Natural Hearing (Aristotle's Physics)
66. Points Cannot Compose a Line: Aristotle's Arguments in the First Reading
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[0:00] Now, when he, I notice the definition given there, of course, is the one from from logic, right? Okay. Continuous whose edges are one. That's the same one basically that we got from the board there, the top one, right?

[0:22] Now, in the second paragraph, how is he reasoning? For neither are the edges or the limits, the boundaries of points one. In fact, you really speak, he's saying, of the what edge of a point, or the boundary of a point, or the limit of a point, or the end of a point. Huh? Does a point have an end? See, a line has an end, right? But does a point have an end?

[0:55] A point had an end would be a point, because the end would be something what different. Different. There'd be something outside of something, right? Wouldn't be, you know, indivisible, right? Okay.

[1:12] Again, can you speak of their edges being together? Then their their ends together, right? See, so the way you're you're excluding both it being continuous there and it's being what touching in that sense that their edges are touching, right? For there is no edge, he says, of the partless, right? There's no boundary, right? There's no limit. There's no end.

[1:35] For the edge and that which it is an edge are other, right? Okay. I think that that's very interesting what he's saying there, and he notices he's going back in a way, I think, to an axiom, right? Okay. I notice when Thomas talks about what an axiom is, if you read Thomas and all my teachers, you know, all follow Thomas, huh?

[2:00] The the standard example of an axiom is the whole is more than one of its what parts, right? Okay. And interestingly, Thomas always uses that as an example, and De Koninck would use it as an example, or a certain teacher. I use an example all the time, right? Um, it's because whole and part in their first sense are tied up with the what continuous, right? Okay. And it's only a very obscure way people see this as being true of other kinds of wholes and parts and the continuous.

[2:34] They see it with the continuous first. Okay. Now, no one has ever made really a list of of all the axioms, to my knowledge.

[2:53] But the whole is more than a part. That's always taken as an example of an axiom by Thomas. Okay. Now Aristotle has, in the fourth book of the Metaphysics, the fourth book of Wisdom, he has a long defense, right, of the. um, you can see it as two axioms the axioms um, about being and non-being, right? Okay, these beginnings about being and non-being, and usually stated as what it's impossible to be and not be.

[3:30] Okay Impossible to. Be and not be at the same time in the same way, okay? And it's necessary to be or not to be, okay? So sometimes I popularly you know, quote Hamlet there, right? To be or not to be, that is the question, right? I say it wouldn't be a question if you could both be and not be, isn't it?

[3:57] Or if you didn't have to either be or not be. You can't really avoid that, right? You can't both be and not be. You must either be or not be, right? That's why it's a question. But kind of you say it's a question because of the two what axioms, right? Okay.

[4:20] Now, I'm a man who's thought a lot about before and after, right? And those are words that you find in the fifth book of wisdom too. And is there an axiom of before and after?

[4:37] Nothing can be before itself. Yeah, nothing is before or after itself, right? Okay. Everybody in a way understands that. Okay. Today can be before tomorrow and after yesterday, but can it be before itself or after itself? No, not strictly speaking. Morning, one part can be before, right? The morning can be before the afternoon, but can't be the morning before the morning. The morning comes after the morning, right?

[5:14] It's always a distinction there, right? Now, although I've never heard them said exactly, when Aristotle is talking in the first book of Natural Hearing, and he's talking about how Melissus and Parmenides say there's only one thing, there's no multiplicity at all. He says they're really doing away with with beginning. He says because the beginning is always the beginning of something, right?

[5:40] Other than itself, right? There's always a distinction between a beginning and that of which it is the beginning. Nothing is the beginning of itself. Would be an axiom about beginning. Yeah, it's always some kind of distinction there, right? Because it's like the axiom there, before and after, isn't it? Now,

[6:10] those two words are related very closely because Aristotle in the fifth book of wisdom distinguishes the senses of before and after. He does so starting with the common notion of beginning. So it's very closely connected the two. And when he gives a common notion of Beginning, he says, what's first in in being or becoming or in knowing? But first is defined by before, right? So the two are very tied together; those two.

[6:39] But then, when you get to the third part of Book Five, right? You come to the famous word end or limit, huh? And Aristotle points out how sometimes we distinguish the beginning and the end. Like this is the beginning, of the day. That's the end, right? The beginning of the day, the end of the day, right? But then sometimes we call both ends, right? Okay, the ends of the day, right?

[7:07] The ends of the day, the limits of the day, right? Okay. But wouldn't be also an axiom that nothing is the end of itself.

[7:30] A surface is the end of a what? Body, right? A line is the end of a surface. A point is the end of a line, right? But a point is not a line, and a line is not a surface, and surface is not a what body, right? Okay.

[7:54] So there's always a distinction there, right? Of the end and that which it is the end, or the limit and that which it is the limit, huh? You see that? Okay.

[8:11] And that's why you got to be very careful. You know, sometimes Aristotle and Thomas will use that phrase causa sui. You got to be very careful that phrase, right? Because in a strict sense, can anything be causa sui? Because every cause is the beginning.

[8:34] Then, if a cause was causa sui, it'd be the beginning of itself. That's impossible, right? Yeah. You can't even speak of God in that sense, right? Yeah. If you understood, if you understood negatively, right? Huh? You see. You know that God has no cause. Yeah. Okay. See, I mean sometimes we state something negatively, not grammatically, in affirmative way, right?

[9:09] You know, like I was saying before, when we talk about statements, some statements are known through other statements. But then there's got to be some statements that are not known through other statements. But instead of saying that they're not known. To other statements, or they're known not to other statements. Right? We say they're known to themselves. If you understood that in an affirmative way,

[9:40] because grammatically it's affirmative, right? They are known to themselves. Then you say it's a statement that proves itself, and that would be what we call circular reasoning in logic, right? Okay.

[9:54] What you're talking about is a statement that is known and truly known, right? But not known to other statements, right? Okay. Okay. You see. So when I say no odd number is even, right? Say. If I know what an odd number is, what an even number is, it's obvious to me that no odd number is even, right? I don't have to prove that statement.

[10:22] I know what a whole is. I know what a part is. It's obvious that the whole is more than the part, right? Say, not to prove that, right? To some other statement, huh? Okay. You'd say it's maybe known to its parts in this example, right? Anyway, you know, but it's not known through other statements, right? Got to be careful that, right? Okay. Have you look at the end of Sartre's work, *There, Being and Nothingness*?

[10:52] You know, of course, he's saying kind of crazy ending because he says a man is the attempt to be causa sui, to be God, which he calls causa sui, right? But that's a contradiction, he says.

[11:11] Therefore, he says the last sentence almost in the book is "Therefore, a man is a useless passion." Say. But if you read, you know, Marx, you know, saying these sort of things like that before him, right? Now Marx is trying to what? Make himself. There's an expression in English, you know, in America, "self-made man," right? Got to be careful what you mean by that. I mean Marx is trying to what?

[11:40] Have man make himself, right? These people that are, you know, fooling around with clones and other such things, right? You know, they want, for instance, for man to really make himself, right? And if I made myself, I don't have to worry about God. I'm my own God, right?

[12:02] But strictly speaking, it's impossible for something to make itself, huh? You see. Got to be careful that expression. There is a legitimate sense if you speak of a self-made man, a man who, you know, wasn't born wealthy and a man who, you know, worked his way up, etc., etc. , right? But strictly speaking, man didn't make himself. And Father and God made you, but you didn't make yourself, right?

[12:28] You see. So nothing is before or after itself. Nothing is the beginning of itself, right? See. Now there's a way which you can say that the foundation of a house is the beginning of a house, right? Some might say, well, isn't the foundation a part of the house? Therefore, isn't the beginning the beginning of itself? No, no. We don't say the foundation of a house is the beginning of the foundation of a house, do we?

[12:59] No, we say the foundation of a house is the beginning of a house. And there's a distinction, after all, between the part and the whole, right? There always has to be some distinction, some otherness there between the beginning and which it is the beginning. Now, if you go to the word end or edge, right, which is the same word here, basically, end or limit, it's always what the end or limit of something, right?

[13:28] But you might say as an axiom that nothing is a limit. Of itself, nothing is the end of itself, huh? Right? And so, if a point really had an end or a limit,

[13:57] then it would have what? Parts. Yeah, it wouldn't be indivisible, right? There'd be some multiplicity there, right? Between the end and that of which it is an end. Okay. There'd be something outside of something, right? It'd have some extension there, right? Okay. So if the point has no end and no limits, how can it have a common boundary, a common edge, a common limit with something else?

[14:29] Doesn't have any, right? Okay. Notice, as Thomas points out at the beginning of the second reading, huh? In a sense, this is also the reason why two nows, right, cannot be what continuous, right? You say. But again, it's a little different meaning there. I know we speak of of one time touching another time, right? Seems to be said more of of the the

[15:01] continuous as a position, like a line, right, or parts of a line, the parts of a surface, right. And that's why he'll go in and try to manifest more in the second reading that motion also is continuous, right? And in the third reading and so on into the fourth reading, the time is continuous, right? Okay. But in a way, the reason here is is is the same for all three, proportionally speaking, at least, huh?

[15:26] Okay. So when he says here at the end of the second paragraph, "for the edge and that which is an edge or other, or you could say the the end and that which it is an end, or the limit and that which it is a limit," those words, that's really an axiom, something that we know without having to what prove it, right? Once you understand what you mean by this, huh?

[15:54] Okay. And I was talking the other day, I think, about how there's something with the idea of limit, like what you showed later on about cause, that there's a cause which has a cause, right? But there's also a cause that has no cause.

[16:17] And the same thing could be said about beginning, right? There's a beginning that has a beginning, right? And then there's a beginning that has what? No beginning. God the Father the Father, right? Okay. The same thing is true about what limit, huh? There's a limit that has a limit. So in a square, the four sides, those four lines, are the limits of that surface, right? But those four lines are limits that have a limit, namely the point, right?

[16:51] Okay. The same thing is true about the surface, right? The surface is the limit of a body, but the surface has limits itself, right? But eventually get to a limit which has no limit, and that's the what? Point.

[17:13] See, strictly speaking, you wouldn't say that the point is unlimited or limited, right? You know, like a line, you know, there's a line going on forever, right? They're also having ends, right? Strictly speaking, the point is a limit, right?

[17:40] So, in limits, it's a little bit like the first cause, right? It's easy to see that, right? You know. Once you realize what a limit is, then you can ask the next question: But can a limit have a limit, right?

[18:05] You know how you proceed. you know, you start with the body, and then you go to the surface, and then you go to the line, and then you go to the point. you're not showing the existence of these things, right? So you see, the surface is a limit before you see the line is a limit, and the line is a limit before you see the point is a limit.

[18:20] Okay. So the limits you first see are limits that have limits, right? And notice, could the the square be limited in size if the four lines containing it were not themselves limited?

[18:44] See, if the lines containing the square went on forever, then the square itself would be unlimited, right? So if the square is a limit, then its limits can't have a limit.

[19:03] Well, be careful about that, Leslie. See. Or you get to the point, Don, you see, Could you say that the point, Don? Could you say that the point, lacks a limit. Is that correct to speak that way? The point lacks a limit. That's what I was going to ask you. could you speak of a point as limitless? doesn't seem so, but because also the point has no magnitude either.

[20:08] Yeah, well, this is an important question because see, if you say the point has no limit, I might, you know, use the term limitless, right?

[20:28] Or unlimited, right? What do I mean? Am I merely negating limit, or am I saying that it's lack of limit? It's not able to have a limit, so it can't lack a limit. Yeah, yeah. In the strict sense of lack, right? Lack is not merely the non-being of something, but the non-being of something you're able to have and should have, right? Especially when you should have it, right?

[21:08] So the chair is sitting on, right? It doesn't hear, right? But is it deaf? That chair. It doesn't see, but is it blind, right? Well, blindness and deafness are lacks in strict sense, right? They're the non-being of something, and that man or an animal is able to have and should have, and when they should have it, right? Then you have a lack in strict sense, right? But the chair is not the sort of thing that is by nature able to have sight or a sense of hearing, right?

[21:53] So it's not not having something able to have and should have when it should have it, right? So strictly speaking, we can say that the chair doesn't see, but strictly speaking, we shouldn't say the chair is what blind, because that would be a what privation. Use the Latin word lack to use the English word, right? Okay. So in Aristotle, you know, in the Categories in the Fifth Book of Wisdom, he'll distinguish the four kinds of opposites, right?

[22:25] And he'll distinguish between being and non-being, that kind of opposition, and between having something and lacking it, right? So. If a straight line had no endpoints, right, then you could say it's lacking, more so, right, because it doesn't have something it's able to have, right? Okay. But the point is not able to have a limit, right? So it's not lacking a limit, right?

[23:09] Okay. So there's an equivocation there in that word "unlimited," right? Okay. That's important when they talk about God, right? Because one of the five attributes of God is that He's unlimited. You usually see the Latin word for that, infinite, right? And Thomas explains how God is said to be infinite. He's always careful to explain it doesn't mean a lack, right? If I say the straight line is infinite, that is a what?

[23:44] Privation or lack, right? But when you say God is infinite, no, it doesn't mean privation, lack. You mean that there is no what? Limit to his perfection, right? Okay, you don't mean he doesn't have something that he's able to have or should have, right? In fact, when you find out what God is, you find out that God is is pure act.

[24:11] There is no ability in God to receive something, right? That's why you can't really give God something in the strict sense, right? You need to think of Thanksgiving, right?

[24:29] Augustine says, "Deo gratias." No prayer shorter or grander. Deo gratias, right? We should thank God, right? Are you really giving God something? Is he receiving something, right?

[24:46] I'll tell you about that. Tell you about that. I was thinking about that. You know, does God rejoice in our thanking Him? Does He enjoy our thanking Him?

[25:02] No, it doesn't give Him more joy. Doesn't give more joy because He rejoices in our thanking Him and our loving Him and so on. Does He rejoice in that? Human nature. What? Human nature. I would say even his divine nature, right? Because it's clear that God loves us, right? And if you love someone, then you rejoice in their good, right?

[25:36] Now it's our good, not adding to His good, right? But it's our good. It's good for us to thank Him, right? It's good for us to what? Love Him, right? It's good for us to honor Him, right? Since He loves us, He rejoices in our good. It doesn't increase His joy, does it? No, because God's love and and His joy they follow upon His what? Knowledge, right?

[26:10] And His knowledge is what eternal, right? So He's not what learning that all sudden when purposes started honoring, right? You know, when it says there is more joy in heaven, doesn't say that somewhere. You know, where one sin, repents, right? Okay. But the joy that God has over my repentance, or my starting God says to love Him, or to honor Him, or to thank Him, right? That joy cannot be new, right?

[26:50] Because He always knew my repentance It's always present to Him in His eternal knowledge, right? And He always loved me. You know, He says in Jeremiah, such a book. You know, a perpetual love, right? Eternal love, right? His love is eternal, like His knowing, and therefore He always rejoices in my repentance.

[27:14] See. That kind of an interesting thing there, you know, this distinction in God, God's pleasure and God's joy, right? And the pleasure is really in Himself, right? The joy is both in Himself and in what? Preacher, yeah, Yeah, very, interesting. So, could you say it belongs to love to sorrow? If someone loves someone, they will be sad over their evil. See, strictly speaking, though, you see, just as joy arises from love, right?

[27:55] So sadness arises from what? Hate. When what I hate is forced upon me, or those I love, right? See, then I have sadness, right? But there is no hate in God.

[28:14] So there can't be any sadness in God. See, But you take the case of our friendship sometime. Okay. But you know, when in us, you know, we have to start with, huh?

[28:29] When we distinguish between love or liking, which is a weaker word for love, but we distinguish between love or liking and and wanting or desire, right? And then joy or pleasure, right?

[28:45] What's the difference between those three, right? Well, in us, wanting or desire or joy or pleasure arises from love or liking, right? And how is that, right? Well, if I love or like something, but I don't have it, right? Then I want or desire it, right? So wanting or desire arises from love or liking in us. When what we love or desire, or love or want, or like, rather, we don't have it, right?

[29:23] But if I get what I love or like, then I have joy or what pleasure, right? Okay. Well, God's not liking anything, right? So there can't be any desire in God, Except metaphorically speaking, but properly. But He always has His good, right? So there's always love in God. There's always what joy in God, right? You know, Psalms talk about that, right? Okay.

[29:57] And these have the same similarity in the other three, right? In the hate, right, or dislike, right, is a weaker word, right? Now, the things I hate or dislike, if they're threatening in some way, I turn away from them, right? Okay. But if they're forced upon me, I can't avoid them. Then I have what pain or sadness, right? Okay.

[30:23] I hate tricky exams because you do. See, so I want to avoid tricky exams But since I can't avoid the exam, okay, so then I have sadness, right? Okay. I wouldn't have sadness unless I hated doing that, right? Okay. Or as a child, you know, or even growing up, I guess certain foods I hate or dislike anyway, and that's being served for dinner, right? Or go to somebody's house and they they're serving that, right?

[30:55] So I'm sad. You know what I mean? But because something that I hate is being forced upon me, So there's no hating God, right?

[31:06] There can't be sadness in God. There is love in God. Why isn't there? Right? There's no hate. Well, we don't get into that too much, but that's first thing you have to see, right? But God is love itself, right? Love itself, there can't be any hate, right? Like we saw back in the first book, of Natural Hearing here right

[31:34] hardness itself be soft? No. Can health itself be sick? No. So can love itself be hating? That's not. God's only good and. Simple, right? Okay. God is whatever He has. Okay.

[32:02] So if He has love, then you know He does, and He's love itself, right? Love itself can't have any hate. Okay. So, the sense in which the point is said to be what unlimited, a little bit like the sense in which God is said to be unlimited, in that it's not a what a lack. But simply a what? Negation, right? Okay.

[32:48] So two points have no end or edge or limit, right? Or boundary, right? How can they have a common edge or limit or boundary? They can't. Nor can their what? Limits be together. They have no end or limit, huh?

[33:11] Now the way Thomas speaks in the third paragraph, huh? He's kind of adding this in a way because of the fact that you want to show that the line cannot be made out of points touching in any way, right?

[33:28] Now what I usually do, you probably remember when I was talking about the fragment there of Anaxagorus. You know, I kind of summarize this a bit there, and I usually run it together, right? And I say if two things touch, they either touch the whole one part of the other, right, or part of one part of the other, or else the whole one touches the whole of the other, right?

[33:51] The other way to touch, it seems at first sight, not, right? But maybe you could distinguish the edge of the thing from its what? Part, right? Right? So you may not want to call the the the four lines, you know, they're the a part of the square, right? Or the edge. So sometimes there's four ways things could touch, then, right? And then I eliminate all four, right?

[34:15] But Aristotle has them separated here, right? Further, it is necessary that the points which make up the continuous be either continuous or touching. There's the same reason in all indivisibles, right? Okay. 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. Since indivisible is without parts, it is necessary that the whole touch the what?

[34:48] Whole, right? Huh? But if the whole touches the whole, it will not be continuous, huh? For the continuous has one part other than another, right? Has part outside of part, right? And is divided into different parts separated in place, huh? In a way, when you say that the two in the continuous two parts have a common limit, do you mean that what? There are another sides of this limit, right?

[35:18] Right? So the outside of each other, right? So it's the idea the continuous they have part outside of what? Part, huh? And I kind of see it a little differently. Sometimes I'll say if two points touch, they you can't have part of one touching part of the other, or part of one touching the whole of the other, because they have no parts, right? So the only way they can touch is in the way that a whole touches a whole, which means to coincide.

[35:45] But if they coincide, they have no more what? Length than one point, which is no length at all. And if ten or a hundred or a million or infinity of points touch, like the mathematician says, the only way they can touch is to coincide. And if they coincide, they have no more length than one point, which is no length at all. So you can't make a line by putting what?

[36:10] Points together, right? You see that? Okay. So is Aristotle here proving the second definition of continuous from the first? Because if it can't be made up of indivisibles, that means it can be divided forever. Well. We'll come down. We'll come down because he's going to do something kind of strange, in my opinion, a little bit. Okay, now the next point he's going to make, and say he wants to show, as I said in the beginning, that you can't have

[36:50] the continuous made out of points that are next to each other, right? Okay. When I was in grade school, you know, in sister school, you know, down in the playground, you got to line up to go into class, and so you line up, you know. So, so you are the next student, and then the next student, the next student, right? Right? Okay. And just like in a row of houses, you got a next house, right?

[37:14] But can you have a next point? Well, if you got two points, you've already got some distance between them, since they can't touch, right? And then you have potentially at least there an infinity of other points between any two points, huh? So you can't put points next to each other either, right? Okay.

[37:47] But neither is a point next to a point, nor a now to a now. So the length of time would be from these, for these things are next to each other between whom there is nothing of the same kind, right? Okay. That was a definition given up there in the first paragraph, huh? So when you speak of the next house, we mean there is no house between this house and and your house, right?

[38:11] My next door neighbor, as we say, right? But there is air and maybe bushes and grass and so on in between my house and the next house, right? So there is something between us, but nothing of the same what kind, right? But these, but in these, there is always a line between points and time between what now, right? Okay.

[38:38] Further, if a thing is divided into those things of which it is, it would be divided into indivisibles. But nothing continuous is divided into that which is without parts.

[39:00] Okay. Now, you know, I don't know if you look at the Latin text there, but in the Marietti text, you know, there is a little confusion there in the thing because

[39:16] it it doesn't give all the divisions that Thomas Thomas Reid is following, right? Okay. And he says Aristotle is first in the beginning here, reasoning from that first definition of the continuous, right? Okay. And then later on, towards the end here, he's reasoning from the second definition, right? Okay. And so he's reasoning from the fact that the continuous is divisible forever, that it's not being made up of what indivisibles, right?

[39:49] Say. And the reason why why the a number is not divisible forever, right, is that it is composed of indivisibles. Now the definition of what number there, the Euclid gives, is a multitude composed of units, right? But the units are altogether what indivisible. They're even simpler than a what point. Sometimes the Greek mathematicians say a a point is a one having position. So it adds something to the idea of the one, right?

[40:21] Okay. So what what's kind of puzzling there is that Aristotle seems to be reasoning from the second definition of the continuous, right, to the continuous not being what divisible into indivisibles. Say.

[40:41] Now as he's reasoning from the continuous being divisible forever, which is the second definition, or that which is always divisible into divisibles, right? Uh no way of saying it, to its not being what composed of indivisibles, right? Okay. Now

[41:08] in his commentary, then Thomas says, and then at the end here he starts to manifest some things he's kind of presupposed, right? Okay. And yeah,

[41:30] in the last two paragraphs, I think it is of this first reading, right? Okay. And the third, from the end paragraph, maybe he's giving that second reason, right? Okay.

[41:43] He says, but it is impossible that look at the next last paragraph. But it is impossible that any other kind of thing be between points and nows, for if there were it would be clearly divisible into indivisible, and if divisible into indivisibles, no, it's always divisible. This serves a continuous.

[42:00] But now in the last paragraph, it is clear that everything continuous is divisible into things which are always divisible, for if into indivisibles the indivisible would be touching indivisible, for continuous things have one edge and touch, right? Okay. Well now if you compare that with the the so-called second argument in the text there that he has, and Thomas gives us the second argument,

[42:25] he's reasoning in the second argument. That because the continuous is that which is divisible forever, therefore it's not composed of indivisibles, right? It's not divisible into indivisibles. But yet, and he gives a reason here later on why the continuous is divisible. The things are always continuous. It is because it can't be made up of indivisibles,

[42:51] right? Kind of paradox there, isn't it? Which came first, right? Which is the reason for which, right? Let's just raise the question a bit, huh? We have this definition: continuous. The continuous

[43:21] is that which is divisible forever. Okay. Or I state it somewhat differently, that which is always divisible into divisibles. Okay, that way of saying it, right? That which is divisible forever. And then you have the other thing that the continuous

[43:53] is not composed of indivisibles. Okay, or the continuous is not divided into indivisibles. My way kind of corresponds to that. We stayed in the first definition, right? Indivisibles. Yeah. As I was saying before, if you reason from a definition, right? Okay. Um, and this is the definition: continuous. You might reason from this to that, right? Okay. But in the last paragraph there, he seems to be reasoning from this to that, right?

[44:47] Okay. I know myself when I'm not in this text, but I'm just teaching the first book here, you know, or teaching the fragments, right? And and we come to the fragment of Anaxagoras. You've probably seen me do it And Anaxagoras says that there's no smallest of the small, right? And I'll say, well, this is true of the mathematical line. I'll draw mathematical line on the board, right?

[45:12] And say you can cut that in half, and you can take the half and cut that in half, and so on. And most students are somewhat inclined to accept that you can keep on cutting it forever into halves, right? Okay. Um, I say, will this ever stop, right? Say, well, Joyce has something further divisible that you cut it into. Say, but they're kind of you know willing to accept that you can always cut into things that are further divisible, in which case they admit it's divisible forever, right?

[45:44] So I try to manifest that. You try to remember the way I manifest that. I say, if he didn't cut it into something that is further divisible, right? Then either you cut it up into nothing, or you cut it up into something indivisible, right? Okay. Say, and I say, can can you cut something up into nothing? Say, well, Anaxagoras says in the other fragment, "What is cannot cease to be by being cut," right?

[46:18] I say, if you could cut something up into nothing, it would be made out of what? Nothing. Nothing, right? I used to quote Nixon's remark one time. You know, he said any. Well, you slice the democratic program is still the same baloney right?

[46:38] Well I say, well, you cut something up into what it's made out of, right? So if you could cut something up into nothing, it'd be made out of nothing, which is absurd, right? Okay. So you don't make that possibility, right? So when you cut a straight line, either you get shorter straight lines, always, right? Unless you could cut it into two points, right? And then you could stop, right?

[47:07] Because the point is not divisible anymore. Well, then I say, well, but could you have a line that's two points long? See, could you have a line in other words composed of points? And then I bring in the other argument. I say, well, if you're going to make a line out of points, the points have to come together in touch, right? And then I say, no, there are these ways things can touch: whole can touch whole, part can touch part, part can touch whole and perhaps you can speak of the edge of things touching, right?

[47:41] Okay. So I draw, you know, you probably see me do that. I draw the circles on the board. I say, you know, part can touch part like these two circles, right? Or part can touch whole like these two here and whole part, right? Or whole can touch whole and I say I can't really draw that very well, but I just go around twice get the idea, right?

[48:00] Maybe speak of a fourth way where they touch at their edge, but the edge is really not a part, right? I say, well, now can two points touch in this way? Well, they don't have any parts, right? So they can't touch in that way. Part touching part, can part touch a hole? No, they're not parts, right? Now, can you make any distinction between a point and its edge?

[48:21] But then you're really imagining the point to be a little tiny circle or something like that, right? And if you had some distinction between a point and its edge, there'd be something inside that's not the edge, right? And then you'd have some kind of extension, and it wouldn't be indivisible. So that way is also impossible, right? So two points can touch only like, and I say like, because if you speak you don't have a hole either.

[48:46] There's a whole in this part, right? But they can only touch like a hole touches a hole. That is to say, coincide, right? And then I say, now two points coincide. How much length do they have? As much as one point, just no length at all. And then I say, if ten or a hundred or a thousand or a million or infinity of points do touch, the only way they can touch is to coincide.

[49:10] And if they coincide, they have no more length than one point. Therefore, they have what? If they have no length, you don't have a line, right? So there's no way you can put points together to make a line, right? So it's impossible that when you cut a line, you end up with two points rather than two shorter lines. So you always end up with two shorter lines, and you cut or bisect a straight line.

[49:36] You end up with two points, and there was nothing. So if you always end up with shorter lines, you can cut again and cut again, cut again. So it's divisible into things that are always divisible. It's divisible forever, right? So

[49:55] I'm showing from the fact that the continuous cannot be put together. The line, in particular, cannot be put together from points that is divisible forever, right? But when Thomas explains where Aristotle's reasoning there in the second argument, and I know I was looking at the Marietti again this morning, and and I noticed a little confusion there in the text. You know, occasionally you'll find that part of the division or one division is left out sometimes.

[50:28] You know, but it's in the original text that you know, perhaps lost or what. You know, but you know Aristotle has two principal reasons, and one is taken from the first definition of continuous, and then later on, now he does the second one, and then he's reasoning from the second definition of continuous to its being what not composed of indivisibles. And also, you could reason from the continuous being divisible forever to its not being composed of indivisibles, because if it were composed of indivisibles, then eventually you arrive at indivisibles, and

[51:11] you go on. But which is really more basic than the other, right? Thomas doesn't say anything in the commentary about apparent what circularity, right? That's interesting.

[51:34] I know a famous place there in the Post Analytics where Thomas points out that Aristotle seems to reason in a circular way because he's showing that

[51:55] statements that are necessary are as such or kath' hauto per se. In one place he reasons from the being necessary to the being statements as such. Another place he reasons the being as such to the being necessary. Of course, Thomas notes this, right? That Aristotle reasons in both directions, right?

[52:24] And especially. The fact that it's in the Post Analytics, Aristotle is saying one cannot reason circularly. You know, you can't use A to prove what B, and then B to prove A, because then A would be more known than B, and since B is more known than A, then A would be more known than itself, and something would be what both before and after itself.

[52:57] You see? So, what is Aristotle doing? Thomas does, right? You know, in the very book where he teaches this, right? And that's interesting because I don't know if you're in the Nicomachean Ethics there, right? But in the first book of Nicomachean Ethics, Aristotle is about to

[53:20] object and argue against a position of Plato, right? And and he gets that point. He says it's difficult for us to attack this because it has. been what? Held by our friends, right? Okay.

[53:38] And then he says, "But being philosophers, right, we should be lovers of the truth, right? " And so he says, "Plato is a friend, " he says, "but truth is a greater friend. " Okay.

[53:58] And Socrates said something like that in the Phaedo, right? If you listen to me, you'll hear little about Socrates and a great deal about the truth, right? Okay. Aristotle said it would be impious to put Plato before the truth. Okay.

[54:17] And of course, Thomas comes and says, "Well, yeah, God is truth itself, right? " Okay. But then Thomas stops and then he says, "But there are other books in Aristotle, like the Metaphysics and so on, where he disagrees in the first book of the Physics, right, where he disagrees with Plato, right? " But he doesn't say, you know, this would be against my friendship with Plato, you know, and so on.

[54:42] Why is he? Why is he stop and make this observation at this point, right? He says, "Well, he's in the Ethics, right? " And the Ethics, friendship is one of the main things to be considered, huh? If you look at the ten books in Nicomachean Ethics, two of the ten books are devoted to what? Friendship, right? This is the main theme there, Nicomachean Ethics, right? And so to be attacking his friend's opinion, right, might seem to be against what?

[55:12] Friendship, right? And Aristotle wants to say this is not really against true friendship, right? To disagree with your friend, right? It doesn't matter like this, huh? Okay.

[55:25] Um. So I think it's kind of appropriate that in the Posterior Analytics, right, when Aristotle seems to reason from A to B in one place and from B to A in another place, right, that Thomas should note this, right? Say, "What the heck is he doing? " Because he's told us in this very book, the Posterior Analytics, that this is bad reasoning to reason in a circle, right?

[55:54] Okay. Now Aristotle is discussing the fact that there are some statements that we know through other statements, right? And then he's going to eventually try to show that there must be some statements that we know, right, not through other statements, right? Because otherwise you'll be going on forever, right? Unless you could reason in a circle, but that's also stupid for the reason I gave, right? Okay. We reason from the more known to the less known.

[56:23] So if we reason correctly from A to B, A has to be more known than B, right? And then if we reason from B to A, B has to be more known than A, so. You say that even weighs more known than itself, right? Makes sense, right? We get some more complicated things, but as he talks about this, he goes on. He says, "Well, the real demonstration," he says, "is from necessary to as such."

[56:58] Okay. Why then does Aristotle reason at this other place from as such necessary? Well, certainly in this academy, right, it's the common opinion that episteme, right, is about the what as such the kath' hauto, right? So you can take that as a probable opinion, right, and then reason from that to its being what necessary, right? Okay. In a way that does show that t here is a real connection between being necessary as such, the fact you can reason from one to the other, right?

[57:35] Okay. But he says the demonstration is basically from necessary to as such, right? And when he reasons from as such to be necessary, he's reasoning from as such as what a probable opinion, right? Back to that, right? Okay.

[57:53] But he doesn't here seem to see this circularity that I seem to see in the way he presents it, right? Because in the commentary, Thomas says he first reasons from the first definition of the continuous, that whose parts have a common boundary, a common limit, a common edges. This translation added, right? And

[58:24] then the second principal reason he reasons from the second definition of the continuous, that which is divisible forever, right? Okay. And then later on, at the end of the reading, he says Aristotle is manifesting some things he presupposed, and then he seems to be reasoning from this to to that, right? Okay. So

[58:55] Thomas doesn't raise the question, so it doesn't answer. But I am raising the question: How would Thomas answer? How would we try to answer that? I think the proper demonstration would be from the definition, but maybe for some reason, like the way you did it, you showed it. The other one seems to be maybe more known. Yeah, it seems to me that we we would prove more than the contingent that which is divisible forever from the fact that it's not composed of what.

[59:35] Indivisibles, right? Okay, like in the way I was indicating where I'm using, I discuss the fragments of an exager, right? Okay. Now, to reason from this simply is something probable, right? You know, solving it the way Thomas solves this here, right? Okay, that's that's one possibility, right? Okay. With the like my students, I was saying, you know, when I ask them, you know, you can take a map line and cut it in half, right?

[1:00:13] Cut it in half, right? And so on, and they're kind of prepared to admit it seems, you know, probable to them, right? That you always get shorter lines, right?

[1:00:25] But to really make it all together certain, I point out that you can't cut it into two nothing's, and you can't cut it into two points, right? Okay. And so I'm really showing the impossibility from the impossibility of a line being composed of points, right? I'm really showing from that that it must be divisible forever. It's always divisible into shorter lines, which in turn are divisible again, right?

[1:00:55] Okay. Is the second would that really be a property of first the definition? I don't know for what I'd say about that, but I'm saying that reasoning from the second here to the first, right? Yeah. But Aristotle apparently is reasoning in this reasoning, right? First from this to there, but at the end he seems to reverse, right? Okay. Now you could you could argue that this first one is, as they say,

[1:01:38] probable. That's one way of trying to solve it, you know. But is there a way of showing that the continuous is divisible forever apart from this down here?

[1:01:58] Could you show that the continuous is divisible forever in some other way, independently of the second thing here, right? And then also be able to reason from it to this, huh? What's what I was using? Oh, wait! How do you

[1:02:25] do it? Take a line, right? Yeah. And you cut it in half. Yeah. Then I zoom in on the one half. Yeah. And to myself, it looks like it's the same length as the original line. But I know that I've zoomed in, right? And so I do that again. Never seems to change, although dividing forever. Yeah.