Logic (2016)
Lecture 18: Conceptions, Abstractions, and the Nature of Number

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sphere, right? Now if you throw a glass ball against the wall, it'll probably, what, shatter, right? If you throw a rubber ball against the wall, it'll bounce back. If you throw a steel ball against it, it'll probably get stuck or go through the wall, right? What about a geometrical sphere now? If you throw it against the wall, it'll bounce off, or it gets stuck, you know? Yeah, yeah, yeah, yeah. It doesn't seem to be, you know, it seems to me, it's something abstract, right? Now not maybe as abstract as two and three, but still, you know, abstract, right? So, now the third thing, huh? Sometimes that which is signified by the name does not have a foundation in the thing, neither a proximate one, like the word man above, right? Or the word cat, or the word dog, right? You know, there's a dog out here that, you know, he came up front to me, you know, I came up, you know, like I'm an old, old, old friend, yeah, I don't know if she knows me or not. If he knew you were a friend of the cat, he wouldn't like it. Yeah, yeah. Like the conception of a chimera, right? That's kind of the word for the imaginative thing, right? See? Or I'd say, like a unicorn or something like that, right, huh? You know? You know? Yeah. Which is neither a likeness of something outside the soul, right, huh? Nor does it follow from the way of understanding truly something, right, huh? And therefore this conception is falsa, right, huh? Okay? Okay. Once it is clear that ratio is said to be in the thing, and so far the signifying signified by the name is in the thing. And this happens when the conception of their understanding is a light sort of thing. You've got to stop at 4.30 or what? Yeah. The unity of number is difficult to understand, and we'll come back to this next weekend. Wow. where he says that the conception of the understanding, look at it again, tripliciter, right, adds itself to the thing which is outside the soul, right? This is a very famous distinction that Thomas gives in a number of places. Sometimes what the understanding conceives is a likeness of a thing existing outside the soul. Just as what is conceived by this name, man, right? Right, and such a conception of the understanding has a foundation in the thing, what? Immediately, right? Insofar as the thing itself, from its own conformity to the understanding, makes what is understood be what? True, right, huh? And the name signifying a thing understood properly is said of that thing, right? Sometimes, however, what the name signifies is not a likeness of a thing existing outside the soul, but it is something that follows from the way of understanding a thing which is outside the soul, right? And of this sort are the intentions which our understanding, right, introduces, right? Just as the signification of this name, genus. Now we're back to the predicables of porphyry there, right, huh? Yes, okay. Okay, it's not a likeness of something outside the soul existing, right? But from this that the understanding understands animal as being in many, what, species. We attribute to animal the intention of what? Being a genus, right? And of this intention, although the proximate, right, the near, the close foundation is not in the thing, right? But in the understanding, right? Because outside the mind, animal is not separated from dog, cat, or elephant, right? Nevertheless, the remote foundation is in the thing itself, huh? Whence the understanding is not false, which introduces these, what, intentions, huh? And then he makes an interesting statement, you know, which he doesn't develop right here. And similar about all the things which follow upon the way of, what, understanding. This is the abstraction of mathematical things and things of this sort, right? That's kind of funny, I was reading in Thomas today, he's talking about the power of God and rereading the De Potentia. And he's saying, God cannot make the, what, genus not be said of a species. He can't make the genus not said of a species. He can't make number not said of seven. He can't make it, animal not said of dog, right? That's pretty confining, right, to even the immortal God, right? Amazing thing, right, huh? But it doesn't correspond to something outside except in a remote way, right? There's a foundation in things, right, huh? For animal to be separated, right, from dog and cat and horse in our mind. The foundation of the things is a thing. Yeah, sometimes we say that the genus is to the difference, like, matter is deformed, right, huh? And the genus is not exactly the same thing as matter, because matter is a part of the thing, right? But it's taken from what is material of the thing, right? And then the difference from what is more the form of the thing, right? So there's a remote foundation, right, in the matter and form of these things. But the immediate foundation is in our way of understanding these things, right? We understand things that get fused first, and in general, right, huh? But our mind separates out what they have in common, huh? But notice that last thing has to be developed sometime, right? Just as the abstraction of mathematical things and of this sort. And I told you about this disagreement that we have with, I don't know, the editors of the Leonine, you know? Okay? When Thomas' commentary in the, what, Premium to Nicomachean Ethics, right? And there he speaks of how reason knows order, right? He's going to distinguish the knowledge of reason by the order it considers, right? And he says, well, order is compared to reason in four ways. Okay? And of course, Thomas there would seem to be validating the rule of two or three, right? But if you look at his words, he says there's an order which reason does not make, although it can consider this order. And then there's the order which reason itself makes. So that's a division of, what, two. And then the order which reason itself makes, he divides that into three, right? Now, it's in comparison to reason, he's doing this, right? So he says, first, the order which reason makes in its own acts. Like the genus is before the difference on the definition, right? Okay? So the first thing you've got to know about a square is that it's a, what, quadrilateral, right? And then that it's equilateral and right-angled, right? Okay? You've got to know that a perfect number is a number before you know it's a perfect number, right? Okay? Then there's the order which reason makes in the acts of the will, right? So if virtue is the order to the road to happiness and vice is the road to misery, I want to, you know, acquire virtue so I'll be happy, right? Vice is the road. And then there's the order which reason makes in, what, exterior matter, like in the table here, right? In the house or in the building, right? Okay? Your clothing, so. Well, it doesn't unfold that here, right, huh? Now he says, human art, mechanical arts they call them, you know, the political liberal arts, civil arts are about the order which reason makes in exterior matter. The art of carpentry, right? The art of Christmas cookies. And then the art, the order which reason makes in the acts of the will, that's studied in, what, ethics and domestics politics, right? Practical philosophy. And the order which reason makes in its own acts, in the thoughts and so on, the words that signify those thoughts considered in logic, right? Then he says, the order which reason does not make is a natural order which is considered a natural philosophy. And then he adds, right, right? And metaphysic is also about an order not made by our, what, reason, right? I didn't make the order of the angels, right? Okay. He doesn't mention mathematics. What order that's about? Now, some people say, well, Aristotle divides looking philosophy into, what, natural philosophy, mathematics, and wisdom, or first philosophy. And they have the famous treatise of, you know, of, what's the famous treatise on that same thing? By Boethius, right? Where he talks about this, huh? Probably in the day Trinitate, right? Yeah, yeah. And he says they pull it out, because it's a more fuller statement of what Aristotle brings out in the metaphysics. So, Thomas must, Thomas must mean, then, when he says, this is the natural order, is this kind of work, and wisdom is like that, he understands what's in the middle there, mathematics, right? So, well, no, he doesn't say it, though, right? You know? Why does he omit that, right? I don't know if it's so clear, right, huh? Well, we point to this text, right? See? What does he mean here, huh? Okay. And simile est omnibus, alis, qui conse qu'untur ex modu intelligente, right? Just as the abstraction of mathematics things and things of this sort. I was reading a beautiful text in Thomas today. You know, God cannot make a square circle. He can't make something that involves a contradiction, right? And so when I was a little boy, you know, they used to try to challenge us Catholics who said God is omnipotent, right? Can God make a stone so big he can't lift it? And it seems you're caught no matter which one you say, right, huh? But there's a contradiction saying a stone too big for God to lift. That involves a contradiction. You know? So God can't, you know, make something that involves a contradiction. Well, some people, of course, challenge that, right? They say God can make something that's contradictory, right? He can make a square circle if he wanted to, right? But anyway, the argument that does any one of the objections there is very interesting, right? When God came, when Christ came in... the apostles after the resurrection, right? Or when he came out of his mother's, you know? My mother always thought that was a wonderful thing, you know, because it's horrible to convert to a child, right? I mean, it's really, really uncomfortable to say the least. And so when he spared his mother that thing, right? And most women, it's worse, I guess, because of the fall, right? Before the fall would have been as easy as Tabitha, you know, when she was getting birth to those kittens, you know? So when Christ came through the wall, right, where the apostles were cowering, you know, and so on, well, you know, in mathematics, right, huh? Between two points, can you draw two straight lines? You see this objection before? Between two points, you can't draw two straight lines, can you? Yeah, yeah. Well, now, when two bodies are together, you had, you know, this point in my body and this point in the wall's body, right? When they're together, those two points, right? And two other points, right? Well, then you have two lines, right? Between what? Two points. Well, that's contradicting what we learn in geometry. Right. So God can make contradictory things, right? Because he didn't come through the wall, right? That's came through the tomb, you know? You know, he wrote, resurrected, right? He didn't, he didn't remove the stones and say, let me out. Well, you know, could have rolled it back. Let him out. No, I think the church won't say that, right? You know, how can you get away at that? Ooh, yeah. Well, Thomas says they're not the same point, right? Because one point is in the body called the wall, and the other point is in the body of Christ. So you've got two points and two points. And two points. And so you don't have, you know, it's not the mathematical one, right? See? Because the mathematical considers what? Line in separation from, as he says here, right? Yeah. Abstractio Mathematicorum, right? You see? So it's impossible in the abstract, you know, to have two lines between what? Two points. It could only be one line. But between two points in the body of Christ that coincide in place, right, with two points in the part of the wall, there could be, what, two different lines, right? Because one is in this body and one is in that body. It's kind of an amazing thing, right? It's still miraculous in any ways. But I mean, it's not a contradiction with mathematics, right? So there's something strange about mathematics abstraction, right? You know? Yeah, yeah, yeah. Thomas actually, is that a contradiction somewhere? Yeah, actually, we've got to be reading it in the, uh, there's a tendency, I guess, I guess now. Oh, is it? Yeah, yeah, yeah. Oh, you see some of that, Thomas, right? Okay. Thomas, you see, I always say, text, oublied, you know, or forgot a text. Something. So you've got to keep on reading these things so you can find things, right? So, at least I'll leave that, you know, there's a question mark in your mind, right? Is the order, right? He puts the order in mathematics, apparently, with the order in what? Yeah, not that they're the same, they're different sciences, right? And one is more in the imagination, the other more in reason, right? But they are, what, uh, not put with, uh, that first one, are they? Mm-hmm. So you've got to be careful. Subtle. Very subtle point, eh? Christel will end us going through all these problems there, right, in the categories, right? Huh? It's kind of the beginning things for people, right? Let's take this little text here from these sentences, huh? To the theory it should be said that aggregation, how would you say aggravatio, flocking together? Grex, gringes, you know, I don't know if other parishes have it, you know, but we have a little computer thing now where the pastor can send us messages, right? And it's called, uh, I think it's called, uh, flocknotes. Maybe I read a flocknotes, yeah. It's kind of nice, too, you see, you know, the pastor went over to, you know, was it, uh, Rome, I guess, yeah. No, no, excuse me, we went to, uh, Palestine, right? You know, people in the parish, right, huh? And they send back, you know, to us from, from the, uh, Holy Land, uh, pictures of the sites they were seeing, you know, and so on. Very, very nice way of sharing, you know, their, the trip of the, you can see the, you know, some of the parishioners you know, and going with Father Rose, you know, and so on. So this is aggregatio, okay. And passions of this sort, right, which follow the, what, division of the continuous, right? As Adesina says in the third book of metaphysics. And it stands that this number is not in God, right, huh? So when you see that there are three persons in the blessed trinity, did you divide something continuous into three? Because then these would be three parts of God, right, huh? And God would no longer be, what? Simple, right? I read another objection today there at the second article, is it, um, whether God's power is infinite. Well, if it's infinite, then it has parts, but it's not simple. Because you have to understand what you mean by saying he's, he's endless, huh? To the fourth, it should be said that in absolute number, there is a plurality which has a certain, what, composition and aggregation, right? So in the number three, there is what? Three ones are brought together, right? It's a composed thing, right? But how does Euclid define number, right, huh? Multitude composed of ones, right, something like that. And it's less certain than one, right? Which is its beginning, right? And therefore, not only as regards the understanding, but also as regards the thing itself, there is the measurability of the multitude, of such a multitude, by what? One. So every number can be measured by one, right? I'm counting, right? Okay. And so he says not only as regards the understanding, but also as regards the thing itself. There's a measure, measurability of a multitude, such a multitude by unity. But in the number of relations in the Trinity, right, or persons, there is not some order of certitude, that one is more certain, right? Or of composition in the thing. And therefore, number in God is not a multitude, what? Measured, huh? Except perhaps according to the understanding only, right, than not in the thing. Which composes right, but things that are not composed, right, huh? According as it understands what? According to also, it forms affirmative propositions, right? So these are all things here that are not, that may be appropriate in studying the categories, but it makes you stop and thinking of it, yeah. Now look at the second in the text here from the sentences. Moreover, just as continuous quantity is made whole from its parts, right, so also number, right? In God, in divine things, there is a number of persons, right? This is the objection is saying. To it, the third, tenereus, right? Of which one part is one, another part is two, right? Therefore, it seems that in God there is there a very integral hall. Integral hall. Integral hall, yeah. I usually call integral hall a composed hall, but it's often called integral hall, right? It's opposed to the universal hall, right? To the second, it should be said that in divine things, unity or duality is not a part of three, right? Except according to the way our reason takes it, right? Because, and the reason for this is, because in another way there is number in the divine things and in what? Created things, right? And in another way, what? Unity, huh? For since the one is that which is undivided in itself and divided from others, each thing that is created, right, huh? Through its essence is distinguished from what? Others. For the essence itself, the thing created, according as it is undivided in itself and distinguishing from other things is its unity, right, huh? And many unities constitute a number of created persons, just that there are many essences congregated according to numbering, thus that there is nothing in one that is in another according to number the same, right? So you and I are really two. Thus, number in created things has the notion of distinction, right, and of a certain piling up, huh, of things distinct in their essence. And from this it has the notion of a what? Yeah. So I was teaching for centuries things called classes, right? What were they, huh? Well, they were all distinct, and they were piles, you know, just piles. Piles, piles. It's kind of funny, this year I had twice in church, you know, at St. Mary's there, somebody come up to me that I had in class, you know, 20 years ago or 30 years ago, I don't know how many years ago, and they reintroduced themselves, you know. And they said, I thought it was you, you know, I haven't seen you in church. And so it's kind of nice to come up, you know, but they can see it's a heap, you know, that one heap after another, you know, that I taught, you know, or tried to teach. Yeah, yeah, yeah. So thus therefore the number in things created has the notion of distinction and of a certain piling up of things distinct in essence. And from this it has the ratio of an integral whole, right? I like the word composed a little bit, it is the word integral. Now, the personal unity, right, is the relative property, right, distinguishing one person from another, like the father from the son and the son from the father. And not the essence of the person. Whence the three persons are not differing by what? Essence. Since one in number is the essence in the three, what, persons. The father and I are one, he said, right? And I say that about my own father. My father and I were one. One, same nature. Yeah, yeah. But not one in number, right? And therefore there cannot be there a piling up. But only what? Distinction. And on account of this, the number does not have the notion of a integral whole or a composed whole. Except perhaps in our mind, right, according as in reason, are united the definitions, you might say, right, of the personal properties, right? There's three properties there, right? But through this, he says, there's no integration of some thing, but in reason only, right? And in this text, now another text here, Thomas speaks of the ones as not being simply one, and therefore a species if they're heaped up. Since they are composed for many unities, now this is more the creature I think he's talking about here, right? Either there's not one simply, but the unities are, what, flocked together by way of a, what, piling up, right? Which does not make something to be simply one, and consequently does not constitute being in some species. Okay, now he's talking about something else here then, right? And thus number is not some species of being. Or if a number is one simply and not by it to itself, it should be said what makes it to be one from many unities, which cannot be assigned, huh? Rest there, I have to think about that some more, huh? Dionysius, as Thomas explains, sees that number, even though it is in some way opposed to the one, partakes of what? Unity. Seven is one number? Isn't it seven ones, though? And he says that all things partake of one. One, he proves through this, about which it would seem less to be so, to wit, by number, which in some ways opposed to one, as the divided is opposed to the undivided, right? For every number partakes one, would the number be taken by itself, as it signified when it said two or three, or it, what, takes, yeah, as when we name it as a half or a third or a tenth. Thomas explains it simply and properly, only those things which come together univocally in something one can be, what? So you and your chair, too? Yeah, we say that you're two. Two what? See, two bodies, I guess. Although God and creature do not come together in something one according to some way of coming together, nevertheless, one can consider the community of analogy between God and the creatures, insofar as creatures imitate him, so far as they're evil, right? Whence in some way he can be numbered with other things, so it'd be said that God and the angel are two things, right? Not, however, simply and properly, right, eh? Just as creatures are, yeah, which come together univocally in something one, right? And from this it does not follow that God is a part of something, or that God and the angel are something greater than God, but that they are simply many things, eh? This raises, incidentally, the question whether the number ten of the categories, or of highest genera, is a number which is a species in the category of quantity. For the ten genera do not come together univocally in something one. In another passage, Thomas adds to coming together in one species or genus, what comes together in some order, right, eh? One existing, which is in the genus of creatures, is able to be numbered. That is, is a certain part of number. Everything that is one in creatures can be numbered with another, which it comes together, either in species or in genus or in some order. That is a rough stuff here. I'm checking you all to this sort of stuff. For many things existing in act, there does come about one thing simply, unless there be something uniting and some way binding them together. For many things existing in act, there is no way binding them together. to what? Each other, right? Thus, therefore, if, according to diverse forms, Socrates was an animal and rational, it would need two things that they'd be united simply, right? Something that would make them to be one. Whence with this, since it's not able to be a sign, remains that man is not one except by what? Yeah. Just as the pile, which is secundum quid unum, one pile, right? But simpliciter, mufta, right? There's a fallacy in the logic of fallacies, and a fallacy of secundum quid and simpliciter, right? Easily mix these up, these two, right? And thus, man would not be a being simply, right? Because each thing, insofar as it is a being, insofar as it's one, right? So I'm a philosophical, what? Grandfather, yeah. Or you can say, I am a white philosopher. And a black man was a philosopher, because he's a black philosopher, right? Is that something, a black philosopher or a white philosopher? You're looking at a white philosopher here, right? Is that something? Philosophical grandfather? Is there any way of becoming a white philosopher? Yeah. See, I was not always a white philosopher, right? So how did I become a white philosopher? Well, I'm a Christian logician. You weren't always. I was not always a Christian. I was not always a logician, right? And I got to be a Christian, I guess, by being baptized, huh? I got to be a logician by reading porphyry, and so I'll still use logic. You know, this granddad there, she's in Fisher College now, right? So she's at Benedictine College there in Kansas. And her logic book was actually his logic book, you know, which is kind of a famous one, right? But Siric, you know, didn't know how great a logician he was, but he says, he's a good businessman. It's still in circulation, you know, huh? You know, also his logic book, you know. Who's making that money now. Yeah. Yeah. And his chapters, you know, the parts of logic and the exercises in the book, too, you know, and so on. One that always sticks in my mind is, you know, he killed his daughter Ruth, therefore he's ruthless. It's kind of a clever thing, right? Some fallacy of an argument, isn't it? It's an accreditation there, right? These senses are ruthless, huh? And according to this way, Democritus speaks rightly, saying that it's impossible for one to come to be from two, and from one to become two, right? It should be understood that two things existing in act never make one thing. But he, not distinguishing, that means Democritus, not distinguishing between potency or ability in act, laid down indivisible magnitudes to be the substances, right? The atoms, right? He wished that just as in that which is one, there are not many in act, so also in either is there, what? In potency. And thus, each magnitude would be, what? Indivisible, right? So it is a straight line, one. It's one in act, but it's many in, what? Yeah, you could divide it, right? There's a theorem in geometry to divide a straight line, and you could divide those lines and so on, right? But for Democritus, Democritus didn't see that distinction of ability and act, right? So what did he have? Everything was... He's supposed to have given a, what, thought experiment, as they call it, Einstein? Suppose bodies were divided in a way they could be divided. No one would be left. Well, either there would be left, huh, these things that have no, what, size at all, right? Or there'd be nothing left, right? It can't be nothing, because you can't get something out of nothing. So if our bodies were divided in every way they could be divided, they would be just having, what, these atoms, these indivisible things. So this is thought, experiment to make you think that everything is composed of these atoms, right? We see in the sunlight coming in, you know, those things moving around all the time, right? Okay. That's why he said, you know, these little ones, circular ones are, you know. What is the other problem there? If you cut a cone, right? Georetically, how do you make a cone? You know, take a triangle, right? Then you rotate around, that right angle, let's say triangle, and rotate it around, and you get a cone, right? Well, now, if you cut a cone parallel to the base, right, and you look it up in the bottom of the top part and the top of the bottom part, right, they're both circles, aren't they? Right? Equal circles or unequal circles? Well, if it's equal, then you're thinking of the cone as being, what, piled up of equal circles, right, which would be not a cone, but a, what? Yeah, yeah, yeah. But if they're unequal, the circle at the bottom of the upper part and the top, then it's, that's not straight down like a cone should be, right? Right. So he had a problem there, right? No. Very subtle, though, these problems, huh? Hey, these Greeks. Socrates would make you think about things, right? That's the way you think. I thank the gods for being born a man and not a woman, for being born a Greek and not a barbarian. And third, for having met, what? Not easily. No, this is Democritus, huh? Not a great mind. Well. Yeah. For rightly, he says, Democritus said, right, supposing his position, by which he laid down that indivisible magnitudes is to be the substances of things, right? And thus always they are in, what? Act, huh? And thus from them there cannot come about, what? One thing, right? And thus, as it is in magnitude, so also in number. If number is a, what? Putting together of units, right? As is said by some. It's necessary either that dwelt is not something one, or that what? Or that unity is not an act in it, right? So are there really three ones and three? Three actually distinct ones? It's really tearing your mind apart here, right? As far as the categories are concerned, we know that you can divide being into, what, the ten? And you can divide quantity, discrete and continuous, and you can divide discrete into number and that, right? And we'll leave this for more advanced parts of philosophy, right? But you've got the fundamental divisions, right? To begin to get really into the difficult questions, right? And thus dwelt, you will not be two units, but something from two units composed, right? Otherwise, number would not be something one per se and truly. But project ends just as, what, things are piled up, right? So there's a pile, you've got a pile of junk in your house. something like that, you know? The least pair of baskets, you know. You come to the least pair of baskets on Thursday, you pick up the trash. And they had a bag of all this junk. And is that something one? That pile? Is that a pile? A pile of rocks, is that something one? What would you say? In some way. Very secundum quid though, right? That's some teaching here one, right, huh? Sometimes you get the modern biologists talking about, you know, that each one of our cells is alive, so we're many, many lives, you know? You know, you have as many lives as you have cells, right? Because each cell is alive, right? And functioning, I mean, you know, occasionally some cells die, brush off some of these things, they probably died, right? You know? Shave off some of them. But it's very hard to understand the unity of a number, right? Something one, or is it a pile? And in the Trinity there, the three persons are one God, right? Puzzle a lot about it. It's a number one, right, huh? We study piles in arithmetic, or what? See? All these wonderful theorems that Euclid has, right? What are these? They've got to be one in some way, per se, and truly, right? Otherwise they'd be cratcheted in some. Now, in the seventh book of the metaphysics here. Compositum, right? Put together, sometimes gathers its species from something one that is a form. As is clear in a, what? Mixed body, right, huh? Because they found a mixed body, right? That the, what? Bodies mixed do not survive and act, right, huh? But they're maybe one by a form. Or composition, as is clear in the house, right? I have one house. Or do I have a pile? No. Put together in some way, huh? It's more closer to the first meaning of composition, though, right? You know, should I say that I'm composed in many ways, or I'm put together in many ways? You know? See, Thomas has a whole article there, right? A whole question, rather. Eight articles that God has not put together in any way. It distinguishes, as I said, how many kinds have been put together, huh? About six ways, right? Or order, okay? As is clear in the syllaba and numeral. And that's going back to the distinction of what can, of a discrete quantity, right? A Shakespearean sonnet has 14 lines. So this is one line, huh? From fairest creatures we desire increase. That thereby beauties we always might never die. So, what do you mean, one line? There's a conglomeration of words and letters, a pile of letters. What kind of unity does a line of Shakespeare have? Yeah, there's an order in the line, right? Distinction of the syllables. Yeah. Fair is foul and follows fair. Hover through the fog in filthy air. But you notice he puts that together, right? Syllaba, I guess. And what? Number, right? And then it's necessary that the whole composed is one. Simpliciter, right? That's a great thing, huh? Sometimes the thing composed gets its species from some multitude of parts collected, as is clear in the pile and in the people, right? As of the sort. And in such ones, the whole composed is not one simply, but one only according secundum quid, huh? Now, in this campaign they're both talking about bringing the country together, right? United the country, you know, behind them. I'm sure everybody would appreciate all this language, you know. Could I ask you a question about this unity that you're supposed to be seeking from the country? Is it some pitchy terrorist that couldn't have quit? No, I'd say, take your questions. Now we get back to some more common sense here, back to the text itself, right? The division of chapter six on quantity, right? You've got the quantities per se, right? And quantities per acidans, right? You have two divisions of quantity into its species, right? The distinction between discrete and continuous, and then that one whose parts are what? They have a position to respect each other and those that don't, right? And those two divisions, are they identical or what? Because what is, why aren't they identical? Because time, the parts don't, what? Have any position to respect each other because they don't exist together, right? Yet time is a continuous quantity, right? So you have those two divisions, right? The first division then, and they're divided, the first ones, huh? Definition of discrete quantity and species, and they were, what, number and speech, right? And the definition of continuous quantity and species, and there were how many there? Line, surface, body, right? Place, and what? Time, remember? Or five, right? Okay. Then species of the second division into those having or not having position. Then quantities per acidans, like, you know, the color that is on the surface, you know, blinding you, right? And I'd help make a wagon there down at my father's factory there, you know. We'd go out to the wood lot there, you know, and it's painted white, you know. And it's a bright July day. Yeah, yeah. So there's too much white, right? But it's white, you know, quantity? Per acidans, right, huh? The surface is really, you know. The surface is all painted white, huh? Then we have the properties of quantity. Does not have a contrary. Is not said more or less. You'll see the connection between those two properties that it doesn't have when you get the qualities, right? Because in qualities you have contraries and you have more or less, right? So hot and cold are what? Contraries, but you can have more or less of them, right? We can be lukewarm and so on. But then the real strict property is equal or unequal, right? Are we up through the text yet? Oh, my God. I thought we were through it. When I got to that, I thought this is it. At least it got back to the text there, you know, putting us around with the physics and all these things. This is not a book or a chapter of one of my books. It's a collection of text, right, you know, that things, huh? Now, some interesting texts here for Thomas here, huh? The philosopher, right, which means Aristotle, in the Predicamentes, right, that's the categories and Latin name for it, distinguishes the species of quantity according to diverse notions of what? Measure, right? So, how long is the surface? Just how long is the line, right? Wait.