De Anima (On the Soul)
149. Aquinas Summa I q.86 a.2–3: Can Our Mind Know Infinitely Many Things? Potential vs. Actual Infinity, Meno's Paradox, and Knowing Contingent Things
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[0:02] So we've got two three articles today. Maybe we can do the next three, you know, next time two, three, and four, right? Okay. Say a little prayer, For our help In the, name of the Father, the Son, the Holy Spirit, Amen. God our enlightenment, guardian angels, strengthen the lights of our minds, order and illumine our images, and arouse us to consider more correctly. Saint Thomas Aquinas, angelic doctor, pray for us and help us to understand all that you've written.

[0:43] We're up to article two, I guess, huh? Question eighty-six. To the second one proceeds thus: It seems that our understanding is able to know an infinity of things. For God exceeds all infinities,

[1:06] but our understanding is able to know God, as has been said above. Therefore, much more is it able to know all other infinities. Moreover, our understanding is naturally apt to know genera and species, but there are infinities of species of some genera. Doctor, second objection, right? Yes, yes, Doctor. Yeah. For example, of numbers, right? And of proportions, ratios, and figures, three-sided, four-sided, five-sided, all the numbers, right?

[1:51] Therefore, understanding is able to know infinities, or an infinite multitude of things. Moreover, if one body does not impede another from existing in one and the same place, nothing prevents infinity of bodies being in one place. But one understandable form does not prevent another from existing at the same time in the same understanding. For it is possible to understand many things habitually or in habit. Therefore, nothing prevents understanding from having in habit a knowledge of an infinity of things.

[2:44] Moreover, our understanding, since it is not a power of a material body, as has been said above, seems to be in some way an infinite what power, huh? But infinite power is able over an infinity of things. It's capable over an infinity of things. Therefore, our, understanding is able to know an infinity of things. This great fragment of Anaxagoras: the first thing he says about the mind is that it's infinite.

[3:20] And Heraclitus had said this before him, right? Of course, there is some signs that there's something infinite about the mind. It's always what learning something new, right? Doesn't seem to have any limit to what it can come to know, right? And it's always inventing something new, right?

[3:43] And if you look inwardly, if it knows universal, universal. Is set of an infinity of things. So, when I say that no odd number is even, I'm making a statement in a way about an infinity of things. But you have to specify in what way it's knowing that infinity. I mentioned the Feuerbach there, the guy between Hegel and Karl Marx, huh? You know, he quotes some theologian saying the infinite is God, and then his minor premise: man's mind is infinite.

[4:20] Therefore, man's mind is what God, right? Infinite is a word that has what many meanings, huh? And it's interesting Aristotle in the first book of the natural hearing of the so-called Physics, he criticizes um, beliefs, for mixing up two meanings of the word infinite. What you know has no beginning and no end. In time, with what has no what

[4:50] whose size has no limits, right? He's confusing those two senses. So that's why we went back to the beginning of philosophy, and way down here in the 19th century, we have Feuerbach making the same kind of mistake, huh? Fallacy of equivocation, mixing up different senses but of the same word, huh? Infinite.

[5:16] Now, against this is what is said in the first book of the physics that the infinite, in so far as it is infinite, is what unknown.

[5:32] I answer: It should be said that since a power or ability is proportional to its object, it is necessary in this way that the understanding has itself towards the infinite as it has itself towards its object. Of course, we find out that the subject is the what it is of a material thing.

[6:00] In material things, however, is not found the infinite in act, but only in what ability, according as one succeeds to another, as is said in the third book of the physics.

[6:19] And therefore, in the in our understanding is found the infinite in ability in taking one thing after another, but never does our intellect understand so many things, but that it could understand what more, right? So it's like when you go through numbers, right? I can go on forever with the numbers, right? But I can never

[6:46] mention so many numbers; I can't mention still more. Okay. When the children are very young, you know, Daddy, can you count to a hundred? Yeah. Wow. A little bit older, you know, Daddy, can you count to a thousand? Yeah. Wow. After a while, you realize you can always count to something larger.

[7:19] But our understanding, huh, in act or in habit, is not able to know an infinity of things. In act is not because understanding can is not able at the same time, right, to know except through the one form that it knows, huh? Okay. Remember how we're comparing the mind was like what to the form which it understands, a little bit like the. Wax or the clay is to the shape it has.

[7:53] It can only have one shape at a time, huh? And so the clay, that's the shape of a sphere, can't be in the shape of a cube at the same time. You can put it in the shape of a cube, but then it ceases to be in the shape of a sphere. Of a

[8:07] sphere. So you don't have two forms of the same kind in pure act anyway, in complete act in the mind. In act, he says, not, however, read that sense again, maybe, because our understanding is not able together at the same time in act to know more than what it knows through one species, huh?

[8:35] But the infinite does not have one what species or one form. Otherwise, it would have the what definition of a whole and what perfect, huh?

[8:52] Okay. This, you know, Thomas is basing himself a lot here upon Aristotle's consideration of the infinite in the physics, huh? So, for example, when we say that a straight line is divisible forever, huh? What does that mean? That you can actually divide it, actually, as much as it can be divided. So, what it means?

[9:26] We've gone through that before. Just, just be cautioned here. If you have a half and half equal line, you have a theorem in geometry that you can bisect the line, right? Now, if you bisect the line and get two shorter lines, then since they are length, they can be bisected too. Now, as you bisect the line, you're getting smaller and smaller lines. But do you ever get a line that you don't cut?

[10:00] You bisect it into two shorter lines. No. Well, if you don't get two shorter lines, there's only two possibilities. You cut a line into two nothings. But something can't be made out of nothing. So a line couldn't be composed of two nothings, right?

[10:24] Now the other alternative, someone says, well, you cut a line and you get two points, and the points cannot be divided, huh? Okay. But that assumes that there could be a straight line put together from two points, huh? And remember the either-or syllogism we use to show that you can't make a line by putting together points. Remember that. You know.

[10:54] One way that two things can touch is for part to touch part, right? One is for part of one to touch the whole of the other. One is for the whole to touch the whole, or to what? Coincide, right? And then you could speak of the edge touching at the edge, because the edge is not strictly speaking a part, but the limit of the thing. Well, the point has no parts, right?

[11:24] So these two ways are impossible for points to touch, right? And is there any distance inside the point there? So you could distinguish between the point and the limit or the edge of it? No. The only way two points can touch is like for the whole of one touches the whole of the other. The only way two points could touch, therefore, would be to coincide. And if two points coincide, you have no more length than one point.

[11:57] There's no length at all. And if ten or a hundred or a million or infinity of points, like the mathematicians say sometimes, were to touch, the only way they could touch would be to coincide. If they coincide, they have no more length than one point. And therefore, you can't put together points to get a what? A line, okay? So the continuous is what? Divisible forever, huh? But notice, what way does that infinity exist, right?

[12:31] In the way that you could always go on and divide further, right? But you never complete the divisibility of the what? Line, huh? So the divisibility of the line is never brought to what? Complete act. It's never perfect or what? A whole, as he says, huh? But it always consists in something what? Further, right? Okay. Now, as Alexander has pointed out, as you divide the line, right?

[13:09] You never come to a shortest line. But the number of lines keeps on increasing, huh? So every time I divide this, I got two, and now I got three, and now I got four, and so on. And so if you can divide forever the straight line, and this is the way they define the continuous, what is divisible forever,

[13:36] then numbers can increase forever. But it's not like you ever what? Complete the dividing of the line. I can divide and subdivide and subdivide and so on, but it's always

[13:55] be more reduced to act, but never completely made actual. So this is the potential. Yeah. See, never you should never actually have an infinity of lines, do you? We define the continuous as that which is divisible forever. Okay. But you don't actually have the what? The infinite, the infinity of lines there, right? You never complete that thing.

[14:30] And therefore, is not able to be understood except by taking part after part, as from its definition is clear in the third book of the physics. For the infinite is that are those taking its quantity right? There is always something further, right? Extra to be taken, huh? Okay.

[14:53] So you could say this is always divisible further, right? Huh? And therefore, also the numbers—they're always what? A greater number is always able, right, to be taken, huh? Okay.

[15:09] And therefore, the infinite is not able to be known in act, right? Unless all its parts were what? Numbered, right? Which is what? Impossible, huh? Incidentally, in the Oxford translation of Aristotle, the English will read, you know, we'd be talking about an infinite number, huh? My old teacher Kasurik who forced me to learn some Greek. He says Aristotle wouldn't speak of an infinite number. That's a contradiction in terms, huh?

[15:43] Look it up in the Greek, Duane, huh? And sure enough, of course, the word "number" is not there. The false translator, right, has put in a word that doesn't isn't in the Greek, huh? Okay.

[16:05] If something can be numbered, huh? Then it's what? Not infinite. That's why Shakespeare will couple them, right? Countless and infinite, he'll say. If there's a number, there's a multitude. Then that multitude is what? Limited, huh?

[16:27] Yeah. I was thinking an interesting objection, you know, in the treatise on the Trinity there from, you know, St. Thomas objects these things, and whether there's only three persons in in the Trinity, and the objection takes what is said in the Athanasian Creed that God is what? Immensus, right? Not measurable, huh? Okay, he's infinite. And well, but here he'd be limited to just three persons. That's kind of interesting objection, huh?

[17:06] Because you know there's all kinds of things you have to understand about numbers there and what numbers mean in the Trinity, you know, huh? But I just got reminded of it here, right? Because of the fact that what can be numbered would seem to be what? Limited, right? And how can the infinite be numbered, huh? Interesting objection, huh?

[17:33] And for the same reason, we are not able to what? Understand the infinite inhabit. For in us, habitual knowledge is caused from what? Some actual consideration. For by understanding something, we become what knowing, right? So it's by understanding the what demonstrations of Euclid that I acquire this habitual knowledge that I now have of what geometry, huh? Okay. So if I can't actually understand one of his demonstrations, I can't have an habitual knowledge of that theorem, can I?

[18:12] And so if I can't actually understand the infinite, I can't have an habitual knowledge of the infinite. Hence, we are not able to have the habit of infinite things by distinct knowledge, unless we considered all the what infinites by numbering them according to succession of knowledge, which is impossible. And therefore, neither in act nor in habit is our understanding able to know infinity of things, but it's so only in what potency, right?

[18:54] Okay. So you compare our understanding to God, and God's mind is what not limited at all to what He knows, huh? But our mind is unlimited in the sense that it's always able to know something more. So, in a sense, the infinity of our mind corresponds to the fact that our mind is always incomplete in its what knowledge, right? So it's a gross confusion of two kinds of infinity that Feuerbach is making, right?

[19:31] This is in Feuerbach's little perverse work called "The Essence of Christianity," huh? Where he maintains, you know, that God did not become man, but man himself is God.

[19:46] And that's one of the arguments he uses, huh? But Karl Marx and Engels read Feuerbach as young men and were convinced by it. You say it takes a little bit of pride to do that.

[19:58] I remember in one class there when I was in college, there someone was describing the French positivists there who were trying to substitute a religion of humanity for for Christian religion, and of course they have days set aside for the worship of humanity. Well, I just broke out laughing in class. Everybody else looking at me, this was a straight face. I just just struck me as so absurd to set aside days for the worship of humanity.

[20:32] I mean, if you must seriously propose this sort of nonsense, but when you look at Karl Marx's in the preface to his doctoral thesis, he says explicitly that the human mind is the highest divinity. So. Okay.

[20:55] Now let's look at the reply to the first objection, which is saying, we know that our mind is capable of knowing God, especially the beatific vision. God is infinite.

[21:11] Of course, God is infinite in a different sense. To the first, therefore, it should be said that has been said above. God is said to be infinite as a form which is not limited by some what matter,

[21:31] but in material things something is said to be infinite by the privation or lack of formal termination. So we speak of a what a line is limited by points. So if it had no endpoints, then the line would be what infinite, right? He would lack something, right? Okay. But when you say infinite of God, it's a mere negation, right? You're negating what the imperfection that would come to from him being in matter.

[22:03] Okay. When Aristotle is talking about perfect, the word perfect there in the fifth book of wisdom, and he distinguishes between the sense in which God is perfect, and the sense in which Homer is perfect, right? Homer is perfect in his kind. Homer is the perfect poet. His plots, his characters, his diction—they're all there, right? Mozart is a perfect musician, right? But in some what genus, some particular kind,

[22:47] why God is perfect, right? Because he's lacking nothing. Okay. It's not just that he's lacking nothing of his kind, some particular kind of thing. He's not limited to any particular kind of thing. You see that? That's the difference.

[23:19] So God is perfect in the sense of a form that is not what limited by matter, not limited by passive potency. While material things to be infinite means to be lacking in form, huh? And through form that something is known. And because form by itself is known, matter without form is unknown. Hence, it is that the material infinite is in itself. Unknown, but the formal infinite, which is God, is in itself known, and only unknown to us on account of the defect of our what understanding, which in the present state of this life has a natural aptitude for knowing what material things,

[24:15] and therefore in in the present we not able to know God except by material effects. So the first proof of existence of God is from motion of of matter, right? You reason from motion to the dependence of motion upon a mover, right? And from the dependence of moved movers upon an unmover. That's the way we reason to God.

[24:40] But in the future, this defect of our understanding will be taken away by glory. And then we'll be able to see God Himself in His very what essence or nature, but without completely what comprehending Him. We never know God as much as He's knowable, but as Saint John tells us in First Epistle, Chapter Three or no, excuse me, First Epistle, Chapter Three, Verse Two, right? We shall see Him as He is, right?

[25:13] Okay, the beatific vision. We'll be talking about that in the later questions. We talk about how we know things above us, right? So, in that first objection, there you're confusing the way in which God is infinite with the way in which material things are said to be infinite. Huh? Material things are said to be infinite because they're never what complete, like the division of the what?

[25:42] Yeah, the ability of the line to be divided is never fully made actual. It's never divided as much as it's divisible, right? See? Why God is is infinite in the sense that His perfection, right, is not limited anyway. Okay, so so His infinity is based upon form. Infinity of matter is a quite different meaning, and it's based upon the lack of what form, right?

[26:17] Just like we said, a piece of clay. You know, take another example, different from this one, but you might say that a piece of clay can be molded into how many shapes? Maybe there's an infinity of different shapes you can mold the clay into, but it can only be put into one of these shapes at a time, right? So when can you shape the clay with all the shapes it can have?

[26:40] You never can. You can give one shape and then give another shape and another shape, but exists only in succession, right? So it's an ability that can never be actualized, right? Completely or all at once. When it has one shape, it's able to have some other shape, right? But if it acquires that one, it doesn't keep the one it has; it loses the one it has, huh?

[27:08] Okay. So as a kind of infinity tied to matter, that is based upon a lack of actuality, right? A lack of form, huh? And through form, that something is actually knowable.

[27:28] Okay. But God's infinity is based upon His perfection, upon His being perfect form, huh? The second objection, talking about some genera being having infinite species.

[27:50] To the second, it should be said that our understanding is apt to know forms by abstraction from images, and therefore those species of numbers and of figures which one has not imagined, it is not able to know either in act or in habit, except perhaps in what? In general, right? Okay. So I can know what an odd number is, right? And that set of an infinity of things, right?

[28:22] But I can't know, right? All the odd numbers in particular, can I? And think of three, five, seven, nine, but I could go on all day thinking about odd numbers and never think about all of them. Okay.

[28:39] Except perhaps in genus or in general, right? And the universal principles, but that's to know in a way in what potency and in a confused or indistinct way. So you could say when I know what an odd number is, in some way I know three, five, seven, nine, all the rest, but an ability, right? And not distinctly one from the other.

[29:10] On the third objection here. To the third, it should be said that if two bodies were in the same in one place or many, it would not be necessary that they successively success success successively enter into the place, so that through this entering into would be numbered a what?

[29:45] The succession of entering into would be numbered things located, but the understandable forms enter our understandings successively because we cannot together in act understand many things, and therefore it is necessary that they be numbered and not infinite the forms that are understanding. So we gradually what? Understand something about the triangle, right? And then I go on to understand something about the parallelogram, right? Then I get to book

[30:23] four, I might be understanding something about the hexagon and the pentagon, right? So the forms in my mind of triangle, of parallelogram, of pentagon, and of hexagon are coming in successively into my mind, right? So do ever get infinity of them in there?

[30:45] Infinity of different figures in my mind. I've thought of no, no, no. Successively, okay. Now the fourth one is talking about the mind being so infinite.

[31:05] To the fourth, it should be said that as understanding is infinite in its power, so it knows the infinite. For its infinite power, according as it is not what limited by some what

[31:23] material body, right, or bodily matter, right, and it is knowing the universal which is abstracted from the individual matter, and consequently is not limited to some individual, but as far as itself is concerned, it extends itself to what infinite individuals, huh? Okay. We can see of the senses, right? Because the eye is limited to colors, and the ear to sound, and so on, right? But my reason can what know color and sound and smell and all these things, right?

[32:03] It's not limited in that sense to what a subject can be, but its ability to know is never fully what actualized, right? At least in this life.

[32:20] Of course, if you know that you can't get something out of nothing, something is completely universal, right? And if everything is something, and knowing what something is, you know everything, right? But in a what very general way, And a very what indistinct way, right? This is something. Say, okay.

[32:46] I know that everything is something, don't you? Sure. And besides everything, there's nothing, right? So I know everything. So my students would pick up in this, you know, I'm going to go home and tell my dad I know everything now. See, but it goes back to this distinction that that we made in in logic, huh? You know, in fact, it's one of the the second kind of mistake outside of language is the mistake of what simply and not simply, but in some limited way.

[33:20] This is a mistake that Meno's making in the dialogue. Meno saying you can't investigate what you don't know. You can't direct your thoughts to what you don't know.

[33:40] Was that true? You were paying all these kinds of people to find the. Cause of cancer or some other disease, right? So, do they know what they're looking for?

[33:58] Secundum quid you have to say, right? Yeah, when there's been you know they discover a body, somebody's been murdered, right? Do police know who they're looking for? They begin the investigation. Well, they don't know who murdered him, right? But they know they're looking for the murderer of so and so. So, in some way, they know who they're looking for, right? Right. Yeah. And that can guide them a bit, right?

[34:31] Did he have any enemies? Anybody threatening him? Anybody stand to profit from his death, etc., etc. Right? You know. And that helps them to kind of what look at some people rather than other people, huh? You see. So you can know what you don't know.

[34:54] Sometimes I play with the students a little bit there with the question, right? And restate Meno's objection. When a student asks a question, does he know what he's looking for? Does he know he wants to know? See? Well, if, the student says he knows what he wants to know. Why bother asking me, right? He already knows what he wants to know, right? If he doesn't know what he wants to know, how can I help him?

[35:22] Does he know what he's asking, right? See. So in order to ask an intelligent question, you have to, in some way, know what you don't know.

[35:34] You see, and that's a good, interesting thing, right? But you see, if you can't see that distinction between what is so simply without qualification, what is so in this limited and perfect way, you couldn't answer his objection.

[35:58] The simple example I always use in class to illustrate the solution to Meno's thing is: I say, "How many students are in class today?" of a fairly large class. Well, I don't know the number of students in class, right? But I know exactly how to get to what I don't know. And so I count the class, and let's say end up with twenty-eight. I count the class.

[36:23] How did I direct myself to twenty-eight? Did I know I was looking for twenty-eight? No. But yet I directed myself to twenty-eight with the greatest of ease, right?

[36:37] Well, how did I direct myself to twenty-eight if I didn't know where I was going? See. Well, I knew twenty-eight in a way already, didn't I? Because twenty-eight is a number of students in class, and I knew I was looking for the number of students in class. And knowing that I was looking for the number of students in class was enough to tell me I should count.

[37:05] And when I counted, I came to know simply and without qualification twenty-eight, right? Which I only knew in kind of a general way to begin with. Do you see that? Yeah.

[37:22] So you could say, say, couldn't count, right? In some imperfect way, man, in knowing what an odd number is, knows inability, you might say, in a confused way, in a general way, an infinity of things, huh? Okay.

[37:44] That's important, you know. in Aristotle's taking up the wise man in the beginning of of wisdom there, and he says wisdom is the knowledge of all things. Right, the wise man knows all things, so far as is possible for man, right? And he can't know all things in particular, but maybe he can know even infinity of things in some way, right? Why God would know all things simply without qualification.

[38:13] Okay, those are two things that our friend Alfarabi is confusing, right? Okay, God is some pigeon there, infinity to his son, but our mind is what, in some way, in some imperfect way, infinite, huh?

[38:34] Its knowledge, huh? Okay, so now we go on to Article Three. To the third one proceeds thus. Incidentally, these things he's asking about our mind. You know, when you take up the mind of God, you ask these same things about God's mind, right? Does God know infinity of things? Does God know contingent things, right? Does God know the future, right? You know, of course, the answer would be somewhat different for God and for us, to say the least, huh?

[39:13] Okay. To the third one proceeds thus. It seems that our understanding does not know contingent things. Now, contingent things are things that can be and what, not be, as opposed to necessary things, right? Which cannot, not, be. and as opposed to impossible things, which cannot be. But contingent things are things that can be and not be.

[39:44] The first objection is taken from the sixth book of the Nicomachean Ethics, where Aristotle talks about these three virtues of looking, reason, understanding, and wisdom, and science or reason out knowledge. Aristotle points out that they are not about contingent things, right? But they are about necessary things. So, if you've done a little bit of scientia in Euclid, right? You know these things are what necessary, right? When straight lines intersect, those opposite angles are necessarily equal.

[40:20] They must be equal. Okay. Moreover, as is said in the fourth book of Natural Hearing, the so-called physics, those things which sometimes are and sometimes are not. Are measured by time, right? But our understanding abstracts from time, just as from the other conditions of matter.

[40:53] Since, therefore, it is proper, or it belongs to contingent things, to sometimes be and sometimes not be, it seems that contingent things are not known by what? Our understanding, huh?

[41:12] Now, but against this, every science is in the understanding, but some sciences are about contingent things, as the moral sciences, which are about human acts subject to what? Free judgment, huh? Subject to free will. And also, natural sciences, as far as that part of them at least that treats of things that are generated and corrupted. Therefore, understanding has knowledge of what? Contingent things, huh?

[41:50] So Thomas makes a distinction here. I answer: It should be said that contingent things can be considered in two ways. In one way, as they are what? Contingent. In another way, as in them is found something of what? Necessity, huh? For nothing is so contingent, but that in it there is something necessary.

[42:23] Just as this, for example, that Socrates is what? Running, right? In itself, is something contingent, right? He can run and not run, right? But the relation of running to motion is necessary. For it is necessary for Socrates to move if he runs. Right. Okay.

[42:55] Now each thing is contingent because of matter, because the contingent is what is able to be and what not be, right? And this ability to be and not be pertains to matter. With necessity follows the what? Notion of form, because those things which follow upon form of necessity are yet. That's why geometry, right? You have complete necessity in geometry because there is no matter there.

[43:30] You have all these interesting things, right? Triangle: the interior angles must be equal to two right angles, huh? Exterior angle of a triangle is always equal to interior opposite. Must be. One of the things that we show, right? But because you have no matter there.

[43:53] Matter, however, is the principle of what? Individuation, huh? We've spoken about that before, right? Okay. Why can you have many individual chairs of the same kind exactly here in this desk or this table? I have enough metal. You have enough metal, right? Why can you have many window panes there? Could be exactly the same kind because you have enough glass, right? So some kind of matter like metal or glass is subject to quantity, huh?

[44:26] That can be divided up, right? So if the woman rolls out the dough and she can make how many cookies? Well, the dough as what extended, and therefore divisible. They can give you many individual what's material cook I got cookies everyone you see. Immaterial cookies? No, we want material cookies'cause this is what he illustrated. The real thing we want here.

[44:56] My grandmother used to be a very great cookie maker, I think that's where I got my first love of cookies. She'd come down after my birthday, maybe come visit or something, and she'd bring some present. But I didn't remember what the present was, but but she'd bring a box of homemade cookies, and they seemed to be the best ones that I ever had I mean'cause you know, I st I still insist that you know her, her sugar cookies and her Scotch bread cookies and the rest of them were the best of their kind, you know, that they were they never equaled in my subsequent life, you know, the ones I'm able to equal them, my grandmother.

[45:27] So there's both our and your grandmother. Yeah, yeah, yeah. You know, her model, her model was I aim to please. What what a motto for a woman, isn't that marvelous?