Volume 20 — Matter. V.

Although continuous matter cannot be proved to exist, yet its existence, as every one knows, is still very commonly believed, even by philosophers, on the ground that it was believed for centuries by all great men, and has never been conclusively refuted. From some hints which we have given in our previous article about the difficulties of this ancient doctrine, the intelligent reader may have already satisfied himself that material continuity is not merely “a philosophical mystery,” as Goudin confesses, but a metaphysical absurdity. As, however, this last conclusion, owing to its paramount importance in metaphysics and in natural philosophy, deserves a more explicit and complete demonstration than we have yet given, we propose to develop in the present article a series of arguments, drawn from different sources, to show the absolute and intrinsic impossibility of continuous matter. The prejudices of our infancy may at first resist the demonstration, but it is to be hoped that they will finally yield to reason.

First argument.—We know, and it is conceded by the advocates of continuous matter, that a finite being cannot involve in its composition an infinite multitude of distinct terms; for evidently the infinite cannot be the constituent of the finite. Now, we have shown in our preceding article that, if there were a piece of continuous matter, it should involve in its continuous constitution an infinite multitude of distinct terms, every one of which should have its own distinct existence independently of the others. Therefore continuous matter cannot exist.

Second argument.—A primitive substance cannot absolutely be made up of other substances. But if there were any continuous matter, a primitive substance would be made up of other substances. Therefore no continuous matter can exist. The major of this syllogism is quite evident; for a primitive substance, if made up of other substances, would be primitive and non-primitive at the same time. The minor can be easily proved. For it is plain that continuous matter, if any such existed, would necessarily consist of continuous parts, substantially distinct from one another, and therefore having their own distinct matter and their own [pg 488] distinct substantial act, and ranking as distinct, complete, and separable substances, as we have shown in our last article. Now, assuming that either of these parts is a primitive substance, it is evident that the primitive substance would be made up of other substances; for such a part, being continuous, is itself made up of other parts, which are likewise distinct and complete substances, as we have just remarked. And since a continuum cannot be resolved into any but continuous parts, the conclusion cannot be avoided that the primitive material substance would always be made up of other substances. To elude this argument, the advocates of continuous matter are compelled to deny that there is any primitive material substance mathematically continuous. But, even so, their position is not improved. For if there is no primitive material substance mathematically continuous, the combination of such primitive substances will never give rise to continuous matter, it being obvious that all the elementary constituents of continuum must be continuous, as all philosophers agree. Whence we again conclude that no continuous matter is possible.

Third argument.—No continuum can be made up of unextended constituents, as we have just observed, and as our opponents not only concede, but also demonstrate most irrefragably in their own treatises. Now, continuous matter, if any such existed, would be made up of unextended constituents—that is, of mere mathematical points. Therefore continuous matter would be a formal contradiction. The minor of our syllogism is proved thus. All the points which can be designated within the dimensions of the continuum are immediately united with one another, and therefore no room is to be found between any two consecutive points; which shows that in the constitution of the continuum we would have nothing but mere points. For let there be a continuous plane and a continuous sphere. The sphere, if perfect, cannot touch the plane, except in a single indivisible point, as is proved in geometry; nevertheless, the sphere may move along the plane, and, always touching the plane in a single point, may measure a linear extension of matter, which, accordingly, would contain nothing but mathematical points immediately following one another. In other terms, the extended matter would be made up of indivisible points; and since all admit that this is impossible, it follows that continuous matter is impossible. Against this argument the objection is made that it proves too much; as it would prove the impossibility of measuring space by continuous movement. But this objection has no good foundation, as we shall show after concluding the series of our arguments.

Fourth argument.—All the points that can be designated in a material continuum would necessarily touch one another in such a manner as to form a continuous extension; hence their contact would necessarily be extensive. But an extensive contact of indivisible points is intrinsically impossible. Therefore material continuity is intrinsically impossible. The major of this syllogism is a mere corollary from the definition of continuum; for, if there be no contact, the continuum will be broken, and if the contact be not extensive—that is, such as to allow each point to extend beyond its neighbor—no continuous extension will result. The minor of our [pg 489] syllogism can be proved as follows:

The contact of a point with a point is the contact of an indivisible with another indivisible; and, since the indivisible has no parts, such a contact cannot be partial, but must needs be total. Accordingly, the second point, by its contact with the first, will be totally in the first; the third, by its contact with the second, will be totally in the second, and consequently in the first; the fourth, by its contact with the third, will be totally in the third, and consequently in the second and in the first, and so on. Therefore all the points which are in mathematical contact will necessarily correspond to the same point in space. Now, to be all in the same point, and to form a continuous extension, are contradictories. And thus it is manifest that material continuity is a mere contradiction.

Some will say that the contact is indeed made in the points, but that the parts, which touch one another in a common point, are quite distinct. But this appeal to the parts of the continuum, though much insisted upon by many ancient philosophers, is of no avail against our argument. For the existence of these parts cannot be assumed, without presupposing the continuity of matter. Such parts are, in fact, assumed to be continuous; and therefore, before we admit their existence, we must inquire whether and how they can have intrinsic extension and continuity. And dividing these parts into other parts, and these again into others without end, of all these parts of parts the same question must be asked—that is, whether and how they can have intrinsic extension and continuity. Hence one of two things will follow: either we shall never find the intrinsic reason of material continuity, or we shall find it only after having exhausted an infinite division—that is, after having reached, if possible, a term incapable of further division, viz., a mathematical point. But in the mathematical point it is impossible to find the intrinsic reason of material continuity, as we have just shown. And therefore the material continuity of the parts has no formal reason of its constitution, or, in other terms, the parts themselves are intrinsically impossible.

Moreover, the very distinction made by our opponents between the points of contact and the parts which touch one another in those points, is altogether irrational. For a parte rei—that is, considering the continuum as it is in itself—there is no foundation for the said distinction, it being evident that in a homogeneous continuum no place is to be found where we cannot mark out a point. Hence it is irrational to limit the designability of the points in order to make room for the parts. In other words, the parts themselves cannot be conceived as continuous without supposing that all the neighboring points which can be designated in them form by their contact a continuous extension, which we have proved to be inadmissible. The aforesaid distinction is therefore one of the subterfuges resorted to by the advocates of material continuity, to evade the unanswerable difficulties arising from their sentence; for it is true indeed, as Goudin remarks, that material continuity is “a philosophic mystery, against which reason objects more than it can answer,” though not because in this question “reason proves more than it can understand,” but because continuous matter is shown to be an absolute impossibility.

[pg 490]

Fifth argument.—It is a known metaphysical principle that “nothing can possibly become actual, except by the intervention of an act”—Impossibile est aliquid fieri in actu nisi per aliquem actum (S. Thomas passim). But no act can be imagined by which matter would become actually continuous. Therefore no actually continuous matter can possibly exist. The minor of our syllogism is proved thus. Acts are either substantial or accidental; hence if any act could be conceived as giving actual continuity to matter, such an act would be either substantial or accidental—that is, it would give to its matter either its first being or a mere mode of being. Now, neither the substantial nor the accidental act can make matter actually continuous. For, first, no substantial act can give to its matter a being for which the matter has no disposition. But actuable matter has no disposition for actual continuity, for where there are no distinct terms requiring continuation, there is no disposition to actual continuity, as is evident; and it is not less evident that the matter which is to be actuated by a substantial act involves no distinct terms, and does not even connote them, but merely implies the privation of the act giving it its first being, which act is one, not many, and gives one being, not many, and consequently is incapable of constituting a number of actual terms actually distinct, as would be required for actual continuity. To say the contrary would be to deny one of the most fundamental and universal principles of metaphysics, viz., Actus est qui distinguit, which means that there cannot be distinct terms where there are no distinct acts.

Moreover, continuity presupposes quantity; hence, if the substantial act gives actual continuity to its matter, it must be conceded that a certain quantity exists potentially in the actuable matter, and is reduced to act by the first actuation of matter. This quantity would therefore rank among the essentials of the substance, and could not possibly be considered as an accident; for the immediate result of the first actuation of a term by its substantial act is not a mere accident, but the very actuality of the essence of which that act and that term are the principles. Whence it follows that so long as quantity remains an accident, it is impossible to make it arise from the substantial act; and, accordingly, no substantial act can make matter actually continuous.

That actual continuity cannot arise from any accidental act is no less evident. For the only accidental act which could be supposed to play a part in the constitution of a material continuum would be some actual composition. But as composition without components is impossible, and the components of continuous matter, before such a composition, are not continuous (since we must now consider continuity as a result of the composition), our continuous matter would be made up of components destitute of continuous extension—that is, of mere mathematical points. But, as this is avowedly impossible, it follows that it is as impossible to admit that matter becomes actually continuous by the reception of an accidental act.

Sixth argument.—In a philosophico-mathematical work published in England a few years ago,108 from [pg 491] which we have already borrowed some plain arguments concerning other questions on matter, the impossibility of continuous matter is proved by the following argument: “A compound which has no first components is a sheer impossibility. Continuous matter, if admitted, would be a compound which has no first components. Therefore continuous matter is a sheer impossibility. In this argument the first proposition is self-evident; for the components are the material constituents of the compound; and therefore a compound which has no first components is a thing which is constituted without its first constituents, or a pure contradiction. The second proposition also is undeniable. And, first, there can be no doubt that continuous matter would be a compound; for continuous matter would be extended, and would have, accordingly, parts distinct from parts; which is the exclusive property of compounds. Now, that this compound would be without first components, can be proved as follows: If continuous matter has any first components, these components will either be extended or unextended. If they are supposed to be extended, then they are by no means the first components; since it is clear that in this case they have distinct parts, and therefore are themselves made up of other components. If they are supposed to be unextended, then they are by no means the components of continuum; since all know and admit that no continuum can be made up of unextended points. And, indeed, unextended points have no parts, and therefore cannot touch one another partially; whence it follows that either they touch each other totally, or they do not touch at all. If they do not touch at all, they do not make a continuum, as is evident. If they touch totally, the one will occupy exactly the same place which is occupied by the other, and no material extension will arise. And for this reason geometrical writers consider that a mathematical line cannot be conceived as made up of points, but only as the track of a single point in motion. We see, then, that a material continuum is a compound, of which the first components cannot be extended, and cannot be unextended. And since it is impossible to think of a third sort of first components which would be neither extended nor unextended, we must needs conclude that continuous matter is a compound which has no first components. And therefore continuous matter is a mere absurdity” (p. 30).

This argument is, in our opinion, altogether unanswerable. Those philosophers, in fact, who still venture to fight in favor of continuous matter, have never been able to solve it. When we urge them to declare whether they hold the first components of continuous matter to be extended or unextended, they constantly ignore and elude the question. They simply answer that the components of material substance are “the matter” and “the form.” But if the matter which lies under the form has no distinct parts, it is evident that the substance cannot be continuous. The composition of matter and form does not, therefore, entail continuity, unless the matter which is under the form has its own material composition of parts; and it is with reference to the composition of these parts of matter, not to the composition of matter and form, that we inquire whether the first components of continuous matter [pg 492] be extended or unextended. To ignore the gist of the argument is, on the part of our opponents, an implicit confession of their inability to cope with it.

Seventh argument.—Material substance, as consisting of act and potency, like everything else in creation, is both active and passive, its activity and passivity being essentially confined, as we have already explained,109 to the production and the reception of local movement. Hence, so long as material substance preserves its essential constitution, it is impossible to admit that matter is incapable of receiving movement from natural causes. But continuous matter would be incapable of receiving movement from natural causes. Therefore it is impossible to admit continuous matter. To prove the minor of this syllogism, let there be two little globes of continuous matter, and let them act on one another. Since no finite velocity can be communicated by an immediate contact of matter with matter, as shown in a preceding article, it follows that the velocity must be communicated by virtual contact in accordance with the law of the inverse squared distances. Hence, since some points of the two globes are nearer to one another, and others are farther, different points must acquire different velocities. Now, one and the same piece of matter cannot move onward with different velocities, as is evident; it will therefore be unable to move so long as such different velocities are not reduced to a mean one, which shall be common to the whole mass. Such a reduction of unequal velocities to a mean one would meet with no difficulty, if the globes in question were made up of free and independent points of matter; for in such a case the globes would be compressed, and each point of matter would act and react according to known mechanical laws, and thus soon equalize their respective velocities. But in the case of material continuity the reduction of different velocities to a mean one is by no means possible. For “in a piece of continuous matter,” to quote again from the above-mentioned work of molecular mechanics, “any point which can be designated is so invariably united with the other points that no impact and no mutual reaction are conceivable; the obvious consequence of which is that no work can be done within the continuous particle in order to equalize the unequal velocities impressed from without. Moreover, in our case the reduction ought to be rigorously instantaneous; which is another impossibility. In fact, if distinct points of a continuous piece of matter were for any short duration of time animated by different velocities, the continuum would evidently undergo immediate and unavoidable resolution; which is against the hypothesis. Since, then, the said reduction cannot be made instantaneously, as we have proved above, nor, indeed, in any other way, and, on the other hand, our continuous particle cannot move onward before the different velocities are reduced to one of mean intensity, it is quite evident that the same continuous particle will never be capable of moving, whatever be the conditions of the impact. And since what is true of one particle on account of its supposed continuity is true also of each of the other particles equally continuous, we must conclude that bodies made up of particles materially continuous are totally incapable [pg 493] of receiving any communication of motion.”110

This argument, though seemingly proving only the non-existence of continuous matter in nature, proves in fact, also, the impossibility of its existence. For, if a substance could be created possessing intrinsic extension and continuity, that substance would essentially differ from the existing matter, and would therefore be anything but matter. Hence not even in this supposition would continuous matter exist.

Eighth argument.—The inertia of matter, and its property of acting in a sphere, might furnish us with a new argument against material continuity. But we prefer to conclude with a mathematical demonstration drawn from the weight of matter. The weight of a mass of matter depends on the number of material terms to which the action of gravity is applied, and it increases exactly in the same ratio as the number of the elementary terms contained in the mass. This being the case, let us assume that there is somewhere an atom of continuous matter. The action of gravity will find in it an infinite multitude of points of application; for it is of the nature of continuum to supply matter for an endless division. Hence if we call g the action of gravity on the unit of mass in the unit of time, the action of the same gravity on any of those infinite points of application will be

g ρ dx dy dz,

ρ being the density of the mass, and dx, dy, dz the three dimensions of an infinitesimal portion of it.

Now, since we know that gravity in the unit of time imparts a finite velocity to every point of matter in the atom, we must admit that the action exerted on the infinitesimal mass ρ dx dy dz has a finite value; and therefore, since the volume dx dy dz is an infinitesimal of the third degree, the density ρ must be an infinite of the third order. But a continuous mass whose elements have an infinite density has itself an infinite density; hence, if its volume has finite dimensions, the mass itself (which is the product of the volume into the density) is necessarily infinite, and will have an infinite weight. Hence the assumption of continuous matter leads to an absurdity. The assumption is therefore to be rejected as evidently false.

We will put an end to the series of our proofs by pointing out the intrinsic and radical reason why matter cannot be continuous. The matter which is under the form is a potency in the same order of reality in which its form is an act. Now, the only property of a potency is to be liable to receive some determinations of a certain kind; and the property of a potency whose form is an active principle of local motion must consist in its being liable to receive a determination to local movement. Hence, as the matter receives its first being by a form of a spherical character, and becomes the real central point from which the actions of the substance proceed, so also the same matter, when already actuated by its essential form, receives any accidental determination to local movement; and, inasmuch as it is liable to local movement, it is in potency to extend through space—that is, to describe in space a continuous line; and when it actually moves, it actually traces a continuous line—that is, it extends from place to place, continuously indeed, but successively; whence it is manifest [pg 494] that its extension is nothing but Actus existentis in potentia ut in potentia Aristotle would say, viz., an actual passage from one potential state to another. Such is the only extension of which matter is capable. Such an extension is always in fieri, never in facto esse; always dynamical, never statical; always potential and successive, never formal or simultaneous. We can, therefore, ascribe to matter potential continuity, just as we ascribe to its active principle a virtual continuity; for the passivity of the matter and the activity of the form correspond to one another as properties of one and the same essence; and whatever can be predicated actively or virtually of a substance on account of its form can be predicated passively or potentially of the same substance on account of its matter.

These remarks form a complement to our fifth argument, where we proved that no substantial and no accidental act could make matter actually continuous. For, since matter cannot receive any accidental act, except the determination to local movement, and since this movement, although continuous, is essentially successive, it follows that by such a determination no actual and permanent continuity can arise, but a mere continuation of local changes. Thus matter, according to its potential nature, has only a potential extension; or, in other terms, it is not in itself actually continuous, but is simply ready to extend through space by continuous movement.

The preceding proofs seem quite sufficient, and more than sufficient, to uproot the prejudice in favor of material continuity; we must, however, defend them from the attacks of our opponents, that no reasonable doubt may remain as to the cogency of our demonstration.

First objection.—The globe and the plane, of which we have spoken in our third argument, though destitute of proportional parts suitable for a statical contact, become proportionate to one another, says Goudin, by the very movement of the one upon the other; and thus our third argument would fall to the ground. For a successive contact partakes of the nature of successive beings. Hence, as time, although having no present, except an indivisible instant, becomes, through its flowing, extended into continuous parts, so also the contact of the globe with the plane, although limited to an indivisible point, can nevertheless, by its flowing, become extended so as to correspond to the extended parts of the plane. For, according to mathematicians, a point, though indivisible when at rest, can by moving describe a divisible line.

To this we answer that a globe and a plane cannot by the movement of the one on the other acquire proportionate parts. For, although it is true that a successive contact partakes of the nature of the successive being which we call movement, it is plain that it does not partake of the nature of matter. In fact, the material plane is not supposed to become continuous through the movement of the globe, but is hypothetically assumed to be continuous before the movement, and even before the existence, of the said globe. The continuous movement is, of course, proportionate to a continuous plane; but it is evident that it cannot originate any proportion between the plane and the globe; because this would be against the essence of both. No part of the plane can be spherical, [pg 495] and no part of the globe can be plane; hence, whatever may be the movement of the one upon the other, they will never touch one another, except in a single point.

That time, although having no present, except an indivisible point, becomes extended by flowing on, is perfectly true; but this proves nothing. For, in the same manner as the act of flowing, by which time flows, has nothing actual but a single indivisible instant, so also the act of flowing, by which the contact of the globe with the plane flows, has no actuality but in an indivisible point of space; and as an indivisible instant by its flowing draws a line of time without ever becoming extended in itself, so also an indivisible point by its flowing draws a line in space without ever becoming extended in itself; and as the instant of time never becomes proportionate to any finite length of time, so also the point of contact never becomes proportionate to any finite line in space.

That a line, therefore, arises from the flowing of a point in the same manner as time from the flowing of an instant, is a plain truth, and there was no need of Goudin's argumentation to make it acceptable. To defeat our argument, he should have proved that the actual flowing of an instant takes up a length of time. If this could have been proved, it would have been easy to conclude that the flowing contact also extends through a length of space. But the author did not attempt to show that an instant of time flows through finite lengths of time. It is evident, on the contrary, that an instant flows through mere instants immediately following one another. And thus the objection has no weight.

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.

[pg 498]

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.

Our answer to this objection is that time and movement are not engendered by a formal contact of a real instant with the instant following, or of a real point with the point following. Duration is not a sum of indivisible instants formally touching one another, nor is the length of space a sum of indivisible points touching one another. We may have points in space, but not points of space; and in like manner we have instants in succession, not instants of succession, though in common language we usually confound the latter with the former. Yet, when we talk of a point of space, our meaning is not that space is made up of points, but simply that a point of matter existing in space marks out its own ubication, thus lending to the space occupied the name of point. Hence no movement in space can be conceived to extend by successive contacts of points, or by the flowing of a point towards other points immediately following; for these points immediately following exist only in our imagination. Nor does a flowing point engender a line of space, but only a line of movement; and even this latter is not properly engendered, but merely marked out in space; for all possible lines are already virtually contained in space, and therefore they need no engendering, but simply marking out by continuous motion.

The same is to be said of the origin of time. Time is not a formal sum of instants touching one another. The instant just past is no more, hence it cannot touch the instant which is now; and the instant which is to follow is not yet, hence it cannot be touched by the instant which is now. Accordingly, as the movement of a single point marks out a continuous line in absolute space, so also the flowing of a single instant extends a line in absolute duration. For, as S. Thomas teaches, in the whole length of time there is but a single instant in re, though this same instant becomes virtually manifold in ratione prioris et posterioris by shifting from “before” to “after.” And in the same manner, in the whole length of a line measured in space by continuous movement, there is but a single point in re actually shifting its ubication from “here” to “there,” and thus becoming virtually manifold in its successive positions. And for this reason both movement and time are always and essentially developing (in fieri), and never exist as developed (in facto esse); since of the former nothing is actual but a point, and of the latter nothing is actual but an instant.

It is scarcely necessary to repeat that, if there were any continuous matter, its parts would all be actual and simultaneous. Its continuous extension would therefore be properly engendered by the contact of indivisible points, not by the shifting of a point from one end of its dimensions to another. This sufficiently shows that from the continuity of movement and of [pg 499] time nothing can be concluded in favor of continuous matter.

Fifth objection.—Between two given points in space infinite other points can be placed. Now, what is possible can be conceived to be done; and thus we can conceive an infinite multitude between the two points. Accordingly, an infinite multitude can be contained within limits; and if so, continuous matter is not impossible, and our first argument has no weight.

We answer that, although an infinite multitude of points can be placed between any two given points, yet nothing can be inferred therefrom in favor of continuous matter. For those innumerable points either will touch one another or not. If they do not touch, they will not make a continuum; and if they touch, they will, as we have shown, entirely coincide, instead of forming a continuous extension. It is plain, therefore, that the distance between the two given points cannot be filled continuously, even by an infinite multitude of other points. And therefore the objection has no force.

Nor is it true that by the creation of an infinite multitude of points between two given points such a multitude would be an infinity within limits. For the two given points are limits, or rather terms, of a local relation, but they are no limits of the multitude, or discrete quantity, which can be placed between them; for, without altering the position of those two points, we can increase without end the number of the intervening points. As volume is not a limit of density, so the distance of two points is not the limit of the multitude that can be condensed between them.

Sixth objection.—All the arguments above given against the continuity of matter are grounded on a false supposition; for they all take for granted that a continuum must be made up of parts—an assumption which can be shown to be false. For, first, in the geometric continuum there are no actual parts; for such a continuum is not made up by composition, but is created, such as it is, all in one piece. Whence it must be inferred that the primitive elements of matter, though exempt, as primitive, from composition of parts, and really simple, may yet possess extension. Secondly, who can deny that God has the power to create a solid body as perfectly continuous as a geometric volume? Such a body, though divisible into any number of parts, would not be a compound; for its parts would be merely possible, not actual; and therefore it would be simple, and yet continuous. Thirdly, those who deny the possibility of continuous matter admit a vacuum existing between simple points of matter. Such a vacuum is a continuous extension intercepted between real terms, and is nothing else than the possibility of real extension. But the real extension, which is possible between real terms, is not, of course, a series of points touching one another, for such a series, as all admit, is impossible. It is, therefore, an extension really continuous, not made up of parts, but only divisible into parts. Hence matter may be continuous and simple at the same time.111

This objection tends to establish the possibility of simple-extended matter. Yet that simplicity and material extension exclude one another [pg 500] is an evident truth; in other terms, material continuity, without composition of parts, is utterly inconceivable. If, therefore, we persist in taking for granted that a material continuum must be made up of actual parts, we do not make a gratuitous supposition.

The three reasons adduced in the objection are far from satisfactory. The first makes an unlawful transition from the geometric extension of volumes to the physical extension of masses. Such a transition, we say, is unlawful; for the geometrical extension is only virtually continuous, and therefore involves no actual multitude of parts; whereas the physical extension of the mass of matter would be formally and materially continuous, thus involving a formal multitude of actual parts perfectly distinct from one another, though united to form one continuous piece. The geometric extension is measured by three linear dimensions, and has no density. Now, a geometric line is nothing else than the trace of the movement of a point; and accordingly its continuity arises from the continuity of the movement itself, which alone is formally continuous; for the space measured by such a movement has no formal continuity of its own, as we have already explained, but is styled “continuous” only inasmuch as it is the region of continuous movement. There is no doubt, therefore, that geometric extension is merely virtual in its continuity; and for this reason it is not made up of parts of its own, but simply corresponds to the parts of the movement by which it can be measured. Material extension, on the contrary, would be densely filled with actual matter, and therefore would be made up of actual parts perfectly distinct, though not separated. To apply, as the objection does, to material extension, what geometry teaches of the extension of volumes, is therefore a mere paralogism. It amounts to saying: Vacuum is free from composition; therefore the matter also which would fill it is free from composition.

We may add that even geometric extension, if real, involves composition. For, evidently, we cannot conceive a geometric cube without its eight vertices, nor can we pretend that a figure requiring eight distinct points as the terms of its dimensions is free from composition. Now, if an empty geometric volume cannot be simple, what shall we say of a volume full of matter? Wherever there is real extension, there are real dimensions, of which the beginning, and the end, and all the intermediate terms are really distinct from one another. Hence in a material extension there should be as many distinct material terms as there are geometric points within its limits. And if this is simplicity, we may well ask what is composition?

The second reason adduced in the objection is a mere petitio principii. For he who says that God can create “a solid body as perfectly continuous as a geometric volume” assumes that such a continuous body involves no contradiction; he therefore begs the question. On the other hand, to affirm that God can create a solid body as perfectly continuous as a geometric volume, is to affirm that God can create a body of infinite density—that is, an infinite mass within finite dimensions. For the mass of a body of matter is the product of its volume into its density; hence, if its volume be finite, and its density infinite, the mass will be [pg 501] infinite. Now, a body materially continuous implies infinite density; for it excludes porosity, and it supplies matter for an endless division. Hence a continuous mass of matter filling a finite volume would be an infinite mass contained within limits. We think we are not presuming too much when we say that God cannot create such a metaphysical monstrosity.

“Such a body,” says the objection, “though divisible into any number of parts, would not be a compound.” This is evidently false; for all that is divisible into parts has parts, and therefore composition. Nor is it true that the parts of a continuous body “would be merely possible, not actual”; for if such parts are not actual, how can the body be actual? No actual continuum can exist without actual parts. The divisibility of continuum is not the possibility of actual parts, but the possibility of their actual separation.

The third reason is based on our admission of a vacuum between material points. Such a vacuum, it is objected, is a continuous (virtual) extension, founding the possibility of some other (formal) extension. This we concede; but when it is argued that this other extension which is possible between the material terms is the extension of continuous matter, we deny the consequence. It is only continuous local movement, not continuous matter, that can formally extend from term to term, as we have proved. When two real points of matter have a distinct ubication in space, the interval between them cannot be estimated otherwise than by the extent of the movement which can be made from one point to the other. We cannot perceive the distance between two terms, except by drawing, at least mentally, a line from the one to the other; and for this reason, as we have remarked elsewhere, the relation of distance is conceived by us as a quantity measured by movement, not by matter, and representing the extension of continuous movement, not of continuous matter. Hence a vacuum intercepted between real points is a real, though only virtual, extension; and that other real and formal extension, which is possible between the same real points, is the extension of local movement. Our opponent concedes that “the real extension possible between real terms is not a series of points touching one another; for such a series, as all admit, is impossible.” Now, this suffices to show that the real extension possible between such real terms is not the extension of continuous matter; for such an extension, as we have abundantly proved, would be made up of nothing but of a series of points touching one another.

Nothing, perhaps, more evidently shows the unquestionable solidity of the thesis we have undertaken to defend than the necessity felt by our opponents of admitting in matter an extended simplicity and a simplicity divisible into parts, as witnessed by this last objection, which we have transcribed from a grave and learned professor of philosophy. Extended and simple matter is such an absurdity as few would admit to be a corollary of their own theories; yet it cannot be escaped by those who consider the first elements of matter as endowed with bulk. For physical simplicity is an essential attribute of all primitive beings; and, if primitive elements are nevertheless supposed to be intrinsically extended, [pg 502] it is plain that their simplicity will be an extended simplicity.

The main reason why some philosophers still cling to material continuity is their fear of actio in distans. We have already shown that such a fear, though very common, cannot be justified. We grant that, owing to popular prejudice and an incorrect notion of things, many are apt to dread action at a distance as a dangerous shoal; but when they resort to an “extended and divisible simplicity,” they steer their ship directly against the reefs.

To Be Continued.