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 elsewhere[52]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, weshall 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 wasgiven 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 eithera priornora posteriori: nota 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 proveda 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, wecome 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.