THE INFINITE.
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CHAPTER I.
TRANSITORY VIEW OF THE ACTUAL STATE OF PHILOSOPHY.
1. In the works on transcendental philosophy which have been published of late years, we find the words infinite, absolute, indeterminate, unconditioned, frequently repeated, and made to play a very prominent part in the explanation of the most recondite secrets which can be presented to the consideration of man. The words finite, relative, determinate, conditional, are easily combined with these; and from this combination they pretend that a ray of light will arise to dissipate the darkness of philosophical questions.
2. In spite of the bad use many make of such words, we must confess that the fact indicated is consoling by reason of the great desire there is to use them. This desire marks an effort in the human mind to raise itself from the mire in which the impious school of the last century has sunk it.
3. What was the world in the eyes of the false philosophers who preceded the French revolution? A mass of matter, subject to simple mechanical laws of motion, the whole explanation of which was given in two words, blind necessity. What was the human mind? Nothing but matter. What was thought? A modification of matter. In what did the difference between thinking and non-thinking matter consist? In a little greater or less subtilty, in a more or less happy disposition of atoms. What was morality? An illusion. What were sentiments? A material phenomenon. What was the origin of man? That of matter,—a phenomenon offered by a quantity of molecules, which at one moment happen to be disposed one way, and a moment after in a very different way. If you inquired if there were a destiny beyond the grave, We argue that question! they would answer with a scornful smile. Have you such a word as religion? The scorn increased and changed into contempt. Do you recognize the dignity of the human race? O, yes! we admit this dignity, and we are of opinion that it is of the same nature as that of the brutes, only it has reached a higher degree of perfection. We do not deny that your form may be more noble and elegant than that of the monkey, nor do we dispute the superiority of your intelligence; but we would have you take good care not to make pretensions to a nobler origin or a loftier destiny. The course of ages may develop and perfect the monkey form, and render it equal with yours; it may develop and perfect his cerebral organs, so that from this very monkey, whose extravagant motions and ridiculous attitudes now amuse, men will be born such as were Plato, Saint Augustine, Leibnitz, or Bossuet.
4. With such a system, it was useless to deal in ideas: they retained only sensations. Whatever could occupy the mind of man, whether the most imbecile or endowed with the loftiest genius, was nothing more than a sensation transformed. The very brutes possessed all the elements of human intelligence; to think was only to feel more perfectly. Such was the last term of their analysis; such the result of their most accurate observation; such the solution their profoundest philosophy gave to the problems of man's understanding. Plato, Aristotle, Saint Augustine, Saint Thomas, Descartes, Malebranche and Leibnitz were nothing but sublime dreamers, whose genius strongly contrasted with their ignorance of the true nature of things. None of them knew any thing about ideology or metaphysics; these sciences were an unknown world until Locke and Condillac came and discovered them.
5. This school, as fatal as frivolous, has involved and stifled mind in matter. The butterfly could not unfold his wings of fair and various colors; he was forced to lay them off and to change into a stupid and filthy worm, entangled in a covering as loathsome and unclean as itself. In this consisted progress. The limit of ideological perfection was to deny ideas; that of metaphysical studies, to deny spirits; that of morals, to deny morality; that of society, to deny authority; that of politics, to establish license; that of religion, to deny God. Thus, human reason, thinking to advance, marched in a retrograde direction; and proposed to raise the edifice of its knowledge, when there was nothing left to demolish: thus they imagined to attain a scientific result by denying every thing, and by finally denying themselves.
6. At present, there is a reaction against so degrading a philosophy. We have only to open the writings of the philosophers of this age to convince ourselves of this consoling truth. We everywhere meet the word idea in contraposition to that of sensation; that of mind to that of matter; that of activity of thought to that of bodily motion; those of cause, order, liberty, of free will, morality, infinity. The ideas which accompany them are sometimes inexact, sometimes extravagant; but at the bottom of all this we distinguish an anxious desire to rise from the abyss down to which an atheistical and material philosophy had dragged the human mind. Some who have contributed to the reaction do not admit a free and intelligent God, distinct from the universe. What we have said above is therefore true, that pantheism is atheism in disguise; nevertheless, the atheism of the pantheist now-a-days is an atheism which is ashamed to confess itself such, and which sometimes, perhaps, deceives itself, being persuaded that it is not.
7. The atheism of modern philosophers unites itself with the infinite: it does not reject those great ideas which as relics of a primitive tradition were common in the old world, and were afterwards fixed, cleared up and elevated by the superior teaching of Christianity. The philosophy of the last century sat down in the darkness and shadow of death and declared itself alone in possession of light and life. Philosophy now still remains in obscurity, but it is not satisfied with it, and gropes about in the dark, seeking some outlet to the regions of light. Hence those desperate efforts to resolve itself, not into matter, but into the focus of intelligence, into the me, that is, the mind: hence the continual use of the words absolute, unconditional, infinite, words which, notwithstanding they ordinarily lead to absurdities, do yet indicate a sublime aspiration.
8. These observations show that we do not confound the philosophy of to-day with that of the past century; that we do not regard the pantheism of to-day as a pure materialism, and that, notwithstanding the atheism of which we accuse the doctrine of certain philosophers, we do not deny that they have, even in the midst of their extravagance, preserved a kind of horror of it, and that, lost as they are in the labyrinth of their speculations, they seek the thread which shall conduct them to the gates of truth.
9. This act of justice we willingly render to modern philosophers, but it will not prevent us from combating their pretension to a merit they do not possess. They style themselves restorers of the spirituality of the soul, and of human liberty; and when they speak of God they almost exact a tribute of gratitude from him for having replaced him upon his throne. Before making such proud pretensions they ought to have considered that they are even yet far from the truth with respect both to God and to man, not only as Christianity has at all times taught it, but also as the most illustrious modern philosophers have professed it. They are ambitious to be called restorers, but their restoration with its licentious frequency is a new revolution, at times as terrible as the evil it attempts to combat.
10. Another consideration ought to have moderated their zeal to be thought inventors, which is that they have said nothing concerning God, the human mind, thought, ideas, the liberty of freewill, which may not be read in all the works of the philosophers who flourished before, or even in the beginning of, the eighteenth century. Open the text-books of the schools, and you will find many things which they would have us believe to be important discoveries. The great philosophers gloried in knowing what they had before learned when children. The philosophical tradition of sound ideas was not interrupted during the past century. In many parts of Europe schools existed which taught them with scrupulous fidelity. And besides human schools, there was that of the God-Man, the Church of Jesus Christ, which, among its supernatural dogmas, preserved even natural truths, notwithstanding the senseless efforts which have been made to obliterate them.
11. To what, then, are the invention and restoration reduced? Invention there is not, either with respect to God, to the human mind, or to morality, for nothing true has been said of them which had not already been said. Restoration, properly so called, there is not; for what does not perish cannot be restored. The truth exists; and has been known and revered during the whole six thousand years it has refused to bow the knee to Baal. Let not deserters say, when they turn and come back to the truth that they have restored it, but that they have recovered it; not that they give, but that they receive it; not that they enlighten the world, but that they are blind, and that it is the goodness of Providence which opens their eyes to the light.
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.
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.
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 = 8, 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.
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, = 8, W also = 8; therefore S + W = 8 + 8 = 2 8. 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.
FUNDAMENTAL EXPLANATION OF THE ABSTRACT IDEA OF THE INFINITE.
49. Supposing that our idea of the infinite is not intuitive but abstract, let us see how its true nature may be explained.
We have the ideas of being and of its opposite, not-being; these ideas considered in themselves are general, indeterminate, and may be applied to every thing which is subjected to our experience.
We may affirm and deny something of every limited being: we may affirm what it is: we may deny what it is not: the limit is only conceived as such when something is denied of it.
50. The activity of our being is unceasing, but it is limited by the absence or the resistance of objects; the external world is an assemblage of beings presenting a great variety of limitations.
Therefore both internal and external experience give us the idea of the finite, that is, of a being which involves some not-being. The brute has sensible perception, but no understanding: it is sensitive, and herein it has being; it is not intelligent, and herein it is limited. Man is sensitive and intelligent; the limit of the brute is not the limit of man. Among intelligent beings some understand more than others; therefore the limit of all is not the same.
51. Since we find a limit in both internal and external experience, it is evident that we can form the general idea of limit, that is, of a negation applied to an object.
52. The same experience teaches that what is the limit of some things is not the limit of others, and that the limit applied to one object must be denied of another. When we compare different beings together, we frequently find ourselves denying certain limits. As our understanding has the faculty of generalizing, it is evident that we may conceive in general the negation of certain limits, and form an indeterminate conception, including the two ideas of negation and limit.
53. I do not see what objection can be made either to the possibility or to the existence of this conception; but as this fact is necessary for the explanation of the idea of infinity, I shall make some further observations for the purpose of confirming it.
We have an idea of negation in general; this is a primitive fact of our mind: without it no negative judgments would be possible, nor could we even know the principle of contradiction. It is impossible for any thing to be and not be at the same time; when we say not be we express a negation, we therefore have the conception of negation. This conception is general, because it involves no determination; we speak of not-being without applying it to any particular object, nor even to any determinate species or genus. Therefore the conception of negation is general and absolutely undetermined.
54. We have the idea of limit; for, as we have seen, it is a negation applied to a being. We have also the idea of the negation of limit; for just as we conceive the limit as applied or applicable, we may and do conceive it as not applied or not applicable. At every moment we deny certain limits; this idea generalized becomes the negation in general of limit in general.
55. After these remarks we may establish what is contained in the idea of the infinite. This idea is a general conception involving the conception of being in general, and the negation of limit in general. The union of these two conceptions constitutes the abstract idea of the infinite.
56. The general conception of the negation of limit gives us an idea of infinity in the abstract, but not any infinite thing. Without the intuitive cognition of an infinite object, and with only a very imperfect idea of it, we may speak of infinity without falling into contradiction, and determine the cases in which it may be applied to a being or to an order of beings, whether real or possible. Man has many ideas of this vague kind, which nevertheless answer his necessities. We shall make this palpable by examples.
57. Suppose we take an uneducated person and point out to him a number of learned men, telling him that one of them knows more than all the rest. The uneducated person has no idea of what the man knows who knows the most, nor the man who knows the least; he has no idea of the degrees of science, nor of what science itself is; but he possesses the general ideas of degree, of more and less, and also of knowledge, and this enables him to speak, without contradiction or confusion, of the greater science of the one and the less science of the others, and even to solve with certainty the questions concerning the science of those individuals, in so far as these questions are contained in the general idea that the science of one is greater than that of all the others.
A servant in an establishment where the most beautiful products of art are collected, may speak of them all without contradiction or confusion, although he may be incapable of knowing their merit, and entirely ignorant of the circumstances which constitute the beauty of the objects. It is sufficient for him to have the idea of perfection or beauty in general, and to arrange by certain arbitrary signs the degrees of perfection or beauty of the objects, in order to be able to point them out to visitors, and talk of the greater skill of one artist, the poorer success of another; the greater effect and value of the works of the former, and the inferiority of those of the second, and to make other remarks of a similar nature, which at first might make us suppose him a consummate artist, or, at the least, an amateur of a great intellect and exquisite taste.
58. It would be easy to show by other examples, how fruitful some general ideas are, and how they may undergo innumerable combinations, without presenting any thing determinate to the intellect. This is precisely what happens with the idea of the infinite: in vain we ask what there is within us which corresponds to it: the conception of being in general and of the negation of limit present nothing fixed, except certain abstract conditions to which we continually reduce the objects which come under our intuition, or are presented to us with certain characteristic properties which permit us to form a less vague idea of the negation of limit.
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.
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.
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.
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.
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.
To undertake to prove that infinite extension is impossible, because every property of a finite substance must be finite, is equivalent to supposing the very thing in dispute; for the precise question is, whether one of these properties, namely, extension, can be infinite. In order to establish the negative proposition, "No property of a finite substance can be infinite," it is necessary to prove this of extension. Hence the argument which we are imposing implies, in some manner, a begging of the question, when they found it on a general proposition which can only be certain when the present question is solved.
86. Infinite extension ought to be the greatest of all extensions, but there is no such extension. From any given extension God can take away a certain quantity; for example, a yard: in that case the infinite extension would become finite, for it would be less than the first; and as the difference between the two extensions is only a yard, it is clear that not even the first could be infinite; for it is impossible that there should be only the difference of one yard between the finite and the infinite.
This difficulty merits a serious consideration: at first sight it seems so conclusive that no possibility of a satisfactory solution is conceivable.
The proposition that the difference between the finite and the infinite cannot be finite, is not wholly correct. We must first of all take notice that the difference between two quantities, whether finite or infinite, cannot be absolutely infinite, in the sense of diminution. Difference is the excess of one quantity over another, and necessity implies a limit; for as the excess only is considered, the quantity exceeded is not contained in the difference. Calling the difference D, the greater quantity A, and the smaller a, I say that D can in no hypothesis be infinite. By the supposition D = A - a; therefore D + a = A; in order that D may equal A it is necessary to add to it a; therefore D cannot be infinite. If we suppose A = 8, we shall have D = A - a = 8 - a, or D + a = 8. Therefore to make D infinite we must add to it a, and we can never have D = 8 unless a = 0; but in that case there would be no true difference, since the equation, D = A - a, would be converted into D = A - 0 = A, and the difference would not be real but imaginary.
It follows from this that no difference between two positive quantities can be absolutely infinite; if it is so in some sense, it is not so in the sense of diminution; and the union of these two ideas of difference and infinity results in a contradiction.[40]
The difference between an infinite quantity and a given finite quantity cannot be another given finite quantity, but it must be infinite in some sense. Let us suppose an infinite line and a given finite line, the difference between them cannot be expressed by a given finite lineal value. For supposing the second line to be a finite and a given line, we may place it upon the infinite line in any of its directions, and from any point in it it will reach a certain point of the infinite line. If we suppose a second given finite line, representing the difference between the other two lines, we ought to place it upon the infinite line at the point where the other terminates; and it is evident that it will terminate at another point determined by its length; therefore it will not measure the whole of the difference between the infinite and the finite lines.
We obtain the same result in algebraic expressions. If A be a given finite value, the difference between A and 8 cannot be another given finite value. For, expressing the difference by D, we shall have 8 - D ± A D. Therefore, D + A = 8; consequently, if both were given finite values, an infinite would result from two given finite values, which is absurd.
Hence, a difference may be in some sense infinite, according to the meaning we attach to the term infinity. If from the point where we are situated, we draw a line towards the north and produce it infinitely, and then produce it, also, infinitely towards the south, the difference between either of these lines and the sum of them both, will be infinite only in a certain sense. This is also verified by algebraic expressions. If we have the infinite value equal 28, and compare it with 8, the result is 28 - 8 = 8.
In general, from any infinite value we may subtract any finite difference in relation to it, so long as the subtrahend is not a given finite value. Let 8 be the infinite value,—I say that we can find in it any finite value; for, 8 being an infinite value, A contains all finite values of the same order; therefore it contains the finite value, A; consequently we may form the equation, 8 - A = B. Whatever be the value of B, the relation of B to 8 is A; for by only adding A to B we obtain 8. The equation, 8 - A = B, gives B + A = 8, and also 8 - B = A; and as A is a given value according to the supposition, and A is the given finite difference between 8 and B, it follows that we may find a finite difference to every infinite value.
We may infer from this that the possibility of assigning a finite difference to an infinite extension, does not prove any thing against its true infinity. The infinite, and because it is infinite, contains all that belongs to the order in which it is infinite. We may take any sure value, and considering it as a difference, and we shall obtain a finite difference. But far from proving the absence of infinity, this confirms its existence; for it shows that all the finite is contained in the infinite.
In this case, the subtrahend would be infinite under a certain aspect; but not in the order of diminution, because it wants the quantity which is taken from it.
87. There is another argument against the absolute infinity of extension, which seems to have more weight than any of those which precede, and I cannot see why it has never occurred to those who argue against this possibility. It is this,—we suppose an infinite extension to exist. God can annihilate it, and then create another equally infinite. The sum of both is greater than either alone; therefore neither of them alone is infinite. This annihilation we may suppose as often as we wish; hence we may have a series of infinite extensions. The terms of this series cannot exist at the same time, since one actual infinite extension excludes all others. Therefore, as the sum of the extensions is greater than any number of particular extensions, the absolute infinite extension must be found, not in the particular extensions, but in the sum, and hence an actual infinite extension is intrinsically impossible.
To solve this difficulty we must distinguish between extension and the thing extended: the whole question turns on the intrinsic possibility of the infinity of extension, considered in itself, abstracting absolutely the subject in which it is found. The difficulty places before our sight a series of successive infinite extensions; but in reality this succession is in the beings which are extended, and the number of which goes on increasing; but not in the extension itself. The pure idea of infinite extension in the one case, is not increased by the new extensions which are produced; the extension appears, disappears, reappears, and again disappears, but is not increased. The succession shows the intrinsic possibility of its appearance and its disappearance, its essential contingency, because it is not repugnant for it to cease to exist when it exists, or to pass again from non-existence to existence. If we examine our ideas, we shall find that we cannot increase the infinite extension which we conceive, by any imaginable supposition; and that whatever we may do, is reduced to a succession of productions and annihilations. The idea of infinite extension seems to be a primitive part of our mind; the infinity which we imagine in space, is only the attempt which our mind makes to express its idea in reality. Created with sensible intuition, we have received the power of expanding this intuition on an infinite scale,—to do this we require the idea of an infinite extension.
EXISTENCE OF INFINITE EXTENSION.
88. The question of the possibility of an infinite extension is very different from that of its existence. The first we answer in the affirmative, the second in the negative.
Descartes maintained that the extension of the world is indefinite; but this is a term which, although it has a very rational meaning when it refers to the compass of our understanding, has no meaning when applied to things. There is no objection to saying that the extension of the world is indefinite, if it only means that we cannot assign its limits; but in the reality, the limits exist or do not exist, indifferently of our power of assigning them; there is no medium between yes and no; therefore there is no medium between the existence and the non-existence of these limits. If they exist, the extension of the world is finite; if they do not exist, it is infinite;—in either case, the word indefinite expresses nothing.
The argument of Descartes proves nothing, or it proves the true infinity of the world. For, if we must remove its limits indefinitely because we always conceive indefinitely an extension beyond every other extension, as, on the other hand, we know that this series of conceptions has no limit, we may at once transfer the unlimitedness to the object which corresponds to those conceptions, and affirm that the extension of the world is absolutely infinite. Unfortunately, the argument of Descartes is without any basis; for it consists in a transition from the ideal, or, rather, imaginary order, to the real order, which is contrary to good logic.[41]
89. Leibnitz maintained, that although God could have made the material universe finite in its extension, it is more in conformity with his wisdom not to have done so. "Thus I do not say," he writes,[42] "as is here imputed to me, that God cannot give limits to the extension of matter; but the appearance is that he does not wish it, but preferred to give it more." The opinion of Leibnitz is founded on his system of optimism, which is open to a multitude of objections, but it is not the place here to examine them.
90. To speak frankly my own opinion, I say that this is a question which cannot be solved on purely philosophic principles; for, as the ideas contain no intrinsic necessity, either for or against the existence of an infinite extension, we must look for its solution to what experience teaches us. All the time occupied in attempting to solve this question is lost. What we can assert is, that the extension of the world exceeds all appreciation; and as the science of astronomy advances, greater depths are discovered in the ocean of space. Where is the shore? or is there any? Reason cannot answer such questions. What do we, poor insects, know, whose life is but a momentary dwelling on this little ball of dust, which we call the globe of the earth?
POSSIBILITY OF AN ACTUAL INFINITE NUMBER.
91. Is an infinite number possible? Does the union of the idea of number with the idea of the absolute negation of limit, involve any contradiction which prevents the realization of the conception?
Whatever number we may conceive, we can always conceive one still greater: this seems to show that no existing number can be absolutely infinite. If we suppose this number to be realized, an intelligence may know it, and may multiply it by two, three, or any other number; therefore the number may be increased, and consequently it is not infinite.
This difficulty is far from being conclusive, if we examine it carefully. The intellectual act of which it speaks, would be impossible on the supposition of the existence of an infinite number. If the intelligence should not know the infinity of the number, it might make the multiplication, but it would fall into a contradiction through its ignorance; for the number being absolutely infinite, could not be increased; its multiplication would be an absurdity, and the intelligence making it, would combine two ideas which would still be repugnant, although not known to be so by the intelligence. If the absolute infinity of the existing number were known to the intelligence, the idea of multiplication could never be associated with it; for the intelligence would know that all possible products already exist.
92. An absolutely infinite number cannot be expressed in the algebraic or geometrical values; the attempt so to express it limits it in a certain sense, and therefore destroys its absolute infinity. If the expression 8, represented an absolutely infinite number, it would not be susceptible of any combination which would increase it: to suppose that it may be multiplied by other numbers, finite or infinite, is to take its infinity in another than an absolute sense.
The fraction a/0 does not express an infinite value in all the strictness of the word; for it is evident that whatever be the value of a/0 it will always be less than 2a/0 or, in general, less than na/0 n representing a value greater than unity.
93. Neither can an infinite number be represented in geometrical values.
Let us take a line one foot long. It is evident that if we produce this line infinitely in opposite directions, the number of feet will be in some sense infinite, since the foot is supposed to be repeated infinite times: the expression of the number of the feet will be the expression of an infinite value. Now, I say that this number is not infinite, because there are other numbers still greater. In each foot there are twelve inches; therefore, the number of inches contained in the line will be twelve times as great as the number of feet; consequently the number of feet is not infinite. Neither is the number of inches infinite; for they in their turn may be divided into lines, the lines into points; and it is evident that the number of the smaller quantities will be proportionally greater than the number of the greater quantities. There will be twelve times as many inches as feet, twelve times as many lines as inches, and twelve times as many points as lines; and this progression can never end, because the value of a line is infinitely divisible.
94. Pushing to infinity the divisibility of an infinite line, we seem to have an infinite number in the elements which constitute it; but a slight reflection will dissipate this illusion. For it is evident that we can draw other infinite lines by the side of the supposed infinite line; and since according to the supposition, each of them may be infinitely divided, it follows that the sum of the elements of all the lines will give a greater number than the sum of the elements of any one of them.
95. If we wish to find an infinite number of parts in values of extension, we must suppose a solid infinite in all its dimensions, with all its parts infinitely divided. But not even then should we have an absolutely infinite number, although we should have the greatest which can be represented in values of extension.
Conceding that an infinite extension existed which is infinitely divisible, the number of its parts would not be absolutely infinite; for we can conceive other beings besides extended beings, and considering both under the general idea of being, we might unite them in a number which would be greater than that of extended beings alone.
96. No imaginable species of beings infinitely multiplied, can give an absolutely infinite number. The reason is the same as that given in the last paragraph: the existence of beings of one species does not render the existence of beings of another species impossible. Therefore, besides the supposed infinity of the number of beings of a determinate species, there are other numbers which, united with this, produce a number greater than the pretended infinity.
97. The existence of an absolutely infinite number requires: first, the existence of infinite species of beings; and secondly, the existence of infinite individuals of each species. Let us see if these conditions can be realized.
98. There seems to be no doubt of the intrinsic possibility of infinite species. The scale of beings is between two extremes, nothing and infinite perfection: the space between these extremes is infinite; and beings may be distributed on it in an infinite gradation.
99. Admitting the intrinsic possibility of an infinite gradation in the scale of beings, the question occurs, whether their possibility is only ideal, or also real, that is, may be realized. God is infinitely powerful; if the infinite gradation is intrinsically possible, God can produce it; for whatever is intrinsically possible falls within the reach of divine omnipotence. On the other hand, supposing, as we must, the liberty of God, there is no doubt but God is free to create all that he can create. If then there is nothing repugnant in an infinity of the species of beings distributed in an infinite gradation, these beings may exist if God will it. Therefore denying all limit to the number of species and of individuals of each species, it seems that the infinite number would exist, since it is impossible to imagine any increase or limitation in the collection of all beings.
On this supposition the most perfect created beings possible would exist, and no more perfect being in the sphere of creatures could be conceived. All that can be imagined would already exist, from nothing to infinite perfection.
100. Still it must be observed that the collection of created beings, whatever be their perfection, are necessarily subject to the condition of dependence on another being; a condition from which the infinite being above is essentially exempt. This condition involves limitation; therefore, all created beings must be finite.
101. Does the character of finite, which is met with in all created beings, involve a determinate limit beyond which they cannot pass? If this limit exists, is not the number of possible species also limited? And if these species are not infinite, is not an infinite number an illusion?
Although the intrinsic possibility of the infinite scale of beings seems beyond a doubt, we must beware of solving too quickly the present question. With respect to indeterminate conceptions, we see no possible limit; but would this still be so, if we had an intuitive knowledge of the species? Are we sure that in the particular qualities of beings, combined with limitation and dependence, which are essential to them, we should not discover a term beyond which they cannot go, by reason of the constitution of their nature? How impotent philosophy is to solve such questions!
102. Whatever may be concluded as to this infinity of species and their respective perfection, I do not believe that an actually infinite number can exist. Among these species must be counted intelligences which exercise their acts in succession. This is evidently so; for in this number are included human minds which think and wish in a successive manner. The acts of these intelligences may be numbered: this we know from consciousness. Therefore there would never be an infinite number, because these acts, being successive, can never be all at the same time.
103. It may be answered that in this case we might suppose that spirits, including our own, have only one act of intelligence and will. To this I reply, that besides contradicting the nature of created beings, which, because they are finite, must be subject to change, it is also open to another objection, inasmuch as it eliminates at once many species of beings, and thus, instead of preserving the infinity, renders it impossible. Who can deny the possibility of that which exists? If, as our experience informs us, there now exist beings of successive activity, why would not these beings be possible on the supposition that the divine omnipotence had exerted all its infinite creative power?
104. This difficulty, which is founded on the nature of finite intelligences, seems to render the existence of an infinite number impossible, and it becomes still stronger if we examine the question under a more general aspect.
The existence of an absolutely infinite number excludes the existence of any other number. That which is numbered is not substance alone, but its modifications also. This has already been demonstrated with regard to intelligences, and is true in general of all finite beings. Every finite being is changeable, and its changes may be counted. The modifications produced by the changes cannot all exist at once, for some of them exclude others. Therefore, an actual infinite number is never possible.
105. Let us apply these considerations to the sensible world. Motion is a modification to which bodies are subject. This modification is essentially successive. A motion, the parts of which co-exist, is absurd. The co-existence of different states, which result from different motions, is also absurd: things that are contradictory cannot exist at the same time, and many of these situations are contradictory, because one of them necessarily involves the negation of others. If a line falling on another line revolve around a point, it will successively describe different angles. When it forms an angle of 45 degrees, it will not form an angle of 30 degrees, nor of 40, nor 70, nor 80; these angles mutually exclude one another. A portion of matter will form different figures, according to the arrangement which is given to the parts of which it is composed. When these parts form a globe, they will not form a cube; these two solids cannot exist at the same time, formed of the same portion of matter.
106. This variety of motion and form can be numbered. At every step we measure motion, applying to it the idea of number; at every instant we count the forms of a portion of matter, as for example, a piece of wax, to which different forms have been given successively: whatever be the number of the beings which we suppose to exist, every one of them will be susceptible of transformations which may be counted. Therefore, in the very nature of things, there is an intrinsic impossibility of the existence of an actual infinite number.
107. I believe that these arguments fully demonstrate the impossibility of an actual infinite number; and if I do not dare to say that I am sure of having given a complete demonstration, it is because the nature of the question presents so many and so great difficulties, it so bewilders and confounds the weak understanding of man, that there is always reason to fear that even those arguments, which seem the clearest and most conclusive, may conceal some fault which vitiates their force, and makes an illusion appear an incontestible truth. Still I cannot but observe that to combat this demonstration, it seems, to me that it would be necessary to deny our primary ideas, the exclusion of being and not-being, and the necessity of succession, of time, to the realization of contradictory things.
108. Perhaps it may be objected to me that contradictory modifications are not a part of the infinite number, which only relates to the possible: but this does not destroy my demonstration; it rather confirms it. For as the absolute infinite number implies the absolute negation of all limit, when, in treating of the realization of this conception, I meet with things that are contradictory, I say that the realization of the conception is contradictory, because the general and indeterminate conception is more extended than all possible number.
109. The origin of their greater conception is, that the indeterminate conception abstracts all conditions, that of time included; but the reality does not and cannot abstract these conditions. Hence arises the conflict between the conception and its realization, and this explains why the conception is not contradictory, although its realization is impossible.
Let us suppose a number realized containing all the species and individuals possible, we may reflect on the conception of the infinite number, and say that the true infinity of the number requires the absolute negation of all limit; but thinking of the collection of things which exists, we can find it a limit, for concerning this collection of units in general, we may add to it another number expressing the new modifications which may be produced. At the instant A, the number of units may be expressed by M. At the instant B, there will be a new collection of units which may be expressed by N. The sum of M + N will be greater than either M or N alone. Therefore, neither M nor N will be absolutely infinite. The indeterminate conception abstracts instants and relates to the sum above; hence it includes things which cannot co-exist.
IDEA OF ABSOLUTELY INFINITE BEING.
110. We are entering on a difficult question. Serious difficulties are found in the idea of the infinite in general; the idea of absolutely infinite being is not less difficult. We have seen that there are different orders of infinities, each one of which is a conception formed by the association of the two ideas of a particular being and the negation of limit. But it is easy to see that none of the infinities hitherto examined can be called infinite in the strict sense of the term: they are all limited under many aspects,—none of them is an infinitely perfect being. The idea of this being is not fully possessed by us while in this life; still it may be analyzed and explained with more clearness than it is by most authors. The great difficulties, which we meet with in this attempt, show the necessity of deep meditation, and the transcendency of the errors which originate in a wrong understanding of the word infinite when applied to God.
111. What is an absolutely infinite being? It might seem that we had said all that is necessary in defining the absolutely infinite being to be that which has no negation of being: but this is a common notion which leaves much to be desired. It is an indisputable truth that the infinite being has no negation of being; but it is a truth so far beyond our reach that it presents to our weak understanding only a gloomy confusion, as soon as we attempt to determine exactly its true sense.
112. If the absolutely infinite being has no negation of being, it seems that nothing can be denied, but that everything may be affirmed of it, for it must be all; in this case pantheism results from the idea of infinity. If a true negative proposition can be established in relation to the infinite being, there is in it a negation of being, or of the predicate which is denied in the proposition.
It cannot be said that when negative propositions are applied to God, only a negation is denied, for in reality positive things are denied of God. When I say that God is not extended, I deny of him a reality which is extension. When I say God is not the universe, I deny of him the reality of the universe. Therefore negative propositions, as applied to God, deny not only negations, but also realities.
It does not seem to solve the difficulty to say that the realities denied involve imperfection, and are, consequently, repugnant to God. This is very true, but we are treating at present of the explanation of the idea of the absolutely infinite, and the difficulty militates against the supposition that the idea of the absolutely infinite is to be explained by the absolute absence of negation of being. If these realities are any thing, when denied of God some being is denied; and since the proposition cannot be true if there is not in God the negation of the being denied, it follows that it is incorrect to say that the absolutely infinite being is that which has no negation of being.
113. It also seems that a being of this nature could have no properties; for some positive properties exclude others: thus, intelligence and extension, freedom of will and necessity with respect to the same thing are positive properties which mutually exclude one another. Therefore the infinite being cannot have all properties, unless we make it a collection of absurdities, after the fashion of pantheists.
114. The infinite being must have all being which involves no imperfection. This is very true, but there still remain serious difficulties to be solved. What is perfection? What is imperfection? These are questions which it is not easy to answer, and yet we cannot advance a step until we have determined their meaning.
115. The idea of perfection implies being: nothing cannot be perfect, a perfect not-being is a manifest contradiction.
116. Not all being is absolute perfection; for there are modes of being which involve imperfection: what is perfection for one being is imperfection for another.
117. In finite beings perfection is relative; a very perfect barn would be a very imperfect church; a painting may be an ornament in a gallery which would be a profanation if placed in the sanctuary. Perfection seems to consist in a property being conducive to its end. This idea is not applicable to the infinite being which can have no other end than itself. Therefore, perfection in the absolutely infinite being cannot be relative, but must be absolute.
118. If perfection is being, it seems that the perfection of the infinite being must consist in certain properties which are found formally in it, and therefore exclude all imperfection. An absolutely indeterminate being, that is, a being without any property, is impossible. What conception can we form of a thing without intelligence, without will, and without liberty? The propositions in which these properties are affirmed of God, are true; therefore these properties really exist in the subject of which they are affirmed.
119. An infinitely perfect being must have all perfection; but in what sense are we to understand all? Does it mean all possible perfections? But what perfections are possible? Those which are not repugnant. To what is the repugnance to be referred? It must be either a mutual repugnance, or a repugnance to a third: if the first, it is necessary to presuppose one of the two extremes, in order that the other may be repugnant to it; in that case, which is to be preferred? If the second, what is the third to which they are repugnant? On what is it founded?
If by all perfection is meant all that we can conceive, the same difficulty remains. For if we speak of the conception of a finite being, the conception is not infinite; if of the conception of an infinite being, it is a begging of the question, because in explaining the perfections of the infinite being we appeal to its conception.
These difficulties can only be solved by determining more precisely the meaning of these ideas.
120. A thing may be denied of another in two manners: by referring the negation to a property, or to an individual. When I say a surface is not a triangle, I may refer the predicate either to the species of triangle in general, or to an individual triangle. In the first instance, I deny that the figure is triangular; in the second, I deny that the figure is another given triangle. When I say God is not extended, I deny a property; when I say God is not the world, I deny an individual.
It is evident that in order to attribute absolute infinity to any being, it is necessary that no being should be denied of it, either with respect to properties or to individuals, and that the predicate should be affirmed without destroying the principle of contradiction. This exception is absolutely indispensable, unless we wish to make the infinite being the greatest of all absurdities, a jumble of contradictions.
I believe that this will explain to a great extent the idea of absolute infinity, not considered in the abstract, but applied to a really existent being.
ALL THE REALITY CONTAINED IN INDETERMINATE CONCEPTIONS IS AFFIRMED OF GOD.
121. We have seen that our cognitions are of two classes: some are general and indeterminate, others intuitive. All the objects which we know, whether indeterminately or intuitively, may be affirmed of God, provided they involve no contradiction.
122. General and indeterminate conceptions are the ideas of being and not-being, substance and accidents, simple and composite, cause and effect. All that is real in these conceptions is affirmed of God.
123. Being or that which really exists, is affirmed of God. That which is not has no property.
124. Substance, or being subsistent in itself, is also affirmed of God.
I do not enter into the discussion of the question greatly disputed in the schools, whether the ideas of being and substance are applied in the same sense, or, as logicians say, univoce, to God and creatures. It is sufficient for my purpose that the idea of being is applied to the infinite being, as opposed to the idea of not-being, and the idea of substance as opposed to accidents, or rather, as implying a thing which contains all that is necessary in order to subsist by itself without inhering in any other.
125. The idea of accident cannot be applied to the infinite being; but this is not to deny it any thing positive, but rather to affirm a perfection; for we say that it has no need of being inherent in another. This is a perfection; it is being: to deny the quality of accident is to remove a negation. To say that a being is a substance is to deny that it is an accident: these two ideas are contradictory and cannot be attributed to the same subject at the same time.
126. Simplicity is affirmed of God. This attribute denies nothing; to be convinced of this we need only recollect what simplicity is. The simple is one; the composite is a union of beings. If the parts are real, as they must be if there is a true composition, the resultant is a collection of beings subordinated to a certain law of unity. When, therefore, we say that God is simple, we say that God is not a collection of beings, but one being. This involves no negation: but on the contrary it is the affirmation of an existence not divided into various beings.
127. The idea of cause, that is, of activity which produces in another the transition from not-being to being, or from one mode of being to another, is also affirmed of God. This involves no negation, but is an affirmation of being; for a cause is not only being, but a being which so abounds in perfection as to communicate it to others.
128. The idea of effect cannot be applied to God; but this is an affirmation, not a negation. Every effect is a thing produced, which has, consequently, passed from not-being to being: to deny the quality of effect is to remove the negation of being, and affirm the fulness of being.
129. What has been said of the ideas of cause and effect, may be extended to the ideas of necessary and contingent. The negative proposition, God is not contingent, is an affirmation; for contingency is the possibility of not-being. To deny this possibility is to affirm the necessity of being, which is the fulness of perfection.
ALL THAT IS NOT CONTRADICTORY IN INTUITIVE IDEAS IS AFFIRMED OF GOD.
130. We have seen that all that is positive in general and indeterminate conceptions is affirmed of God. Let us see if the same is true of intuitive ideas. These ideas, in all that touches our understanding, may be reduced to these four; passive sensibility, active sensibility, intelligence, and will.
131. Passive sensibility, or the form under which the objects of the external world are presented to our senses, cannot be attributed to the infinite being. This negative proposition, the infinite being is not passively sensible, is strictly true.
Does this proposition deny any thing positive of God? Let us examine it.
The form of passive sensibility is extension, which necessarily implies multiplicity. The extended is necessarily a collection of parts: to deny extension of God is to affirm his simplicity; to deny that he is a collection of beings, and to affirm the indivisible unity of his nature.
132. Besides extension, there is in the passive sensibility of objects only the relation of causes which produce in us the effects called sensations. This causality can and must be affirmed of God: for it is certain that the infinite cause is capable of producing in us all sensations without the intervention of any medium.
133. The negative proposition: the infinite being is not material, means nothing more than the other; the infinite being is not passively sensible. We do not know the intrinsic nature of matter: all we know is, that it is presented in intuition to our sensibility under the form of extension, as an essentially multiplex object. When we deny that God is material or corporeal, we deny that he is passively sensible, or that he is multiple under the form of extension.
134. The other properties of matter, such as mobility, impenetrability, and divisibility, relate to extension, or to a particular impression caused on our senses. The difficulties that may be raised on these points are solved by the preceding paragraphs.
Inertness, or indifference to rest or motion, is a purely negative property. It is the incapacity of all action, the absence of an internal principle productive of change, the purely passive disposition to receive all that is communicated to it.
135. It therefore remains demonstrated that to deny to God passive sensibility, or corporeal nature, is to affirm his undivided nature, his productive activity, and the impossibility of his suffering any kind of change.
136. Active sensibility, or the faculty of perceiving, presents two characteristics which must be defined. There are in sensation two things: the affection caused in the sensitive being by the sensible object, and the internal representation of the sensible being. The first is purely passive, and supposes the possibility of being affected by an object, and, consequently, of being subject to change. This cannot be attributed to the infinite being: to deny it is to affirm immutability, or the necessity of remaining always in the same state. The second is a sort of inferior order of cognition, by which the sensitive being perceives the sensible object. The representation of all objects must necessarily be found in the infinite being, consequently all that is intuitively perceptive in the sensitive faculty must be contained in the perception of the infinite being; that is to say, all that sensibility presents to us of external objects, all that it transfers to our intuition of external existence, must be contained in the representation which the infinite intelligence has within itself. Man cannot know under what form objects are presented to the intuition of the infinite being; but it is certain that all the truth contained in sensitive representation is presented to this intuition.
137. Intelligence, or the perception of objects without the forms of sensibility, implies the perception of beings and of their relations, which is something positive. In us it is often accompanied by the negative circumstance, of the absence of determinate objects to which the general conception may be referred. The infinite being sees in a single intuition all that exists and all that can exist, and contains all that is positive in intelligence, without what is negative, which is an imperfection.
138. It is evident that will must be affirmed of God; for we cannot deny the infinite being that internal, spontaneous activity which is called to will, and the nature of which involves no imperfection.
139. The will of God, although one and most simple, is distinguished into free and necessary, according to the objects to which it is referred. This gives rise to various negative propositions, which it is well to examine.
We say: God cannot will moral evil; this proposition, apparently negative, is, logically considered, affirmative. God cannot will moral evil, because his will is invariably fixed on good, on that sublime type of all holiness which he contemplates in his infinite essence. The impotence of moral evil is in God an infinite perfection of his infinite holiness.
140. The divine will may be referred to external objects, which, being finite, can be combined in different manners, and the existence or non-existence of these combinations depends on the end proposed by the agent which produces or modifies them. The will of God exerted on these objects is free; and to say that he has no necessity of doing this or that is to deny nothing, but to affirm a perfection, namely, the faculty of willing or not willing, or willing in different manners, objects which, on account of their finite nature, cannot bind the infinite will.
141. Hence all the reality contained in general ideas, whether indeterminate or intuitive, that is not contradictory, is affirmed of the absolutely infinite being. As to individual realities, it is evident that those which are finite cannot be affirmed of the infinite being without contradiction. The proposition: the infinite being is the corporeal universe, is equivalent to this: the infinite being is an essentially finite being. The same contradiction will be met with in every proposition where the subject is the infinite being, and the predicate an individual reality distinct from the infinite being. This remark will suffice for the present: they will be more clearly understood when we come to treat of the multitude of substances, in refuting the error of pantheists.
INTELLIGENCE AND THE ABSOLUTELY INFINITE BEING.
142. The infinite being is not a vague object presented in the general idea of being, but is possessed of true properties which, without ceasing to be real, are identified with its infinite essence. A being which is not something, of which some property cannot be affirmed, is a dead being, which we conceive only under the general idea of thing, and is presented to us as something which cannot be realized. Such is not the conception which mankind form of the infinite being; the idea of activity has always been associated with the idea of God: this is not a general, but a fixed and determinate activity; internally, it is the activity of intelligence; externally, the activity which produces beings.
143. The idea of activity in general does not exclude all imperfection: activity to do evil is an imperfect activity: the activity by which some sensible beings act on others, is subject to the conditions of motion and extension, and is, consequently, not exempt from imperfection. Pure, internal activity, considered in itself, involves no imperfection; this is intellectual activity. It is an inoffensive activity, and of itself does no harm; it is an immaculate faculty, and of itself is never stained.
144. To know good, is good; to know evil, is also good; to wish good is good; to wish evil is evil; here is a difference between the understanding and the will; the will may be defiled by its object, the understanding never. The moralist considers, examines, and analyzes the greatest iniquities, and studies the details of the most degrading corruption; the politician knows the passions, the miseries, and the crimes of society; the lawyer witnesses injustice under all its aspects; the naturalist and the physician contemplate the most filthy and loathsome objects; and in all this no stain attaches to the intelligence. God himself knows all the evil there is or can be in the physical or in the moral order, and yet his intelligence remains immaculate.
145. Created beings abuse liberty as such; for it is essentially a principle of action, and may be directed to evil; but the intelligence, as regards itself alone, cannot be abused. It is essentially an immanent or intransitive act in which are represented real or possible objects; the abuse does not commence until the free will combines the acts of the intelligence and directs them to a bad action; there is no evil knowledge until the act of the will is introduced into the combinations of the understanding. A collection of stratagems to commit the most horrible crimes, may be the innocent object of intellectual contemplation.
146. A wonderful thing is intelligence. With it there is relation, order, rule, science, art; without intelligence there is nothing. Conceive, if you can, the world without the pre-existence of intelligence; all is chaos; imagine the order which now exists, destroy intelligence, and the universe is a beautiful picture placed before the extinguished sight of a corpse.
147. We conceive beings as more perfect accordingly as they are higher in the order of intelligence. Leaving the sphere of the insensible and entering the order of sensitive representation, a new world commences. The first degree is the animal in which sensations are limited to a small number of objects, and the summit is intelligence. Morality flows from intelligence, or, rather, is one of its laws, it is the prescription of conformity to an infinitely perfect type. Morality is explained with intelligence; without intelligence it is an absurdity. The intelligence has its laws, its duties, but they proceed from itself, as the sun enlightens itself by its own light. Liberty is explained with intelligence; without it, liberty is an absurdity. Without intelligence causality is presented to us as a farce operating without an object or a direction, without a sufficient reason, and is consequently the greatest of absurdities. When some theologians said that the constitutive attribute of the essence of God is intelligence, they expressed an idea which contains a wonderfully profound philosophical meaning.
148. By the intellectual act being does not go out of itself: intelligence is an immanent act which may be extended to infinity, and exercised with infinite intensity without the intelligent leaving itself. The more profound its understanding is, the more profound is its concentration on the abyss of its consciousness. Intelligence is essentially active: it is activity. See what happens in man: he thinks, and his will awakes and acts: he thinks, and his body moves: he thinks, and his strength is multiplied, all his faculties are subject to his thought. Let us imagine an intelligence infinite in extension and in intensity, an intelligence in which there is no alternation of action and rest, of energy and abatement, an infinite intelligence which knows itself infinitely, and knows infinite, real, or possible objects with an infinitely perfect knowledge; an intelligence, the source of all light without any darkness, the origin of all truth without any mixture of error; we may then form some idea of the absolutely infinite being. By this infinite intelligence I conceive an infinitely perfect will; I conceive creation, a pure act of will calling into existence, from nothing, the types which pre-existed in the infinite intelligence; I conceive infinite holiness, and all the perfections identified in that ocean of light. Without intelligence I conceive nothing: the absolute being, which is in the origin of all things, seems the old chaos, and I try in vain to induce some order into it. The ideas of being, of substance, and of necessity are knocked about in the greatest confusion in my understanding; the infinite is not a focus of light for me, but an abyss of darkness: I know not whether I am immerged in an infinite reality, or lost in the imaginary space of a vague and empty conception.
SUMMING UP.
149. The examination of the idea of the infinite is of the greatest importance, because it is inseparably united with the idea of God.
150. We have the idea of the infinite; but the disputes concerning its nature, and even its existence, denote its obscurity.
151. The finite is that which has limits.
152. The infinite is not the same as the indefinite. The infinite is that which has no limits—the not-finite; the indefinite is that to which no limits are assigned—the not-defined.
153. The difference between the infinite and the finite is founded on the principle of contradiction: the finite affirms limits; the infinite denies them: there is no medium between yes and no.
154. Limit is the negation of a being, or of something real, applied to a being: the limit of a line is the point which terminates it; the limit of a force is the point beyond which it does not extend.
155. The idea of the infinite, denying limit, denies a negation; therefore it is an affirmative idea: the idea of the finite is negative, because it affirms a negation.
156. The idea of the infinite is applied to many orders of beings, and presents strange anomalies, which seem contradictions. A line produced to infinity in only one direction appears infinite, since it is greater than all finite lines; and it is not infinite, because it has a limit in the point where it starts. The same thing is verified in surfaces and solids. To explain these anomalies we must attend to the following observations.
157. The idea of the infinite is not intuitive. We have no intuition of an object either absolutely or relatively infinite.
158. The idea of the infinite is an indeterminate conception formed by the union of the two indeterminate ideas of being in general, and the negation of limit in general.
159. The indeterminate conception of the infinite gives us no knowledge of any thing infinite.
160. The anomalies and apparent contradictions, which we find in the application of the idea of the infinite, vanish when we reflect that the difference of the results depends on the different conditions under which we apply the idea of the infinite. Things which would be infinite under one condition cease to be so when considered under other conditions: the apparent contradiction is caused by one not remarking the change of conditions.
161. We have the conception of infinite number, for we can unite in our mind the two indeterminate conceptions of number and the negation of limit.
162. We have the conception of infinite extension, for we can unite the two indeterminate ideas of extension and the negation of limit.
163. The possibility or non-contradiction of conceptions in the purely ideal order does not prove their possibility in the real order. When the conceptions are realized, their reality is not in an abstract extension or an abstract number, but in individual extended beings, or individual numbers: the determinateness implied by the reality may involve contradiction to the true infinity, although it be impossible for us to discover any contradiction in the indeterminate conception, which abstracts the conditions of their realization.
164. Although we have the conception of infinite extension, it is impossible for us to imagine it.
165. No extrinsic or intrinsic repugnance can be discovered in the existence of infinite extension.
166. We cannot know by purely philosophical means whether the extension of the universe is infinite or finite.
167. Although an absolutely infinite number may be indeterminately conceived, it is not susceptible of any arithmetical or geometrical expression: no series of what mathematicians call infinite expresses an absolutely infinite number.
168. The intrinsic impossibility of an actual infinite number may be demonstrated from the intrinsic repugnance of the co-existence of certain things which may be numbered.
169. The idea of the absolutely infinite real being cannot be indeterminate: it necessarily involves positive and formal perfections.
170. All that does not imply a contradiction must be affirmed of the infinite being. That which is absurd is not a perfection.
171. Analyzing indeterminate and intuitive ideas, we find that all the reality contained in them is affirmed of God.
172. The absolutely infinite being must be intelligent.
173. Intelligence is a perfection which does not imply contradiction.
174. Will and liberty must also be found in the absolutely infinite being.
175. The indeterminate idea of the infinite is favored by the combination of the ideas of being and not-being.
176. The idea of an absolutely infinite being consists in the idea of a union of all being that involves no contradiction.
177. The indeterminate idea of a real infinite being, or of God, is formed from the idea of an absolutely infinite being, combined with the intuitive ideas of intelligence, will, liberty, causality, and all others that can be conceived without imperfection, in any infinite degree.
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