Natural Hearing (Aristotle's Physics)
63. Physics VI: The Continuous Is Not Composed of Indivisibles, and Democritus' Cone
https://berquistcourse.com/course/philosophy/3-natural-hearing/063-physics-vi-the-continuous-is-not-composed-of-indivisibles/

[0:00] So let's come back now to the text here at hand. We'll touch upon these things again, but I want you to see kind of the fact, you know, that that these lowly things have to be understood to understand even the what higher things, so far as we can, huh?

[0:20] So the continuous is that whose edges are one, touching whose edges are what together, but they're not one and the same. Next to which there is nothing of the same kind between.

[0:35] Now notice in a way things that are continuous or things that are touching are next to each other too, aren't they, right? But sometimes you might keep the word next for something that is not continuous or touching, but it has nothing of the same kind in between, right? Okay. So my house and the neighbor's house are not continuous, and they're not contiguous, they're not touching, right?

[1:06] But there's no house in between my house and the next house, right? Okay. The same way you two are not what continuous, right? You're not what do you call them Siamese twins or something, right? Okay. You're not touching, right? But you're the next, okay, next, okay. Like they say at the doctor's office, next. They're all lined up there, right? Whatever offices it might be, huh? Okay.

[1:39] You see that? And in the strict sense, thoughts are more what? There's a next thought, right? And maybe a next topic, a next thing to be done, and so on, right? But not a continuous thing, maybe, or touching thing, in many cases.

[2:00] I says, given those definitions, it is impossible for anything continuous to be from what? Indivisibles, right? As a line from points, right? If by the line we mean something continuous, and by the point what? Something indivisible, right? Okay. Now, as Thomas explains in the commentary, he's going to be developing this reason for everything that is continuous, not just for a line, right? But also for the what?

[2:38] Motion down that line, right? He's going to argue that's continuous, and the time it takes to go down that, right? Okay. But kind of the emphasis here in the first reading is upon the the line, right? The magnitude being continuous, huh? And not being composed of indivisibles, right? And he's going to kind of elaborate, you know, more about motion about time in the next readings, right? Although really the basic reason is is seen here too, right?

[3:08] Okay. He's going to have extra ways of seeing it, just so you can you can be sure anything continuous, right, is not composed of indivisible, son. Okay.

[3:22] Now, sometimes when I do this, I'll say, you know, well, if you want to put something continuous together from indivisibles, right? That were possible. If you wanted to put a line, for example, together from points, the points would have to what?

[3:44] Come together, right? Okay. Okay. But can they come together at their edges? Say. There are no edges. Could two points have a a edge or limit in common, right? Say, like the two semicircles, right? Have that diameter, you might say, right? It's the end of one and the beginning of the other, right? They have that. Or could they have two edges together, two points?

[4:24] Why not? Say. Why not? They have no parts. Okay. So is there a distinction between a point and its edge? No. Well, in that case, you're imagining the the point to be like a little circle where the the circumference is what not the same as what's inside, right? See, but then you're giving the point some what, yeah, yeah, yeah.

[5:01] Now, in a way, you could say. Sometimes I have a student in class who denies that there is such a thing as a point,

[5:13] and you know, Zeno is supposed to have argued, you know, some of the ancient Greeks that if you add a point to a line, is the line any longer?

[5:25] No, so the point seems to be nothing, right? See, but in in a way that that is assuming that what is is what continuous, right?

[5:40] Okay. If I mention how Thomas uses that though, in his De Potentia, for instance, Magna... there when he's talking about how the goodness of the creature adds nothing to the goodness of God, because the goodness of the creature is to the goodness of God, not like a shorter line is to a longer line.

[6:05] In which case, if you add the goodness of the creature to the goodness of God, you'd have a what greater good, right? But there's an infinite distance between what God's goodness and that of the creature, so the creature's goodness is to God like a point is to a what a line, and a point added to a line is no more.

[6:30] Okay, that's kind of an amazing thing to see, right? That how purely gratuitous on his part, right, and how generous on his part, Th the the great uh Arab philosopher Avicenna there, you know, Thomas will quote him. Avicenna, you know, God. he's talking about the virtues of God, right? God alone, in a way, is liberal, right? Because when you talk about the virtue of liberality or generosity in in ethics, it's now when I give you something, expecting something in return, right, or demanding something in return, right?

[7:05] But just what, you know? Yours, you know. You know, I don't expect you to pay me for it. No, I expect in return just it's yours, really? Yeah, yeah, you know that's generosity, right? But I mean, God gets what nothing out of what He gives us, huh? You know, He doesn't even get a good act out of it, right? You see, if I give you something, you know, gratuitously today or tomorrow or something like that.

[7:40] At least he got a good act out of it, right? See, but when God gives us something, he doesn't get a new act out of that. he's completely liberal, he's completely generous, right? astounding right? you see, a lot of times students have a hard time understanding, you know, God, right? You know how utterly independent he is in the way of us, right?

[8:20] Anyway, so certainly, as I say, students would deny that there is such a thing as a point. I told. you how I tried to lead them to see that in a way there is a point.

[8:41] Well, I start off with that there are bodies, right? And they know there are bodies, right? Okay, they don't doubt that. That follows a Berkeley or something.

[8:54] Okay. And secondly, they know that there are bodies that don't go on forever. Like this table comes to an end, right? So there's an end of bodies, right? And what do you call the end of a body? Surface, yeah. Okay, so now you know that there are surfaces, right? And then I say to them, now,

[9:21] the surface has length and width, right? But does the surface of the ice cube does it have any depth? Because if you give it some depth, you're taking part of the body, aren't you? Right? So you haven't quite come yet to the end of the body. So there is an end of the body, right? Namely the surface, but it has no depth, right? So now they're admitting that there is something that has length and width, but no depth, right?

[9:58] Okay. And now I say, now, in this finite cube, let's take that as an example, right? The surface, which is squares, right? They have length and width, but no depth. Now do those go on forever? If they went on forever, then the body would have gone on forever, right? So there's an end to surfaces too. And now I say, now, does the end of a surface of a square, for example, does it have any width?

[10:35] No. In that case, you would not have come to the end of the what? The surface, right? But that's an end, right? It doesn't go on forever. So now they've got something that has length, but no width, right? Now they admit the existence of that. And then finally, I say, now,

[10:55] is there? Is that line that is the end of the square, any one of those lines? Do they go on forever? If they went on forever, then the then the square would have gone on forever, right? So they have an end, right? Now how long is the end of a line?

[11:16] No. If you gave it any length, you hadn't quite come to the end, right? But it does have an end. So now you've got something that has neither length nor width nor depth, right? So it has no what? Magnitude at all, right? So how can you distinguish between the point which exists by this argument and and its edge, right? Because then you'd be giving it some kind of what length, right?

[11:41] And even width. wouldn't. so two points couldn't really what have an edge in common, and that's it. The rest not right. Or you could have two edges touching, right? But the rest of them not touching. Yeah. Now I would make another little side thing, here with an interesting one, I think.

[12:15] Aristotle talks about the four kinds of causes. He argues that there's a first cause in every kind of cause, okay. And we'll see these arguments eventually. He gives them individually in the different particular parts of philosophy, but universally in the second book of wisdom, right? Okay. But he also says in the beginning of the fifth book of wisdom that every cause is a beginning, but not every beginning is a what cause, right?

[12:46] Okay. And then when he gets to the word end or limit, he says that we first you know speak of the beginning as here and the end at the other end, right? But sometimes we speak of both as being a what end, the endpoints of a line, right? Okay. So in some way, beginning is more general than cause, and end or limit in some ways what more general than beginning?

[13:14] Okay. What's kind of interesting? you have, something like what you say about causes that are on the limit. Just as there is a cause, it has a cause, right?

[13:28] So there's a limit that what has a limit, right? So the surface is a limit of a body that has a limit, right? So the square that is the limit of a cube has limits which are lines, right? The lines are also a limit that have a limit because the four lines that are the limits of the square each of them has a limit, which is the point, right?

[14:01] But now is the point a limit that has a limit? No, no. It is a limit, just like God is a cause. But just as God is a cause, it has no cause. The point is a limit that has no limit. Kind of interesting, right? Something something similar to what you have there. You sort of see that here.

[14:29] So he says there is not an edge, and some other part. Of the indivisible, or a limit, you could say, right? So you can make some distinction between the end of a body and the body, right? They're not identical, right? You can make some distinction between the end of a surface and the surface, right? And between the end of a what line and the line. The end of a line is a point.

[14:58] It's not a line. But can you speak of the end of a point? And the point to make some distinction between them. Kind of marvelous, right?

[15:15] There's not an edge, he says, in some other part of the indivisible. But there's no difference between the indivisible there, and and the limit. You can't make that. It is a limit. Nor are the edges together, for there's no edge of the partless. For the edge and that of which it is an edge are other, or the end and that of which it is an end are other, right?

[15:38] Or the limit and that of which it is a limit are other, right? Okay. Now, setting that aside, he's going to give a kind of either-or syllogism in the next paragraph here.

[15:57] He's going to reason against the idea that that a line can be put together from points, either by the points being continuous, right? Or by the points what even touching, right? Okay. And he says there is the same reason all indivisibles. And Thomas will note that, right? Okay. But it's a little more explicit here about the line and so on here, so that he'll kind of to more fully satisfy the mind, right?

[16:30] Gives some more particular reasons for thinking that the motion, right, over the line, right, and the time it takes are also continuous, right? But it's basically the same reason here.

[16:45] It's interesting. He does something like that with the causes, because he he gives a reason that's common really to all four kinds of cause, right? Although maybe it's a little more explicit with the cause in the sense of mover, right? But then later on he gives special reasons for the form and special reasons for the end, right? And so on, right? And look what he's doing here, right?

[17:11] See if it's the limit. You know, another little analogy there between the the two treatments, huh? Okay. So he says there is the same reason all indivisibles. They will not be continuous on account of the aforesaid reason. In everything touching, either the whole touches the whole, or the part the part, or the whole the part, huh? So, like two circles, right? Two circles can touch.

[17:50] Part can touch part, right? Or the whole can touch part of the other one, and vice versa. Or the whole can touch the whole. I just draw the circle twice. That's really okay. Now, sometimes when I I put that earlier part with this, and I'll say maybe there's a fourth way of touching, which is what to touch at their edge, right? Okay. Okay. Then you have to eliminate the piece of the point, right?

[18:20] Because the point has no parts, right? So that's impossible. And this is impossible, right? Therefore, the two points touch they're going to what Coincide coincide. And therefore, they have no more length than what one point. There's no length at all. So how are you getting any line out of two points touching, right? And if you know a hundred or a thousand or a million or infinity of them touched, the only way they could touch would be to coincide.

[18:53] And if they coincide, they have no more length than one point. There's no length at all, so you can't be put together. You could have a series of points, but that wouldn't be continuous, right? But then, now look at this like we said before, because you can't have any distinction between the point and the edge, or in their limit of the point, huh?

[19:18] Okay. That's the second one with the small circle inside that represents what again? The whole of one is touching part of the other, right? And vice versa, right? Okay. But both of those first two ways imply that we have parts, right?

[19:38] So he says, since indivisible without parts, it is necessary that the whole touch the whole. But yeah, you're kind of careful when you say that because whole might imply that you have parts, right? But it means that the two would have to coincide, right? Okay. Which would be like the whole touching the whole, coinciding, huh? But if the whole touches the whole, it will not be continuous.

[20:03] For the continuous has one part other than another, right? And is divided into different parts, separated in place, one part outside the other, right? Because that's part of the understanding of what continuous is, right? That is part outside of part, huh? Okay. So that's the way he develops it there, a little different than the way I was developing it there, right?

[20:36] Then he goes on to point out, but neither is a point next to a point, nor a now to a now, huh? Okay. So that length or time could be from these. For those things are next to each other between whom there is nothing of the same kind.

[20:58] But these maybe that should be there. But there is always a line between points, huh? And time between nows, huh? So in a straight line,

[21:16] the endpoints are the points that are furthest apart, right? But is there a point on this line that is the closest to, and not the furthest apart? Say.

[21:31] Or at any point you take, there's always another one closer to that, huh? So is there a next point on the line? Is there a next now in time?

[22:25] So if you try to imagine two points being next to each other, you are falsely imagining, right? I think, aren't you? So do you have a next thought, though? But not next point. Professor says my next point. You

[23:14] heard the definition of reason that Shakespeare gives, right? Okay. Now in English, there's another meaning of the noun reason. Now if I let me just put a couple of statements before here, you know, imitating my mother there a little bit. You know, she'd say, I see, said the blind man, but he couldn't see at all. You know, you have to see what there's two different meanings of the word to see there, right?

[23:52] Okay. Now if I say that reason is able to give a reason, does reason have the same meaning in those? Okay. Now which of those is the ability for large discourse, looking before and after? Which is that? First one. Okay. So this here, the genus,

[24:25] is ability, right? Okay. Differences are what? It's ability for, right? It's ability for large discourses, looking before and after. Okay. How about this here? Reason. Okay, reason over here is not an ability, is it? Yeah. Okay. Reason is the reason why

[24:55] man is more than a beast. Reason is the reason why man is more than a beast. Same two meanings, a reason there? Same to me. In that sentence, so with the first one, first one, yeah. Yeah, two different meanings here. Reason, right? Yeah. Reason is the reason why man is more than a beast. Okay, and which is the ability? First. Okay.

[25:39] So the reason why man is more than a beast is that he has this ability called reason. But Shakespeare defined right for us, huh? Now, what's this reason? What is it? It's a cause. It's a cause the genus, of it? No.

[26:32] It's a statement. You've got to define it as a statement. Yes. You complete your definition then? Is every reason a statement, but not every statement a reason? Yes. Okay, so what separates the reason that is a statement from other statements? Okay. Which gives a cause. Well. do you always say, do you have a cause as your reason,

[27:48] I think so. Some kind of cause. Do you like a species making difference or something? Yeah, we're looking for the for the the difference. Yeah, species making difference here, right? Well, sometimes I say to students, you know, I'll say to students that

[28:12] the best reason, I think, ever right, is the reason why something must be so. And Socrates and Aristotle teach us that, right? And I usually exemplify. I'll take, you know, something with geometry, and I'll say, when straight lines intersect. Are those angles equal? And they'll say yes. Okay, nice to know. What's your reason for saying they're equal?

[28:47] Well, they look equal. But is that the best reason you could give, right? Because I might say, you know, they look unequal, right? That's probably not the best reason you could give for saying that they're equal. I might say, well, if you measure them, you find them equal. But would that tell you it's so always?

[29:34] Euclid says that they're so that they're equal or is equal. But that's a reason, right? Saying they're equal, but is that the best reason you could give? Euclid said so. The geometry says so.

[29:54] Well, in this case, I go and I say because this is the right angle. Excuse me, because this is the right line, this is the right line meeting it. These two angles have to be equal to what? Two right angles, right? Okay.

[30:16] And because this is a right line meeting a straight line, straight line meeting a straight line, a plus x must be equal to right

[30:28] And quantities equal to the same must be equal to what? Each other, right? And equals subtracting equals to itself too. But notice the reason why these must be equal is that these lines are straight, right? Because these lines are straight, a plus x must equal two right angles, and b plus x must be equal. That's a consequence of that, right? And because equal to the same, they must be equal to each other, and the equals subtracting equals to is also equal.

[30:59] Just the axioms. But the basic reason why in this case is what the straightness of the line, right? The intersection of the line is the cause of there being angles there, right? But the straightness of the lines is the cause of the angles being equal, right? Okay. So that's the best reason you can give. The reason why it must be so, right? But then not every reason is a reason why it must be so, right?

[31:26] Say. But maybe there is a way that you could put cause in the definition, though of every reason, but not cause in the sense of the reason why something is so. Right?

[31:50] How about a statement through which you know or accept another statement, or through which you come to know? Yeah, yeah. Now, myself, my first inclination, maybe you could use statement because you probably stated the form of a statement, right? But I would probably want to put first to say thought, right? Okay.

[32:18] It's a thought of why, right? A thought of why someone thinks, right? The statement is true or false. Maybe that thought would take the form of a statement, but I don't know if it's necessary to spot out at this moment. Right? It's a thought of why someone thinks the statement is true or false.

[33:10] So the reason why someone thinks the statement is true or false may not be the reason why it is true, right? So there are some theorems in geometry that I might not know the proof of, but I have this respect for Euclid, right? And other people who studied geometry, right?

[33:32] So if I didn't know the proof of the Pythagorean theorem, I do know that at least. But but if I didn't know the proof of the Pythagorean theorem, I might think that it's true nevertheless, right? Because all the geometries say it is so, or because Euclid says it is so, right? You know. So my reason for thinking that that statement is true would not be the causality being true, right?

[34:03] But in the most perfect, like in this example I gave here, the reason why I think this is true is also the reason why it is true. You see, and then you see the perfection of that reason, right? Or someone whose reason for thinking these are true is that a couple that he measured they were, right? And that's why he thinks it's always true, right? That's not really such a good reason, is it?

[34:35] But me, who knows that their straightness makes them have to be equal, right? For me, the reason why I think they are equal is the reason why they are equal, right? In this case, right? Don't you have two different senses of reason? from that point of view? Well, in either case, it's a thought of why someone thinks the statement is true or false, right? Why do I think that these angles will be equal?

[35:11] Well, because those lines are straight, right? And that's why all this follows, right? Someone else might have a weaker reason for thinking they are so, right?

[35:29] Doesn't Thomas say there, you know, that the argument from authority is what the weakest in say philosophy, right? Strongest in theology. So, doesn't this person have a reason for thinking this?

[35:50] You know, if you ask me, you know, there's two Chinese restaurants in town, right? A and B. Which one should I go to? And I say, well, I went to A and I had a wonderful meal there a month ago. Went to B two weeks ago and had a lousy meal. Okay. Your reason for going to A, I assume you better go to A. unless you find the price of all this.

[36:22] What's your reason for going to A rather than B? In your experience. Yeah, It's because I went to A and I couldn't be, though, right? And went to B and had a lousy meal, right? Okay. So that's an argument by example, right? But is that you know the best reason? Maybe the best reason I can have in this case, right? But it's not a reason why I must get a better meal at A, right?

[36:54] You see. I might have changed cooks. Who knows what? Was on the cook's mind when he when he poisoned the made a bad meal, right? Who knows? Might have been the problem in the you know the raw materials too, right?

[37:11] You know. We can buy in the grocery store, or, you know, a piece of meat or something that's defective, right? Not Noah. Used to be at a restaurant I'd stop in on the way up to Quebec, you know, and it's kind of a handy place there. And I don't know if they don't call it steakhouse, but the steaks always had kind of a fishy smell. They were the same compartment of fish.

[37:37] or something. Did you ever tell what happened at the restaurant? Did I tell you a funny story that happened there? Huh? Not, sure. Well, one day I goes in there, right, and I sit down, and then I went to the bathroom, right. Went to the bathroom, right, and the the sink was up like this. Oh. And you can kind of see they must have, you know, lowered the floor or something, you know, in some kind of construction, right?

[38:09] And rather than change all the pipes, right, they left the sink where it was. So it was kind of usual. I mean, you could do you wash it. It was kind of you know usual like that, right? You see afterwards, right? Okay. I've never been in a public restroom like that. where that happened to be so, right? It was kind of okay. So anyway, I go back and I sit down in my booth there and I'm ordering my meal, right?

[38:30] When comes a man and his wife, see. And they sit down in the next booth, and the man is complaining about some place they've been, and all he does is complain, one complaint after another. And I can kind of watch his wife there, like she must be a long-suffering woman. Yeah. Just having to deal with everything, you know, and so on, see. And she must hear her complain all the time.

[38:53] But finally, waitress comes over to take their order, see. And you know how you want your steak done, and so on, right? And he says, "If it isn't done exactly this way," he says, "I will send it back." And he, you know, you know, why he's so excited? He doesn't just happen. He doesn't send them out, you know, and you know, just kind of, you know, you know, has to put up with this man, and the waiters do that.

[39:17] Man arrived, and and so on. So I mean, he's spelled out ahead of time, right? But he's he's reading the plates, and he thinks it isn't exactly what he wants it. So I'm kind of, you know, I'm just throwing something. Hearing this kind of situation, then finally the man says, "I didn't go to the bathroom." Oh. I knew he went in there. But the question of woof woof, he complains about everything, right?

[39:41] Yeah. He comes out. I knew. I could just see it coming, you know. How terrible, but the way that it was, he says, it was up like this, he says.

[39:51] I'm sure she didn't believe him, you know. She thought he must be exaggerating. I think it was kind of ridiculous. It's just so funny to see them.

[40:05] So anyway, the best reason, you know, you know, for for for going there may not be the best reason, you know. In all reasons that I can give for things, right? Okay. We have to use those reasons to decide all we got to go on, right?

[40:28] I can't give you a reason why you must have a better meal at A than B, right? And as you know, I mean, it's possible you get a better meal at A one week and a better meal at B the next week, right? See, but if all I know is the two times, the one time been at each one, and I had a good meal at one place and another one, so I got to go on.

[40:52] You're going to guess, you're going to get a better meal, right? But you're really guessing, right? But it's a reasonable guess, isn't it? Huh? Wouldn't be reasonable to go to the restaurant that you know someone you know had a wonderful meal at, and and and rather than one he had a lousy meal at, right? You see. See. So we have a reason, but it's not a reason of the caliber of that.

[41:17] It's not a reason that makes me say you must necessarily then, right? See, I'd probably say, "Well, I guess I'll go to A then." You know, I would say, "Well, then I must have a better meal at A than B," right? That would be that would be attributing more power to the reason than I than I actually should have for it. You see.

[41:41] So, of course. Nevertheless, that's the thought of why I think, right, that I should go to A, right? Maybe that thought is stated in the form of a statement, right? This is my new statement here again. I don't like to say twice. I don't have to, you know. It's a little bit like the question when you define a a statement itself, right? You know, what's the genus of statement?

[42:16] Speech, yeah, speech signifying the true or the false. It could be one of those things. A sentence signifying the true or the false, right? But not necessary, really. You know, say a sentence.

[42:34] So there we have the other reason defined. So I put statement into its definition. I put cause into its definition, but not exactly the way you did.

[43:06] So this is the last paragraph now. But it is impossible that any other kind of thing be between points and nouns. If there were, it would be clearly divisible or indivisible. And if divisible, either into indivisibles or into what is always divisible. This, however, is the continuous.

[43:33] It is clear that everything continuous is divisible into things which are always divisible. Why? For if into indivisibles, the indivisible would be what? Touching the indivisible, right? For continuous things have one edge and touch.

[43:56] So obviously, is the form of the argument there good? If you divide a line into two points, then the two points would be what? Touching, right?

[44:21] But two points cannot touch, right? Of course, I. You mean two? So you're negating the consequent, right? Then you have to negate the antecedent, right? Okay.

[44:40] I think I mentioned that before, like in Meno, right, where Socrates reasons from if then statements, right? But he reasons from the affirmation of the antecedent to the affirmation of the consequent, and then he reasons from the denial of the consequent, the denial of the antecedent. He argues that virtue can be taught. If virtue is knowledge, then it can be taught. Virtue is knowledge, therefore, can be taught.

[45:04] And then he reasons that virtue can be taught. There are teachers of it, but there are no teachers of it. Therefore, cannot be taught. You see? But you know, he doesn't you know teach formally or universally the the form of the if then syllogism, but exemplifies right the two forms that are in fact syllogisms, right? But he doesn't reason from the denial of the antecedent to the denial of the consequent, or from the affirmation of the consequent to the affirmation of the antecedent.

[45:38] I know from experience when I when I talk there and you put the the forms on the board, if A is so, B is so, right, and A is so or A is not so, and B is so or B is not so, and you know, I teach logic. I just get four students to ramble with the board, and I say, if anything follows necessarily, right, you know, right under the line, what follows necessarily?

[45:58] If nothing does, right, you know, invalid. And somebody always gets one of them wrong at least. And sometimes, one time, it was perfect. I mean, they got all four wrong. In other words, where something followed, they thought nothing followed, and where something didn't follow, they thought something did follow. I say that now. If that isn't a clear enough example of the need for logic, right? I mean, just imagine, you know, your mind is in the state of thinking that when a conclusion does follow, it doesn't follow, and when it doesn't follow, but it does follow, what what harm is going to do to your whole the whole life of your mind?

[46:41] You see, You know, they're not struck by the significance of their mind being in that situation, huh? You're going to miss out an awful lot of truth because you're not going to see what follows from what you do know, and you're going to run into a lot of mistakes and errors in your life because you think something follows when it doesn't. You are in serious trouble. I know.

[47:12] Is your doctor? Your mind's not functioning as it should, right? Yeah. Okay. Now. So over there in the second reading area, Aristotle has kind of hinted that this reason is common to all of them.

[47:51] He says there is the same reason for composing magnitude, like a line, time, and motion indivisibles, and dividing them into indivisibles or none of them. Right

[48:10] now he's going to go try to manifest this a bit more for motion in this particular reading, right? And then in the third reading he's going to be manifesting it again for time in particular, right? Okay, and

[48:29] yeah, um, but. Let's just stop on something maybe that also could be manifested, not done or yet. But is there also the same reason in a sense for saying that a point is not composed of, I mean, the line is not composed of points,

[49:00] and a surface is not composed of what lines, and a body is not composed of surfaces. I don't know if he explicitly says it here, but I mean, wouldn't it be like a similar reason, huh? If they were, the surface was the lines would have to touch, and

[49:40] they have no width. You know, if you imagine a a square or something like that to be made out of what, you know, all these lines lined up,

[49:54] then the lines would have to what come in touch with each other, and the two straight lines when they touch, could they touch at their edge and not coincide?

[50:16] Because they have no what. Okay. Yeah, no width. Because then you you give it some width, you'd be making a little bit of a what. Pencil stick, isn't it?

[50:38] So you can't really make a what surface out of what lines, huh? Now the same thing would be true about a body being a layer, being a pancaking surfaces up, so to speak, a pile of surfaces, a pile of pancakes. Because the surfaces have no depth.

[51:02] Yeah. So you can't get any depth by adding what what has no depth to what has no depth, right? Kind of getting something out of nothing almost. She puts some this way and then a few that way. I mentioned I think before, but come back to it now in this context here. You know, the Democritus It's in the it's in the Loeb edition of Greek mathematics. There, right?

[51:36] The Loeb Greek. The the the one of Democritus raising this about the cone. And he's saying, suppose you had a cone here, right? And you bisected it parallel to the base, right? And then you took it apart. Well, you have a circle at the bottom of this pile of circles, and a circle at the top of this pile, right? Okay. Now he says, you look at those two circles.

[52:22] Are they equal or unequal? Equal. Okay, they're equal. But he says, if they're equal, and as if you do the same thing, you cut it, the circle above the one below would be equal, right? Then this pile of pancakes, circles, is a pile of what? Pancakes that all have the same what? Size, right? So then this whole thing should be not a cone, but a what? A cylinder.

[52:56] But a cylinder, right? Okay. That's the problem that comes if you say the one above the one below is equal. Well, now, now other alternatives say they're unequal, right? But they're unequal. Would this be straight? Because it isn't a cone.

[53:21] In fact, you generate a cone. Is this way to do it? You know, kind of Euclid says with a circle, you take the diameter, you rotate the circle around the diameter, you get a sphere, right? Well, doesn't you take a triangle and kind of rotate it around, and then you get a what? A cone, right? Okay. So this should be a straight line because the way it's been done.

[53:47] It's straight and all around, right? So if if the one below is a little bit bigger, it's going to be. not straight, right? So it's got to be equal or unequal. Either way, you've got

[54:11] a what? Contradiction, right? That's what we have in the fragment of Democritus, right? See. But isn't that a problem because you're what? Circle doesn't have depth. Trying to compose it. Yeah, you're thinking of this of this of the thing here as being what layered, right? See. Now. If we cut the cone parallel to the base, right?

[54:50] What would we say is there? Yeah, two circles there, okay. The one above and one below are they equal or unequal? What would you say? Equal. Equal, yeah, yeah. Now, why don't you get into the problem, therefore, saying that this is a cylinder instead of a cone? Because they

[55:12] they coincide. Yeah, when you're together. Yeah. Again, if you cut, you know, anywhere along there, they're going to be the same. You say, So why isn't this a cylinder then.

[55:29] Because I'm not I'm piling on top. I'm not composing. It won't be there. The top cut won't be the the circle won't be the same as the bottom cut circle. Yeah, if you cut it here, and the circle above and below would be equal, right? Right. Cut it here. The one above and below be equal. Right. But these two wouldn't be equal to those two, right? Right.

[55:56] Okay. But how close can you bring those unequal circles? See, somebody might say, "Well, let's take the one right below this, right? Right. Okay. Let's take the next one, right? The next cut. Is that equal or unequal, right?

[56:28] Well, there you'd seem to be back in this problem because if you make it the equal or unequal, the next one is is equal, then the next one of that is equal, and all the way down they're equal, right? You say the next one is unequal, then it's jagged all the way down, right? Okay. And you seem to have an insoluble problem like Kant's antinomies, you know, that terrible thing.

[56:55] What's the solution to this then? This this there's a seems to be a contradiction whichever you say, right? What's the solution to it? No next. Yeah. See. Remember that?

[57:06] Just as there's no next point in the line. So I mean, I think you could talk about bisecting. I mean, you know, cutting the the cone, can't you? Parallel to the base and then to the base to the base Cutting it, what Uh, that's where they get conic sections, don't they? So I'm going to cut these things, okay? You can cut it, right? And then you'd have to say that on either side you have a circle, right?

[57:37] You have to say they're equal, right? Okay. Well, it depends on the material, though, too. If it's wood, you actually lose that space in between. Yeah, Yeah. How wide your blade is. Yeah, that's true. Then it would be unequal. But in geometry, you don't have that problem. You see?

[57:56] But it wasn't really actually there? Those two circles until you cut it, see? Okay. So I think, in Euclidean geometry, you'd have to, in Euclidean geometry, have to say that the circles are equal. So if, in fact, there was the next one below or the next one above,

[58:18] then you would get a problem, right? That's it. But there is no next one, right? You think? But the same, you know, if you just, you know, may t take it, make this, you know, just a triangle now, right? See? Okay.

[58:47] And forget about the three dimensions; it's simpler it's always easier to work in two dimensions. Okay. Now, let's say I have a, you know, sausage triangle. Okay. Now, I cut it. I draw a line through there, parallel to the base, right? Okay. Now the, I lift this up. So the bottom line here, and the top line here, would they be equal or unequal? Equal, yeah.

[59:21] Now, what about the, the next cut? See. There's no next cut. No, no. See. But if someone admits falsely, right, that there is a next cut, right,

[59:42] then, he's got a problem, right? Because if you cut, take the next cut above that, and they are both equal, right? Well, then, the same as the one above that, and all the way up, and all the way down, right? So now you've got a what? Not a cone, but a what? Rectangle. A rectangle rather than a triangle, right? See.

[1:00:12] So. No, so every little bit of the truth there is important. So it's interesting that that Democritus presents that difficulty, right? Now, where you got to the solution of that difficulty is another question, right? But but the way he has it, I mean, you're inside the difficulty, unless you understand the continuous, as Aristotle understands it.

[1:01:08] You get this difficulty because you're thinking that the divisible is composed of indivisible, right, in some way, and or because you're thinking that there's a next. Well it's kind of marvelous? That thing I think of Democritus. I mean, for especially for people who are thinking that way, right? Because then you put them in that difficulty, and there's no way out.

[1:01:40] They think there's a next one? Because they're imagining like a pile of pancakes, and there is a next pancake, right? There's a next one above that, next one above that, right?

[1:02:04] If you want your pile of pancakes to resemble a cylinder or a cone?