The Quantity of Bodies | Catholic Philosophy

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Quantity is the proper accident of body: it gives the body its extension in parts outside of parts and its consequent divisibility and measurability. Continuous quantity (the parts are actually united — a body, a line, a surface) is distinguished from discrete quantity (the parts are counted as distinct units — a number, a collection). From quantity derive the properties of bodies: divisibility (into parts of the same nature), measurability (capable of being compared with a standard), impenetrability (one quantity excludes another from its place), and the capacity for spatial position and place. Space is relative (defined by the mutual distances of bodies), not an absolute container independent of them. Time is the measure of motion with respect to before and after; it too is relative to the changing realities whose succession it measures.

The Latin word quantum is fundamentally an interrogative word and means “how much?” Anything of which the question “how much?” can be rightly and literally asked, has quantity. Quantity, therefore, involves, fundamentally, a notion of amount, extent, bulk, size, content, parts, number.

Notice, in the description just given, the important words rightly and literally. For quantity, properly speaking, ts referable only to bodies. When we ask “how much?” or “how many?” of things other than bodily substances, we use the terms of our question in an extended or metaphorical sense, and the answer to those questions has the same character,— that is, it is not literal, but analogical. Thus when we speak of an amount of learning, or of a number of ideas, or of the extent of wisdom or piety, we are using words that express quantity, but not literal quantity. Quantity, literally taken, is a proper mark (a proper accident) of bodily substance and of no other thing whatever.

We notice in the world around us that many bodily realities are present. These have their respective places here. And we notice that a bodily substance extends itself, so to speak, to fill its place; we see that it is “part here, part there.” Now, the property which spreads out or extends a bodily substance so that it is “part here, part there” is called external quantity. External quantity has been defined as “‘the property whereby a bodily substance has parts outside of parts, with reference to its place” (accidens extensivum substantiae corporeae in partes locales, or, proprietas qua fit ut partes corporis sint extra partes in ordine ad locum).

Now, external quantity is not “the heart and soul” (that is, the very essence) of quantity itself ; external quantity is the complement and external manifestation of the inner, essential thing called internal quantity. And we learn what internal quantity is by considering closely the character of external quantity or external extension. We must notice that a bodily substance could not spread itself out to fill its place (that is, it could not have actual external or local quantity) unless, as a fact, it possessed some inner aptitude for such extension. In a word, a bodily substance must have parts in itself (or must be internally quantified) if it has parts in @ place (or is externally quantified). The inner extension or internal quantity of an existing body consists in “the position of parts outside parts in the substance.” Now, internal quantity is quantity in the strict sense. It is important that we remember this fact. For when we speak of bodily quantity we usually think of external quantity,—in fact and effect—and we are apt to be led by this common thought into the assumption that the very essence of quantity lies in externals. And so we may be ready to conclude, upon this mistaken assumption, that certain phenomena (such as mtultilocation and compenetration) are absolute impossibilities.

We may define internal quantity as “the property whereby an existing bodily substance has, in itself, parts outside of parts,” or “the property whereby an existing body has actual parts in itself”; (accidens extensivum substantiae corporeae in partes sutipsius ; or positio partium extra partes in ordine ad se).

Quantity is a proper mark or proper accident or simply a property of existing bodily substance. It is not to be identified with bodily substance itself. It is something which a bodily substance has, not something which such a substance is. A bodily substance is constituted by the union of its essential substantial parts, its matter and form, its physical and elemental constitution; and these are not quantity-parts (or quantitative parts), even though quantity is a requisite condition for their natural existence in the com- pleted and existing body. A drop of water and a gallon of water, are alike in their essential parts, that is, in their constitution; each is a body made of prime matter and substantial form; each is the same kind of bodily substance ; each is truly and completely water ; but the drop and the gallon are different quantities. Quantity, then, is not the constituting element, the basic essence, of a bodily substance itself. It is something in the order of accidents, not of substances. Quantitative parts are, therefore, not essential parts; they are integral parts and even substantial integral parts inasmuch as they belong to the unbroken and undivided substance and bring it a kind of “perfection.” The point to remember is that they are not substantial essential parts. Even where definite quantities effect the physical essence of a reality (as, for example, two parts hydrogen and one part oxygen effect the essence of water) it is never the quantities themselves or as such that constitute or effect the essence; it is the quantified substances concerned that do the work, even if they must be conditioned by such and such quantity.

The history of philosophy, and of science, lists many erroneous theories about the nature of quantity; and costly errors these have often been; for a mistake in so fundamental a matter as the quantity of bodily substance is sure to lead to further errors in other fields. René Descartes (1596-1650) taught that the essence of a natural body is its extension or dimensions; thus he identified bodily substance and its quantity. Others,—like Babenstuber (1660-1726) and de Aguirre (d. 1699),—taught that a bodily substance has essentially a spread of parts, and that the accident called quantity merely sets these in order. The great Suarez (1548-1617) made a clear distinction between a bodily substance and its quantity, but he held that a body, antecedently to the effect of quantity upon it, has in its own being distinct substantial parts, and that it is the function of quantity to render these impenetrable. These erroneous views come of failure to hold clear concepts of what is meant by substance as distinct from accident (and from proper accident), and from a confusion of mind on the distinction between internal extension and external extension.

We may sum up sound doctrine on the nature of quantity in the following sentences. A bodily substance is not identified with its quantity; for substance is substance, and quantity is an accident. A bodily substance has not, antecedently to its quantification (that is, before it is affected by quantity) any entitative extension of distinct integral parts; but it is one reality, integrally uncomposed though radically requiring parts and extension. When a body is actually affected by quantity, then it has substantial integral parts in itself (that is, it has internal extension or internal quantity), but the formal cause of these parts is the substance itself which has them, and with this substance the accident of quantity concurs as a required condition.

To put the matter in another, and perhaps simpler, way: The essence of quantity consists in internal extension. For the external extension of a body is consequent upon its internal extension; a body cannot have parts in a place unless it have parts in itself; therefore, internal extension and not external extension is the root and essence of quantity. Now, while the essence of quantity (which is an accident) is found in internal extension, the essence of the quantified substance (that is, of the body which has quantity) is not constituted by extension, internal or external. Indeed, the essence of bodily substance is, in itself, independent of extension (for it is, in itself, integrally one and non-composed), although it has a natural requirement for extension; extension is a condition required for the natural existence of a body in the world of actual substances. When the body actually exists in the natural way, it has internal extension; it has integral parts which are parts of itself; the bodily substance itself constitutes these parts (that is, the parts consist of the substance), and this is saying that the substance itself is the formal cause of its integral parts.

In the world around us, we see that bodies have external as well as internal extension. External extension renders bodies impenetrable, mensurable, divisible, and determines their location. Now, as we have noticed, external extension is a secondary effect of quantity; a body must have internal quantity in the first place or it cannot be externally extended in the second place. But it is at least conceivable that the secondary effects of quantity might be prevented or removed without destroying the actuality or the primary effects of quantity in its essence. In other words, a body might conceivably exist with its internal extension (that is, its internal quantity) even if it had no external extension. Nature, of course, offers us no instances of such a thing, and our natural knowledge of bodies is always bound up with their external extension. But reason sees no contradiction, no impossibility, in the existence of a bodily substance without external extension. Philosophy has nothing further to say on the point ; it merely indicates the truth that such an existence is not intrinsically impossible or unthinkable. Implicitly, philosophy concludes that, if a bodily substance is to have existence and internal quantity without external extension, more than natural power or forces will be required to give it actuality. For purposes of illustration we may borrow from our Faith an actual example of the thing of which we are speaking. In the Blessed Sacrament, Christ is present,—not only as God, but as Man with His true Body. The Body of Christ in the Eucharist has actuality; it has integral parts internally extended; that is, the Body has internal extension or quantity. But the Body has no external extension or quantity. The parts of the

Body are not codimensional with corresponding external parts of the host; we cannot say that part of Christ’s body is in one part of the host, and another part of the Body in another part of the host, and so on. Nor can we say that the Body is dwarfed, or held in miniature, within the actual external dimensions of the host. Nor can the Body of Christ be locally confined by the quantity of the host, nor measured or divided with the measurements or divisions of the host. The entire Body of Our Lord is present (in mature and perfect being) in each host and in each part of each host. In the Eucharist, the secondary effect of quantity,—that is, external extension,—is blocked out by supernatural power, and the Body of Christ, with its true internal quantity, is here present without external extension or external quantity. It is plain, then, that the essence of quantity lies in internal extension, and that the actual extension of a bodily substance in a place is a secondary effect of quantity and not its essential expression.

b) PROPERTIES OF QUANTITY

We now come to consider quantity, not only in its essence, but in its normal and natural actuality involving extension both internal and external. In a word, we consider bodily substances as they naturally exist in the world.

We seek to determine the properties of quantity, or of quantified bodies. Now, a property (that which is proper to or belongs to) is something that belongs to a reality by natural necessity, so that, when the reality is fully and naturally constituted, this “something” will always be found in it. A property is said to “flow from” the fully constituted essence of a reality. Thus, the power of speech is a property of a human being. When a man’s nature is fully and completely constituted ; when all its essential, and all its integral elements are present in full development; when nothing interferes or intervenes to block or thwart their normal function; then, inevitably, the man will have the power of speech. This power is not a constituent part of a man (for he would be a man in essential completeness even if he lacked the power in question) but it comes from or “flows from” the nature of man equipped with all essential and integral parts and unhampered in their exercise. Thus it is something proper to man; it belongs to him by normal and natural necessity ; it is called a property of man. Again, to illustrate further : infallibility is said to be a property or an attribute (a word synonymous with property) of the Church. The Church is an institution established by the Almighty and Infallible God to lead men to truth; it follows of necessity that the Church cannot lead men to error. Being what it is, that is, having a divinely given nature and commission for the teaching of truth, the Church possesses infallibility. We do not call infallibility a part of the Church, or a constituting element of the Church: in- fallibility “flows from” the rounded and perfect nature of the Church, and is therefore a property or an attribute of the Church.

When we seek to determine the properties of quantity, or, in more precise terms, to list the properties of bodies as quantified, we look for those characteristics which belong by natural necessity to quantified matter. We find that these properties are four, viz., external extension, incompenetrability, divisibility, mensurability. We do not include internal extension among the properties of quantity, for, while it ts a property of bodily substance, it is the essential constituent, and not the property, of quantity itself. We must say a brief word on each of the four properties :

I. External extension belongs by natural necessity to bodies, and will always be found in bodies unless supernaturally excluded, as, for example, it is excluded in the Body of Our Lord in the Holy Eucharist. External extension is that property of natural bodies by which they are extended in space and occupy place. The place of a body consists in its external extension or location in space. Place is discovered or determined by the distance-relation of an externally extended body to surrounding bodies. If there were only one body in existence, we could not properly speak of its place; of course we could speak of the place of its parts, considering each of these as a body, and viewing each with reference to the sur- rounding parts. But the existence of many bodily realities is a manifest fact of experience, and we inevitably notice that this body is “here” and that body is “there”; the same fact is observed when we consider the distinct parts of a single body. Now, the “here” and “there” relationship of existing bodies is what makes manifest their place; and the fact that bodies have external extension accounts for their being “here” and “there” in the natural sense of these terms. Place is proper or common. The proper place of a body is its position with reference to the body or bodies that immediately surround it and come in contact with it at every point. Consider a sphere or ball poised motionless in the air. The ball is surrounded by air; there is a perfectly fitting pocket of air, the inner concave surface of which is codimensional with the outer convex surface of the ball. This inner and concave surface marks the proper place of the ball. A ball that is flying through the air is not properly located or placed; the notion of place suggests immobility; but at any given instant of its flight, and considered statically in that instant, the ball has its place determined by the immediately surrounding body (in the example, the atmospheric air) which perfectly encloses it and is codimensional with its outer bulk or external extension. Thus bodies in the world have their proper place. But what we have been considering so far is the proper external place of bodies. Now, a body has also its proper internal place, and this consists in its being enclosed in its own dimensions. A baseball flying through the air may be said to pass through a continuous series of places (considering it statically at each instant of flight) and, in this sense, the proper external place of the ball is constantly changed,—or rather, the ball passes from one to another of places which do not change. But the baseball (as long as its bulk and external extension remains the same) does not pass from one internal proper place to another at all; wherever it is in its flight, its proper internal place is the same and is wholly motionless—In addition to proper place (internal and external) a body has, by reason of its external extension, a common place, that is a place shared with other bodies. Thus, a book is on the shelf, in the bookcase, in the library, in the house, in the town. Shelf, bookcase, library, house, town, indicate common locations or common places, for these may be assigned to more objects than the one book in question. A body shares its proper place with no other body; it shares its common place with one or more bodies. External extension is the property of natural bodies by which they are said to occupy proper and common place in the normal and natural sense of these terms. Now, even a body without external extension, as well as bodily powers which are not in themselves immediately subject to exten- sion, are also said to be placed or located. This leads us to the consideration of the question: how may a reality be in a place? The answer is fourfold:

a) A body with external location is said to be in its place, or to be localized, in a circumscriptive manner. The term circumscriptive is from the Latin circumscriptum which means “written around” or “marked round about.” Just as a coin placed on paper may be “written around” by drawing a sharp pencil about its circumference, so a body in the world is enclosed by surrounding bodies. A natural body in the world is circumscriptively located, first in point of its proper, and then in point of its common, place or location or ubication. This latter term, ubication, is from the Latin ubi, which means “where,” and the term may therefore be translated as “whereness” or “having its whereabouts.”

b) Any determinant of a body (size, shape, beauty, temperature,—or the essential and substantial determinant which makes the body an actual body of this specific kind) is called a form. A determinant which makes a body the actual substance that it is, is its substantial form; a determinant that marks or characterizes a body as to size, temperature, or other accident, is an accidental form. Any existing natural body is a single substance, and has only one substantial form ; it usually has many accidental marks, qualifications, characteristics, (in a word, accidents) and thus it has many accidental forms. Now, the form

(substantial or accidental) is said to be present, to be located or placed, in the substance that has it, whether the substance be constituted in actuality by the form (substantial form) or be merely marked and qualified by the form (accidental form). And the form,—substantial or accidental,—is manifestly present in some manner other than that in which a body is in its place in the world of bodies. A form is not present circumscriptively. For beauty is not present in a beautiful object as a coin is in a purse or a baseball in the air. Nor is the temperature of a body in the body in the same sense as the body is in the house or inacorner or in water. We define the presence or location or place of a form as informative presence, for the determinant (or form) is said to in-form the substance which has it and in which it is thus located. Thus, the soul (the substantial form of the human substance) is in the body informatively; thus beauty or coldness or shape or flavor is in a bodily substance informatively; thus knowledge is in the mind informatively, and, indeed, we speak more accurately than we realize when we call knowledge by the name of information—A substantial form is not, in itself, dependent upon external extension, although a natural body normally requires some quantity of externally extended matter for existence in the material universe. Some accidental forms require for their existence a substance externally extended; indeed, all sheerly bodily accidental forms require such a substance.—In passing, we must notice that when we say that each single substance has only one substantial form, we do not consider substances as single in virtue of their mere external appearance. A bar of iron or a block of marble is one kind of substance, but the bar and the block are really collections or amassings of particles or minimwm-amounts of iron and of marble. And each minimum-amount (that is, the amount requisite for the natural existence of iron or of marble) has its substantial form which makes it that kind of substance, actually existing. The greater or lesser number of particles caught together to make a bar or a block is something accidental to the iron or the marble as such; it is a point of their quantity.

c) A form (substantial or accidental) may be a working force or active power, and it is said to be located in, or have place in, the bodily substance which operates by its means. This type of form is present (or located or placed) not only informatively, but operatively. And, if the operative power is capable of activity in but one place at a time,—that is, within one substance at a time,—it is said to be there definitively, that is, its activity is limited or defined by the limitations of the one substance which it affects. All the active forces or powers of natural bodies are located in their respective substances informatively, operatively, and definitively, but not, of course, circumscriptively. The power of seeing, for example, is thus present, or thus has place, in a man. The substantial form of a man (that is, the spiritual soul) which in-forms the human substance (and is thus present there informatively), is active or operative, and each soul operates only within the individual man whose substance it makes human; thus the soul is present in a man, or in a man’s body, informatively, operatively, and definitively. Of course, the soul, having no extension of its own, is not present circumscriptively. The infinite power of God (creating, preserving, providing) is exercised in the world, and is said to be present in the world, operatively; but it is not present definitively, for the infinite power is in no wise limited but operates everywhere. Nor is the divine power present in the world informatively, for God is not the form or soul of the world (as the old Stoics thought), nor is He the accidental form of anything in the world; God does not enter into creatures as a substantial constituent (substantial form) nor as an accidental determinant (accidental form). We say that God is present everywhere operatively, ubiquitously (or non-definitively), and essentially.— For the normal exercise of powers that belong to natural bodies, some minimum of externally extended matter is required; external extension or external quantity is thus a requisite condition for powers operatively present and naturally active in bodily substances.

d) A fourth mode of presence (that is, a fourth mode or manner in which a reality may have place or location) is exampled in the presence of Our Lord in the Holy Eucharist. This is called sacramental presence or location. It may be described as a mode of presence in which one substance has place through the mediation of the dimensions (or external extension) of another substance, but without making these dimensions its own. Thus Our Lord is present in the consecrated host sacramentally; He uses the external quantity and dimensions of the host as the medium, so to speak, of His actual presence, but He does not make the dimensions of the host His own dimensions.

Incompenetrability or impenetrability is that property of a natural body (consequent upon its external extension) which prevents another body from occupying its place while it is present there itself. The simultaneous location of two or more bodies in one and the same place is called compenetration. In the natural order, no compenetration of bodies is possible. Of course, one body may be inside another, as contents in container (thus water is in a glass) ; and one bodily substance may interpenetrate with another (as ink in a blotter or water in a sponge) ; but externally extended bodies cannot compenetrate, or occupy exactly the same place at the same time. However, reason sees no intrinsic impossibility in the very idea of compenetration, and therefore the thing might be a fact under the action of supernatural power. The Body of the Risen Savior passed through the great stone that closed the Sepulcher, and also, on at least two occasions, it passed through the doors of the chamber where the disciples were gathered together. True, the Body of Christ is a glorified Body, but it is a true body, an actual bodily substance. It cannot, indeed, be known whether the obstacles (the stone, and the doors) were miraculously and momentarily withdrawn, or rendered tenuous so as to admit the passing of a body through their interstices. Thus we cannot point to the miraculous passing of the Lord through bodily substances as a certain example of compenetration. If the cases mentioned were true instances of compenetration, we may say that, at the moment of compenetration (that is, at the moment when the Body of Christ and the substance of stone or doors actually occupied the same place) , these diverse substances were not present in the same way. One of the substances could have had its normal and natural circumscriptive presence, consequent upon its external extension; the other must have lacked, at least momentarily, its own external extension, and must have been extended internally only. In the concept of compenetration under these conditions, there is no absolute or metaphysical impossibility—The topic of compenetration suggests to the mind the question of multilocation or the simultaneous presence of one and the same body in a plurality of places.

Here, as in the question of compenetration, we must affirm that nature affords no instances of such a phenomenon. And again, the mind sees in multilocation of bodies no intrinsic or absolute impossibility if causes beyond nature be set to work. After all, the natural location of bodily substances is a thing consequent upon external extension, and this, in turn, is a secondary effect of bodily substance. But a supernatural power might suspend secondary effects, leaving essence and primary effects intact. Thus, there is no contradiction or conflict in the very thought of a single body being in several places at the same moment. Such multilocation is more readily conceivable under the condition that the body be present in different places in different ways,—circumscriptively in

. one place, non-circumscriptively in the others. Indeed, our Faith affords us a certain instance of multilocation in the presence of Christ in the Blessed Sacrament. The Body of the Lord is present in an extended manner in Heaven, and in an inextended manner in every consecrated host (and in each part of every host) and in every drop contained in the consecrated chalice—Sometimes we read, and notably in the lives of certain Saints, of the simultaneous presence of a man in two or more places. But perhaps this is no example of multilocation, but of a real presence in one place and an apparent presence (or apparition) in the other places.

Divisibility is that property of a natural body (consequent upon its extension) which renders it capable of being “taken apart” or divided into an indefinite number of parts. We say, an indefinite number, for the parts of any body can be divided and subdivided without coming to a point where further division is unthinkable. Of course, physical partition has its limits; the instruments by which we cut and divide a body into parts are clumsy tools at best, and their work is soon done. But mathematically there is no definite point at which further division becomes impossible. Suppose one should say, “Yes, there is such a point. Here we have the last possible division or part of a substance.” Might we not reply, “Let us consider an amount just half of the bulk of this socalled indivisible part”? Yes, and if we chose we might consider one-millionth of the so-called ultimate part, or one-billionth of it, for that matter; and we might consider the billionth part of that billionth part, and so on endlessly. Does this mean that the number of parts conceivable in any bodily substance is actually infinite? No, the word is indefinite. Or, if you prefer, you may say potentially infinite, but never actually infinite. Actual infinity is absolute boundlessness; it involves impossibility of increase or diminishment. If we could think of a number so great that it could not be increased no matter how many times we multiplied it by itself, and could not be diminished no matter how we halved and quartered it, then we should have the concept of an infinite number. But the very idea of number is the concept of a thing actually made up of units, and increasable and decreasable by units. In a word, the very idea of number involves limitation or finiteness, and, conversely, blocks out the possibility of actual infinity. But a number is potentially infinite (or indefinite) in the sense that it can be increased or divided and diminished, and that the process never comes to a point where further increase or diminishment is unthinkable. You may go on for a lifetime multiplying a large number by itself, and the result by itself, and so on; you may bequeath the task to your heirs and assigns; the work may run through centuries and cover continents of paper. But, at any instant in the process of multiplication, the number is finite, and, after centuries of labor, the vast result is still as finite a number as that with which the whole process started. The point is, however, that further multiplication is always possible, and this is the sense of the term potentially infinite. Manifestly, the case is the same if we consider division of a number into smaller and smaller fractions. Number cannot be actually infinite. This being so, numbered parts cannot be actually infinite. And thus we say that the divisibility of a body is a property which renders it resolvable into an indefinite (or potentially infinite) number of parts. Divisibility is, in other words, a property of naturally existing bodies, consequent upon extension, and, normally, upon external extension.
Mensurability is that property of a natural body (consequent upon its external extension) which renders it capable of comparison with the extension of other bodies, and so discerned as greater, or lesser, or equal ; further, mensurability renders a body capable of being comparatively numbered in the extent of its divisible parts. Indeed, mensurability is seen to be a kind or aspect of divisibility itself. The noting of divisible parts in the terms of units of extension, and the numbering of such parts, is the basis of the mensurability or measurability of a bodily substance. Length, width, thickness, units of bulk or content, units of surface, weight, specific gravity,—these are familiar terms which indicate measurements (and mensurability) of bodily substances.

c) VARIETIES OF QUANTITY

Quantity in natural bodies is either continuous or discrete.

Continuous quantity is unbroken, undivided quantity. A bodily substance which has continuous quantity is called a continuum. A continuum is either perfect or imperfect. A perfect continuum has no pores or interstices or vacuoles between or among its parts. An imperfect continuum has pores, interstices, or intervals, but its own unbroken substance com- pletely surrounds them. If we suppose for the moment that a plate of polished steel is a perfect continuum, then the same plate with several small holes bored through it is an imperfect continuum.

Discrete quantity is broken or divided quantity ; it is non-continuous quantity. A drop of water on a pane of glass is, or at least illustrates, a continuum. Three drops, lying separately on the glass, but considered as one quantity or amount of water, are a discrete quantity. A grain of sugar illustrates a continuous quantity; a spoonful of sugar illustrates a discrete quantity. Discrete quantity is contiguous if the items that make it up come into immediate contact with one another ; if there is no such contact, the discrete quantity is non-contiguous or separate. A few pebbles held closely in the hand, each pebble touching one or more of the others, make a discrete contiguous quantity; they constitute @ contiguum. The same pebbles held loosely on the palm so that none of them touches any other, make a discrete separate quantity.

Each of the pebbles is a continuum, perfect or imperfect. But is it truly so? The science of physics tells us that a bodily substance is made up of ultimate particles caught up in a kind of amalgam. If the bodily substance is chemically simple, that is, if it is one of the chemical elements or chemically uncompounded substances (of which ninety-four are now recognized ), its ultimate particles are called atoms. If the bodily substance is a chemical compound, its ultimate particles are called molecules. The smallest existible particle of water (that is, the smallest body existible as water) is a molecule of water. And, since water is composed of two parts hydrogen and one part oxygen (hydrogen and oxygen being chemically simple or elemental), it is manifest that the molecule of water consists of three atoms, two of hydrogen and one of oxygen.

Now, suppose our pebbles are limestone pebbles. Limestone is a chemically compounded substance, the chief elements of which are calcium and carbon. UItimately, then, limestone consists of atoms of calcium and atoms of carbon. Would it be correct to say that each molecule of each limestone pebble is a contiguum, a discrete contiguous quantity consisting of atoms of calcium and atoms of carbon lying closely together? No; for the structure of the pebble resembles rather the sieve-like plate of pierced steel than the spoonful of sugar. Though there be interstices or intervals in the limestone substance, one can “go around the holes” without stepping off the continuous substance of the stone; and, indeed, the imponderable matter which fills up the intervals or vacuoles (improperly called so) is itself to be regarded as part and parcel of the structure of the substance called limestone. Thus we are justified in regarding each pebble as a continuum, and perhaps as a perfect continuum. Limestone is a substance with its true substantial character ; it is not a mere heap of atoms of calcium and atoms of carbon mingled together as sand might be mingled with salt; the atoms which ultimately compose limestone are substantially united to constitute a substance which is neither calcium nor carbon, but a third thing called limestone. Thus, each pebble and each molecule of each pebble must be regarded as continuous quantity, whether perfectly so or imperfectly so.

An imperfect continuum must contain in itself, and of itself, a reach of perfect continuity. Consider the plate of pierced steel,—or the metal top of a saltshaker,—as an illustration of an imperfect continuum. The substance that “lies between the holes” is not an imperfect, but a perfect, continuum. Hence, the basis of quantity in bodies is always perfectly continuous matter.

Is the atom perfectly continuous? Formerly it was universally thought to be so, for the atom, before the present century, was regarded as perfectly unified and physically indivisible. Indeed, the name atom is a direct derivative from a Greek word which means “that which cannot be cut.” But now it is known that the atom can be cut. Thus it loses its strict right to the name atom, although we continue to call it so. The atom can be, and has been, divided or split into its parts. Indeed, the atom is not only a thing made up of parts, but it has a kind of porosity, so that the sub-atomic parts are held, not in perfect solidity and compactness, but with relative looseness. The atom has its core or nucleus which consists of a particle of matter, or several welded particles, bearing an electrical charge ; indeed the nucleus itself may have some of its constituent particles charged negatively (electrons) and some positively (protons). Around the nucleus, and spatially distinct from it by a greater or lesser reach of imponderable matter, are other electrons. Thus it appears that the atom itself may be regarded as an imperfect continuum ; but it is a continuum, since protons, electrons, and imponderable matter all unite in its unbroken structure ; and indeed, so truly unbroken is this structure, that it seems more just to call the atom a perfect continuum than to regard it as an imperfect one.

Ultimately, then, whatever the future discoveries and achievements of the scientist may be, matter consists, in the quantitative aspect, of fundamental continua. And, as we have observed, continua are radically perfect continua, even though any tangible

Modern physics distinguishes a good many different particles of charged matter which may have place within the atom. These particles differ from one another in charge or in mass or in both. The particles are sometimes called, poetically, “the building-stones of the atom.” Some of these are listed as follows: electron, positron, negatron, proton, neutron, deuton, alpha-particle. But, in a briefly descriptive account of the atom, all these particles may be classed with sufficient accuracy as protons and electrons according as they bear, respectively, positive or negative charges of electricity. Perhaps special mention should be made of the neutron which has both a positive and a negative charge which balance each other. Or it may be said that the neutron lacks electrical charge altogether.

quantity of them be regarded as constituting an imperfect continuum.

Mathematically, we may carry on our division of matter to an indefinite extent. Even the minimumquantity of matter has extension and hence has divisibility. Therefore a continuum, even though it be the smallest existible amount of a material substance, is not only one in itself ; it is also potentially multiple or many. For the smallest continuum may be regarded as two halves of its quantity, and one hundred hundredths, and so on indefinitely. The physically existible minimum of any kind of matter has, consequently, a capacity for endless mathematical division into fractions or parts; it is said to have these parts not formally or as such, but fundamentally; not actually, but potentially.

To illustrate : Let us suppose that each of five slates in a blackboard is a perfectly continuous quantity. The whole blackboard, viewed as a totality or unit, is a discrete and contiguous quantity ; for the slates are so aligned that they “touch” ; each slate comes in contact with one or two of the others. But each individual slate is a continuum, and we are supposing, for purposes of illustration, that it is a perfect continuum, a stretch of substance with absolutely no intervals or interstices in its quantity. Manifestly, the slate can be divided; it can have its parts designated (as, for instance, “the upper portion,” “the central area,” “the lower left section,” “the four square inches in the upper right corner”), and it can be subjected to actual physical partition: the slate may be cut into quarters, tenths, sixty-fourths, hundredths, or it may be broken up with a hammer into thousands of irregularly shaped parts, or it may be ground into millions of tiny grains. Now, each of the parts (designated or broken off) is itself a continuum; and as such it is capable of division into further parts, each of which will be a continuum. The original and unbroken slate can, therefore, be discerned as made up of designatable parts, and the slate can, as a fact, be divided into actual parts. Therefore, even while yet unbroken, the slate may be said to have these parts in some manner. For the designating or breaking off of parts does not add anything substantial to the slate, or bring an increase or diminishment of its total original quantity. The substance of the divided parts is neither a new substance nor a new total amount of substance. Still, the unbroken slate does not have its parts formally or as such; these come with designation or actual partition. The slate has rather the capacity or capability for distinction and division into all its possible parts. In a word, it has these parts potentially, not actually.

The illustration just offered should help clarify the definition of a continuum, viz., “An extended quantity which has no parts in itself”; or “An extended quantity which has within itself no limits or boundaries of (actual) parts.” Further, the illustration should help to explain the statements: (a) That a continuous quantity is actually one and potentially multiple; (b) That a continuous quantity is made up of divisible components; (c) That the division of a continuum (or continuous quantity) results in other and smaller continua; (d) That a continuum is capable of indefinite division. Much of all this is summed up in the terse sentences of Aristotle (Physics, vi, ¢ 1; and v, c 3): “It is not possible to form a continuum out of indivisibles” ; “It is clear that a continuum is divisible into parts which are themselves divisible.”

The following points, adapted from the philosopher Lepidi and others, are worthy of note in this place:

I. In every extended reality there must be some continuous quantity. If, in natural bodies, there were no perfectly continuous quantity, we should be forced to accept one of two impossible conclusions: (a) we should be compelled to deny the actuality of extended matter (as Sir James Jeans seems to do in his The Mysterious Universe) ; or (b) we should be forced to conclude that actual extension is the product of inextension, that is, that extended bodies are made up of non-extended parts. Consciousness, experience, and reason concur to make us reject both conclusions as self-contradictory and absolutely false.

While the ultimate quantitative parts of an extended body must be perfect continua, it is plain that a complex body in its larger portions as well as in its full structure may be only imperfectly continuous. There is a kind of porosity in bodily substances. Some bodies are manifestly marked with interstices or open intervals,—a sponge, for instance, or a piece of coke or of slag,—but even those bodies in which human vision can detect no break are seldom perfectly continuous. In many cases, the microscope reveals “holes” in substances we normally regard as perfectly compact, solid, smooth, and continuous. Indeed, we have noticed that the atom itself has intervals between and among its parts. If the porosity of matter were denied we should have great difficulty in explaining the phenomena of expansion and contraction, and the absorption of one substance by another (as ink is absorbed by a blotting-pad or water by dry wood). Further, we could hardly explain the vibration of particles of matter which give rise, under due conditions, to perceptible sound, color, or heat. However, we must be careful not to confuse the truly substantial unity of a quantity of bodily substance with the mere aggregation of molecules or atoms, particularly in living bodies. A living body is a manifest continuum throughout its organic structure, a fact that is proved by its unified vital character and function. Hence, though Sir Arthur Eddington says, in The Nature of the Physical World, “The atom is as porous as the solar system. If we eliminated all the unfilled space in a man’s body and collected his pro- tons and electrons into one mass, the man would be reduced to a speck just visible with a magnifying glass,” it is pertinent to remember that this microscopic man would be just as truly a continuum, involving components in perfect continuity, as he is in his normal and natural state as a bulky human adult. Nor, indeed, is it right to “eliminate the unfilled spaces,” since such spaces are not, properly speaking, “unfilled” at all; there is, at least among the smallest ponderable quantities of a substance, an imponderable matter which may justly be regarded as “constituent” of the substance, together with the ponderable components.
Though physical science knows nothing of the so-called ‘‘open spaces” between and among the subatomic particles of matter, and pays no direct attention to them, it is certain that these vacancies are not perfectly empty. They are not vacua, in the strict sense. For a vacuum, strictly speaking, is absolute absence of all bodily reality; a vacuum is, in the material sense, a complete nothing. But the bodily world, throughout its structure, appears to be always a something, and not nothing. When we speak casually of a vacuum (as, for example, when we say that there is a vacuum under a laboratory-bell, or when we speak of “vacuum-packed coffee”) we use the term vacuum in a relative sense, not in an absolute one. We indicate the absence of atmospheric air; we do not indicate the absolute absence of all material substance. It is universally admitted that no such thing as an absolute vacuum is known within the limits of the bodily world. Scientists agree that the holes or pores or interstices in the most compact quantities of bodily matter (such as the atoms or molecules of matter) are perfectly filled with some tenuous and imponderable matter, even as the holes in a dry sponge are filled with air and those of a submerged sponge are filled with liquid. Whether this imponderable matter is properly to be called “aether” (as was the fashion until recently), or whether it is an amalgam of various unknown substances that should have a better and more accurate set of names, is a question for mere academic discussion. Whatever the imponderable matter may be called, it is undoubtedly true matter ; it is truly a bodily substance, truly extended. For bodies act upon one another, and material action requires a material medium of activity. The particles of molecule or atom adhere closely together, yet it is recognized that their ponderable quantities do not come into immediate contact. Nor can we, in our present state of knowledge, apply any force sufficient to produce such immediate contact. Between and among the most closely adhering ponderable particles there is a film of imponderable matter which is the medium and channel of the contact and adhesion. And there must always be such a medium or channel for the activities or influences which bodily quantities exercise upon one another; bodily activity cannot leap the void; it cannot be exercised across an absolute vacuum. Such phenomena as the transference of light and of sound, radioactivity, gravitation, chemical affinities, physical attractions and repulsions, indicate the truth that there are actual channels of media-of-contact among interacting bodily quantities, and that these channels or media are material, and are continuous all along the line of influence or interaction. We are thoroughly justified in the assertion that there neither is nor can be actio in distans, that is, action upon a material object by a material object across an absolute void. Just as there can be no flowing of a river without a river-bed, so there can be no flowing of material influence or activity from body to body without some material medium or channel of communication between the bodies. And the medium must be material; that is, it must be proportioned to its function which is a material influence or activity. The science of ontology (or fundamental metaphysics) discusses the question of actio in distans as to its metaphysical or absolute possibility; cosmology merely indicates the physical impossibility of bodily interaction without a bodily or material medium-ofcontact between the interacting bodies. We must conclude that our bodily universe, which manifests such close-knit and constant complexities of bodily influences and interactions, presents no true vacua to our knowledge. It appears to be strictly true that “nature abhors a vacuum.” We do not say that no tiniest vacuum could exist within the limits of the material universe ; we say that, so far as we can make out, no such vacuum does exist. It might be possible to defend some theory of vacuum-intervals which would be skirted by corporeal action and interaction, and this might be done without any appeal to current theories of curved space or bent light. But no such theory is exacted by the phenomena observable in the world. It is the general agreement of philosophers and scientists that vacua do not exist in the world, or, at least, that they are utterly unknown.
Our bodily universe is a vast contiguum made up of a multitude of bodily substances which are, at least in their essential existible elements, respective true continua, perfect or imperfect. Each living body is a perfect continuum throughout its actual organic structure. Each quantity of non-living substance appears to be an imperfect continuum, with its ultimate quantitative components perfectly continuous.

d) SPACE AND TIME

Quantity means extension, first internal and then external extension. Now, the quantified world,—that is, the world of natural bodies,—is characterized by external as well as internal extension. For a body to exist in a purely natural manner, external extension is a requisite condition, even though in itself it is a secondary effect of bodily substance. A natural body, then, has external extension. And by virtue of this a body occupies a place. Now the sum-total of all places,—or, in other words, the sum of all actual external extension,—constitutes real space.

We might put the matter the other way about and say that space is the extent of the bodily universe, and that each place is a part of space. But, whether we describe space as the sum of places, or say that space is the unit of which places are fractions, we mean that space is the whole sweep of extension included within the boundaries of the existing bodily world.

An externally extended body is always characterized by change or motion. The world about us, and all bodily substances in it, are constantly “on the move,” not only in the sense of local movement (although this type of motion is universal and continuous, and we have instances of it everywhere, from speeding stars and galaxies to swirling atoms and electrons) but in the sense of mutation or change. We have already seen, in our study of the outstanding characteristics of bodily substances, that these are always mutable, and that they may be described with accuracy as entia mobilia or “mobile beings.” Everything in the bodily world evidences a procession and succession; things come into actuality and pass on to new actuality; bodies act and interact, running through a ceaseless series of influences received and influences imparted. Motion or change is a constant phenomenon in the world of bodies. Now, motion or change is essentially a matter of succession (not in the sense of gradualness as opposed to instantaneousness, but in the basic sense of one thing,—being, state, condition,—and then another), and even in things that we speak of as lasting or enduring we find a succession of items, points, elements, or instants of lastingness or endurance itself. In a word, the world is a world marked universally by motion. And, since motion necessarily involves this state or condition and then that, it is a thing that can be somehow measured or numbered. The mensurable or numerable motion in the bodily world is the entitative basis of the thing called real time.

The description of real space and real time just given should be diligently studied as a preliminary to the more detailed discussion we are now to undertake.

I. Space is, as we have said, the actual external extension of the bodily world viewed as a whole. Thus it is something real. But the human mind inevitably makes its own contribution in forming the concept of space, and regards it as a kind of receptacle, as a sort of container of all bodies and as the field which encloses bodily movements. Literally, however, space is not a container, for a literal container is always really distinct from the thing it contains, just as a sack is distinct from the sugar that is in it, or as the glass is distinct from the wine which it holds. But space, while it is thought of as holding or containing all bodily extension (that is, external quantity) is not really distinct from that extension; it ts that extension regarded in totality.

Immanuel Kant (1724-1804) stressed too forcibly the mind’s contribution to the concept of space, and came to deny all reality to it. He said that space is all in the mind; that it is an inborn “form” of the mind, a kind of mental groove through which a person is forced to pour the findings of the senses. But, as Kant’s followers quickly demonstrated, one cannot deny all reality to space without soon coming to a denial of the substantial reality of the world itself. “That way madness lies” ; the madness of skepticism, which is an utterly impossible doctrine.

Modern physicists like Albert Einstein (1879- ) and Arthur Stanley Eddington (1882- ) find difficulty in the objective concept of space, but their difficulty is bound up with the actual measurement of space, which does not concern the cosmologist at all. It is not space, but distance (which is a partial space) and the difficulty of defining distance in extent and “shape,” which has upset, to a considerable degree, both Euclidian measurements and Newtonian physics. Now, the physicist, like every scientist, is forced by his very nature as a rational being to step across the frontiers of his science and invade the field of philosophy. Unfortunately, most scientists, while rightly sticklers for accuracy (“religious in it”) in their proper domain, are given to the loosest sort of generalizing as philosophers, and do not seem to be conscious of their presumption in tossing off definitions that do not define and in making conclusions that do not conclude. Thus Professor Eddington, that wholly admirable scientist, bogs down when he philosophizes on space. He says (in The Nature of the Physical World, p. 13), “Space is an empty void; or it is such and such a number of inches, acres, pints.” Of course, space is nothing of the sort. It is not an empty void, else the world is not here; in which case, Professor Eddington is not here, and his statements are not made. Nor is space “‘such and such a number” of units of measurement. Space (that is, real space) is the actual extension of the material universe. If our units of measurement are relative things ; if they hold no absolute value ; if they give us faulty notions about the size of things or their distances from one another, it by no means follows that the reality we try to measure by their means is not a reality at all, or that there is no size and no distance to be measured. The cosmologist is not concerned with measurements as such, while the physicist and the mathematician undoubtedly are concerned, and properly so, with these things. What the cosmologist asserts is that the universe is extended, and that it is finite. In these fundamental points the cosmologist finds, with much happiness, that the physical scientist perfectly agrees with him. He knows that the bodily world is extended, for he cannot otherwise avoid the self-contradiction of skepticism, and the imbecilic silence which it imposes upon all theorists, scientist and philosopher alike. And he knows that the bodily world is finite, not infinite, because it is made up of finite realities, and the sum of finite things can never reach to actual infinity. But when the physicist says, “We have different frames of space to which we refer the location of objects. The frame of space used by an observer depends only on his motion. Observers on different planets with the same velocity will agree as to the location of objects in the universe, but observers on planets with different velocities have different frames of location,” the cosmologist answers, “Go your way in peace, and God be with you. We have no quarrel. But our roads part here. Go on, follow your own path, do your own work, and may success attend your efforts. I, meanwhile, must be getting on with mine.” For whether the bodily universe be capable of accurate measurement or not so capable; whether the universe be expanding or contracting or holding to a constant size; whether bodies in the universe can be absolutely located or only relatively placed; whether distances between bodies, and rates and directions of moving bodies, can be determined with rigid correctness or are incapable of such determination, it still remains a fact that the bodily universe is here, that it is extended, that it is finite. Now, the finite extent of the universe (whether it changes or remains constant) is the cosmologist’s concept of real space.

Real space, then, is the relation of extent in, between, and among actually existing bodies. If we conceive of this space as the actual container of all bodies, we are well aware that we do so as a matter of mental convenience, and that space, in this view, takes on the character of ens rationis or ens logicum (that is, of rational or logical being as contrasted with real being). We do not deceive ourselves, and project our concept of “‘space the container” into the world of nature as though it were actually there like a great bag full of stars. No, we are clear upon the point that real space is the relation of extension (not the measurement of extension) among actually existing bodies, and that it is coterminous with the external limits of these bodies. So space, while a logical being inasmuch as it is viewed as a container, has its basis in reality; it is an ens rationis cum fundamento inre. .

Ideal space is the concept of possible space. It is the intellectual grasp (not the attempted picture in imagination) of the fact that the actual limits of the bodily world might be indefinitely extended. The world has its limits, and these are the limits of real space; but the Creator might create any number of new worlds, thus expanding the limits now really imposed. The idea or concept of such a possible new expanse of space, is the idea or concept of ideal space.

Imaginary space is the space which fancy pictures as extending beyond the limits of the actual bodily universe. Imaginary space is manifestly not to be confused with either ideal space or real space. Yet such confusion is not infrequently found, even among men of real prominence in the scientific world. Sir James Jeans (1877— ), for instance, has this to say, in The Mysterious Universe (p. 166f): “Anyone who has written or lectured on the finiteness of space is accustomed to the objection that the concept of a finite space is self-contradictory and nonsensical. If space is finite, our critics say, it must be possible to go out beyond this finite space, and what can we possibly find beyond it but more space, and so on ad infinitum?—which proves that space cannot be finite. And again, they say, if space is expanding, what can it possibly expand into, if not into more space?—which again proves that what is expanding can only be a part of space, so that the whole of space cannot expand. The twentieth-century critics who make these comments are still in the state of mind of the nineteenth-century scientists; they take it for granted that the universe must admit of material representation. If we grant their premisses, we must, I think, also grant their conclusion—that we are talking nonsense—for their logic is irrefutable. But modern science cannot possibly grant their con- clusion; it insists on the finiteness of space at all costs. This of course means that we must deny the premisses which our critics unknowingly assume. The universe cannot admit of material representation, and the reason, I think, is that it has become a mere mental concept.” This lengthy citation calls for comment on several points, and we may as well list these under numbers:

(1.) Sir James Jeans, mistakenly judging the logic of the critics as irrefutable (whereas it is really nonexistent) is like a timorous man who rolls under the bed to escape purely imaginary burglars, and then calls out loudly, “Go away; there’s nobody here!” In terror because of a little meaningless noise, he is ready to deny his own reality and reduce himself to the status of a mental concept—in whose mind, one wonders?

(2.) Sir James is correct in affirming the finiteness of space, that is, of real space, but he loses his right to be correct when he assigns the mere stubbornness of scientists as the reason by which we must regard space as limited. Further, he cannot logically hold to material finiteness, when he asserts in the same breath, that the world of space is not itself a material thing, which must be the basic meaning of the phrase ‘“‘does not admit of material representation.” For a material thing can be materially represented. But that which admits of no material representation surely may, for all we can know, be actually infinite ; indeed, the presumption must be that it is actually infinite.

(3.) Sir James thinks the logic of his critics irrefutable. That is because he, like the critics, makes a muddle of the concept of space, and mixes up real space with imaginary space and ideal space. The critics say, “If space is finite, it must be possible to go out beyond this finite space, and what can we possibly find there but more space, and so on ad infinituum ?—which proves that space cannot be finite.” It proves nothing of the sort. If space is finite (and real space, being the actual extension of real bodies, is certainly finite), it is possible to go out in thought or in fancy beyond the limits of space. Granted. But the thought of space beyond the real limits of the universe is merely the thought that further extension of these limits is always possible, and this is true; this is a question of idcal space. And the fancy or imagination-picture of space as extended beyond its finite and real limits to farther reaches (still with limits) is a matter of imaginary space. What really lies beyond the actual limits of space (that is, beyond the actual limits of all bodily creation) is nothingness, that is, nothingness in the material sense. The mind can understand this, but the concept of it cannot be made with the adequateness and clarity of positive concepts; for we only get at the idea of nothing by removing the idea of everything, and there is admittedly a mental difficulty in conceiving absolute negation. Yet the mind can have a sufficient understanding of material nothingness to know that such a thing can be, and even that such a thing must be. The mind that can conceive of a limited vacuum within the confines of the material world, cannot be said to be powerless to represent an unlimited vacuum outside it. If the mind confuses the mere possibility of farther space (beyond the confines of the bodily world) with the actuality of space; or if the mind takes the imagination-picture of extended space as a real extension of space, then we have not “irrefutable logic” but only muddled thinking. And of this sort of thinking, Sir James Jeans and the critics he quotes must, in all charity, be flatly accused.

(4.) If it be said that Jeans, in declaring that the universe does not admit of “material representation,” is only saying that we cannot form a wholly adequate and comprehensive concept of the actual extent of the material universe; or if it be said that Jeans means merely that the extent of the world cannot be set down in definite measurement-units, and that the effort to express the size of the world in cubic miles or cubic inches is always bound to be a failure; then we are prepared to assent, and to say that Sir James is right. But such an interpretation of his words appears to be an unwarranted expanding of the limits of charity, and even of common logic. From the terms of his expression, it appears that Jeans is not far removed from the position of the idealistic pantheist who makes the material universe only an unfolding concept (or image or dream) in the mind of the Deity.

(5.) The concept of finite real space is no more self-contradictory and nonsensical than the concept of a finite real elephant, or of a sparrow, for that matter, or an amoeba. But if I allow myself to become confused about the actual quantity of the elephant and its possible quantity, or if I permit myself to become foggy about the real elephant because I persist in imagining him to be much bigger than he is, then I may rightly conclude that it is nonsensical (in my state of befuddlement) to attempt to say just where the limits of the elephant are, or even to assert with certitude that he has any limits at all. Ideal space and imaginary space are potentially infinite, or indefinite,—which only means that I may go on thinking the elephant larger and larger, and may go on imagining him bulkier and bulkier, and I never reach a point where I must stop my thought or my fancy, even though my ideal or imaginary elephant should block out the sun with his head and scratch his back against the remotest stars. But real space is definitely finite—which only means that the real elephant has his real quantity or extension, no matter what my thoughts or imaginings about him may be. And if the universe is expanding, this can be no more puzzling than the fact that the real elephant is growing. He does not lose finiteness or even dimensions by growing; only his dimensions are progressively larger. There is no occasion for wonder or worriment about what the universe is expanding into, if it is expanding. It is expanding into nothing; it is increasing its size; it is getting bigger; it is widening its outer limits. But it has, throughout the process, an extension that is definitely finite, and, had we the instruments to measure it, and a stable position in which to apply them, we could measure the universe at any instant, and express it in terms of measurement. To say that the expanding of the universe disproves its finiteness, is to say that the baby-elephant is infinite because he is growing bigger.

(6.) The critics mentioned by Jeans are talking nonsense and are guilty of self-contradiction in their assertion that an expanding universe “can only be a part of space,” which, in their view, is infinite. For the concept of a part of infinity is a simple absurdity. What has parts is finite, and necessarily so. If infinity could have parts, then (a) either these would be finite and their sum-total would be finite; thus the infinite would be both infinite and finite at the same time—a neat contradiction: or (b) the parts of the infinite would be themselves infinite, that is, each of the parts would be infinite; thus each part would be equal to the whole; nay, each part would be identical with the whole, since a plurality of infinities is impossible. This would mean that the bodily world and each of its parts would be infinite, and each part would be infinitely indentified with each other part. Each stone and tree as well as each galaxy and nebular mass would be infinity, and would be the same infinity. In other words, objects which are admittedly finite would be also infinite,—again, we have a nice contradiction in thought and terms. And these objects would not be individual objects at all, but all one and only infinite object,—and this plumps us right into a world of illusion, a negation of the very reality with which the critics come to such confident grips.

(7.) That the bodily world is of tremendous size, no one will deny. But we must not be overpowered by mere size. A sight of the lordly Alps does not distress the school-boy to such an extent that he is unable to pick up and measure a stone; nor does it befog his mind to the truth that enough stones of the size he can handle would make a heap the size of the Alps. After all, size is truly a relative thing. Professor Einstein, with his doctrine of relativity, has upset many minds and many theories; yet he is far from teaching that there is no absolute value in anything material; he asserts the existence of certain “invariants” or absolutes; he does not make everything relative. And although we cannot go along with his philosophy we cannot brusquely deny the value of much of his scientific work. But, long before Einstein, the human mind recognized many things as relative, and size is just such a thing. If there were only one bodily object in existence, it would be neither big nor little; it would have no size. You must have at least two things to compare before you can speak of size, even if one of the things is the Matterhorn and the other a foot-rule. Bishop John Vaughan says (somewhere in Faith and Folly) that if the material universe were suddenly contracted in size; if there were a general reduction in strict proportion throughout the world, so that, for example, everything from the remote heavenly bodies to the ash-tray on the desk were reduced by half, there would be no means available to the human mind for knowing that a reduction had taken place at all. Nay, if the whole universe were reduced to such a size that it could be enclosed within a tea-cup,—keeping the reduction strictly proportionate in every detail,—no human being would notice any change in the world at all. Everything would go on precisely as it goes on now. And there would be no change whatever in our true concept of real space. Real space, then as now, would be the actual extension of the bodily universe, neither more nor less. And all the scientists would be busy, then as now, in computing the “light years,” in millions and billions, required to bring us the light of those stars out near the rim of the tea-cup; other scientists, then as now, would be peering through microscopes, and working with expansion-chambers, to find out the internal economy of the myxomycete, and to observe the activity of electrons. No one would be conscious of any change, because the change took place all along the line, in strictly proportionate reduction of everything. And philosophers, then as now, would labor over the difficulties of disentangling, and aligning true relations of natural philosophy and natural science, cosmology and physics. But, clear of all entanglements and doubts, philosophers, then as now, would assert the ringing truth that real space is the actual extension of the bodily universe, whether this be “big” or “little,” expanding or contracting or standing constant.

Time, as we have seen, is entitatively based upon the fact that there is motion in the world, and that one motion can be compared with another, and numbered in terms of the other. (Similarly, the size of things is a matter of comparing one with another, and of taking one size as a unit in which to express the size of other things.) But, while the basis of time is real motion in the material world, the complete concept of time involves the mind of man making comparison of motion with motion, and measuring one by another. So also with size. If one little block of wood rests upon a larger block, these are not measured until someone notices how many times the smaller block can be placed on the surface of the larger until it has rested on all the completely separate, yet contiguous, area-spaces of the larger: only thus, for instance, could it be discovered that the smaller block has an edge that must be set to the edge of the larger eight times before exhausting its length; then if we call the smaller block an inch on each edge, the larger is eight inches on each edge, and the area of any face of the smaller block is one square inch, and the larger (expressed in terms of the smaller, as of a standard) is sixty-four square inches on each face or surface. Measurement, which gives size, is a matter of an intelligence making comparison and of adopting one object, in any or all dimensions, as a standard for expressing other objects in their corresponding dimensions. Thus with time. It is a matter of intelligence noticing motion or movement, and of laying hold of some regular movement as a standard by which to measure other movements. Hence, the ancient saying is true: “If there were no mind, there would be no time.” Still, as we have seen, time is not wholly a projection of the mind (an ens rationis or ens logicum, simply) ; it has an entitative basis in real motion in the real bodily world; it is an ens rationis cum fundamento in re, that is, as a measure of motion, it is “a logical being with a foundation in reality.”

Immanuel Kant, as with space, erred in his concept of time. He made time a form in the knowing-power of man, a kind of mould or groove, through which the findings of the senses are necessarily received and by which these are “conditioned.” Modern physicists often make time a “fourth dimension,” and distinguish in bodily objects not only the directions or dimensions of up-down, right-left, forward-backward, but they add the direction of before-after. It is a convenient device for science to deal with a fourdimensional universe; just so, it is a convenience for the statistician to multiply a thousand men by the hours each works each week, and so present a compact account of 40,000 ‘‘man hours” of labor as the week’s tally for that group. But, after all, men and hours are different things. Whatever the physicist may find useful is surely within his right to employ ; but he must not turn muddled philosopher and declare that his device for practical simplification represents the inmost nature of things. There is much in the current talk of time as a dimension to suggest the loose descriptions one reads in newspapers about “visible sounds.” Sound, of course, is audible, not visible. But the effect of sound upon a vibrating surface can be recorded electrically, and this, in turn, can be expressed visibly; but the fact remains that the sound as such is never visible. When the visitor to Radio City is told to speak or sing into a microphone, and to watch the movements of lines of light in a box placed conveniently for his entertainment, he is told that he ‘‘actually sees his voice.” But, as a fact, he doesn’t. He sees the lines of light, and these are affected in their movements by the vibrations which his voice sets in motion; that is all. He sees what is visible, not what is audible. But the convenient and startling phrase “you see your voice” is intriguing; it is good “salesmanship”; it has the attraction-value of modern advertising. But it leads the unwary to muddled thinking. This, in some measure, is also the case with time as a dimension or direction. Of course, the box that stands in the corner is not only two feet long, one foot wide, and one foot deep; it is also in the corner now. But the dimensions of the box, and the time of its existence or its measurement, are in different categories of things.

Space and time have something in common: they are quantities. Space is a matter of what is called permanent quantity (that is, all the quantified reality is present at once), and time is a matter of flowing or successive quantity (that is, its parts follow one another into being in a continuous series that carries one in as another is carried out and none lingers). In passing, it should be noted here that space and time, being quantities, are, in their literal meaning, strictly confined to the universe of material things. When we apply the terms of space or time to things non-material, we use these terms in an extended or analogical way.

Three elements enter our concept of time: past, present, future. The present, that is, the now, is an indivisible instant. If it could be divided (which it cannot) it would fall into three parts,—part would be past, part future, and between them still would lie an indivisible now. Thus no further division of time, beyond the threefold division mentioned, is even thinkable. The now is indivisible and cannot be confined or captured, for even as we name it, the now becomes then—what was the present moment is aligned with the past. Our concept of infinity, of boundless perfection boundlessly present, is thus seen to exclude time; an infinite being is necessarily eternal, without past or future, without flow of time. God is outside time; He is timeless, eternal. But our concept of finiteness is inescapably bound up with time, and even the endlessness of what we call the “eternal life” for which we hope, does not include beginninglessness, nor the simultaneous possession of boundless life, which is the definition of eternity in strict sense.

We may describe time as the spread or extension of an unbroken series of motions. Further, our mind persists in regarding time as a measure of these motions. The manifest movement which we call,— unscientifically, and indeed incorrectly, but with sufficient practicality—the “movement of the sun across the heavens,” was taken at a very early date in human history as a standard movement. It is a regular movement, dependable, recurrent. It was but natural that men should use it with reference to other movements, changes, events. It was inevitable that man’s inventiveness should produce mechanical devices to indicate regular portions of this regular movement. Yet the measurement of time is not the essence of time. This essence is in the actual movement and in the mind’s recognition of the movement as progressive, as not all there at once, as consisting of what has gone before, what now is, and what is to come. This essential time is called intrinsic time. Time expressed in terms of measurement; time that we read from our watch-dials; time that we call “lunar” or “sidereal” ; all measured time, is called extrinsic time. Extrinsic time is either general (as solar time, sidereal time, lunar time) or particular (as hours, minutes, seconds).

But the most important division or classification of time (as of space) is that which distinguishes it as real time, ideal time, imaginary time.

Real time is the actual succession (in unbroken series) of movements, events, changes,—in a word, motions,—which a finite intelligence can recognize in the material world. Ideal time is the mind’s understanding of the possible duration of real time. Imaginary time is the duration of successive motion as depicted in the human fancy or imagination.

Real time is necessarily finite. Since it is (in strict literalness) referable only to material motion in a material world, and since the material world is necessarily a world which had a beginning, time had a beginning. That is, time had an initial boundary, and is not infinite. Further, while the material world may be thought of or imagined as indefinitely enduring, as “reaching on unto eternity,” this is a matter of ideal or of imaginary time, not of real time. Even so, such time is only potentially endless, for each instant of it finishes it “so far”; each instant is a point, a now, a then, a limitation. Even should the material universe with its time-movements be continued endlessly in existence, each instant of it would be a point for beginning or ending measurement; thus, while relatively or analogously eternal, it would still be finite. Indeed, more noticeably than in the case of space, time is a matter of parts (of numerable successive motions) ; and what has parts is, by definition, a thing finite and limited.

To certain philosophers, like Stephen Alexander (1806-1883) and A. N. Whitehead (1861- ), the world is a single reality called space-time. What we know as bodily substances in the world are merely events which evolve out of the bosom of space-time. Thus the substantial character of the world is denied. The space-time philosophy (not the science with time as fourth dimension, which is sometimes referred to as the theory of space-time) is a lamentable development of Einstein’s relativity. It tells us that we move in a world of illusion; it sets our feet directly in the path to monism and idealistic pantheism. It spells futility and silence; it is the suicide of philosophy and of science. Consider the dismal summary of this situation, as it is presented in the words of Sir James Jeans (The Mysterious Universe, p. 135 f) : “A soap- bubble with irregularities and corrugations on its surface is perhaps the best representation, in terms of simple and familiar materials, of the new universe revealed to us by the theory of relativity. The universe is not the interior of the soap-bubble but its surface, and we must always remember that, while the surface of the soap-bubble has only two dimensions, the universe-bubble has four—three dimensions of space and one of time. And the substance out of which this bubble is blown, the soap-film, is empty space welded on to empty time.” When more meaningless words are to be coined, we may count upon the scientist-philosopher to coin them!

SUMMARY OF THE ARTICLE

In this lengthy Article we have studied the meaning of quantity in a substantial and bodily world. We have distinguished internal and external quantity or extension, We have seen that quantity is not to be identified with bodily substance, but that it is a requisite condition for the existence of bodily substance, and is formally constituted by the substance itself. We have discerned the essence of quantity in internal extension. We have listed and discussed the properties of quantity, viz., external extension, impenetrability, divisibility, mensurability. In this discussion, we have studied place, and the different ways in which a reality may be located ; we have considered