Natural Hearing (Aristotle's Physics)
Lecture 73: The Continuous: Foundational to All Philosophy

Transcript
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You can't think of a work on the continuous by the modern philosophers, can you? I can't think of a book by Louis de Broglie there, the great French physicist, right? The continuous and the discrete in modern science. When you stop and think about the philosophy of the continuous, how fundamental is this for philosophy and for our knowledge in general? Well, as you may recall, there are two chief kinds of philosophy. Looking philosophy, right? And practical philosophy. And in looking philosophy, you have three particular kinds. You have mathematical philosophy, then you have natural philosophy, and then you have wisdom, or first philosophy, huh? Okay? Now, the philosophy of the continuous is altogether fundamental for natural philosophy, because motion and place or distance and time, which are the things that we saw in the beginning of book three are about to be considered in natural philosophy. They're all continuous, huh? So it's absolutely fundamental for a natural philosophy, to understand the continuous. But isn't it important for mathematical philosophy? Yeah. Geometry. Because geometry is about continuous quantity, huh? Uh-huh. And some of the things that are shown here are assumed in geometry, right? Uh-huh. Like in geometry, we say, for example, between any two points, you can draw a, what, straight line, right? Uh-huh. Now, two points have come up and touched without coinciding, and still remain two. Could you draw a straight line between these two points? No. But it's here that we meet, you know, in this first reading, in fact, that two points cannot, what, touch without coinciding, right? Of course, geometry is always assuming you can, what, get smaller and smaller and smaller, a line, you can keep on bisecting a line, right? Uh-huh. You can always inscribe a square and a circle and a circle and a square, which case they get smaller and smaller and smaller, right? You can always bisect an angle, right? In the theorem, you know, giving a straight line to bisect it, that's a universal theorem, right? And then you can keep on doing that. So it's absolutely fundamental for geometry, huh? And in a way, it's important for even arithmetic, the science of numbers, not because they're continuous, but because number arises from the division of the continuous. We touched upon that when we looked at the fragment of Anaxagoras, where we talked about there being no smallest of the small, and then there'd be no greater, greatest of the great. So what is the connection between those two, right? Well, if you start off with one line, you divide it, you have the first number, which is two, right? And you divide again, you have three, and you can divide forever, then the numbers can increase forever, huh? So in a way, the infinity of numbers there, or the potential infinity of numbers, corresponds to the infinite divisibility of the, what? Continuous, right? So the philosophy of the continuous is fundamental for a mathematical philosophy as well as for, what? Natural philosophy. But then when you get into the philosophy of the soul, and at a fortiori, when you get into wisdom, and you're going to be studying eventually immaterial things, right? Like the angels, huh? Or God himself. Now, is the philosophy of the continuous important for these things? Well, not if we knew these things as they are, right? If we saw God face to face, right? The philosophy of the continuous would not be necessary to consider God, right? Although in seeing God face to face, we would understand the continuous. Okay? But neither God nor the angels are continuous, huh? But in this life, we understand more what God and the angels are, what? Not, right? Okay? Like in the Summa Karajantiles, when Thomas shows that there are understanding creatures, creatures that understand and have will and so on, then he shows they're not a, what? Body, right, huh? They're not continuous, huh? Okay? So you have to understand the immaterial substances in God in part by the negation of the continuous. And that's how you also understand the immateriality and large part of our own reason, huh? Okay? And consequently, the immateriality of our soul. So even for the study of those immaterial things, those things that are not continuous, the philosophy of the continuous is altogether, what? Basic, right? Okay? When Aristotle points out, and we will see it eventually when we look at the philosophy of the soul, when he takes up the human reason, he points out that the human reason, proper object is that what it is, is something you can sense or imagine. So what it is, therefore, is something continuous, or terribly continuous, huh? And so that reason never understands in this life without imagining in some way, huh? And that's why in the books on the sense and sensible Aristotle, and Thomas explaining the commentary, say that we understand nothing without the continuous and time, although time itself is continuous, huh? Because the images are all involved in continuous and time, huh? If you read Boethius' work in the Didi Twinitati there, and Thomas' commentary or his articles on it, and one of the objections, you know, is saying, well, we understand nothing without the continuous and time, but God is not continuous or in time, right? Therefore, we don't understand God. Well, the reply to the objection is that we understand God by the negation of the continuous, huh? And we saw in eternity the negation of time, things that are found in the definition of time, okay? So, in that sense, we don't understand even God in this life, as far as we can, without the, what, continuous, huh? Okay? But the fact that we understand nothing without the continuous shows how basic that is, huh? To all our thinking, huh? And that's found, then, as we've pointed out before, in the words that we use everywhere, huh? The most basic words, huh? Now, when you study the most basic words, like Aristotle does very fully in the fifth book of wisdom, fifth book of metaphysics, you see that these words are the words used, to some extent, everywhere, but especially in the axioms, huh? The statements which are known by themselves, by all men, right? But they're also the words used most of all in wisdom, huh? And, of course, those words are divided into three groups, but the first group begins with the word beginning, right? And the first meaning of the word beginning is the beginning of this table. It's tied up with the, what, beginning that is in the continuous. Of course, every cause is a beginning, but not every beginning is a cause, right? So our very understanding of the word beginning starts with the, what, continuous, huh? Eventually, we see other meanings of the word beginning, right? They're not tied to the continuous, but they're seen at first by certain likeness to the beginning that is in the continuous. But that's an extremely basic word, beginning, huh? Every cause is a beginning, huh? Not every beginning is a cause. It's even more general than cause, right? His first meaning is tied to, what, the continuous, huh? The curve is the beginning of the... The campus, they always say to the students, right? They curb out there on Salisbury Street, right? Stay in the campus. First thing, beginning, right? Okay. And in the third part of the words, you've come to the famous word end or limit, right? And the first meaning of end or limit is the end or limit of the continuous. So we speak sometimes the beginning of a line and the what? End of a line, huh? The beginning of the table and then the end down there. But Aristotle says sometimes you use the word end for both. We can say the two ends of the table, right? The two ends of a line, the two end points and so on, right? So end in some ways is even more universal than what? Beginning, yeah, yeah. But the first meaning of end is again in the continuous, huh? The first meaning of end is the end of the table. And then the next meaning of end is the end of emotion. Which is also something continuous, right? And then comes the sense of end, which is purpose, that for the sake of which, huh? And then last of all, end or limit in the sense of a definition, huh? So that's a key word in our thinking. But the first meaning is tied to the what? Continuous, huh? You take the key words of wisdom, the most universal words like being and one, huh? Well, continuous is one of the fundamental meanings of the word one for us. I told you that little joke of Socrates, I think, when he asks somebody to define something and they give him what? Many examples rather than a definition. And he's kind of, you know, playing the fact that he asked for one thing and they gave him many. And in some of the places where Socrates does, one place at least, he seems to, I think it must be kind of a standard joke the Greeks had, if I hand you a plate, right, and you, what, drop it or something, right, or I drop it in handing it to you, and you say, well, I asked for one plate, and now you've got it. But no, what happened, the thing is no longer continuous once it's been broken, right? So it's now many instead of one, right? That's kind of the first meaning, it seems, almost of the word one for us. The, what, continuous, huh? And everything that is, in some way, is one. If it's simple, it's very much one. If it's composed, it doesn't exist unless its parts are, what, united, huh? So one is very important for understanding, or continuous is very important for understanding, what? The meaning of one, right? And, of course, it's obviously important for being in the sense of, what, quantity, right? Which is so close to substance that Descartes confused the continuous or extension with, what, substance, huh? Okay. Now, that is a little manifestation of the importance of the philosophy of the continuous for all of our, what, thinking, huh? Okay. Now, you might say, well, if geometry is about continuous quantity and so on, why is the philosophy of the continuous this basic consideration of the continuous here, which is especially in the first four readings, right? Before it goes into the division of motion in some detail. Well, why is this philosophy of the continuous here that you have, especially in the first four readings, why does that belong to natural philosophy rather than to geometry, huh? See? Uh-huh. You know, geometry is a science of continuous quantity, right? Uh-huh. You know, lines and surfaces and bodies in the sense of the three-dimensional continuum, right? Uh-huh. Why does it belong to natural philosophy to determine, basically, that the continuous is, huh? That it's divisible forever and it's not divisible into indivisibles, right? It's not composed of indivisibles and these things that he's been saying here. Why does it belong to natural philosophy to do that? That the geometry, in a way, depends upon him, right? Or he assumes, right? What he's shown here. One reason is that continuous is broader than just the continuous study in geometry. For example, time is continuous. Okay. In motion. Okay. Assume it belong to geometry to determine what time is, right? Or what motion is, right? Uh-huh. Okay. But you could also, perhaps, go back to what we show in logic, right, huh? And the highest part of logic is in the prior and posterior analytics where we're talking about demonstration, huh? Which produces reasoned-out knowledge in the strict sense, huh? And the highest kind of demonstration, of course, is a demonstration poked or quit, right? Giving you the reason, right? In the sense of the cause, huh? And when Aristotle is talking about this, he ties it up with what we know about, or say about cause, that you have to compare causes and effects that are, what, proportional, right? Okay? Um, now, that means that you have to, uh, find, uh, exactly why, you know, something belongs to something, right? And sometimes we're not exactly sure, right? And sometimes we're not exactly sure, okay? Let me give an example of this, right? You've all read the, uh, you've all read the, uh, the Mino, haven't you? Okay? Okay. Now, the Mino, in a way, has, in Socrates' demonstration, the geometrical demonstration he gives to the slave boy, um, in a way, it's, it's a particular, uh, case of the, what, Pythagorean theorem, right? Socrates is saying, suppose you have a square, and you want to get a square that is, what, twice as big, right? What will be the side of a square twice as big? And as you know, the slave boy answers, twice as long, right? Okay? And Socrates will first show the slave boy that if you take a square whose side is twice as long as the original one, you won't get a square twice as big, with one, what, four times as big, right? Okay? Then he wants to show the slave boy that if you take the diagonal original square, that would be the side of a square twice as big. That's a kind of interesting theorem, right? And if you just look at a square and you say, okay, now I want to find the side of a square twice as big, and Socrates, or someone tells you, well, it's going to be the square in the diagonal, that's very interesting, isn't it? But, um, how would you show me that that is so, right? See? Now, if you knew, or if the slave boy knew, you knew the Pythagorean theorem, you can see this is a special case of the Pythagorean theorem, right? Because this is obviously a right-angled, what, triangle, and the square on the side, opposite the right angle, right, is equal to the squares on the sides containing it. Now, obviously, the squares and the sides containing it, together, are twice the, what, original square. And so if the square in the diagonal is equal to the squares on these two sides, the square in the diagonal would be exactly twice the original square, okay? Socrates can't assume that the slave boy knows the Pythagorean theorem, that would be too hard to show, right? So he actually shows something more particular. He says, suppose this is your original square, he said, right? Now, if we put another square exactly equal to it right here, he says, right? And then put another one exactly equal to it right below it, and then a fourth one exactly equal to it throughout the corner here, right? We have a kind of bigger square, composed of, what, four squares, and therefore, four squares of the original size, and therefore four tens is big, right? Okay? And then he starts to draw these, what, diagonals, right? Now, it's not too hard to see that the diagonal cuts the square into exactly two parts, right? And of course, you know that formally from the fourth theorem, but the fourth theorem is almost obvious, huh? The fourth theorem says, what, that we have two triangles with an equal angle, and they're contained... the equal angles, the equal sides, the two triangles are equal, right? You can see that if you lay this on there at that point, right, and drop this line and that, because they're equal, they would coincide, and because the angle's equal, this side would fall on this side, and because the lines are equal, they would coincide. And you can't draw two points, right? Excuse me, two lines, straight lines, that is, between the same two points, but only one, right? So the third line would coincide. So, now this here, okay, now it's also not too hard to see that these two triangles will be equal, and therefore these two angles will be equal, and these two will be equal, right? And therefore, each of these angles will be exactly one half of the right angle, and so two of them will be a right angle, right? Okay. So now you've got a square within a square, right? And it's made up of four halves of four squares, and therefore it's exactly equal to what? The half of four is two, so it's exactly twice as big, right? That square is the original one. And it's on, in fact, to what? Diagonal, right? Now you've shown that, right? Now, be careful what I'm going to say right now. Now, that's a good argument, right? Okay. Nothing wrong with that argument at all, right? Okay. But notice what you're showing here is a particular case of the, what? Right angle triangle, huh? In fact, you want to be very precise, you could say it's a, what? Isosceles right angle triangle, right? Yeah. Okay. Now, is it a property of the isosceles right angle, right triangle, right? To have that property, or does it belong to every right angle to have that property? Every one. Yeah. Yeah. And if you've gone through the whole book one of Euclid, right, in the 47 theorem, I guess it is, right, at the end of the first book, it's appropriate to take this as a culminating point of the first book of geometry, it shows it about any, what? Right angle triangle, right? Okay. So it'd be true of, let's say, the famous, the first one in whole numbers, the one whose sides are three and four, and the diagonal is, what? Five, I'm not diagonal, but the side opposite of the right angle is five, right? Okay. Four times four is sixteen, nine times five, okay? But this is not an isosceles triangle, is it? No, see. So is it really, insofar as it's an isosceles triangle, that it has the angle, or the square on that side, opposite of the right angle, equal to the ones on the two sides? No. It belongs to it insofar as it's a, what? Right angle triangle, huh? Not insofar as it's an isosceles right angle triangle. It's a kind of subtle thing there, right, huh? Okay. So, number 47, huh? That's how it would be, it would be a more perfect demonstration, really, than this one over a year, in the sense that you're assigning the property to what it's proportional to, right? It's something that belongs to this, not as an isosceles right angle triangle, but something it has because it's a, what? Right angle triangle. Yeah, yeah, yeah. You see that? But no, it would be easier to follow this thing with the slave boy than to get to understand number 47 in book one, right? And so it may be, even historically, that men saw some of these theorems for something in particular first, right, before they saw the more universal thing, huh? Okay? Now, let's take a little example of those, a simple example in some ways, huh? Take the fifth theorem of the book one of Euclid, huh? That's the theorem that says an isosceles triangle, these two sides equal, right? Then you prove that the angles at the base are, what? Equal, right? So is it a property of the isosceles triangle to have the angles, right, opposites, equal sides, equals it a property of the isosceles triangle? See, the theorem, I guess, is enunciated in Euclid with the name one isosceles triangle, right? Right? Yes. Yeah. But when he first defines, you know, the various particular kinds of triangles, equilateral isosceles and scaling, right? The isosceles is a triangle that has only two sides equal, right, as opposed to equilateral which are all three sides, huh? Well, does this property belong to the isosceles triangle in that sense as opposed to the equilateral triangle? No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. You could make explicit something, I don't remember you could make explicit, but I mean... No. No. No. No. It's so obvious to him, right? You could take that theorem and you could show from that that all the angles of a, what, equilateral triangle are equal, right? Okay. And you could take that in the next theorem and show that none of the angles of the, what, scaling triangle can be equal, right? You know, the next theorem is the reverse of that. It says that the angle is equal, the side is equal, right? So since the scaling triangle has no sides equal, if any two angles are equal, by the sixth theorem, the next one and after that, that would have the opposite sides equal, but that contradicts what a scaling triangle is, right? Okay. So, but when you show that an equilateral triangle has all its angles equal, from that theorem number five, it shows that you're not, what, understanding that fifth theorem really in a way that is private to the isosceles triangle is distinguished from the, what, equilateral, or you can apply it to it. Right? Okay. Do you see that? So, with Aristotle, or not Aristotle, when Euclid, or when Pythagoras, let's take Euclid's demonstration we have, the actual reading of it, when he demonstrates in Proposition 47 that it's a property of an equilateral, not an equilateral, but a right-angle triangle, right, to always have the square on the side opposite the right angle equal to the squares on the sides containing the right angle, right? He's showing that of what? He's giving you a reason why any right-angle triangle, right, would have this, right? Not a reason why the isosceles would have it, right? The isosceles right angle triangle. The reason is more common than that, right? Well, you can say that the reason why a line, or the motion over a line, or the time it takes to go that motion, the reason why all of those things are divisible forever, right, and none of them is composed of indivisibles, right, or is divided into indivisibles. It's really the same reason for all three, proportionally speaking, huh, at least, huh? Okay? And therefore, like in the 47th theorem there in Book 1 of Euclid, it's appropriate to, what, bring them all together, right, and show it for all of them together, right? Because there's kind of a common reason, at least proportionally, for all of them, right? Okay? Well, if you showed it in geometry, you'd be showing it just for the line, and not for the motion or the time, right? You see? It's a little bit like what Socrates is doing there in the other one, right? Wouldn't be a bad argument, but wouldn't be the most, yeah. But you have a solid in that sense in which Aristotle's taking it there, right, huh? Okay? It's a property of whatever it is, right? As such, right? Okay? Okay? Is the continuous equivocal instead of time and motion and magnitude? It's hard to say. There's something, perhaps, yeah, but it's something not equivocal by chance, right? Equivocal by reason, huh? Yeah. Certainly, it's clear in Thomas and in Aristotle that the word beginning, right, is equivocally said of what? The beginning of the magnitude and the beginning of the motion, the beginning of time, right? In the first book of natural hearing there, the first book of the physics, Aristotle's examining the reasoning of Melissus, right? Melissus is saying, you know, that being could not have come to be, could not have been generated, right? It had no beginning, right? If it has no beginning, he says, it has no end. Therefore, it goes on forever. And Aristotle says he's going from, what? A premise that it has no beginning in time, right? And perhaps, therefore, no end in time, right? To it's having no beginning or end in its, what? Magnitude and size, right? And therefore, he's committing the, what? False equivocation. He's making this mistake for mixing up two senses of the word, right? I don't know, did we talk in here, in the introduction of philosophy the other day there, and we were talking about the disagreement between Socrates and the Athenians in the Apology. Did we talk about that in here? I don't know if we did one time. I'm too sure. And you know what, if you look at Socrates' speech in the Apology, in the first part of his speech, he explains how he got in the habit of going around examining people, right? Yes. And often finding out they didn't know what they claimed and all, but why he started doing that. But then the second part of the speech, he says that he's been examining the Athenians especially about their seeming to prefer the goods of the body and exterior goods, outside goods, to the goods of the soul. Okay? And Socrates doesn't develop there at that point the reason why the goods of the soul are better than the goods of the body and exterior goods. But he does touch upon this in other dialogues and so on, see? So I stop at that point and I say, now, who's right, huh? Socrates or the Athenians, huh? And we've already shown earlier in the Course that something is not good because you want it, right? So you can syllogize then that something is not better because you're what? Want it more. Yeah. See? So I say, you can't say to this, you know, well, Socrates, if you like the goods of the soul so much, go for them, right? We want these other goods, right? You can't say that the goods of the soul are better for Socrates because he wants them more. And the goods of the body and the outside goods are better for the Athenians because those are the goods they want more, right? Because then you'd be saying that something is better for you because you want it more. But if that was so, then it would be good because you're what? Want it. But we'd already shown earlier in the Course that something is not good in case you want it done. See? So as I say to the students, I say to give them a simple example, is something sweet because it's white? No. And so it's not sweet because it's white, it's not sweeter because it's whiter. While someone was saying, you know, something is sweeter, the whiter it is, you'd have to say it's sweet because it is white. And if that isn't so, as everybody knows, then it can't be true, right? So he says, the ball game's over, that ball game's over, you can't say that. Right? Okay? Now you're going to have to give a reason for what you're saying, right? And I used to have a colleague who used to quote the Republic, did I ever tell you this? In the Republic, I think it is, Socrates says, an opinion without a reason for it is an ugly thing. Oh, yeah. And he'd say, he'd quote this to students, he says, and I said, don't pawn your uglies off on me, he'd say. you know, you know, you know, you know, what I did was to develop, you know, the, I get to, first of all, to admit that the Americans agree with the Athenians rather than Socrates. Okay? And then, I developed, you know, all the reasons that you give for Socrates being correct, right? Okay? And I say, no, can you give any reason to the other side? You can't give the reason that they want more, right? see? Well, they can't really come up with a reason, see? So, then I try to help them a bit, see? And my favorite example, you may have heard me use it, I say, which is better, philosophizing or breathing? And, see, now think about that a little bit, which is better? See? Okay, now, how many think philosophizing is better? Raise your hands. Nobody raised their hand. How many think that breathing is better? They're all raising their hands, see? I say, okay.