idea, then, of time, does not demand that of the universe; it existed before it, and will survive it: but without time the universe is inconceivable.

17. The idea of time seems to be independent of the idea of any being; of all duration in it; every thing may endure in it; but it does not begin or end with what endures in itself; it is applicable to all that endures, but it is not itself an endurable thing. We imagine it to be one in the multiple, uniform in the various, fixed in the movable, eternal in the perishable; and it even seems to contain some features of the attributes of Divinity; but it is, on the other hand, essentially despoiled of every property excepting that of succession in its abstractest signification. It is essentially sterile, has no power of its own, no condition of being or action, and consequently leads to the highest imaginations of what a pure idea really is, an abstraction, which, like space, we have imagined in the presence of things.

18. The points of similarity between time and space are worthy of our attention. Both are infinite, immovable; both are a general measure; both essentially composed of continuous and inseparable parts. Limit them you cannot, determine any limit you chose, and beyond it you will see an ocean extended. Your powers are impotent; beyond the highest heaven are unbounded abysses of space; before the beginning of things there was a long chain of interminable ages.

In vain would you undertake to move space; you can only move yourself in it, or survey its various points. Its points are all fixed; you may mark out distances and directions with respect to them, but you cannot change them. The result will be analogous if you attempt to move time. The present instant is not the one just past, nor the one next to succeed; they are of necessity distinct, and of necessity exclude each other. Their very nature is to succeed each other. If their place be changed with respect to time, it ceases to be the same. Imagine, if you can, that to-morrow is to-day, that to-day is yesterday. It is impossible for that which was at a certain time not to have then been; but this would not be impossible if time could be moved; for in order that what was yesterday may not be, it is necessary to convert yesterday into to-morrow; but this would be an absurdity. The past, the present, and the future, are essentially distinct things.

A simple space, a space without parts, is no space at all, it is a contradiction; neither is a simple time, a time without parts, a time, but is a contradiction.

A space whose parts are not continuous, is not a space; neither is a time whose parts are not continuous, a time. The parts of space are inseparable; you may distinguish them one from another, count them one after the other, compare them one with another, and consider them one after another, but you cannot separate them. All imaginable bodies may exist in the apartment where we write, one or many, at rest or in motion; but the space which we conceive is one, fixed, and always the same; we can estimate its extent in cubic feet, if we choose, but these feet are fixed and inseparable; we cannot separate one cubic foot from another, even if we would; for even while we annihilate it, it is present to us, and in the same distance that we need in order to conceive separation. We cannot conceive separation, if we do not conceive distance; nor conceive distance, if we do not conceive space. We separate bodies from each other, but not one space from another. Space remains with the same continuity when bodies are separated, and it is by this continuity remaining unalterable that we measure the extent of their separation. The same happens with time; it is a chain which cannot be broken. Can we conceive three successive, immediate instants, A, B, C, and then suppress B? Certainly not; such a suppression would be impossible, or it would be a poor diversion. We destroy B in our caprice, and A and C are continuous; since being only separated by B, when it disappears the extremes meet. But in this case it is no longer A, but B, for B is the instant which precedes C. We have no other distinction than that of priority with respect to C, and continuity with A. When, then, by the imaginary disappearance of B, A is brought into contact with C, it is converted into B. Moreover, A is not only connected with C, but is preceded by others; if, then, by the disappearance of B, it makes a step, so also must the whole infinite chain which precedes it. Each one is then a soldier, or rather no soldiery is possible, for we have taken an instant from the infinite chain, and so rendered it finite. Or, more distinctly; can we conceive yesterday or to-morrow without to-day, a future or a past without the present? Evidently we cannot. Time, then, is essentially composed of inseparable parts.

19. This similarity between time and space naturally leads us to believe that time is an abstract idea just as space is. What we have said of space is applicable to time, only with a few modifications exacted by the very nature of the thing. It can in no case be without utility, in scientific investigations, to approximate and compare these great ideas, which are as immense receptacles wherein our mind deposits its treasures. The actual corporeal universe, and all possible universes, are included in the idea of space; and all finite beings, corporeal or incorporeal, are included in that of time.

20. We may well suspect that these ideas, so intimately united to our perceptions, are formed in a similar manner; for it is probable that they belong to the order of those primitive laws which govern the development of our intellect.

21. The similarity between space and time must not make us ignore the differences which distinguish them.

I. All the parts of space are co-existent; otherwise, that continuity which is essential to them, would be inconceivable. Time is composed of successive parts; to imagine them co-existent, is to destroy the essence of time.

II. Space refers solely to the corporeal world, under only one aspect, that of continuity. Time extends to all that is successive, corporeal or incorporeal.

III. Consequently, the idea of space exists only in the geometrical order, of which it is the basis. The idea of time is mingled with every thing, and more especially with our own acts.

IV. Our soul, when reflecting upon itself, can totally prescind space, and forget all its relations with extended objects; but it cannot prescind time, which it finds necessary even to its own operations.

This last difference is a great help to the understanding in what the idea of time consists; and we venture to recommend it to the attention and memory of the reader.