Fundamental Philosophy, Vol. I (of 2)

155. To solve this difficulty, which is really a serious one, it is necessary to recollect the distinction on which I insisted so earnestly between extension as sensation and extension as idea. The former can become subjective only in a sensible being; the second may be, and is, subjective in a purely intellectual being. Extension as sensation is something subjective, it is an appearance; its object exists in reality, but without including in its essence any thing more than is necessary in order to produce the sensation. Extension as idea is also subjective; but it has a real object which corresponds to it, and satisfies all the conditions of the idea.

156. Does not this theory seem to establish two geometries? We must distinguish. The scientific and the pure ideal geometry will remain the same, save the difference of the intelligences which possess it. But notwithstanding this difference, what is true in one is true in the other. Empirical geometry as the representative part of geometry will be different: we have the idea only of our own.

157. In fact we observe two parts in geometry even in ourselves; the one purely scientific, the other of sensible representation. The former includes the connection of ideas; the latter the images and particular cases by means of which we make the ideas sensible: the first is the ground; the second is the form. But although the two are different, we cannot separate them entirely: we cannot have the geometrical idea without the sensible representation, we understand it only per conversionem ad phantasmata, as say the scholastics. Thus the two orders of geometry, the sensible and the intellectual, though different, are always joined in us; whether because the pure geometrical idea arises from the sensible, or is excited by it, or because this is perhaps a necessary primitive condition imposed on our mind by its union with the body.

158. This shows how the pure geometry may be separated from the sensible, and how it may exist in pure intellectual beings, without any of the forms which represent the geometrical idea in sensible beings.

159. But what becomes of extension in itself and stripped of all sensible form? When we speak of extension stripped of all sensible form, we do not mean to deprive it of its capacity to be perceived by the senses, we merely abstract the relations of this capacity to sensible beings. Extension is then reduced, not to an imaginary space, nor to an eternal and infinite being, but to an order of beings, to the sum of their constant relations subject to necessary laws. What then are these relations? I know not. But I know that they exist and that these necessary laws exist. That they exist in reality I know by experience, which gives testimony of their existence; that they are possible, I know on the authority of my ideas, the connection of which forces my assent to their intrinsical evidence.

160. That this evidence touches but one aspect of the object, is true; that there are many things in the object which we do not know, is likewise true; but this only proves that our science is incomplete, not that it is illusory or false.

161. It is difficult for us to conceive the pure intelligibility of the sensible world, both because our ideas are always accompanied by representations of the imagination, and because we try to explain it by simple addition and subtraction of parts, as though all the problems of the universe could be reduced to expressions of lines, surfaces, and solids. Geometry plays an important part in all that regards the appreciation of the phenomena of nature; but when we want to penetrate to the essence of things, we must lay aside geometry and take up metaphysics.

There is no more seductive philosophy than that which reduces the world to motions and figures, but at the same time there is none more superficial. A slight reflection on the reality of things shows the insufficiency of such a system. For, though the imagination be satisfied with it, the understanding is not, and it takes a noble revenge on its unfaithful companion, when, forcing the imagination to fix itself upon objects, the understanding sinks it in an ocean of darkness and contradiction. Those who laugh at the forms, the acts, the forces, and other such expressions used with more or less exactness in different schools, ought to reflect that even in the physical world there is something more than is perceived by the senses; and that even sensible phenomena cannot be explained by mere sensible representations. Physical science is not complete until it calls to its aid metaphysics.

The best proof of this will be found in the next chapter, where we shall see the imagination entangled in its own representations.

CHAPTER XXII.

INFINITE DIVISIBILITY

162. The divisibility of matter is a question that torments philosophers. Matter is divisible because it is extended, and there is no extension without parts. These parts are extended or are not: if they are, they are again divisible; if they are not, they are simple, and in the division of matter we must come to unextended points.

This last consequence can be avoided only by recourse to the infinite divisibility of matter, and even this is a means of escaping the difficulty rather than a true solution. I intimated elsewhere52 that infinite divisibility seems to suppose the very thing which it denies. Division does not make the parts, it supposes them; that which is simple cannot be divided; therefore, the parts which may be divided pre-exist in the infinitely divisible composition.

Let us imagine God to exert his infinite power in dividing, will he exhaust divisibility? If you say no, you seem to place limits to his omnipotence; if you say yes, we shall have arrived at simple points, as otherwise the divisibility would not be exhausted.

Even supposing that God does not make this division, his infinite intelligence certainly sees all the parts into which the composite is divisible; these parts must be simple, or else the infinite intelligence would not see the limit of divisibility. If you answer that this limit does not exist, and therefore cannot be seen, I reply that we must then admit an infinite number of parts in each portion of matter; there would, in this case, be no limit of divisibility, because the number of parts would be inexhaustible; but this infinite number would be seen by the infinite intelligence, as it is, and all these parts would be known as they are. The difficulty still remains; these parts are simple or composite; if simple, the opinion which we are opposing does, at least, admit unextended points; if composite, the same argument may be repeated; they are again divisible. We shall then have a new infinite number in each one of the parts of the first infinite number; but as this series of infinities must be always known to the infinite intelligence, we must come to simple points, or else say that the infinite intelligence does not know all that there is in matter.

It does not mend the matter to say that the parts are not actual but only possible. In the first place, possible parts are existing parts, because, if the parts are not real, there must be real simplicity, and consequently, indivisibility. Secondly, if they are possible, they may be made to exist by the intervention of an infinite power; and then what are these parts? they are either extended or unextended, and the matter returns to where it was before.

163. Some say that a mathematical quantity, or a body mathematically considered, is infinitely divisible, but that natural bodies are not, because their natural form requires a determinate quantity. This is the explanation which was given in the schools; but it is very clear that there is no ground for affirming that these natural bodies require a certain quantity, beyond which division is impossible. This cannot be proved either a prior nor a posteriori: not a priori, because we do not know the essence of bodies, and cannot say that there is a point where the natural form requires the limit of divisibility; neither can it be proved a posteriori, because the means of observation at our disposal are so coarse, that it is impossible for us to reach the last limit of division and discover a part which cannot be divided. Besides, when we reach this quantity beyond which division cannot go, we have a true quantity, by the supposition; if it is quantity it is extended; if it is extended it has parts; if it has parts it is divisible. Therefore there is no reason for saying that there is any natural form which limits division.

164. The distinction between a natural and a mathematical body is not admissible in what relates to division. This is a result of the nature of extension, which is real in natural bodies, and ideal in mathematical. That the parts in natural bodies are not actual but possible, may be understood in two ways; it may mean that they are not actually separated; or, that they are not distinct. That they are not separated has no bearing on the question; for division may be conceived without separating the parts. But, if they are not distinct, the division is impossible; for it cannot even be conceived where the things are not distinct.

165. This distinction seems to have originated in the attempt to avoid the necessity of admitting infinite divisibility in natural bodies. But the difficulty still remaining with regard to mathematical bodies, the philosophical mystery still subsists. It consists in this, that no limit can be assigned to division so long as there is any thing extended; and, on the other hand, if, in order to assign this limit, we come to simple points, then it is impossible to reconstitute extension. The difficulty arises from the very nature of extended things, whether realized or only conceived; the real order escapes none of the difficulties of the ideal. If ideal extension cannot be constituted out of unextended points, neither can real extension; if ideal extension has no limit to its divisibility until we come to simple points, the same is also true of real extension; for in both it is a result of the essence of extension, and inseparable from it.

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.