Aristotle’s Teleology: Motion, Goal-Directed Action and the Unmoved Mover

1. Review of Aristotle’s metaphysics


Good evening, ladies and gentlemen. Last week we surveyed Aristotle’s epistemology and some of the essentials of his metaphysics. In regard to metaphysics we said that reality, for Aristotle, is this world, the world in which we live, the world of concrete, particular individual things as revealed to man’s senses. We said that for Aristotle, each particular, each primary substance, is if you recall comprised of two elements—a universalizing element, which constitutes the basis for our putting it into a certain class and ascribing to it a certain nature; and an individuating element, an element which constitutes the basis of its uniqueness, that which makes it a “this.” And you recall Aristotle’s specialized technical terms for these two elements, “form” and “matter.” Matter, we went on, is the stuff or material comprising a thing; form represents its structure or organization.


And in these terms change, we said, is the process of matter taking on new form, so the change in no way involves a contradiction; it is eminently logical, rational, scientifically intelligible. Or, using the other terminology that we developed, we could say that change is the passage from potentiality to actuality, a process we saw which occurs in orderly, predictable, lawful ways.


And every change, we saw, involves four essential factors, four causes as they are called—the material cause (that is, the material from which the change proceeds), the formal cause (which is the new structure imposed on that material), the efficient cause (which is the action of the agent which gives the new structure to the matter), and the final cause (the end or goal or purpose of the process, the final answer to the question “Why does it occur?”).


2. Final causation


Now against that two minute recapitulation, let us pick up from this point without further ado and continue with Aristotle’s metaphysics. And the first question is, Is this four cause analysis of change really put forward by Aristotle as universal, as applicable to all changes of every kind? Because, as applied to human action, you might say his analysis is obviously sensible, as against for instance the mechanism of the Atomists who deny the reality of purpose. But you will probably ask, what about unconscious biological change—for instance, an acorn changing into an oak? And what about non-biological, inanimate change—for instance, a case from the realm of physics such as upsetting a bucket of water on the top of a hill and then the water flowing down the hill? How do the four causes operate in these areas?


Well consider as examples the two cases I just mentioned: the acorn becoming the oak and the water flowing downhill mechanically. The first three causes (the material, formal, and efficient) still apply obviously. In both these cases, you start from something—the acorn, or the water on the top of the hill—which are respectively the material cause of these two changes. In both these cases, you go to a new form—the oak, or the water at the bottom—and that is the formal cause. And in both these cases, the change is effected by some means. Now here I don’t have to specify; there are various biological and/or mechanical processes of different kinds at work on the “matter” effecting the change, and those are the efficient causes. But the big question is: What about the final cause? Does it apply to such processes also? According to Aristotle, the answer is yes.


Now why did he hold this? Well, I have to point out that Aristotle’s favorite subject was biology. Plato’s favorite subject was mathematics; Aristotle’s was biology. He was, of course, not only a great philosopher, but also a great biologist, and he tended to use biological examples and then make metaphysical generalizations from them. And in biology the doctrine of final causes at least has a considerable plausibility.


For instance, look at the growing acorn—watch the little acorn become a sprout, and then a young plant, and so on through all the intermediate stages until it becomes a fully mature tree. Now, asks Aristotle, can you explain this progression of stages as simply a blind reaction to outside forces which has no inherent aim or end toward which it is striving? Observe a plant’s actions—they are unconscious, but nevertheless the plant turns toward the sunlight, it sends its roots out reaching for water; if you put a rock in its way, within appropriate size limits, it will push against the rock to try to go around it. It seems apparent from these and countless other such facts, says Aristotle, that the plant has a goal—to live, to grow, to reach its full development, its form, its actuality. It does not seem to be simply an indifferent reactor to external stimuli.


Or consider the self-repairing actions of an animal body. You break your arm; now, within certain limits, the bones knit (of course beyond a certain point there’s nothing the body can do); you cut your finger and the body forms a scab, and we say why does it do that?—in order to keep the germs out. That’s a final cause, an end, a goal. You contract a disease (to take a modern example) and the body manufactures antibodies, and we say why?—in order to fight the disease. Look at the organs of an animal body—each of them has a function, which is often described in terms of its end or goal. What are the lungs for? In order to take in air. What is the heart for? In order to pump blood. Etc.


It seemed to Aristotle obvious that the organs and the actions of living entities have ends or goals, that their structure and functioning does not seem to be just the result of an indifferent reaction to outside factors. It seemed to him obvious that living things aim at an end, a goal, that they strive for that as far as they can, and their goal is to develop, to grow, to reach their full form or actuality. And their completed form he calls their entelechy, that’s the final completed form of a living thing, the oak tree in relation to the acorn. This goals seems to him to be the primary factor determining the actions of the living entity.


Now, he asks, how would you account for all this behavior except by reference to an end or goal guiding the living thing? Suppose we ask the Atomists, he says, how they would account for it. They would say it’s simply a blind mixing and unmixing of the atoms owing to mechanical forces. Well, Aristotle says, if we granted such a thing as a blind mechanistic mixing of atoms, that might produce a few cases of acorns becoming oaks, but why does it happen regularly? What keeps the process on the track so many times?


On the theory of atomism, why isn’t it the case that sometimes by mechanistic reactions the atoms making up an acorn are reshuffled and come out as carrots, or playing cards, or Hegel? Why do they repeatedly, regularly come out as oak trees? Now, Aristotle granted, they do not always come out as oak trees because there are stunted acorns. His expression for this is, “They happen always or for the most part.” But such regularity, he says, implies an aim inherent in the process to keep it on the track. And therefore, Aristotle is a teleologist, that is to say a universal teleologist—he believes that everything that exists, every change has a final cause.


 Now as far as the inanimate world, we won’t labor that point (his physics), but it seems that he generalized from human and biological behavior to the inanimate world as a whole. In his view, the inanimate world is ultimately reducible to four basic elements: earth, air, fire and water; that he just took over from early Greek physics. Each of these elements, he believes, has its own natural place, its own proper location in the universe, and that location represents its true form or actuality; and therefore, the final cause of each mechanical change is ultimately reducible back to the aim of the elements to reach their natural place. Now for instance, the natural place of water is next to the earth, and therefore if you take water way up in the air and turn it upside down, turn the pail upside down, the water is on its way back to its natural place, and that’s its final cause. On the other hand, the natural place of fire is up near the heavens, and that’s why when you light a match the fire goes up instead of going down, and so on.


For Aristotle, therefore, everything has an end or goal, whether it’s human, biological, or inanimate. And the ultimate natural goal of a thing is to reach its form. In this sense, the formal cause and the final cause of every change become the same thing, the same form. For instance, when the acorn becomes the oak, the formal cause is the new structure, and the final cause is to acquire and develop that same new structure. And the way this is usually put is, the formal and the final causes are for Aristotle the same single fact— the same form, regarded from two different perspectives. You call it the “formal cause” when you regard the form as already attained; you call it the “final cause” when you regard the form as being aimed at but not yet attained. And this is known technically as "the Aristotelian doctrine of the identity of the formal and final causes." It’s simply a way of expressing his particular version of universal teleology.


3. Objections to teleology


Now there are of course (I here interject a critical comment), there are many objections to this view of Aristotle’s. Obviously purpose or goal-directed behavior he does not intend in a conscious sense in the case of unconscious entities, and yet it’s very difficult to know what exactly it would mean to talk of an “unconscious goal-directed action.” If “goal-directed action” is taken at its face-value, it implies an entity with the capacity to be aware of a future state and to pursue it. What exactly it would mean to pursue a future state by an entity deprived of the capacity for awareness of anything (including of the future) remains a mystery. And of course Aristotle does not want to say that it’s an unconscious striving, nor certainly that it’s a conscious striving in these cases, and ultimately it seems that he must leave the mechanism of such goal-directed behavior in such cases unintelligible (at least from the surviving writings that we have).


 And of course, many people have argued that you can give an alternative explanation of the apparent purposiveness of biological phenomena, an explanation which in fact denies that there is such a thing as purposive unconscious action. People say frequently, “Couldn’t living entities be built in such a way, have such a nature, that no matter what happens to them (within the appropriate limits), their necessary reaction is a pro-life course of behavior, so that it would look as though they are pursuing an end, but in fact they are simply expressing their nature?” And sometimes the example of a thermostat is cited in this connection—it is so structured that whatever the forces operating upon it (within certain limits), it will react in order to produce a certain temperature. Someone unfamiliar with this mechanism might say the thermostat has an end because it systematically acts to achieve a certain goal, but in fact it is simply expressing the laws of its nature without itself pursuing an end. In other words, people have argued (and you can see the plausibility in this) that you can use Aristotle’s own concept that the nature or actuality of a thing determines its behavior, in order to explain the biological phenomena that he refers to, and that you do not need reference to final causation to keep such phenomena on the track, simply an appropriate kind of efficient causation.


Now, I want above all to be fair to Aristotle, so I should mention that the issue of Aristotle’s teleology, and how precisely to interpret it, is a very controversial question. I’ve given you in effect the standard traditional interpretation, but others are possible and have some basis in the writings which have come down to us. In particular it’s possible to interpret Aristotle’s teleology as in no way implying any unconscious striving or yearning for a goal on the part of non-conscious entities.


Now of course that would raise the question, “Well what then does teleology consist of, and how would you defend this interpretation of Aristotle?” This, however, is a technical question entirely beyond the scope of this course. For those of you who are interested, I might mention a doctoral dissertation which is being written on this exact subject by Professor Allan Gotthelf, titled Aristotle’s Conception of Final Causality, and I understand it will be available in the stacks at Columbia University sometime in the spring of 1973. And I refer those interested in this subject to that work for a thorough discussion of the complex issues involved in this point.


4. Teleology and natural law


Before we leave the subject of Aristotle’s teleology, however, I do want to mention one unfortunate effect of Aristotle’s teleology on his viewpoint, namely that it prevented him from grasping explicitly the idea of a universe run by absolute natural laws. You recall I said last week that although Aristotle laid the basis for cause and effect, he himself seemed, from the surviving fragments, to have no clear idea of a universal reign of cause and effect, because he observed that sometimes acorns don’t become oaks, they are stunted; sometimes little babies don’t grow up into healthy men, they become monsters (and here I mean not moral monsters, but metaphysical freaks). In other words, sometimes the teleological process seems to be interfered with or break down. And consequently, for Aristotle, what happens in the physical world is not absolutely necessary. Certain things, he says, are necessary if the end, the form, is to be achieved; but he maintains there is such a thing as accidental or chance factors, which can interfere occasionally, and thus breech the absolute universality of natural law.


 Consequently, for Aristotle, laws are always expressed in the form “such and such happens always, or for the most part.” And the exceptions being cases where final causality break down, in his view cannot be scientifically understood; they are outside the province of science in Aristotle’s view. These accidental facts, he says, are simply brute contingent facts (that’s the later word for it), that is to say, facts which cannot be ultimately explained, brute data which we simply have to accept as facts. And thus, you see, even Aristotle (for those of you who know the later Kantian philosophy) accepts a form of the necessary vs. contingent dichotomy, and that of course feeds very nicely into Kant’s later analytic-synthetic dichotomy.


 And if you ask Aristotle, well what is his explanation for such accidental or chance phenomena, characteristically he says that in those cases, the form was thwarted in its development by matter, by the resistance of the material element. Now that is an obvious Platonic carryover in him, a legacy of Plato’s myth of the Demiurge (if you remember) who tried to shape matter to the perfection of the forms but met a certain resistance. And this kind of element does exist in Aristotle. Needless to say, this is a very bad limitation on science, it prevents the world from being wholly intelligible if you accept this doctrine. And that’s why I stressed last week that although Aristotle laid the basis for causality, he did not himself have any clear idea that every event is necessitated in accordance with strict universal laws. I might mention another root of his belief in chance or contingency is that he apparently believed in free will, but seems to be unclear how to reconcile free will with the universal reign of cause and effect; and that is another element feeding his view that there is contingency, chance, at work in the universe.


Now a last word on Aristotle’s teleology—it is what is known as immanent. In other words, each thing is metaphysically egoistic so to speak. It is not striving to achieve an outside cosmic purpose, as for instance in the Christian version of teleology everything is striving to fulfill God’s purpose, or for Plato everything is striving to satisfy an external Form of the Good. In Aristotle, the end of each thing is immanent within it, namely, each thing is striving to reach its own fulfillment, actualize its unique potentialities, reach its own form. Everything is striving to realize itself, and this is therefore very often referred to as the "metaphysics of self-realization," in the broadest sense encompassing now water going downhill and acorns becoming oaks. And as you’ll see this evening, that becomes the metaphysical basis of Aristotle’s ethics. It is a universe of development, in which everything is striving to develop itself, fulfill itself, ascend the latter from matter to form, become fully and in actual reality what it has in it to become.


5. The cause of motion


Now let’s ask the question: What keeps it all happening? What keeps things striving to actualize their forms? What keeps things on the go? Why are the acorns out to become oaks, and the baby busily changing into a man, and the water flowing downhill, and the sculptor shaping his statues, etc.? Why does the universe not run down, stop dead, become motionless? In a word: what is the cause of motion? And by “motion” in this question, we mean any change, any happening, any occurrence. Now please note that for Aristotle, motion always existed; motion is eternal; there was never a time, he insists, when there was not motion. And his proof of this is that time itself is simply a measure of motion. A year for instance, if we take modern astronomy, is the period of the revolution of the earth around the sun; a day, a period of the rotation of the earth on its axis. If we stopped all motion completely, then of course there would be no years, no days, no seconds, no time. And if so, to speak of a time when there was no motion would be to speak of a time when there was no time, since time is simply the measure of motion. And that would be a contradiction. Consequently, Aristotle concludes, time is eternal as the measurement of motion, and therefore motion is eternal.


 Consequently, the cause of motion that we are looking for is not something which starts motion at a particular point in time. No, it’s the eternal factor (whatever it is) which underlies all motions and explains why there is such a phenomenon as motion in the universe. Any particular motion can be explained by an earlier motion. Why did this thing happen?—because this one did; and why this?—because this one; and so on. But what we want to know is what explains the fact of motion as such?


Now let us engage in a chain of reasoning here with Aristotle, and let us call this factor (whatever it is) that is responsible ultimately for motion "the Mover" (and if you want to anticipate, you can give it a capitol “M”). What can we infer about it? Well of course, the first thing is, it must be an eternal existent, since it is the cause of motion, and motion is eternal. Well let us ask the question: Can the Mover itself move? Answer—no; this Mover must be itself unmoved and even immovable.


Why? Well, if the Mover itself were capable of motion, as soon as it moved, the question would obviously become: “How do you explain its motion?” We’d have an infinite regress. If we are trying to explain the phenomenon of motion, we obviously can’t do it by appealing to something which either moves or is capable of moving, because we would be going in a circle, we would be begging the question, we’d be assuming the thing we’re trying to explain. If you want the ultimate source of motion—the Prime Mover, you see—then it, whatever it is, must be beyond motion, it must be immovable.


Well, from that we can infer right away that it has no potentialities at all, because anything with potentialities is capable of change when its potentialities are realized or actualized. A thing which is immovable is a thing which must be devoid of potentiality. What then would it be? Well the only other category is actuality. So this must therefore be pure actuality, or to use the other term, it must be pure form. It will be an individual thing, not a Platonic universal, but it is not a stuff or matter organized in a way which can be differently organized. It, therefore, will be an exception to the metaphysical principle that I told you last time of no matter without form and no form without matter; it will be pure form, pure actuality. And therefore, of course, it will not be material or physical in the modern sense either, because anything material or physical is capable of change.


Well, let’s observe another thing about this Mover (we’re slowly sketching in a little biography or character sketch of this Mover)—whatever it is, and we don’t know yet fully, it must be perfect, completely perfect.


Because we know that everything is striving to realize its potentiality, to achieve the higher state of actuality. And we know that it must be better to be actual than potential; that is why everything strives, to achieve the actuality. And this is inherent in teleology, at least as advocated by Aristotle. Everything is striving for the best state, the state of fulfillment. Now here in the Prime Mover, we have a being that has no unrealized potentialities, a being which is pure actuality. And it, therefore, has hit the metaphysical jackpot; it must be perfect.


Well, you might ask yourself: how does such a Mover cause motion? Could it reach out and push the world? No, because it can’t move. It can’t push the world, it can’t pull it—it can’t even desire, because desire is a form of motion, a mental motion. It can’t even say to itself in Prime Mover language “Let there be motion,” because it cannot speak, it cannot will, it’s motionless, it is truly immovable. Well, what is the solution to this dilemma? How does it function to cause motion?


6. The cause of the motion of celestial bodies


Now to understand Aristotle’s answer you have to know something of his astronomy, which was not original with him but was a standard Greek view which he simply took over from the scientists of the time the way a modern philosopher might take over Einsteinian physics from the physicists. For Aristotle, the universe is a nest of hollow, crystalline, transparent spheres, connected along an axis; and embedded in the sides of these spheres are the various heavenly bodies—the sun, the stars, the planets, etc. The earth, he believed, is at the center, and it is stationary. These various spheres revolve around the earth, and the rotation or the revolution of the various spheres is what’s responsible for the motions that we observe, of the sun and the planets and the fixed stars across the sky.


 There is, according to Aristotle (and this also is not an original view with him) a soul, or an intelligence, connected to each of the spheres. These spheres are in effect semi-divine in the Greek view; they are regarded as living entities (the reason seeming to be that the heavenly motions were so perfectly ordered and so long known that it seemed to suggest to the Greeks that an intelligence of some sort had to be guiding them and keeping them on their perfect course). Well in any event, as I said, the various revolutions of these spheres are responsible for the motions we observe, which are communicated along the axis.


Now the problem of motion, therefore, reduces to the problem of getting the outermost sphere moving, the sphere in which according to Aristotle the fixed stars are embedded. If we could get the outer one moving or explain its motion (of course, it always has been moving, but if we could explain its motion), that motion of course would be eternally communicated along the various axes to the rest of the spheres and ultimately to the things on the earth. So we have this nest of spheres, with the outermost one guided by an intelligence (many of them have intelligences—I can’t recall offhand if all of them do or not, but there’s more than one intelligence), and beyond, there is the perfect Immovable Mover. Well now, if you have any power of imagination, you should be able to figure out the solution.


 The intelligence connected to the outermost sphere is capable of awareness as an intelligence, and particularly is capable of being aware of the Prime Mover; it is eternally aware of the Prime Mover. And it is aware of the perfection of the Prime Mover; it wishes with all its might to emulate this perfection; in other words, to do the most perfect thing that it can do. Now you simply ask yourself: what would you do if you were an intelligence connected to a sphere and you wanted to do the most perfect thing? Well it’s obvious that you would go in circles. You would engage in circular motion. This is the best motion because that’s the only eternal motion ultimately; if it went in a straight line, since the universe is finite, it would have to turn around at a certain point, so the motion wouldn’t be unbroken; so the best motion is going round and round.


 Consequently, the famous line that you have heard is an actual description of Aristotle’s metaphysics—“It is love that makes the world go around.” There is, in effect, a cosmic if one-sided love affair between the intelligence which moves the outermost sphere and the Prime Mover, and its motion once it’s going around is then communicated along the axes to the rest of the spheres and the earth. So the Prime Mover causes motion in the same sense in which a beautiful woman put at the front of the room, who was herself entirely motionless, might produce motion in her direction on the part of certain members of the class who might wish to emulate or participate in her perfection. The Prime Mover is the cause of motion in the sense of the final cause.


7. The nature of the Prime Mover


Now what is the nature of the Prime Mover? It is not matter, and it must, says Aristotle, we must think of it as a mind, so we can now start saying “He.” What do minds do? They think. But this must be a very special kind of thought process, because no motion is allowed. And so it must be a kind of motionless contemplation; not a process of sensing, or inferring, or reasoning; you can get as close to it as we as human beings can get if you look at the tip of my finger but motionlessly; don’t blink, and don’t draw any conclusions; just, without a flicker of mental activity, look at a motionless finger. Now of course, as soon as I move my finger, that wipes out, that introduces mental motion in you, and changes your mental state, so it’s disqualified. But if you can grasp a motionless contemplation, that’s what the Prime Mover does.


Well what is the object of its contemplation? Well, it can only contemplate something which is motionless, obviously. And the only thing which is motionless is the Prime Mover. And consequently Aristotle draws the conclusion that the Prime Mover thinks or is conscious only of Himself.


He describes it as pure self-consciousness, thought thinking about itself. Now this eternal, immutable, perfect, utterly self-absorbed mind responsible for the motion of the universe Aristotle frequently calls theos, “God.” And this is therefore regarded as Aristotle’s God.


 Now you see that there is a strong element of Platonism here; the idea of a pure immutable perfect form is a pure Platonist idea. And it of course represents the Primacy of Consciousness in an obvious, blatant way—here’s this Prime Mover, He’s a pure consciousness detached from reality, responsible for the activities of things on earth. Now this is a glaring contradiction to Aristotle’s distinctive approach, and represents a Platonic carryover.


 But you see that even when he’s a Platonist, he’s also an Aristotelian; even his Platonism is modified, because this is a God that would not do a religious person very much, if any, good. This God didn’t create the universe; remember, He is unmoved, the universe is eternal. You couldn’t pray to this God, because He couldn’t hear you. This God couldn’t perform miracles even if He could hear you; He’s completely impotent. He has no plan; He does not even know that the world exists. He has therefore neither knowledge nor power. He is utterly ignorant and impotent. And indeed, Aristotle discusses Him primarily in his physics, and it’s sometimes said that the God for Aristotle is simply a footnote to physics; it is not a central concept.


 And Aristotle is frequently attacked, in spite of the Prime Mover, on the ground that he lacks any real religious feeling or interest. And that is true. The best illustration of this fact is that after he arrived at the Prime Mover theory, the astronomers came back and reported to him that one component of motion would not be enough to account for the observed motions of the heavenly bodies, and that as they had calculated it, we need either forty-seven or fifty-five separate components of motion to account for the heavenly bodies, at which point Aristotle appended a chapter saying there is not one Prime Mover, but either forty-seven or fifty-five of them. And some books seriously therefore classify Aristotle as a polytheist on this point. Now that shows how seriously he took it.


Now of course the answer to this argument is that its basic question is misguided; the question “How do you explain motion?” in the sense Aristotle asks it is an illegitimate question. Motion must simply be regarded, the fact of motion as such, as an irreducible primary, in the same way that the fact of existence as such is an irreducible primary. You can explain any particular existent in terms of the actions of other existents; but the phenomenon of existence as such, as Aristotle understands, is simply there; that’s where you start. And an equivalent account would have to be given of the phenomenon of motion as such. If you attempt an explanation of motion, Aristotle’s is the only one. The only explanation of motion would have to be in terms of an unmoving thing. And consequently, his is a perfectly logical answer, if the question is permitted. I have to say that it is undoubtedly, this argument, the sweetest argument for God ever offered. It is called the "cosmological argument," the argument from the cosmos; it takes many forms in later philosophy, deriving back ultimately from this point in Aristotle and from certain suggests in Plato, and we will see it again in Aquinas.


8. Answering Zeno's paradoxes


All right, let’s continue. We’ve seen so far in what ways the concepts of “form” and “matter” are central to Aristotle’s metaphysics—they’re the basis of his view of universals and particulars, they’re the basis for his explanation of change, the basis for his view of causality and of an ordered universe (though he himself was not consistent on this point), and the basis for his definition of “God.” Now let’s take about a minute to see how Aristotle used the concepts of “potentiality” and “actuality” to answer Zeno. And we’ll take just one of Zeno’s paradoxes because they all raise essentially the same issues.


 Remember the idea that you can’t cross a room because first you have to cross half of it, and then half of it, and so on infinitely, and that therefore there’s an infinite number of distances that you have to get across, and of course that’s impossible to do. Now the whole paradox depends, as all of Zeno’s paradoxes do, on the idea that there can actually exist an infinite number of subdivisions of the distance. What is Aristotle’s answer? He says this is impossible. Nothing, he says, can actually (now notice the word), nothing can actuallybe infinite. Now here we must distinguish between the infinite and the very, very large. The infinite is not simply ten billion, or twenty trillion zillion. The infinite is that which is greater than any particular quantity, which means it is no quantity in particular, which means it is a quantity which has no identity, which means it is forbidden by the law of identity. Whatever actually exists, Aristotle concludes, will always be finite, limited, specific in its amount.


 In what sense then can we speak of infinity? Only, he says, as a potentiality (and there’s another use of his concept). For instance, we can keep dividing a line further, and subdividing, and subdividing; and as a potentiality, there are no limits; we can keep on doing it. In this sense, the line is infinitely divisible as a potentiality. But Aristotle’s key point is that no matter how much we keep subdividing it, we will always actually have only a finite number of parts—two parts, or four parts, or eight parts, or twenty drillion zillion parts—but always some specific number of parts. And the same is true of the number series. As a potentiality, it's infinite, you can keep adding new numbers. But actually, whether you’re going through it in your mind or writing it down on paper, you always actually have some finite specific number, even if you’re counting by intervals of a zillion.


In a word, there is no such thing as the actual infinite, and therefore Zeno’s paradoxes collapse. Now some of you would be curious to know how does Aristotle apply this to space and time? Why doesn’t he regard either of them as actually infinite? And I will be glad to answer that in the question period if anyone asks.


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