Fig. 14. Motion in Aristotle.
Aristotle, Physics
Part 3 (Book III, Chapters 2–3; Book V, Chapters 1–3)
MCKEON: In our next discussion we shall go on to Galileo and start our consideration of the beginning of the modern formulation. Let me merely give you a few suggestions. In reading Galileo, read him as you would Plato and Aristotle. If you understood either of them, you should not have trouble with Galileo. The mere fact, moreover, it’s a dialogue and therefore in line with the Platonic experience, the fact that every once in a while one of the speakers in the dialogue runs through a few theorems and propositions, that should not increase the difficulty. Moreover, do not try to do it in terms of historical erudition; do it, rather, in terms of what he’s talking about. And it’s here, in point of fact, that your historical commentators lead you astray. For example, in the edition that we will be using, we’ll be reading the Third Day. For the Third Day, the title in Latin is De Motu Locali, “local motion.” From what we have said about Plato and Aristotle, what this means is perfectly clear; but to make it clearer, the English translation of De Motu Locali is “Change of Position.” [L!] Why is that a better modern expression? Halfway down the same page the expression “free motion” occurs, and in parentheses the Latin is given: the Latin is “natural motion.” Remember, we have distinguished between natural and violent motion in both Plato and Aristotle, and it makes perfectly good sense. The footnote explains why we will be talking about “free” motion instead of “natural” motion: ‘"Natural motion’ of the author has been translated into ‘free motion’—since this is the term used to-day to distinguish the ‘natural’ from the ‘violent’ motions of the Renaissance” [153]. I hope this is clearer to you than it is to me! [L!] In any case, you will find if you watch yourself, there will be many curious things that will come to light in terms of the discussion.
Among other things, let me suggest—and then I will stop making suggestions—that on the second page of the Third Day when your first postulate set is established, you’re given one definition and four axioms, with the intervening statement that the four axioms follow from the definition, that is, the axioms are deduced from the definition. If you think this is odd, I merely tell you that this is exactly the way Descartes talks, Descartes being a contemporary of Galileo; this is exactly the way Huygens talks; this is the way Fermat talks; this is the way Pascal talks. If any of you are interested in postulate sets today, bear in mind that one of the things that comes out of a careful reading of what is going on is that the conceptions change. Two things are worth noting, though neither of them will really come into our discussion; they are, therefore, more illustrative things of side importance. If we were dealing with the postulates that Huygens uses in physics, he would expect that he could lay down his basic definition, deduce the axioms from the definition, and then deduce the propositions from the axioms. Secondly, he would expect—and remember, I said there were four kinds of method—that this was not merely proof; he would expect that he would discover something, that the geometric deduction was, in fancy language, a heuristic method. One of the things he did was write a Dioptics1, which contributed a great deal to our knowledge of light, of the motion of light and related subjects. It was based on Descartes; and in his letters he makes the statement that his great contribution there is that whereas Descartes had apparently fiddled around with observations—and they are observations that couldn’t be made—, he, by contrast, had showed in his Dioptics that all of the propositions—remember, this is an empirical science—could be deduced from propositions drawn from the first six Books of Euclid. As I say, if this claim were being made by a crackpot philosopher of science, you could have discounted it; but this was a man who knew his philosophy and who invented the science he was talking about. It’s an empirical science, and he got a lot things straight that Descartes didn’t get straight. We have since verified them.
I mention all of this as a guide as to what you should keep in mind as you read. Read twenty pages, from page 153 to 172, because then you’ll come to another subtle theorem. As I say, I hope you will learn that theorems are read in the same way as you read dialogue.
This means we can turn back to finish Aristotle now. Let me recall to you what we have done. We have readings from three books of the Physics, a book which we have learned also managed to reach Galileo. In the second book, we talked about principles and nature, and we differentiated nature as a principle from other kinds of principles. We didn’t read more than the first two chapters of book II, so we jumped to book III, where in chapter 1 we differentiated kinds of change and motion. We analyzed them into the varieties that are possible and clarified the definition that we set up in the course of this, that is, motion is the actuality of the potential qua potential. There are two more chapters that I want you to pay some attention to, but I suggested we would do these rapidly because the remaining third of your selection, which is from book V, goes into problems of some interest and difficulty. Let me, therefore, continue on in the manner in which I’ve been proceeding. Mr. Goren?
GOREN: Yes?
MCKEON: Would you turn to chapter 2 of book II. I’ve already told you what the chapter is about; that is, having given his definition of motion, he wants to compare it with earlier definitions which he shows are inadequate. What I would like you to do, therefore, is to make some sense of what these other definitions are and indicate why they’re inadequate.
GOREN: He talks about Plato’s definition as one definition that is inadequate.
MCKEON: Which one is that?
GOREN: This is the definition by which motion is identified with becoming or inequality, and then his reasons are given why it is inadequate.
MCKEON: Do you remember how that came out in Plato?
GOREN: Well, Plato had the two levels; and since motion was on the second level, which was the level of becoming, it wasn’t fully real.
MCKEON: Yes, but notice what we’re talking about here. Here are three people, one who says motion is defined as difference, a second who says it’s inequality, a third who says it’s not being. And it’s not historical reconstruction that I want; rather, I want some sense of why anyone would say that motion is not being, why anyone would say that motion is inequality, why anyone would say that motion is difference.
GOREN: Motion is indefinite, and a principle like unequal is also indefinite because it’s privative. And, therefore . . .
MCKEON: What do you mean by indefinite? Let me give you a warning that would hold for any writer. Eventually in the chapter Aristotle will say that these people did this because they realized that motion was something indefinite. Aristotle frequently says something about his predecessors that we’re told on good authority doesn’t do them justice: it’s simpleminded or absurd or in error.
GOREN: Aristotle seems to think it’s indefinite because it can’t definitely be classed either with potentiality or with actuality.
MCKEON: But he does that.
GOREN: He doesn’t. I thought that he wasn’t certain about classing it with one or the other.
MCKEON: What is his definition of motion?
GOREN: I don’t remember it at this time.
MCKEON: In strict language, just taking it word by word, that is, in very few words, it’s the actuality of the potential qua potential; so he identifies it with an actuality. But in this chapter what’s he saying about it?
GOREN: Still, he says at one point that it’s a sort of actuality. He isn’t fully confident.
MCKEON: Well, what sort of actuality? . . . Let’s not get into this too far; there are some questions I could ask you about this which would be beyond the scope of what anyone hitting this for the first time should answer. All I’m asking is, what am I talking about? . . . Any hypotheses?
STUDENT: I’m not so sure, but doesn’t it follow from not being? Aristotle here is talking about motion as looking at the subject before and after the motion, so that that is what anybody is saying who says that motion is difference.
MCKEON: They would be doing that; that is, they would take the stance that what you have at the end is different from what you have at the beginning. But is he doing that?
STUDENT: No, Aristotle’s saying that to contrast it to what he’s doing.
MCKEON: Yeah, that is, the mistake that these people are making is that, as is obviously the case in modern motion, there are different things at the beginning and at the end. His stand is that this isn’t what motion is. What about inequality?
STUDENT: There you would look at the subject after the motion and say it was not the same as the original subject was.
MCKEON: Remember the word that Plato uses. He didn’t say “inequality”—this is Aristotle—; he said “equipoise” or “lack of equipoise.” If you have equipoise, you have rest; and if you don’t have equipoise, then you have motion. This is a little different than the first one, isn’t it? Obviously, if you are standing at the edge of the step, balancing yourself, you are not moving; if you lose your balance, you move. Aristotle is saying this isn’t the definition of motion, either. As I say, I don’t want to get into this too deeply. If what we are talking about is a process which moves from point a to point c, from the privation to the form with the matter continuing, then what we’ve got to talk about is just that: the connection (see fig. 14). This connection, whatever else you want to say about it, is an actuality, that is, it’s in process; therefore, we ought to be able to talk about it. It is not the beginning point, it’s not the difference from the beginning point, it is not the nonbeing that would separate the beginning point from what you end up with. Obviously, if we were talking about the lightening of the blackboard that I used as an example before, and you asked, Well, now, what does the light mark have before you put it on? if I were careless, I would think that it’s nonbeing, but that is only being careless. All of these would be relevant. They’d be relevant only in the sense that motion is not itself determinative in advance of everything that’s there; consequently, what you need to do is to define it in such terms, for instance, like the buildable, thatwhat is going on at any given moment is the motion. If, in the example in figure 14, you’re stuck at b and you haven’t gotten to c, the motion that you defined the moment before you got to b is still motion. Consequently, this is what we are saying in criticism of our predecessors.
Well, in this chapter he does go on to say that motion is itself an actuality; therefore, the point he is making needs to take the actuality into account. In this connection, what is the problem that’s involved with the mover and the moved, which comes in the last paragraph, starting at 202a2? Mr. Flanders? . . . Well, let me put it another way; that’s too much like a guess question. Aristotle ends with a new definition of motion at 202a7–8. How does that differ from the one he had before?
FLANDERS: Well, it’s put in terms of a mover and a movable.
MCKEON: Well, the first part is really the same; that is, “the actuality of the mobile qua mobile” is the same as “the actuality of the potential qua potential.” We have added, however, the phrase, “the cause being in contact with something that can be moved.”2 . . . Well, let me answer the question or we’ll get into philosophic subtleties here. Notice, we’re defining motion in terms of principle, and this would include violent motion as well as natural motion. Now, in the case of natural motion the principle is inside and, consequently, it’s an actuality of potential qua potential. If you take into account the instances when the cause is external, you have the same definition plus the fact that the cause which would start the motion is in contact with the body and the body which will be affected by the cause could be affected by the cause. In other words, you could have an immovable object moved through contact with a mover, but it would be immovable simply in a sense that was irrelevant to the motion in question.
In the same sense, chapter 3 adds a series of questions. I’m going to recite now because I want to get you on to book V and if I did it dialectically I might get you into details that would keep us from getting there. When we talk about motion, what is it that’s moved? Notice, our new definition gave us the movable and the mover. Is the motion in the movable or is it in the mover? If it’s in both, which is Aristotle’s answer, then you have a dialectical difficulty: how can the same thing be in both or, in his language, in the agent and the patient? For instance, the chalk is being pushed, therefore, an action; but the finger is acting, the chalk suffering or being a patient. You have an apparent absurdity, which he then brings home to you by taking the example of teaching. In teaching, you have an agent and a patient: you have the teacher who moves and the student who learns, and the point that he is making is that it’s a single process which belongs in both. That is, it is not that the teacher is doing something totally different; on the contrary, taking teaching as a motion, the motion that goes on in the teacher and in the learner is continuous in the sense of being the same. If you can go through the same physical motion but no teaching occurs, then you’re not talking about teaching. Remember, teaching occurs when the mover is in contact with a movable that can be moved [L!]; then it is the case that exactly the same thing goes on in the agent and in the patient. From this he goes on to consider the particular types of motion; but unless you have had difficulties in regard to the larger question, I’d like to go on to book V because therein we’re going to be asking another set of questions about motion.
In book II we asked about the principle of motion. In book III we differentiated the kinds of motion. Now we’re going to begin book V by saying, “Everything which changes does so in one of three senses” [224a21]. The question I want to ask, Mr. Henderson, is, What are these three senses?
HENDERSON: How do they change accidentally? How do they change in part? How do they change in essence?
MCKEON: All right. Give me an example of each.
HENDERSON: I assume you don’t want me to build on the examples he gives here.
MCKEON: Well, he gives examples which are a bit cryptic, so even students with beards need to interpret them. [L!] Consequently, if you—I mean with gray beards; I didn’t mean you [L!]—if you could substitute a like example and indicate what it is that he’s driving at and why it’s important to make this distinction, I will accept that.
HENDERSON: Well, for the accidental change, let us suppose there’s an object moving from A to B . . .
MCKEON: Well, let’s not take that kind of change; I think it will get you into difficulty. Let’s take the teaching example. Give me an example of accidental teaching.
HENDERSON: Maybe I better ought not do that. [L!]
MCKEON: No. All I’m trying to say is let’s get out of locomotion because in locomotion you don’t have much elbow room. Take another example, if you like. Or if you prefer, tell me what he’s talking about when he says that the musical walks?
HENDERSON: Well, to use your example of accidental teaching, why is it thought that teaching is going on when the teacher made a motion comparable to reading a text?
MCKEON: IS this a good example?
STUDENT: By accidental teaching, I think he means, for instance, the teacher tries to teach a student something and then he uses an outline; and the student, instead of understanding—well, perhaps he understands what the teacher’s trying to teach, but he also perceives that the outline is a good way to present it.
MCKEON: No, no. You see, this is one of the reasons why I’m nailing this down. What you are saying would hold for the modern conception of accident. An accident, for Aristotle, is a quality which has nothing essential to do with the agent. Suppose I were to say that in the present situation—this example obviously involves Americans—the fellow with the check coat on is who is teaching, and that’s how you know who is teaching. [L!] What kind of a change would I be talking about?
STUDENT: Potential?
STUDENT: Accidental.
MCKEON: What?
STUDENT: Accidental.
MCKEON: Accidental. That is, the fact that I have a check coat on has absolutely nothing to do with whether I am teaching or not. Yes?
STUDENT: Is that accidental teaching or accidental motion?
MCKEON: No, no. All the way through, this is what we mean when we say that changes occur accidentally; that is, if you identify the mover in terms of an accident which is sufficient to identify that mover, and, therefore, it’s not wrong, this is an instance of accidental change. Here, I’d have to describe the man in the coat a little bit more fully because others are wearing coats, too; but if I get a unique characteristic, say, the fellow in the coat who is not wearing a bow tie, and state that he is teaching, this is an accident which would identify the mover, but the mover is not moving in virtue of that characteristic.
Next, how would I state the essential cause? And here it doesn’t require any very great ability.
STUDENT: The teacher.
MCKEON: Yeah, the teacher is teaching. That is, in other words, X qua teacher performs the teaching; X qua wearing particular clothes, sitting in a particular place, living in a particular age, having a particular degree, and all the other possibilities, all of these would be accidental in nature.
What about the middle one, the one that we haven’t talked about, namely, what is moving not accidentally or essentially but in part?
STUDENT: Could it be like a hand writing on the blackboard?
MCKEON: But why would this be in part?
STUDENT: Because the hand is only part of the process of teaching?
MCKEON: I know, but it isn’t the hand that does the writing. In other words, the error has to be of exactly the same sort as the accidental. No, let’s go to locomotion here. Suppose that I were to say, stating this as clearly as possible, that as soon as I start moving my lips, I will be in violent motion. How could I justify this speech?
STUDENT: Well, you’re in violent motion because you’re breathing at the same time, too.
MCKEON: Yes, I am breathing. Is anything else going on? My pulse is going pretty rapidly; a lot of blood is being pumped around inside of me at a great rate. There are a whole series of things that are going on. I’m not sure I’ve finished digesting my lunch, as a matter of fact. The point would be that I would not be saying here that the man is in motion. He’s in motion if you consider certain parts of him, different parts of him; but he isn’t really moving qua individual: he’s sitting still. Or to take the teaching example, if you were to say, as the description in the University’s catalogue of courses does, that the University is teaching you in this course, this would be correct insofar as the University is a whole of which the faculty is a part and I am part of the faculty. But in the strict sense, the University doesn’t teach; it does it with respect to the parts.
All right. So we will be talking about motion in what sense?
STUDENT: In three senses.
MCKEON: Well, we will try to keep away from the partial motion, but it is both accidental motion and substantial motion we will focus on. As a matter of fact, we will come to find that as he goes along, sometimes he will classify a motion as a kind of accidental motion in order to bring it in.
If this is the case, we now move on to a second aspect of this question in chapter 1. We have been talking about the three kinds of motion. We go on to talk about the factors that are involved in motion, beginning at 224a33. What things over here do we talk about dealing with motion? Figure 14 practically has them.
STUDENT: Aren’t there five?
MCKEON: The direct cause of motion, what is in motion, that in which it is, or the time, that from which, and that to which. There are five factors that he enumerates in this transition of pages; and one of them, he says, “the starting-point,” we’re going to leave out at the beginning. What we’re going to talk about, therefore, is the mover—the cause, if you like—, the moved, and the goal. As we go along in this book, let me call your attention to the fact that some of his decisions here are exactly the reason why he didn’t come out well in early modern physics; that is, his decision led in the opposite direction. But part of what we are talking about has to do primarily with the difficulty of seeing this point, and those are the questions I want to raise as we go along.
Mr. Stern, in the first part of the next paragraph, he makes the statement quite flatly that the goal of every motion, “whether it be a form, an affectation or a place, is immovable” [224b11–12]. What does that mean? . . . We’ve indicated the things we’re going to talk about. What he talks about first is the end, that is, teleology, which is one of the things that we don’t esteem in his physics. Then he says that for any motion you can think of, the end of the motion is immobile.
STERN: He says the end determines the motion, and I thought . . .
MCKEON: I know. This is all right. I’m perfectly willing to consider that to get on the IC and get off at Van Buren,3 this determines where I am going; but isn’t the Van Buren station mobile as I’m carried down along the tracks, getting closer and closer? . . . Or am I being simpleminded?
STERN: Well, what changes is not the goal you’re going toward, but the motion itself.
MCKEON: I drew a white line on the blackboard—and he uses practically that example.
STERN: It’s the whitening that’s the motion.
MCKEON: It’s the whitening which is the motion. When I get all the way through, I’ve got all those nice white marks. . . .
STERN: Well, but the goal, the whitening, is the thing which is constant.
MCKEON: I know, but in what sense? Do you see what sense that makes?
STUDENT: Well, let’s talk about the house. The house is the actuality of the actual. That is, the actual can only exist in the sense that it has been actualized, whereas the motion itself is the possibility of the actuality.
MCKEON: Well, we have exactly the same trouble that we did before. What’s the use of saying that the house is immobile? It’s highly possible that, as in the case of one of the suburbs of Chicago, as soon as they get it built, they have to tear it down because they’re going to run an expressway through it. But the house isn’t immobile.
STERN: I was going to say that what would describe the motion itself would be a process, whether or not the end is going to help you and in spite of the fact that it’s actual. Then, I also think this distinction would be related to Plato, where you’re trying to describe motion in terms of the end, trying to identify the extremes, either the one or the other, its beginning or its end, rather than describing motion as though it was a continuous process.
MCKEON: What do the rest of you think? That this is a justification for these odd lines? . . .
STUDENT: Well, the end’s a form, it’s a form for what is changed.
MCKEON: Yeah.
STUDENT: In that sense it is immobile.
MCKEON: In other words, you would say that in analyzing any motion, if you give the characteristic which would mark the end, this is what the change is. As a matter of fact, if you take a look at the words which he’s using, he says, “To this we may reply that it is not whiteness but whitening that is a motion” [224b15–16]. That is, if you take any picture that I have put on the blackboard, what I would do would be to put black in by erasing the board and then put white in by writing with chalk; those are the processes. If I wanted to identify the motion, I would use the verb, “whitening”: that’s the process of motion. I get white as a result of this process, and if we’re in any position in which we’re doubtful about what we’re doing, you need to bring in a criterion. The criterion is formulated in terms of what constitutes whiteness; and whatever the variations that may occur to the white that I put on the board, what constitutes the white object must be the defined only after I’ve written, when the change has taken place. Is this all right?
Let’s go on to the next place, and I want merely to bring in an indication of why he raises the earlier question of what kinds of changes there are. He says at 224b27–28, “Now accidental change we may leave out of account: for it is to be found in everything, at any time, and in any respect. Change which is not accidental on the other hand is not to be found in everything.” Consequently, we have eliminated accidental change, just as we have eliminated partial change. Where is it to be found? Mr. Milstein?
MILSTEIN: IS this to be found in the same contrary to the beginning and to the ending, the contrary of contraries?
MCKEON: What does that mean?
STUDENT: Well, he talks about contraries in the case of the in-between contraries . . .
MILSTEIN: That’s right. But why is he saying apparently the same thing?
STUDENT: . . . that is, there are contraries which are contrary to the contraries at either extreme.
MCKEON: Well, what’s involved in this? I mean, he goes through an enumeration of the three possibilities, and there’s a difference. Consequently, putting down these three would run through it. But what I’m chiefly interested in is how the analysis is going. What is it the man is doing?
STUDENT: He’s giving those differences in which the definition of a given motion does not apply. Isn’t that what it is?
MCKEON: No, this would be . . .
STUDENT: The analysis of these differences would help us say that accidental motion can be eliminated.
MCKEON: No, no. The reason why accidental motion can be eliminated is that—we’ve already taken the motion of teaching—if you wanted to examine, to do a Ph.D. on, any aspect of the teaching process, and you said, “O.K., we will make a list of all the accidental aspects of any given process of teaching,” there would be an infinite number of them, and they’d have in common that they don’t have anything to do with teaching. Consequently, if we were taking a serious interest in analyzing motion, we would leave them out. Is that clear? It will frequently be the case that if we move from, let’s say, the pedagogic analysis of teaching to the sociological aspect, what was accidental from the pedagogic point of view may turn out to be essential from the sociological; but this is also a shift . . .
STUDENT: . . . by which we no longer talk about the actual.
MCKEON: No. The point I was driving at—I don’t want to get bogged down here—is that we’re laying down the way in which to analyze motion. A good deal of it sounds—and this is always the case—as if we’re doing semantics, but it’s a semantics in which our words are identifying things. We have said that what we’re going to focus on is the process. The principles of the process are privation, form, and matter. Consequently, the important thing is where you start, where you get, what the cause is, what moves, what the time is: these are all the questions. Therefore, the first thing we’d want to ask is, How are these two things on the blackboard related? The two I have on the blackboard are what?
STUDENT: Black and white.
MCKEON: I know, but what are black and white?
STUDENT: These are contraries in color.
STUDENT: But he said there are no contraries in colors.
MCKEON: This is a different question. What we are dealing with here is that if there weren’t contraries and contradictories in colors, there’s no qualitative change in colors. Let me put it this way. Suppose I wanted to describe the same motion in terms of contraries and of contradictories: what words would I use? It’s obviously the case that I’ve got to end with white. What’s the other word by which I can deal with this relation?
STUDENT: Nonwhite.
MCKEON: Nonwhite. What’s the difference between the relation of nonwhite to white and black to white?
STUDENT: Black would be the opposite of white.
STUDENT: You would think that black is only one possibility for nonwhite. Nonwhite is more inclusive in terms of colors.
MCKEON: Which is contradictory and which is contrary?
STUDENT: Nonwhite is contradictory of white.
MCKEON: Nonwhite is contradictory, and these, the black and white on the blackboard, are contraries. What does that mean about the relations that are involved? Is there anything between nonwhite and white?
STUDENT: Aren’t there many intermediaries?
MCKEON: No, it’s either white or it isn’t. Consequently, there are no intermediaries. Is there anything continuable between white and black?
STUDENT: Yes, there are intermediaries.
MCKEON: Gray, for a change. [L!] In fact, in the Middle Ages, it was held—the Arabs started this—that there are two sets of intermediates between black and white: one is the series of gradations of gray, and the other is the spectrum, that is, all of the colors come between black and white. So in either case, you have intermediaries. Bearing in mind that this is more than merely semantics since I am indicating things by these words, what we’ve got, therefore, is that the two terms, contrary and contradictory, are the same; that is, this is one of the reasons why we can leave out the beginning because we could describe it different ways. I want to get to the process which ends with the production of white—notice, the matter is the same. I can describe the motion, I can conceive it and analyze it, in terms of the transition, on the one hand, from the nonwhite to white, in which case we’re dealing with contradictories, or, on the other, from black to white or from any of the intermediaries in between to the white. Consequently, this is the reason for his interjection at this point.
At 224b35–225a7, he raises questions about a change of four kinds in which the analysis is somewhat similar. If you take subject and nonsubject, they can be related to each other four different ways. Since there’s no change from a nonsubject to a nonsubject, one kind we drop at once. Next, two of them give you generation in their extreme form, although they may not be absolute generation. Nonbeing he discusses beginning at 225a20. Aristotle frequently will say that there are as many senses of being as there are meanings you can give to the word to be. Here, too, he says there are many senses of nonbeing; and there’s motion in each sense because that’s what we’ve been doing with our contraries and contradictories: we were considering the ways in which you can begin with something which wasn’t what you’re going to end with. Finally, motion always falls under a change from a subject to a subject, or a qualified form of generation. Therefore, we end chapter 1 with the enumeration of the categories of motion, which, again, are quantity, quality, and place.
In chapter 2, beginning 225b10, he talks about the classifications of motion in itself, motion per se. Again he is eliminating: substance has no motion; substance is a kind of thing which is instantaneous, that involves only generation, not motion. This is a good paragraph to focus on because it is one of the places where the discussion of basic principles in science would make a big difference. What he is doing here is, among other things, demonstrating that there isn’t any motion with respect to motion; that is, there isn’t any motion of a motion, there isn’t any becoming of a becoming. In a significant sense, what Galileo’s analysis does is precisely that. It puts emphasis on the concept of acceleration, which is a motion of a motion, a change of a change; and you can begin to write your equations in ways that will give you squares and cubes, which would indicate what is going on.
What is it that Aristotle is saying here? Well, I’m only going to make two points; if you want to meditate about this, this is beyond the level of our present discussion. The first of the two points I want to make here is that Aristotle is not entering into direct contradiction with this other possibility, which was not unknown to the Greeks; there were other philosophers who were going in this direction even at this time. He is saying, rather, after the careful analysis which we have carried on, including the elimination of generation, which brings a body or a thing into existence, that it is quite clear that any motion is the change of a quality of a thing; and motion, although it is itself an actuality, does not have any qualities. At worst, you can say this was an unpromising approach if it didn’t get into a mathematical formulation. But then there’s a second thing: in the paragraph beginning at 225b17, he gives a series of careful considerations—this obviously had him worried—: “For in the first place there are two senses in which motion of motion is conceivable”—having just eliminated them. What is the first of these senses? If the motion itself is conceived as a subject. This is exactly what Galileo was saying; therefore, even within his own framework Aristotle could have gotten into it. In other words, the framework itself did not exclude the possibility. The second sense is when “some other subject changes from a change to another mode of being” [225b22–23]; that is, if I’m talking about the way in which the movement of my hand dies down when I stop the movement of my hand, this also would be a motion of a motion.
Although the period is up and many of you have another class to move on to, let me merely turn to the beginning of chapter 3. Having set up this schematism in which the whole analysis of motion has to do with the continuity from a to c (see fig. 14), he now makes a list of the things that need to be defined, and he then proceeds to analyze them: “’together’ and ‘apart’, ‘in contact’, ‘between’, ‘in succession’, ‘contiguous’, and ‘continuous’” [226b18–19]. This is a list, if any of you go on to the study of the history of science, which has a continuing importance. Let me merely give you one instance. Bertrand Russell wrote the Principles of Mathematics—he called it the Principia Mathematica, though we put it into English—, which is comprised of a series of propositions, and in it he analyzes the relation between infinity, continuity, contiguity, and succession. His argument is that you can’t tell what continuity is unless you know what the idea of infinity is; in other words, if you know what a dense series is, which an infinite series is, then you can know what continuity is. What Aristotle is doing here is going in exactly the reverse direction. That is, suppose you begin with two things: they’re apart, but if I put them in a context, they are together—notice, I’m going down the list. If I move one over to the other, they’re either in contact or, if they’re not in contact, there’s something between. If I put them in a process, they are in succession, and in succession they may be contiguous. Suppose, finally, they are continuous rather than merely contiguous; that is, suppose I were dealing with the line ac in figure 14 instead of my two objects. The lines ab and bc would be continuous because the end of the first line would not only be in contact with the beginning of the second line but the point b would now be identical for the two. The continuity results from this. And if you have a continuum, Aristotle argues, then you can go on and explain what infinity is because a continuous line like the one I’ve just indicated can be cut indefinitely; therefore, infinity is possible. Out of this discussion of continuity, consequently, and the earlier discussion in the Physics when he gets into the question of infinity4 comes the opening up of this whole series of problems. A shift in principle and in method will reverse the process. What Bertrand Russell was doing was moving through this same list of terms in the opposite direction. One of the things you would need to examine, then, apart from the fact that Bertrand Russell had the advantage of knowing a lot of things about physics that Aristotle didn’t know about, is what happens when you reverse the direction in terms of the things you can know and do in the region of physics.
Well, we will jump now almost two thousand years from Howard to Galileo. As I said, when you are reading the first ten pages of the Third Day, try to do it in terms that would raise the question of what it is that Galileo is doing that Plato and Aristotle didn’t do? How does the method of approach change?