CHAPTER IV.
REALITY OF EXTENSION
27. We now come to more difficult questions. Is extension any thing in itself, abstracted from the idea of it? If any thing, what is it? Is it identified with bodies, or is it confounded with space?
I have proved39 that extension exists outside of ourselves, that it is not an illusion of the senses; and this solves the first question, whether extension is any thing.
Whatever may be its nature or our ignorance on this point, there is in reality something which corresponds to our idea of extension. Whoever denies this truth must be content to deny every thing except the consciousness of himself, if indeed he does not experience doubts even of this too. Whatever idealists may assert, there is not, nor ever was a man who in his sound judgment seriously doubted the existence of an external world. This conviction is for man a necessity against which it is vain to contend.
This external world is for us inseparable from that which is represented by the idea of extension. It either does not exist, or else it is extended. If we could be persuaded that it is not extended, it would not be difficult to convince us that it does not exist. For my part, I find it just as difficult to imagine the world without extension as without existence, and if I could be made to believe its extension an illusion, I should easily believe its existence also an illusion.
28. It is to be observed that although we confess our ignorance of the internal nature of extension, it is still necessary to admit that we know something of it; its dimensions, namely, and what serves as the basis of geometry. The difficulty is not in knowing what extension is geometrically considered, but what it is in reality. We know the geometrical essence, but what we want to ascertain is, whether this essence realized is something which is confounded with some other real thing, or is only a quality which we know without knowing the being to which it belongs. Without this distinction we should deny the basis of geometry; for, it is evident that if we should not know the essence of extension in the aforesaid manner, we could not be sure that we are not building in the air when we raise upon the idea of extension the whole science of geometry.
29. Thus then under this aspect, we are certain that extension exists outside of us, and that there are true dimensions. This idea is a necessary consequence of the idea of the external world, as we said before. The dimensions in the external world must be subject to the same principles as those which we conceive, or the very idea which we have formed of the external world is reversed. I do not mean by this that a real circle may be a geometrical circle, but only that what is true of the second must be true of the first also, in proportion as it is constructed with greater or less exactness. Beyond what can be formed by the most perfect and exact instruments, I can conceive, without passing from the order of reality, a circle or any other figure, as near as I please to the geometrical idea. The sharpest instrument can never mark an indivisible point, nor draw a line without breadth; but this surface, on which the point is marked, on the line drawn, being infinitely divisible, I can conceive a case in which the reality will come infinitely near to the geometrical idea.
30. Astronomy and all the physical sciences rest on the supposition that real extension is subject to the same principles as ideal extension; and that experience comes closer to theory in proportion as the conditions of the second are more exactly fulfilled in the first. The art of constructing mathematical instruments, which has been brought in our day to a surprising perfection, regards the ideal as the type of the real order; and progress in the latter is the approximation to the models of the former.
Theory directs the operations of practice, and these in their turn confirm by the result the foresight of theory. Therefore, extension exists not only in the ideal order, but also in the real; and it is something, independently of our ideas; and geometry, that vast representation of a world of lines and figures, has a real object in nature.
How far the real corresponds with the ideal, we shall examine in the next chapter.
CHAPTER V.
GEOMETRICAL EXACTNESS REALIZED IN NATURE
31. The disagreement which we discover between the phenomena and the geometrical theory makes us apt to think that reality is rough and coarse, and that purity and exactness are found only in our ideas. This is a mistaken opinion caused by want of reflection. The reality is as geometrical as our ideas; the phenomenon realizes the idea in all its purity and vigor. Be not startled by this seeming paradox; for it will soon appear to you a very true, reasonable, and well-grounded proposition.
We shall first prove that the ideas which are the elements of geometry have their objects in the real world, and that these objects are subject to precisely the same conditions as the ideas. This proved, it clearly follows that geometry in all its strictness exists as well in the real as in the ideal order.
32. Let us begin with a point. In the ideal order, a point is an invisible thing, it is the limit of a line and its generating element, and it occupies a determinate position in space. It is the limit of a line; for when we take away its length, we have a point remaining which we are forced to regard as the limit of the line unless we destroy it entirely so as to have nothing left. The more the line is shortened the nearer it approaches to a point, yet can never be identified with it until its length is wholly suppressed. The point is the generating element of the line; for we form the idea of lineal dimension by considering a point in motion. The occupation of a determinate position in space is another indispensable condition of the idea of a point, if we wish to use it in geometrical figures. The centre of a circle is a point in itself indivisible, it fills no space; but in order that it be of any use as centre, we must be able to refer all the radii to it, and this is impossible unless it occupy a determinate position equidistant from all points of the circumference. As a general rule, geometry acts upon dimensions, and these dimensions require points in which they commence, points through which they pass, and points in which they end, and by which distances, inclinations, and all that relates to the position of lines and planes, are measured. Nothing of all this can be conceived unless the point, although not extended, occupies a determinate position in space.
33. Does there exist in nature anything which corresponds to the geometrical point, and unites all its conditions with as great exactness as science in its purest idealism can desire? I believe there does.
Philosophers have adopted different opinions as to the divisibility of matter. Some maintain that there are unextended points in which the division ends, and that all composite bodies are formed of these. Others assert that it is not possible to arrive at simple elements, but the division may continue ad infinitum continually approaching the limit of composition, but never reaching it. The first of these opinions is equivalent to the admission of geometrical points realized in nature; the second, though apparently less favorable to this realization, must come to it at last.
Unextended molecules are the realization of the geometrical point, in all its exactness. They are the limit of dimension, because division ends with them. They are the generative elements of dimension, because they form extension. They occupy a determinate position in space, because bodies with all their conditions and determinations in space are formed of them. Therefore, from this opinion, held by eminent philosophers like Leibnitz and Boscowich, it follows that the geometrical point exists in nature in all the purity and exactness of the scientific order.
The opinion which denies the existence of unextended points, admits, as it necessarily must admit, infinite divisibility. Extension has parts, and therefore is divisible; these parts, in their turn, are either extended or not extended; if unextended, the supposition fails, and the opinion of unextended points is admitted; if extended, they are divisible, and we must either come at last to unextended points, or continue the division ad infinitum.
I remarked above that, although less favorable to the real existence of geometrical points, this opinion as well as the other does acknowledge their realization. The parts into which the composite is divided are not created by the division, but exist before the division, and without them the division would be impossible. They do not exist because they may be divided, but they may be divided because they exist. This opinion therefore, does not expressly admit the existence of unextended points, but it admits the possibility of eternally coming nearer to them, and this not only in the ideal, but also in the real order; because the divisibility is not affirmed of the ideas, but of the matter itself.
Although our experience of division is limited, divisibility itself is unlimited. A being endowed with greater powers than we possess, might carry the division further than we are able to do. Our ability to divide is limited, but God, by his infinite power, can push the division ad infinitum, and His infinite intelligence sees in an instant all the parts into which the composite may be divided.
Omitting the difficulties which attend an opinion which seems to suppose the existence of what it denies, I will ask if geometry can require more rigorous exactness than is found in the points to which infinite power can come, if we suppose it to exercise its eternal action in dividing the composite; or, in other words, can there be any more strictly geometrical points than those seen by an infinite intelligence in an infinitely divisible being? This not only satisfies our imagination and our ideas of exactness, but goes even beyond. Experience teaches us that to imagine an unextended point is not impossible; and to think it in the purely intellectual order, is only to conceive the possibility of this infinite divisibility, and to be suddenly placed at the last limit, – a limit which must still be far distant from that to which, not abstraction, but the sight of infinite intelligence can reach.
If the geometrical point exists, the geometrical line also exists; for it is only a series of unextended points; or, if we are unwilling to acknowledge these, a series of extremes to which division infinitely continued at last arrives. A series of geometrical lines forms a surface; and a union of surfaces forms a solid, the ideal order agreeing with reality in its formation as in its nature.
34. This theory of the realization of geometry extends equally to all the natural sciences. It is an error to say, for example, that the reality does not correspond to the theories of mechanics. It should rather be said that it is not the reality that is at fault, but the means of experimenting; the blame should not be imputed to the reality, but rather to the limitation of our experience.
The centre of gravity in a body, is the point where all the forces of gravitation in the body unite. Mechanics supposes this point to be indivisible, and in accordance with this supposition, establishes and demonstrates its theorems, and solves its problems. Here stops the mechanician, and the machinist begins, who can never discover the strict centre of gravity supposed in the theory. Experience disagrees with the principles, and we ought to correct the former by adhering to that which is determined by the latter. Is this because the centre of gravity does not exist in nature with all the exactness which science supposes? No; the centre exists, but the means of finding it are wanting. Nature goes as far as science; neither remains behind; but our means of experience are unable to keep up with them.
The mechanician determines the indivisible point in which the centre of gravity is situated, supposing the surface without thickness, lines without breadth, and the length divided at a determinate point of space, which has no extension. Nature entirely fulfills these conditions. The point exists, and the reality should not be blamed for the limitation of our experience. The point exists in either of the hypotheses mentioned above. The first, which favors unextended points, admits the existence of the centre of gravity in all its scientific purity. The other is not so decided, but it says to us: "Do you see this molecule, this little globe of infinitesimal diameter, the smallness of which the imagination cannot represent? Make it still smaller, by dividing it for all eternity, in decreasing geometrical progression, and you will always be coming nearer the centre of gravity without ever reaching it. Nature will never fail; the limit will ever retire from you; but you will know you are approaching it. Within this molecule is what you seek. Continue to advance, you will never reach it, – but what you want is there." In this case I do not see that the reality falls short of scientific exactness; no mechanical theory imagined or conceived can go farther.
35. These reflections place beyond all doubt that geometry with all its exactness, and theories in all their rigor, exist in nature. If we could follow it in our experience, we should find the real conformed to the ideal order, and we should discover that when experience is opposed to theory, it is not the latter which is wrong, but the limitation of our means makes us lay aside the conditions imposed by the theory. The machinist who constructs a system of indented wheels finds himself obliged to correct the rules of theory, on account of friction, and other circumstances, proceeding from the material which he employs. If he could see with a glance the bosom of nature, he would discover in the friction itself a new system of infinitesimal gearing which would confirm with wonderful exactness those very rules which a rude experience represents to him as opposed to reality.
36. If the universe is admirable in its masses of gigantic immensity, it is not less so in its smallest parts. We are placed between two infinities. Man in his weakness, unable to reach either one or the other, must content himself with feeling them, hoping that a new existence in another world will clear up the secrets which are now veiled in impenetrable darkness.
CHAPTER VI.
REMARKS ON EXTENSION
37. If extension is something as we have proved; what is it?
We find extension in bodies and also in space because in both we find that which constitutes its essence, which is dimension. Is the extension of bodies the same as the extension of space?
I see and hold in my hand a pen: it is certainly extended. It moves, and its extension moves with it. The space in which its motion is executed remains immovable. At the instant A the extension of the pen occupies the point A′; at the moment B the same extension of the pen occupies the part B′ of space which is distinct from the part A′; therefore neither the part A′ of space nor the part B′ is identified with the extension of the body.
This seems to have all the force of a demonstration; but to make it more clear and more general, I will put it into the form of a syllogism. Things which are separated or may be separated are distinct; but the extension of bodies may be separated from any part of space; therefore the extension of bodies and the extension of space are distinct. I said that this reasoning seems to have all the force of a demonstration, but it is nevertheless subject to serious difficulties. These difficulties cannot be understood without a profound analysis of the idea of space, and therefore I shall reserve my opinion until this has been treated of in the following chapters.
38. Is the extension of a body the body itself? I cannot conceive a body without extension, but this does not prove that extension is the same thing as the body. My soul has acquired a knowledge of the body by means of the senses. These senses have awakened in me the idea of extension; but they have told me nothing of the intrinsic nature of the body perceived.
In those beings which we call bodies we find the power of producing in us impressions very distinct from that of extension. From two bodies of equal extension we receive very different impressions, therefore there is in them something besides extension. If extension was their only quality, this being equal, the effect would be the same; but experience teaches us that it is not so.
Moreover we conceive extension in pure space where there is no body. The idea of body implies the idea of mobility, while space is immovable. It implies the power of producing impressions; the extension of space has not of itself this power.
Therefore the simple idea of extension does not include even in our cognitions the whole idea of a body. We do not know in what the essence of body consists; but we know that in the idea which we have of it there is something more than extension.
39. When it is said that a body is inconceivable without extension it is not meant that extension is the constitutive notion of the essence of body. This essence is unknown to us, and therefore we cannot know what does or does not belong to it. The true meaning of this inseparability of the two ideas of extension and body is this: As we have no knowledge a priori of bodies, but whatever we know of them, their existence included, we derive through the senses, all that we think or imagine concerning them must presuppose that which is the basis of our sensations. This basis, as we have already seen, is extension; without it there is no sensation, and consequently without it a body ceases to exist for us, or is reduced to a being which we cannot distinguish from others.
I will explain my ideas. If I strip bodies of extension and leave them only the nature of a being which causes the impressions which I receive; this being is the same, so far as I am concerned, as a spirit which should produce the same impressions. I see this paper, and it causes in me the impression of a white surface. There is no doubt that God could produce in my mind the same sensation without the existence of any body. Then supposing that I knew that no external extended object corresponded to my sensation, which was caused by a being acting upon me, it is evident that there would be two distinct things in my mind. First, the phenomenon of sensation, which under all hypotheses is the same; and secondly, the idea of the being which produced it, which is only the idea of a being distinct from myself, acting upon me, which in relation to the external world, would involve two ideas; those of distinction and causality.
I now take from the paper extension, and what remains? The same as before. 1. An internal phenomenon, made known by consciousness. 2. The idea of a being the cause of this phenomenon.
I do not know whether this must always be a body; but I know that the idea of a body, as I understand it, includes something more than this. I know that being is not in relation to myself distinguishable from other beings, and that if there is any thing in its nature to distinguish it from them, it is something unknown to me.40
40. This is the sense in which I say that we cannot separate the idea of extension from the body. But from this it must not be inferred that the things themselves are identified; perhaps, even, a more profound knowledge of matter would show us that instead of being identical, they are entirely distinct. We have seen that it is so with their ideas, and this is a sign that it is so in reality.
41. We have few ideas as clear as that of extension geometrically considered; every attempt to explain it is useless; we know it more perfectly by mere intuition than whole volumes could make it known to us. It is so clear an idea, that on it is founded a whole science, the most extensive and evident which we possess, that of geometry. Therefore there is reason to believe that we know the true essence of extension, since we know its necessary properties, and even base a whole science on this knowledge. Yet we do not discover in this idea, either impenetrability or any of the properties of bodies; but rather on the contrary, we find a capacity indifferent to them all. We conceive extension penetrable as easily as impenetrable, empty or full, white or green, with properties by which it can be placed in relation with our organs, as easily as without them. We can conceive extension in a body acting on another body, or in pure space; in the sun which enlightens and warms the world, or in the vague dimensions of an empty immensity.