Wisdom (Metaphysics 2005)
43. Metaphysics VIII, Reading 3 (Part 2): Definitions and Forms Are Like Numbers
Class of 2005-10-13
https://berquistcourse.com/course/philosophy/7a-wisdom-2005/043-metaphysics-viii-reading-3-part-2-definitions-and-forms-are/

[0:00] So let's begin here now. The second part of the third reading. Now, what Aristotle is going to do here is compare definitions, which are especially the formal thing, with what numbers, right? Okay.

[0:37] It is clear also why, if substances are in some way numbers, he's saying in likeness there, they are so thus and not as some say as units. For a definition is a number in a qualified way. For it is divisible and indivisibles. For definitions are not unlimited, and number is also such. Well, as Thomas explains in the commentary,

[1:07] Aristotle will compare. the third, chapter of the first book about the soul, in the first book about the soul. He compares thoughts and definitions to numbers, right? Okay. Rather than to the continuous. Now the continuous

[1:30] is divisible forever, right? Okay. You recall the argument that, right? But the geometrists assume that Now the geometrists say you can always bisect the line, right? Now, you bisect the line, you get two lines that are shorter than the original line, huh? Now you bisect those lines into two even shorter lines, and does this go on forever or does it ever come to an end?

[2:16] It goes on forever, and you can kind of show that it never comes to an end because the only way to come to an end is would be if when you divide a line you've got either two nothings, in which case the line would be made out of two nothings, and that's obviously absurd, right? Or else you had a line that was two points long, right? Remember that?

[2:44] Okay. Now, if a line was two points long, when you divide that line, you got two points, and they have no length, so you can't divide anymore, right? Okay. But that's assuming that you can put together a line by points, huh? And If you try to put a line together by points, like the mathematician sometimes says, a straight line is composed of an infinity of points. Composed is really the last word for put together.

[3:12] Well, the points have to come up and touch each other, right? And you may recall the argument that we use, Aristotle uses, that two things touch either part of one touches part of the other, right? Or part of one touches the whole of the other, right? Or the whole of one touches the whole of the other; they coincide. Can't draw a circle twice. And perhaps you distinguish a fourth way they could touch at their edge, right?

[3:48] Okay, because the edge is not really a part in the strict sense, right? Well, two points can't touch part touching part because the point has no parts, right? So that eliminates that particular way. The same reason part of one cannot touch the whole of the other because one has no parts, right? So you eliminate that. Can you distinguish between the point and its edge, such as some part of the point within that is not the edge?

[4:22] Well, then you'd be giving it as a little circle, not as a point. So that way is impossible. So the only way the two points could touch would be like when the whole of one touches the whole of the other, which is to say they coincide. Well, two points coincide; they have no more length than one point, which is no length at all. If a hundred or a million or infinity of points touch, the only way they could touch would be to coincide.

[4:52] And if they coincide, they have no more length than one point, which is no length at all. So you can't put together a line from what points, huh? So when you cut a line, you always get two smaller lines, huh? And then they can be divided. Or if you divide them, you divide a line, you always get two smaller lines. So this goes on forever, right? Okay.

[5:18] Aristotle has other arguments that he does where he combines some the infinite divisibility of time, just continuous, and indivisibility of the line. And he takes something we all know about that one body is faster than another. So the faster body, if you represent one of these lines, say, as distance, another one as time. Well, the faster body, at the same time, that the faster body covers this distance, the slower body is going to cover a less distance.

[5:56] Well, that less distance the faster body would have covered in what a certain time. But in that lesser time, the slower body covers a lesser distance. So, just by alternating the fact, right, that the faster body covers the same distance in less time, and the slower body covers less distance in the same time, both time and distance can be seen to be what divisible forever. To act divisible forever, then the faster body would cover only the same distance in the same time, which means it cannot be faster, or vice versa.

[6:33] Right, the slower body covers the same distance in the same time. So, given what the faster and slower body are, then time and distance must be divisible what forever. Now, our thoughts like that, right? Because Euclid will define square, right? So you divide the definition. You divide square into the parts of the definition. It's an equilateral, one right angle quadrilateral, and then he defines quadrilateral, right, as a rectilinear plane figure contained by four lines, and so on.

[7:12] And then you define rectilinear plane figure, and then you define plane figure, and then you define figure. So you keep on dividing, right, the definition, and the part of the definition to a definition, and so on. But does that go on forever?

[7:31] See, if it did, then the division of thoughts of definitions would be like the division of what continuous. It would be divisible forever. But you know how we're showing in logic that not every what genus is a species, that not every genus has a genus above it. That

[8:08] eventually would come to a what the highest genus, right? Now there's a number of ways that they show that, right? But one way that we show that not every genus has a genus is that if that were so, you'd have to know infinity of genera, right? Before you know anything, in which case you would know nothing by definition. But you know, you came to know what a rhombus is or what a square is by definition.

[8:38] But also, you can say that if what every genus had a genus above it, it'd always be a more universal, more universal name than any name, right? But is there something more universal than being or or something?

[8:54] No. Could it be something more universal than something? Could there be? You know. Something more universal than being, huh? Okay. So definitions are not like what lines, huh? But now if you take a number, say, take the number seven, well, you can divide it into two and five, or into three and four, or you can divide it. You divide three into two and one, but then you divide two into two ones.

[9:26] Well, then it stops, right? Unless you're one of these crazy mixed-up modern mathematicians. But the one, which is the beginning of number, is even simpler than the than the point, because the point is what Our position, right? In addition to being simple, but the one at the beginning of the number has no position, right? And of course, if you can divide three into say two one and a halfs then you could divide three points into you know 1 and. a Half points, and 1 and a half points.

[10:03] So numbers are not divisible forever. So Aristotle compares definitions to numbers, and so likeness there. And the definition is tied up with the what form, which completes what the thing is. The definition signifies what the thing is. So that's what he makes a connection, right, between form and numbers through definition. The definition of like numbers not being divisible forever, and the definition expresses what the thing is, and the form is what completes what the thing is.

[10:39] So forms are like what definitions. Definitions are like numbers. They're not divisible what forever. Instead of the same thing is true about what statements.

[10:56] In fact, those things are really proportionate. So there must be something which is known without what definition, and there must be some statements that are known but not known through what other statements. And those are the first statements in there naturally known.

[11:22] Notice another thing about this here is that because of the infinite divisibility of the continuous between any two lines, say of unequal length, right? You can always find a line what in between, right? So between, say, this line and this line here, which is longer, right? There is a what line that is longer than the first but shorter than the second. Between these two, there's always a line.

[11:55] Because there's a divisibility, right? You can say between two unequal lines, there's always a line in between in length, okay? And if you're going from one to the other, you come to that in between before you come to the the greater or lesser line. But is that true about thoughts? You see? Well, isn't that true about numbers, right? Between six and eight, there's a number seven. But now between six and seven, is there a number?

[12:28] Well, it goes crazy, mixed up, models I don't say six and a half, but between six and seven points, there's no point. Now our thoughts like numbers in that respect too. There's

[12:47] what you could say a next thought, right? There's a next bigger number, right? You take any number, say six. There's a next bigger number, which would be seven, right? And a next lesser number, which would be what five. Well, what about thoughts? Is there a next thought?

[13:13] So, take the example of the syllogism that I had there a minute ago. What changes is composed, right? God is not composed, therefore. Yeah. Now, is there any thought between the premises and the conclusion?

[13:34] Yeah, yeah. See, so there's a next thought, right? Okay. And you go to the definition and you say, well, what's a square, Well, it's a quadrilateral, right? And then you add what? Well, maybe it's right-angled, right? You get a rectangle, right? And then you add what? Equilateral, and then you're at square, right? Nothing in between that. Yes. That's it. Okay. So definitions are like what numbers, huh?

[14:11] Okay. That's the first comparison he gives, but especially in so far as the just as a number is not divisible forever, so definitions and thoughts are not divisible forever, right? Okay.

[14:33] Now, in the third paragraph on page five, he has a second comparison. If you add or subtract even one from a number, you have a different what? Number, huh? And he's saying the same thing as what? Definition and the what was to be the what it is. You add or subtract anything, you're on the same number, huh? Now, notice the way of speaking there, huh? And Shakespeare uses it in the definition of reason when he's touching upon man, huh?

[15:08] And he says, "What is a man? If his chief good and market of his time be but to sleep and feed, a beast no more, right? He'd be no more than a what beast, right? Okay. Now, sometimes to give the likeness between definitions there and numbers, I say, "What is a five if it be half of eight?

[15:39] Four, no more, right? " I say. Because half of eight is to four, like the chief good of the beast is to the beast, right? And half of ten would be to five, right? Okay. The chief good of man is to man, huh? But notice they use the word more there, right? Okay. So,

[16:07] if you have a body and you add to body life, you've got a what? Living thing. A plant. See. Now, if you add to body life, and then you add a sense, then you have what? An animal. See. But animal is the sense of a beast, right? Okay. Then you add to body and life and sense, you add reason, and now you have a what? Man. To take away reason, you have nothing but a what?

[16:51] Beast. To take away sense, you have nothing but a what? Plant. Take away life, growth, and so on, you just have a body, a stone, right? So it's it's like a what? Numbers, right? You add and subtract something, you have a different kind of thing, right? Just like you add and subtract one from a number, and you have a different what? Number. Okay. Um.

[17:22] That's very important to see, huh? It's funny, Thomas was was going back to this in the Trees of the Trinity, huh? When he says to, you know, in the article that was the article where the Holy Spirit proceeds from the Son as well as from the Father. There has to be some order between them, right? You have to have order in immaterial things because you're like numbers.

[17:54] Interesting. You. You come back to the text for that. So that's the second comparison. Now the third comparison here to again two definitions here. And is necessary that a number be one by something which these thinkers cannot state, by what it is one if it is one. Now some people think of a number as just a collection of what? Ones. In which case a number would be like a heap or a pile, right?

[18:29] It wouldn't be one thing. Okay. What is it that makes a number to be one? We've got to see, and not as being simply a collection of ones, but you have to see each one as making actual what the what previous number was able to be. Okay. So six is able to be seven by the addition of one. So one makes actual the ability six to be seven, and that's why you have a unity, and it's like the unity of what?

[19:07] Act and ability, matter and form, and that's the way a definition is what is one, because the difference, strictly speaking, determines the ability of the genus. huh? Remember the famous difficulty there of John Locke that we maybe mentioned before, and John Locke was having some difficulty

[19:38] with the understanding of general idea, right? Okay, we've talked about that before, and he's trying to understand the general idea of a triangle, right? And he's a little... Confused as to whether it's a thought or a what image, right? Okay. He's thinking of it kind of as being an image, huh? But now, any triangle you imagine will be what equilateral or isosceles or scaling, but it won't be all of these, right?

[20:18] Now, what's the definition of triangle in general? What's a plane figure contained by three straight lines, something like that? And are those three lines equal, or just two of them, or none of them? Well, if you take just one of these, then you seem to exclude the other ones, right?

[20:41] So it seems that it's all none of these, he says. Okay. Well, then Berkeley, the next British empiricist, says that's a very silly thing. It can't be all none of them. Therefore, there are no general ideas. Okay. So they're all messed up. These empiricists, huh? They can't distinguish between an image and a like thought, huh? Because any triangle you imagine will be one or the other, but you understand what they're having in common, even aside from the differences.

[21:15] But now, going back to the original question, there, you see, what's the definition of triangle in general? Well, it's something like a plane figure contained by three straight lines, huh? That's even

[21:37] a reasonable question to say: Are those three straight lines equal, or just two of them, or none of them, right? See. But in the definition of triangle in general here, what do you say?

[21:51] It's indifferent to it; it doesn't determine that. Well, you could say it's all of them in ability, none of them in act. See. But Locke can't see that distinction, so he just says they're all none. Well, unless it's all of them in one way and none of them in another way, you've got a contradiction, and Berkeley can't see his way out of the contradiction either, so he says there are no general ideas.

[22:23] You know, it's like talking about a square circle, right? You know, you can't obviously, you know, who knows? He's a square circle, right? So there's no triangle that is all of these things and none of them. Okay.

[22:41] So when you add then, right, equilateral to triangle, right, you are determining what is already. In the definition of triangle, inability. It's not to bring in something entirely what other.

[23:10] Not to bring in another individual, right? But equilateral is determining those three straight lines, making actual something they're able to be. Okay. So he said he said these three straight lines equal or unequal. What's able to be both, right? And but none of them enact. So what is able to be, or one thing it's able to be, is made actual by the what difference, huh? And that's like matter again in form, isn't it?

[23:48] So the piece of clay is able to be a sphere or a cube or a pyramid, right? But its ability to be one another of these, one of these abilities is made actual by the form it receives from my hands or something.

[24:10] So it's not like you're bringing in two things, right? Not like when my hand picks up the chalk, right? Well, my hand is not actualizing some ability of the chalk to be something, is it? No, you have kind of a collection of chalk in my hand now, and or it's like a heap or a pile, you know, where you got a a book here and I got a glass here and I got a piece of chalk and I got something else, right?

[24:36] You're bringing one thing to another thing. No, one is the what actuality of the other, right? So what makes seven to be seven? It's the seventh one that makes it to be seven, right? And strictly speaking, it's six that is able to be seven, but it becomes actually seven by the addition of one. So that's the connection he's seeing there, right? Of ability and what act matter and form definitions are like numbers in that sense, huh?

[25:14] Okay. Just like an animal is able to be a man by the addition of what reason, and a living body is able to be an animal by the addition of what A sense, huh? And a body is able to be you know a living body or a plant by the addition of some kind of life, huh? Okay.

[25:36] So six is able to be seven by addition of one, and seven is able to be eight. So it's like act and ability, matter and form.

[25:47] So that's another subtle comparison he's making between the two and definitions like numbers, huh? But you can say that the positive thing is that the just as. The number, if it's really one thing, is not simply a collection of ones. So the definition of it is really a definition, right? It's not just a collection of names, or not just a collection of different things. But one is the act of the other's ability,

[26:21] okay? Just like form is the act. Now Aristotle mentions, you know, in the study of the soul, there people will say, "What makes a soul and body one?" Right? And he says, in a way, once you realize there is matter and form, as act and ability, then it's kind of stupid to ask what unites them, right? Because one is the actuality of the other's ability, so you don't need a glue or a nail or a rope or something to hold them together.

[26:52] But say the different parts of the chair, are one is not the act of the other's ability, right? So you got to screw them together, and nail them together, or something, or glue them together, or something, right? Need some third thing to unite them. But matter and form, you don't need a third thing to unite them because form is the actuality of the matter's ability. So the clay, say, in the shape of the clay, you don't need a third thing, you know, to screw them together.

[27:26] You know, I don't usually screw together the shape of the clay and the clay. I don't know what you guys do, but here, I suppose it's the same as us, right? You see, and

[27:40] the same way you don't bundle together the ones and the number, right? But the last one actualizes the previous number, right? Makes actual the next number. He has something like the definition, right? But the difference actualizes the genus, right? Makes actual what is there in ability.

[28:03] So, in a sense, you have to understand the genus first before you can see the differences of what contract that, right? Okay. Well, because the differences are determining something intrinsic to the genus, the ability that's already intrinsic to the nature of the genus. See that?

[28:30] And I got my number right. Got the number. It's a good, good old expression, huh? I may not be exactly that to get the nature of the thing is to get its number, right? But there's a likeness, right, between getting the number of a thing and getting the nature of a thing. And this is the third what comparison he has, huh? Okay.

[29:04] That number and definition are not divisible forever. Was the first one, right? And the second one was that you add or subtract something, however small, from a number, you have a different number, right? And so you add or subtract something, definition, and you have a different what? Definition, right? Different nature. And so Thomas was often back to this: the natures of things are like what numbers. We say you go from the stone to the plant to the animal to man, right?

[29:39] It's like going from one to two to three to four, and then that the parts are as built in act, right? In the definition, like with matter and form, and this is necessary if the number is going to be one thing. There'd not be a collection of units, but that the unit. That completes the number, complete it right Making actual what it's able to be, right? Okay

[30:10] and It's interesting how how, you know, when we there's one way of naming things kind of equivocally that I've spoken about, where sometimes the one that adds something noteworthy gets a new name, and the other one keeps. Now, I like the example. You know, how many fingers do I have in one hand? And you hold it up like this. You got five fingers, I'll say, right? And somebody just say, "No, you have."

[30:41] And they'll say, "You got four fingers and a what? Thumb." So somehow the thumb stands out. They call it the opposable thumb, right? Which means it has some distinction. So it's something special. So these four keep the name, and this one gets the what? Yeah. Okay. Now, maybe a more clear example of that. Sometimes the word "animal" is kept as a name for the beast, and so we distinguish man against the animals, right?

[31:17] And man gets no name because he adds something, right? Okay. What's a little bit like, you know, let's talk about five and five plus one. Well, it's going to keep the name five. The one that is just five, right? But the one that is five plus one, it's got something more, right? Something additional. Give it a new name which happens to be a new name, which happens to be six, right?

[31:41] Okay. I told you how my mother didn't like me calling man an animal, and I didn't like that way of speaking. And I said, "Well, Mama, I don't mean he's just an animal." I say, "Okay, that's better," she said.

[32:00] But it'd be very much like saying, you know, to call man an animal might seem a bit like calling six a five right five, right? You know, and I said, "Well, it's not just a five; it's a five plus one." Okay, that's better. You know, you see, see the likeness there.

[32:24] Okay. Now the last comparison next to the comparison of definitions and so on. He says, "And as number does not have the more and the less, neither does substance according to the form. But if any does, it is one with what matter, right? " And one time in in class, quoting the one of our founding documents, we hold these truths to be self-evident that all men are created equal, right?

[33:01] I say now, if I got up in slave owning Athens or slave owning Rome and said, "We hold these truths to be self evident, that all men are created equal," what what do you mean? Some are free, some are noble born, some are slaves, huh? But even apart from those, you might say that some men are born more healthy than others, right? Some men are born stronger, right?

[33:26] Some babies are kind of weak. Some are born more handsome, better looking, right? Some are born more intelligent, right? Okay, some are born more courageous. I think some more maybe temperate, some more mild, some more irascible. Right? What do you mean we're all born equal? What the heck is this meaning, right? You know, it seems like a fiction, there maybe, huh?

[33:59] But is one man more a man than another? Now, if I mention courage, yeah, but that's not what a man is, right? That's that's a that's something added, that's a a virtue or a disposition. But is one man more what a man is? Or is one dog? huh? One cat would be more beautiful than the cat, right? And I went to get a cat there for my daughter there, so let's get a male cat.

[34:31] I would be by the way kittens, you know. All the male kittens are kind of ugly. And if he's a cat, he's all good looking, you know. And it's kind of funny because even I remember the trash men coming, you know, but didn't have any too great a aesthetic sense. as they say But everybody was, you know, kind of you know admiration. What a beautiful cat he said, you know.

[34:54] Now I was, you know, I say, see, it's not just not just enough, you know. Tabitha was was beautiful, so but was Tabitha more a cat than the other cat? We used to give her the title of the Queen of Bumblebee Circle, you know, the neighbor there, you know, neighborhood because she was obviously superior cat in the whole neighborhood. But

[35:18] Tabitha was not more a cat than the other cats, huh? And that's the what a thing is, right? If you add or subtract anything, you'd have a different kind of thing, right? Wouldn't be a cat anymore, right? Okay.

[35:37] So he says about the generation corruption then of the four said substances, how it can happen, how impossible. This kind of like a little summary, and it was touched upon a little bit here.

[35:53] We saw that right in the first group of comparison, right, where form is not generated and corrupted as such, but the composite. And about the reduction of things' number, right? Bring them back to a limitless number. Let it be determined as far as they say. So that's kind of an epilogue to the to chapter three. But you'll find as you read through Thomas and even theology, he's always coming back to this thing when he's trying to explain things.

[36:25] You're talking about, you know, how God can what by knowing Himself know everything else, right? Well, I mean, Thomas will go back to comparison numbers, right? If ten really knew itself, right, would it know nine and eight and seven and six and five and four, right? Because in a sense, they're contained, right, in the perfection of ten, right? But they're something less, huh?

[36:58] So, but we kind of know animals, huh? Because they're like what one unit less than us—Without reason, right? But they have something, right? Like what we have, huh? The plants, you know a little bit less because they're less like us. I feel closer to the plants and to the stones myself.

[37:23] And people want to talk to the plants, and they they they help them to grow. And you know, they there was this spat of experiments, you know, where they played Mozart and rock and roll, and the plants grew much better with the Mozart and grew straight, and so rock and roll together get all twisted. I don't know how how scientific these experiments were, but but the results at least kind of pleased me, you know.

[37:57] Tell the students what this music is doing to them, you know, and what the good things it do for them, huh? So I guess we'll have to stop there, huh? Because we just went into through chapter. uh do the third chapter on this.