Aristotle on the Infinite
Aristotle (384-322 B.C.E)
Aristotle is Plato’s best student and accepts many of his teachers ideas while at the same time criticizing and reinterpreting his central idea: the theory of Forms. Before exploring Aristotle’s thought about the infinite we can first examine his philosophical presuppositions and then take a closer look at his model of the cosmos to see how the infinite relates to finite existence. We saw that Platonism presents a systematic philosophy founded on a consistent metaphysics with a theory about both the status of the world and our relation to it. Aristotle also wrote about metaphysics (‘Meta’ = ‘after’ or ‘above’ and ‘phusis’ = 'nature', so ‘meta-physics’ deals with what is above or comes after natural being) which he called first philosophy.
Unlike Plato, Aristotle will reject the radical separation of Forms as the basis of the study of reality. According to Aristotle if the Forms make things actual then the particulars that are modeled after them must somehow have contact with their causes. Plato argued that this contact with the world of change was one-sided. The Forms are the reason why things are, but while they bring about transient things the Forms are themselves transcendent of the spatio-temporal particulars that come to be and pass out of existence. Aristotle rejects this idealism pertaining to substance and argues that the formal cause of a natural thing is always immanent in a material substratum which it actualizes. The Physics presents Arisotle's second philosophy, i.e., observations about the natural world, undergirded with philosophical (i.e. metaphysical) explanations of the nature of sensible substances. He also there discusses the nature of movement, time, space, divisibility, and infinity. Aristotle’s Platonism is present throughout his thought about nature but he also challenges his teacher in a number of important ways.
The world of nature, as Plato taught, is characterized by motion. According to the doctrines presented by Aristotle in the Physics moreover the natural world is also a world of self-moving things. Since Forms are immanent in the world, the end or telos of everything can be discovered through reflection on sensible objects and their development. Aristotelian nature is dynamic and intelligible but not, however, evolutionary in the modern sense since there is no temporal movement towards any end-point nor complexification of living beings in time that changes their essential or formal aspects. Aristotle concieved of time as cyclical and eternal and of natural species as fixed. Circular movement, therefore, defines both life and living nature for Aristotle. Nature always exists, always existed and is an uncreated living thing. This assumption will have consequences for the Aristotelian understanding of natural laws, forces, matter and space. Although Aristotle is not a materialist (since not all being is said to be reducible to nature or natural being) there is nonetheless a sense in which all natural ends are self-sufficient and immanent to the natural order.
Aristotle’s articulation of the four causes and his concepts of ‘potency’ and ‘act’ all inform the natural world in the sense that they must be seen as affecting and making intelligible the processes of sublunary (i.e. natural) change. This leads him to interesting criticisms of Plato. For example, Aristotle critiques the Platonic conception of being as dyadic. [Plato's dyad. as we've seen, consists of multiple particulars ‘the many’ all participating in singlular and self-identical eide or Forms]. We conceive the world of nature inevitably in a false and erroneous way according to Plato. This is because the senses perceive only what changes [the many] and true knowledge is of what does not change (the Forms). Recall that the world of change cannot be said to fully possess being according to Plato. This means that it’s not “really real” but more like a cheap carbon copy of the Forms. Aristotle, on the other hand, thinks that the senses can provide knowledge of reality. To secure the above knowledge, matter and privation must be distinguished. According to Aristotle only privation is by nature not-being. Matter has a reality. What kind of reality? A reality “as potency”, that is matter is nearly, in a sense, real being or substance. The Platonic triad is different- Plato overlooks privation with the consequences that, according to Aristotle, he wrongly views the Forms or formal causes as always perfectly actual and matter as merely a kind of privation of form. The One or ‘Form of the Good’ (described by Plato in his dialogue The Republic) is said by Plato to be the only cause of becoming. Aristotle responds that: “admitting [with the Platonists] that there is something divine, good, and desirable there are two other (real) principles (the contraries)” (Physics 192a 17-19).
The consequence of the Platonic view, Aristotle argues, is that the contrary desires its own extinction. This must be so on Plato's account because Forms cannot desire or actualize themselves- as mentioned they are purely and already actual [and recall, for Plato, the Forms are not defective or lacking in being but said to be more real than the natural or material world]. But this is a distortion of the nature of matter according to Aristotle. Matter is not privation of form but has its own (albeit mysterious) reality. Matter therefore can be said to desire Form and to do so in an instinctually basic way (Aristotle gives a biological metaphor: “as the female desires the male” see Physics 192a 23). According to Aristotle some things are potentially substances; some are actually so (cf. also Metaphysics Books V & IX). Things can have being in two basic ways: (1) accidentally or (2) substantially (by its own nature). Later Aristotle adds ‘being as true’ (predication and logical being), and ‘being as potency and act’. Aristotle illustrates this point by explaining that we say both of what sees potentially and actually that both “see” or “are seeing”. This position leads him to a singular view of the origin of natural movement and change. Since, change permeates nature, Aristotle develops his cosmosology into many-sphere model with his physics investing these spheres with physical and etheric reality. The earthly sphere is the center of the universe and the planets [for Aristotle these include: the Moon, Mercury, Venus, Sun, Mars, Jupiter, and Saturn] are attached to the translucent spheres that rotate around the Earth. More than 50 supporting spheres are said to be necessary in the Physics to account for the motions of the planets and stars. Not only is the Aristotelian cosmos finite, it is also much, much, smaller than what our current models tell us is the enormous size of our universe. According to Aristotle however, the terrestrial elements of earth, water, air, and fire that compose the Earth and the inner sublunary sphere, along with a fifth element, called the aether (later known as the quintessence) are the basic building blocks composing the outer celestial spheres. The natural motion of these terrestrial elements is argued to occur in ‘up’ and ‘down’ movement in a way that they always seek to find their proper 'natural' places (this is, in effect, Aristotle’s theory of gravity). The natural motion of the etheric elements is described an endless revolution around the Earth. But the process of circular motion cannot explain itself. Therefore Aristotle introduces the Prime Mover a power that sets the world in motion while itself remaining unmoved (see Physics Book VIII). This geocentric Aristotelian solar-system can be pictured as follows, see below, (the blue sphere at the center is the Earth):
The Aristotelian Cosmos
In Aristotle’s universe all motion requires the continual application of forces. Bodies move because they are pushed and pulled by direct contact with other bodies that, in turn, do the pushing and pulling. The void (a vacuum) cannot exist, argues Aristotle, because isolated bodies would be completely motionless. In the same way atoms (the smallest particles of matter) also cannot exist. This follows from the basic assumptions of his physics- because if physical atoms existed they would be isolated from one another by a plurality of inter-atomic voids, therefore, if they actually existed in space the atoms could not appeal to any force in order to have direct contact with and be the cause of their motion. If atomism were true, Aristotle argued, all bodies would be perpetually motionless. But movement is a self-evident fact about the natural world, therefore atomism is false.
The Infinite and Nature
Aristotle denies the reality of any infinite magnitude as actually existing and present in nature. The main text where we find his discussion of the infinite is his Physics, especially Book III, Chapters Four to Eight. In Book III Chapter Six, Aristotle examines the position that no infinite properties or realities exist at all - in any sense. Unfortunately, he says, if no infinite exists, this leads to many impossible consequences.
- A beginning and an end of time [Aristotle doesn’t think this is possible],
- No infinite numbers (magnitudes) would be permitted, which would be mathematically disastrous.
[We] must not construe potential existence in the way we do when we say that it is possible for this to be a statue—this will be a statue, but something infinite will not be in actuality. Being is spoken of in many ways, and we say that the infinite is in the sense in which we say it is day or it is the games, because one thing after another is always coming into existence. For of these things too the distinction between potential and actual existence holds. We say that there are Olympic games, both in the sense that they may occur and that they are actually occurring. (206a 19-25)
When we say this marble is potentially a statue, for example, we mean it could actually become a statue (if carved in the right way by a trained sculptor). This cannot be so with the potential infinite. Why not? Because, according to Aristotle there cannot exist (and can never be) an actual infinite thing [he argued for this in Book III, Chapter 5 of the Physics]. Therefore, there is no actual infinity in nature. The mode of existence that the infinite has, according to Aristotle, must therefore be as follows: “the infinite can be viewed as one thing taken after another”.
Accordingly, the infinite exhibits itself in different ways:
…—in time, in the generations of man, and in the division of magnitudes. For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different. [Again, ‘being’ is spoken of in several ways, so that we must not regard the infinite as a ‘this’, such as a man or a horse, but must suppose it to exist in the sense in which we speak of the day or the games as existing—things whose being has not come to them like that of a substance, but consists in a process of coming to be or passing away, finite, yet always different.] (206a 26-34).
Here the distinction between a process and an activity is relevant. Elsewhere Aristotle describes how a process takes time to happen but an activity is instantaneous. When you open your eyes and see, the process of light traveling to the retinae is one that takes time (albeit, light travels so fast that we experience it as omnipresent and instantaneously arriving from all directions). The activity of seeing, however, is always complete in itself. Every moment your eyes are open and functioning, you are seeing. The infinite, therefore, can never have being as an actualized unit, i.e. there can never be an infinite substance or an actually existing active infinite series. But the infinite can be viewed instead as more like a day: the passing of a day is one second, minute, hour, etc. at a time - forever. Or, Aristotle says, like the Olympic Games which take place as a grand event, but not all the games occur at once and at the same time. The Games, like the passing of time, are a process. The infinite is akin to a process, but with a special qualification. Natural processes can be completed and actualized but the infinite never can be.
This is effectively Aristotle’s answer to Zeno. We can, like Zeno, break down existing magnitudes into definite infinite series. Achilles can be said to have an infinite number of places to pass before he can reach the tortoise. If we take the reality of the infinite in nature seriously, then Achilles will NEVER beat the tortoise. According to Aristotle, the infinite must be understood as real in another way. Space and time, for example, are viewed as continuums by Aristotle. In the case of a continuum what is real about it is that it is a series and a whole, not the existence of indefinite or definite real parts within it. Therefore, taking space as a continuum we can conceptually break it down into infinite places but in reality it is unified and connected (remember: Aristotle rejects atomism and he thinks that space is different from place determined by substances). Therefore, Achilles naturally always catches up with the tortoise and beats him in the end and the arrow reaches its target.
…in spatial magnitudes, what is taken persists, while in the succession of time and of men it takes place by the passing away of these in such a way that the source of supply never gives out. (206a 35-206b 2).
By the above, once again, what Aristotle means is that the infinite cannot be an actual active presence in nature. The being of the infinite therefore consists in a process of ‘coming to be’ or ‘passing away’ and it is definite at each stage, yet it is always different (think of time and temporal flows). The source or supply of something infinite "never gives out” (206b 2). We will return to address the problems of an ‘infinite source’ below when we deal with Aristotle's theory of time. For the present, we can say that, according to Aristotle, finite things (substances) have an independent existence but the infinite does not and cannot. And even if, by addition, there is a potential infinite- as Aristotle will say is the case, but manifesting a sui generis kind of potency-no actual infinite is allowed in Aristotelian metaphysics.
Aristotle argues that in a finite magnitude, the infinite by addition comes about in a way inverse to that of the other: "For just as we see division going on ad infinitum, so we see addition being made in the same proportion to what is already marked off. For if we take a determinate part of a finite magnitude and add another part determined by the same ratio (not taking in the same amount of the original whole), we shall not traverse the given magnitude. But if we increase the ratio of the part, so as always to take in the same amount, we shall traverse the magnitude; for every finite magnitude is exhausted by means of any determinate quantity however small." (Physics 206b 3-13)
According to Aristotle: The infinite exists potentially and by reduction but not independently:
It exists in fulfillment in the sense in which we say ‘it is day’ or ‘it is the games’; and potentially as matter exists, not independently as what is finite does. (206b 14-15).
When we count or add natural numbers there is always a potential infinity there. A mathematical infinity. But this potential is never actualized. No one can count up to infinity and if they could (if someone were immortal say) in order for them to do it, it would take time [like a process], but they would never actually accomplish the task. This is what Aristotle means when he writes that:
…it will always be possible to take something ab extra. Yet the sum of the parts taken will not exceed every determinate magnitude, just as in the direction of division every determinate magnitude is surpassed and there will always be a smaller part. (206b 18-20).
This may sound like a prejudice on Aristotle’s part. Modern mathematics frequently deals with transfinite quantities, and we are tempted to view infinity as mathematically valid. We also have an intuition that God is able to count to infinity and to do so without effort or time. But think about the following:
What if you met an immortal being in your neigborhood and you asked what they were doing and they answered “I’ve just finished counting to infinity” -would you believe their claim? Aristotle would deny it had happened. Moreover, if the immortal being had a clock, and he said it had ticked an infinite number of times, again, Aristotle would argue, there is no way that any finite clock can actually have done this. As a potential action, the ticking can-in theory- continue indefinitely, but there will never be an actual infinite number of ticks. The main reason for why this could not happen, on the Aristotlian theory, is that ticking, like counting, is a process that takes time. To put this into context: if the immortal clock wielding mathematician said to you: “I’m going to count backwards starting now, from infinity and down to zero”. Do you think they could ever finish?
As Aristotle writes: “…in respect of addition there cannot even potentially be an infinite which exceeds every assignable magnitude, unless it is accidentally infinite in fulfillment, as the physicists hold to be true of the body which is outside the world, whose substance is air or something of the kind. (Physics 206b 21-24). And, given tight-knit the unity of his cosmos, Aristotle doesn't think that an extra-universal body can exist. Therefore the infinite, according to Aristotle, is merely a qualified kind of potentiality. Aristotle notes that the infinite turns out to be “contrary to what it was said to be” (Physics 206b 35).
Also, the infinite is what always has something outside it. Aristotle’s formal definition of infinity goes as follows:
something is infinite if, taking it quantity by quantity, we can always take something outside. (207a 8)
Aristotle describes the bezel ring, it has nothing outside it but it is, in a sense, infinite. As a more modern illustration we can take the lemniscate, the inverse folded ribbon pattern.
This is our mathematical symbol for infinity (first introduced by the British mathematician John Wallis in 1655), and it is an appropriate symbol since if we follow the figure, we can travel endlessly around its curves. Aristotle is basically rejecting that these closed off figures are genuinely infinite. Imagine if we form, as we can easily do, a lemniscate out of two pieces of string. Aristotle writes:
For to connect the infinite with the universe and the whole is not like joining two pieces of string; for it is from this they get the dignity they ascribe to the infinite—its containing all things and holding the universe in itself—from its having a certain similarity to the whole. It is in fact the matter of the completeness which belongs to size, and what is potentially a whole, though not in fulfilment. It is divisible both in the direction of reduction and of the inverse addition. It is a whole and limited; not, however, in virtue of its own nature, but in virtue of something else. It does not contain, but, in so far as it is infinite, is contained. Consequently, also, it is unknowable, qua infinite; for the matter has no form. (Hence it is plain that the infinite stands in the relation of part rather than of whole. For the matter is part of the whole, as the bronze is of the bronze statue.) If it contains in the case of sensible things, in the case of intelligible things the great and the small ought to contain them. But it is absurd and impossible to suppose that the unknowable and indeterminate should contain and determine. (207a 18-30).
In other words: The infinite is a whole and limited; not, however, in virtue of its own nature but in virtue of what is other than it. It does not contain, but insofar as it is infinite, it is contained. The infinite is, finally, unknowable qua infinite, because it is best viewed as being like matter without form.
In chapter Seven of the Physics Aristotle deals with the topic of substance and substantial being versus numbers and logical being or entities. There’s an important difference made here between modes of being and how we can say something exists. For Aristotle, it will be recalled, substance is being in the primary and central sense. That means that other modes of being are derivative on how substances exist. These other modes of being include: being in the sense of accident (i.e. a part of a substance), being in the sense of potentiality and actuality (which are said of substances) and being in the sense of what is true or false (logical being). Aristotle begins this chapter by saying:
It is reasonable that there should not be held to be an infinite in respect of addition such as to surpass every magnitude, but that there should be thought to be such an infinite in the direction of division. (Physics 207a 33-4).He says this because magnitudes apply to primary substances, whereas what he calls division is an abstractive mathematical process that we apply to a finite substance. In other words, Aristotle is distinguishing between the mathematical (negative) versus metaphysical (actual) infinities. The concepts that shape Aristotle’s discourse here are the important distinction between matter and form with matter being associated with the principle of potency and form with that act or actual and determined being. Aristotle writes:
For the matter and the infinite are contained inside what contains them, while it is the form which contains. It is reasonable too to suppose that in number there is a limit in the direction of the minimum, and that in the other direction every amount is always surpassed. In magnitude, on the contrary, every magnitude is surpassed in the direction of smallness, while in the other direction there is no infinite magnitude. (Physics 207a 35-207b 3)
Numerical versus magnitudinal modes of existence are here discussed by Aristotle and the concept of magnitude or formal being is always something, and so –he writes- always surpassed in the direction of smallness but never in largeness [or else it would have no determinable boundaries and would not be an existing thing].
And, as Aristotle has said already and repeats here: …what is one is indivisible whatever it may be, e.g. a man is one man, not many. Number on the other hand is a plurality of ‘ones’ and a certain quantity of them. Hence number must stop at the indivisible; for ‘two’ and ‘three’ are derivative terms, and so with each of the other numbers. But in the direction of largeness it is always possible to think of a large number; for the number of times a magnitude can be bisected is infinite. (207b 6-11).
For Aristotle, numbers have an abstract mode of being. Numbers exist primarily in the mind of a thinker, but since he views space as a continuum and a limit no sensible magnitude is actually infinite. A man is one man, not many. Numbers are different from things. This is an anti-Pythagorean point of view. But we can nonetheless criticize Aristotle’s limited view of mathematics. He views only positive cardinal numbers (the natural numbers) and their ratios as properly suitable for mathematical calculation. We know that negative numbers, irrationals, etc. are all useful. Does this vindicate Pythagoras? Not exactly, it makes mathematics less like the real world not more like it- since we never see negative or infinite bodies. Does it vindicate Plato? Some think it does. We will take this issue up again when we look at the infinite in modern mathematics. In Chapter Seven of his Physics, Aristotle also reiterates his definition of the mathematical infinite and describes how it is different from an actual infinite series that applies to existing beings, Hence this infinite [the mathematical] is potential, never actual: the number of parts that can be taken always surpasses any definite amount. But this number is not separable, and its infinity does not persist but consists in a process of coming to be, like time and the number of time. (207b 12-14). When Aristotle says that “this number is not separable” he means it does not exist apart from things as substances. Numbers are only real in the sense that, as quantities, they are abstractions that we form in our minds by taking intelligible forms from existing things in lived experience without altering the matter of the substances we abstract from. Time is infinite however, but in a different way than movement or the being of substances. Time, like space, is a continuum with boundaries, and according to Aristotle it can be viewed as a non-sensible magnitude:
With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every definite magnitude; for if it were possible there would be something bigger than the heavens. (207b 15 – 20).
Every sensible magnitude (the technical term is hylo(-matter)morphic(-form) substance) is of a definite magnitude. Nothing can be bigger than the heavens, hence there can be no actual infinity in nature. But time is the only exception, since:
The infinite is not the same in magnitude and movement and time, in the sense of a single nature, but the posterior depends on the prior, e.g. movement is called infinite in virtue of the magnitude covered by the movement (or alteration or growth), and time because of the movement. (207b 21-25).
This account, Aristotle says, does not: …rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that a finite straight line may be produced as far as they wish. It is possible to have divided into the same ratio as the largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will make no difference to them whether the infinite is found among existent magnitudes. (Physics 207b 29-34).
Mathematicians don’t actually use infinite quantities (in ancient Greek science, they didn’t even use negative numbers), so denying the reality of infinite sets that exist in reality does nothing to alter mathematics or mathematical poof. As regards causes however, the infinite is a cause in the sense of the material cause. The essence of the infinite is said to be privation. Aristotle writes:
In the four-fold scheme of causes, it is plain that the infinite is a cause in the sense of matter, and that its essence is privation, the subject as such being what is continuous and sensible. (Physics 207b 35- 208a -2).
Earlier thinkers were therefore “inconsistent” to treat the infinite as a body or substance that contains things- i.e. to associate the infinite with actuality or form- this is according to Aristotle a mistake. In chapter Eight of the Physics, Aristotle writes: “It remains to go through the arguments which are supposed to support the view that the infinite exists not only potentially but as a separate thing. Some have no cogency; others can be met by fresh objections that are true.” (208a 5-7). Having given an account of the infinite, and treated of what it can and cannot be, Aristotle also wants to provide objections to arguments that say the infinite is different from how he conceptualizes it.
Aristotle disposes of the remaining arguments that attempt to make of the infinite an actual and existing thing as follows. We might think there is an infinite amount of matter- since the process of generation and corruption takes place in time (and Aristotle can conceive of no beginning and no end to time). “In order that coming to be should not fail, it is not necessary that there should be a sensible body which is actually infinite. The passing away of one thing may be the coming to be of another, the universe being limited. (208a 9-11). But Aristotle thinks there is no need to posit an infinite sensible body or substance. On the one hand, if we think of Prime Matter or pure matter as recyclable, then Aristotle’s position makes sense. But, on the other hand, it’s hard to see how he can reconcile his finite universe with an infinite series of temporal succession. If there is no change ever in the heavens (which are eternal) and there is no ‘coming to be’ or ‘corruption’ of either time or celestial nature, then this implies that nature IS in a sense infinite. How so? Well, nature must be as infinite as the eternal divine and immaculate celestial sphere posited in Aristotle’s cosmology. Furthermore, because of the original motion of the Unmoved- Prime Mover, Aristotle thinks that the celestial sphere is always in motion.
Here we can ask: if there is no first moment of time, and if time is only real when motion exists, doesn’t that mean that the past existence of time is an actually infinite thing? Aristotle will counter that the universe is finite because it is limited: “There is a difference between touching and being limited. The former is relative to something and is the touching of something (for everything that touches touches something), and further is an attribute of some one of the things which are limited. On the other hand, what is limited is not limited in relation to anything. Again, contact is not possible between any two things taken at random.” (Physics 208a 11-14).
But an eternal world seems to present problems to this finite position. In conclusion, Aristotle thinks that the problem of infinity is one that is produced more by thought than by nature. We can always think of things as never ending and infinite- but nature is definite and exists beyond our thoughts and concepts. “To rely on thinking is absurd; for then the excess or defect is not in the thing but in the thought. One might think that one of us is bigger than he is and magnify him ad infinitum. But it does not follow that he is bigger than the size we are, just because someone thinks he is, but only because he is the size he is. The thought is an accident.” (Physics 208a 15-19). By saying “thought is an accident”, Aristotle means that thoughts are activities that belong to existing beings. Nothing in nature changes or becomes what it is because we think about it in a particular way. Nonetheless, since the Prime Mover thinks (it's basically all it does), thinking is in a sense infinite. And Aristotle concedes that time and movement are, likewise infinite in this qualified way: “Time indeed and movement are infinite, and also thinking; but the parts that are taken do not persist.”
He goes on:
Magnitude is not infinite either in the way of reduction or of magnification in thought. This concludes my account of the way in which the infinite exists, and of the way in which it does not exist, and of what it is. (Physics 208a 19-24).
Aristotle advances our understanding of the problem of infinity in several ways. First, his conceptual clarity and attempt to seek clear definitions are a marked improvement over the work of his predecessors. Second, Aristotle’s rigorous attempts to make sense of the natural world and interpret being as connected by causal processes that we can understand through their effects is the beginning of what we can call the ‘scientific’ study of the natural world. Aristotle gives us a physics that starts from empirical observations. His speculation builds on what he can observe and, as a result, he develops a systematic understanding of the natural world that accepts the forces and entities that affect our senses as primary and real.
Regarding the notion of infinity, Aristotle’s views can very squarely be placed in the ancient Greek tradition since he shares the predominant Greek view that infinity is mysterious and suspicious. According to Aristotle what exists primarily are substances or concrete things, and he holds that space, time and motion are all continuous but dependent on primary substances. Although Aristotle strongly rejects the idea of an actually infinite substance- he does accept the reality of the infinite as a qualified and sui generis type of potency in the world. Since it is a qualified potency, the infinite is not like the potency that informs either primary substances (metaphysical potency) or motion (physical potency), instead the potential nature of the infinite is said to be one that can never become actual in a metaphysical or substantial sense.
The proof of this, according to Aristotle, is that every finite space or time can be subdivided into an infinite number of divisions (see Zeno). If we understand consecutive and continuous to mean ‘without end’ and also think that infinity can be actualized, then we will never be able to make sense of the physical world. Infinity as actual, in other words, can never be quantified or measured or understood. But if we understand continuous division as that which is ‘without limit’, then we begin to understand the existence of the infinite in our world. Primary substances, as actually existing things, come first and must exist before motion, space and time can be possible. Since primary substances are of finite magnitudes and quantities [a horse is one horse not many] the mode of existence for primary being is finite.
This metaphysical postulate automatically makes motion, time and space finite for Aristotle. Beginning with motion, which is the most important concept in physics. For something to be in motion it must first exist (and see the arguments outlined above). If nothing existed, there would be no motion. However, what primarily exists, as already stated, are bounded and finite individual bodies. However, in the case of space and time, since they also ultimately apply to bodies or substances, we are twice removed from actuality since they can only be understood and applied when we already have (1) an existing body, and (2) that realization of the potency which is the existing thing in motion. We can understand the necessarily finite sense of the universe for Aristotle if we accept that all matter- in order to actually exist as something- must be actualized by an immanent or substantial form. Since what is structured and organized and exists is a definite thing and not a non-thing. Experience proves that things are related to each other by space and present in time. According to Aristotle, if we were unable to understand and relate to things in the world, we would not be able to live in time and space. Space, time, and motion can therefore be viewed as phenomena of the world that are intelligible because things exist outside of us. Since space and time can be infinitely divided, they can however be said to be infinite in a sense. This sense is a secondary sense, mathematical if you will, since no actually infinitely divisible space or actually infinitely divisible time can be made actual. For example, if a distance is a measure of length between point A and point B circumscribing a space, then it makes no sense to say that an infinite time exists such that I can infinitely travel a journey leaving from point A towards point B but never actually arrive at point B. Likewise, if the length between A and B is infinite, according to Aristotle, then no amount of time will allow me to arrive, but all of this is viewed as unacceptable. If A is real and B is real, then the space and time between them must exist by virtue of their prior existence and they are both connected as real parts of the [finite] universe. All this is to say that, as continuums, space and time are both wholes, limited by what exists but infinitely divisible. Motion is also limited by what exists in a finite way, but like space and time, all the phenomena of nature are continuous and form continuums that make sense of reality. The mistake of viewing the infinite as actual would be the mistake of thinking that motion, space and time are real independent of the primarily existing substances.
In this way, time has a real ‘now’ point, and space has a real location ‘here’, and motion a real beginning and real end. But the reality is that substance is real and time, space and motion are infinite only in the way that an unlimited conceptual series is limited- whatever we say is the end, we can always point to the next step in the series.
Critical Evaluation of Aristotle's theory
The main problem with Aristotle’s position would be addressed by later thinkers, specifically the Neoplatonist and Christian/Islamic monotheists. Aristotle believed that substance was real insofar as it could act and be bounded, but the ultimate substance in his universe is the Prime Mover or God. God is, according to Aristotle, pure act but he is also the cause of all motion. Since the world of nature and the celestial sphere are eternal, then time begins once a soul can count and separate the moments of motion - past from present (time 1 from time 2, etc.). However, given these assumptions, it would appear that the generations of life on earth –actualized by the natural order of reproduction- is actually infinite. Time itself (since the Prime Mover contemplates the universe instantaneously but eternally) is also infinite as Aristotle admits. But he fails to see any problems here since it is a potential infinity with only a select group of parts ever being actualized at any one time. This works for the future, which hasn’t happened, but how can the infinity of past time in an eternal universe not be an actual infinity?
Aristotle might argue that only one set of living beings or process of events changing in time can exist at a single bounded division of a temporally processed series of events. Death might also be held to effectively recycle the matter of the organisms that had once been born (since Aristotle seems to also have rejected the Pythagorean/Platonic notion of personal immortality). Nonetheless, later thinkers, such as the Neoplatonists, believed in the immortality of the soul and found it impossible to reconcile the finite number of bodies with an infinite number of souls.
Their argument went something like this: Every living soul has a body, and every dead soul seeks out a body or a place in the afterlife, but if the past generations of living being’s is infinite that would mean that an infinite number of bodies has already lived and died. This, again, seems to introduce an actual infinity, not a potential one. Furthermore, according to later Christians and Moslems, if every soul must be preserved by God after death of the body- and the world is eternal then the number of souls in heaven is actually infinite. This makes infinity a real and actually existing property- at least as measured by the souls who have been born and died or generations of creatures that have lived on the earth.
From a more modern perspective, if the reader holds to the tenet that physics and theology should be methodologically separate, the above critiques might not be a convincing refutation of Aristotle. After all, we know -after Darwin-that the generations of human beings are NOT infinite. However this does not completely place the Aristotelian account in the clear. There are also intrinsic problems with Aristotle’s physics that render his account of nature equally problematic, beginning with its account of material objects and due to the fallacious account of movement as dependent on natural place in the Physics. Since the central doctrine of his physics can be doubted, Aristotle's natural philosophy itself should also be viewed as questionable. While Aristotle may be correct that the Platonic Dyad is wrong for radically separating Formal causes from material substrates, the theory of the contraries that he replaces it with is not exactly an improvement. Giving precedence to the evidence obtained by the senses, Aristotle dismissed atomism as speculative. Plato, we can recall, accepted a Pythagorean interpretation of the ultimate structure of matter. The four elements are basic building blocks but whereas Aristotle says no level deeper than the elements can be maintained, Plato conjectured that the elements themselves were built up out of smaller particles. Although Plato was wrong to say that the structure of these smallest particles is triangular he was, as later advances in physics showed, right to say that matter has an atomic structure. Aristotle insisted that after the elements comes only matter, but both his conception of so-called 'prime matter' and his account of substances containing inherent contrary principles and in this way making possible natural motion is fraught with conceptual difficulties.
Further proof of the problems with Aristotelian contraries can be seen upon closer examination of the role they are supposed to play in facilitating motion in natural objects. According to Aristotle substance is what underlies contraries and no substance per se manifests motion. Aristotle claims that: “substance has no contrary among things that are” (Physics 225b 11). Aristotle then maintains that there are two conceivable senses of ‘motion’. He gives the examples of: (1) A man changing from fair to dark and (2) A man changing from falling ill to getting healthy.
In case 1 above the change is not a subject, in case 2 the change is only accidental and therefore not true change. If we admit that the change from health to sickness must simultaneously be changing from ‘this very change to another’ – from motion to motion, then we can end up with problems, since an infinite regress seems to arise. In an infinite series there is no first term and so no first stage of change and no following stage, etc. Infinite series need not be pernicious Aristotle thinks, but in this case it may be since things capable of becoming must also be capable of perishing. But if we admit a becoming of becoming—we would have to say that that which is in the process of becoming is also, and simultaneously, in the process of perishing at the very moment when it reaches the moment/stage of becoming. The terms would lose their intelligibility. There must, therefore, be a substance underlying all processes of change and becoming. Aristotle’s examples are the following: The becoming of learning cannot be learning (the learning takes place in a subject). Likewise the becoming of becoming cannot be becoming—since becoming likewise happens to a substance. Since there are three kinds of motion, the substratum and the goal of motion must be one or another of these three: (1) accidental, (2) partial or (3) essential. The accidental and accidental change is not relevant here. Motion cannot belong to being, Aristotle thinks, because relation or agent and patient required to actualize motion can only happen in respect of: (i) quality, (ii) quantity, or (iii) place. Each of these gives us a pair of contraries.
- Alteration: includes both contraries. Quality here does not mean a property of substance (not the difference in the predication) but a passive quality in virtue of which a thing is said to be acted on or to be incapable of being acted on.
- Increase or decrease—motion in the direction of complete magnitude= increase and motion in the contrary direction = decrease
Some things are immovable, i.e. absolutely incapable of being moved [the Prime Mover]. Also there are things moved only with difficulty or after a long time whose movement is slow at the start (things very hard to move such as large rocks fit this picture). But both these kinds of motions are examples of Aristotelian locomotion or natural movement from one place to another. These are said to be different from alteration or accidental change on the one hand, and generation or the coming into being, and passing out of existence [of a substance] on the other. What is naturally designed for and capable of motion, Aristotle maintains, but is not in motion at the present time, is a being at rest. Rest, subsequently is a privation of motion in a thing that can change in intrinsic and extrinsic ways. There is a real ambiguity manifested here between the sense of change as motion and change as alteration, trading on the basic assumption that substance does not move. The logical form of Aristotle’s argument for natural motion is as follows:
- Hypothesis: Natural substances are what primarily exist
- Premise: (1) That which is moved, must exist [before the motion].
- (2) That which is generated, cannot exist [before the generation].
- Therefore, (3) generation is not motion.
But as it stands this argument is of questionable validity.
It can only be viewed as valid if the moving thing itself (which is moved or generated) stands for the same thing in both (1) and (2). If it stands for different things then Aristotle is committing a fallacy of equivocation. As we’ve seen, on the standard Aristotelian analysis, motion and generation are both types of change. However if this claim is true then generation is not a process occurring in a natural place defined by a static balance of celestial causes and forces. Can we show the above to be true? Insofar as Aristotle views motion as the central characteristic of all of nature, and insofar as he understands it as outlined above, we can indeed. Understanding motion is central to Aristotle’s project in the Physics. Since motion can also be seen in the celestial sphere, Aristotle developed a rule for explaining how constant motions continue at constant speeds against an invariant horizon. The stars and planets, he thinks, turn slowly and evenly on trajectories following spherical surfaces (the celestial spheres) (See image above). For this reason motion on earth would never come to rest if the celestial propulsion that set motion going was not met with resistance. What we see on earth, of course, is precisely limit cases where endless motion is resisted and the resulting speed of any object or thing can be subsequently understood as a ratio between its propulsion and resistance. For an illustration of the above take a team of rowers. Two rowers can row a boat more quickly than one rower can, double the amount of rowers and the boat can potentially move twice as fast. Speed can be further increased if you eliminate resistance, which is why the shape of racing boats is designed with a suitable smoothed hull to move more quickly in the water. Now according to Aristotle the above is a universal principle but the reason why it’s valid is due to the essential nature of the substances affected and his account of matter as potency allowing for natural motion. However the weakness in this account was clearly shown in experiments conducted by natural philosophers who more closely explored the behavior of falling objects (for example Galileo).
The actual behavior of falling objects does not easily fit the Aristotelian model. For one thing, falling objects seem to pick up speed as they descend through the air. Aristotle was unable to coherently explain how this behavior came about. According to his physics it meant one of two things: either resistance was decreasing or propulsion was increasing. Later thinkers, trying to maintain Aristotle’s account of motion, debated this point fervently but no consensus could be arrived at regarding what was actually going on.A direct consequence of Aristotle’s position regarding free fall was that if weight can increase propulsion then heavy objects should fall faster than lighter ones. Galileo’s experiments with inclined planes and his subsequent formulation of the principle of relativity definitively showed that this had nothing to do with intrinsic nature of the objects themselves. Light objects can fall almost as fast, or even just as fast as, heavier ones and Aristotle’s account of natural place is therefore erroneous. The ultimate undermining of Aristotle’s theory of natural place, however, is found in Galileo’s experiments showing that motion is relative to a frame of reference. On this theory, still maintained by modern physicists, there can be no absolute rest of natural things since there is nothing that the object at rest can be resting against,i.e. there is no measureable absolute framework of substances. If the celestial sphere bounding the heavens made of immaculate prime matter could be confirmed- it would serve as an inertial and absolute frame of reference. More detailed evidence about the make up of heavenly bodies shows that no such spheres exist. The earth is, in fact, and contrary to the assumptions of Aristotle, itself in motion around the sun. The sun moves in our galaxy, the milky way. And the milky way itself is moving. With these falsifications of his central assumptions Aristotle’s notion of natural place goes out the window-even if it wasn't replaced with any competing intelligible theory until the time of Newton who introduced his laws of universal motion, postulating gravitation as a universal force of nature.
The upshot of these tensions, notwithstanding his thoroughness and the analytical rigor of his treatment of the topic, is that Aristotle's account cannot be the last word on the nature of the infinite.