CHAPTER XXIII.
UNEXTENDED POINTS
166. There are two strong arguments against the existence of unextended points: the first is, that we must suppose them infinite in number, for otherwise it does not seem possible to arrive at the simple, starting from the extended: the second is, that even supposing them infinite in number they are incapable of producing extension. These arguments are so powerful as to excuse all the aberrations of the contrary opinion, which, however strange they may seem, are not more strange than the simple forming extension, and the smallest portion of matter containing an infinite number of parts.
167. It does not seem possible to arrive at unextended points unless by an infinite division. The unextended is zero in the order of extension, and in order to arrive at zero by a decreasing geometrical progression it must be continued ad infinitum. Mathematical calculation presents a sensible image of this. When two parts are united they must have a side where they touch, and another where they are not in contact. If we separate the interior side from the exterior we have two new sides, one which touches and another which does not. Continuing the division the same thing happens again; we must, therefore, pass through an infinite series in order to arrive at the unextended, which is equivalent to saying that we shall never arrive there. To continue the division ad infinitum we must suppose infinite parts, and consequently the existence of an actual infinite number. From the moment that we suppose this infinite number to exist it seems to become finite, since we already see a limit to the division, and also other numbers greater than it. Let us suppose that this infinite number of parts is found in a cubic inch; there are numbers which are greater than this which we suppose infinite; a cubic foot, for example, will contain 1,728 times the infinite number of parts contained in the cubic inch.
Thus the opinion of unextended points seeking to avoid infinite division, runs into it; just as its adversaries trying to escape from unextended points are forced to acknowledge their existence. The imagination loses itself and the understanding is confused.
168. The other objection is not less unanswerable. Suppose we have arrived at unextended points, how shall we reconstitute extension? The unextended has no dimensions; therefore, no matter how many unextended points we may take, we can never form extension with them. Let us imagine two points to be united, as neither of them alone occupies any place, neither will they both together. We cannot say that they penetrate each other; for penetration cannot exist without extension. We must admit that these parts being zero in the order of extension, their sum can never give extension, no matter how many of them we may add together.
169. It is certain that a sum of zeros can give only zero for the result, but mathematicians admit that there are certain expressions equal to zero, which multiplied by an infinite quantity will give a finite quantity for the product. 0 + 0 + 0 + 0 + N × 0 = 0; but if we take 0/M = 0, and multiply it by the expression M/0 = 0, we shall have (0/M) × (M/0) = (0 × M)/(M × 0) = 0/0 which is equal to any finite quantity, which we may express by A. This is shown by the principles of elementary algebra only; if we pass to the transcendental we have dz/dx = o/o = B; B expressing the differential coefficient which may be equal to a finite value. Can these mathematical doctrines serve to explain the generation of the extended from unextended points? I think not.
It is evident that, multiplication being only addition shortened, if an infinite addition of zeros can give only zero; multiplication can give no other result, although the other factor be infinite. Why then do mathematical results say the contrary? This contradiction is not true, but only apparent. In the multiplication of the infinitesimal by the infinite we may obtain a finite quantity for product, because the infinitesimal is not regarded as a true zero, but as a quantity less than all imaginable quantities, but still it is something. If this condition were wanting, all the operations would be absurd, because they would turn upon a pure nothing. Shall we therefore say that the equation, dz/dx = o/o, is only approximate? No; for it expresses the relation of the limit of the decrement, which is equal to B only when the differentials are equal to zero. But as geometricians only consider the limit in itself, they pass over all the intervals of the decrement, and place themselves at once at the point of true exactness. Why then operate on these quantities? Because the operations are a sort of algebraic language, and mark the course that has been followed in the calculations, and recall the connection of the limit with the quantity to which it refers.
170. Unity which is not number produces number; why then cannot points without extension produce extension? There is a great disparity between the two cases. The unextended, as such, involves only the negative idea of extension; but in unity, although number is denied, this negation does not constitute its nature. No one ever defined unity to be the negation of number, yet we always define the unextended to be that which has no extension. Unity is any being taken in general, without considering its divisibility; number is a collection of unities; therefore the idea of number involves the idea of unity, of an undivided being, number being nothing more than the repetition of this unity. It belongs to the essence of all number that it can be resolved into unity; it contains unity in a determinate manner. But the extended can not be resolved into the unextended, unless by proceeding ad infinitum, or else by some process of decomposition which we know nothing of.
CHAPTER XXIV.
A CONJECTURE ON THE TRANSCENDENTAL NOTION OF EXTENSION
171. The arguments for or against unextended points, for or against the infinite divisibility of matter seem equally conclusive. The understanding is afraid that it has met with contradictory demonstrations; it thinks it discovers absurdities in infinite divisibility, and absurdities in limiting it; absurdities in denying unextended points, and absurdities in admitting them. It is invincible attacking an opinion, but its strength is turned into weakness as soon as it attempts to establish or defend any thing of its own. Yet reason can never contradict itself; two contradictory demonstrations would be the contradiction of reason, and would produce its ruin; the contradiction can, therefore, only be apparent. But who shall flatter himself that he can untie the knot? Excessive confidence on this point is a sure proof that one has not understood the true state of the question, and such vanity would be punished by the conviction of ignorance. With all these reserves I now proceed to make a few observations on this mysterious subject.
172. I am inclined to believe that in all investigations on the first elements of matter, there is an error which renders any result impossible. You wish to know whether extension may be produced from unextended points, and the method which you employ consists in imagining them already approached, and then trying to see if any part of space can be filled by them. This seems to me like trying to make a denial correspond to an affirmation. The unextended point represents nothing determinate to us except the denial of extension; when, therefore, we ask if this point joined with others like it can occupy space, we ask if the unextended can be extended. Our imagination makes us presuppose extension in the very act in which we wish to examine its primitive generation. Space, such as we conceive it, is a true extension; and, as has been shown, is the idea of extension in general; to imagine, therefore, that the unextended can fill space, is to change non-extension into extension. It is true that this is precisely what is required, and in this consists the whole difficulty; but the error is in attempting to solve it by a juxtaposition which makes these points both unextended and extended, an evident contradiction.
173. In order to know how extension is generated, it would be necessary to free ourselves from all sensible representations, and from all ideas which are in the least degree affected by the phenomenon, and to contemplate it with an eye as simple, a look as penetrating, as that of a pure spirit. It would be necessary to take from all geometrical ideas all phenomenal forms, all representations of the imagination, and present them to the imagination purified from all mixture with the sensible order. It would be necessary to know how far extension, real continuity, agrees with the phenomenal. It would, in fine, be necessary to eliminate from the object perceived, all that relates to the subject which perceives it.
174. In extension, as we have already seen, there are two things to be considered; multiplicity, and continuity. As to the first, there is no objection to supposing that it may be the result of unextended points, since number results from various units whether they are simple or composite. But the difficulty is with regard to continuity, which sensible intuition clearly presents to us as the basis of the representations of the imagination, but the nature of which is a puzzle to the understanding. It may perhaps be said that continuity, abstracted from the sensible representation, and considered only in the transcendental order, is, in its reality and as it appears to a pure spirit, nothing more than the constant relation of many beings, which are of a nature to produce in a sensitive being the phenomenon of representation, and to be perceived in the intuition which we call the representation of space.
According to this hypothesis extension in the external world is real, not only as a principle of causality of our impressions, but also as an object subject to the necessary relations which we conceive.
175. But, then, it will be asked, is the external world such as we imagine it? To this we must answer, in accordance with what we have said when treating of sensations, that it is necessary to take from sensations all that is subjective, and which by an innocent illustration we look upon as objective; we may then say that extension really exists outside of us and independent of our sensations; considered in itself, it exists free from all sensible representation, and in the same manner in which a pure spirit may perceive it.
176. We see no objection which can reasonably be made to this theory which affirms the reality of the corporeal world, at the same time that it settles the difficulties of idealism. To give my opinion in a few words, I say: That extension in itself, exists such as God knows it, and in the cognition of God there is no mixture of any of the sensible representations which always accompany man's perception. That which is positive in extension is multiplicity, together with a certain constant order; continuity is nothing more than this constant order, in so far as sensibly represented in us, it is a purely subjective phenomenon which does not at all affect the reality.
177. We may even assign a reason why sensible intuition has been given to us. Our soul is united to an organized body, – that is to say, a collection of beings bound together by constant relation to each other and to the other bodies of the universe. In order that the harmony might not be interrupted, and that the soul which presides over this organization might rightly exercise its functions, there was need of a continued representation of this collection of the relations of our own and other bodies. This representation must be simultaneous and independent of intellectual combinations; for otherwise the animal faculties could not be exercised with the promptness and perseverance which the satisfaction of the necessities of life demands. Therefore it is that all sensible beings, even those which have not reason, have been endowed with this intuition of extension or space: which is like an unlimited field on which the different parts of the universe are represented.
CHAPTER XXV.
HARMONY OF THE REAL, PHENOMENAL, AND IDEAL ORDERS
178. We may consider two natures in the external world, the one real, the other phenomenal; the first is particular and absolute, the second is relative to the being which perceives the phenomenon; by the first the world is, by the second it appears. A pure intellectual being knows the world as it is; a sensitive being experiences it as it appears. We can discover this duality in ourselves; in so far as we are sensitive beings, we experience the phenomenon, but in so far as intelligent, although we do not know the reality, we attempt to reach it by reasoning and conjecture.
179. The external world in its real nature, abstracted from the phenomenal, is not an illusion. Its existence is known to us not by phenomena only, but by principles of pure intelligence which are superior to all that is individual and contingent. These principles, based on the data of experience, – that is, on sensations the existence of which we know from consciousness, assure us that the objectiveness of sensations, or the reality of the external world, is a truth.
180. This distinction between the essential and the accidental, and between the absolute and the relative, was admitted in the schools. Extension was considered not as the essence, but as an accident of bodies; the relations of bodies to our senses are not founded immediately on their essence, but on their accidents. Matter and substantial form united constitute the essence of bodies; the matter receiving the form, and the form actuating the matter. Neither the matter nor the substantial form can be immediately perceived by the senses, because this perception requires the determination of figure and other accidents distinct from the essence of body.
Therefore the scholastics distinguished sensible objects into three classes; particular, common, and accidental, proprium, commune, et per accidens. The particular is that which appears immediately to the senses, and is only perceived by one of them, as color, sound, taste, and smell. The common is that which is perceived by more than one sense, as figure, which is the object of sight and of touch. The accidental is that which is not directly perceived by any of the senses, but is hidden under sensible qualities, by means of which it is discovered, as are substances. The sensible per accidens is connected with sensible qualities; but they do not present it to the understanding as an image presents the original, but as a sign the signified. Hence they did not consider the sensible per accidens as proceeding from the species and reducing the sensitive faculty to act: it was intelligible rather than sensible.
181. In the corporeal universe considered in its essence, there is no necessity of supposing any thing resembling the sensible representation, but we must suppose the object to correspond to the idea; for otherwise we should have to admit that geometrical truths may be contradicted by experience.
182. Although extension is an order of beings of which we cannot form a perfect conception, because we cannot purify our ideas from all sensible form, still this order must correspond to our ideas, and even to our sensible representations, so far as is necessary to prove the truth of the ideas.
It is evident that although the phenomenal order is distinct from the real, it depends on it, and is connected with it by constant laws. If we suppose that there is no parallel between the reality and the phenomenon, and that the reality has not all the conditions necessary to satisfy the demands of the phenomenon, there can be no reason why the phenomena should be subject to constant laws, and why experience should not suffer continual confusion. Without a fixed and constant correspondence between the reality and the appearance, the world becomes a chaos to us, and all regular and constant experience becomes impossible.
183. Let us examine this at greater length. One of the elementary propositions of geometry says: "When two straight lines intersect each other, the opposite or vertical angles, which they form, are equal." In order to demonstrate this, I must have the internal intuition of two lines intersecting each other. But the geometrical proposition is not confined to any particular intuition, but embraces all that can be imagined, without any limit to their number, or any determination as to the measure of the angles, the length of the lines, or their position in space.
Here the pure idea extends to an infinity of cases, whereas the sensible intuition represents them only one at a time, and isolated if represented successively. The understanding is not limited to the affirmation of this relation between the ideas, but applies it to the reality, and says: Whenever the conditions of this ideal order are realized, that which I see in my ideas is true in reality, and the relation expressed will be more or less exact in proportion to the exactness of the realization of the conditions; the more delicate the real lines are, that is, the more they approach the condition of right lines, the nearer will the relation of the two angles approach to perfect equality. This conviction is founded on the principle of contradiction, which would be false if the proposition were not true; and it is confirmed by experience, so far as it touches the conditions of the ideal order.
184. What is there in reality which corresponds to this proposition? An existing or real line is an order of beings; two lines which intersect each other are two orders of beings with a determinate relation; the angle is the result of this relation, or, rather, it is the relation itself; the equality of the opposite angle is the correspondence of these relations in the ratio of equality by the continuation of the same order in another sense. These relations between the orders and the beings, and the correspondence of these orders to each other, is what corresponds in reality to the pure geometrical idea, or to the idea separated from all sensible representation. Since the relations of the idea have their corresponding objects in the relations of the reality, geometry exists not only in the ideal order, but also in the real. Since the phenomenon or sensible representation is subject to the same conditions as the idea, because the order of phenomena presents certain relations of the same nature as the relations of the idea and the fact; the idea, the phenomenon, and the reality agree, and it is explained why the intellectual order is confirmed by experience, and experience receives with confidence the direction it gives.
185. This harmony must have a cause; we must look for a principle which is the sufficient reason of this wonderful agreement between things so distinct. Here new problems arise which overwhelm the understanding, but at the same time expand and invigorate it by the grandeur of the spectacle presented to its view, and the immensity of the field opened to its investigations.
CHAPTER XXVI.
CHARACTER OF THE RELATIONS OF THE REAL ORDER TO THE PHENOMENAL
186. Is the agreement of the idea, the phenomenon, and the reality necessary, is it founded on the essence of things, or has it been freely established by the will of the Creator?
If the world had no other reality than that expressed by the sensible representation, if the appearances were an exact copy of the essence of things, we should have to say that this agreement is unalterable, that things are what they appear, and that if we suppose them to exist, it is absolutely necessary that they should be just what they appear; for nothing can be in contradiction with its constitutive notion. That which now is extended, would be necessarily extended, and could not but be extended in the same manner in which it appears to us, and under the same conditions; the relation of bodies to each other would be necessarily subject to the same phenomenal laws, and all which does not come under this order would be a contradiction, and beyond the limit of omnipotence.
187. Bodies are presented to us in the sensible intuition with a determinate magnitude, and in a certain fixed relation which we calculate by comparison with an immovable extension, such as we imagine space. By magnitude, bodies occupy a certain space, determinate, though changeable by motion; by the relation of magnitudes they occupy a greater or smaller place, and mutually exclude each other; this exclusion is called impenetrability. The question to be examined here is, whether the determination of magnitudes, and their relation in respect to the occupation of place, are things absolutely necessary, so that their alteration involves a contradiction, or not. I answer that they are not.
188. Relation to place considered as a portion of pure space, means nothing; for I have already shown that this space is only an abstraction of our understanding, and that in itself it has no reality, – it is nothing. Therefore the relation to it must be nothing also, because the relation is destroyed if one of the terms is nothing. Therefore, the relations of bodies to place can only be the relations of bodies to one another.
189. This is the principal thing to be noticed in this question. The understanding gets confused when it begins by supposing space an absolute nature with necessary relation to all bodies. We must remember the doctrine of the chapters,53 where we explained how the idea of space is generated in us, what object corresponds to this idea in reality, and how; and we shall easily perceive that the absolute and essential relations which we think we discover between bodies and a vacant and real capacity, are illusions of our imagination, in consequence of our not sufficiently purifying the ideal order by separating from it all sensible impressions. We cannot understand so much as the meaning of these questions, if we do not make an attempt at this separation as far as is possible to our nature. If this is done, then the questions proposed in the following chapters will appear very philosophical, and their solution will seem probable, if not true; but they must seem absurd, if we confound the pure intellectual order with the sensible. We cannot admit the idealism which destroys the real world; but the empiricism which annihilates the ideal order, is equally objectionable. If we cannot rise above the sensible representations, let us renounce philosophy, give up thinking, and confine ourselves to sensation.