Logic (2016)
Lecture 17: Perfect Numbers, Proportions, and the Properties of Quantity

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Perhaps it could also be divided into perfect and imperfect, right? What's a perfect number? Well, it's equal to the sum of its parts, the numbers that measure it, right? So 6 is the first perfect number, right? It's measured by, what, 1 and by 2 and by 3, but not by 4 or 5. And 1 plus 2 plus 3 equals 6, right? Then the second perfect number is, what, 28, right? And that's measured by, what? 1, yeah, I forget. But it gets pretty high math after a while, you know. Yeah, some math teacher has a chart. They would do a prime number. They would propose the next prime number. That's all they would do all day, just think of the next prime number. I don't know how they figured it out. Yeah, yeah. I saw this interesting thing that I was kind of just thinking about by the students there. I was talking about proportion, right? And if four numbers are proportional, then the first is to the second as the third is to the fourth. But if the first number is to the second as the third is to the fourth, the product of the first and the fourth will always equal the product of the second and the third, right? You can see it even at 28 there, right? So if two and 14, right? Well, I can't get four numbers, right? So if you say two and 14, right, make 28, right? And four and seven, mid-28. So it must be proportional to those numbers. So two is to four and seven is to what? Interesting. It's going to work out that way, you know? So if you take a big thing like here, you take a hundred and say, what two numbers can equal any pair of numbers, right? And that equal hundred, right, multiplied together, laid out with the lowest to the highest, would give you four numbers that are proportional. Kind of beautiful to see that, you know? But it's good for people to see that, right, huh? Because when I was a student, I mean, a teacher, an assumption, and I'd say, you know, I'd say two is to three as four is to, always somebody say five, you know? And so if you don't know what a proportion is with numbers, how are you going to know what a proportion is with these other things? You know, proportions in math are actually the clearest in some sense, right? But, I mean, you've got to see that two is the same parts of, I mean, four is the same parts of six, that two is a three. So if you think of six as being three twos, then four is two of the three parts, right? So it's practically proportional. We have seen two divisions by contradictories in the second chapter give rise to what? Four parts. What was that division? You know, I mean, here now, I mean, in the categories, the book itself, you had the universal substance and the particular substance and universal accident and particular accident. So you've got four, right? But when we crisscross two divisions by contradictories, we do not always get four parts, right? When Aristotle, for example, divides the parts of a plot by the contradictories before and not before, and after or not after, does he get four parts? There could not be a part of the plot which is either before anything or after anything. It wouldn't fit in at all, right? Likewise, when we divide quantity by two contradictories, we do not get four parts. A quantity either does have parts that have a common limit or not, right? It is parts that either have a position towards each other or not. Line, surface, body, and place have parts that have a common limit and also that have a position toward each other, right? Number and speech have parts that have neither a common limit nor position toward each other. What is time? Time has parts that have a common limit but not positioned toward each other, right? But is there a quantity whose parts have a position toward each other but no common limit? This does not seem to be a real possibility, right? Okay? Just like the example before and after, right? The definition of... If Aristotle had divided quantity a second time by the natural differences, and I call them the natural differences because they're done in natural philosophy, but what's the reason why it's more appropriate for the natural philosopher to speak of continuous quantity as a quantity who's just divisible forever? Why is it more appropriate for the logician to say, no, it's one whose parts meet a common boundary, right? And parts are like matter, right? It's divisible forever, I'm going down. You know? And, but the whole is like the form, right? It's a logician, right? It's about the forms of things, right? Species, sorry, English word for species is form, right? I used to love to talk about the forms of fiction, right? Why does Aristotle divide quantity twice and quality only once? Perhaps there is a special reason for emphasizing quantity having parts of position. We generally say that accidents are individualized by their subject or substance in which they exist. But quantities that have position are individualized by themselves. Now, what sacrament does that illuminate? See, are the dimensions of the bread, right, in the Eucharist, right? Are they individual? Not, that's not a universal thing, is it? It's individual, right? But are they individualized? No, you see? And why is that? Because quantity is a little bit like what? Substance, right? That it can be individualized in a sense from its own, by its own very nature, right? If it's that kind of quantity that is what? Asposition, right? So it's beautiful they're still helping us. I think I was knowing it. It shows you how quantity, and again, it's another example of how quantity is close to substance in some way, right? See? And so the color of the bread, right, or the taste of the bread, right, that's individualized by being in the, what, quantity of the bread, right? It's kind of amazing, right? Quantity is acting a bit like a substance, right? I mentioned before how people confuse, you know, my friend Descartes there, you know, body, which is a species of quantity, with the body, which is a species of what? Substance, right? In Plato and the Bethlehem, it's not the numbers of the substance of things, right? But there's an affinity there between quantity, right? And Aristotle gives that right after quantity, right? Right after substance, I mean. What's the cause, what's the cause of the individuation of material substances, right? How is it possible we can have five men in this room here? How is it possible we can have five men in this room here? How is it possible we can have five men in this room here? How is it possible we can have five men in this room here? Yeah, because we've got enough flesh and blood and bones to go around. I don't need your flesh, and you don't need my flesh. There's enough flesh. But why can't we have all these chairs that are more or less the same kind, right? The simplest answer is you have enough wood. See? Hards that a grandma can make, you know, a couple of dozens of nice cookies at Christmas time. My grandmother, great cookie maker. Because you've got enough dough, right? Yeah, yeah, yeah. Yeah. So it's matter as subject to what? Quantity. That is the beginning of individuation in material things. Not in the angels, right? But the difference between one angel and another angel is wholly what? Formal, right? That's why Thomas says that no two angels are really the same in what? Species in kind. Interesting, right? I can find out something about distinction of the angels, right? You don't have that composition we have, right? So why are there many chairs of the same kind in this room? Yeah, in the hood, I thought, yeah. You know that's a carpenter, right? That's a carpenter, right? Now, look from this great book here, Summa Concentile, as I recommend this book to you. Thomas says, Hobbit altum et hopproprium. It's proper to quantity among accents, I think he means, right? Quantitas dimensia. That's what his name of, he says, mean for what? Continuous quantity, right? But not maybe what? All continuous quantity, right? It doesn't fit time, right? Interaccidentia radicalum, among the other accents. Quat ipsa, secundum same, in itself or by itself, is what? Yeah. And that is because positio, which is the order of parts in a whole, right, is in its what? Yeah. Put it in its definition, right? A quantity having, what, position, right, huh? Okay. That's a, I want you to distinguish this positio from positio, which is a category, right? I'm laying down, I'm sitting down now, right? That's a different kind of position, right? This again brings out the closeness of quantity to substance. And this also perhaps explains why the order of the species is reversed in second division, because according to this difference, continuous quantities having parts in position toward each other have a special connection with or likeness to what? Substance, huh? And in another passage here, Thomas touches upon this closeness of quantity to what? Substance, huh? He distinguishes the modes of quantity and about this. There's a C missing there. This does three things, right, huh? First, he distinguishes quantity into that which is what? Quantum per se, right, huh? To it align, and that which is quantum procedens, as musicum, right, huh? And secondly, he distinguishes quantum per se, which is twofold, huh? For some signify per modem substantiae subiecti, right? There's some likeness to that, right? Just as line or surface or what? Number, right, huh? So I talk about numbers if there's something existing by themselves, don't I? You know, I'm all crazy about these numbers, right? Each of these substantially is quantified because the definition of each of them is place quantity, right? For a line is a quantity that's continuous according to length, and it's divisible, right? Finite and so on. And similarly about other ones, right? There's infinites, you couldn't measure it, right? But some things per se pertain to the genus of quantity, and are signified by way of what? Habitus, huh? Or the passion of such a substance, huh? It should be known, Thomas says, that quantity, among other accidents, propinquior, right? It's nearer to substance, whence some, like Descartes, and, you know, some of the Pythagorean and so on, think quantities to be what? Substances, right? Terrible mistake, right? But understandable. As line, and even number, right? And surface, and what? Body, right? For only quantity has division in its own parts, right? After substance, huh? For whiteness is not able to be divided, and consequently neither is it understood to be individuated, except through its what? Subject, right? And therefore is only in quantity, in the genus of quantity, that some things are signified as subject, and others as, what? Actions, huh? Moreover, that the divisions include the same species with the exception of time, show a special place for time. That's what Tim was talking about before there, right? Although time is a continuous quantity, it is also very much the number of something, right? And like number, it's parts of order, but not, what? Position. It's interesting, huh? Let me anticipate something here, but I was just thinking about it the other day. I was thinking about how am I going to torture these guys, you know, with study of statement, right? And, well, one division we make a statement is into its parts, right? And sometimes you divide it into nomen and verbal, right? Noun and verb, right? Onamon, rhema in Greek, right? Now, can you have a statement without having a noun and a verb, right? Something like that. Those are kind of the basic ones together, one. So you say, man thinks, right? Man plays, huh? Man eats, okay? One is a noun, one is a verb, right? Now, noun and verb have in common that they both are names, right, in the sense that they are vocal sounds, right, that signify by human agreement, no part of which signifies something by itself, right, huh? But what's the difference between a noun and a verb, huh? Yeah. The time, the verb signifies with time, and the noun signifies, what, without time, right? Now, kind of interesting, huh, in, I think, in Latin and in Greek, huh, they have the same name for noun and name, but in English we have the two words, name and noun, but if you look in the Greek and the Latin, it's like you have name instead of both, and then name instead of one in distinction from the other, right? Well, we've talked about this way of naming things, right? The name is now of Nomen, or the name of Onoma, has become, what, equivocal by reason, right? It's been kept by one of the two things it's said of as its own name, and the other gets a, what, a new name. A new name, I'm sorry. Yeah. Now, why does one of them, in this case, get a, a, its own name, right? And the other keeps the common name. Yeah. Something that kind of stands out in some way, right, huh? I was very struck by this the other day I was thinking about it. You know, um, we sometimes in philosophy distinguish between doing and making. You know, um, you know, um, you know, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh And other times we say, when you're making something, you're doing something. So doing can be said of doing and making, right? And then there's a different meaning when you're juxtaposing or, you know. Now the distinction between doing and making lies behind the distinction of two virtues of reason. Foresight. So art is right reason about making. I had right reason about making cookies. Oh, she did. I never tasted as good as cookies as my aunt. My grandma's motto in life was I am to please. That's her motto. She's a great woman, great woman. You achieved her goal. Oh, yeah, I said great woman. So this is a very important distinction. But why does making get the new name, right? Yeah, because it stands out, right? And what do you got to show for it, right? I used to say to the kids, I said, sex without children is exciting, it's fun and so on, pleasurable, right? Well, what do you got to show for it? So when you have a cookie or a child or a chair or whatever it is you made, it stands out, right? So it gets the new name, right? It doesn't mean that making is necessarily better than doing, right, in this case. But there's something in addition to it, right? So why does verb get the new name and gnomon keep it in Latin and in green? Yeah, yeah, it signifies with time and it doesn't, right? Well, time is a measure of what motion, right? And things in motion, sooner catch the eye than what not stirs, right? Okay, that's very striking, isn't it? That shows, you know, what I was struck by, I was thinking about the first sense of looking before and after is in time, right? And was Aristotle right in putting that first? Well, it's so much so that you can't make a statement without using the verb, right? And the verb signifies with time. It's like, you know, the question is, can we get beyond that? Not only is it the first sense, but how can we say anything, right? How can we talk about God who's not in time, right? You know, you know? If Abraham was, I am, huh? Good grammar. But you're stretching the language, right, huh? To express something, right, that's not in time, right? That's not in motion, not subject to motion, right? Our mind is so much tied to motion, right? Yeah, yeah, yeah. So I was thinking, you know, when we start talking about statement, you know, we'll talk about that again. But it's kind of a confirmation of, I think, of what you're saying, right? That the first sense of reason looking before and after is in time, huh? We're always saying, well, what can we do in time? We're going to do this if I do that, you know? You know, no big deal. But we're living that way all the time, right? When Shakespeare says, you know, what is a man if his chief good and market of his time be but to sleep and feed a beast no more? But what does he say market of his time? What does time mean? It means his life, doesn't it, right? But why should he use the word time for our life in place of the word for our life there? Because our time is so measured by time, right, huh? It's so known by time, right, huh? To write up your day, don't you, in some way, you have to, right? You've got to be at the... Bell ring. Yeah, you've got to be at the thing, yeah. Yeah. Yeah. Now, teaching us always in the arts, I don't want to get to know at all the time. It's nice. Yeah. You don't get the time, you know, but, you know, you realize you've got to get through this, so you've got to go on to their class or something, you know, class and time, you know. That's where you go. Students get the impression that you have. Yeah, yeah. Moreover, that the divisions include the same species, right? The two divisions, right? They kind of coincide, don't they, right? Just like if you were to divide the quantity into quantity that is divisible forever and quantity that is not divisible forever, that would give you the same division as discrete and, what, continuance, right? But now, it has parts that have position in respect to each other and does not, right? That almost seems to be the same, again, but with this oddball thing called, what? And some people wanted to say, well, you should really say time is a number. Huh? You've got to be careful about that, no? Well, I'm going to sit right there for, you know, about eight days. It's a number, isn't it? Eight days? Eight days. It's not the abstract number, but it's the numbered number, right, huh? One of the first things, because Sirius is in his doctoral list, I guess. But the distinction between numbering number and numbered number, right? Yeah. Well, time is numbered number, right, huh? You know? It's like ten grandchildren, right? That's a numbered number, right? But the abstract number is the numbering number, ten. So he says, that's pretty beautiful what Aristotle does there, right, huh? He brings out time, right, huh? What does Richard II say there? He's in prison there, right? He's covering his tears, you know. And he says, now is time made, and many heard, what? Numbering clock, right? More that the divisions include the same species, the exception of time shows a special place for time, right? Although time is a continuous quantity, it is also very much the number of something, huh? And like number, it's parts have order, but not position. That the parts of time have order is important in the categories. Because as we have seen in Thomas, this is the reason why in the last six highest genera, there are two genera connected with place. Where? And position, right? And only one with time. When the idea of time already is the idea of before and after, right? It's the number of time, the before and after in time, right? So you already have the idea of order there and the idea of time. By place is where you are, right? Where you are, your body might be in, what, sitting or laying down or standing or something, right? And so you need this other category, which is the order of parts in place, right? But you don't have to have that in time, right? It's already in the idea of time. So notice how that helps to understand the last six, right? Why are there two ones for place, right? You know, that would be on your final exam, right? Why are there two ones for place but only one for time, right? See how wonderful this is? But even more important is that before and after have their first meaning in time. And Aristotle exemplifies the second meaning with, like, number. Remember how the second sense of before and after was, what? I'd say in being, right, huh? But Aristotle's example, one is before two. Why does he take number, right? One can be without two, but two cannot be without, what, one? Can five be without one? Yeah. There's one man here, right? One man here, one man here, one man here. And each one of us could be without the five of us. But the five of us couldn't be here without each one of us being here. So each one of us is before all five of us. Aristotle's even simpler than that. One is before two, huh? Okay. There are very important connections between time and reason. And Aristotle points out, time would not be fully without reason. So, I mean, how can I count three days? Because yesterday is gone, so I can't, you know. I can count my folders here, you know, because they're all here. But, not all my folders, but my thing there. But I don't have the three days here, yesterday and today and tomorrow, right? So how can I count them? Yeah, I can remember yesterday, right? I can anticipate tomorrow. I can go to mind you up there. Aristotle raises after he takes up time. Would time be without the mind, huh? Not fully. See, how can there be a number of days without reality? Time hardly exists, right? I just love to tell people, you know, that God never changes. He's always the same. Sounds awfully boring in his life, doesn't it? Absolutely, yeah. No, right, right is the spice of life. You know, I grew up with that proverb, you know, being sent by adults and so on. You know, you know, the same thing for dinner every night. It's driving my wife crazy, you know. I come out, you know, use a bookshop, another edition of Shakespeare. You know, you know, the same thing over and over again. And when I retired from Assumption, I said, I'm going to, like, get all these books here. I don't know. Don't worry, these books are modern philosophy, you know. I'm not going to read those in my retirement. But here I'm reading the Subaconda Gentiles and then the Prima Paras and so on. We read the Depotentia next, you know. It's powerful. Disputed questions ever written. Marvelous things. Very important for the Trinity, too. So there are very important connections between time and reason. Time would not be fully without reason, right? That's a very important connection, right? And we've seen before the definition of reason as reason is through before and after, right? As Shakespeare has taught us, reason is the ability to look before and after, right? And the first sense of before and after is in time, right? So when Christ said, before Abraham was, I am, right? He's using the word before and now in the later sense, right? And that strikes us kind of strange at first, right? But he expressed himself as well as our language allows him to express himself, right? Before Abraham was, I am. He didn't say I was. I was thinking of this the other day and I didn't have a chance to look it up when John the Baptist says, he who comes after me is before me. At least that's one translation. Yeah, yeah. Exactly the words were used in the Greek. Yeah. I was thinking of that. Yeah. But he's thinking probably of everything before in excellence, you know? Yeah. But it's kind of interesting. You have to know the sense of before to really expound what he's probably saying there, right? Let me do this. Further, the categories is ordered to the perihermeneus, the logic of the first act, the perihermeneus, and Thomas explains that, right? That's the book on statements, right? Perihermeneus. And the truth of statements whose formal part is the verb that signifies with time. Well, you know, another thing we'll do when we think out a statement, we'll say it's speech signifying the true or the false, right? Well, that's very important for reason, right? But you can't express this without the verb. And the verb signifies the time, right? So, I mean, you tell those connections between time and reason, you know? You say, hey, this is really interesting. After Aristotle has distinguished the species of quantity, he distinguishes quantities through happening as opposed to per se, right, huh? From quantities through themselves. The above species are quantities through themselves. And other things are quantified through them and not through them, what? Cells, right, huh? So, white is the white wall there when it's big bother you in the summer, you know? It's size, right? But it's size, does it belong to whiteness as such? Or is it because whiteness is in the wall, right? Thus, as Aristotle says, there is much white when the surface which is white is large. Or a trip is long when the time it takes is what? Long, huh? Why does he distinguish per se and parochidins, quantities, but not substances or qualities, thus? Well, part of the reason is that quantity is the subject of certain qualities, right? And therefore, they are what? Quantified parochidins, right, huh? Because their subject is spread out, right? If color is in the surface of a body, right? And surface is a, what? Continuous quantity, right? Then the color is, what? Spread out, right, huh? That's because of the surface, okay? And this again brings out the order of these two highest genera, huh? Now, the properties of quantity does not have a contrary, which you'll find with some of these qualities, right? Is not said more or less. Is one three more three than another three? See? One sweet thing is sweeter maybe than another sweet thing, right? But is one three more three? It's not said more or less, huh? It doesn't have, does it have a contrary? Equal or unequal, right? That's a real nice property. You know, say we're all equal, you know? We hold these truths to be self-evident that all men are created equal, right? Well, I mean, some men are taller. Some, the basketball players are taller than me. And some guys are shorter than me, you know, huh? And for some reason, I guess, you know, a man doesn't like to marry a woman taller than himself. A woman doesn't like to marry a man shorter than herself, right? And once in a while, we're in a couple where there is this, you know? And it's kind of interesting, you know? I don't want to, I don't want to, don't tear ass at all, they resolve this, you know? You know? But, uh, uh, let's go to the Royal of the Pack and Margo. Roy was a tall guy, the tallest guy in the class, and Margo was a tall girl, right? So she had to, you know, get, go with. Poor Roy is getting in charge, being pursued by Margo all the time because of his height, the poor man. I think he became a priest behind me. He drove into the priesthood, I don't know. Being pursued by Margo. I could have felt sorry for Margo, but I felt sorry for Roy, too, you know? Yeah, I see. We're not all equal, are we, then, huh? In quantity, right? We're equally healthy, equally strong. Equally beautiful? Equally virtuous? Equally vicious? The first property of quantity given by Aristotle is that it does not have a contrary, huh? Aristotle spends the most time in his property which quantity shares with, what? Substance, huh? Some confuse relations that follow upon quantity with quantity. And Aristotle speaks at some length to refute this mistake. Large and small, for example, might seem to be contrary, what? But they belong to the genus, what? Relation or towards something, right? Remember later on, okay, Aristotle says here a little bit here, but Aristotle would take up relation or towards something, right? After quantity before what? Yeah. Well, when he enumerates the 10th January, right, huh? He gives quality before or towards something, right? What happened? Did the guy, you know, get mixed up in the order of these things? Well, no, you're going to find out, right, huh? And that Plato and, you know, people had defined relation as something that's said to be of a... another, right? Or towards another in some way, right? Well, knowledge is said to be knowledge of something, right? Whether it is knowledge, a relation, you see? Aristotle's got to distinguish between what is said relatively, in a way, right? Like, knowledge needs to be knowledge of something, and what is essentially a relation, right? So he's got to clear that first, you know? But here he's already, in a sense, pointing out that you don't want to confuse quantity with relation either, right? Large and small, you might say, well, these are what? Quantities, yeah. Are they quantities or are they relations, huh? You see? When I learned, you know, these quantities are divisible forever. You can get something that's very large compared to anything. And something that's very small, right? Because it gets smaller and smaller and smaller, right? You know? So you take any line, right, you can divide it forever and you can get, you know, all the way down. One is larger, one is smaller. It's just small compared to the whole line, but it's, you know, larger than this other part of the line and so on. Thank you. Some confuse relations that follow upon quantity and quantity. And Aristotle speaks of some length to refute this mistake, huh? Large and small, right? For example, might seem to be contrary qualities, but they belong to the genus relation, or to speak in the teaching of Suritome, towards something, right? The same quantity can be called large and small, but toward other things, right? If they were contraries, contraries would belong to the same thing at the same time. And something would be contrary to itself, right? Aristotle seems to exclude place from this property. Indeed, the definition of contrary seems to get its words from place, because we say that contraries are the species furthest apart in any genus, right? So in the genus of color, black and white are furthest apart, right, huh? And the genus of habit is virtue and vice, right? So conant and incontinence are not as far apart as virtue and vice, right, huh? Okay? Because incontinent man, he eats too much, he drinks too much, but he's kind of ashamed of himself or something, right? Why the drunkard, right? That's the way to live. The second property of quantity is also shared with substance, huh? It's not said more or less. This, again, will show that quantity is closer to substance than quality, right? And it follows upon the first property. One quantity is not more or less another. One three is not more or less three than another. One three-foot line is not more or less three feet long than another. The third property of quantity is to be equal or unequal. This is a property in a more strict sense of the word than the first two, what, properties, because they're common to many things, right? There's an appendix here, right? You get all this stuff that's, you know, gathered, you know. I wasn't writing this in a book. This is just for my notes, you know, in my computer there. So, what are you throwing all these things at us? There's a connection between number and the continuous, huh? According to Avicenna, right, in his metaphysics, unity in number, which the arithmetician considers, right, are not that unity in multitude which are found in all, what, beings, huh? But only according as they are found in material things, according as plurality is caused from the division of the, what, continuous, huh? And from this we are able to, oh, there was suffering in that. Um, why is it that numbers can, what, increase, huh, forever, right? What, because the continuous is divisible forever, right? So if a line is divisible forever into lines, right, then the number of lines can grow beyond any, what, is that possible for the species of quantity to grow beyond any number? Well, if you get down to numbers, the species of numbers are, right? But if you take the more general division, the Aristotle makes it discrete and continuous, right? That's it, there's two, there's two of them, right? Yeah, two or three, right? What kind of multitude is that, right? Oh, you see, there are ten categories, right? Did that ten categories arise from division and continuous? What did you divide up and you got into ten? If it arose from the division and continuous, it would go on forever, right? It'd be an infinity of highest gender, right? And it is, it turns out, there's only ten of them. Yeah, yeah, yeah. When Thomas says, you know, why there's not more than three, right, huh, he goes back to the fact that there's only really two, what, kinds of intrinsic operations of God, right? Understanding and loving, right, huh? One proceeds by way of God understanding himself, and the other by way of loving himself, right? But that's a much different multitude than that of what you have in number, right? The multitude in number arises from the division of the continuous, huh? From this one can find all those passions and numbers which mathematicians demonstrate, there's multiplication and bringing together and things of this sort, which are founded upon the infinite division of continuous, right? Once there is infinity in numbers, according to the philosopher in the third book, the physics, and therefore such a unity is in potency every number. Is the number which is the species of quantity numbering number or numbered number? Well, if numbering number is the same as numerous absolutus in the following text, and absolute number is not what? It's not absolute for a number of things except in the understanding, right? And the categories are a division of being outside the sore reason, then the number which is a species of quantity would be a numerous which is in... Yeah. Time is also a numbered number, right, huh? Let's see if this text is from the Summa. Further, wherever there is number, there is a whole and what? Parts, huh? If therefore in God there is a number of persons, there will be in God a whole and part, which is a pugnant to the divine simplicity. To the fourth, it should be said that number is twofold. To wit, simple number or absolute number as two or three or four, right? And number which is in things numbered as two men and two what? Horses. If therefore in divine things one takes number absolutely or abstractly, nothing prevents their being in him whole and part, right? But thus it is not except in the way our mind what? Takes it, right? For absolute number is not what? For number is not... ... Separate from numbered things, except in the understanding, right? If we take number in so far as it's in numbered things, then in things created, one is part of two, and two of three. As one man of two, and two men of three, right? But it's not thus and gone. Because as much as is the Father, so much is the, what? The whole trinity has become clear below, right? On the other hand, a number like two men or two horses seems to involve more than one highest genus, right? It's like white man involves more than what? One genus, right? What's a white man? What genus do you put a white man in? Well, man belongs in substance, and white belongs in quality, right? So where do you put two men? Yeah, I wonder if that's right, they'll say. Moreover, logic considers things insofar as they are in reason. An abstract number is in reason only. Another text, Thomas compares the foundation things, which that should be probably with, huh? With what is known by mathematical abstraction. It has logical intentions. It's a little bit confusing. Yeah, by, probably. My text, right? Okay. Now I get this, this is a very important text, but it's, you know, so you have to go to the sentences sometimes, right, huh? Well, I've seen it, of course, you know. Passage, you know, forgotten about, you know, in sentences, you know. Wait for me to read this text. Yeah, yeah, yeah, yeah. But sometimes something is taken up a little bit, you know, and it's said about in a little different way, you know. Okay, this is like one of these texts here. This is not the text that he was talking about. I find a text like this, I've got to put it in here, because I might. This is it. Yeah, this is the most subtle text, though, you know. The conception of the understanding, huh, in three ways has itself towards the thing which is outside the soul, right? Did you know that? Three ways? Three. A distinction of three, yeah. So, compared to Thomas' mind, you have a mind of. Oh. For sometimes what the understanding conceives is a likeness of a thing existing outside the soul, just as one conceives about this name man, right, huh? And such a conception of the mind, thought of the mind, has a foundation in the thing immediately, right? Insofar as the thing itself, right, from its conformity to the understanding, makes what is understood to be true, right? And such a name signifying that thing understood. Yeah. Yeah. Sometimes what the name signifies is not a likeness of a thing existing outside the soul, huh, but something that follows upon the way of understanding a thing which is outside the soul. So, and this are the intentions which our understanding introduces as the signification of its name, genus, right? We met genus there in our portrait, right, huh, but it was taken from Aristotle's fifth book of wisdom. Genus is not a likeness of something outside the soul existing, right, huh, but from the fact that the understanding understands animal as something common to many species, right, huh? It attributes to that, if anything understood as common to them, the intention of being a genus, right? And this kind of intention, although the proximate foundation is not in the thing, right, but in their understanding, understanding what's common to them, right, huh? It makes you a play test, right, and you think there's a man himself, right, huh? Nevertheless, there's a remote foundation in the thing itself, right? Once the understanding is not false, which introduces these intentions, these thoughts, huh? And this is the interesting text, and it's similar about all other things which follow upon the way of understanding, as is the abstraction of what? Yeah. It's a very subtle thing, right, huh? And that's why there's a controversy now about that text in the premium to the Nicomachean Ethics. Remember that text we talked about before? Thomas says order is compared to reason in four ways. He's going to distinguish the knowledge of reason by the order that it's a knowledge of, right? So he wants to distinguish order in comparison to reason, right? Now, you know, distinction of four is not understandable, of course, but if you look at what Thomas does there, he says there's one order which is not made by reason but can be known by reason, right? And then there's the order made by reason, right? And the order made by reason he divides into three, right? That's how he gets his four, right? There's the order which reason makes in its own acts and its own thoughts. There's the order which reason makes in the acts of the will and therefore in our actions, right? There's the order which reason makes in exterior matter, right? In a chair or something, right? In wood. And then he says, okay, now, the order which reason makes in its own acts belongs to logic to consider, right? The order which reason makes in the acts of the will belongs to practical philosophy. And the order which reason makes in exterior matter belongs to the mechanical arts, he calls them, right? That's a technical term. But the art of carpentry and the art of cooking and so on, right? Art of making cookies. Now, the art which reasoned, he considers the order which reasoned does not make, right? This is what natural philosophy is about, this order, right? And he says that wisdom also belongs there, right? He doesn't say mathematics. I see some people want to say, well, yeah, but does he divide, you know, looking sciences into these three, you know? And should mathematics be understood in there, you know? He just mentions, you know, you know, well, you've got to be careful, maybe, maybe there's some kind of, what? Some kind of abstraction that's peculiar to mathematics, right? I keep on thinking, I'm worried about that one that is the beginning of number, right? One what? It's something abstract to me, right? Does that one really exist outside my mind? Yeah, yeah, yeah. I would find number very difficult to understand. Yeah, yeah. It seems continuous quantity is easier to understand. Yeah, yeah. And does two or three exist? I think it can use quantities of stuff, you know? I mean, if you eat the whole pie, I'll complain about it. But does two and three really exist outside my mind? I mean, two or three men do it, two or three cherries do it. That's right. I was like, go get me a three. Yeah, yeah. Yeah, yeah, yeah. For real life. Yeah, yeah. You know, there's some kind of, you know. Peculiarness. So, Warren Murray and myself, we both think, you know, that there's something about this text, you know, that makes you got to read that other text carefully. Maybe it's significant that it isn't, you know? You know, it's such a mess, it's like, but it doesn't give us an example, right? It gives natural philosophy, it's about an order, right? And then wisdom is also about that, right? Okay? It's closer to geometry in that sense, it seems closer, yeah. Yeah, yeah. It's not pure blood. Yeah. Even there, there's kind of a separation, there's a kind of an abstraction there, because it's not sensible matter anymore, right? Yeah, yeah. You see? Okay, see, yeah. Yeah. That doesn't even stop outside my mind. Yeah, yeah. But notice what he says there in that second paragraph. And simile est de omnibus allis qui consequentur ex modu intelligendi, sicude as abstractio mathematicorum in gius modi. You see, a certain likeness between, what, logic, right? And, what, mathematics, right? That it's dealing with something that has a remote foundation in things, right? But the proximate foundation, maybe, is in the, what, the mind, you know? And these are the two kinds of abstraction, right? The abstraction that you have in mathematics, which is from sensible matter, right? So I always say to students, I used to take the example of a geometry. 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