Natural Hearing (Aristotle's Physics)
Lecture 69: The Continuous: Definitions, Divisibility, and Becoming

Transcript
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So, why talk about the continuous, right? Well, continuous is common to those things, right? The motion and place and time, right? These things are what? Continuous, right? And because the distance from here to there is continuous, the motion over it and the time over it, right? So they're all, you see, connected to that. Yeah, but another reason why I want to talk about the continuous here is that it seems to be the most, almost the most basic thing in our knowledge. I'll bring in the text there from the books on Sense and Sensible and the book on Memory and Reminiscence. But Aristotle says, you know, we don't think without an image and therefore we don't think without the continuous in time, right? Okay. And if you take, you know, these basic words like beginning and before and after and things like this, they all seem to have a connection with the, what? Continuous, right? You know, the beginning of the table here, right? Beginning of time. The before and after for us to see it in time and so on. You don't realize how fundamental the continuous is in our, what, in the very words that we use. And when you study later on in the three books about the soul, you study the, what they call sensible, right? And Aristotle will distinguish between the sensible as such and then the accidental sensible. And someone says, I can see you're angry. Does he see your anger? Have you ever seen anybody's anger? That's accidental, right? If you're not really sensing it, right? But you know it by something else in you when you sense, right? Sitting things, huh? But then the sensible as such, he divides into what? Two. And one is what you might call the proper or the private sensible. It's private to one sense. Like color is to the eye or sound to the ear and smell to the nose and flavor or something like that to the taste and so on. And then he speaks of what he calls the common sensible. Something that's sensed by more than one sense. Like, for example, what? Surface, right? I can know the surface through my, what? Sense of touch, right? I can know the surface by my eye. And, um, or the shape, right? I can know the shape of this table without my eyes, right? I can feel the shape of this glass, right? I don't know what I'm talking about. I can feel the shape, right? It's round, you know, if you put it in my hand, right? But I can also see it, right? Because he calls those the common sensibles. So the common sensibles are tied up with the, what? Continuous, right? Okay. Like the surface here is continuous, and even the shape is kind of the limit of the continuous, huh? Okay. So, how many words are tied up? The basic words that I'm thinking are tied up with the continuous, huh? As opposed to the private sensible, huh? Take something like the word red or green, right? Well, I might call communist or red or something, but... Does red get extended much to anything else than red? Hmm. Doesn't seem like it. No. But now take the word like beginning and the word end. The first meaning of beginning and end is like, this is the beginning of the desk and that's the end down there. Mm-hmm. But how many senses later on does the word beginning take on? Aristotle talks about the foundation of the house as the beginning, right? And he talks about how the prince and the city is the beginning, right? The principle is the beginning. This is Sister Carolyn I was talking about. Mm-hmm. But principle comes from the lack of a word for beginning, right? And then you speak of what? The axioms as being the beginning of geometry or something. The definitions being the beginning of a science, huh? You have all these different meanings of beginning, but they all involve carrying over that first meaning of beginning, which is the beginning of something continuous, right? Mm-hmm. And then we speak of end in the sense of purpose, right? And then even the Greek word for definition, the Latin word, they use horos, terminus, limit, end, huh? You see? So we take words that name something in the continuous, right? And we carry them over to other things, huh? I'm always talking, I'm going to tell you the Greeks here, when I say this, I'm using the word road all the time, right? The first road in our knowledge is the road from the senses into reason, huh? Mm-hmm. The second road is the road from reasonable guesses towards reasoned-out knowledge, huh? Then there's a private road of each reasoned-out knowledge and so on. But this word road first comes from something, what? Continuous, the road upon which I drive here, or walk here, right? See? Let me carry it over and apply it to the mind, huh? So, in and out. Well, in is the first meaning of the place, right? The place, again, is something, what? Continuous, right? You'll stop and realize how fundamentally continuous is in all I think. That's why in the De Trinitacta, you know, one of the atheists is talking about these things and so on. Well, if we never think without an image, and therefore without the continuous, how can we think about something that is not continuous? Well, we have to negate the continuous, right? In corporeal, he's not a body. He has neither link nor death, right? Now, another way that's going to be very important is when we get to the three books on the soul, because there we want to eventually talk about whether man's understanding or his reason is a body or not, right? And if we can show that reason is not continuous, right, then we can syllogize that it's not a body, right? So you have to understand what the continuous is before we can see that reason is not continuous, right? And you'll see it even in the Dialectic of the first book about the soul, that Aristotle is arguing against thinking, right, being something continuous, right? And notice, we do speak of a line of thinking, don't we? There's some likeness of thinking to a line, right? But is it really continuous, like a line is, right? Well, you have to understand the continuous before you can see the reasons for thinking that thinking is not continuous. You have to see that before you can see why we think that reason is not, what? A body, right? Okay? We have to know that before we know that the human soul has a power or an ability that's not in the body, right? And that's a sign that its existence is not entirely, what? Immersed in the body, right? Okay? If it had existence only in the body, right? It would only, what? All right. All right. All right. All right. All right. All right. All right. ...operate in the body, right? Okay, right. Okay? So, if thinking is not in the body, if understanding is not in the body, then the soul's existence is not only in the body. So you have to, in order to see if the understanding is not in the body, you have to see how understanding is not something like continuous, right? And how even in understanding a continuous thing, we understand a continuous thing in a non-continuous way. My mind is not false in doing that, right? Because it's not saying the continuous thing is not continuous, that would be absurd, right? But the way in which it knows the continuous is not continuous. Just as the way I know the past, which is now, is not the way the past is. But I don't see it, the past is not doing it. Past is past. That's gone. That's over. Right? My youth is gone. You know? But I can remember my youth and my way back to kindergarten, right? Each had a little drawer in the kindergarten, right? And it was supposed to, you know, each student's going to have a drawer, so the sister said, you'll bring a little toys in the night from home, and we'll tie it to your drawer, so you don't go to recognize your drawer, right? I brought some toy soldier in the home and tied to the thing. But then I wanted to pair that soldier, but I couldn't because it had to stay at school, because that was the education. Yeah, I hadn't foreseen that. Yeah. So, I mean, that frustration, that's, you know. You know who our teacher was? Did I tell you who she was? Her name? Sister of kindness. Yeah, yeah, yeah. So, sometimes I tell you what my reference, you know who our teacher was? No, it was kindness. You are, though. I was getting a, what do you call it, a coconut, you know? You know, a little thing of coconut. But it's very hard to open a coconut, right? You're dark. You couldn't get the thing open, you know? I tried breaking it. But I just had broken it. So I can know the past now, right? The now is in my what? I know it, right? Not in what I know. I don't think that Sister Aquinas is still trying to get that thing open. That cooking open. That's it. My toy soldier is still in Chibi Great School in the kindergarten. Chibi Great School, right? Everyone pronounce your name. So again, here, you'll find out that our reason knows the continuous in an uncontinuous way, right? It doesn't say that the continuous is not continuous, but the way in which it knows it is not continuous. That's where I saw the stubborn, right? The recognition there. That the way we know doesn't have to be the way what things are, right? All right. And you'll notice, I'll take a little different example here, but something like that. I could know the shape of this glass, right? By the sense of touch, right? And I could know it by what? My eyes, right? I could know the shape of this table here by my sense of touch. But the way we know this is different, isn't it? The eye and the touch. The eye knows shape through what? Color, right? Color extends us far and no further. The sense of touch knows shape through hardness, right? So if the way of knowing had to be exactly the way the thing is, you could have two ways of knowing the same thing, could you? Sometimes people think, you know, hey, the theologian talks about the immortality of the soul. The philosopher can't talk about that. But it's possible sometimes the same thing be known in two different, what, ways, right? Did you see that already in the senses, huh? The senses. So the better you understand the continuists, the better you can understand later on what? What is not continuous, right? Just like we saw with the example there of what? Time. The better you know time in the now of time, the better you can understand the definition of eternity, right? Okay, so there's a lot of reasons why it's important to know the what? Continuous, huh? Now when you looked at the definition of reason before, it's the ability for what? Large discourse, huh? Looking before and after. And then we had a text in our style, remember, before and after. But the first meaning of before and after is in the continuous, right? And then the second one he gave, he took an example from the discrete, right? One is before two, remember that? You know? So you have to, you know, understand the first meaning of before and after and see how the word is moving forward according to certain connection with these meanings. So you don't realize how basic and continuous is in our thinking. And most people can't transcend it at all. Whatever it is must be somewhere. They can't transcend it and continue this. See? I mean, I think if you ask most people, you know, what eternity is, they'd think it's a what? Endless time, right? When endless time, like an infinite line, there is such a thing, it's still something continuous, isn't it? They can't transcend the continuous. And the imagination, you see, is tied to the continuous as are the senses too, huh? You know, in that it senses the imagination, right? You can't sense or imagine without the continuous. It's so fundamental in our knowing. So the philosophy of the continuous is important for understanding, motion, place and time, right? And if you ever look at, you know, the Summa Theologiae, it's not as complete as the Summa Contra Gentiles as far as the reasoning from motion to God, right? Now, if you look at the Summa Contra Gentiles, it's much more complete, huh? And when he's arguing, you know, to the proposition, say that whatever is in motion is moved by another, right? You know? He has, you know, three different Solgisms to show it in the Summa Contra Gentiles taken from the Aristotle and in the Summa Theologiae just as one, right? But some of those are drawn from the sixth book, we'll see. Some of the reasons for saying that something can't move itself are drawn from an understanding of the continuous character of what motion, right? I think we talked about that problem that Decai wrote the article on, paradox of Duvenir of Parla-Khandra-Xion, right? And how do you have this change between, what? None being and being, right? Suppose what is not a circle becomes a circle, right? Well, there's a period of time in which it is, what? Not a circle, right? Right. There's a period of time in which it is a circle, right? Now, is the last instant in which it is not a circle the first instant in which it is a circle? Well, if you say that, that it both is and is not a circle, that's what Hegel says, right? Now, if they're not the same instant, right? Last instance which it is, then there's two different instances, two different nows, right? And just like with two different points, you have to have a what? Timing between them, right? Okay, unless two what? Nows could be what? Adjacent. Yeah, nothing between them, right? Okay. But is that possible? Or is it like two points, see? Well, that's something you learn when you study this, right? That if two points touch, they coincide, it would be the same one, right? Right. So if you really have two distinct points, there's got to be a line between them. I can always draw a line between them. So if you have really two distinct nows here, the last instance in which it's not a circle, in which it is a circle, then there has to be a time in between, right? And I either are a circle or are not a circle, so what do you get in this time in between? Well, if it's not a circle, well, then this pertains to that time over here, right? If you are a circle in this time, it pertains to this time, right? So you can't have any time in between the two. So it seems you're going to be forced to say that the last instant in which it is not a circle is the first one which it is. And then you've got the contradiction, right? And that's what Hegel argues, you know, that becoming is a contradiction, right? Okay. Now, Christel is going to solve that problem in the, what, sixth book, right? He's going to bring out that there is not a last, what, instant in which it is not a circle. But there is a first instant in which it is a circle, I don't think. And that might seem arbitrary at first, but it's not, as we'll see when I'm going to get into the book. There's a reason why it makes sense to say that there is a first instant in which what is becoming a circle is a circle, right? But there's not a last instant in which it is, what, not a circle, right? Just like when you're dying, right, huh? See? Okay. See, so-and-so is dying, right, huh? Takes some time to die, apparently. Sometimes, anyway. Okay? You're dying, right, huh? Okay? Now, is there a first, is there a last instant in which you're still alive? Or just a first instant in which you are alive? Yeah. Yeah, yeah. But the end of the time in which you are dying, right, at the end of that time, you're dead. So the end of that time is an instant, right? But the end of the time in which I'm becoming, when I'm dying, I'm dead. So there's a first instant in which I'm dead, but not a last instant in which I'm still, what, alive, right? But at the end of my life, when I was coming to be, right, was there a last instant in which I was not? But there's a first instant in which I am, right? That's what we're just about in there, but I think that's very important, right, for understanding becoming, right? And for understanding it as being without a, what, kind of addiction, right? Now, the Zeno, you also have probably heard of Zeno's paradoxes, and they'd be texted upon some of the sixth book, but, right, he comes back to them in the eighth book, but Zeno had certain problems. Zeno was the people of Parmenides, right? And Parmenides was denying change because it seemed to involve a, what, contradiction, right? Okay, the one that we saw solved earlier, right? In day becomes night, right, the sick become healthy, right? We all say that one opposite becomes the other, right? Right. Now, if one did come to be the other, you'd have contradiction, right? Day would be night, because day becomes night, night becomes day, right? Well, this is another example, you know, this one that Hebron made use of, where you seem to have a contradiction in becoming, right? But Zeno raised other problems. Zeno, because people of Parmenides wanted to show the difficulties, right? Well, his problem was, what, since the continuous is divisible forever, right? Before I can get out the door, I've got to go half the way to the door, right? Before I can go half the way to the door, I can go half the way to the door. Yeah, if I can go half the way to the half, I've got to go, what, half of that, right? And maybe you can get out the door, then, in this case. Or other paradoxes, Zeno, you know, where can the Achilles, who's a fast runner, catch the turtle or something, you know? It's very slow, right? Well, now that the turtle has a head start. Because if the turtle has a head start, before he can pass it, he's got to catch up to where he was, right? And in that time, you walk all a bit further, right? And before he can catch up to that, he's got to catch up to where he is, and that kind of thing will go a little more, right? See, in various ways, you know, dealing with this continuous character emotion to say that it can be, right? And he's kind of defending his master of primenities, and people are saying, well, you're ridiculous, right? Denying that most exists. He said, well, you're saying that it does exist involves all these impossibilities. So, this is a very fundamental thing for understanding change, and for understanding it as something possible, rather than something that involves a big contradiction, huh? So it's an awful fundamental consideration, you know, this philosophy of the continuous. So you've got to reproduce some cameras for us, right? We'll be spending some time on that. Continuism is going to be a key thing, huh? All right. Sounds like also the argument against, or to show that it's a miracle to have a death-bed conversion, because there is no last moment. Well, he does, before the last instant comes in, right? And they go down. When the tree falls, there's a lot. Okay? We extend more on the words that are from the common sense. Yeah, yeah. We don't extend green or red so much. Sure. Except metaphorically, you know. A word like sweet is extended, you know, you use it metaphorically, right? Taste and see how sweet is the Lord, right? Thomas explains that metaphor, right? But the words drawn from the continuous are extended, what? All of us say. Why is that more common? Is that... Well, I think part of it is that it's more understandable than continuous, right? More proper sensible. Yeah. Because you'd say in the modern, so, you know, that, you know, how would you define the redness of red, right? It's kind of hard to define red, isn't it? I can define the continuous in a number of ways, right? It seems to be more understandable, the continuous, huh? You can see that, too, you know, in how geometry is to the clarity, right? What is red? I mean, I can point to it, but what is red? Of course, you know, the moderns, we may talk about a wavelength, but then they're talking about something continuous, right? But if you don't go to it mathematically, that continuous aspect, what is red? I can't, you know, over there. You know? I don't think it's red. Why is that that it's not so just that I can imagine red? I can, just like I can imagine it continues. Yeah, yeah. Well, it's, again, you know, go back to the distinction they make between math and natural philosophy, right? Sensible matter is something more, what, material, right? But it's a sign, man. It's more tied to matter, the sensible qualities. Sometimes, you know, we'll say that mathematics abstracts a matter, right? Other times, Aristotle will speak of an understandable matter, right? As opposed to a sensible matter, but that seems to be not matter in the... in the fundamental sense, right? So just, you know, if you look at the first, excuse me, the first chapter there in the fifth book of the metaphysics there, just in the word beginning, take the one, you know, or look at the word before and after in the before in the categories, right? It comes up again in the fifth book, the wisdom, too. See how important those words are, right? Look at the word in, you know. Look at the word in, like we were talking about, right? They're all tied up with the continuous, huh? See, and how those words are so extendable, right? But you can't... You know, we've had eight meetings, have you ever been, remember, the other day? You know, and they're all ordered, right, from that first meeting, you know, by certain likeness and proportion and so on. And, but can you do it with the word red, you know? Huh? Yeah, can you? We're in this room, right, huh? Teeth in my mouth, genus and the species, species and the genus, right? All these things, right, that come, you know? I can move the word in to all those eight meanings, huh? I left my heart in San Francisco, right? Where can I move the word red to? That's because it's less sensible. Well, it's that part of the relationship, yeah. I mean, I think it's easier to see that it can't be moved, right? And exactly why, you know? But it has something to do with the fact that the sense qualities are more, what, tied up with matter, right? Even in the continuous, huh? And that's why you have one species of motion is alteration, right? But alteration is always spoken of as being the third species of quality, right? Sensible quality, so... Nothing in the fourth species figure, shape. But don't worry too much about exactly why it is so, you know, green at first, but just see how important, you know, that is, all right? How we can extend that word in in a way we can't extend the word red or green, huh? I wonder if you can, you know, the words... I'm going to tip off a little bit, I'm just going to sip you up there. But the words like, like, for the sense of touch, like hard and soft, right? They seem to be almost more movable than red or green, don't they? Yeah, and soft-headed, you know, soft-hardy, right? It's a touch, of course, of the most basic kind. You kind of used to point that out, you know, because you have a famous talk about assumption, the way back, the way back, I sit, therefore I am. And how it works, so kind of the sense of science, you know, and other basic notions like substance and nature of the good are kind of the sense of, like, touch, eh? You get the idea of good more from the sense of touch than from the sense of sight, right? From the sense of nature, the sense of the interior, right? Touch is a sense of the interior, right? With the eye superficially. The way it is that, in heaven, everyone does the will of God, it's never a contrary to God, so like the angels, so the prayer is that, as it is in heaven, may it be on earth. Okay. So since those who are in heaven, their wills are completely conformed to God's will, right? They're asking something, you know. Approximately asking for our wills down here on earth, right? But as St. Augustine says, and Thomas Aquinas says as well, in the sense of the letter, even the sense of the letter, not only in the spiritual sense, but in the sense of the letter, there can be more than one, what, meaning, yeah. Now, going back to what my old teacher, Kassir, used to say, you know, he says, you're going to give glory to God willy-nilly, right? Yeah. If you'd be good, you're going to give glory to God's mercy, right? And if you're bad, you're going to glorify His justice in the end, huh? So you can't really avoid glorifying God, huh? So maybe another sense in this petition, huh? You can say that God's will is done not only in heaven, but also, in another way, His will is done in hell, right? Okay. And in heaven, it's His what? His mercy, right? That is being, His merciful will that is being fulfilled there, right? But in hell, His what? Just will, right, huh? Okay. So when we say, that will be done on earth as is in heaven, we're asking for His what? Mercy rather than what we deserve. Which is asking that His will be done on earth as is done in hell, right? In other words, punish us for, you know, as much as we deserve or something like that, right? And, right? In a way of understanding it, no? Huh? Okay. So that's extra today, okay? Okay. Let's look now here at the philosophy of the continuous here. Aristotle begins with the statement here, if the continuous and the touching and the next are as has been determined before. These have been defined earlier in the book, back in the fifth book. Now, how are those three things related? The continuous, the touching, and the next? Boundary. Not necessarily. Necessarily, huh? The one adds to the other. Yes. The next, the adds touching, adds something. Yeah. Continuous adds something to the touching. Now, notice, huh? The continuous is, in one definition, the continuous, that whose parts have a, what? Common boundary, right? Okay. So if you imagine, say, a circle and the diameter of a circle, right? The two semicircles are continuous, huh? And that diameter could be considered to be the end of one semicircle and the beginning of the, what? Other, right? But it's one and the same end or limit that is the end of one and the beginning of another, huh? Presumably, the border there, huh? The strict sense between Canada, say, and the United States, that border there, huh, wherever exactly it is, all of the United States is south of the border. As you think of Mexico, when you sing south of the border, but the Canadians were south of the border. And everything above that line is, what? Canada, yeah. Now, if there's no, no man's land, as he's having the military, yeah, then that line that divides the two countries is the, what? Upper limit of the United States, but at the same time, it's the, Yeah. And strictly speaking, what is that line called the horizon? The back of the sun in the sky. Yeah. And it's the same, what, line that is the upward limit, so to speak, of the earth, and the lower limit of the, what? Yeah. Yeah. Now, as I say, sometimes we use the word continuous in an equivocal way, right? But by reason of his own likeness. For example, when the Arab philosophers say that man's soul is on the horizon, or man is on the horizon in a way, between the material world and the immaterial world, in a way he's the, what, end of the material world, right, and the beginning of the, what, the immaterial world. Like when Democritus said, man is a microcosm, a little universe, huh? Because he seems to have, you know, one foot in the material world, so to speak, and another foot in the immaterial world. He's right on the, on the border. That's sort of interesting. Or sometimes when you're talking about natural philosophy and metaphysics, huh? Or wisdom. Aristotle usually calls first philosophy. But metaphysics in Greek is actually three words, huh? Meta, meaning after, and ta, which is the article, in Fusica, right, after the books in natural philosophy. But natural philosophy is about things that involve matter and motion. But when you study motion, and later on in this seventh and eighth books, when you see that motion depends upon a mover, but the movers that we are familiar with most of all are mood movers, right? And then you see that mood movers depend upon an unmoved mover, huh? And then you reason out that the unmoved mover cannot be a body, because a body doesn't move other things without being moved itself, huh? Well, you're at the end of natural science. And in a way, at the beginning of what? Yeah, science which is going to be about immaterial things, huh? So you could say the proof of the existence of the unmoved mover and of the incorporeality, it's not being a body, that in a sense is the border, like between the United States and Canada, right? Do you see that? Yeah. And the same way in the third book about the soul, when Aristotle proves the immortality of the human soul, and that the human soul can exist without the body, after death and so on. Where are you? Well, you're at the end, in a way, of natural science, huh? One of the borders, right? Not the same border as the other one was, but it's another border. Just like we had a border up here in Canada, and one down there with Mexico and so on, right? But you're at the borders, huh? Between, though, in this case, at the same time, between the, what, natural philosophy and, what, first philosophy. And then, as we said before, sometimes, even in talking about numbers, right, which are not continuous, as they're going to be defined here, but we sometimes speak of a continuous, what, proportion, right? Okay? So, four is to six, as six is to nine, right? That is distinguished from saying two is to three, as four is to six, right? What, two is to three, as four is to six? Well, you don't find three with the four and the six at all, right? But if you have four is to six, as six is to nine, right? The end of one, six, is the beginning of the next. So there's a certain, what? Lightness there, okay, to the continuous, but it's not, what? As we'll see, continuous in the strict sense here, right? Because numbers don't have any limit in that sense, huh? As Aristotle would say, the number seven, for example, the three and the four, or the two and the five, whether you want to divide it, they don't meet anywhere, right? So it's not a continuous quantity, but there's something in that four is to six, six is to nine, that makes us call it a, what? Tours. Proportion, yeah. And a little bit like, you know, if they double the dyad, right? Two, four, eight, you know? So, you know, but you have this two is to four, as four is to eight, as eight is to so on, so on, so on. You may have it going on for some time, right? And then I think you can, in the same way, speak even in definitions and in divisions and in demonstrations or syllogisms or arguments. You can speak of, what, continuous arguments or continuous syllogisms. What does that mean? Yeah, yeah. So you have two syllogisms and the conclusion of one is a premise in the other. You might, by certain likeness here, huh, say that they are, what? Continuous, yeah. Yeah. Because the end of one is the beginning of the other. And if what is defined by one definition is a part of another thing's definition, then we'd say that those definitions are, what? Continuous, right? So the definition of quadrilateral and the definition of square, we could say are, what? Continuous, right? And maybe there'd be many things. You know, the definition of plane figure and rectilineal plane figure and quadrilateral and square, they're all, what? Continuous, right? And so, but like when you say that a chain is no stronger than its weakest link, huh? And so it seems to be something like that, right, huh? Okay? With the arguments and the definition, you could have the same thing with divisions, huh? So if I divide, you know, rectilineal plane figure into triangle and quadrilateral and so on, then I divide quadrilateral into the square and oblong and rhombus, rhomboids, it depends if those divisions seem to be, what? Continuous, right? And in most, I would say really in any reasoned-out knowledge, you're going to have continuous definitions and continuous divisions, right? And continuous syllogisms, or sometimes, you know, an induction will be continuous, right? Very often, I mean, we have an induction whereby we arrive at a general statement that is used then in a syllogism, huh? And so you could say in that case the induction and the syllogism are continuous, huh? And so just as we transfer words like square and cube to numbers, huh? We speak of a square number and a cube number. And if you've done the theorems in Euclid for those and for squares and so on, you see a certain likeness even in the arguments, right? But I think we tend to transfer the words from geometry to arithmetic more than the reverse, huh? Because geometry is more sensible and it's continuous, right? As well as the fact the number arises maybe from the division of the continuous. So, but now in this text, you want to say continuous in this strict sense, huh? Okay? But nevertheless, there's an order among these three, right? And in the case of the continuous, the two what? The end of one is the beginning of the other. They have a common what? Boundary, a common limit, right? you want to say you want to say you want to say you want to say you want to say you want to say you want to say