Basic Issues in Natural Philosophy

Reviewing Aristotle’s Physics in Light of Modern Science

Daniel J. Castellano (2015)

Part II

[Full Table of Contents]
8. Movement
9. Mover and Moved
10. The Infinite
   10.1 Reality of the Infinite
   10.2 The Infinite in Natural Philosophy
   10.3 The Infinite as Something Separable from the Sensible World
   10.4 The Infinite as an Actual Physical Being
   10.5 Infinity as Potentiality of Process

8. Movement

Physics is specially concerned with a type of change known as movement or kinesis. Aristotle defined kinesis as a change (metabole) that is a continuous process between definite starting and end states. That is to say, a movement passes through a continuum of intermediate states between two definitely distinct states. “Change” more generally may be considered as process without reference to starting or end states, and may be continuous or discontinuous.

Movement, rather than change in general, is more properly the domain of physicists, as virtually all physical changes can be modeled as changes of this sort; i.e., as continuous transitions across all intermediate states between two states. Quantum mechanical “measurements” seem to be an exception, but even these are modeled by a continuously developing time-dependent wavefunction. At any rate, the study of kinesis is a good starting point for physics.

Movement can be between states of location, quality, quantity, or substantial being. Thus Aristotelian kinesis is a broader concept than what we ordinarily understand by motion, i.e., a change in location. Kinesis can refer to qualitative alteration, increase or decrease in quantity (i.e., bulk or amount), generation and destruction. These last two mean coming into or out of being as a substance, without necessarily implying that there is such a thing as absolute creation or annihilation. A substance may come into being though it is made of some prior matter, and may cease to exist as that substance even if that matter persists.

All these processes are conceived as a continuous passage through intermediate states, so that each state seamlessly grows out of the previous. It is for this reason that the term “physics” comes from the Greek word for growth (φυη), and this seamless continuity gives us confidence in the reality of causality. Without such continuity, there would be only a succession of disconnected happenings, undermining the possibility of physical explanation.

Aristotle seems to have ignored two ontological categories, time and relation, as possible end-points for movement. Time, we will later see, is the measure of movement, not a set of states to move from or toward. If an object could “change” only in time and in no other respect, we should say it has not changed or moved at all. It is questionable whether we could truly say that time had passed for that object, if it has not changed in any other respect.

The neglect of changes in relation is a serious omission. Aristotle only considers the possibility of correlatives being endpoints of movement. Dismissing this possibility, he ignores other types of relation. To anyone with a cursory knowledge of mathematical physics, it seems obvious that there can be continuous changes in relations, such as higher order derivatives, which we will discuss later, when considering the possibility of “movements of movements.” Nonetheless, as discussed previously [Introduction to Ontological Categories, Part IV], a relation can admit variation in degree only if two or more of its predicates have magnitude, so movement in relation is ultimately referable to movement in one of the other categories of accident, though without requiring absolute values of the magnitudes of each predicate. Not all formally defined relations among properties constitute relational accidents that deserve to be called physical, i.e., essential attributes of some principle of change.

There is at least one relation essential to analyzing physical movement, namely that between the mover (that which can cause movement) and the moved (that which can move). There is no logical reason why the mover and moved cannot be the same physical object, though they would be formally distinct in terms of function. Since nothing comes out of nothing as such, the beginning of a movement must have some existent causal agent, which we call the mover. The moved is the object actually undergoing the movement in question, changing its physical state.

The mover and the moved might be different real entities, but the movement itself cannot be a tertium quid, at least not in the sense of having independent existence. At most, the movement may be considered an ontological accident of the moved. It can have no existence apart from its subject, since the very notion of movement presupposes beginning and end states, which in turn requires some suppositum that holds those states, and this subject must persist throughout the movement if we are to say that something has moved. This last clause is denied by process ontology, which would make the process or movement itself the suppositum. In this case, there is no distinction between the “moved” and the movement, so process ontology does not make movement a third thing, but the only thing.

Although there can be movements in different ontological categories (place, quality, quantity, substance, and perhaps relation), movement is not thereby something that transcends the categories. In other words, there is no “higher” or more generic movement with respect to something that transcends the categories; movement is always with respect to some determinate category. We have shown previously that the categories are conceptually primitive modes of being that have no common predication, so one cannot construct some more generic meta-category, at least not as a concept corresponding to a mode of being with a common predication. For example, there is no “being a quantity-quality” that encompasses “being a quantity” and “being a quality,” such that would enable us to construct ontological statements that could handle quantities and qualities with indifference. Any formal logic that ignored such distinction would be fallacious as an ontology.

Aristotle’s initial descriptive definition of movement is inadequate for physics, since it does not explicitly bring out the causal mechanism we find essential to scientific explanation. If a movement is just a continuum of states between two endpoints, then we might be describing a locus in some phase space, without showing any real dynamism.

Realizing this deficiency, Aristotle employed his concepts of dynamis (power; potentiality) and energeia (activity; actuality) to develop a fuller definition of movement. In the earlier discussion of change (Part I), we ignored these concepts, finding that the reality of change was sufficiently accounted by the presence of persistent and mutable aspects of natural beings.

The metaphysical concepts of dynamis and energeia are eminently useful for explaining movement. When we situate a movement in time, it becomes meaningful to speak of each state as actual or not yet actual. At the beginning of the movement, the final state or form is not yet actual, though it will become so when the movement is completed and we have advanced in time. If we consider that the movement, or succession in states, is driven by a causal mechanism, then we may conceive of the movement as containing or manifesting a power (dynamis) to bring the subject to the final state. In short, it has the power to make the final state actual. The mere presence of the movement, even if it is not yet completed, is evidence that the final state can be attained. The final state then may be said to already exist as a possibility or potentiality, which is “a thing that can be.” Since we are here considering physical potentiality, we require not merely that a thing can be thought of without logical or metaphysical incoherence, but that a physical power is present that can produce it as an effect.

Clearly, any account of movement that is both situated in time and causal will need some concepts of potentiality and actuality. It is less obvious, however, that such concepts need to be formulated as Aristotle did. One of the odder conventions of Aristotelianism, from a modern mathematical perspective, is that potentiality and actuality are defined to be mutually exclusive. We would consider something that exists with probability 1 is also possible; indeed its actuality is the most incontrovertible proof of its possibility. Why, then, did the Aristotelians consider potentiality to vanish upon completion in actuality? Should not actuality or activity be considered the perfection of power, rather than its dissolution?

First of all, actuality is something more than existing with probability 1. When we assess the probability of anything, evaluating it from 0 to 1, we are considering it prior to (or abstracted from) its actualization. Just because something will certainly occur does not mean it has yet occurred, so its mode of existence as possibility, even with probability 1, is distinct from its eventual mode of existence as actuality.

When we say object A exists as potentiality, we mean that the power to produce A presently exists, not that A itself exists as some ghost from the future. In other words, A “exists” only as power (dynamis). Once A is produced, it can no longer be said that the power to produce A is present in that instance, for that particular object A is already produced. The “power” has been spent. Physical potentiality or possibility should not be confused with abstract mathematical possibility, which is a merely formal quantification of likelihood, without overt reference to actuality or its absence.

We are squarely within the realm of physics, not metaphysics, when we speak of potentiality and actuality in terms of physical powers and operation. Indeed, the classical terms dynamis and energeia simply mean “power” and “operation” in their ordinary meanings. (The latter term is unattested before Aristotle, but literally means “in doing.”) Analyzing physical changes in terms of potentiality and actuality means that we consider physical states not as mere successions of events, but as products of real powers. The produced objects themselves may be considered as being “in operation,” so that they may power the production of other things. Those things yet to be produced exist only as “power” in some other agent.

It should be clear from this discussion that potentiality and actuality are not two distinct kinds of stuff (which is why they are not ontological categories), but rather two modes of existence that an object may have with respect to a given state of affairs. A currently actual object may manifest another yet-to-be-realized object as potentiality, not that there are two things, actuality and potentiality, in the first object; rather the only thing that exists is the nature of the first object, which is both its own actuality and the potentiality of the second. There is no strong contradiction, then, with Nietzsche’s contention that there is only operating force, without “power” as some distinct substance. If we examine the world, we only see forces at work. These actual forces are “powers” only as considered with respect to something that is not yet actual.

Movement is “the activity of what exists potentially, insofar as it exists potentially.” Specifically, alteration is the activity (energeia) of what is alterable qua alterable, and so on for the other categories. Being moveable, in this conceptualization, means being potentially. In other words, movability implies a capacity to come to be in some other state than the current state, and the basis of this capacity is some physical power already present. Movement is the actualization or operation of this power; the realization of a potential to become what one is not yet.

To give a concrete example, the actualization of bronze as bronze is not movement. Rather the actualization of bronze as something malleable would be a movement. Movement makes use of some capacity for change in bronze. If it were merely the capacity for bronze to be what it already is essentially, it would not be movement at all.

The “actualization” that is movement is different from that of a completed or final state. Since movement includes an entire continuum of intermediate states, there is actualization even in these intermediate states, as each becomes present in its turn. Aristotle describes this intermediate actualization by making movement the actualization of the potential qua potential. He illustrates this with the example of house-building.

The actuality of the buildable as buildable is the process of building. For the actuality of the buildable must be either this or the house. But when there is a house, the buildable is no longer buildable. On the other hand, it is the buildable which is being built. The process then of being built must be the kind of actuality required. But building is a kind of movement, and the same account will apply to the other kinds also. [III, 1]

Movement operates on the potential (e.g., the buildable). It obviously cannot operate on the completed house, for then there is nothing to be done. It is only insofar as the object is not what it will be, yet at the same time is capable of becoming something other than what it is currently, that it can be in movement.

As an alternative to Aristotelian potential, one might say that movement consists simply in the object not being what it is already, so that movement is a kind of privation. This has the apparent advantage of making movement univocal across categories. Yet this would be an inadequate definition, for the mere absence of some determinate being does not suffice to move one towards what is absent. Just because X is not Y, it does not follow that X is in the process of becoming Y.

Note that both potentiality and actuality are needed to account for movement. Something that is merely capable of undergoing movement need not be undergoing movement, and that which has actually attained some state is no longer in motion toward that state, so movement cannot be pure potentiality or actuality. Movement might be an incomplete actuality, as may be said without contradiction if the incompleteness is due to an incompleteness in the potential. Aristotle recognizes that this is hard to grasp, but holds that there at least is no strong logical objection, making his definition advantageous over the alternatives.


9. Mover and Moved

Physical movement is fundamentally causal, if we accept that changes in the sensible world are explainable. We call the cause of a movement the mover, and the subject of movement the moved. These need not be bodily distinct entities. Since movement involves change from one state to another, where each state is defined by some quality, quantity, location, relation, or substantial species, we may define each movement by some form that characterizes the final state. This is only consistent with what we know of change in general, where persistent matter may adopt different forms over time. The mover, then, may be said to impart a form onto the moved.

Natural philosophers from antiquity through the early modern era have mostly agreed that movement is mediated by direct contact between the moved and the mover. The ubiquity of this opinion is astonishing, given how far from self-evident it is. We have noted that continuity is the strongest evidence of causality in physical changes, yet a temporal continuity might suffice, without need for spatial continuity (i.e., direct contact). With relativistic insight into the interdependence of space and time, however, the point is moot, since temporal continuity implies spatial continuity.

One might object that Newton’s Third Law abolishes any real distinction between “mover” and “moved,” at least with respect to local motion. Yet even Aristotle acknowledged that the cause of movement is “contact with what can move so that the mover is also acted on.” Action by direct contact entails action in both directions. Newton’s law only adds that the reaction must be equal in magnitude to the action.

Modern physicists are accustomed to regarding the distinction between “action” and “reaction” as formally arbitrary. Yet for Newton himself, who believed in absolute space, and therefore absolute rest and motion, such a distinction could be physically meaningful. When a moving body A strikes a resting body B, it makes sense to think of A as initiating the collision, so A “acts” on B and B “reacts” to A. Action and reaction are nonetheless simultaneous in time and equal in magnitude. Even on the supposition of absolute space, however, the distinction between “action” and “reaction” is obscured if both bodies are initially in motion. If we further introduce a principle of relativity (even with a Galilean transformation), then the distinction collapses for all interactions, since there is no absolute rest.

The lack of a real distinction between “action” and “reaction” does not mean there is no distinction between mover and moved. “Action” and “reaction” are two distinct actions, each of which having a “mover” and “moved.” Body A moves body B, and body B moves body A. The physical reality of one causal action does not abolish the reality of the other. Still, the principle of action and reaction implies that nothing can be a mover without also being moved. This will have bearing on “prime mover” arguments. Any “unmoved mover” would have to be exempt from Newton’s Third Law.

Newtonian reaction applies only to local motion, not to other kinds of movement or change. Mechanistic physicists have held, however, that all qualitative alterations, increases, decreases, generations and destructions are explainable in terms of the local motions of material constituents. Even so, changes in quality, quantity and being as such would not necessarily have corresponding reactions, though the micro-actions underlying them would have micro-reactions. Recall, however, that there is some sort of reaction in all kinds of movement mediated by direct contact. If an agent effects change by imparting something of itself, we should expect that the product conversely affects the agent. At any rate, this is what we always find in nature: anything that effects change is itself changed thereby.

A ubiquitous principle of reaction creates a conundrum, as you always have causality operating in two directions. This chicken-and-egg scenario leaves us with no apparent hope of explaining reality in a linear regress of causes. If absolutely all changes (i.e., not just movements) obey a principle of reaction, the cosmos could have no definite causal origin. This is not problematic on Aristotle’s assumption of an eternal universe, but it becomes so as our universe appears to have a definite origin in time.


10. The Infinite

Physical movements involve magnitudes (of quality, quantity, or space) and time, which can be regarded as finite or infinite, i.e., with or without limit. It is therefore within the domain of physics to inquire whether there are infinite substances or properties.

Direct observation or measurement of the infinite is impossible, for we can sense a physical object only by finding some limit differentiating it from other objects, and we can measure a magnitude only by determining its quantitative limit. Nonetheless, it might be possible to infer the existence of the infinite, or to disprove it, from the aspects of reality we can observe.

10.1 Reality of the Infinite

Various ancient Greek philosophers thought they could infer the existence of the infinite, from one or more of the following considerations.

  1. Time never ends, so it is infinite.
  2. Magnitudes are infinitely divisible, as shown by mathematics.
  3. Things ceaselessly come into and out of being, so what underlies their generation is eternal.
  4. That which is finite is limited by something else beyond it, and this may continue without end.
  5. There is no stopping-place in thought for ever greater numbers or magnitudes, or for the outer boundary of the universe.

Aristotle recognized that none of these considerations suffice to establish an actually existing infinite, but at most they might suggest an infinite potentiality. (1) Time (and associated movement) can be infinite only because one thing becomes actual after another ceases to be, yet at no point do you necessarily have an actual infinite being. (2) Magnitudes of physical beings need only be infinitely divisible in potential, not actually divided ad infinitum. (3) An eternal succession of generation and destruction need not depend on an infinite substance, for each thing may be produced out of a previously destroyed finite substance, and the cosmos as a whole may still be finite in extent. (4) Not all finite things are limited by something surrounding them, nor are all finite things necessarily in contact with everything else. (5) Lastly, the infinite in thought proves nothing about physical reality.

Modern physics and mathematics raise further objections to these arguments. The use of non-Euclidean geometry in relativistic cosmology allows a space-time that is finite in magnitude, yet without a boundary, being curved in on itself, like a closed loop or surface. Thus Lucretius’ question about where a spear would land if thrown from the edge of the cosmos is rendered inapt. The universe need not be infinite in space or time.

None of these rejoinders, however, disprove the existence of the infinite as potentiality. Even if the universe is actually closed and finite, it may be physically possible for it to have been greater in extent, even “open” and expanding indefinitely. Thus the universe might be at least potentially infinite. To illustrate, if you have a closed loop with circumference C, there is at least the theoretical potential of a circumference greater than C, even if there is no visible wall or boundary to your loop path. It is likewise with finite time; the supposition that time actually curves in on itself in a “big crunch” with no definite endpoint does not abolish the possibility that time might have extended further.

If the world really is finite in space-time, this would not abolish the infinite absolutely, but at most within the cosmos. The “curvature” of space-time is intelligible only in the context of some metaspace, even if coordinates in that metaspace are computationally unnecessary for the physics of our world. At most, Einsteinian relativity pushes the question of infinity out of physics and into metaphysics.

The metaphysical necessity of the eternal is likewise not abolished by modern physics. A finite universe is necessarily both absolutely mortal and absolutely created. Such a universe must be the creature of something else, though this is beyond the domain of physics.

Even if all the extensive and numerable aspects of the physical world are actually finite, they might nonetheless be potentially infinite. It seems, however, that an infinite potential, expressed for example as an infinite void, may imply the existence of an infinity of substance. After all, if there is an infinity of space or metaspace, time or metatime, then why should substance exist only in the cosmos we know? There must in fact be infinitely many worlds. Whatever may be the merits of such reasoning, this goes beyond the domain of physics, insofar as it postulates worlds we may never sense.


10.2 The Infinite in Natural Philosophy

The term “infinite” (or “unlimited”) can be used in various senses, only some of which are relevant to physics. “Infinite” might characterize the magnitude of some physical property, be it a quality, quantity (i.e., how much of some substance), or spatial (e.g., distance, area, volume). Magnitudes of physical relations are ultimately referable to these categories. A physical magnitude may be modeled mathematically as a discrete number or a continuous extension. (From measurements of the latter, we get the so-called real numbers.) If physical magnitudes could be infinite, then we might have infinite movements, since the bounds of a movement are defined by the magnitudes of its starting and end states. There cannot be infinite movement without infinite magnitude. Likewise, there cannot be infinite time unless there is an infinite movement, or an infinite number of successive movements. If, as we have suggested, physical magnitudes are only potentially infinite, then it follows that movement and time are also only potentially infinite at most.

A couple of objections may arise. First, it seems possible to have an infinite relative magnitude derived from finite magnitudes. For example, you might have “infinite resistance,” defined as a ratio of some finite voltage over zero current. In quantum mechanics, you can have “infinite temperature” in a spin system where the ground state and excited state are made equipopulous by population inversion.

Yet these formally defined relations fall short of being physical entities. If there is no current to resist, then there is no physically real relational property between voltage and current. Zero does not correspond to a finite physical quantity, but to an absence, so an essential term of the relation is missing, and the ratio does not correspond to a physical existent. “Infinite resistance” is just a formalism. Insofar as resistance corresponds to a positive physical property, i.e. resistivity, it is always finite in magnitude.

Likewise, you can only have infinite or negative temperatures when you define temperature in a way that is divorced from average kinetic energy. The formalism of infinite temperature arises from taking the Boltzmann distribution for thermal equilibrium and applying it to a small number of particles (to guarantee exact equality of numbers), ignoring other properties (e.g., local motion). Yet such a probability distribution was designed to be a statistical description of large ensembles of particles. It is inappropriate to use a statistical model to compute a physical variable (temperature) under conditions where that model does not hold (i.e., non-equilibrium states, small groups of particles).

When we restrict our discussion to physical rather than formal relations, it is evident that these can have no infinite magnitude unless there is some real physical property of infinite magnitude underlying them. Thus we can restrict our discussion to the magnitudes of quality, quantity, and space.

A second objection is that it seems possible to have an infinite movement without an infinite magnitude, as in the case of a ring, loop, spheroidal or toroidal surface, or any other closed manifold. One may move around and around a ring without end, though the ring is finite in spatial extent.

We may describe this situation by saying the ring is metrically finite, but parametrically infinite. In other words, the ring itself is finite in extension, since it would be invalid, when measuring it, to count the same part over and over again. Yet if we were to trace along it or any other closed manifold in a fixed direction within it, this process would never end, as it would encounter no limit or boundary. The variable measuring the duration of this process is a parameter, since it does not really measure the manifold, but rather some process of traversing it, quickly or slowly, at a constant or uneven rate, once or many times. This variable is not spatial, though its value can have some correspondence to spatial coordinates. Since we are dealing with physical movement, we can call this parameter t, to represent time, the measure of movement. Yet we should not call such cyclical movement infinite, since physical movement is defined by traversing physically distinct states. Rather the same finite movement can be repeated infinitely many times. If the totality of physical movements in the cosmos merely repeated themselves cyclically, we should say that time follows a finite cycle, like Nietzsche’s eternal recurrence, rather than call it infinite in the sense of transcending all magnitudes. Infinite time requires infinite movement, just as infinite movement requires some infinite physical magnitude.

In natural philosophy, we may apply the term “infinite” to (a) some physical property with unlimited magnitude, (b) the subject bearing such a property, (c) a process or movement across such magnitude, or (d) the time such a process takes.

If what is “infinite” is some physical property (a) or its subject (b), we mean that its extent (i.e., range of magnitude) cannot be traversed by any process that terminates. Here we assume that some category of magnitude is applicable to the infinite thing. If this were not so, it would be “infinite” only in an accidental sense, i.e., it is “not finite” since magnitude does not apply to it at all. Another equivocal use of “infinite” would involve supposing that the thing in question can be traversed by some process or movement, but no such process has actually occurred.

The above sense of “infinite” accords with the mathematical notion of “infinity” as an unbounded quantity greater than every real number (i.e., every extensive magnitude), yet another notion of infinity arises when we apply it to a process.

A converging series of quantities, by addition, subtraction, or a combination of both, is infinite when considered as a process, though its sum may be limited or finite. In antiquity and in the Middle Ages, “infinity by addition” and “infinity by subtraction” were conceived geometrically. The latter may be illustrated by taking a line segment, removing half, then removing half of what remains, and so on. Such a process can continue without end, since continuous magnitudes are infinitely divisible. Suppose we take each of these segment fragments, and add them to some other segment of unit length, so that its total length is 1 + ½ + ¼ + ⅛ … Although this process can continue forever, it will never bring us to a quantity greater than 2, so it is only infinite as a process, not as a magnitude. Infinity by addition and subtraction can exist even in a world where all physical quantities are finite. Thus the physical reality of “infinitesimals” has no bearing on our present inquiry as to whether there are physically real infinite magnitudes.


10.3 The Infinite as Something Separable from the Sensible World

An existent Infinite may be considered either as (1) something separable from all sensible substances, such as “the One” or “the All” proposed by some philosophers, or (2) as something that is causally connected to the sensible world, and so a principle of sensible change. Against the first possibility, Aristotle argues as follows (paraphrasing):

(1a) If the Infinite is a substance, it must be indivisible, for only magnitudes and aggregates are divisible, and these are attributes, not substances. Yet that which is indivisible cannot be infinite, except in the first sense.

(1b) If the Infinite is an attribute, then it cannot be a constituent element of any substance, any more than “invisible” is an element of speech.

Here (1a) Aristotle uses a restrictive definition of substance, excluding aggregates, though elsewhere in his corpus he allows that aggregates and parts of substance fall under the category of substance. Strictly speaking, an aggregate qua aggregate depends on its constituent substances for existence, so it has no being other than that of the constituents. A physical substance is formed not by mere aggregation, but only by combinations that create new physical properties or capacities. Physical substance as such is indivisible, though a substance may be transformed into a plurality of substances.

A substantial Infinite that is separable from sensible substance is “infinite” only in the generic sense of “not finite,” since it is without magnitude or number, like all substance as such. Thus when pantheists and theists speak of “the Infinite,” “the One,” or “the All” as something separable from sensible substance, they mean something that has no limit only because extension does not apply to it, not something physically infinite in extent.

Suppose we allowed that the Infinite could be an aggregate, and said the Infinite exists because there are infinitely many substances. This statement would have no physical significance, because the Infinite does not account for why there are infinitely many substances if it is nothing more than their aggregation. Here the Infinite would have a purely nominal existence.

If the Infinite is an attribute (1b), as seems necessary since number and magnitude are attributes, then it cannot be an element of substances, as some natural philosophers held. Attributes depend on substances for their existence, so they cannot account for the existence of the substance to which they pertain. They might account for the formation of secondary substances, but this is not what advocates of the Infinite in natural philosophy claim. The notion that the Infinite is the material or substantial source of all physical reality is contradicted by the necessity that the Infinite is an attribute of quantity. The only way the Infinite could be the ultimate foundation of physics is if “Infinite” does not mean something of unlimited extent, but something to which the category of quantity does not apply.

Aristotle discards the idea of “the Infinite” as something hiding behind physical reality. The notion of actually existent infinite extension necessarily entails connection to the quantitative, sensible world.


10.4 The Infinite as an Actual Physical Being

Supposing, then, that the Infinite is connected to the causal structure of the physical world, it should be able to act as a principle of sensible change, or an attribute of such. First, (2a) consider if the Infinite were a physical (natural) substance; i.e., a principle of sensible change. Second, (2b) consider if it were the attribute of a physical substance. Aristotle argues against both options as follows (paraphrasing):

(2a) Since, for a substance X, “being X” and “X” are the same, so “being infinite” and “the Infinite” would be the same if “the Infinite” were a substance. If the Infinite has parts, these parts must be infinite, just as a part of air is air. Yet the same thing cannot be many infinites. So the Infinite must not have parts, which means it is indivisible. Yet if the Infinite is an actual substance; i.e., something in full completion, it must be a definite quantity, and therefore (virtually) divisible.

(2b) If Infinity is an attribute of a physical substance, then it cannot be a principle of sensible change, but rather the substance of which it is an attribute is a principle.

The first argument (2a) allows that substances can have parts, which seems to contradict the earlier denial (1a) that an aggregate can be a substance. Yet the “parts” of a substance allowed by Aristotle are merely virtual. He does not consider these to be a plurality of substances, on account of homogeneity. A volume of air can be virtually divided into any arbitrary number of parts, but none of these parts qua part is a physical principle. The various parts of the volume of air are physical principles only insofar as they partake of the nature of air. Thus there is only one principle or substance, air, actualized over various parts of space.

Supposing that the Infinite were a physical substance like air, it might also be virtually divisible into parts. Yet while we can allow that a part of air is air, Aristotle finds it impossible that a part of the Infinite can be infinite, for then the same thing would be many infinites. The problem is not mathematical, for obviously there is no contradiction in a subset of some infinite domain being infinite. Rather, the problem arises specifically from conceiving the Infinite as a physical substance. If it were some homogeneous substance like air, then every part of it, even those of finite extent, ought to exemplify its essence. Its essence (or being) is “the Infinite,” for, as a substance, “being infinite” and “the Infinite” are one and the same, so all of its parts must be infinite. Yet it is absurd for a finite part to be infinite, so the Infinite, if it is a physical substance, cannot be virtually divisible into parts.

Aristotle holds that all actual substances must be divisible, which seems to contradict his earlier assertion (1a) that substances are indivisible. From the rest of the present argument (2a), however, it is apparent that he is speaking only of virtual divisibility. Any substance that really exists as a principle of sensible change in the world must operate on some volume of space, and therefore be virtually divisible into parts. Yet this virtual divisibility of the Infinite as physical substance has already been contradicted, so it is impossible for the Infinite to be a physical substance.

Note that this argument uses the Aristotelian notion of substance as a kind of being. “Air” and “being air” are one and the same thing. This ontology is unobjectionable to most modern logicians, who view any reification of “being” with distrust. At the same time, it does not preclude the introduction of a metaphysical distinction between essence (being) and existence (“to be”), though this is beyond the scope of natural philosophy.

A possible exception to the above argument would be to suppose that the Infinite were operable only at some indivisible point. Yet in this case, we would be dealing not with infinite magnitude, but the infinitesimal. The reality of infinitesimals seems problematic in view of the requirement that an actual substance, being existentially complete, must have definite quantity, but there is no obstacle to admitting their reality as potentiality. Although modern physics frequently assumes point-like particles and singularities, it may be that these, much like the infinitesimals in differential calculus, are just useful approximations, not precise accounts of physical reality.

Some philosophers, such as Plato and Anaxagoras, treated the Infinite as an elemental substance or first principle, yet admitted it could be physically divided into parts. Yet this would make those parts primary, according to Aristotle, who considers particulars rather than universals to be primary substance in his ontology. For the purposes of physics, he is correct to insist that the parts are primary, as long as these parts contain distinct principles of sensible change.


The supposition that the Infinite is an attribute of a physical substance (2b) is not refuted here, but it suffices to remark that an attribute as such cannot act as a principle of change, unless it is a property of something that really exists, i.e., a substance. Again, this ontology does not require that substance is corporeal stuff. Substance is merely the suppositum which is presupposed by the real instantiation of some property. (In trope theory, the property itself is the suppositum.)

It remains to be seen whether infinity could at least be an attribute of some actual physical substance. (2b) Since infinity is an attribute of number and magnitude (themselves attributes, further proving that infinity is not a substance), it follows that it can be physically actual only on something extended in space, since space is the principle of pluralization in physical reality. That is to say, if infinity is the attribute of anything physical, it must be the attribute of something corporeal, i.e., something with spatial extension or numerability.

Aristotle’s arguments against the existence of an infinite body (not shown) rely on the faulty physics of his time, which supposed that elements must be balanced contraries, and that substances have a naturally preferred proper place. We should consider whether there is anything in modern physics that would preclude an infinitely extended body.

In modern cosmology, it is still an open question whether the universe is (1) “curved into itself” (like a toroid surface, analogically), in which case it would be certainly finite in extent, or (2) “flat,” so that it can continue expanding indefinitely. Even if it could be proved that the universe is flat, it has only been expanding for a finite duration, so it will always be finite at any given point in time. It would be only potentially infinite; for it to become actually so, an infinite amount of time from the Big Bang would have to elapse.

If everything is causally connected in spacetime, and perhaps this is what it means to be a universe, then there must be spatiotemporal continuity from the origin of the universe. So you could not have anything “starting off” an infinite distance from some other part of the universe, nor could you have infinitely distant objects any finite amount of time after the origin. If anything, modern physics raises stronger objections against the existence of an infinite body than those discussed by Aristotle.


10.5 Infinity as Potentiality of Process

Given that there is no actually infinite body, it does not follow that infinity has no physical reality whatsoever. Aristotle admits a potential reality to the infinite, recognizing, for example, that finite magnitudes are infinitely divisible, and that time may never end. Yet he denies that there can be any infinite magnitude even in potentiality, for potential means the power to realize some actual state. If nothing can ever be actually infinite in magnitude, then there cannot be any physical potential for such. The potential infinity of divisibility refers to a virtual process that can be continued without end, so there is no contradiction here with the impossibility of infinite magnitude.

What, then, of the infinity of time? Accustomed as we are to representing time spatially, it might seem evident that time is an infinite magnitude. Yet this is only a convenient mathematical model, not a true picture of reality. In reality, time is not actual all at once, but one moment replaces the next, so that only the present is actual. The potential infinity of time does not imply an infinity of substances, nor a substance of infinite magnitude, since each thing replaces its predecessor in the succession of time.

Still, it would seem that the potential infinity of time at least implies a potential infinity of movement, either as a movement of infinite duration, or an infinite succession of finite movements. Yet movement is an activity (i.e., of the potential qua potential), so it must be regarded with respect to some endpoint or state of completion in order to be actual movement. The notion of infinite movement is self-contradictory, since the infinite is never complete or actual. This does not prevent us from admitting an infinite succession of movements, which is not problematic, since this does not make the infinite actual, as there is no “infinity of movements” existing at once.

The potential infinity of time does imply that things do not merely exist, but are also always in a state of becoming. Though each physically real movement is defined by some finite end state, this may always be replaced by yet another movement and its end state. In modern physics, we do not ordinarily speak of “movements” in discrete units, but as a seamless continuum of change. Given our spatial representation of time, it seems practically unavoidable to admit the potential infinity of movement. Yet recall that movement is partly defined by the physical difference between start and end states. Since there can be no actual infinite magnitude, neither can there be an actually infinite movement, which would require an infinite difference between magnitudes. Accordingly, there cannot be a potentially infinite movement, insofar as this is understood to mean there can ever be an infinite magnitude. As long as time persists, objects in the universe will be a finite distance apart, and have finite differences in all physical properties. The infinity of time implies an infinity of process, not of magnitude.

Likewise, the infinity by addition (i.e., convergent series) or subtraction (i.e., infinitesimal divisibility) of magnitudes implies only an infinity of process, not an actual, or even virtual, infinite magnitude. This is obvious in the case of infinity by addition, since the convergent series never transcends some finite limit. Yet infinity by subtraction or division always takes us to quantities smaller than any finite quantity we can define. Why is this reasonable? As Thomas Aquinas explains:

... since the infinite is like matter it is contained within just as matter is, while that which contains is the species and form. Now it is clear from what was said in Book II (l.5) that the whole is like form and the parts are like matter. Since, therefore, the division of a magnitude proceeds from the whole to the parts, it is reasonable that no limit be found there which is not transcended through infinite division. But the process of addition goes from the parts to the whole, which is like a form that contains and terminates; hence it is reasonable that there be some definite quantity which infinite addition does not exceed. [Commentary on Physics, Lectio XII, 391]

The infinite is a material principle, not a formal principle, since it does not tell us what kind of a thing something is, but only how much of it there is. Its essence is privation, namely the absence of limit, and its subject is that which is of continuous magnitude. As mentioned previously, matter and form do not exist by themselves, but rather a substance exists as matter defined by some form. Thus form is a sort of limit or bound to matter as an existent, making it act one way rather than other. The easiest visualization of this principle is when the form is simply the shape or spatial distribution of a corporeal substance. The form here literally defines the boundary containing the matter. Since the process of subdivision goes from the whole to its parts, it seems reasonable that the process should go to infinity, since this infinity is contained by the form, and so remains a material principle. As such, the infinite remains merely potential or virtual, not the actuality of form.

The fact that the infinite is a material principle in physics, not a formal principle, means that we should not regard it as some mathematicians do, i.e., as that which has nothing beyond it. On the contrary, the infinite is that which always has something beyond it. Aristotle writes:

A quantity is infinite if it is such that we can always take a part outside what has been already taken. On the other hand, what has nothing outside it is complete and whole. For thus we define the whole—that from which nothing is wanting, as a whole man or a whole box. What is true of each particular is true of the whole as such—the whole is that of which nothing is outside. On the other hand that from which something is absent and outside, however small that may be, is not 'all'. 'Whole' and 'complete' are either quite identical or closely akin. Nothing is complete (teleion) which has no end (telos); and the end is a limit. [III, 6]

The infinite is a quantity that is never completed; there is always something beyond it. Mathematicians may, for convenience, define “infinity” formally (ignoring the self-contradiction implied by “defining infinity”) and treat it as though it were a kind of magnitude with special properties, but this artifice should not be confused with a physically coherent account of reality.

It might be thought that so-called “prime matter,” i.e., matter abstracted from every form, ought to have potentially any quantity, including the infinite. Aquinas responds:

...the claim of some that in prime matter there is a potency to every quantity is false; for in prime matter there is a potency only to determined quantity. [Commentary on Physics, Lectio XII, 396]

This statement relies on the metaphysical thesis that signate matter (i.e., some determinate matter), not matter in general, is the principle of individuation, making something “this one” rather than “that one.” Thus prime matter’s existential potentiality to quantity is always to a determinate quantity, and infinity is not such.

Aquinas continues by distinguishing infinity of number and magnitude:

It is plain also from the foregoing why number does not have to be as great in act as it is potentially, as is said here of magnitude: for addition occurs in number as a consequence of the division of the continuum, by which one passes from a whole to what is in potency to number. Hence one need not arrive at some act terminating the potency. But the addition of magnitudes arrives at act, as was said (no. 391). [Commentary on Physics, Lectio XII, 396]

It may seem strange to admit that numbers are infinite while denying that there can be an infinity of magnitude, since we are accustomed to conceiving of whole numbers as points on a line of infinite extension, so that there is a correspondence between extensive magnitude (length) and number. Yet Aquinas is not here discussing such mathematical abstraction, but referring only to what is physically numerable or physically extensive in magnitude. In order for some collection of entities to be numerically distinct in a physical sense, not merely in formalism, it is necessary for some physical extensive magnitude to be subdivided into parts.

For example, if a collection of objects is spatially distinct and numerable, we must be able to subdivide some spatial magnitude in as many parts. There cannot be five apples unless some magnitude of space is divisible into at least five parts. The same holds even for spatially overlapping entities distinguishable by some other property. We cannot say there are two electrons in the same orbital unless the charge and mass of that orbital are physically divisible into two, or unless the “spin space” admits at least two distinct states. Since physics deals with what is sensible or measurable, therefore determinate and at least capable of actuality, we must start with some finite magnitude from which we may derive distinctions in physical number.

Nothing can be infinite in magnitude, though there can be infinity of process, as a potentiality for indefinite succession. If there is infinity in the sensible world, it is to be found not in being, but in becoming.

Continue to Part III

© 2015 Daniel J. Castellano. All rights reserved.