Second objection.—If a material continuum is impossible, all continuum is impossible, and thus we are constrained to deny the continuity of both space and time. For space and time—as, for instance, a cubic-foot and an hour—include within their respective limits an infinite multitude of indivisible points, or indivisible instants, just as would continuous matter include within its limits an infinite multitude of material points; for it is clear that space and time cannot be made up of anything but points and instants. Hence, if, in spite of this, we admit continuous space and continuous time, we implicitly avow that our first argument against continuous matter is far from conclusive.
We reply that there is no parity between the continuity of space and time and the continuity of matter; and that the impossibility of the latter does not show the impossibility of the former. The continuity of space and of time is intimately connected with the continuity of local movement. Movement, though formally continuous, or rather owing to its formal continuity, is necessarily successive, so that we can never find one part of the movement coexisting with another part of the same movement; and consequently there is no danger of finding in such a movement any actual multitude, whilst we should necessarily find it in continuous matter. Time also, as being nothing else than the actuality or duration of movement, is entirely successive; and consequently no two parts of time can ever be found together; which again prevents the danger of an actual multitude of coexisting instants. As to space, we observe that its continuity is by no means formal, but only virtual, and that space as such has no parts into which it can be divided, whatever [pg 496] our imagination may suggest to the contrary. We indeed consider space as a continuous extension, but such an extension and continuity is the property of the movement extending through space, not of space itself. Space is a region through which movement can extend in a continuous manner; hence the space measured, or mensurable, is styled continuous from the continuity of the movement made, or possible. We likewise consider the parts of the extension of the movement made or possible as so many parts of the space measured or mensurable. And thus space is called continuous, extended, and divisible into parts, merely because the movement by which space is, or can be, measured is continuous, extended, and divisible into successive parts; but space, as such, has of itself no formal continuity, no formal extension, and no formal divisibility, since space, as such, is nothing else than the virtuality, or extrinsic terminability, of divine immensity, as we may have occasion hereafter to show.
Hence neither space, nor time, nor movement is made up by composition of points or of instants; but time and movement owe their continuous extension to the flowing of a single instant and of a single point, whilst space, which is only virtually continuous, owes its denomination of continuous to the possibility of continuous movement through it. But if there were any continuous matter, its formal extension would arise from actual, simultaneous, and indivisible points constituting a formal infinite multitude within the limits of its extension. Hence there is no parity between continuous matter and continuous space or time; and the impossibility of the former does not prove the impossibility of the latter.
Third objection.—Accelerated movement is a movement the velocity of which increases by continuous infinitesimal degrees—that is, by indivisible momenta of motion. It is therefore possible for a quantity of movement to arise from the accumulation of indivisibles. Why, then, should not the quantity of matter arise in a like manner from the accumulation of indivisible points? That which causes the acceleration of movement is, in fact, continuous action—that is, a series of real, distinct, and innumerable instantaneous actions, by which the movement is made to increase by distinct infinitesimal degrees; which would show that it is not impossible to make a continuum by means of indivisibles.
We reply, first, that there is no degree of velocity which can be styled indivisible; for however small may be the acceleration of the movement, it may become smaller and smaller without end, as we shall presently explain.
But, waiving this, we reply, secondly, that intensive and extensive quantity are of a very different nature, and, even if it were true that intensive quantity can arise from an accumulation of indivisibles, the same would not be the case with extension. The degrees of intensity never unite by way of composition; for all intensity belongs to some form or act, whilst all composition of parts regards the material constituents of things. Hence movement, though increasing or decreasing, by continuous degrees, is not composed of them; whereas the continuum of matter, if any such existed, should be composed of its indivisible elements. In movement the increased velocity [pg 497] is not a multitude of distinct acts, but a single act, equivalent to all the acts which we may distinguish under the name of degrees of velocity. Hence such degrees are only virtually distinct, and do not constitute a formal multitude; whence it follows that there is no absurdity in the notion of accelerated or retarded movement. But with a material continuum the case is entirely different; for such a continuum would be an extensive, not an intensive, quantity, and would have parts not only mentally or virtually, but entitatively and formally, distinct, and making an actual infinite multitude within the limits of a finite bulk.
As to the continuous action which causes the acceleration of movement, it is not true that it consists of a sum of distinct instantaneous actions. The action may be considered either in fieri or in facto esse. The action in fieri is the exertion of the agent, and the action in facto esse is the determination received by the patient. Now, the exertion of the agent is successive; for its continuity is the continuity of time, and is therefore continuation rather than continuity. Hence nothing exists of the action in fieri, except an instantaneous exertion corresponding to the moment of time which unites the past with the future. All the past exertions have ceased to be in fieri, and all the future exertions have still to be made. Accordingly, continuous action is not made up of other actual actions, and, though passing through different degrees of intensity, is not an actual multitude.
On the other hand, if we consider the action in facto esse—that is, the determination as received in the patient—we shall find that, although such a determination is the result of a continued exertion, and exhibits its totality under the form of velocity, nevertheless this result consists of intensity, not of continuity, and therefore contains no formal multitude, but is, as we have said, a simple act equivalent to many. Hence accelerated movement is one movement, and not many, and a great velocity is one velocity, and not a formal multitude of lesser velocities. In a word, there is not the least resemblance between continuous acceleration and continuous matter.
Although the preceding answer sufficiently shows the flimsiness of the objection, we may yet observe that actions having an infinitesimal duration are indeed infinitesimal, but are not true indivisibles. For the expression of an accelerating action, in dynamics, contains three variable functions—that is, first, the intensity of the action at the unit of distance in the unit of time; secondly, its duration; thirdly, the distance from the agent to the patient. Hence, in the case of an action of infinitesimal duration, there still remain two variables, viz., the intensity of the power, and the distance from the patient; and their variation causes a variation of the action in its infinitesimal duration. Thus it is manifest that actions of infinitesimal duration can have a greater or a less intensity, and therefore are not true indivisibles of intensity. If, for instance, two agents by their constant and continuous action produce in the same length of time different effects, it is evident that their actions have different intensities in every infinitesimal instant of time; hence such infinitesimal actions, though bearing no comparison with finite quantities, bear comparison with one another, and form definite geometric ratios.
Fourth objection.—If the contact of one indivisible with another cannot engender a continuum, we must deny the existence of time and of local motion. For time is engendered by the flowing of an instant towards the instant immediately following, and movement is engendered by the flowing of a point in space towards the point immediately following. If, then, indivisibles cannot, by their contact, give rise to continuous extension, neither time nor local motion will acquire continuous extension.