Kitabı oku: «Fundamental Philosophy, Vol. 2 (of 2)», sayfa 20

CHAPTER II.
IMPORTANCE AND ANOMALY OF THE QUESTIONS ON THE IDEA OF THE INFINITE

12. The examination of the idea of the infinite is of the highest importance, not only because we meet it in various sciences, the exact sciences among others, but because it is one of the principal characteristics by which we distinguish God from creatures. A finite God would be no God; an infinite creature would not be a creature.

In the scale of finite beings we discover a gradation, by which they are interlinked; the less perfect, as they are perfected, go on approaching the perfect; and there are, preserving the limits of each one's nature, points of comparison by which we may measure their respective distances. Between the finite and the infinite there is no comparison; all measures are inadequate and as nothing. We pass from an imperceptible drop to an immense ocean; from the atom which escapes observation to the abundance of matter diffused through all space; and much as these transitions express, they are as nothing to the transition from the finite to the infinite; these oceans, compared with the infinite truth, become in their turn imperceptible drops, and thus an interminable scale baffles the efforts of the mind in search of something to correspond to its idea. The examination of the idea of the infinite ought to occupy an important place in the study of philosophy, although it served for no other purpose than the contemplation of infinite greatness.

13. The disputes on the idea of the infinite, not only in relation to its nature, but also to its existence, present a strange anomaly. If it exists in our mind it ought to fill it entirely, so that it must be impossible to cease to perceive it. Yet it is well known that philosophers dispute even on the existence of this idea; although it is an infinite treasure, those who possess it doubt its reality – just as the heroes in romance, when they find themselves in a castle richly and splendidly adorned, imagine it the effect of enchantment.

14. The mere dispute as to whether the idea of the infinite be positive or negative, is equivalent to the question of its existence. If it is negative, it expresses an absence of being; if positive, the plenitude of being. What question can be more vital to an idea than the dispute whether it represents the absence or the plenitude of being?

15. Here again we meet the fact which we have observed in the preceding discussions. Reason, after digging at its own foundations, is threatened with death under the ruins of its loftiest edifices.

CHAPTER III.
HAVE WE THE IDEA OF THE INFINITE?

16. If we had no idea of the infinite, the word would have no meaning to us, and when used it would not be understood.

17. Whatever may be the nature and perfection of our idea of the infinite, it is certain that it involves something fixed, and common to all intelligences. We apply the idea to things of very different orders, and it is always understood in the same sense by all men. Even the difficulty we find in attempting to explain it, in itself or in its applications, proceeds from the idea itself; it is a difficulty which we all meet with, because we all conceive in the same manner what is understood by the infinite, taken in general.

18. Infinite and indefinite express very different meanings. The infinite implies the absence of limits; the indefinite implies that these limits retire continually from us; it abstracts their existence, and only says that they cannot be assigned.

19. Whatever exists is finite or infinite; for it either has limits or it has not: in the first case, it is finite; in the second, infinite: there is no medium between yes and no.

20. Hence, properly speaking, there is in reality nothing indefinite; this word only expresses a mode of conceiving things, or rather a vagueness in the conception, or indecision in the judgment. When we do not know the limits of any thing, and, on the other hand, do not dare to affirm its infinity, we call it indefinite. Thus, space is called indefinite by those who see no way of assigning a limit to it, and yet are unwilling to say that it is infinite. Even in ordinary language we call a thing indefinite which has no limits assigned to it; thus, we say "a concession has been made for an indefinite time," although it is limited to some time which has not been determined.

21. The idea of the infinite does not consist in conceiving that another quantity may always be added to a given quantity, or that a perfection may be made more intense; this expresses only the possibility of a series of conceptions by which we endeavor to approach the absolute idea of the infinite. It is easy to see that the absolute idea is something distinct from those conceptions, because we regard it as a type to which the series of connections is referred, but which it can never equal, no matter how greatly prolonged.

22. Let us consider the words in which we naturally express what passes within us when we think of the infinite.

What is an infinite line? A line which has no limits. Is it a million, or a billion miles in length? There is no number to express its length; it will always be greater than the number. But do we not approach the infinite in proportion as we prolong a finite line? Certainly, in so far as approaching means only placing quantities which are found in what we approach; but not in so far as it means that this difference can be assigned. There is no comparison between the finite and the infinite; and therefore it is not possible to assign the difference between them. Would an infinite line be formed by the addition of all finite lines? No; for we can conceive the multiplication of each of the terms of the addition, and therefore an increase in the infinite, which would be absurd. Would the infinity of the line consist in our not knowing its limits, or not thinking of them? No; but in its not having them.

23. Thus, we see, that the idea of the infinite, is in the reach of the most common intellects, and expresses only what any person of ordinary understanding would say, even though he had never occupied himself with philosophical studies; that the idea of the infinite is in our understanding, as a constant type, to which all finite representations are unable to arrive. We know the conditions which must be fulfilled, but at the same time, we see the impossibility of fulfilling them. When any one tries to persuade us of the contrary, we reflect on the idea of the infinite, and say: "No; it is a contradiction of infinity; it is not infinite, but finite." We distinguish perfectly well between the absence of the perception of the limit and its non-existence. If any one tries to make us confound these two ideas, we answer, "No; they must not be confounded; there is a great difference between our not perceiving an object and the non-existence of that object, and we are not now examining whether we conceive the limit, but whether it exists." Though the limit retire and hide itself, so to speak, from our eyes, we are not deceived: it exists, or does not exist. If it exists, the condition involved in the conception of infinity is not fulfilled, and the object is not infinite, but finite; if it does not exist, there is true infinity, – the condition is complied with.

24. When the idea of the infinite is considered in general, it can never be confounded with the idea of the finite. There is a line which divides them, and which prevents all error; for it is the principle of contradiction itself; it is the distinction between yes and no. When we say finite, we affirm the limit; when we say infinite, we deny it. No ideas can be clearer or more exact.

CHAPTER IV.
THE LIMIT

25. The word infinite is equivalent to not finite, and seems to express a negation. But negations are not always truly such, although the terms imply it; for if that which is denied, be a negation, the denial of it is an affirmation. This is the reason why two negatives are said to be equivalent to an affirmative. If I say, it has not varied, and you deny it, you deny my negation; for it is the same thing to deny that it has not varied, as to affirm that it has varied. In order, therefore, to determine whether the word infinite expresses a true negative, we must know what is meant by the word finite.

26. The finite is that which has a limit. A limit is the term beyond which there is nothing of the object limited. The limits of a line, are the points beyond which the line does not extend; the limit of a number, is the extreme where the number stops; the limit of human knowledge, is the point to which we may arrive, but which we cannot go beyond. A limit being a negation, to deny a limit, is to deny a negation, and is consequently an affirmation.

27. It is easy to see from these examples, that a limit in the ordinary sense, expresses an idea distinct from what mathematicians define it. They call a limit every expression, whether finite, infinite, or a nullity, which a quantity may continually approach without ever reaching. Thus, the value 0/a is the limit of the decrement of a fraction, the numerator of which is variable x/a; because, if we suppose X to be constantly diminishing, the fraction will approach the expression 0/a, without ever being confounded with it, so long as X does not entirely disappear. If we suppose (b + x)/a an expression in which X is decreasing, the expression will continually approach (b + 0)/a = b/a, which will be the limit of the fraction. If we suppose the expression a/x, in which X is decreasing, we shall continually approach the expression a/0 = ∞, an infinite value which the fraction can never attain, until X becomes 0, which cannot happen, because X is a true quantity. These examples show that mathematicians admit limits which are finite, infinite, or a nullity, and prove that mathematicians employ the word limit in a different sense from its ordinary as well as philosophical meaning.

28. A limit, therefore, expresses a true negation, and the word finite, or limited, necessarily involves a negative idea. That which is not, is not limited; therefore the finite is not an absolute negation. An absolute negation is nothing, and we do not call the finite nothing. Therefore, in the idea of finite are contained being, and a negation of another being. A line one foot in length, involves the positive value of one foot, and the negation of all value of more than a foot. Therefore, the finite, in so far as finite, involves a negation relatively to a being. If we could express this idea in the abstract, using the word finity, as we have the word infinity, we should say that finity in itself expresses only the negation of being relatively to a being.

29. Hence, the word infinite is not negative; for it is the negation of a negation. The infinite is the not-finite; it is that which has no negation of being, consequently that which possesses all being.

30. We have, therefore, an idea of the infinite, and this idea is not a pure negation. But it must not be supposed that we have arrived at the last term of the analysis of the infinite. We are still far from it, and it is even doubtful whether we shall obtain any satisfactory result after long investigations.

CHAPTER V.
CONSIDERATIONS ON THE APPLICATION OF THE IDEA OF THE INFINITE TO CONTINUOUS QUANTITIES, AND TO DISCRETE QUANTITIES, IN SO FAR AS THESE LAST ARE EXPRESSED IN SERIES

31. One of the characteristic properties of the idea of the infinite is application to different orders. This gives occasion to some important considerations which greatly assist to make this idea clear in our mind.

32. From the point where I am situated I draw a line in the direction of the north; it is evident that I may prolong this line infinitely. This line is greater than any finite line can be; for the finite line must have a determinate value, and therefore, if placed on the infinite line, will reach only to a certain point. This line, therefore, seems to be strictly infinite in all the force of the word, because there is no medium between the finite and the infinite, and we have shown that it is not finite, since it is greater than any finite line; therefore it must be infinite.

This demonstration seems to leave nothing to be desired; yet there is a conclusive argument against the infinity of this line. The infinite has no limits, and this line has a limit, because, starting from the point from which it is drawn in the direction of the north, it does not extend in the direction of the south.

33. This line is greater than any finite line; but we may find another line greater still. If we suppose it produced in the direction of the south, it will be greater by how much it is produced towards the south; and if it be infinitely produced in this direction, its length will be twice that of the first line.

34. By the infinite prolongation of a line in two opposite directions we seem to obtain an absolutely infinite line; for we cannot conceive a lineal value greater than that of a right line infinitely prolonged in opposite directions. But it is not so: by the side of this right line another may be drawn, either finite or infinite, and the sum of the two will form a lineal value greater than that of the first line; therefore that line is not infinite, because it is possible to find another still greater. And as, on the other hand, we may draw infinite lines and prolong them infinitely, it follows that none of them can form an infinite lineal value, because it is only a part of the lineal sum resulting from the addition of all the lines.

35. Reflecting on this apparent contradiction in our ideas, we discover that the idea of the infinite is indeterminate, and consequently susceptible of different applications. Thus, in the present instance, it cannot be doubted that the right line, prolonged to infinity, has some infinity, since it is certain that it has no limit in its respective directions.

36. This example would lead us to believe that the idea of the infinite represents nothing absolute to us; because even among those objects which are presented the most clearly to our mind, such as the objects of sensible intuition, we find infinity under one aspect which is contradicted one by another.

37. What we have observed of lineal values is also true of numerical values expressed in series. Mathematics speak of infinite series, but there can be no such series. Let the series be a, b, c, d, e, …: it is called infinite if its terms continue ad infinitum. It cannot be denied that the series is infinite under one aspect; for there is no limit which puts an end to it in one sense; but it is evident that the number of its terms will never be infinite, because there are others greater; such, for instance, is the series continued from left to right, if continued from right to left at the same time, in this manner:

..... e, d, c, b, | a, b, c, d, e, …

In this case the number of terms is evidently twice as great as in the first series.

Therefore the series which are called infinite are not infinite, and cannot be so, in the strict sense of the term.

38. But what is still more strange is, that the series is not infinite, even though we suppose it continued in opposite directions; for by its side we may imagine another, and the sum of the terms of both will be greater than the terms of either; therefore neither will be infinite. As it is evident that whatever be the series, we can always imagine others, it follows that there can be no infinite series in the sense in which mathematicians use the word series to express a continuation of terms, not excluding the possibility of other continuations besides the supposed infinite continuation.

39. The objections against lineal infinity apply equally to surfaces. If we suppose an infinite plane, it is evident that we can describe an infinity of planes distinct from the first plain and intersecting it in a variety of angles; the sum of all these surfaces will be greater than any one of them. Therefore the infinite extension of a plain in all directions does not constitute a truly infinite surface.

40. A solid expanding in all directions seems to be infinite; but if we consider that the mathematical idea of a solid does not involve impenetrability, we shall see that inside of the first solid a second may be placed, which, added to the first, will give a value double that of the first alone. Let S be the empty space which we imagine to be infinite; and let W be a world of equal extension placed in it and filling it; it is evident that S + W are greater than S alone. Therefore, although we suppose S to be infinite, = ∞, W also = ∞; therefore S + W = ∞ + ∞ = 2 ∞. And as this value expresses the size, the first is not infinite because it can be doubled. If we take the impenetrability, the operation may proceed ad infinitum.

Therefore the first infinite, far from being infinite, seems to be a quantity susceptible of infinite increase.

CHAPTER VI.
ORIGIN OF THE VAGUENESS AND APPARENT CONTRADICTIONS IN THE APPLICATION OF THE IDEA OF THE INFINITE

41. The difficulties in the application of the idea of infinity, seem on the one hand, to prove that either this idea does not exist in us, or is very confused; and on the other hand, that we possess it, and in a very perfect degree. Why do we discover that numbers are not infinite, although at first they seem to be? Why do we deny the infinity of certain dimensions, notwithstanding their infinite prolongation in one sense? Because, on examining these objects, we find that they do not correspond to the type of infinity. If this type did not exist in our mind, how could it be possible for us to make use of it? How could we compare beings with it, if we did not know it? Is it possible to know when any thing arrives at a turn, if we have no idea of that turn? It is comparing without a point of comparison; that is, it is exercising a contradictory act.

42. Although these arguments in favor of the existence of the idea of the infinite, if we examine our own mind, we cannot deny that we find there a certain vagueness and confusion which inspire strong doubts as to the reality of this idea. What is presented to our mind, when we think of the infinite? The imagination abandoned to itself, extends space, expands dimensions, multiplies numbers indefinitely, but it offers nothing to the intellect which has the marks of infinity. If we leave the imagination, and regard the understanding only, it gives a type by which to judge of the infinity or not-infinity of the objects presented to it, but if we reflect on the type itself, it loses the clearness it possessed before, and we even ask if the type really exists.

43. Do we, therefore, deny the existence of this idea? are we going to renounce our intention of explaining it? We do neither. I believe that it is necessary to admit the idea, that it is not impossible to explain it, and that we may even point out the reason of its obscurity.

44. Before passing further, I wish to observe, that one of the causes of the difficulties in the explanation of the idea of the infinite, arises from our not distinguishing the intuitive from the abstract cognition.³⁶ Many difficulties would be avoided by attending to this distinction. When we say that the idea of the infinite is not intuitive, but abstract, we give the key to the solution of the principal objections brought against it.

45. We have no intuitive idea of infinity; that is to say, this idea does not present to our mind an infinite object; we can have this intuition only when we see the essence of God, which will happen in a future life.

46. If we had now the intuition of an infinite object, we should see its perfections as they are, with their true marks; or rather, we should see how all the perfections dispersed among limited beings, are united in one infinite perfection. We could not refer the idea of the infinite to determinate objects, as, for example, to extension, because these objects contradict the idea. It would be impossible for us to modify the idea in different ways, and apply it first in one sense, and then in another very different sense. The idea is one, and simple; it would, therefore, always relate to an object which is also one and simple, not vague and indeterminate, as now, but with the determination of a necessary existence and an infinite perfection. We should have intuition of infinite being, as we have intuition of the facts of our consciousness: our cognition of it would be that of an object eminently incommunicable, as predicate to any order of finite beings; and it would be as manifest a contradiction, to apply the idea of this infinity to any number or extension, as it would be to identify an act of our consciousness with external objects.

47. The indeterminate character in which the idea of the infinite is presented us, and the ease with which we modify it in various ways, and apply it to different objects, in different senses, proves that this idea is not intuitive, but abstract and indeterminate, that it is one of those general conceptions, by the aid of which the mind obtains a certain knowledge not afforded by intuition.

This will explain the origin of the vagueness of our idea of infinity. Indeterminate conceptions, and because they are indeterminate, relate to no particular object, or quality, which may be conceived by itself alone, as something which may be realized; they do not contain those determinations which fix our cognition in an absolute manner. The indeterminate manner in which they present any property of beings, causes a difference in the application, accordingly as the particular properties, which are combined with the general, are different. If we take a right-angle triangle, in which we know the measure of all the sides and angles, the determinateness of the idea avoids the vagueness of the intellect, and prevents the application of this idea to cases different from that which is determinate and fixed. But if we take a right-angle, in general, without determining the value of its sides and angles, its applications may be infinite. The more general and indeterminate the idea of a triangle becomes, the greater is the variety of its applications.

48. Indeterminate ideas, in order to represent any thing, must be applied to some property which is the condition of their actual or possible realization. Until this application is made, they are pure intellectual forms, which represent nothing determinate. I do not mean by this, that these ideas are empty conceptions, which cannot be applied outside of the sensible order, as was maintained by Kant;³⁷ but only that granting them an universal value, I deny that they have by themselves alone a value representative of any thing that can be realized, beyond the property which they express. The idea of a pure triangle can not be realized, for every real triangle would contain something more than is in the idea: it would be a right-angled or oblique-angled, etc., all which, the pure idea abstracts. The object will be indeterminate, in proportion to the indeterminateness of the properties contained in the conception; consequently, that which is presented to the understanding will also be more vague, and the applications which may be made of the idea, will be more varied and numerous, as is the case in the ideas of being, not-being, limit, and the like.