Platonism – Lecture 34

Table of Contents

• Platonism, concepts, and disputes
• Disputes about reality, the Sages, and the myth of “there are no disputes about reality”
• Plato and Aristotle on ideas versus categories
• The example of convex shapes: definition, proof, and the gap between intuition and formalization
• Intuition, language, and the diagnostics of definitions
• Bertrand Russell, direct reference, and description
• Definition by extension versus definition by content, and the application to a democratic state
• “A correct definition” according to Plato versus Aristotle

Summary

General Overview

The text argues that disputes over concepts and changes in the definitions of concepts are an indication of a Platonic outlook, because they assume that there is a “concept” that exists beyond the collection of characteristics and linguistic agreements. It distinguishes between a dispute about facts or intentions and a dispute about a concept, and insists that in a dispute about reality, one side is right and the other is wrong, even if sometimes the Talmud resolves it by saying “both are right” in the sense that each one grasped only a partial aspect. It illustrates this through a mathematical statement about the intersection of convex shapes, showing how a formal definition can make a proof trivial, yet still does not “prove” that the definition fully overlaps with ordinary intuition, and therefore the proof is about a mathematical correlate and not necessarily about the world. It connects this to Bertrand Russell’s distinction between direct reference and reference by description, and concludes that according to Plato there are correct and incorrect definitions in relation to the idea, whereas according to an Aristotelian-conventionalist view there is no “correct/incorrect” definition, only consistency and usefulness. Still, our actual feeling is that there are definitions that are simply wrong because they fail to capture the intuitive classification.

Platonism, Concepts, and Disputes

The text states that a dispute over the definition of a concept, and changes in a concept over time, point to Platonism, because they assume there is a conceptual entity that continues to be “the same concept” even when the definition changes. It argues that if concepts are only a communal, conventionalist agreement to group a set of characteristics under a term, then there is no point in arguing, because you could simply define two different terms and be done with it. It describes an essentialist view in which concepts are like tables: the concept is not the definition, but an abstract thing that the definition merely describes, and a definition is understood as an observation and formulation of the essential characteristics of an idea.

Disputes About Reality, the Sages, and the Myth of “There Are No Disputes About Reality”

The text rejects the yeshiva claim that there are no disputes about reality and calls it a “myth.” It brings examples of factual disputes in the Talmud and in history, such as “Did he find a hair on her, or did he find a fly on her?” in the story of the concubine in Gibeah, disputes about the beams and the wagon in the Tabernacle, and the question in Ketubot 6 whether “the blood is merely stored up or whether it is an active discharge.” It says the reason this myth was invented is the desire to “save both sides” and avoid assuming error on the part of the Sages, because in a factual dispute one side is mistaken. It accepts that the Sages can make mistakes even where there is no dispute, and explains that when people say “both are right” in a factual dispute, what that really means is that each side grasped part of the truth, not the full truth, so in practice each one is wrong relative to his exclusive claim.

Plato and Aristotle on Ideas Versus Categories

The text presents the dispute between Plato and Aristotle as the question whether things in the world are expressions of abstract ideas that themselves exist, or whether they are merely categories that we invent. It argues that the debate is not about the existence of concrete things like a table, but about the existence of a “table-idea” in the world of ideas. It explains that Plato sees a definition as an attempt to describe an existing idea, whereas an Aristotelian position might see definitions as useful decisions that do not have to correspond to something “out there.”

The Example of Convex Shapes: Definition, Proof, and the Gap Between Intuition and Formalization

The text brings a mathematical distinction between convex shapes and concave shapes, and presents the statement that the intersection of two convex shapes is convex. It describes the ordinary-person difficulty of proving this intuitively, and then gives a formal definition: a convex shape is a shape in which any two points inside it are connected by a straight line segment that lies entirely within the shape; by contrast, in a concave shape one can choose two points such that part of the segment goes outside the shape. It shows that the proof then becomes simple: if two points lie in the intersection, they lie in each of the convex shapes, and therefore the segment between them lies in each of them, and so it lies in the intersection as well.

The text argues that this move still does not fully solve the original problem regarding the intuitive world, because there is no proof that the formal definition “represents” the ordinary intuitive concept of convexity. Therefore the proof is about a defined concept and not necessarily about what we meant in the first place. It presents this as a division between a “physical” step of choosing definitions/axioms and a “mathematical” step of proving things from those definitions, and argues that the whole move merely reduces the unproven area by pushing it into the definition. It formulates the position that when one tries to prove claims in mathematics about the world, what one actually proves are claims about a model or mathematical correlate of the world, and the question of whether it matches the world cannot itself be proved.

Intuition, Language, and the Diagnostics of Definitions

The text argues that every verbal description of intuition is already a step of conceptualization, so even wording like “a belly sticking outward” is a kind of definition. But it emphasizes that the intuition itself is the thing the words are trying to capture. It says that in mathematics there will always be basic concepts that cannot be defined within the system, because every definition rests on other concepts. It clarifies that for him, “capturing” an intuition is meant in a diagnostic sense: a definition is considered good if it classifies objects exactly as intuition classifies them into groups, even if it does not convey all the connotations of the experience.

The text uses the example of love and the example of color to show that one can build useful correlative criteria without saying “what the thing itself is,” and it emphasizes that in the case of convexity, the feeling is that the formal definition gives the same intuitive classification. It says that even in the world of proofs one can always question the assumptions themselves and refine things further, so there is no end to questions about the basis of the definition.

Bertrand Russell, Direct Reference, and Description

The text gives the example of “the center of mass of the Milky Way galaxy on Friday, 27/9/2024 at 9:55” to show that a complex description can point to a single point in space. It attributes to Bertrand Russell, in his 1905 article “On Denoting,” a distinction between direct reference such as “here, this point” and reference by description, and argues that both ways can refer to the same object even though the description requires a great deal of knowledge. It gives the parallel example of “David Ben-Gurion” versus “the first Prime Minister of the State of Israel,” and distinguishes between the ability to identify something diagnostically and a rich acquaintance with the person himself.

Definition by Extension Versus Definition by Content, and the Application to a Democratic State

The text presents a distinction between extensional description by extension, such as a list of democratic countries, and intensional description by content, such as properties like separation of powers, freedom, and elections. It states that the two methods can overlap diagnostically and yield the same list of countries, but a definition by extension does not teach us “what” democracy is and does not help us decide about a new country without extracting the common content. It says that a descriptive definition is more productive because it enables work and inference beyond the cases that were listed, just as the formal definition of convexity made possible a proof that was not available from feeling alone.

“A Correct Definition” According to Plato Versus Aristotle

The text argues that some mathematicians say there is no such thing as a correct definition, only consistency, whereas from a Platonic perspective there are correct and incorrect definitions because the definition is supposed to capture an existing idea. It explains that only if there is a concept prior to the definition can one say of a definition that it is “incorrect,” because it fails to hit the concept. It gives examples of flawed definitions of convexity such as “a round belly” or “a belly sticking outward,” which do not fit everything that is intuitively included in convexity. It concludes that the common feeling that there are mistaken definitions is itself a sign of a Platonic outlook that assumes a concept existing beyond the definition, whereas in an Aristotelian view the definition is not tested against an idea but established as a decision.

My Zoom, after all the updates, things have changed here. Okay, let’s begin. In the last few sessions I talked about changes in concepts and about disputes over concepts. And I said that these two phenomena are basically an indication of Platonism. Because if we relate to concepts in a conventionalist way, meaning that it’s some agreement of the community, some collection of characteristics that we call by one general term, then there’s nothing to argue about. Whatever was agreed upon is the concept, and if we have a disagreement then we’ll define two concepts or two terms for them and part peacefully. So disagreement is an indication of Platonism. The same goes for a change in a concept. If I change the definition of a concept over time or propose a different definition for it, then I’m basically claiming that the new concept continues, in some sense, the previous concept, or that this is the updated appearance of the previous concept for the present time. So that basically means there is such a thing as a concept beyond the collection of characteristics, because otherwise, the moment the collection of characteristics changes, it’s a different concept. In what sense is it the same concept if it’s defined differently? If it’s defined differently, then it’s a different concept. Unless you understand that there is something in the concept beyond its definition. And what is that? It’s that very thing that we’re trying to define. The concept is some abstract thing, a Platonic idea if you like, and when we define it, we’re basically observing it, looking at it, and trying to formulate its essential characteristics explicitly. Therefore, the definition is not the concept but a description of the concept. Exactly the same as when I define a table: the definition of the table is not a table. The table is what stands before me. The definition is an attempt to describe the essential characteristics of the thing standing before me. The same applies to concepts. The essentialist view basically understands that concepts are like tables. That is, they are existing things, and definitions are only an attempt to describe that thing. Just as I describe objects, I can also describe ideas or concepts. That is the Platonic view. I have a question. There are disputes where it seems, at least, that it’s a dispute about the facts. A person buys something—did he think it included the roof, did it not include the roof? No, those are not disputes about the definition of a concept. Those are disputes about the intention of the parties when they signed a contract. Okay, but I’m talking about disputes about the existence of a concept. I’m not talking about, say, a dispute between socialism and capitalism or communism and capitalism. That is, in principle, a value dispute. We talked about the fact that valid morality also has to be grounded in some moral ideas. And then there is a dispute here over what the moral idea says. Does it speak about equality, about freedom, about this model or that model? It seems like there’s a dispute over what the situation is. There are—I don’t remember right now—but there are all kinds of disputes. But a dispute over what the situation is talks about the same situation, and the two sides describe that same situation differently. If they were talking about different situations, then it wouldn’t be a dispute. If I say that now it’s day and you tell me, what are you talking about, yesterday it was night—that’s not a dispute, right? You’re talking about yesterday and I’m talking about today. Yes, why is there a dispute in the Talmud that seems to be a dispute about reality or something? Okay, so there’s a dispute about reality because they don’t agree what reality is. In the story of the concubine at Gibeah, yes, was it that he found a hair on her or a fly on her? That’s not some essential Platonic issue or something, that dispute. No, I didn’t say every dispute is a Platonic matter. I said again: a dispute about a concept assumes that the concept exists. There are other disputes. There are disputes where one person says I love so-and-so and another says I hate so-and-so. In my eyes that’s not really a dispute at all, never mind, but we can tussle over that issue. That doesn’t mean there’s something objective here that we’re arguing about. But a dispute about the definition of a concept is indeed an indication that the concept is some existing entity. Okay, I’m thinking more from the Brisk side, where they always want to describe all kinds of… Right, with that I completely agree; I’ve even written about it more than once. This yeshiva myth that says there are no disputes about reality is nonsense of the highest order. Of course there are disputes about reality, and anyone who claims there are no disputes about reality is disputing reality. Meaning, obviously there are disputes about reality. There’s a dispute about what happened with the planks and the wagon in the Tabernacle, as Rabbi Yitzchak Hutner famously said. He discusses a dispute about reality and brings that as an example. There’s a dispute whether the blood is flowing or deposited in Ketubot there on page 6. And there are disputes—there are lots of them. Whether he found a hair on her or a fly on her in the story of the concubine at Gibeah. What happened there? A historical dispute. Okay? Lots of things of that kind. Of course there are disputes about reality; it’s just a myth. Therefore I don’t want to claim there are no disputes about reality. I do want to claim—and that’s what bothers those who invented this invention—that in a dispute about reality one side is right and the other is wrong. One of the reasons they invented this myth that there are no disputes about reality was in order to save both these and those—to save the idea that error among the Sages is impossible. In a dispute about reality, certainly one side is mistaken; therefore, they say, there are no disputes about reality. If I remember correctly, in the Talmud precisely about the Gibeah example, there’s something there saying both are right. She said both this and that. Really? The Talmud says that. The Talmud on Gittin 6b. Even there, in that dispute about reality, both are really right. Yes, so what does it mean both are right? In other words, both are wrong. Meaning, what happened there was that there was both a fly and a hair. I said there was a fly—that’s mistaken, there was also a hair. Someone else says there was a hair—that’s mistaken, because there was also a fly. Meaning each of us grasped one aspect of the truth. But it’s not that both are fully right. There’s no such thing. On a factual question, there is no question: one is right and the other is wrong. Does the Rabbi really think that the tanna’im or the amora’im—I don’t remember which—who discuss whether he found a hair or a fly on her meant seriously to say, I received a tradition, or I thought, or I discovered through some research that this is what happened historically, and the other one said no, it was a hair? Doesn’t that sound really absurd to the Rabbi? I don’t know—if he doesn’t discover it historically, he can also discover it interpretively. But you’re right, it could be a parable, that doesn’t matter. I’m only saying that in principle, to claim there are no disputes about reality is basically an attempt to save both sides, to save the idea that no one is mistaken. But I agree that if there is a dispute about reality, one is right and the other is wrong. I just don’t agree that there are no mistakes among the Sages. There are mistakes among the Sages. There are mistakes even where there is no dispute. And the story with the rebbetzin, where the rabbi said to this one, you’re right, and to that one, you’re right, and she told him, they can’t both be right—and he told her, you’re also right. Anyway, so I return to our matter. There are always disputes—in Hebrew, there are disputes about the shape of the Temple. Was the menorah like this, where was the altar? They’re talking two generations or something after the destruction of the Temple and they don’t remember exactly where the altar was, a bit north, a bit south, whether the menorah was in this direction or another. How can it be that… Right, that’s a good example. A good example of a dispute about reality. Okay. Fifty years later they don’t… Fifty years, from a historical perspective, fifty years seems to us like nothing. But you don’t remember everything that happened fifty years ago, right? Meaning fifty years is not a short time. We look historically backward two thousand five hundred or two thousand years and say, well, what is fifty years—before that, fifty years earlier, it was there. Okay, but fifty years is enough time to forget things. At least it could be. Okay? Of course, there were many people and everyone would have to forget, so fine, that’s… But there was a dispute, so one group remembered it this way, another group remembered it that way. Of course, for every such place you can find excuses. You can say the dispute is about what ought to have been, not what actually was. Okay? Like the dispute between Rashi and Rabbenu Tam about tefillin. Rashi was his grandfather. So they ask Rabbenu Tam: tell me, what did your grandfather do for tefillin? Don’t you know? Open your grandfather’s tefillin and see. So the claim is no. Rabbenu Tam argued that maybe his grandfather did it that way, but he was mistaken. Meaning, I think tefillin should look different. So in various places there are explanations for such disputes in a way that avoids calling them disputes about reality. There are such explanations for all kinds of these disputes, but basically I do not accept that there are no disputes about reality. There are disputes about reality. Fine, in any case. Rabbi, Rabbi, Rabbi, if Aristotle and Plato had met, let’s imagine—how would the Rabbi imagine a dispute between them about the concept of love? Yes, Plato says love is a concept, what’s called an idea. What would Aristotle say? Let’s make up something else. After all, they know they’re talking about the same thing. They don’t need to say a word to know they’re talking about exactly the same thing. The question is what it is. Suppose one felt inside himself such a feeling—so what? Doesn’t he feel that there is such a feeling within us? So maybe that’s Plato’s whole idea? No, that’s not Plato’s idea, because if it were Plato’s idea then there wouldn’t be a dispute. Plato argued something else. Meaning regarding feelings, Aristotle agrees that this is some idea or something existing? It’s not an idea; it’s a fact. Like tables. What is the claim against Plato? Look, there’s a table here before me—does that prove there is an idea of table? Fine, obviously there’s a table before me; Plato agrees too. The question is whether there is an ideal table in the world of ideas, of which all the tables in the world are some expression. That’s the dispute between him and Aristotle. The existence of a concrete thing everyone agrees exists, except again solipsists. But Plato and Aristotle are not arguing about the existence of the world; the world certainly exists. The question is whether the things in the world are expressions of some abstract ideas that also exist, or whether they are merely categories that we basically invented. So now I want to bring another nice example of this issue. I think it illustrates well the claim I’ve made here. Look. In mathematics they distinguish between convex shapes and concave shapes. Now, I’ll talk about this in a layman’s way, yes—my apologies to any mathematicians here—but with convex shapes, what is this thing running around here… Shmuel, you’re drawing on the board and I don’t know exactly how that’s happening, I’m not familiar with this, this format here is new. Fine, anyway—so what are shapes, what are convex shapes? Hello, what is this fellow doing to me here? I scribbled that. Ah, no no, then don’t make a mess for me here. It seems I need to do this in some way so that only I can draw here, but I don’t really know how to do that, this format is new, I… On the left side there’s the pencil. What? No, I know how to draw. I don’t know how to stop you from drawing. At the top in the middle there’s some whiteboard tab and there’s apparently an arrow there—if they didn’t change the version—and there you can control the settings of who can touch it. Say don’t and that’s it. What? The question is where to say the don’t. Wait, one second. Like when you do share, so there are options for share at the top. No, I can’t manage to see it. Fine, then don’t play with it now because you’re… it bothers me here, I don’t know how to disable it. No, that’s it, don’t touch it. Fine. Anyway, so look. In mathematics they distinguish between concave shapes and convex shapes. Concave shapes—from the word bowl, yes? So it’s something like this. And convex shapes are something like this. Okay? Now when I’m talking about concave shapes, I’m talking now about closed shapes. Closed shapes. So convex shapes, for example, are something like this—not exactly convex, but let’s say, let’s say this is a straight line, okay? So that’s a convex shape. Why? Because all of its sides are not concave, okay? Think of it as a straight line for the purpose of discussion. So all of its sides are not concave. In this case, you see here some little bowl, so that is concave. But the assumption is that a convex shape is a shape that is convex in all directions, where convex means either straight or convex. Fine? Anything that is not concave is called convex. In contrast, a concave shape is, for example, this shape. This shape is concave. Why? This shape. The opposite. The opposite. Let’s fill this in—like a banana or a crescent moon. Why is it concave? It’s convex. It’s concave. Because it has one side—you see, it has one boundary that is concave. Why is it convex? No, it’s concave. The concavity or convexity is relative to the center of the shape. Look, if the thing is like this, then it’s convex. When you look from here in all directions, there’s a belly outward. Okay? So that’s convex. But when you take a banana, then here it’s not convex. Why? Because when you look from here, you see there’s a bulge inward. Fine? Since there is a bulge inward, it’s not convex. So it’s enough that there be one bulge inward at one place, and then it’s not convex but concave. Those are the definitions, okay? Now, now there is a theorem in mathematics that says the intersection of two convex shapes is also convex. Okay? Instead of erasing, I’m just reopening it. I still don’t know all the mess here. So look—for example, let’s say there is one shape like this and it’s convex. Okay? Now there’s another convex shape. Fine? Their intersection—you see this region here, this region is convex. Also convex. And so if you take any collection of convex shapes, any two convex shapes have an intersection that is convex, and then of course a larger number of convex shapes whose intersection you take will also be convex, because any two give a convex shape, and their intersection with the next shape is also convex, and so on. So every intersection of two convex shapes, or of several convex shapes, is convex. Fine? That’s a theorem in mathematics. Now the question is how do you prove this theorem? Have I brought this example once before? I don’t remember. How do you prove this theorem? Okay. So the question is how to prove this theorem. Now when you try to prove this theorem, I believe—because at least that’s how it was for me—I believe you’ll run into not-so-simple difficulties. It’s not easy to prove this theorem. I tried, made attempts, tried to think what possibilities there were, tried to check—after all, it’s obvious that every such line will be convex, but the question is what happens at the intersections. How can I prove that the intersections also never break the convexity of the intersection of this shape? In short, it’s not easy to prove, and eventually I found it once in some book in some used bookstore at Harvard there in the United States, in Boston. A little topology book for amateurs or something like that. So he says, now he moves on to prove this theorem, and he says as follows: in order to prove this theorem, first of all, of course as mathematicians we need to define the concept of a convex shape. Because otherwise, when we thought about it like laymen—laymen can’t prove anything. Meaning in order to prove something you need to define the concepts involved, otherwise you’ve done nothing. How do you define the concept of a convex shape? That is the key question. Now it turns out there’s a pretty simple definition, and it says this: suppose we have here two points inside the shape, and now I connect them to one another. You see that the entire line connecting the shapes to one another lies inside the shape, right? The points. In contrast, in a concave shape this does not hold. Oy, what did he do. Wait. Here, this is a concave shape. And within such a shape this does not hold. Why not? Take this point and this point. Now let’s connect them. You see that part of the line lies outside the shape? That means the shape is concave. Okay? So now we have a definition. The definition of a convex shape is a shape such that every two points you choose inside it, if you connect them by a straight line, the entire line lies within the shape. That is the definition of a convex shape. Okay? Now, wait, let’s use the advanced technology again of reopening it. So now let’s look, for example, at the intersection of these two convex shapes. Okay, these are two convex shapes, and I want to look at their intersection. What do I need to prove in order to prove that the intersection is a convex shape? I need to prove that every two points inside this shape, if we connect them, the whole line will be inside the shape, right? That’s the proof that the shape is convex. Yes. Now the proof is very simple. Why? These two points are in the intersection, so they’re in both this shape and this shape, right? The fact that they’re in the intersection means they belong to both this shape and this shape. That’s the meaning of intersection. An intersection is the set of points that are in both of the intersecting shapes. So if I take these two endpoints, if they’re in the intersection then they’re also here and also here. Now since each of these two shapes is convex, then if there are two points inside it and I draw a straight line between them, the line must be in this shape because it is convex. It must also be in this shape because it too is convex. And if it’s in both here and here, then it’s in the intersection. Which is what had to be proved. Did you understand what I did? Who said that every two points will satisfy this? Because I’m showing you. Because if you have two points in the intersection, then first of all I know those two points are in both this shape and this shape. Otherwise they wouldn’t be in the intersection, right? Yes, but I need to try all the possible points. No, no, no, no—you don’t need to try anything, you need to prove it. So now I take two points. I can claim: every pair of points I choose in the intersection belongs both to this shape and to this shape—is that correct? For any two points. Yes. Right? Now I connect those two points with a line. Now this line must belong to this shape. Why? Because it’s a convex shape and there are two points inside it, so the line connecting them is entirely inside it. Right? The assumption is that it’s convex. Same thing here—this too is convex. So if the two points are in this one, then the line connecting them is also in this one. So that means the line is both in this shape and in this shape, or in other words, the line belongs to the intersection, because it’s in both shapes. So I proved that if there are two points in the intersection set, then the line connecting them is also in the intersection set, which means the intersection set is convex. Nice. Fine? Now look what an amazingly simple proof this is—a child could find this proof. So why did I get so tangled up before? I got so tangled up before because I hadn’t defined the concept of a convex shape. The moment I defined the concept of a convex shape, suddenly the proof becomes trivial. Now notice: this is a little lesson in simple mathematics. But now notice the significance of this. The significance is this: did I really solve the problem? After all, I asked a layman’s question, right? I took two convex shapes and asked whether their intersection—right, every normal layman asks himself, when you take two convex shapes and check their intersection, is it also convex? There’s no one who hasn’t asked himself that at some point. So what do you answer to that? I don’t know. Now the mathematician comes and advises me how to solve the problem. How do you prove the problem? Let’s define the concept of a convex shape in the way we defined it, that every two points, the line connecting them lies inside the shape. And after we define that, you’ll see the proof is very, very simple. Only one problem remains: who says this definition really represents the layman’s concept of a convex shape? It makes sense, right? Everyone understands that it fits very well with the concept of a convex shape, to say that for every two points inside the shape, the line connecting them is entirely inside the shape. Do you have a proof that this really describes all the shapes I would call convex in an ordinary layman’s way? There is no proof; it’s a definition. Okay. Now some mathematicians, if you ask them where the definition came from, they’ll say it came from nowhere—we decided that this is the definition, and from here on we begin to operate. It’s arbitrary, it requires no justification. A definition requires no justification—some mathematicians will say. But—but if that’s so, then you didn’t prove that every intersection of two convex shapes is convex. Because I asked a question about the layman’s world, and you answered me about the concept of a convex shape as mathematicians define it. I don’t know whether it’s the same concept, or at any rate you can’t tell me you have a proof that it’s the same concept. And after all, mathematicians demand proofs. Meaning, you didn’t really prove the theorem I was looking for. This is a very important point, notice—it’s a bit subtle. You didn’t really prove the theorem I wanted to prove. You proved another theorem that seems to me probably equivalent, it’ll be the same thing—but “it seems to me” isn’t good enough for mathematicians; they need proofs. And therefore what I really want to claim is that this route we took didn’t actually help me prove what I was looking for. In the end, the result remains unproved even after the mathematical move. Because when I ask the layman’s question within my world of simple concepts, I have no answer to that question. I have an answer in the world of mathematical concepts. After they defined the concept of a convex shape, regarding that definition I can prove the theorem. But who says that that definition overlaps with my simple understanding of the concept of a convex shape? Okay, but I don’t exactly agree. We already know that in everything in physics there’s no such thing as proof. I can’t say Newton’s theorem or something is true. Well, so why don’t you agree? You just gave another example of something in the world of physics. There are no proofs in the world of physics. Thank you. I agree with everything you say except the first sentence. You said you don’t agree—why don’t you agree? You’re repeating what I said. I completely agree: there’s nothing in physics that has a proof, and what I only want to claim is that the question I asked is a question in physics, not in mathematics. Therefore, when a mathematician thinks he proved it for me, he’s mistaken. He proved something parallel in the mathematical world. The question whether the thing in the mathematical world is parallel to what I call in the physical world, what happens in the physical world—that is a question for which there can be no proof, and therefore you won’t succeed in presenting this to me as a proof about the world of physics. It’s a proof in mathematics. There are things that are axioms. Okay, make it an axiom, do whatever you want, but there are no proofs here. What I really want to say is that when we moved from the layman’s concept of a convex shape to the mathematical concept that is sharply defined—yes, that between any two points the line connecting them belongs entirely to the shape—we actually made some sort of leap without noticing. For us, it seemed the same. We had only put into words the intuitive feeling we had about a convex shape. But that’s not accurate. It’s not certain that you truly captured completely the layman’s feeling, yes, your simple intuition regarding convex shapes. Maybe it seems right, the feeling is that it’s right, but mathematicians don’t go grocery shopping with feelings—physicists do. But the point is that if you claim you proved something here, you proved nothing. So what did the mathematical move help me with? You can say it helped me because I published a paper on it. Fine, that helps mathematicians. But how did it help me as a physicist or as a philosopher? How did it help me? The answer is this. It basically reduced the unproved space that I have. It didn’t eliminate it. There will always be some part of our world that is unproved. All this process I described does is reduce it. Let’s put all the unproved part into the definition. The definition is that I have a convex shape—what is a convex shape? Every two points inside it, the line connecting them also belongs to the shape. That’s the definition. Now within that definition is hidden everything I don’t know how to prove about the world. From that point onward I can proceed with mathematics and prove things simply and necessarily. Okay? So what did I do? I took a problem I didn’t know what to do with, because I don’t know, I hadn’t defined it, I don’t know how to prove things, it’s just some intuition that the intersection of convex shapes is convex but I have no way to advance to a proof. So what do I do? I reduce it, or divide it, into a part that is physics and a part that is mathematics. The part that is physics says: let’s define a convex shape in the following way—for every two points inside the shape, the line connecting them belongs to the shape. And now the part that is mathematics comes and says: okay, that’s the definition; from that definition onward I’ll prove the theorems for you. That’s what I do as a mathematician. But people don’t notice that there is a physical step here in the middle. How we built our definitions or our axioms—the same thing—but our definitions and axioms are actually steps that do not belong to mathematics but are steps of a physicist. The physicist tells you: for me this is the definition of a convex shape. Now the mathematician will come and prove theorems about convex shapes. But all the time in the background there is the assumption that the mathematical concept of convexity is parallel or equivalent to the layman’s or physical concept of convexity, and that we will never manage to prove. And therefore what I really want to claim is that when people try to prove in mathematics claims about the world, they are not really proving claims about the world. They’re proving claims about some mathematical correlate of the world. I build something parallel to what happens in the world, something well-defined mathematically, and I prove all kinds of things about it. By the way, many people claim that this is basically what physics does too—that we formulate physical phenomena mathematically. Everything we then do with the mathematical form, we do on the theoretical correlate. But I don’t know if that’s true in the practical world; I assume it’s the same thing and therefore it’s true. But what I proved about the physical model is not necessarily a claim about the world. That’s a major question in the philosophy of physics. But for our purposes, what I really want to say is: let’s notice the meaning of this step. I made this step and took this intuitive concept of a convex shape and translated it into the definition that every two points inside it have the straight line connecting them also inside the shape. There’s something I don’t understand. Okay. What does intuition say? For each of us, what is the definition of—the word definition isn’t the right word—how do we intuitively understand a convex shape? When I give you an answer to that question, that answer will be a definition. Fine, but my intuition and your intuition could be different. Not in the form of a definition—I’ll tell you, it’s something with the belly out. Mathematicians turn over in their graves when I say something like that, but yes—it’s a shape whose belly is always outward from all directions. So if its belly is outward in all directions, that is in fact a definition. So why can’t I prove using our definition? First of all, maybe you can prove on the basis of that definition, I don’t know. I think it would be very hard. I tried and didn’t succeed. But maybe it’s possible on that basis too. Because in fact the intuition you call intuition is really a definition. No, it’s not a definition. Its verbal expression is a definition. “Belly outward” is a definition. Its verbal expression is a definition. The intuition is that thing which the verbal expression is trying to capture. It’s a thought in the head, I don’t know—once you translate it into words, you’ve already made some step of conceptualization. And when you make the conceptualization in an unambiguous way—because the concept “belly” is not well-defined, therefore mathematicians won’t like the definition of “belly outward.” They won’t like it, fine, but in the end it is a definition. Therefore I say: whenever you turn it into a verbal description, it will indeed be a definition. But what are you describing verbally in that way? You are verbally describing some kind of thought or intuition you have in your head about the concept of a convex shape. Any verbal description you give of it will be some sort of definition, more successful or less successful, but some sort of definition. And then you can begin to work. What is the definition of a straight line, for example? That too is intuition. I have an intuition of what a straight line is. Right. But that is exactly the serious point here. But it is defined. It is defined so that I can use it in mathematical proofs. You can use it in mathematics… I don’t understand the difference. And I’ll give another example. Another famous example: all of Euclid’s theorems for two thousand years about parallel lines and so on. And a hundred, a hundred and fifty years ago they suddenly thought: what happens if I live on a sphere or something like that? Then all those theorems are not true. If I’m not in Euclidean geometry, there are other possible geometries. And the assumption that we live in Euclidean geometry is an assumption. Yes. I’ll get to non-Euclidean geometries in a moment, but I just want to sharpen this point. My claim is: when you define something—and I say again, “belly outward” is also a definition, as long as the concept “belly” is clear enough to you, okay?—and maybe on that basis too you can prove this theorem, then you made a move parallel to the one I made here. It won’t change anything. In the end my claim is… You’re right, I agree. In the end my claim is that the move from an intuitive, abstract, unformulated concept to a formulated conceptualization or a definition is basically some kind of move that tries to capture in words an abstract idea. Now what is this abstract idea? So that is true of everything in mathematics. A straight line—I gave the example of a straight line. That too is a translation of it. It’s true of everything, not only in mathematics. It’s true of everything, period. We’re talking about mathematics now. Right. No, I completely agree—it’s just that I brought an example from here, but it’s an example. Clearly all concepts are like this. And therefore, in mathematics too, when you define things, at the base there will always be a set of concepts that you cannot define. Because every definition is based on concepts. So why was all this pilpul about the proof of this needed? You could simply have said “straight line.” A straight line—how do we define it? That’s exactly the point. Once you say the concept “straight line,” I can show you there will be theorems in mathematics that you won’t be able to prove. You won’t be able to. But after I define for you clearly the concept of a straight line, you’ll see that I can prove them, just as I did here. Fine, okay. It’s an example, but what you’re saying is very general. Yes. Does this relate to the concept of “it is impossible to be exact”? How do you understand that? There are various ways to understand it. One way maybe is exactly this, that it’s an abstract concept and doesn’t belong to reality. I don’t know why this relates to “it is impossible to be exact.” I agree with your last sentence. It’s an abstract concept and belongs to the Platonic world, let’s say to the world of ideas, and when I want to put it into words, I’m basically turning it into something concrete. I connect the idea to its appearances in the world. The idea of the triangle is something that exists in the world of ideas, or the convex shape, and in our world there are specific convex shapes. This shape and that shape, and they are all convex. And then I ask myself what all these shapes have in common. What they have in common is basically the properties of the idea in the Platonic world of ideas. No, but there’s another way to understand “it is impossible to be exact”: not that it’s abstract and you can’t really find that exact point in reality, but rather that the point really exists, only a person can’t be so precise. No, so that’s why I say I don’t understand the connection to the issue of “it is impossible to be exact.” On that issue I agree with you, and it’s a dispute among the medieval authorities (Rishonim) on that question. It’s not only among the medieval authorities; it already begins in the Talmud—whether exact precision is possible in the hands of Heaven or in the hands of man. There’s a dispute whether exact precision is only possible by man or also by Heaven. And that is exactly the question: whether it doesn’t exist at all, or whether it exists but human beings simply don’t know how to be that precise. But I’m speaking about a different question. The question of definition is not the question of exact precision. The question of exact precision is a question about measures—can you hit exactly fifty cubits, as in Rabbi Yirmiyah’s question. But that’s not what I’m speaking about here. I’m speaking about the question of the definition versus the defined concept. It’s not a question of… whether it is or isn’t possible to be exact. But of course the dispute between Aristotle and Plato is equivalent to the dispute you described regarding whether exact precision is possible or impossible. Because Aristotle claims that this abstract thing we are trying to capture doesn’t really exist. There is no such thing, really. It is our invention. We take a collection of shapes like these, see that in our eyes they seem to have some common property, and define that property as convexity. It’s convenient for us to talk about it. Whereas Plato claims that through these concrete shapes we understand that there is a Platonic concept in the world of ideas called convexity, and then we are really observing it. Maybe through the shapes, but we are observing it and trying to understand what it says. And that is exactly the point: Plato sees the process I described here as a process of observation. Meaning, I look at the world, at this idea of convexity in the Platonic world of ideas, and try to describe it in my own words. That is how I create a definition. A definition is the result of observation. It is a description of an object I am observing. Where is the practical implication? The implication is, for example, often if you ask mathematicians, tell me, is this a correct definition or not? He’ll tell you, what do you mean? There’s no such thing as a correct definition. What is a correct definition? You can ask whether it is consistent, whether it does not include a contradiction within itself, but what is a correct definition? If that’s your definition, that’s your definition. There’s no such thing as right or wrong. In the Platonic perspective, that’s not so. In the Platonic perspective there are correct definitions and incorrect definitions. If the definition doesn’t properly capture the Platonic idea, then it is not a correct definition. And that is exactly the question: does the claim that between every two points in the shape, if you connect them by a line, the whole line will be inside the shape—does that fully capture the Platonic idea of a convex shape, which is the intuitive feeling with which I started out, what I had at the beginning? That’s the dispute. If it fully captures it, then you proved it, but you cannot prove to me that it fully captures it. It’s a feeling that it fully captures it. Yes. Rabbi, Rabbi, Rabbi, if I understand the Rabbi, that basically it’s impossible to prove whether the definition captures the entire concept or the entire intuition of the one defining it, then maybe one can say something even sharper. Since I think, for example, that intuition is also a kind of feeling, and a feeling after all cannot be defined. In no way—the Rabbi demonstrated for us in the past about color and so on. It cannot be defined. You can say it’s such-and-such a light wave. Suppose a person tries to define love, so he says let’s say we decide that whenever we checked, every time the pulse rises when he remembers it. So now we’ll start proving and discussing and say that the definition of love is a pulse that rises by 10 percent. Then obviously that doesn’t capture the thing itself, but maybe in terms of proof it will be practical and useful, but there’s no chance that anyone—not even a reasonable person—will say that the definition captures it. There’s something to what you’re saying. But with a convex shape our feeling is that it does capture it. That this is the whole essence of a convex shape, that the two points inside, the line connecting them? Yes, yes. Every intuitive property you think of regarding a convex shape, I can show you that it holds for a shape defined in this mathematical way. No, but that it captures the feeling that this defines it—that’s what we feel when we speak about convexity. Yes, yes. And that’s equivalent—what does “captures” mean? It’s completely equivalent to the diagnosis that your feeling would give. Definition by extension and definition by intension? So here, I’ll take all the shapes in the world as convex shapes and not—but still, Rabbi, still I say: if, suppose, everyone who loves has his pulse rise by 10 percent—that’s what we checked in billions of people, that’s what happens. Then would someone come and claim that this is the definition of love? That the pulse rises from 90 to 120? No, no, that’s already a question of what you’re looking for in a definition. I agree with you. Meaning, obviously you can say that this thing is merely correlative. Meaning, I can diagnose every person as either loving or not loving according to the pulse, and still I haven’t really understood what love is. That I agree with. But in the mathematical sense, when I say that this definition captures the intuitive concept, I mean that the classification of shapes that a person equipped with intuitive feeling would make will be exactly the same classification as the classification of shapes a mathematician will make according to that definition. But always, even in mathematics, what is a proof? You can always keep going further. You need more assumptions. Meaning every proof has assumptions, axioms, on which you base your proof. And you can say okay, you have an assumption, but is that assumption really itself based on further assumptions? What is a point? What is a line? What is a set? There are always assumptions and you can always be more precise. So I’m only saying that when you say that the assumption fits the feeling people have, it’s always like that—even when you reach a mathematical proof, you can also say okay, but what about the assumptions of mathematics, I want to be more precise—there always are. That is exactly the claim I made here. I don’t understand—that’s what I said here. I said: here, I have now proved to you that every intersection of two convex shapes is also convex. And then we ask: wait, is that really a proof? Who says your definition of a convex shape is really exact or completely suited to the intuition? Maybe you need to go deeper and define it differently, and suddenly you’ll discover there are differences. That is exactly what I claimed. But another example of the same thing is color. It’s impossible to define what red is. Each person—but if I translate it into the frequency of some wave, I can prove all kinds of things. But is that wave exactly the red I see with my eyes? Maybe yes, maybe no. No, obviously. The color red is a concept in consciousness, not a concept that exists in the world. What exists in the world is an electromagnetic wave at best, but colors are what exist in our consciousness—that’s clear. But that’s why I said that when I speak about whether the definition captures the concept, I mean in the diagnostic sense. Meaning, every decision I would make about the world intuitively would also be made by the definition; it would be the same set of determinations. We spoke about definition by extension and by intension of concepts; we talked about it sometime in the past. Yes, when you define a democratic state. If we mix red and blue or something, I get purple or something like that. Anyone who paints knows these things. Okay. I want to prove it, so I have to start with waves and frequency. Exactly, that’s exactly the same thing, right, exactly the same thing. But everyone who paints knows what the results will be. But this is exactly what I’m saying. Meaning, you need to define things in a diagnostic way. What does that mean? The definition is supposed to give the same diagnosis as the intuitive feeling. That does not mean that someone who has the definition understands the full intuitive feeling, like the example of love from before. But it does mean that diagnostically, the formal definition will give you the same diagnoses as the intuitive feeling. That is what I here call capturing the intuitive feeling. It does not mean understanding its meaning and its connotations. That is a separate issue altogether, another topic, which we also discussed in the past, but I’m not going into that here. When I speak about whether the definition of a convex shape captures the intuitive concept of a convex shape, I mean only on the diagnostic level. If someone classifies shapes into convex and non-convex by intuition, and someone else classifies shapes into convex and non-convex by the formal definition, do they get the same classification? Do they get the same sets? If yes, then for me this definition is excellent. Fine? That’s the claim. So what I wanted to show through this example is that the transition from the intuitive concept to the definition is, according to Plato at least, basically a transition that is a kind of observation. I look at an idea in the Platonic world of ideas and put it into words, and that is how I create a definition, where the definition is collecting the set of essential properties of that idea I am observing. You know, maybe I mentioned this—I don’t remember anymore—someone once said, brought in analytic philosophy an example they sometimes bring, maybe Kripke brought it, I don’t remember—suppose we’re talking about the center of mass of the Milky Way galaxy on Friday the 27th of 9, 2024, yes, today, at 9:55, yes, that’s the current time. Now in order to understand that definition you need to understand lots of concepts. You need to understand hours, what 9 is, what 50 is, what 5 is, what a galaxy is, what the Milky Way is, what center is, what center of gravity is—lots and lots of things you need to understand in order to understand what I just said. But do you understand what this definition is really pointing to? Some point in space, right? Some point in space is the center of gravity of the Milky Way galaxy on Friday at such-and-such a time and date. That one. But I described it with all sorts of descriptions that require a lot of knowledge to understand. But what does all this description really do? It points to a particular point in space. I could have pointed at that very particular point with my finger: there, that point. And that is the same thing as the definition. The difference, according to Bertrand Russell in his article “On Denoting,” 1905—the year Einstein made his revolutions—was also a revolutionary year in philosophy. In 1905 Bertrand Russell’s article “On Denoting” was published, considered the opening of analytic philosophy. Anyway, in his article “On Denoting” he speaks about two ways of pointing to things, or representing things if you like. There is direct pointing—I simply say, “that point there”—and there is pointing by means of a description. I describe: that point is the center of gravity of the galaxy on such-and-such a date at such-and-such an hour and so on. The description can require a great deal of knowledge, but it merely points me to the same point I could have pointed to with my finger. These are two different ways of pointing to that point, and Bertrand Russell discusses the question of what, despite that, the differences are between the two modes of pointing. It’s like referring to a person. So I can say: this is David Ben-Gurion—a way of referring to him by his name. Or I can describe him: the first prime minister of the State of Israel. Both those expressions, the name and the description, refer to the same person. Their content is ostensibly the same content. But you understand that the description requires a lot of knowledge: what “head” is, what “prime minister” is, what a state is, what the State of Israel is, what the prime minister of a state is, and so on. You need to know all that in order to know who David Ben-Gurion is. But there is another option: I can point to him—there, that is David Ben-Gurion. I can point at him. Now when I describe David Ben-Gurion as the first prime minister of the State of Israel, that description captures David Ben-Gurion because there is no one else who fits that description, right? There is no other person of whom that description is also true. If I said he is a prime minister, there would be other human beings. There are other people who are prime ministers. If I say the first prime minister of the State of Israel, then it is only one person. It is only David Ben-Gurion. There is no one else who fits that description. But still, if I tell you that, you still do not know David Ben-Gurion. You don’t understand who David Ben-Gurion is, you do not grasp the person, you know who he is. That is the meaning I spoke about here when I said that the definition captures, as it were. I mean captures in the sense of diagnostic classification. Now, someone equipped with that definition will go through all the people in the world and divide them up, and he will discover that David Ben-Gurion satisfies the definition and all the rest do not satisfy the definition. I, as someone who knows David Ben-Gurion, can also say: this is David Ben-Gurion; all the rest are not David Ben-Gurion. We will arrive at the same classification of people. In that sense, that definition is a good definition. That does not mean that someone who says “the first prime minister of the State of Israel” can describe David Ben-Gurion with all his characteristics and his history and his biography and so on. No, he doesn’t understand who David Ben-Gurion is, but he knows how to point to him. But on the other hand, if I speak about David Ben-Gurion, presumably there are other people with that same name. No, leave that aside for a second. Yosef—or I’m speaking about the first… No, it doesn’t matter—then I’ll make it… There are two Yosef ben Shimon in one city. So what do they do? They write Yosef ben Shimon ben Levi. Find the chain that leaves only one person. It’s not important, it’s not essential. I’m talking about a name that is already unique to one person. If you need to add more fathers in the chain, then add them. So exactly, then essentially you’re adding a definition. Maybe not a full definition, but maybe that’s still pointing. A definition is a collection of properties. A name is not a property; after all, they could have called him something else. They happened to call him that. A name is a way of pointing directly to the person, not describing him—or to the concept, not describing it. It’s like when I say “a democratic state.” I can say “a democratic state” is a state like England, the United States, Israel, Spain, France, I don’t know—give the list of democratic states. And I can say “a democratic state” is a state that has such-and-such properties: separation of powers, freedom, voting, democratic elections, and the like. This is a description by extension, extensional, and that is a description by content, intensional. These two descriptions are identical in the diagnostic sense; both will give me the same list of states as democratic. But obviously they do not have the same content. The definition by extension tells me nothing about what a democratic state is. It only points—it gives me a list, and diagnostically I can know which state is democratic and which is not. If it’s in that group, then it’s democratic, and if not, then not. So diagnostically it does the job. But ask me a moment: but what is a democratic state really, in essence? I have no idea. I can tell you it’s Spain, France, England… Fine, but that’s not—you understand? And a definition, an essential definition, is parallel to a name, as it were. A definition by extension is the name. In contrast, what we really call a definition is a definition by content, a definition by characteristics. That is not a name; those are descriptions. Fine? So in that sense, say, “convex shape” is a name. If I understand it intuitively, then I understand what a convex shape is. A definition of the concept “convex shape” is a collection of characteristics of the concept “convex shape” that will give me the same diagnosis as the intuition of a convex shape. But the transition from this to that is a transition from a Platonic idea to an Aristotelian definition. Therefore, if you claim that you did not define convex shape correctly—say, if someone tried to propose to me a definition: always with the belly outward. For example, the definition Shmuel suggested earlier. Then I would find some shape that we all understand is convex, but there I don’t see the belly outward. By the way, in fact you don’t see it, because a straight shape is also called convex. So it’s not “with the belly outward”; rather “with the belly not inward”—that would be a more accurate definition. But let’s say I said “with the belly outward” and found a straight line. Then it doesn’t fit, it doesn’t fit. So someone will tell me: look, that’s not a correct definition. What do you mean not correct? That’s my definition. What does “not correct” mean? Not correct because it does not hit the intuitive concept of a convex shape. The intuitive concept of a convex shape gives me a different diagnosis than that definition. So the definition is not correct. What does “not correct” mean? It means that there is some concept like this in the world of ideas to which the definition is trying to correspond, and if it does not correspond to it, then it is not correct. But according to Aristotle there is no such thing as an incorrect definition. A definition, by virtue of being a definition, is the definition. It is not supposed to hit anything else. So there’s no such thing as right or wrong. Meaning only according to Plato can there be incorrect definitions. According to Aristotle, the definition by virtue of being a definition is what determines; it doesn’t need to fit anything outside it. So there’s no such thing as an incorrect definition. But all of us feel that there is such a thing as a correct and incorrect definition of a convex shape. If I proposed to you: a convex shape is always something with a round belly—all of you would say: what are you talking about? Not true! The belly can also be elliptical, can protrude this way, that way. The definition is incorrect. What do you mean incorrect? That’s how I define a convex shape—what’s incorrect about it? That’s my definition! No, because you understand that the definition is trying to capture some concept that already exists before it, and if it doesn’t manage to capture it, then it is an incorrect definition. That means there is actually here a Platonic view. Okay, this took me much longer than I thought, but we’ll stop here. Any comments or questions? What is Bertrand Russell’s conclusion in that article? Not a conclusion—he makes a distinction between two forms of pointing. How far do they overlap? And how far really is one less… You can’t answer such a question. You can answer such a question if you give me a definition for a certain concept and its name, and then I can see whether it fits or doesn’t fit. You can’t say in general how much they fit. Exact definitions fit completely. But what is an exact definition? I don’t know. The Rabbi still claims that there could be a definition that encompasses the entire concept. Diagnostically. No, not diagnostically. We already said there’s no dispute there, no, that’s not interesting. Not diagnostically. But diagnostically of course that’s true, I understand that, but the question is whether there is a possibility that it encompasses everything encompassed by our intuition? I think not, but I certainly didn’t claim it. I’m speaking only diagnostically. Diagnostically that’s obvious. Okay. But there’s a difference: if this is a pointing by description, then you can also prove similar questions. For example, if now I ask what is a democratic state and you give me a list, then tomorrow I’ll ask you what is a democratic state—maybe there’s some new state, and what you answered yesterday isn’t relevant today. Correct. But if it’s a description, then that too will help me today. Absolutely. More than that, I’ll tell you—I think we discussed this, I think we discussed definitions by extension and by content. And I said that if you ask me now about a new state whether it is democratic or not, the definition by extension won’t help me. I’ll need to take the definition by extension, extract from it the content-definition, try to understand what is common to all the objects in that extensional definition, and only then will I be able to try to answer the question about the new state. Meaning, a definition by extension won’t help me with a new state. I’ll need to perform this conceptualization process, exactly what we did with the convex shape. Suppose you are in a curved world, in a curved world, and you ask yourself whether this shape is convex or not—a curved space, yes? A convex shape—you won’t be able to answer with the answer “belly outward.” You’ll have to understand what convexity means in the mathematical sense and then apply it to a curved space, and only then answer. So that means that definitional pointing is more basic. It’s more productive, more efficient. Yes. Yes, that’s why we do it. Otherwise, what’s the problem? Stay with the source, stay with your feeling. After all, all you want is to define your feelings well, so why define them? Stay with the feeling. No, because many times when I define it, I succeed in doing things that with the feeling alone I can’t do, like the proof I gave here. With the feeling I couldn’t prove that the intersection of two convex shapes is convex. The moment I put it into a definition, I succeeded. Okay. Sabbath peace, good news. Sabbath peace. Sabbath peace, thank you very much.