Fundamental Philosophy, Vol. 2 (of 2)

CHAPTER VIII.

THE DEFINITION OF INFINITY CONFIRMED BY APPLICATION TO EXTENSION

59. We have explained the idea of infinity in general, by the indeterminate conceptions of being and the negation of limit. In order to assure ourselves that the explanation is well grounded, and that we have pointed out the essential marks of the conception, let us examine whether their application to determinate objects corresponds to what we have established in general.

If the idea of infinity is what we have defined it to be, we may apply it to all objects of sensible intuition or of the pure understanding, and we shall obtain the results which we ought to obtain, including the anomalies already referred to.38

60. The anomalies, or, rather, the contradictions which we seem to find in the applications of the idea of the infinite, when any thing is presented to us as infinite which we afterwards discover not to be so, originate in the application of this idea under different conditions. This variety would not be possible if the idea represented any thing determinate; but as it only contains the negation of limit in general joined to being in general, it follows that we subject this negation to particular conditions in each case, and therefore when we pass to other conditions, the general idea cannot give us the same result.

61. A line drawn from the point where we are situated in the direction of the north, and produced infinitely, gives us an infinite and a not-infinite. This contradiction is only apparent; there is really only the difference of result caused by the condition under which the general idea is applied.

When we consider a line infinitely produced towards the north, we do not apply the idea of the infinite to a lineal value in the abstract, but to a right line starting from a point and produced only in one direction. The result is what it should be. The negation of limit is affirmed under a condition; the infinite which results is subject to that condition. It may be said that there is no medium between the infinite and the not-infinite; but it is easy to solve this difficulty, if we observe that yes and no, to be contradictory, must be referred to the same thing, which is not the case when the conditions of the object are changed.

62. If instead of a line produced in one direction only, we had wished to apply the negation of limit to a right line in general, it is evident that we should have been obliged to produce the line in the two opposite directions: which would have given us another infinite under a new condition.

We have before seen that not even in this case can we have a lineal value strictly infinite; because this right line only forms a part of the sum of lines which we can imagine. Is it then infinite, or is it not? It is both, if we make the proper distinction. It will be infinite, or we shall have the idea of infinity or negation of limit, applied to a right line alone; but if instead of one right line alone, we take a lineal value, without any condition, the supposed line will not be infinite; the negation of the limit is not applied under that condition; the result must therefore be different.

63. We find the same anomaly, if we take two lines alone. Let us suppose a right line infinitely produced in both directions, and by its side let us describe a curve with continual undulations extending infinitely in a direction parallel to the right line. Both lines will be infinite if we consider only their direction, abstracting their lineal value; but if we regard this value the curve is greater than the straight line; for it is evident if we take a part of the curve corresponding to a part of the straight line, and extend or straighten this part of the curve, it will be greater than the corresponding part of the straight line; as this may be done throughout the whole length of the lines, the lineal value of the curve must be greater than that of the straight line in proportion to the law of its undulations.

64. This may suffice to show how the idea of infinity may be applied under different conditions and produce different results, without any contradiction. What is infinite under one aspect is not so under another aspect; hence we have the orders of infinities which figure so largely in mathematics; but I say again that these contradictions are not susceptible of any explanation if we attribute an absolute value to the idea of the infinite, instead of considering it as the abstract representation of the negation of limit.

65. Is it possible to conceive in a right line or curve an absolutely infinite length or lineal value, to which we may apply the negation of limit absolutely? I think not: for whatever be the line under consideration we can always draw others, which, added to the first, will give a value greater than that of the first above. This is a case in which there is a contradiction between the negation of limit and the condition to which it is subjected. You demand a lineal value to which the negation of limit may be applied absolutely; and on the other hand you require that this lineal value should be found in a determinate line, which by the fact of its being determinate, excludes the absolute negation of limit. The problem supposes contradictory data; therefore the result must be a contradiction.

66. What must we suppose in order to conceive an absolutely infinite lineal value? We need only suppose no condition which excludes the absolute negation of limit. We must here distinguish between the pure conception and the sensible intuition in which it is expressed. The conception of infinite lineal value exists from the moment that we unite the two general conceptions of lineal value and negation of limit. But the sensible intuition, which may represent this conception, is not so easy to imagine, even in general. To arrive at it we must imagine a space without any limit; and then considering in general all the lines whether right lines or curves, which may be drawn in it, in all directions, and under all possible conditions, we must take the sum of all these lineal values; and the result will be an absolutely infinite lineal value; for we shall have applied the negation of limit without any restriction.

67. We may obtain in the same way an infinite superficial value; for it is evident that we may apply to it all that we have said of lineal values.

68. In all these cases we apply the negation of limit to extension considered only in some of its dimensions. If we wish to obtain an absolutely infinite extension, we must abstract no dimension; consequently the absolutely infinite of this order, is extension in all its dimensions with the absolute negation of limit. But it is also to be observed that we must presuppose an absolutely infinite value of extension in order to obtain an absolutely infinite value of lines or surfaces; because it is equivalent to presupposing an infinite space in which the lines and surfaces may be drawn in all directions and under all possible conditions.

CHAPTER IX.

CONCEPTION OF AN INFINITE NUMBER

69. Can we conceive an infinite number? On one side, it seems not; because we doubt its possibility, and if we possessed this idea we should have no doubt of its existence. On the other side, it seems that we can conceive an infinite number; for we know immediately when a number is not infinite, and we could not know this if we had not the idea of infinite number.

Our observations on infinite series would seem to prove that the idea of infinite number is an illusion; for we find those numbers which we believed infinite, not to be so.

I think this question may be solved on the same principles as those of the last chapter. I see no difficulty in admitting the idea of an infinite number, nor how any contradiction can proceed from it.

70. Number is a collection of units; it is a general idea, because to conceive the number, we do not need to know of what class, or how many the units may be. The idea of number in general abstracts absolutely all such determinations. It is evident that, whatever number we imagine, we can always conceive another still greater, and if we assign a limit to a number, we can always remove it indefinitely, so that the limit of one is not the limit of the other. To the idea of number, we unite the idea of a limit and of the negation of another limit. Therefore, if we unite to the idea of number in general, the idea of the negation of limit in general, we shall obtain the idea of an infinite number.

71. What does this idea represent? It represents nothing determinate: it is an entirely abstract conception, formed of two other abstract conceptions, those of number and the negation of limit. No determinate object corresponds to it; it is a work of our understanding referred to objects in general, without a determination of any sort. We may now solve the difficulties previously intimated.

72. Why is a series of terms presented to us as infinite, which, when we examine it closely, we find wants some of the marks of infinity? Because, in the first instance, we apply the negation of limit under a condition which we take no notice of in the second instance.

Set us the series a, b, c, d, e, .....

It is evident that we may continue it infinitely, and conceive the negation of all limit of this continuation: in this sense, the number of terms is infinite; for the idea of the negation of limit is really applied to the series. When we ask if the number of terms is absolutely infinite, we abstract the condition under which we had united the negation of limit. That, therefore, which is infinite in one instance is not so in another. Still there is not any contradiction because the yes and the no refer to different suppositions.

73. Let us take a line and measure it by feet. Producing this line we multiply the number of feet; and we may conceive the negation of all limit of this multiplication. The number of feet will then be infinite. If instead of a foot we take an inch as the unit of measure, we shall have a number twelve times as great. This number would also be infinite, and thus we should have two infinite numbers, one of them greater than the other. Is there any contradiction in this? Certainly not: there is only a different combination of ideas. In the first case, the idea of the negation of limit was subordinated to the condition of the division of the line into feet: whereas, in the second case, we introduce a different condition; the division of the line into inches.

74. But, it may be said, these numbers, considered in themselves, abstracted from their relation to feet or inches, are equal or they are not equal; consequently they are infinite or not infinite. The objection vanishes as soon as we correct the error which supports it. When we abstract all relation to determinate divisions, we consider number in general; on this supposition there are not two cases, but only one; there cannot then be a relation of greater or less. We have only the conception of number in general combined with the idea of the negation of limit in general; therefore the result must be an infinite number in the abstract.

The difficulty consists in a contradiction which escapes our sight at first. We abstract particular conditions in order to know if the numbers are in themselves infinite or not; and at the same time we do not abstract them, because it is only in reference to them that the objection has any meaning, since it supposes the division into various kinds of units. When, therefore, we speak of particular numbers, and at the same time pretend to consider them in themselves, we fall into a contradiction, because we take the numbers both with and without particular conditions at the same time.

75. From all that has been said, we may conclude that the conception of infinite number, abstracted from the nature and relations of the things numbered, involves no contradiction, since it contains only the two ideas of number, as a collection of beings, and of the absolute negation of limit; but we cannot affirm from this alone, that an infinite number can be realized. Infinite number cannot become actual without an infinite collection of beings; and these beings, when realized, cannot be abstract beings, which contain nothing else but being; they must have characteristic qualities, and must be subject to the conditions imposed by these qualities. As we absolutely abstract these conditions in the general conception, it is not possible to discover, from the conception alone, the contradiction which they may imply. Hence, although there is no contradiction contained in the conception, there may still be in the reality. In the same manner, certain mechanical theories are perfectly conceivable, but they cannot be reduced to practice on account of the opposition of the matter to which they should be applied. Finite beings are the matter on which indeterminate and metaphysical conceptions are to be realized; the possibility of the conceptions does not absolutely prove the possibility of the beings. The reality may draw with it certain determinations involving a contradiction which was latent in the general conception, and is made manifest by the reality.

CHAPTER X.

CONCEPTION OF INFINITE EXTENSION

76. Is infinite extension conceivable? This conception includes two ideas: the idea of extension, and the idea of the negation of limit. The idea of extension is a general conception, referring to the intuition which, whatever may be in itself and in its object, represents extension and the union of the three dimensions, the pure form of which is space. It is evident that we can unite, in one conception, the two ideas of extension in general and the negation of limit; and if this is what is called the idea of infinite extension, it is clear that we have this idea. This conception of infinite extension, abstracts all conditions of the reality; we do not know whether there be, in the nature of extended things, any thing which prevents the absolute infinity of their extension; consequently, we are ignorant whether there is or is not any latent contradiction, which the general conception does not reveal to us.

77. It must be remembered that I am speaking of the idea and not of the sensible representation of extension; for although I hold that it is possible for us to have the conception of an infinite extension, I do not think the same with respect to its sensible representation. The latter may be indefinitely expanded, but it cannot become infinite.

Reason demonstrates this impossibility which consciousness makes known to us. Internal sensible representations are only the repetition of the external, or at least are formed from the elements which these latter furnish. Sight and touch are the two senses which produced the representation of extension, and they both imply a limit. Touch only reaches that which is immediate to it, and sight cannot see with a limit which sends the rays of light to it. Internal sensible representations must always retain this limitation; their object may be expanded, or the limit removed to a greater distance, but to destroy this limit would be to destroy themselves. Therefore, the imagination of an infinite extension is impossible to every sensitive being.

78. I have proposed above (§ 40) an objection against the infinity of extension, in so far as we may represent it as a size without limits.

The objection was, that as the idea of impenetrability is not contained in the conception of a solid, we may imagine an infinite series of infinites placed one inside of another. This difficulty is only conclusive when speaking of the conception of a solid which contains something more than the pure idea of extension. The idea of extension necessarily implies that some parts are outside of others, and it is not possible to conceive extension otherwise. It is certain that a body may be situated in a part of space; taking from this body its impenetrability, we may put another body in the same place, and so on to infinity; but in that case we conceive something besides pure extension, we unite something, although in a general and indeterminate manner, to the idea of things situated in space; otherwise we should not distinguish the space, representing pure extension, from the solids placed in it, nor should we distinguish these solids from one another, if we did not recognize in them some difference, although general and undetermined.

79. It seems most probable that the pure idea of an infinite extension is contained in the idea of an infinite size, which is nothing more than the idea of space. Whatever else is introduced into the idea is a foreign element, adding to pure extension something which does not belong to it, such is the difference between extended beings, although conceived in an indeterminate manner.

CHAPTER XI.

POSSIBILITY OF INFINITE EXTENSION

80. What are we to think as to the possibility of the infinities which we conceive? Let us examine the question.

Is an infinite extension possible? There is no incompatibility between the idea of extension and the negation of limit, at least, according to our way of conceiving them. It is more difficult for us to conceive extension absolutely limited, than to conceive it unlimited: beyond all limit, we imagine space without end.

81. Neither do we discover any impossibility in the existence of an unlimited extension, if we consider the question in relation to the divine omnipotence. Beyond all extension God can create another extension; if we suppose that he has applied his creative power to all the extension possible, he must have created an infinite extension.

82. Here a difficulty arises. If God had created an infinite extension he could not create another extension; his power would be exhausted, and consequently it would not be infinite.

This difficulty proceeds from understanding infinite power in a false sense. When we say that God can do all things, we do not mean that he can do things that are contradictory: omnipotence is not an absurd attribute, as it would be if applied to things that are absurd. An absolutely infinite extension is contradictory in relation to another distinct extension; for, being absolutely infinite, it contains all possible extensions. If we suppose it to exist, no other is possible: to affirm that God could not produce another, is not to limit his omnipotence, but only to say that he cannot do a thing which is absurd.

83. We will make this solution clearer. The intelligence of God is infinite; and he cannot understand more than he now understands; all progress would suppose imperfection, because it would involve a change from a less to a greater intelligence. If, then, we say that God will never understand more than he does now, do we limit his intelligence? Certainly not. He cannot understand more, because he understands all that is real and all that is possible, and we cannot, without contradiction, conceive that he can understand more than he now does: this is not to limit his intelligence, but to affirm its infinity: it is not susceptible of perfection, because it is infinite. This will enable us to understand the expression cannot, as applied to God. What is denied is not a perfection, but an absurdity: wherefore St. Thomas very opportunely observes, that we should much better say that the thing cannot be done, than that God cannot do it.

CHAPTER XII.

SOLUTION OF VARIOUS OBJECTIONS AGAINST THE POSSIBILITY OF AN INFINITE EXTENSION

84. The discussions on the possibility of an infinite extension are of a very ancient date. How could it be otherwise? Must not the glorious spectacle of the universe, and the space which we imagine beyond the boundaries of all worlds, naturally have given rise to questions as to the existence or possibility of a limit to this immensity?

Some philosophers think an infinite extension impossible. Let us see on what they found their opinion.

85. Extension is a property of a finite substance, and that which belongs to a finite thing cannot be infinite; therefore it is impossible to conceive infinity of any kind in a finite being. This argument is not conclusive. It is true that an extended substance is finite, in the sense that it does not possess absolute infinity such as is conceived in the Supreme Being; but it does not follow from this that it cannot be infinite under certain aspects. Neither is it correct to say that no finite substance can have an infinite property, because the properties flow from the substance, and the infinite cannot proceed from the finite. In order that this argument may be valid, it is necessary to prove that all the properties of a being emanate from its substance: figures are accidental properties of bodies, and yet many of them have no relation to the substance, and are mere accidents which appear or disappear, not by the internal force of the substance, but by the action of an external cause. We see extension in bodies; but as we know not the essence of corporeal substance, we cannot say how far this property is connected with the substance, whether it is an emanation from it, or only something which has been given to it and may be taken from it without any essential alteration.39

Moreover, when we say that the infinite cannot proceed from the finite, we do not deny that an infinite property may proceed from a substance finite in its essence.

When we admit the infinite property, we admit at the same time all that is necessary in the substance in order that this property may have its root in it, so long as we do not deny the character of finite which essentially belongs to every creature. When we deny that creatures are or can be infinite, we speak of essential infinity, of that infinity which implies necessity of being and absolute independence under every aspect; but we do not deny them a relative infinity, such as that of extension.