[Full Table of Contents]
11. Place and Space
11.1 Place as Substance or Inherent Property
11.2 Place as Matter or Form
11.3 Place as Extension or Limit
11.4 Criticism of Place as Limit
11.5 Problems with Aristotelian Place
11.6 Place and Space as Relational
11.7 Revised Account of Place, Space and Position
12. The Void
13.1 Existential Problems of Past, Present and Future
13.2 Time Distinguished from Movement
13.3 Direction of Time
13.4 Plurality of the Present
13.5 The Problem of Simultaneity across Space
13.6 Time as Measure of Existence and the Mobile
13.7 Unity of Time
Location or place is closely bound to our notion of physical reality. If something is physically real, it is somewhere, and this somewhere is real. We do not expect a fictional character to be in a physical location, while on the other hand, if you assert that Bigfoot is real, you are expected to point to some definite place it inhabits. A sensible substance is discernible as actually existent by virtue of being in some determinate part of space, thereby distinguishing itself from other sensibles.
Those who think sensible or physical reality is the only reality find it impossible to believe in ghosts, spirits, souls or deities precisely because these occupy no location, and therefore are outside of sense. Yet we have noted previously that physics is necessarily an incomplete account of reality, and there is no logical obstacle to the existence of things beyond the sensible world, having no location. For example, our own rational thoughts considered as thought (as opposed to their neural representations) have no location. Fascination with locationless ideas and concepts motivated Plato to envision a bodiless reality beyond the world of sense. Space, in his view, was that which participated in universal ideas to instantiate them as individual objects.
Seeing that place or space is so closely bound to the sensible aspects of the world, it is incontrovertible that this concept is relevant to physics, and not merely a metaphysical concern. It serves as a sort of background or anchor upon which we may detect the motions of sensible substances. As the ground of physical reality, we expect it to be real, but in what sense? Is it a substance, or a property or relation of such? Does it have an existence distinct or independent from the objects that occupy it? Above all, we wish to consider the physical relationship between space and mobile objects. As such, we cannot be content with a purely mathematical description of space, but need to consider it as something physically real.
The Scholastic distinction of locus, situs, and spatium is useful for keeping our discussion of place and space unconfused. Locus is our intuitive notion of “place,” i.e., the simplest answer to the question, “Where is X?” We will not yet specify whether the locus ought to include the interior, as in mathematics, or if it should be considered only as the boundary or limit. In either case, locus is that which makes an object “here” rather than “there.” It need not be definable absolutely, but at least within some specified frame of reference. Since we are dealing with physics, a locus necessarily contains or has a finite volume.
Situs is the relative position of parts of an object (i.e., how it is situated or oriented) within its given locus. For example, a cube painted a distinct color on each face may be oriented twenty-four different ways in the exact same locus. It is at least debatable if situs ought to be differentiated when it entails no difference in physical state. If a cube’s six sides are all physically identical, perhaps its situs is irrelevant to physics. Even then, the situs might be useful if we were keeping track of how many physical rotations of the cube have been performed. Situs corresponds to what Aristotle called keisthai (lie) in the Categories, commonly interpreted as “position,” and we remarked in the Introduction to Ontological Categories, Part V that position is really a type of relation among substances or their parts.
Spatium or space is the extension between some definite limits of place. In one dimension, it is the extension between two points. In two dimensions, it is the interior of any figure defined by a closed loop. In three dimensions, it is a volume enclosed by a surface. In all cases, spatium is defined with respect to some locus or loci, so it may be considered a relation of place. In the late sixteenth century, spatium took on a different meaning, that of a locus internus, i.e., the collection of all places within some boundary. While this is geometrically coextensive with spatium in the old sense, we are here no longer considering spatium merely as extension, but also as a locus (whence the modern mathematical term). When Descartes, Newton and Leibniz spoke of spatium, they really meant a kind of place, a “where,” rather than a substantial thing.
Since physics deals only with finite magnitudes, physical space must always be finite in extent, and thus should have finite limits at a given point in time. This need not be a visible surface boundary, if we allow that space could be non-Euclidean and curved into itself. An absolutely uniform, infinite space belongs strictly to mathematics, not physics, since such a space would provide no basis for distinction in location. We can only know that something is “here” if “here” has a limit, so that there is some other place that is not “here.” Place must be differentiable, at least in a given frame of reference, in order for it to fulfill its function of differentiating instances of substance.
Where am I? “I am in Boston.” This is not taken to mean that I am coextensive with the city of Boston, but rather that the locus of that city contains me. Yet if I say, “I am here,” and this is interpreted in the strictest sense of exactly where I am, we would take that to mean a space coextensive with my body, or the boundary of such. That space or its boundary may be called my “proper place,” while a larger containing place, such as Boston, North America, etc. may be called a “common place,” since it is shared with other bodies.
Since place pertains to corporeal objects (or their higher or lower-dimensional analogues), the question of where I am can only answered by reference to my body’s location. My thoughts per se (as opposed to their neural representations) have no corporeality, hence no location, so they may be said to be in my body’s location only incidentally, by attribution.
The physical reality of place or locus must be distinct from that of the substances that occupy it, for otherwise it would be senseless to speak of one substance physically displacing another, as when water poured into a vessel displaces the air that was once there. The very notion of displacement requires specifying that water now occupies the place (in a given reference frame) where the air once was. If there were only water and air, without place as a tertium quid, air could be substituted by water only by substantial transformation, never by displacement. Accepting that place or locus has a physical existence distinct from the substances that occupy it, there remains the question of whether it ought to be considered something completely separable from that which it contains. Further, we may ask if place has a substantial existence, i.e., not intrinsically dependent on the existence of some other entity. If instead we judge it to be an ontological accident, we may ask if this is an intrinsic or extrinsic property of some sensible substance, or if it is a relation between substances or other accidents.
A locus cannot be simply identical with the corporeal substance it contains, for then it would be impossible for that substance to change place, and it would be senseless to speak of local motion. If we grant that all sensible substances are capable of local motion, then it would follow that place cannot be a sensible substance. This means it is either an insensible substance or an accident.
If place is an accident, it cannot be one whose existence is intrinsically dependent on any determinate mobile object. If it were so linked, then this so-called place would cease to exist when the physical object is destroyed. Even if physical substances were indestructible, this so-called place would move along with its object, making it useless as a reference for the motion of that object. To account for the reality of local motion, we should dispense with this mobile pseudo-place as a vacuous intermediary, and instead seek the places where the pseudo-place moves to and from.
Still, a locus might be existentially dependent on a mobile object contained, perhaps not any particular object, but whatever object it happens to contain at the moment. If there is such a dependence, of what sort could it be?
Might it be the dependence of a part on its whole? Yet if that were so, a place would be as inseparable from its occupant as a part from its whole. We could save the thesis only by saying a place becomes first a part of one thing, then another, depending on whatever occupies it. Since any kind of substance can occupy a given place, place could only be “part” of the most generic substance, in which case it is not really a part, but prime matter, as Plato seemed to suggest.
To be an intrinsic property of corporeal substance, a place would have to be “in” (i.e., contained by) its corporeal occupant, rather than the other way around, which may seem plausible since, after all, a body and its proper place are coextensive. Yet if the body were truly the physical receptacle of a place, it follows that the place would move wherever the body moved, just as wine is carried along with its bottle. Such “place” would be useless as a reference frame for motion, which is precisely what we seek when we speak of place. We might save this thesis by making the receptacle of place whatever body happens to coincide with it at the moment, so that the place is first in one body, then in another. This bizarre account of local motion would make place move from substance to substance, rather than substance moving from place to place. Yet we would fail to account for local motion in this way, since substance as such contains no data about location. We would need to know the positional arrangement of places, so again this requires some aspect to the reality of place that is not intrinsically dependent on corporeal substances.
From the above considerations, it seems that place cannot be a corporeal substance nor an intrinsic part or property of any such determinate or indeterminate substance. This leaves the possibilities that it is an insensible substance, or an extrinsic or relational accident of substances.
Another ontological possibility to consider is that place might not be a physical existent, but one aspect of a physical existent, such as its matter or form.
Plato, who identified place with space (extension), seemed to regard space as material when he defined it as that which participates; i.e., that which instantiates some universal. I have said elsewhere that space is the medium of pluralization, which is intelligible regardless of where we stand on the reality of universals. Yet space need not be the only medium of pluralization, if we admit that numerically distinct physical objects, or at least properties, can occupy the same place. It is the medium of pluralization for corporeal objects, allowing there to be numerically distinct members of the same genus or species, which is impossible for incorporeal objects. If there are more dimensions of pluralization than the three ordinary spatial dimensions, we might consider these other dimensions as defining a sort of space, as physicists do when they speak of “phase space.” Nonetheless, we will confine our discussion to space in the ordinary sense of length, breadth and height.
By Plato’s account of space as participator or instantiator, it would seem that space is a material cause. Indeed, it would seem to be the sole material cause, in agreement with Descartes’ definition of matter as extension. This might be a fair definition if we regard space as the dimensions of a particular object, but not if we regard space as place, for place must be separable from its occupant, while matter is an inseparable principle of substance. The same objection holds for characterizing space as the form of corporeal substance.
We might salvage the Platonic thesis by saying that a place, though not the matter of any determinate thing, might be the matter of first one thing, than another, depending on what occupies it at a given moment. If place is physical material, and therefore extensive (so place = space), it may, without contradiction, adopt different forms, becoming different kinds of substance. In this case, what we perceive as local motion would actually be the transformation of different parts of matter, which itself remains in place. Space or matter (one and the same) would be a medium through which forms propagate like waves. If this were true for all local motion whatsoever, then when I seem to move my hand from left to right through water, what is really happening is that different parts of matter are being transformed from water into my hand and from my hand into water. Yet this is clearly not what occurs, as the motion of water around my hand can be seen and felt.
Further, recall that matter is supposed to be the persistent aspect of a substance, which may adopt different forms. It therefore seems senseless to say that space is the “matter” of whatever sensible substance occupies it fleetingly, since there is nothing persistent about such matter with respect to the substance in question. If space is matter, it can only be the matter of some substance distinct from the subjects of sensible motion. Otherwise, we would deny that any sensible substance can really change place, which is to deny the basic testimony of the senses and leave physics without any subject matter.
If place has the function of “containing” a sensible substance, making it existent here rather than there, this alone would suffice to make it different from matter. The bronze of the statue is no more nor less truly the matter of that substance if the statue moves from here to there. Thus place, qua containing, is different from matter, even if we define place to include space, i.e., the interior extension. It belongs not to matter, but to form, to contain or impose definite limits on a substance’s existence.
It might seem, then, that place should be reckoned as the form of some sensible substance. After all, the proper place of a thing, considered as its boundary, contains the thing primarily and per se, as opposed to a common place, which contains it per accidens, i.e., by virtue of containing some larger thing which contains it. Thus the city in which I reside is not what contains me per se, but rather the outer boundary of my body does so. Yet if place is that which delimits or bounds a substance, and form is likewise the boundary of a thing, does it not follow that place is form?
As Thomas Aquinas remarks [Comm. on Physics, 424], such reasoning involves the fallacy of the consequent, i.e., a syllogism in the second figure with two affirmative premises.
A is B (Proper place is the boundary of a thing)
C is B (Form is the boundary of a thing)
Therefore: A is C (Proper place is form)? Non sequitur.
The same genus may be predicated of two species without the two being identical. In this case, proper place and form are two different kinds of boundaries. Form is the boundary of matter, the latter being substance stripped of all its determinate qualities. Matter “has the nature of the infinite,” i.e., that which is without limit, since without form it would be limitless. In fact, matter cannot exist without form in the sensible world, where we always find finite magnitudes.
The form of a substance can be more than a mere spatial boundary, since it can bound or define qualitative properties or relations as well. Still, let us consider only a specific kind of form, namely the geometric form or shape of a corporeal object, since that alone resembles place. There is no question that shape is a form; indeed the primary sense of morphe is “shape,” from which the concept of substantial form is generalized to other categories. Even so, a geometric boundary or shape has physically distinct roles when considered as form or as place.
As form, a boundary is that which prevents the matter from exceeding a certain quantity or bulk (volume), and defines the internal relations among its outer limits (e.g., the angles between sides of a polyhedron). An object of a given form could change its locus end even its situs, yet retain the same form (shape), proving that form is not the same as place.
Place is what prevents an object from being there rather than here, abstracting from its size or internal geometric relations. The spatial boundaries of a proper place coincide in location with the external boundaries of an object at a given point in time. The boundary of place has a physically distinct function from the boundary of form. Form is intrinsic to substance, while place tells us where the substance as a whole stands in relation to external objects.
Suppose that place were not the form of any determinate substance, but of whatever substance happened to occupy it as long as it occupies it. All kinds of substance may occupy the same place at different times, in which case the “form” would do nothing to specify what kind of substance is there, making the form absolutely generic and indefinite. This is utterly contrary to what we mean by substantial form, being more akin to matter. It might be that place is one of many characteristics of a form, in which case we would have to say a substance becomes a different kind of substance by virtue of being here rather there, for every different place. This would be a complete repudiation of the notion that physical substances obey universal laws, having the same nature regardless of their location. The idea that a substance’s nature changes with every change in location (relative or absolute) is irreconcilable with what we observe, and would make general physical explanations impossible.
If we allowed that place could be the matter or form of a sensible substance, we would also have to allow that place can have a place. Wherever a substance moves, its matter and form must go with it, since these are actually inseparable from each other and constitute the substance. If place were matter or form, then place could occupy different places! Then we would have to define some deeper sense of place, which the “substantial” place occupies, but this is what we intend all along when we speak of place, so we might as well dispense with any intermediate concept as superfluous.
All of the above considerations only prove that place cannot be the matter or form of any sensible substance, i.e., a substance whose properties we may detect in principle, and is the subject of motion. This does not eliminate the possibility that place or space might be an insensible substance, or the matter or form of such. If space is substantial, its existence can never be detected, but only inferred. Still, insofar as place or space must be posited as existing in order for our observations of motion to be intelligible, it may be considered within the domain of physics, notwithstanding its unobservability.
If space is insensible matter, it ought to be considered immovable, even though we cannot prove it is absolutely immovable. Rather, we define sensible motion with respect to it, so it is immovable with respect to the motions we can observe. Whether space itself can move with respect to some meta-space seems to be beyond the domain of physics, though some interpreters of general relativity have tackled this question.
The “matter” or “form” of space as insensible substance must be understood in a more generalized sense than these terms as used in physics. Metaphysically, matter and form are potency and act in the order of essence. If we suppose space, considered as substance, to be divisible, then it must have a material aspect. If space were indivisible, being considered as the Infinite or as something beyond corporeality, it might be pure act or form. We can see why Newton’s uniform infinite space seemed to be an aspect of Divinity, since it would be indivisible into matter and form in the order of essence, like God, who is pure act in order of essence.
Newton’s concept of space seems fatally flawed, however, since space cannot be space without extension, and if extension is indivisible, uniform and infinite, it must be purely formless matter, which is to say potency without act, impossible for anything existent. Newton thought to avoid contradiction by conceiving of extension not as connected with material reality. Rather, he conceived of extended geometric figures having prior reality, which God may endow with material determination. Space or extension belongs to the realm of formal geometry, which is an actual reality underlying the material world. Whatever we may think of this theological metaphysics, it omits an important aspect of extension, namely that it is pluralized and individualized in the actual material world. Newton’s formal extension leaves the problem of individuation unanswerable in his ontology, with no real distinction between “here” and “there” in a given frame of reference, which makes local motion unintelligible.
General relativity permits a more credible account of space in this respect, since it allows that the insensible “matter” of spacetime, finite in extent, can adopt different geometric forms. This makes space composite and changeable in the order of essence, consistent with being an existent, since it has form.
In sum, place or space cannot be the matter or form of any sensible substance, though it might be an insensible substance or the matter of such.
If place is not a sensible “this thing” (matter) or “this kind of thing” (form), then it must be something accidental, at least with respect to sensible substances. The question remains as to how place might be defined with respect to movable substances, which are the only substances we observe. Setting aside metaphysical questions (i.e., the possibility that place is an insensible substance), we now inquire as to the relation between a place and its substantial occupant.
The proper place of a corporeal substance might be defined in one of two ways, either as the boundary of the volume coextensive with the substance, or as the extension or space within that boundary. In modern mathematics and mathematical physics, we are accustomed to visualizing the locus as the extension or volume occupied by a body. Aristotle found this conception to be incoherent, however, when considering place as physically existent.
To illustrate the concept of place as a limit or boundary, consider a floating log. The wood (A) is in contact with the air (B) and water (C) as shown. Since we are considering place as something physical that contains the substance (A), it makes sense to think of the boundaries of the surrounding substances (B and C) as fulfilling the role of place. Since Aristotle believes there is no such thing as a void, this is generally not problematic, for there is always some external containing substance (though he must address the problem of the “place” of the whole cosmos). In our example, the place of the log (A) consists of those parts of the boundaries of air (B) and water (C) that are adjacent to it. Such “place” is firmly within the realm of physics, as we can clearly apprehend how boundaries of substances may act physically to contain another substance.
If a void is possible, however, then place will not always be the boundary of some containing substance, but of the containing void. Yet if the void is absolutely nothing substantial, it cannot physically act on anything, and so cannot contain the substance to here rather than there. We mean by proper place whatever primarily accounts for why a substance is in a particular location rather than another. Place can only fulfill this physical function if it is a property (essential or accidental) or relation of sensible substances. This is implicitly recognized even in modern physics, for whatever else may be said about “spacetime” or “vacuum fields,” they cannot be a “void” in Aristotle’s sense (place without substance), for they are ascribed properties, relations, and even a capacity for action.
We avoid insisting that place necessarily acts as a physical cause, as even Aristotle acknowledged that place does not seem to act as any of the four types of cause. If anything, it seems to be a property per accidens of a sensible substance, or else a relation between substances or their attributes. Place per se does not cause determinate motions or even contribute to them, according to Aristotle. Instead, it is the background against which motion occurs, according to the powers and actions of sensible substances.
This notion of place as acausal seems to be contradicted by his belief that each kind of substance has a “natural place” toward which it tends. Even there, however, the place as such does not effect motion, but rather it is in the substance’s form or nature to tend to that place. Likewise, we do not invoke the “curvature” of spacetime as a kind of force acting upon massive bodies, causing them to move certain ways. Rather, we consider the intrinsic inertia of each body, plus whatever external forces from sensible substances are acting upon it, as causing it to move along a geodesic of spacetime or to deviate from it.
It remains to be seen, however, why the notion of place as a boundary surrounding a substance should be preferred over the notion of place as the space or extension within a substance. For one thing, if place were simply the extension of the contained substance itself, it would just be matter (of sensible substance), a possibility we have already discarded. Suppose, instead, that place was the space or extension not of the contained substance, but an extension that has independent existence, persisting even after the substance departs. What problems could this conception have?
Aristotle gives a subtle argument that is elaborated by Thomas Aquinas, yet best explained with the aid of diagrams. Suppose that place is just space, namely the extension within a substantial body considered as existing even if the body should depart. In that case, we may speak of the dimensions of the space or place as something distinct from the dimensions of the body. By “dimensions” (dimensiones), we mean the extensive magnitudes of the body, such as length, breadth and height, spanning from one boundary to another. Consider a substance contained in that place. Now every part of that substance is contained by the whole of the substance as when a thing in place is contained by a vessel, except that the part is not separated from the whole, whereas the thing in a place is separated from its place.
If a part of the substance should be actually divided from the whole (say we slice a block of ice as shown), then that part’s dimensions are actual (upper left of diagram), as are the dimensions of its proper place (i.e., the space exactly coextensive with it, lower left of diagram). Yet since the division of magnitude does not create new extensions or dimensions, but divides dimensions already existing, it follows that the proper dimensions of that part already existed before the division took place (lower right of diagram). In other words, slicing a substance into parts does not cut space into parts, since space is considered to exist independently of substance. Thus the proper place or space of each part must have already existed prior to cutting. The same would be true for any other part we took. From this it follows that the undivided whole contains an infinite number of overlapping places of all possible shapes and sizes of its parts.
From a mathematical perspective, there is nothing problematic about this. It is only problematic when we regard space as a physical existent. There cannot be infinitely many actual magnitudes, as discussed previously. While there is nothing physically incoherent about substance being infinitely divisible, since this is only a potentially infinite divisibility, the same is not the case when you treat spatial magnitudes as physically actual loci. Further, there cannot be a plurality of overlapping places, if place is considered as a physical existent, for then more than one place could be the same place, which is absurd. Place, if it is anything, distinguishes here from there, yet such a distinction cannot be reliably made by place if places may overlap.
There are several possible ways out of this conundrum. The most obvious, following Aristotle, is to admit that place cannot be extension or space. Alternatively, we might remove the requirement that space exists independently of substantial bodies. This raises the problem that place would move along with its body, which is not what we intend by place. Yet there is another option, not considered by Aristotle, namely that space might be defined as a relation, rather than as some determinate extension or expanse. While such a concept might remove objections against space as a physical existent, it is still not what we mean by place. In fact, a relational notion of space presupposes bodies or points having loci.
This leaves us with the conclusion that proper place is the boundary containing a substance. Aristotle was careful to specify that place is the boundary of the containing substances, not of the contained substance, in order to indicate that place is not an intrinsic property of the body contained. This may seem like hairsplitting, since the limit of the contained body is geometrically identical to the limit of containing bodies adjacent to it. Indeed, Aquinas recognized that the line or surface of a receptacle must be the same as that of the thing contained. [Comm. on Physics, 417] This geometric identity does not make the distinction physically meaningless, however. By considering the limit as approached from without the body, we indicate the ontological independence of place from its contents. If we were to instead approach the limit from within, we would be treating it as some intrinsic property of the substance, which leads to the absurdity of place changing its place. It is not allowable that a geometric limit should be considered as physically existent independent of any substance, since that would lead to the same paradox of infinitely many overlapping places discussed earlier.
Since place is geometrically identical with the boundary of a substance, it may have whatever form is allowable to corporeal substances. In our examples, we have assumed, for simplicity, that the boundary is a closed surface, i.e., a compact connected 2-manifold without a boundary within its two dimensions. Conceivably, however, a substance might have hollows within it, in which case its place would be defined by some finite set of closed surfaces. A truly fractal body or surface would require infinite iterations of an algorithm, so it cannot exist actually, though an unlimited process might tend toward it. Thus we should not expect there to be physical surfaces or places of fractional dimensionality.
Surfaces with a boundary could not be the place of any body in Euclidean space. Given that space (in the modern sense, which, we shall see, encompasses locus, spatium and situs) is non-Euclidean, however, more exotic surfaces, including a Möbius strip or a Klein bottle, might be mapped onto spacetime, as long as their Euler characteristic is 0. In general relativity, spacetime is still locally homeomorphic to Euclidean space, which imposes considerable constraints on topology.
Aristotle’s notion of place is universally applicable only if there is some sort of plenum. Since modern physics has demonstrated that there is no plenum of massive or sensible substance, any plenum would have to be insensible, and possibly identical with space or spacetime. Physicists regularly theorize as though spacetime itself were a substantial plenum, even if they do not state it in these terms. This means a plenum of sorts, and therefore a universally applicable notion of place as a physical existent, is still viable.
Aristotle defines proper place as “the immobile boundary of that which contains first.” Though the boundary as such is immobile, it may be the limit of movable substances, much as an anchored boat in a river may be surrounded by a fixed boundary of water (viewed from above) with respect to the banks, though the water itself is continually flowing past and around the boat. It is not necessary for proper place to be absolutely immobile, but only in a specified frame of reference within which sensible motion is observed.
For Aristotle, place could be defined absolutely by reference to the place of the whole cosmos, which was the outer surface of the sphere of fixed stars. It might be wondered how the whole cosmos could be said to have a place if there is nothing outside to contain it. Aristotle admitted that the whole cosmos, of itself, does not have a place, yet each part of the outermost heaven is contained by other parts, so they have place in some sense. Thus it is no contradiction to allow that the outermost heaven can really move in situs (e.g., rotating around the center), even though there is no place external to the cosmos against which to measure change in locus. Further, since all the parts of the universe have a place within the outer limit of the universe, then the whole universe must be somewhere, for its parts are somewhere. Still, it seems problematic to speak of the locus of the whole universe, a difficulty that was not lost on Aristotelians from Theophrastus to Avicenna. The notion of each part of the outer heaven being “contained” by other parts relies on the potential divisibility of space. The “places” of these virtual parts cannot be actually existent, lest we arrive at the same paradox of infinitely many overlapping places.
Aristotle’s definition of place as limit rather than extension met much resistance even among classical and medieval commentators. The most notable opposition came from John Philoponus (a.k.a. the Grammarian), who held that Aristotle’s refutation of place as extension was not cogent. Against the argument about divided dimensions, Philoponus counters that there is no reason why the division of a substance into parts should also divide space. Yet Aristotle makes no assertion that space is cut by the act of dividing substance. Rather, if space is place, where place is something physically actual (as opposed to a mere mathematical abstraction), and every substance has a proper place, then the actually divided part must have a proper place. Yet this proper place must have already existed before the division, precisely because cutting a substance does not cut space.
If the notion of having two sets of existent extensions, one belonging to the substance and one belonging to empty space, seems like nonsense, that is precisely the point. Any notion of place as a physically existent extension that is distinct from the extension of the occupying substance requires this double set of extensions. Thomas Aquinas illustrates the absurdity of this position vividly. Let us abstract the dimensions (i.e., the extensions) of a wooden block from all its other attributes. Then these dimensions, of themselves, are no different at all from the dimensions of the same space if it were empty, since two magnitudes of equal quantity can differ only in situs. If we allow, nonetheless, that there is no contradiction in treating the dimensions abstracted from the wood block and the dimensions of empty space as numerically distinct, and that they may overlap, then there is nothing preventing us from abstracting the dimensions of another substance and superimposing them, so that two bodies or more may occupy the same place.
If it is said that bodies are prevented from occupying the same place on account of their matter rather than their extension, we should point out that place has no direct reference to matter.
For we cannot say that it is because of matter that some other sensible body cannot exist together with the wooden cubic body, for place does not belong to a body because of its matter, except in the sense that the matter is contained under dimensions. Hence the impossibility for two bodies to be together is not on account of the matter or of the sensible qualities, but only on account of the dimensions. [Comm. on Physics, 541]
Matter has place only insofar as it has dimensions. The wood as such does not require the block to have place, but only insofar as there is some quantity or extension of wood.
Philoponus also criticizes Aristotle’s identification of place as “surface,” saying that the argument of infinite divisibility and overlap should just as well apply to it. Yet Aristotle consistently prefers to call place a “boundary” or “limit” containing a body rather than a surface, even though it seems clear that it should be represented geometrically as a surface. The reason for this terminology is that Aristotle does not consider surface to be place insofar as it is extension (in two dimensions), but only with respect to the body it bounds. With respect to that body, a surface is dimensionless (i.e., a point’s breadth in cross-section), so there is nothing of it to be subdivided, as it is insubstantial. If the physical world were two-dimensional, then place could be represented as a closed loop, considered not as the extension of its circumference, but as the infinitesimal boundary surrounding the thing contained.
Aristotle’s point is that place is not extension, i.e., quantity, but something categorically different. Extension tells us how bulky something is, not where it is. The ontological role of place is fundamentally different from that of spatial quantity. The latter, of itself, tells us the magnitude of substance; i.e., how much of it there is. Place has the role of containing this magnitude, so that we can say it is here rather than there, distributed in this shape rather than another. Although both place and extension can be represented by similar mathematical objects (e.g., a limiting surface in three dimensions may be treated as an extension in two dimensions), this does not suffice to establish physical identity. Extension is not the same as being somewhere. Further, considered as a physical existent, extension is essential to body or corporeality, while place is an extrinsic or accidental attribute of a body. I can conceive of a body having an extension, being “this wide,” without specifying where it is, yet I cannot conceive of a body at all without giving it some dimension (which is the same as quantity, as Philoponus admits in his Commentary on Aristotle Physics, 4.2).
Place is extrinsic, as it gives a body’s relation to other things. This is why it is fitting that Aristotle should make place the boundary of some external, containing substance, rather than the boundary of the contained substance. Though there is zero distance between the two boundaries, the choice of definition remains significant as it emphasizes the separability of place from its contents.
Although Aristotle’s definition of place as a boundary rather than extension is justified, other aspects of his concept are problematic. First, he supposes that there must be an absolute reference frame for location. If there really is such a thing as local motion, it must be possible for there to be really distinct states of location, hence really distinct places. Following a general belief among Greek physicists, Aristotle thinks physical change requires contraries or opposites, so the reality of local motion requires the reality of the contraries “up” and “down.”
This reasoning, though basically logical, does not require there to be absolute location or direction, for the reality of local motion would be no less assured if locations and directions could only be defined with respect to a chosen reference frame, not absolutely. This would be allowable if local motion were itself relative rather than absolute. The modern theory of relativity has made us accustomed to the idea that there is no such thing as absolute place or direction. Strictly speaking, however, relativity has not disproved their existence, but only shows them to be physically irrelevant, notwithstanding the proposals of some quantum theorists.
Avicenna already recognized the inadequacy of requiring real contraries to account for motion in space. By using the Aristotelian category of “position” (akin to situs) instead of place, he noted that there could be motion with respect to this even though there are no contraries. Indeed, this was an early step to the relative concept of locality that would be more fully developed by Galileo and finally Einstein. Still, as Aquinas noted, any supposed motion with respect to situs is really reducible to a change in locus, since situs implies an order of parts in place. [Comm. on Physics, 475]
A second problem for Aristotelian place, related to the first, is that of defining the “place” of the whole cosmos. This was difficult enough with classical cosmology, and modern physics accentuates the problem by allowing that the space of the whole cosmos might be itself a closed manifold without a boundary.
Another difficulty resulting from the presupposition of absolute place is Aristotle’s belief that place is some sort of immobile reference point. This seems to be contradicted by general relativity, which treats spacetime as a curvable, and therefore mobile, manifold. These views may be harmonized if we consider that the Aristotelian immobility of place need only be with reference to sensible motion, carrying no logical implication that place is absolutely immobile, i.e., in the metaspace in which spacetime may bend. General relativity does not require that any part of space moves within space so that two extended places overlap, which is all Aristotle finds objectionable.
Further, it is difficult to account for place in substance-accident ontology, which is a problem either for that ontology or for the Aristotelian account of place. As Thomas Aquinas comments, place cannot be an element of a body (i.e., a constituent part), since such elements must be corporeal, “because the elements are not outside the genus of their compounds.” [Comm. on Physics, 418] This would seem to leave only the possibility that place is an accident of bodies, yet we have seen that place must exist independently of any determinate occupying body if it is to account for local motion. Aquinas resolves this tension by noting that “place is the measure of the mobile” [Ibid., 435] In this way, place is an accident of bodies, while remaining extrinsic to them.
This account of place as an extrinsic accident of bodies becomes problematic when we consider that bodies are “in” a place, and it is hardly characteristic for a substance to be in its accident, rather than the other way around. As Aquinas notes, Aristotle holds here and in Metaphysics Books IV and V that the other ontological senses of the term “in” are derived from the sense in which something is in a place. A part is “in” a whole as it is surrounded by it. A form is “contained” by the potency of matter, as a part with respect to the whole it might have actualized. [Comm. on Physics, 436] If an accident or property is likewise “contained” by its substance, how can place be the accident of a body?
Such difficulty might be circumvented if we supposed that place were an insensible substance (or the matter of such), rather than an accident of the sensible bodies whose motions are defined by place. Another option, equally compatible with modern physics, would be to make place or space a relational accident, so there is no expectation that it should inhere in a single subject. We could then account for “empty space” not as some self-subsisting extension, but as a relation between objects on which it is existentially dependent.
We are inclined to think of place as an accident of substance because we cannot coherently think of place except as the place of some substance. If there were no sensible substance, we would have no reason to suppose there is physical place. Location or place depends on substance at least in the order of reason, though this does not prove it is really an accident or property of the occupying substance.
The alternative of treating place or space as a physical substance, while upholding the existential independence of place from mobile objects, perhaps makes this independence too absolute, and runs into the problem of redundant sets of dimensions for the void and bodies.
The fact that place has a mode of existence that does not fit with those of substance and other known accidents perhaps only proves that it deserves its own ontological category, and should not be reduced to any other. We should be content with noting that it acts as a boundary or limit on sensible bodies, yet its existence is not dependent on the body it bounds. If anything, its existence depends on a body being in relation to some other body, in the sense of Heidegger’s Dasein. If there were only one body, we could hardly say it is in place. For this reason, the cosmos as a whole cannot be in place, though its parts may be. This paradox ceases to involve contradiction when we understand place as relational.
Place as such cannot be a substance, since a substance is that whose being need not be predicated of anything. It makes no sense to think of place without being a place of something, whether that something is potential or actual. Place differs from qualitative accidents in that it does not inhere in a substance, and might even be said to exist when its occupant is only potentially present. Place is an accident that is separable from its subject, unlike quantity or quality, which are inherent accidents.
In the Introduction to Ontological Categories, Part IV, we noted that “place” could be an inherent accident only if space were absolute. Otherwise, it must be a relational accident. Judging from the discoveries of modern physics, place must be a relational accident, at least from the perspective of the sensible world. It is possible that place is absolute in some meta-space, but the position of place in any meta-space beyond observable spacetime is extrinsic knowledge that has no bearing on any dynamics within the sensible world. Absolute place is beyond the realm of physics, which accordingly treats place as relational.
Nonetheless, we refrain from saying simply that place is a relation, since the existence of a relation is entirely dependent on the things related. Place, by contrast, must be presupposed in order for individualized corporeal substances to exist, so it cannot be existentially dependent on the sensible things related. It must depend on some insensible substance, which we may perceive only indirectly through mathematical models of physics.
Since we have no absolute reference point, and modern physics deals also with particles separated by distances, not just contiguous masses of substance, it often makes more sense to think in terms of space rather than place. The Aristotelian objection against spatial extension as a physical entity vanishes when we consider space relationally, rather than as something that inheres in a sensible body or as something that exists as an independent set of dimensions coextensive with those of a sensible body. Space, unlike place, is truly a relation, being just the distance spanned between objects or their parts, or more generally, any such extension spanned, including area and volume.
The measurement of space, it turns out, is inseparable from activities involving the passage of time, since some finite time must elapse to move from one place to another. Consequently, the measurement of space is inseparable from the measurement of time, which we express mathematically with the four-dimensional construct called “spacetime.” The “space” and “time” measured in spacetime are not place and time themselves, but relational accidents.
Absolute place, if it exists, cannot be defined by physical observation, so it is extrinsic to physics. Local motion can be fully understood in terms of relative position; absolute location, even if it existed, would affect nothing sensible. Accordingly, even though place (locus) is ontologically indispensable to any account of physical motion, this “place” need not be defined with respect to some absolute reference frame. Relative “place” suffices, in which case we might equivalently have recourse to position (situs) and space (spatium), the latter considered not as an inherent accident of substance, nor as a set of dimensions substantially existing, but as a relational attribute of sensible substances. Place, position and space are all extrinsic accidents (i.e., they do not affect the substance as such, but denominate it by way of externals), and the last two are also relational.
Thus far we have found that place:
While place may pertain to some insensible substance, this need not be directly relevant to physics. From a physical perspective, place may be considered as something accidental to sensible substances and extrinsic to them. Place is a limit containing material substance, yet it is independent of what it contains, so it is not merely the form of a material substance, though it geometrically coincides with the shape of its contents.
Place is not the form of a sensible substance, but the form of a sensible substance qua mobile, where mobile refers to that which moves without change in quality, and by implication, without change in form. Place might be called the form of “material movement,” i.e., the aspect of movement that pertains to matter. Material movement may involve a change in quantity or bulk, which is called increase or decrease. If there is no change in quantity, then movement must take place with respect to another category that is neither quantity nor quality, which we call place. In the modern atomic model of sensible substance, all expansion and contraction is really reducible to the local motion of its parts. Even if this were not so, increase and decrease would have to be accompanied by some local motion, since a larger substance would occupy a place it did not when it was smaller.
Local motion of a substance is only a change in relation to other things, involving no intrinsic change. The relations of place include distance, which is the measure of one-dimensional spatium between two points, and orientation, which is an aspect of situs or position. These relations suffice to define relative place, which is the only kind of place that has computational relevance to physics, the study of sensible changes.
Relative place is only definable if two or more things exist some distance apart. This supposition at first seems difficult to reconcile with the modern relativistic doctrine against simultaneity at a distance. Yet we should keep in mind that special relativity does not deny the possibility of distant objects existing simultaneously, only that there is a univocal definition of simultaneity.
Place has two roles with respect to matter. First, it limits the quantity or geometric bulk of a material substance. Second, it individuates material substance, providing a medium upon which signate matter can act as a principle of individuation.
Place would suffice to fulfill these roles if the cosmos consisted of discrete monads or points, but the continuity of motion, upon which we base our comprehension of physical causality, implies that matter may occupy a continuum. Thus space or extension accompanies place as a dependent relation.
It is through its spatial or extensive aspect that place quantifies matter and acts as a medium of individualization, making substances numerable. This medium defines relations among material substances, such as distance (the magnitude of spatium in one dimension) and orientation (an aspect of situs or position). Though we call it a medium, place is not a sensible substance. Neither can it be an accident of a determinate sensible substance, for its existence does not depend on “this substance,” but on substance in general.
Situs or position relates places to each other, showing how they are connected and arranged in a manifold. Given that “space” (really place) is not geometrically uniform, situs has physical significance, and may be represented by a curvature tensor.
The continuity of space might be taken to imply that physical substances must exist as continuous volumes of matter. In fact, we find that matter is highly localized in force at the molecular level and below. Further, the fundamental particles do not seem to be adequately modeled on the assumption that they are solid spheres of material. Rather, they are point-like perturbations or centers of force in some field. Still, particles may act anywhere in space, moving freely through the continuum. More importantly, the fields themselves seem to permeate space and act upon particles locally, i.e., by direct contact, as substances. Thus it might still be the case that every place is the boundary of some mobile substance.
A coherent definition of place was necessary in order to render intelligible any discussion of whether there is a void. If a void were absolutely “nothing,” it trivially could not exist. If it were “emptiness,” it would be a property of something, rather than something substantive. If it were spatial extension or magnitude, we would have the incoherent scenario of there being two numerically distinct but otherwise identical sets of dimensions for the same space, discussed previously. We may still speak of a void coherently if we define it to be a place which has no substance occupying it. Aristotle does not dismiss the possibility of a void on a priori logical grounds, as indeed there is no basis for doing so. He does, however argue against it by invoking some principles of natural philosophy. It remains to be seen whether his suppositions are sound, or if they are grounded in the flawed physics of his time.
At a minimum, a void must be a place that contains no corporeal substance. By corporeal, we mean not an abstract mathematical body, but something palpable, or capable of receiving what is palpable. The importance of “touch” is that physical movements are presumed to be mediated by direct contact, so we are limiting consideration to physical bodies. We might allow that some more ethereal substance beyond physics could exist in the void, but that is outside our consideration.
Aristotle has already discounted the possibility of a spatial interval having existence apart from any substance. Insofar as the void is considered to be such an interval, it cannot exist, and arguments that the void is necessary for motion are really grounded in the necessity of place for motion, and the confusion of place with spatial interval.
Still, even if the void is not logically necessary for there to be motion, we may still inquire whether in fact it happens to be present (at least as a privation) in some places. Even if we deny existence to the interval per se, we may at least ask if there are places (i.e., boundary limits) that contain no corporeal substance.
Several of Aristotle’s arguments show that void, far from being necessary to local motion, does nothing to account for it. If a void is perfectly uniform, how can it account for the tendencies of bodies to move to a preferred place? While we would today discount Aristotle’s assumption that there is an absolute “up” or “down,” his basic point still holds that space as such cannot account for motion insofar as it is uniform.
Going further, we might hold that a uniform void is altogether irrelevant to motion. If one spot in the void is no different than another, how can we say that anything has moved? If we answer that all physical locomotion is purely in relation to other substances, then we are asserting that the void as such has no role in local motion, and so has no place in physics.
Next, Aristotle argues that speed through the void ought to be infinite, on account of an inverse proportion between speed and the density of a medium. Medieval philosophers knew this linear relationship to be empirically false. Philoponus noted that the difference in speeds between falling objects of different weights was much smaller than a linear relation would suggest. Aquinas remarks that velocity is proportionate to mobile energy (i.e., momentum or impetus, the mayl of Avicenna), even if there is no obstacle. Aristotle is correct about air drag being linearly proportional to velocity, but this is only one term in the equation of motion. Aquinas charitably considers that Aristotle was making an argumentum ad hominem, not an absolute demonstration, adopting his adversaries’ position that a void was necessary for motion, and showing that this is not so. In that case, we still have come no closer to showing that there is in fact no void.
If we suppose that there is an independent void that can be occupied by bodies, we may encounter the same problem discussed previously when considering that place might be the spatial extension coinciding with a body. We saw that the dimensions of the body and the dimensions of the space could not be considered physically distinct things. Thus the volume of a wooden block is numerically identical with the volume of the void it supposedly occupies. If void is nothing more than extension, then it cannot be something in addition to the occupying body, which already has extension. If void is something that has extension, then it is a sort of substance (i.e., a being whose existence is not predicated of something else). If we allowed that the extension of the void and the extension of a body may overlap, then why not any number of volumes, of metal, marble, etc., since they are all identical? We should provide some reason why the void is a fundamentally different kind of substance, that should be exempt from the rule against bodies coinciding in extension.
Aristotle goes on to show that the compression and rarefaction of substances does not prove the existence of a void within bodies, since these phenomena can be accounted for without a void. We set aside his arguments that a void cannot account for the upward motion of light bodies, as these assume the erroneous notion of a natural place.
Modern physics finds that the vast majority of space is unoccupied by any particle that has mass. Does this establish that space is a void in the sense of a place without corporeal substance? Further, the apparently corporeal substances with which we are familiar consist of atoms, which in turn consist of subatomic particles with highly localized concentrations of mass and electric charge, with the vast majority of the volume being apparently empty space. Even the omnipresent microwave background radiation is equivalent to only 400-500 photons per cubic centimeter. Given the formidable empirical evidence in favor of the reality of empty space, how do we reconcile this with the problem of coincident magnitudes?
First of all, it is far from clear that what we call “empty space” is really a void in the sense of a place containing no corporeal substance. Physical corporeality entails the ability to act by direct contact over some volume of space or the limiting surface of a volume. Yet it seems that all particles are capable of interacting with so-called empty space. If physicists may speak of a “vacuum electromagnetic field” and “particle-vacuum interaction,” then their so-called vacuum might not be void of corporeality after all, since it is capable of physical action by direct contact.
This raises another set of problems. Is so-called “empty space” really a physical substance? It would seem so, on account of it bearing physical properties. Yet how can more than one substance overlap in place? What we improperly call “empty space” has real physical existence only because it is not mere extension. Yet if it is an existent, with definite qualities at definite locations, does this not undermine the relativistic claim that there is no absolute place? Even within relativity theory, we see a certain incoherence, as spacetime is sometimes treated as something substantial, which can act and be acted upon by massive objects, yet at other times it is described as a purely relative attribute of sensible bodies, an artifact of one’s choice of reference frame. Without pretending to decide such difficult matters here, or even presuming that they could be resolved by abstract reasoning alone, we may at least admit that the existence of a void has not been proved.
Time is ordinarily considered the measure of movement, a measure that conceivably could be abstracted from any determinate movement, allowing us to speak of the passage of time itself. In classical mechanics, time is treated as a parameter of changes in position or other properties modeled by continuous variables. In relativity, it is modeled as a pseudo-metric extension, a fourth dimension of “spacetime.” These mathematical treatments of time may capture its function as a measure, but seem to ignore other essential aspects. Classical mechanics, for example, is indifferent to whether time moves forward or backward, but this is hardly the case in reality. The spacetime model, if taken too literally, would make time a mere extension, yet extension as such contains no notion of past, present and future. Worse, if we treat time geometrically, as a set of curves or points, we are using static objects to describe something fundamentally dynamic.
In order to evaluate modern models of time properly, we must first establish precisely what we mean by time, and ascertain what aspects of physical reality, if any, may correspond to this concept. Only then can we determine if our models fully account for time.
Time is a primitive notion, built into our language and our psychological experience, a fact which may make it suspect as a description of objective physical reality. Perhaps time has no physical existence, and it is nothing more than a way we speak about the world, a characteristic only of our perspective. On the other hand, man has long seen the objectivity of time attested by the marvel of celestial motions, which each follow their own clock with precise regularity, since long before humans have gazed upon them. The presence of periodic motion in nature, confirmed with much greater precision in atomic vibration, gives us confidence that time is something objectively real, or at least corresponds to something real, even if its magnitude may be dependent on perspective.
Our terms “before” and “after,” which come from “fore” and “aft,” show our deep-rooted habit of metaphorically representing time in space. These terms consider the past as that which is in front of or ahead of the present, while the future is lagging behind, much as a racer in front arrives first and the one behind arrives afterward. We can also use a reverse model, making the past something “behind us,” i.e., something we have “passed,” while the future is in front of us, something to which we look forward. The equal suitability of both metaphors only shows that the “direction” of time does not really correspond to spatial direction.
A comprehensive account of time should include not only its role as an extensive measure of movement, but also its succession from past to future, and its role in the determination of existence, as the potential becomes actual. At the same time, we should justify including these aspects in physics by showing that they correspond to extramental physical reality.
Time is relevant to our understanding of basic logical principles as applied to ontology. The principles of identity and non-contradiction (in an ontological context) require specification of time. The predicate statements “x is A” and “x is not A” are mutually exclusive only if they are understood as referring to the same time. The identity of a thing with itself over some period of time supposes some existential continuity across time, and that position in time is something accidental to the persistent thing’s existence.
The connection between time and existence seems ambiguous, even tenuous. Should we say that only the present actually exists, while the past is non-existent? Or should we instead say that everything that has happened up until now is fixed in existence, “in the books,” so to speak? What about the future? Is it utterly non-existent, only potentially existent, or every bit as actual as the present? If there is no absolute simultaneity, should not our “future” be the present in some other frame of reference?
If “past,” “present,” and “future” are treated as parts of time, problems arise, for a whole thing does not exist unless all its parts exist. Yet the supposed “parts” of time do not all exist at once, at least not in the same way, but one after the other. If we allowed that the parts of time actually exist at once, then the principle of non-contradiction would be thoroughly violated, and there would be an actual infinity of objects, to name just two egregious impossibilities. Further, the “part” we call the present is not really a measurable part, but apparently some infinitesimal point, if we represent time as an extensive magnitude or timeline.
Is the present really infinitesimal in duration? Consider the psychological present, the archetype from which we extrapolate various notions of past, present, and future. My present cannot be infinitesimal in duration, if I can understand statements only through their grammatical representations. It takes some finite amount of time to think a sentence verbally, yet my understanding is not restricted to this or that part of the sentence, but extends to the judgment as a whole. To understand the statement “Leaves are green,” I must contemplate both leaves and greenness simultaneously, linking the two. I could not arrive at “Leaves are green,” by thinking only of “leaves” and then only “green.” Somehow I must integrate the two thoughts. Even to think, “This is now,” takes some considerable time longer than our quickest reaction time. Similarly, we could not appreciate music unless we could consider several notes at once, since the sequence of notes is important to melody. Yet we hesitate to give the “present” any definite duration, as this duration depends on the thought presently occurrent. There is no sharp boundary between past, present and future, though the far future and far past are definitely existentially dissimilar to the present, at least from our perspective.
If there is an objective present, which is infinitesimal in duration, it could hardly be the case that each “now” is supplanted by the next, since there is always some finite distance between any two points. On the other hand, if the objective present has some finite duration, then you would have violation of non-contradiction and other impossibilities, unless there is something preventing physical changes from occurring faster than this minimal duration. Yet if there is no change within this minimal quantized duration, how can such a duration be a passage of time at all?
Time may be quantified by counting periodic movements (local motions or qualitative alterations) of natural bodies or machines, each of which is reckoned as a unit of time. Yet it would be a mistake to identify time strictly with such movements, for time certainly has elapsed even when only a portion of an oscillation or revolution has been accomplished. Moreover, the period of motion varies among bodies, so that no one of them represents the unit of time, though they all coexist in the same reality, and are related to each other by measurable ratios. All this suggests that, although movements may help us quantify time, time itself is something distinct from any determinate movement.
Time must be something distinct from sensible change, or else it could not serve to measure the speed of change. How could we say a movement is “fast” or “slow” if time, being nothing other than the movement itself, also increased or decreased in speed by the same amount? By “fast” we mean that which moves a great distance (or alters to a great degree) in a short duration of time, while “slow” moves a small distance in a great duration of time, or rather, the distance to time ratio is high for “fast” movements and low for “slow” movements. Clearly, time is distinct from movement, for the latter is defined by the difference between start and end states, i.e., the distance moved, or the degree altered. Yet a movement may be “faster” or “slower” without any change in its defining endpoints. Speed may be a property of movement, and time may serve as a measure of this property. At any rate, time is distinct from kinesis, and indeed from any change (metabole).
This need not imply that time is existentially independent of movement or change. We do not notice time to have passed except by perceiving change, and it is doubtful whether it is even sensible to speak of time where there is no change at all. Yet it is possible in a given environment for one object to change while another remains unaltered. Should we say that time has elapsed only for the one object and not the other? Then why should we not say that time has elapsed twice as quickly for an object that changes twice as fast? Still, it seems that time depends on the existence of movement in general, even if it does not depend on any particular movement.
Since we are dealing with physics, Aristotle only treats time insofar as it measures movement (kinesis), without considering whether there is some metaphysical time that could mark more general kinds of change (metabole). This movement (kinesis) could be qualitative alteration, substantial generation or corruption, which do not require changes measured with respect to space. Thus we may perceive the passage of time even without a change in place, though there must always be some sort of sensible movement.
If time is existentially linked to physical movements outside the mind, it follows that time is not a purely psychological phenomenon, but an aspect of physical reality. The fact that we are incapable of perceiving time without perceiving movement at least suggests this existential link, though it does not prove it. For example, when we fall asleep, we perceive no movement, so we have no sense of how much time has elapsed. While awake, we may perceive time only if we perceive some external sensible movement, or at least some movement or change within the mind.
This raises a problem as to whether psychological time is the same in kind as that of external change. Granting that sensible phenomena really do succeed one after another, not just in our perception, it follows that the duration or number of such succession can be given some measure (though perhaps not univocally), which we call time. If physics deals with the sensible world, how can psychological time be the same as physical time, if succession in thought, as Aquinas observes, really deals with insensible change?
What has been just said about the perceiving of time and of motion raises a difficulty. For if time follows upon some sensible motion outside the mind, it follows that whosoever does not sense that motion, does not sense time; whereas the opposite of that is said here. And if time depends upon some motion of the mind, it follows that things are not connected to time except through the medium of the mind: thus time will not be a thing of nature but a notion in the mind like the intention of genus and species. But if time follows upon any and every motion, then there are as many times as there are motions—which is impossible, for there cannot be two times together as we said above.
In order to clear up this difficulty it must be remembered that there is one first motion which is the cause of every other motion. Hence whatever is in a transmutable state possesses that state on account of the first motion, which is the motion of the first mobile being. Whosoever, therefore, perceives any motion, whether it exists in sensible things or in the mind, is perceiving transmutable being and consequently is perceiving the first motion, which time follows. Thus anyone who perceives any motion whatsoever is perceiving time, although time follows upon just the one first motion by which all other motions are caused and measured. Consequently, there remains only one time. [Comm. on Physics, 573-4]
This “first motion” does not mean some motion (in the generic sense of “movement” defined previously) that occurred a long time ago, subsequently causing all present motions, but the motion of the primum mobile, which moves even today, powering all other motions. In antiquity, the primum mobile was visualized as the outermost sphere moving the visible heavens, but this is not essential to the concept. Primum mobile may be thought of as akin to “prime matter,” that is, it is motion in the most generic sense. A primum mobile so conceived means that the various movements observed in the universe are not causally disconnected, but are all related to each other by movement. Thus there is only one generic physical movement, of which all observed movements are particular instances.
If it be granted that all movement is of the same causal network, then it follows that there can only be one time, since time is the measure of movement. Otherwise, one might suppose there could be multiple times, e.g., one for the solar period, another for the lunar, and so on, none of them having any connection. We may readily concede that all sensible local motions may be related to each other as ratios, and likewise with their durations or periods, so that there is only one time. This just means that all time is common, not necessarily that there is an absolute measure of time. However, if we really believe that all physical movements, including the movements of the mind, are in the same causal network, then it follows that “psychological time” cannot be categorically distinct from the time of sensible motion.
The apparent contradiction in Aristotle’s two accounts of psychological time may be harmonized. When he says that a sleeping person does not notice the passage of time, this is because the conscious mind is not active, so it experiences no internal movement that would enable it to perceive time. While awake, a person may perceive the passage of time even if he observes no external sensible motion, since it suffices for him to note the internal movements of his mind, as when a series of words passes through his thoughts.
It seems clear that time is dependent on movement, yet distinct from it. Recall that movement is a continuous change between two endpoint states. The movement goes with the continuous magnitude spanning all intermediate states. If the magnitude is continuous, so is movement, and if the movement is continuous, then so must be time, if it measures movement. Aristotle simply accepts that the amount of time that has passed is assumed to be proportional to the movement; i.e., that this is what we mean by time. Indeed, it is difficult to see what we could mean by time without reference to movement, or how we could quantify it without referring to some quantity of movement.
Nonetheless, the assumption that the quantity of time corresponds to the quantity of movement seems to be problematic. After all, we can conceive of the same movement as occurring quickly or slowly, whether we are speaking of local motion or qualitative alteration. The magnitude of change, then, would not seem to be a good measure of time. Aristotle acknowledges that movements can happen at different speeds, which is why he holds that time is only “proportionate” to the amount of movement, with the ratio depending on the speed. The speed, of course, may vary over time. Still, it seems unassailable that any increase in movement must be accompanied by an increase in the quantity of time elapsed. There is no way to measure the magnitude of time independently of some movement assumed to be uniform in speed. Once we appreciate this, the nonintuitive results of special relativity become less surprising, since the observation and quantification of time duration are utterly dependent on physical movements.
The “before” and “after” of time follow from the “before” and “after” of the states in each movement. For local motion, “before” and “after” refer to a succession of places in space. For changes in quantity (increase and decrease) and alteration, states are sequenced not by spatial distribution, but by their well-ordered magnitudes or degrees. Most generally, then, it is the sequence of “before” and “after” in magnitudes that is the basis of the “before” and “after” in time.
A movement may occur in either direction, forward or backward, yet time is always reckoned as moving forward. A common direction for all local time can be defined with reference to a prime motion, which is the revolution of the outermost sphere in ancient Greek cosmology. We do not yet pretend to determine what the prime motion is in modern physics. The expansion of the universe seems a likely candidate, yet we would not say that time is reversed if cosmic expansion eventually reverses into contraction. For want of a better description, we say that the direction of time is that in which an effect succeeds its cause (when they are not simultaneous). This direction is never reversed in one temporal frame with respect to another.
Aristotle considered the direction of succession to be essential to time, so he included it in his definition: Time is the number of movement with respect to before and after. Here “number” is meant in the sense of what is counted, not that with which we count (i.e., the mathematical object).
It seems strange that Aristotle should say time is a number (which in Greek meant discrete quantity, i.e., positive integers), when it measures continuous movement. Aquinas gives an example of ten measures of cloth as a discrete number of continuous movement. This saves Aristotle from absurdity, but does not really explain the preference for number. It cannot be that he regarded continuous magnitude as being applicable only to space, for he allows that various kinds of movement, including qualitative alteration, can have continuous magnitude. In the Categories, Aristotle says time is the only non-geometric continuous quantity, in apparent contradiction with his present characterization of time as number.
Whether we quantify time discretely or continuously, it is clear that time must be an ordered magnitude, to distinguish “before” and “after.” Arithmetic number is ordered by greater or lesser value, while extensive magnitudes can be ordered by position (e.g., line segments to the left or right of each other). Time measures movement, and a movement can have magnitude only by situating that magnitude positionally, i.e., in a sequence of states along some continuum. “First” and “last,” or “before” and “after,” are built into the definition of a determinate movement, and this same definition is adopted by time. Whether a movement is from state A to state B or from state B to state A is built into what it means to be that particular movement. There is no mystery about the direction of time, since this is adopted from the direction of existing movements. The only mystery, if any, would be why certain movements favor a given direction, which requires us to get into the specifics of empirical physics. As Aquinas notes, Aristotle’s definition of time avoids circularity precisely because he situates “before” and “after” as characteristics of movement, not as something defined primarily by time.
Time is necessary for grounding the determination of physical existence, for without specifying a position in time, there is no ontological principle of non-contradiction. From the perspective we call the present, all prior events, known as the past, are determinate, while subsequent events, known as the future, may be indeterminate. The limiting boundary we call the present is the interface between the determination and indetermination of physical existence. We may understand the present in two ways. There can be a succession of distinct “nows” replacing one another, or we may speak of “the present” as a single thing that is carried along forward with the passage of time.
The present in the latter, generic sense is analogous to a mobile substance, accepting different attributes or determinations of being. Just as motion, though imperceptible per se, is perceived through the mobile object, so we are able to perceive time only by our experience of the mobile “now.”
Yet there is also some truth in the first sense of the present, as a succession of distinct “nows.” Since time is not identical with movement, the “now” I am experiencing is distinguished not only by the movement states of various ongoing processes, but also by the fact that this moment in time is really distinct per se from previous moments that used to be “now.” This plurality of the “now” succinctly expresses duration itself, so that it may be used as a naturalistic definition of time.
Aristotle defines time as the number of movement with respect to before and after. The meaning of this “number” has confused many modern commentators, but Mark Sentsey has correctly deciphered it in: “Time as the Number of Motion in Aristotle’s Physics” Proceedings in the Kent State University May 4th Philosophy Graduate Student Conference, 2008. For Aristotle, “now” may be envisioned as a limit or point dividing time into extensions of past and future. In order for there to be time, there must be more than one such now, i.e., one “before” and one “after,” yielding an ordered duration we call time. He is not saying that time is composed of “nows,” but that the plurality of now is what generates time. This definition of time is truly naturalistic, not merely formal, for it expresses how time is generated.
I submit that, instead of saying “number,” the definition is better expressed thus: Time is the ordered plurality of termini in (partial) movement. A movement consists of a succession of states, so any movement or partial movement has two such states as its termini. Each terminus constitutes a limit of movement, to which corresponds a limit in time (the measure of movement), which we call “now.”
With this clarification, other pieces fall into place. We now see why Aristotle considers the minimum number to be two. Thomas Aquinas’ example of measures of cloth does not merely show the self-consistency of Aristotle, but expresses exactly how the latter conceived time as numbered by intervals between “nows,” which Aquinas understood correctly. The “now” is a boundary; it is not time, but an accident of time. [See Comm. on Physics, 592]
Modern number theory shows the consistency of the notion of time as the plurality of now with the continuity of time. Aristotle’s present or “now” can be represented as a Dedekind cut, dividing a continuum of number into two parts. Though we need only a discrete number of such cuts to define the duration or time of some movement, there is no obstacle to admitting the successive existence of uncountably many “nows.”
Applied to physics, time always accompanies some movement. For this reason, we cannot regard the point-like “now” as time, for there would be no movement, and we would paradoxically regard time as static. Aristotle, no less than Henri Bergson, recognized that time is essentially dynamic, so he does not allow time to be a point. For there to be movement, there must be some measurable continuous change in quantity, quality or place. For it to be measurable, it must be finite, and therefore numerable. Thus time accompanies movement insofar as it is countable, as Aquinas correctly notes. [Comm. on Physics, 580] The temporal order of before and after likewise comes from movement, so that the correspondence of the direction of time with that of causality arises naturally.
The “now” is not time or a part of it, but a determination or accident of time. It bounds or limits time, but can do this only if it is multiple or plural. The “now” is pluralized not by time (for then we would have a circular definition), but by the mobile objects (i.e., the object as determined at each terminus of movement) it numbers. [Ibid., 592] Non enim est tempus numerus simpliciter, sed numerus numeratus. [Ibid., 594]
The unity of a movement comes from the object being moved, insofar as the latter is considered to subsist throughout movement. Analogously, the unity of time comes from the “now,” considered as something carried forward in time. Yet there is also a sense in which we can consider an object in each successive state as distinct entities, and likewise with time. Time is rendered multiple by the plurality of the “now.”
Time is not number in the sense of plain quantity, for that would contain no information about sequence, which is no less essential to time. Considered as pure number, a time period in the remote past would be indistinguishable from a period of equal duration in the far future. [cf. Comm. on Physics, 596] Instead, we consider that each duration of time has a definite, unique position in sequence. This sequence is determined by what is prior and posterior in movement.
Recognizing that the plurality of now is essential to time, Aquinas remarks that eternity, by contrast, is a state where the “now” is always in the same modus, not mobile, not flowing with a before or after, but a unity. This notion of eternity is not the same as infinite time; in fact they are diametric opposites. Strangely enough, this idea of eternity has a possible application in modern physics, when considering the supposed “timelessness” of photons.
Time may always be compared with local motion (i.e., movement in place). Local motion is the first of all movements in the sense that quantitative and qualitative changes are always preceded or accompanied by some local motion, but the reverse is not always true. This is the case in modern physics no less than that of the ancient Greeks, since all of our four fundamental force interactions effect change by a change in relative location. Gravity and electromagnetic forces are inversely proportionate to distance, while weak and strong nuclear interactions require particles to approach or move away from each other in order to effect a change in state. Accordingly, the time associated with any physical movement can always be situated with respect to the time of some local motion. Thus, when examining the question of the universality of time, we can simplify discussion by considering only local motion.
If we accept that all local motions are in the same causal network, it follows that they can all be temporally positioned with respect to one another. Aristotle posited a first motion, that of the outermost sphere, which serves as the origin and reference point for all other motions. We have noted that the primum motus need not be the motion of a determinate body, but rather, in analogy to “prime matter,” there could be a “prime motion,” which is physical motion in its most generic form, abstracted from determinate mobile objects. This does not mean that there can really be motion without a determinate object, just as we may speak of prime matter without asserting that there can exist matter without form. The reason for positing prime motion is that, since all motions are in the same causal network, they can be placed in a common network of continuous succession of states, which may be considered a generic movement.
We are careful to call this a network of succession, rather than suggest that all successions in movement can be referred to the succession of a single determinate movement. If the latter were true, then Aristotle would be justified in his assertion that time is the same everywhere at once, since all motions can be measured with respect to the first motion. In that case, we could speak of a universal time, which is the number of the first motion.
Relativity has taught us that there can be no universal measure of time. Time is the same everywhere only in the sense that all events are connected in the same temporal network, not that there is a uniquely definable simultaneity across distant parts of space. This is sometimes expressed by saying that there is no such thing as simultaneity at a distance, but such an assertion yields a highly problematic ontology. Since relativity holds even for small distances, though its effects are negligible, we would be left with the no less paradoxical result that nothing larger than a point can exist. Determinate existence presupposes simultaneity, or else we have no basis for speaking of anything as a substantial unity, or for applying the principle of non-contradiction. As we will see later, the death of simultaneity has been greatly exaggerated.
Still, relativity provides the important insight that the “now” cannot be abstracted from location, as if time were separable from space. This insight is consistent with Aristotelian claims that time is inseparable from movement and all movement is linked to local motion. Since local motion is inseparable from space, it follows that time is inseparable from space. Taking this further than Aristotle could have envisioned, it follows that we cannot know the “when” without also specifying the “where.”
The present is the boundary between determinate and indeterminate existence, and we have found that it is location-contingent, and therefore perspective-contingent. This implies that the determinacy or indeterminacy of existence is also dependent on perspective.
If determinate existence depended on being in a definable present, we would have the further complication that an infinitesimal present is not a part of time, but a limit of time. Thus being “in the present,” if such a thing were possible, would not be the same as being in time. We have already noted that the psychological present evidently spans some finite duration. Accordingly, the “now” should not be considered something that contains determinate existents, for these exist only in time, or in parts of time, and the “now” is neither of these.
These problems may be resolved by adopting some form of Bergson’s thesis that what we colloquially call “the present” is actually the recent past, recalled by perception and memory, not an infinitesimal “now.” With this view, it is unavoidable that we attribute determinate existence to the past. This should be interpreted as existence in continuous succession, in order to avoid paradoxes of identity and contradiction.
When we say some body exists, this refers to the reality that all parts of that body have been fully determined in the recent past. Even though relativity ensures that there is no absolute simultaneity among the loci of the parts, we may nonetheless consider the whole body as an existent insofar as the parts endure long enough to guarantee overlapping simultaneity. Duration is what permits large bodies to exist, despite the problem of simultaneity at a distance.
With these clarifications, we see that the problem of the present does not require time to be “grainy,” as in some interpretations of quantum field theory. It is wrong to say that there is no “now” simply because it is impossible for anything to exist in an infinitesimal present. The “now” is a limit, not a part of time. As for the fact that the psychological present, or any other pseudo-present that would contain physical existents, is actually a finite duration, this only shows that determinate existence is contained by past durations, not that there needs to be a minimum unit of duration.
In any event, quantum field theory makes time grainy only in the sense that particles interact with fields at discrete intervals. This need not really abolish the continuity of time. In fact, the notion of duration is unintelligible without such continuity. If you define your minimum duration to be some fraction of a second (e.g., the Planck time or some smaller unit), you are effectively ascribing so much continuous measure to the duration. Further, if it is really impossible for there to be physical change within some duration, it should be impossible to detect that any duration has elapsed.
Still, we need not dismiss the possibility that time consists of discrete parts, since continuity is preserved as long as the end of one part is identical with the beginning of the next. Within each minimal part, an existent endures without actually changing, yet it may prepare for the next step in movement by changing in potentiality, i.e., by continuous time evolution of the wavefunction. Thus time would remain continuous with respect to potentiality, but discrete with respect to change in actual, determinate states of reality.
This grainy notion of time resembles what some Scholastics called “instants” or “moments.” The problem of how there can be duration without succession (as there is no internal order of actual states within each minimal part of time) is answerable using the Thomist account of eternity mentioned previously, where the now is not in succession, but contains all in a self-subsisting unity. This notion may be modified so that a duration without internal succession, rather than being unbounded like eternity, has an extrinsic beginning and end, giving us a bounded quasi-eternity, known to Scholastics as an aevum. Bergson similarly posits a durée réelle in which a thing possesses itself by an act of identity that is distinct from Aristotelian movement. This allows us to resolve the determinate existence of self-identical substances with the continuity of time. Although time may not be continuous in the sense of measuring uninterrupted movement, it is still continuous insofar as each moment grows out of the previous, just as each corresponding movement grows out of the potentiality developed in the previous instant.
It should not be too surprising that we resort to the infinite (eternity) as a model for instants of time in which existents may self-identify. As Bergson remarks in An Introduction to Metaphysics (1903), the infinite and (individualized) existence are both absolute, in the sense of perfection (i.e., completeness). To really get at the thing that exists, it is not enough to multiply perspectives or representations of it, or to identify what it has in common with other things. The existent qua existent can only be intuited, not analyzed or represented. If anything expresses our intuition of what it is to exist, it is our own sense of perduring. Duration, then, is a measure of existence or self-identity, and as such it has nothing to do with spatial extensions, which are distinct and juxtaposable. Thus our representation of time as a kind of space, while computationally and visually helpful, is fundamentally misleading. Time “measures” something that is not extensive, but self-identical, i.e., existence.
The metaphysical distinction between essence and existence, first clearly articulated by Thomas Aquinas, is unnecessary to account for time as the measure of movement, but is useful insofar as time “measures” (in the aforementioned equivocal sense) the mobile.
Motion is measured by time both as to what it is, and to its duration, which is its existence (esse). Other things are measured by time as to their duration or their existence (esse) as this is changeable, but not according to what they are in themselvesto the latter the “now” of time corresponds. [T. Aquinas, Comm. on Physics, 601]
Insofar as it “measures” a mobile object, time measures its existence, i.e., how long it endures or persists, yet it does not measure what a thing is. A rock is not more fully a rock by virtue of existing for a longer duration. It is the kind of thing that it is regardless of how long it endures. I do not become qualitatively “more human” by living longer. Long life adds to my existence as a human, but not to my humanness.
Aristotle situates the essence (“what it is”) in the “now.” This agrees with Aquinas’ account of the eternal as now. When we think of something in its essence, we think of it as unchanging and eternal, abstracting from the passage of time. “Now” is time abstracted from duration or existence.
Other philosophers have said similar things in different ways, according to their metaphysical schemes. Duns Scotus asserts that an essence is constituted by a self-moving process, but for him “essence” is an individualized, potential existent. This agrees with what was said immediately above, except it is expanded to include potentiality in process. Pietro di Giovanni Olivi (c. 1248-1298) likewise held that self-existence is possible not in an infinitesimal instant, but only in a duration. Bergson took this further, making a strong identification between existence and process, dispensing with universal essences. Regardless of the metaphysical rationalization invoked, some ontological dependence between time and existence is indicated.
For some substance X to exist “in time” is not the same as X simply existing when time exists, i.e., “co-existing” with time. A more intimate relation is suggested. The existence of X is not incidental to the existence of time, as is the case with co-existing objects in general. Time necessarily follows upon the existence of X, as the measure of that existence. Thus time might be called a property or attribute of the persistent (i.e., existent) thing.
Considered in another way, however, time seems to be antithetical to existence. Things suffer under time, which seems to make things waste away. Material substances may decay, and minds may lose their memories, which gradually become less vivid and complete, from the passage of time. No positive perfection or improvement comes from the mere passage of time as such, but rather some other physical power must be brought to bear. We learn not from the mere passage of time, but from active studying. Things grow not from time, but from some vital process. Yet it seems that, in the absence of any positive natural process, time alone will cause things to deteriorate.
Is it fair to impute to time itself the process of decay? Might there not be natural processes of decay? Alternatively, cannot new creative processes themselves incidentally (or necessarily) effect the decay of other objects? Aristotle does not say that all decay is produced by time, only that some of it is. We might express this stochastically or entropically. Left to itself, without any contrary process, a complex substance disintegrates with the passage of time as a result of purely statistical considerations. This “thermodynamic arrow of time” is unidirectional.
The invocation of statistical mechanics, however, suggests that such stochastic decay is really a property of movement, not of time. Against this, one might contend there are cases where a thing is destroyed without moving at all (as when movement is necessary for creation and growth). Aristotle replies: “Still, time does not work even this change. Even this sort of change takes place incidentally in time.” As Aquinas clarifies, time is the number of movement, and change is per se destructive (since something must cease to exist in order for the new thing to appear), so time is destructive only per accidens.
While place is the measure of the mobile qua mobile, time measures the mobile incidentally, via its motion, rather than its proper quantity (i.e., its bulk or volume). Accordingly, things measurable by time must be capable of motion and rest.
Since time does not measure proper physical quantity, it need not be finite. Indeed, by its very constitution, time will not fail to continue, because the now is always at a beginning (while at the same time an end of something else).
The fact that time is a measure does not imply that it is merely psychological, a product of our subjectivity. It is a number in the sense of the thing numbered rather than of that by which we number. Numbers of objects (e.g., seven sheep) may exist even if there is no one to count them. Since the countable or measurable can exist even if there is no one to count or to measure, time qua countable or measurable may be objectively real. Nonetheless, the objective reality of the measurable does not imply that the magnitude of a measure is absolute, i.e., the same in every frame of reference.
The concept of time as a number (of termini) or measure (of movement) should be understood univocally, even when applied to different movements. Time is specifically one, so there is not one kind of time for this motion and another kind for that motion. When there are seven horses and seven dogs, there are not two different numbers called “seven,” but they are one and the same number, applied to different species, and the same holds for continuous measures. We affirm that time is one in kind because there is one and the same time for all motions that are equal and simultaneous, not a plurality of times. In such cases, time is also numerically one.
Aristotle held that time is one for all equal and simultaneous motions, no matter where they are. This last specification is untenable, since we have discovered that time is not neatly isolatable from space. Thus time is known to be numerically one only for movements in identical loci (e.g., two alterations in the same place) or sets of loci (e.g., two bosons moving simultaneously along the same trajectory). We cannot necessarily make univocal comparisons in magnitude for times in different parts of space.
Still, it may be possible for time to be numerically one if all motions are referable to the same primary motion. In ancient Greek cosmology, the motion of the outer heaven, the first and most regular of motions, served as the clock by which all other time could be measured. While we now identify this motion with the diurnal rotation of the earth, we may still appreciate the basic insight that uniform circular motion is most regular, since each part of that motion is identical, resulting in a constant period. The same applies to simple harmonic motion, since it is a projection of uniform circular motion. Astronomical periods are not quite as unchanging as formerly believed, due to gravitational perturbations, yet we may use smaller cycles, such as the oscillations of strontium atoms in a lattice, to mark time without losing a second over billions of years.
In order to prove that time is numerically one, not just specifically one, it is not enough to show that all motions are interrelated, but they must all be referable to a single motion. In the example of seven horses and seven dogs, there is only one kind of “seven” in both cases, so “seven” is specifically one, but it is not numerically one, since the “seven” for the horses is not identical with the “seven” for the dogs, since to be seven horses is not the same as to be seven dogs. There are two “sevens,” i.e., two sets of seven things.
We may easily accept that time is specifically one, but it is far from obvious that time is numerically one, unless we appeal to a shared causality with a common source. Yet even if time is numerically one by this commonality of origin, this does not eliminate the possibility that time may branch off in different directions, as the parts of the universe spread away from each other, and that subsequently it is no longer possible to define simultaneity univocally across the reaches of space.
An important consequence of the relativity of simultaneity is that the magnitudes of space and time measurements are interrelated. While the metric aspects of place and time can be integrated into “spacetime,” any truly physical interpretation of relativity must take into account the ontological differences between place and time that we have discussed. Outlines of such interpretations will be shown in the next chapter.
Continue to Part IV
 In this chapter on place, “corporeal” refers only to sensible substances with some geometric extension or continuous quantity of matter. Here we do not pronounce on whether it is possible for metaphysically real, insensible substances to be corporeal.
 In fairness, Newton did not think space was an essential attribute of God. Rather it was “where” God could act, and therefore unlimited in extent by virtue of the limitlessness of divine power.
 Newton correctly shows, against Descartes, that extension cannot be identical with prime matter, since it is a definite kind of quantity. Still, his adoption of the other extreme, that extension does not at all pertain to the material aspect of substance, results in the unsolvability of the problem of individuation.
For an extensive discussion of this shortcoming, see: J.E. McGuire, “Space, Infinity and Indivisibility: Newton on the Creation of Matter” in: Zev Bechler, ed. Contemporary Newtonian Research, Vol. 9 of Studies in the History of Modern Science, pp. 145-90.
In particular, McGuire notes that Newton identifies material phenomena by their operations, which depend on their intrinsic powers. Yet these powers have no intrinsic correction with their extension, so extension cannot account for the instantiation of matter. This leaves unanswered: “But what are those powers powers of?” [p.178] i.e., what is the individual, determinate existent?
Newton claims the apparent permanence of matter moving across space comes from God continuously acting on space in the same way. God creates matter first here, then in the next place, along a continuum. While this doctrine of successive creation seems highly unparsimonious, Newton thinks otherwise, since he refers similar effects in separate regions of space to a single divine cause. In his view, persistent material is only apparent, the result of a similar form imposed over time and space. Yet this denial of persistence destroys the unity of the existent. Applied to humans, the problem is especially acute, as we must assert that our persons do not really persist over time.
McGuire rightly concludes: “There seems no way of analyzing the spatio-temporal continuity of any bit of matter without quantifying over places and times. This means that the criteria of identity over time for a figured part in space depend on some bit of matter remaining numerically one and the same at different times. Equally, however, criteria of identity over time for a bit of matter presuppose that places in extended space remain numerically one and the same at different times.” [p.180]
 Philoponus. On Aristotle Physics, trans. Keimpe Algra and Johannes van Ophuijsen (London: Bristol Classical Press, 2012), p. 40.
 In 1999, investigators of the muon g-2 experiment at Brookhaven National Laboratory were asked by a theoretician to look for a diurnal effect, which would indicate alignment with respect to a cosmic tensor with absolute directions. No such effect was reported. The author was then a graduate research assistant on the project.
 For further discussion of the distinction between intrinsic and extrinsic accidents, see: Charles Coppens, SJ. Logic and Mental Philosophy (New York: Schwartz, Kirwin and Fauss, 1891), “Mental Philosophy”, Bk. I, Ch. iii.
 Heidegger would disagree that this is the most fundamental notion of “in,” instead regarding Dasein as being-in-the-world, i.e., in contact with other things, not necessarily spatially contained.
 Leibniz argued that points could serve as loci, since “space,” the locus internus of the cosmos, was constituted (though not composed of) points, which have situs but not extension. This radical separation of point-loci from extension is possible in mathematics, as proved by Dedekind’s construction of the continuum from purely arithmetic considerations. In physics, however, Leibniz later recognized that point-loci must be “not exact,” i.e., not truly infinitesimal.
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