We have thus exhausted all the sources from which any à priori or à posteriori argument in favor of material continuity might have been drawn, if any had been possible; and the result of our investigation authorizes the conclusion that the hypothesis of continuous matter is both scientifically and philosophically gratuitous.
False reasonings in behalf of continuous matter.—But some philosophers, who are afraid that the denial of material continuity may subvert all the scholastic doctrines (to which they most laudably, but perhaps too exclusively, adhere in questions of natural science), contend that the existence of continuous matter can be established by good philosophical reasons. It is therefore our duty, before we proceed further, to acquaint our reader with such reasons, and with our answers to them.
The first reason is the following: Geometry is a real, not a chimerical, science; and therefore it has to deal with real bodies—not indeed inasmuch as they are substances, but inasmuch as they have a quantity which can be considered in the abstract. Hence we must admit that the geometric quantity is a quantity of matter considered in the abstract; and accordingly, if the geometric quantity is continuous and infinitely divisible, as no one doubts, the quantity of matter in the bodies must also be continuous and infinitely divisible.
We reply that bodies have two very different kinds of quantity—the quantity of the mass and the quantity of the volume—and that geometry deals indeed with the latter, but has nothing to do with the former. Hence the geometric quantity is a quantity of volume or bulk, not a quantity of matter; and therefore to argue that, because the geometric quantity is continuous and infinitely divisible, the same must be true of the quantity of matter, is to make an inexcusable confusion of matter with space. The argument might have some value, if the quantity of the volume could be measured by the quantity of the mass; but no one who has studied the first elements of physics can be ignorant that such is not the case. Equal masses are found under unequal volumes, and unequal masses under equal volumes. Volumes preserve the same geometric nature and the same geometric quantity, be they filled with matter or not. A cubic inch of platinum and a cubic inch of water contain different amounts of matter, since the former weighs twenty-one times as much as the latter; and yet they are geometrically equal. Geometry is not concerned with the density of bodies; and therefore geometrical quantities are altogether independent of the quantity of matter, and cannot be altered except by altering the relative position of the extreme terms between which their three dimensions are measured. These dimensions are not made up of matter, but are mere relations in space, with or without interjacent matter, representing, as we have already observed, the quantity of continuous movement which is possible between the correlated terms; and their continuity depends on the continuity of space, not of matter.
The author from whom we have taken this objection pretends also that the geometric quantity possesses no other attributes than those which belong to all quantity, and are [pg 277] essential to it; whence he concludes that whatever is predicated of geometric quantity must also be predicated of the quantity of matter. But the assumption is evidently false; for it is not of the essence of all quantity to be continuous as the geometric quantity, it being manifest that discrete quantity is a true quantity, although it has no continuity. The general notion of quantity extends to everything which admits of more or less; hence there is intensive quantity, extensive quantity, and numeric quantity. The first is measured by arbitrary degrees of intensity; the second is measured by arbitrary intervals of space and time; the third is measured by natural units—that is, by individual realities as they exist in nature. It is therefore absurd to pretend that whatever can be predicated of geometric quantity must be predicated of all kinds of quantity.
The second reason adduced in behalf of material continuity is as follows: To deny the continuity of matter is to destroy all real extension. For how can real extension arise from simple unextended points arranged in a certain manner, and acting upon one another? The notions of simplicity, order, and activity transcend the attributions of matter, and are applicable to all spiritual beings. If, then, extension could arise from simple unextended elements by their arrangement and actions, why could not angels, by meeting in a sufficient number and acting on one another, give rise to extension, and form, say, a watermelon?
This argument has no weight whatever; but, as it appeared not many years ago in a Catholic periodical of great reputation, we have thought it best to give it a place among other arguments of the same sort. Our answer is that to deny the continuity of matter is not to deny real extension, but only to maintain that no real extension is made up of continuous matter. And we are by no means embarrassed to explain “how real extension can arise from simple unextended points.” The thing is very plain. Two points, A and B, being given in space, the interval of space between them is a real interval, really determined by the real points A and B, and really determining the extension of the real movement possible between the same points. Such an interval is therefore a real extension. This is the way in which real extension arises from unextended points.
Nor can it be objected that nothing extended can be made up of unextended points. This is true, of course, but has nothing to do with the question. For we do not pretend that extension is made up by composition of points—which would be a very gross error—but we say that extension results from the simple position of real points in space, and that it results not in them, but between them. It is the mass of the body that is made up of its components; and thus the sum A + B represents a mass, not an extension. The geometric dimensions, on the contrary, consist entirely of relations between distinct points intercepting mensurable space. The distinct points are the terms of the relation, while the extent of the space mensurable between them by continuous movement is the formal reason of their relativity. And since this continuous movement may extend more or less, according as the terms are variously situated, hence the resulting relation has the nature of continuous [pg 278] quantity. This suffices to show that to deny the continuity of matter is not to destroy all real extension.
And now, what shall we say of those angels freely uniting to form a watermelon? It is hardly necessary to say that this bright idea is only a dream. There is no volume without dimensions, no dimension without distance, and no distance without terms distinctly ubicated in space and marking out the point where the distance begins, and the point where it ends. Now, nothing marks out a point in space but matter. Angels, as destitute of matter, mark no points in space, and accordingly cannot terminate distances nor give rise to dimensions. Had they matter, they would, like the simple elements, possess a formal ubication in space, and determine dimensions; but, owing to their spiritual nature, they transcend all local determinations, and have no formal ubication except in the intellectual sphere of their spiritual operation. It is therefore owing to their spirituality, and not to their simplicity, that they cannot form themselves into a volume. Lastly, we must not forget that the “angelic” watermelon should have not only volume, but mass also. Such a mass would, of course, be made up without matter. How a mass can be conceived without matter is a profound secret, which the author of the argument very prudently avoided to reveal. But let us come to another objection.
A third reason adduced in favor of continuous matter is that we cannot, without employing a vicious circle, account for the extension of bodies by the notion either of space, distance, or movement. For these notions already presuppose extension, and cannot be formed without a previous knowledge of what extension is. To think of space is, in fact, to think of extension. So also distance cannot be conceived except by imagining something extended, which lies, or can lie, between the distant terms. Hence, to avoid the vicious circle, it is necessary to trace the origin of our notion of extension to the matter we see in the bodies. And therefore our very notion of extension is a sufficient proof of the existence of continuous matter.