Platonism – Lesson 2
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Table of Contents
- [0:00] Introduction to Platonism: Plato versus Aristotle
- [1:18] The existence of abstract concepts: the photon as an example
- [4:35] Why adopt a Platonic view?
- [6:05] Aristotle and arbitrary classification
- [7:35] Plato and tangible ideas
- [9:46] Scientific classification: Aristotle or Plato?
- [17:30] Newton and the unification of phenomena under the law of gravity
- [23:53] The dependence of description on the coordinate system
- [25:52] Zeno’s paradox and the concept of motion
- [27:43] Defining velocity: intervals rather than points
- [29:49] Defining continuity and a line through small segments
Summary
General Overview
The text presents the dispute between Plato and Aristotle over the status of generalizations and abstract concepts, and emphasizes that the dispute is also about how we know them: for Plato, generalizations are perceived as the result of contemplation with the “mind’s eye,” while for Aristotle they are the product of intellectual abstraction. It argues that people’s initial intuitions tend to be Platonic because they experience certain divisions as natural and essential rather than arbitrary, and connects this to a philosophy of science that sees theories as possible descriptive systems and not necessarily as revelations of the structure of the world. Later, it uses Zeno’s arrow paradox to show how a linguistic-conceptual shift changes our understanding of the problem, and develops the idea of the “position picture” versus the “momentum picture” in quantum theory as equivalent world-pictures that cannot be merged. Finally, it brings Borges (“Tlön, Uqbar, Orbis Tertius”) to illustrate the absurdity of idealism with respect to objects, and compares this directly to the Plato-Aristotle dispute regarding concepts such as “horseness” and “democracy.”
Defining Platonism versus Aristotelianism in abstract concepts
The text defines Platonism as the question of how we relate to generalizations, abstract concepts, properties, and abstract objects. Plato argues that generalizations are existing entities, like the Idea of “horseness,” which exists in the “world of ideas,” and physical horses are its realizations. Aristotle argues that “horseness” does not really exist; rather, a person abstracts from concrete horses a useful general concept, and the generalization is an intellectual process rather than an external entity.
Knowledge as contemplation versus knowledge as abstraction
The text presents that the difference is not only what these abstract things are, but also how we know them. According to Aristotle, knowledge of generalizations comes from a thought process that creates the concept. According to Plato, perception of generalizations is the result of observation with the “mind’s eye,” as Maimonides calls it at the beginning of the Guide for the Perplexed, and the idea is “imposed” in the sense that it is perceived rather than invented.
The intuition of natural divisions and Aristotelian skepticism
The text argues that in principle one can group objects in any way, for example by grouping a donkey, a sheep, and a horse together as “four-legged animals,” yet there is a feeling that the division into horses, donkeys, ducks, and sheep is a natural division—the sort science deals with. Within an Aristotelian framework, it is hard to explain why one division is more “essential” than another, because any property can serve as a basis for grouping, and there is no “more correct and less correct,” only “what seems essential to me.” Plato explains this by saying that in the world of ideas there is an Idea of horseness, and no idea of “animals found in New Zealand,” and therefore some divisions are perceived as natural while others are fictitious. The text describes Aristotelianism as producing a kind of skepticism, whereas the Platonic answer is presented as adherence to the immediate perception that natural division is a perception of reality and not merely a product of the structure of consciousness.
Classification in science, empirical criteria, and Kant
The text argues that every classification has empirical criteria for testing it, but the very possibility of testing does not prove that the division “exists” or is “truer.” It presents a position according to which even evolutionary theories and historical accounts of the branching of species can be seen as a human way of organizing reality rather than as statements about “the world itself.” It attributes this framework to a philosophy of science “following Kant,” which centers on the phenomenon—the world as perceived—and not the world as it is in itself, and emphasizes that no scientific experiment can decide between Plato and Aristotle because this is a philosophical question.
Examples of alternative classifications and descriptions: Newton, the periodic table, Copernicus
The text explains that before Newton, phenomena such as tides, the paths of stars, and the falling of bodies were familiar, but were not assigned to one law, and Newton united them under the law of gravity in a non-trivial way. It argues that even such a unification does not force us to say there is “one correct division,” but shows theoretical fruitfulness that makes it possible to solve problems. It compares Mendeleev’s periodic table to the division into “fire, water, wind, and dust,” and emphasizes that the modern division is not “truer” but “more scientifically fruitful.” It even suggests that there could be a creature whose thought-structure would let it reach the same results with the Aristotelian division. It argues that Copernicus did not “discover” a new truth but proposed a simpler descriptive system, similar to choosing a polar coordinate system instead of a Cartesian one, and adds the qualification that physics also deals with fictitious forces and discussions such as “Mach’s principle,” but on the kinematic level this is a matter of choosing an origin for the coordinate system.
Zeno, velocity at a point, and the distinction between “being located” and “standing still”
The text presents Zeno’s arrow paradox as the claim that motion is a fiction because at every moment the arrow “stands” in a different place, and asks when exactly it moves. It argues that the solution is not differential and integral calculus as an evasive language, but a linguistic distinction between “being located” and “standing still,” because standing still means being located together with zero velocity, whereas being in a place does not negate motion. It argues that there is velocity at a point in time, and that “change of place” is not identical to “velocity,” so that the formula of difference in places divided by difference in times is a way of calculating, not a definition of the concept, and velocity is a basic property from which change of place over time is derived.
Static versus dynamic perception and quantum theory
The text argues that human beings are built for static perception, and that is why a film is made of frames; motion is perceived through a sequence of static states rather than as motion “itself.” It describes the possibility of a creature that directly perceives velocities rather than positions, so that for such a being place would be the integral of velocity and not the other way around. It connects this to the “position picture” versus the “momentum picture” in quantum theory, and argues that the uncertainty principle says that the two pictures cannot be merged simultaneously, so that one world-picture excludes the other as a unified perceptual fusion. As an illustration, it mentions radar and the Doppler effect as an instantaneous measurement of velocity without sampling two points in space, to show that velocity is not necessarily perceptually derived from place.
Feminist criticism of physics and Gadi Taub
The text recounts that Gadi Taub, in his book The Slouching Rebellion, brings a feminist critique claiming that physics is “masculine,” and quotes his response that he would not want to fly in an airplane based on “feminine laws of physics.” It argues that Taub is mistaken, because the critique can be interpreted not as proposing a different physics, but as offering a different description of the same physics, similar to different coordinate systems. It argues that one could in principle develop another equivalent description of the same physical relations, even if its conceptual language were completely different, and therefore one could also “build airplanes” from a different description if it is equivalent and fruitful.
Borges: Tlön, Uqbar, Orbis Tertius and linguistic idealism
The text presents Borges as an Argentine writer, a “crazy genius,” and mentions the Hebrew collection In the Dunes and the story Tlön, Uqbar, Orbis Tertius, as well as a book by Leo Corry of Tel Aviv University with a chapter on the connection between Borges and Umberto Eco’s The Name of the Rose. It describes how the planet Tlön is inhabited by idealists who do not believe in an objective world of objects, and therefore in the southern hemisphere nouns are not used, and instead of saying “the moon rose above the river,” one says something like “it mooned above the ongoing flowing.” It describes that in the northern hemisphere there are nouns, but any arbitrary collection of properties can become an “object,” such as “the cry of the bird in the distance together with the color of the cloud that passed over my head at seven in the morning,” which is called “screecloud” or the like. In this way, a literary illustration is created of the idea that if an object is only a bundle of properties, then there is no essential difference between a “telephone” and any random combination of properties.
Borges as parallel to the Plato-Aristotle dispute: objects and concepts
The text argues that the realist will say that a telephone is something that exists and has properties, and therefore the properties are grouped together in a distinctive way because they belong to the same object, whereas that invented bundle groups properties of different objects and therefore is not an object. It presents two possible responses to the idealist absurdity: an Aristotelian response according to which the distinction is a matter of convenience and useful fruitfulness without any principled difference, and a Platonic response according to which there is an object or idea that establishes the unity, and therefore not every collection of properties is a natural unit. It sharpens the point that the Platonic-Aristotelian dispute is the same dispute as realism versus idealism, but with respect to concepts such as “horseness” and “democracy,” not with respect to the individual horse. It argues that Plato derives the properties from the existing concept, so that “the path is from the concept to the properties,” whereas for Aristotle “the collection of properties” is the source of the concept and the distinction is measured by usefulness and fruitfulness rather than essential existence. It concludes that Platonism is perceived as a strong basic tendency and not as “some weird mysticism,” but as the natural way human beings experience certain divisions as natural.
Full Transcript
[Rabbi Michael Abraham] Okay. Last time we began the series on Platonism, abstractions, Platonic generalizations in Jewish law and in general, and right now I’m still in the conceptual introduction. I started with what is considered the dispute between Plato and Aristotle. The definition of the concept of Platonism in our context concerns the question of how we relate to the generalizations we have, to abstract concepts, properties, abstract objects, and the like. Plato claims that these are a kind of existing entities. When we look at the collection of horses we encounter in life, we identify in them an appearance or realization of a shared idea. He calls that idea horseness, and according to him that idea is something that exists in the world of ideas. But no matter, it exists. The world of ideas is not located anywhere, which is why it’s called the world of ideas. It exists in some sense. I talked about how what often bothers people is: in what sense does it exist if it’s nowhere, if it weighs nothing, right? It’s not a tangible object, but we already know that—there are other non-tangible things that we usually do regard as existing. In the religious world, of course, that would be the Holy One, blessed be He, but in other worlds too—if we talk about a photon, about light in some sense, or things of that sort—those too are abstract things. A pure photon, right, doesn’t even have a location in space, so it’s not… and we can still talk about it as something that exists. The fact that it’s an abstract thing and doesn’t have the properties of tangible objects doesn’t have to bother us. So that’s the Platonic view. Opposed to it stands the Aristotelian view, which sees these generalizations as some kind of abstraction or intellectual process that a person performs. In other words, there really is no such thing as horseness. We see horses, and from the concrete horses that we see, we abstract some general abstract concept called horseness, and we use the concept of horseness not because horseness actually exists in some sense, but because it is convenient for us to see this group as a group that has shared features. So the fundamental difference between Plato and Aristotle is the question whether these abstractions, these abstract things, exist, or whether they are our abstractions. But as a derivative of that point, you have to remember that the difference is also in the question not only of what these things themselves are, but how we know them. Because according to Aristotle, we know them as the result of a thought process. We think about the horses we have encountered, perform some thought process, and from that… turn it over like this with the corner,
[Speaker B] I fixed the corner.
[Rabbi Michael Abraham] Noises. By contrast, according to Plato, our grasp of generalizations, of abstractions, is the result of observation. We simply contemplate the world—not with the eyes, of course—but contemplate the world with what Maimonides called at the beginning of the Guide for the Perplexed the mind’s eye. The intellect has some kind of capacity for contemplation, and with the mind’s eye we understand the idea of horseness. So the idea of horseness is imposed on us; it is the result of perception. We understand it, we see it, or discern it in some sense. For Aristotle, it is our creation. It doesn’t exist outside us. We perform a thought process by means of which we create the concept of horseness, not perceive the concept of horseness. Right? The question is whether this is thinking or cognition. Now at the end of last time I talked about the question—why actually adopt some mystical view like Plato’s? Why do people become Platonists? So I said that I think there’s a very good reason, and it lies in our most basic intuitions. When we look at objects in the world, in principle, if we were tabula rasa, a blank slate, we could group any set of objects within one common framework. Say I could take, I don’t know, this donkey and that sheep and that horse and group them into the class of four-legged animals. Right? That too is just a group, and they all share that feature, and so that too is basically some kind of slicing up or division, a classification, of the objects in reality into groups or subgroups. But we have some sense that the division into horses, donkeys, ducks, sheep—that’s a natural division. Not for nothing science deals with it. It’s not literature that deals with it, right? It’s science. That means we perceive it as some kind of correct perception of the world. It’s a fact that there is such a thing as horses, or donkeys, or ducks. Now within Aristotle’s conceptual framework it’s very unclear why. On the face of it, it seems like a property of our way of thinking. It’s not really a statement about the world. We are built in such a way that, for some reason, we have the feeling that the division into horses and donkeys is more essential, more real, let’s call it, than the division between horses that are one meter eighty and horses that are two meters ten. Or I don’t know, animals found in the Mediterranean basin versus animals in New Zealand. I don’t know—I could divide into all sorts of groups, but we have a very clear feeling that science will not deal with things like that; that’s subjective. You can of course write about the animals of New Zealand, but if you are a scientist writing about the animals of New Zealand, you’ll write about some property that characterizes them—the real property—not about the fact that they are in New Zealand, but maybe that they are all dark-colored, I’m just saying, okay? So you’ll write about why they are dark-colored. But you will always be seeking some essential classification of those things. Now what counts as an essential classification? From Aristotle’s standpoint, there is no such thing as essential and non-essential. Whatever seems essential to me is essential, and whatever doesn’t, isn’t. All these things are just properties of objects. I can group sets of objects by any property I choose or any set of properties I choose. There is nothing more correct or less correct. By contrast, with Plato I simply look into the world of ideas and I see horses there; being in New Zealand isn’t there. There is no such idea. Now I can define it, of course, in a fictive way, but it will be clear to me that it’s a fiction. It’s a fictive definition. By contrast, horses seem to me like something natural, right? It’s obvious—there are horses in the world. Who would argue about that? What does that really mean? That I grasp the concept of horseness, and as a result of that the horses, right? I take all the horses—why do they belong to the same group? Because they are all embodiments or realizations of the idea of horseness, and that is true, right? As opposed to other divisions that are technical or fictive, the division into horses, donkeys, ducks, and the like—human beings, chairs, I don’t know what—seems to us like a correct division, a natural one, I don’t know what to call it. Now Aristotle will say: yes, that’s how it seems to you because that’s how you are built, but it is still arbitrary. In a certain sense, that’s a kind of skepticism. And Plato says no—it doesn’t seem that way to me because that’s how I’m built, but because that is actually how I perceive reality. Just as we wouldn’t say—if I say it seems to me that the door I see here in front of me really exists, it’s not just some fantasy in my head. Suppose someone says, not at all—you’re just built in such a way that for some reason, in certain places, you see doors. I wouldn’t take such a claim seriously. Why not? Because it’s immediately obvious to me: there is a door here. Stop confusing me. There is a door here; I know there is a door here; I perceive it directly. In a certain sense, my grasp of the idea of horseness is like that too. Someone tells me, no no—Aristotle tells me, no no, that’s fiction, that’s an abstraction that you yourself are making, there really is no such thing as horses. It’s one division; you could have made any other division. That’s not how it feels. That is a skeptical claim that says: throw away what you feel, because it’s fiction. It’s imposed on you; it comes from your nature. That’s the claim skeptics make about everything, and Plato will answer what one answers skeptics everywhere: stop talking nonsense. I feel that this is how it is. What?
[Speaker D] Is today’s scientific classification closer to Aristotle or to Plato?
[Rabbi Michael Abraham] That’s a philosophical question, you know—a question of how you relate to it. If you relate to classification…
[Speaker D] Today there’s something defined. How do I know what a horse is, a donkey, and so on?
[Rabbi Michael Abraham] That doesn’t prove anything. I can also tell you how I know that this animal is in New Zealand—just look. If it’s in New Zealand, then it’s there. Every classification has criteria or indications that can be checked empirically. That doesn’t mean it exists. Every classification is like that. The difference between horses that are one meter eighty and horses that are two meters ten—no problem at all, take a measuring tape and measure. That will give you the scientific determination whether this is one type of horse or another type of horse—or not even horse, one type of animal or another type of animal. It’s just a matter of definition. Now someone will come… now you look…
[Speaker D] Today’s science says it’s history or something, that once there was some original horse and it split into several types and so on. It’s not arbitrary.
[Rabbi Michael Abraham] That’s an interesting question, but still, the fact that it happened in some way—I can also explain that regarding animals that are in New Zealand. Science will explain to me that they arrived in New Zealand on a raft from South America before the continents drifted apart, and then it separated, I don’t know, all sorts of things… that too has a scientific explanation. Everything that happens in the world is supposed to have, and probably does have, a scientific explanation. But the fact that something has a scientific explanation does not mean that this property or this division is truer than another scientific explanation or another scientific division. So that… I don’t think those are criteria. More than that: the more extreme views will tell you that this scientific theory of how it happened and how it divided and evolution and so forth—that too is a statement about us. We are not really claiming that this is what happened historically; this is how we organize reality for ourselves so that it will be easier for us to deal with it. Religious people will really love this, because it means evolution didn’t really happen—it’s just our way of looking at the world. But there are also philosophers of science who understand scientific theories or scientific laws that way. Their claim is that these are claims about us, not about the world. We are accustomed to arranging the phenomena in a way convenient for us, because that’s how we think. And scientific theory too, following Kant mainly, right, deals with the phenomenon—with the world as I perceive it, not the world itself. So this is really a function of how I think and not a function of how the world is structured. So that is a philosophical dispute that will be hard to settle with scientific tools. I can’t say, okay, what scientific experiment shall we do to know whether Aristotle was right or Plato was right? That’s a philosophical question. But I think the Platonic view, contrary to the initial impression that it seems like some strange mysticism, is in my opinion closer to common sense, closer to our primary intuitions. We are used afterward… we have these primary intuitions that horses are a natural division—yes, that’s obvious—but then the philosopher comes along, the Aristotelian philosopher, and says wait, actually why? Why not make a different division of all animals—or all objects, why only animals, all objects in the world. Make some completely different division, and it’s no less real; this is one division and that is another division—the question is which characteristics you choose. And then a person may come and say: true, I have some primary Platonic intuition, but the philosopher convinced me, so despite having that intuition I give it up. And I say: don’t give it up, because that’s not right. If you feel that this is natural and that isn’t natural, then apparently these aren’t just abstractions that we make, but we really do see the horseness; we see the horse-like dimension in these objects as opposed to the donkey-like dimension, and it seems to us more essential than whether the thing is in New Zealand or in Australia. But wait, one more question.
[Speaker C] If we, or science, think that these divisions really are correct and true, does that also obligate us to think that they exist in the sense Plato spoke about?
[Rabbi Michael Abraham] Otherwise it’s hard to see in what sense they are correct. What does “correct” mean? As I said earlier to Professor Turkel, in the end every division you make is correct. Even the division between animals in New Zealand and animals in Australia is correct. There are animals in New Zealand, and we also have a scientific tool to check it—just look where they are. So why does the division into horses and donkeys seem more essential to us? Not more correct, but more essential. They’re all correct. What’s the difference? This is a characteristic and that is a characteristic—why did you decide this characteristic matters and that one doesn’t? So Plato offers an explanation. This characteristic matters because it simply exists in the world of ideas—there is an idea of horseness. There is no idea of animals found in New Zealand. But clearly being in New Zealand is a scientific property; it can be checked with scientific tools and there are criteria and everything is fine. So I don’t think “correct” and “existing” are the same thing. Maybe I used the word correct before, and if so then I’m qualifying it a bit. But I think the notion of correct is not precise here; something like natural is better. Let’s call it that. Okay? Everything is correct. So I think that this consideration definitely underlies the Platonic intuition, if one doesn’t give it up too quickly, because there’s something in Aristotelianism that convinces people tremendously, so much so that they give up their initial intuitions. But notice: our initial intuitions are Platonic, not Aristotelian. We have some tendency to feel that these are natural things, existing things; the division is a real division, clear, correct—I don’t even know what to call it—different from New Zealand and Australia. So I claim that if I have such an intuition, I do not necessarily have to give it up. Aristotle claims that you can define things that way too—true, you can define them that way, but you can’t perceive them that way. In other words, that you won’t be able to perceive, while this you can. I think I spoke about this in one of the workshops—I don’t remember if it was Thursday, probably it was—about the division of the sciences into different categories. For example, why divide between physics, chemistry, biology, I don’t know, psychology? Why not divide between phenomena that happen in Asia and phenomena that happen in Africa? Because this division of scientific fields seems right to us in some sense. What does “right” mean? Natural. Or I talked there about Newton. Yes, Newton: before we knew the law of gravity, everyone knew the phenomena. There is the phenomenon of tides, everyone knew the paths of the stars, everyone knew that bodies with mass fall toward the earth. All of these phenomena were familiar to every person in the street. No one imagined that all these phenomena reflected one law—the law of attraction, gravity. Tides, elliptical stellar orbits, and the fall of bodies to the earth. Then Newton comes and says, wait a second, all of these phenomena are really particular cases of the law of gravity. There is attraction between every two masses, and that explains all these phenomena and many others. Now before Newton proposed this, there was no reason at all to assign all these phenomena to one scientific field. Why assume they belong to the same scientific field? Tides, falling to the earth, and stellar orbits. I would have said: this belongs to astronomy, this to earth science, and this to, I don’t know what, something else; properties of the sea, tides, oceanography. Fine? So the claim that all these things belong to one scientific field is a non-trivial claim. In principle I could have classified it in a completely different way. We classify it this way because we found a common law that explains all these phenomena. But even that doesn’t say much. I found such a law—maybe I’ll find another law that will explain a completely different set of phenomena, especially if the law is a descriptive law, because I can describe phenomena in many ways. I gave—I think I may have given this example—the periodic table in chemistry. Today we divide substances as though they are built from elements in Mendeleev’s table, and every substance is built from a different mixture of those elements that appear in the periodic table. But in ancient physics they divided objects differently: into four elements—earth, wind, water, and fire; fire, water, wind, and dust. Okay? And there too they said that every object has a different mixture of earthiness, fieriness, airiness, and so on, and that too is a way of describing objects. By the way, it’s not a less correct way. It’s just another way. You can compare it to choosing coordinate systems. I can choose one coordinate system, I can choose another. There is no correct and incorrect coordinate system. I can describe reality on this coordinate system or that one, whichever is convenient. And therefore it is not true that the chemical description of the periodic table is more correct than the Aristotelian description. It is more scientifically fruitful—it works better. It lets us solve problems; it is more useful to us. But theoretically there could have been a creature whose way of thinking is built differently from ours, and perhaps it could have done the same thing with Aristotle’s division. It could have reached practical consequences and sent missiles to the moon with Aristotle’s division, not Mendeleev’s. And it could be that Mendeleev’s division is simply a function of the human mode of thought. It is convenient for us, and therefore we divide reality that way. Or the same regarding, yes, Copernicus. There is a very widespread statement that Copernicus discovered that the earth revolves around the sun and not the sun around the earth, as people thought before him. But that’s nonsense. Copernicus didn’t discover anything. Copernicus simply proposed a simpler way of describing our astronomy. That’s all. It’s not a truer description, just a simpler one. In the previous description I say the earth revolves around the sun and the paths are very very complicated. Copernicus says, forget it, let’s place the center of the coordinate system at the sun, not the earth. And suddenly you see that this is a much more natural coordinate system. You understand that if you have a circle, then if you describe it in a Cartesian coordinate system it’s complicated, right? x squared plus y squared equals r squared. That is the description of a circle in a Cartesian coordinate system. By contrast, in a polar coordinate system the description is r—r equals a constant. That’s it. Much simpler. Okay? So what does that mean? That the polar description of a circle is more correct than the Cartesian one? No. It’s simpler, not more correct. With Copernicus too that’s the case. Now from the standpoint of physics, what I’m saying now isn’t so simple. There is Mach’s principle, for those who know, meaning it may be that you can distinguish between the question whether body A revolves around B or body B revolves around A, depending on whether fictitious forces appear or don’t appear. I’m not getting into the physical definitions right now, but at the kinematic level, when I ignore forces and everything else and simply look at the motion, when I look at one body revolving around another there is no preference for that description over the description in which I now take this as the origin of the coordinate system and that thing that was standing still—actually, look at it—it is really revolving around this.
[Speaker C] After all, you started talking about Mach’s principle.
[Rabbi Michael Abraham] Okay, sorry, my internet somehow disconnected for a moment. The claim is that Copernicus—and I don’t even know up to where you heard me—Copernicus didn’t discover a finding that was more correct than what people thought before him, but rather a finding that was more convenient than what people thought before him. On the physical level, physicists have a definition of who is rotating, meaning a definition that prefers one description over another: that body A rotates around B, or B rotates around A—through fictitious forces, through all kinds of things of that sort. But that’s a physicists’ definition; it’s not really interesting. On the kinematic level, when you look at the motion itself, there’s no preference at all for the claim that this rotates around that or that rotates around this. It’s only a question of where you place the origin of the coordinate system. If you place the origin on this one, then it stands still, and you’ll see that one rotating. If you place the origin here, then this one will stand still and you’ll see that one rotating. So all of this is only a question of how you define the coordinate system. There’s no more correct and less correct here. So there are philosophers of science who argue that the laws of nature are basically some kind of coordinate system. We could have found a different coordinate system and described nature in a completely different way that would be no less correct. Say, a creature that thinks differently from us wouldn’t use our conceptual system. Say, it wouldn’t understand at all what velocity is, what acceleration is; those concepts wouldn’t make sense to it, it has no perceptual tools for those concepts. That doesn’t mean it would grasp reality less well than we do. It would grasp it differently. Yes, like someone who sees everything in green or everything in red—there’s no right and wrong here. The question is what you have over your eyes, this kind of cellophane or that kind of cellophane. In other words, it’s only—once I wrote about Gadi Taub, in his book The Stooping Rebellion, where he brings various feminists who criticize physics. They say physics is masculine physics. It’s constructed in a masculine way, and therefore men do better than women in that field. Okay? And then he says he wouldn’t want to fly on a plane built on the laws of feminine physics. Meaning yes, it’s preferable to use a plane built on physics; it usually works better, even though sometimes the charm of political correctness is greater than the charm of physics. Now, I think he’s mistaken. He’s mistaken because he understands this feminine criticism—yes, what’s called the new criticism—as proposing a different physics. But that’s not true. Not necessarily true. It proposes the same physics, a different description of the same physics. Someone built differently will see the same physics in a completely different way. Maybe sometime I’ll use here an example I once wrote about in an article. There’s Zeno’s paradox—actually several of Zeno’s paradoxes—which are meant to prove that the concept of motion is a fiction. What we see in the world as motion is a fiction, it doesn’t really exist in the world; it’s some kind of illusion of ours, or I don’t know exactly what to call it, a way we have of grasping reality. He has a series of paradoxes meant to show this—not only motion, but more generally. The most famous is Achilles and the tortoise, yes, how will Achilles catch the tortoise? He has his proof. A somewhat less famous paradox is the paradox of the flying arrow; I once wrote an article about that. He argues, basically: look at an arrow in flight. At every single moment that you look at it, it is standing in a different place. Now you look at it—it’s here. A moment later you look—it’s here. Another moment later—it’s here. So at every moment it’s standing in a different place. When does it pass between the places? When does it move? Okay, somehow at every moment it’s in a different place, standing in a different place. But then how does the motion take place? That’s basically a question, yes, a question about the concept of motion. Yes, frames, exactly. Usually this paradox is handled in terms of infinitesimals. I won’t go into the details now, but the claim is that he didn’t have infinitesimals—yes, differential and integral calculus—so he couldn’t understand the matter, but we have a coherent description of motion and everything is fine. In the article I argued that this solves nothing; all you did was find a language to describe this phenomenon, but you didn’t explain the difficulty. And I explained it there through the following point: look at how we define velocity in physics. Velocity in physics is a difference in positions. A body moves from here to here and it takes two seconds to do it, so its velocity is the distance between the two places divided by the time it took to cover that distance. Yes, the difference between points in space divided by the difference between points in time—that ratio is velocity. All right? Now, if the velocity changes, not constant velocity, then you have to do this at very, very close points, yes, that’s what’s called a derivative. When you make these differences at points that are extremely close to one another, tending almost to coincide, that’s called a derivative, and then you find the instantaneous velocity of the body. And at every moment it can have a different velocity, if it’s a body that doesn’t move at constant speed; then there’s a velocity function and every moment has a different velocity. Okay, now the concept of velocity—if you ask people, and I think even if you ask physicists—the concept of velocity is usually understood as something that doesn’t really exist at a point in time. A body can’t have velocity at a single instant, because velocity always requires going to a tiny, tiny, tiny interval—but still an interval. Okay? You have to show what distance it traveled and how long it took, but where the spatial interval is very, very small. Okay? But not at a point. What is velocity at a point? At a single moment in time it changes position? There’s no such thing. Rather, you can talk about an interval—yes, an infinitesimal or a differential—a time interval or a spatial interval as small as you like, but still an interval. You know what the difference is between a differential and a point in the simple view—yes, there are other mathematical definitions—but the accepted definition is that a point is zero-dimensional, and an interval as small as you like is one-dimensional; its length is just zero, tending to zero, but its dimension is one. So there is an essential difference between a point and an interval as small as you like, and therefore when you talk about velocity you need to talk about intervals, not points. Okay, and then yes, in mathematics when continuity is defined, they say that a line is not composed of points but of very, very tiny intervals—or what mathematicians sometimes call the property of density, not important right now—but it’s not just points points points lying next to each other very, very densely and that’s how a line is formed. Such points do not form a line; they remain points. And therefore the claim is that if you describe continuity correctly, you don’t run into Zeno’s paradoxes. But in my opinion that doesn’t really solve the paradox. It doesn’t solve it because it’s perfectly clear to me that the concept of a point in time is well-defined. There are points on the time axis just as there are points on the space axis. The fact that you found a description that manages to bypass the concept of a point, or not to use the concept of a point but rather an arbitrarily small interval, means you found a language that lets you escape the paradox, but you didn’t solve the paradox. It’s like in the introduction to Principia Mathematica, yes, by Russell and Whitehead. They propose a solution to all the self-reference paradoxes. They say: let’s build a language—the theory of types, yes, descriptive types—let’s build a language in which every sentence in the language can refer only to sentences lower than it in the hierarchy. You understand that in such a language there will be no self-reference. A sentence cannot refer to itself because it can refer only to sentences lower than it in the hierarchy. Is that a solution to the paradoxes of self-reference? Nonsense. Obviously not. At most it’s a language in which those paradoxes won’t appear. But that is not a solution to the paradox. A solution to the paradox is when you explain to me what is wrong in this self-referential sentence. For example, one indication that this is an ad hoc solution—the theory of types—is that the theory of types also forbids non-paradoxical sentences. Sentences that have, say, “the sentence I am saying right now is also a sentence.” I just said a sentence that refers to itself, but there’s no paradox here at all; it’s a completely legitimate self-reference. We all understand what it means, and Russell’s theory of types forbids saying this. In his language, such a sentence may not be said. Why? It seems that you are adopting something that has no intrinsic justification, only in order to solve the paradox. Yes, that is an ad hoc solution. And it’s not a solution. It’s finding a language, yes? Leibniz dreamed of an ideal language such that if we found it, no philosophical difficulties and no paradoxes would arise. The problem is that this ideal language would be a linguistic-logical achievement, but it would not be a solution to the paradoxes. Because that language would merely forbid me from expressing the paradox; it would not solve it. Because I can still express it in another language, and you haven’t explained to me what is wrong there. This is Stalin’s favorite solution to paradoxes. Whoever raises a paradox gets his head chopped off, and the paradox is solved. No problem, that’s how we solve paradoxes. What does that mean? If you raise a paradox, you’re using an illegal language. You mustn’t use that language; use only the legal language, and lo and behold there will suddenly be no paradoxes. It doesn’t work that way. Okay? So therefore I think infinitesimal calculus is some sort of language, but it isn’t really a solution to the arrow paradox. And I want to argue that the solution to the arrow paradox lies—and in a moment you’ll see the connection to us; I’m going on a bit, but I think it’s a good example—the solution to the arrow paradox, in my opinion, can be seen through a linguistic distinction. I think what Zeno lacked was not mathematics. What Zeno lacked was a linguistic distinction between two concepts: between “is located” and “stands still.” When I say that a certain object is located in a certain place, and when I say that a certain object stands still in a certain place, that is not the same statement. Those are two completely different claims. The claim that the object stands in a certain place means that it is there and also not moving. Its velocity is zero; both the derivative is zero and the position is fixed. When I say that it is located in a certain place, I mean it is in that place but in motion; it is moving. Now what do I mean by that? I mean that contrary to what Zeno argued, the arrow does not stand still at every moment in a different place; it is located at every moment in a different place, but it does not stand still—it moves. Zeno asked: then when does it move, if at every moment it stands in a different place? The answer is: when it is there, it is also moving. Motion does not contradict being located somewhere; motion contradicts standing still. Because standing still is being at zero velocity; motion is being at a velocity different from zero. Okay? So when you say that something is in a certain place, that does not mean it is not moving. But notice the price. The price is that I am talking about velocity at a point in time. Meaning, there can be an object that I define as moving even though I am speaking about a point in time, not a tiny interval. When one performs the derivative operation—again, I don’t want to go into the mathematical details, only to understand the principle—in the end the result is velocity at every point in time. Give me a point in time, I’ll tell you the velocity. But physicists see this as a fiction. Because velocity belongs only to an interval. As small as you want, but still an interval. It has to change position in a certain very small time, and then it has velocity. At a point in time there is no such thing as velocity. So you say: that is the body’s velocity in a very small interval around that point. I want to argue that this is not true. There is such a thing as velocity at a point. A body can have velocity while it is at a point in time. What cannot happen at a point in time is a change of position. That cannot happen, because you are in the same place. A change of position at a point in time is not even infinite speed. Because infinite speed is a change of position in a time of zero length, a time interval of zero length. But when you are at a point in time, it is a logical problem, not a physical problem of infinite speed. There is no change of position at a point in time because change of position at a point in time means being in two different places at the same point in time. That’s a logical problem. It’s not a problem of speed being too high, like the speed limit in relativity, the speed of light. You can’t exceed the speed of light—that’s a problem in physics. I’m talking about a problem in logic. A change of position at a point in time cannot exist, but velocity at a point in time can. And my claim is that velocity and change of position are not synonymous. Saying that a body has velocity and saying that a body changes place are not the same thing. My claim is that having velocity is the basic property, and change of place is an expression of the fact that it has velocity. It is not the definition of velocity. And this calculation of difference in positions divided by difference in times is not a definition of the concept of velocity but the way to calculate it. It’s an operational definition, not a definition of the concept itself. A body has velocity at a point in time. I don’t know how to calculate velocity at a point. In order to calculate it, I need to look at a certain very small interval around the point, say what distance it covers, divide by the time it takes, and then I discover the velocity. But that is only the way to calculate it. The concept of velocity itself characterizes the body at a point in time—that’s what I want to claim. And that basically means that the concept of velocity is not derived from the concept of place. It has an independent existence. There is a connection between them. The claim that velocity is derived from place is not a definition; it is a claim. Okay, there is a connection between these two things—call it a discovery of physics, if you like—it is not a definition. And therefore my claim is that Zeno’s question—when does the arrow actually fly?—is not the same thing as when it changes place. It changes place all the time: at one time it is in one place, at the next time it is in another place. When does it have velocity? At every moment it has velocity. Even when it is at some point at some specific moment, it has velocity. And having velocity does not contradict being in a place. It contradicts standing still in a place, because standing still means being there at zero velocity. That is what having velocity contradicts. Okay? And therefore I think this paradox does not arise at all. So I continued from there, and now I’m getting to the important point. My claim was that in principle, if we were built differently, then the concept of velocity would be grasped by us naturally, and we wouldn’t need to derive it from the concept of place or changes of place over time. We are built in a way of static perception. So the natural thing we perceive is the location of the body. When I ask whether it has velocity, I can’t perceive that it has velocity; I simply see that it changes place. And if it changes place, that is an indication that it has velocity. How do I calculate velocity? Difference in positions divided by difference in times. But that is all because that’s how I’m built. We are simply built in a form of static perception of location. But think of a completely different creature, one built in such a way that its perception is dynamic. Say, to illustrate this: with us, when we make a film, a film is built from frames. Right? These are pictures that differ slightly from one another and are projected at a very, very high speed, and it appears to us as continuous motion. Why can’t one make a film of motion itself? Why do we need to build it from static pictures at high speed? Because we are built in such a way that the basic thing we perceive is a static image. And in order to perceive motion, I need many static images one after another, and that creates for me the illusion of motion. But even actual motion we perceive that way, as with the arrow in flight. We perceive the arrow here, and then here, and then here, and then here. We don’t perceive it moving. But theoretically there could be a creature whose sensory system is not like ours. It looks at bodies and sees velocities, not positions. For it, position is the integral of velocity, not velocity the derivative of position. Because for it velocity is the thing it sees, and place it understands as some kind of integral over velocity. Again, the definitions of the concepts are not what matters right now; what matters is that velocity is the fundamental thing and from it one derives place, and not the other way around as we perceive it. That is a different perception. By the way, I explained there that the uncertainty principle in quantum theory is actually about this. The uncertainty principle is really about two ways of perceiving the world. You can perceive—say, between position and momentum, one of the uncertainty principles—one possibility of perceiving the world, and physicists call this the position representation. What is the position representation? That everything described in the world is a function of position. This body is in this place at this time, in that place at that time, and so on. If I want to know the body’s velocity, I have to differentiate the position and divide by—yes, difference in position divided by difference in times. Differentiate the position. Okay? But there is a completely different representation; it’s called the momentum representation. Momentum is velocity, yes, more or less. Let’s call it the velocity representation for the sake of discussion. What is the velocity representation? Everything in the world is described as a function of velocity, not as a function of position. This body, at every moment, has such-and-such a velocity. I don’t know what place it is in; I have no idea. I see velocities; I do not see places. If I want to know its place, I will integrate the velocity. That is a different representation. And quantum theory speaks about these two representations, and the beautiful thing it says—the uncertainty principle in quantum theory says that these two representations can never be combined simultaneously. They cannot be combined at the same time. You cannot see a body’s position and velocity simultaneously. Why? I claim it is because the position perception contradicts the velocity perception. The point is: when you perceive velocity, there is no place in your world. There is velocity; you only see how fast the body is moving. You don’t grasp where it is. Think of a camera, yes, photographing some object passing very fast, so you see a kind of trail of the object. When you focus on velocity, you cannot define exactly where it is. And when you focus on place, you have no idea what the velocity is, because you’re looking at it at an instant. Think, for example, of someone familiar with radar—yes, there are radars based on the Doppler effect. What is the Doppler effect? A lot of physics today, but it’s just a good illustration of what I’m talking about. Doppler effect: I send an electromagnetic beam toward a moving body, like a car. The beam returns to me at a slightly different frequency. The difference between the frequency I sent and the one that returned is proportional to the speed of the car. And that’s how we find out which of us is the criminal. Okay? The simple way of looking at the Doppler effect is basically that I see, I perceive the car’s speed, but I perceive it instantaneously. The beam hits the car for a second and comes back, and that’s enough to know the speed of the car. Usually when we measure speed, we say: the car is here at this time, here at that time; the difference in positions divided by the difference in times is the speed. But in the Doppler effect—this isn’t exact, I’m cheating a bit here—but in the Doppler effect, you supposedly grasp the car’s velocity at a point in time. You don’t need to sample it at two points. One point. And this means that the concept of velocity does not necessarily have to be calculated or grasped by differences in position divided by differences in times. There is such a thing as velocity at an instant. And I might perhaps even get to it directly, not through calculations of place divided by time. It’s just that we are built in such a way that it is more convenient for us—or that is how we are built—to look at static images of things, and dynamics is always some derivative of static states. But there could be another creature that works with integration, not differentiation. For it, the dynamic properties are the basic thing, and the static properties are an integral over the dynamic properties. Just a different creature. We cannot grasp this because we are not like that. But there is no problem defining such a creature. Quantum theory defines this entirely and regards it as completely equivalent to the first creature. And what it also tells us is that these two creatures can never reside in one head. If you are this, then you are not that, and vice versa. You cannot be both a camera and a camcorder at the same time. There’s no such thing. If you are a camcorder, then by definition you are not a camera, and vice versa. Yes, by camcorder I don’t mean a camcorder made of frames. I mean a camera that films motion itself. A camcorder, for me, means something that photographs the motion itself, not something that produces it. Yes, I don’t know exactly what those things are, the Doppler effect and so on, but it sounds like something close to what I’m talking about here. I have no idea what it is. The claim is that these two ways of looking give you a completely different picture of the world. Now who is right? There is no such thing as right. There are two equivalent pictures. You can see the world in the momentum representation; you can see it in the position representation. It’s not—there’s no right and wrong here. The whole question is how you are built, or what measuring instrument you use in quantum theory. In quantum theory it depends on the measuring instrument you use. Philosophically I would say it depends on your cognitive mode, yes, on how your perception is built. And therefore, I return to Gadi Taub’s claim about the feminist critiques. My claim is that there may be something to it, because physics was developed by men. It may be that it is built in a language that is more accessible or more convenient for men. And if women had been at the forefront of science at the right time, maybe they would have developed a completely different, but no less correct, description of the same physics. And it would have been possible to build airplanes with feminine physics. Such feminine physics would not have been built on velocity, acceleration, energy, momentum, and all the masculine concepts, but on—I don’t know—empathy, kindness, and nursing children. I don’t know, and breastfeeding squared divided by empathy equals velocity. I don’t know exactly how they would formulate it, but it is definitely possible that there is a completely different description of the same physics that is no less correct. It is simply a different coordinate system, as I said before. And therefore it is often easy to argue—and now I return to our own line of thought—that our way of looking at the world is a function of how we are built. There is no right and wrong here. If someone else were to describe the world differently, then it would be different. It would not be any less correct or more correct than the way I grasp the world. And from here it is a short road to saying that all scientific descriptions and the laws of science and everything are basically a result of how we are built. There is nothing more natural about the division between horses and donkeys than about the division between animals in New Zealand and animals in Australia. We are built in such a way that one seems more natural to us, and with that we can make progress. In principle one could create a completely different taxonomy of animals, built according to location and height and I don’t know exactly what, and arrive at the same results. Meaning, to understand nature in a completely different coordinate system from the one accepted today, but no less correctly. In principle, that could be. Now we have to decide: do we accept this description? It could be; that’s true. The question is whether it is true, not whether it could be true. When we look at the division between horses and donkeys, the feeling is that it bears fruit because it is true in some sense, and not merely because that’s how we are built. There is something more natural here. Now, I have no way to prove that. I am only trying to draw your attention to the fact that Platonism is deeply ingrained in us. It is not some invented mystical notion. This is how we see reality. Now, the example I want to bring on this matter is a very beautiful literary example from Borges. Borges, the well-known Argentine writer—a wildly brilliant genius, in my opinion. I don’t think there is any other writer, maybe aside from Kishon, about whom I can say that he really was a genius, obviously a genius without any doubt. Borges has short stories. In Hebrew there was a collection called Fictions, and the first story in the collection is called “Tlön, Uqbar, Orbis Tertius.” And he goes completely wild there—it’s absolutely insane—with imagination and astounding philosophical understanding, with amazing knowledge in many fields. A truly unique person. By the way, there is an interesting book by Leo Corry from Tel Aviv University, from the institute for the history and philosophy of science and ideas, a book dealing with Borges. It has a chapter devoted to the connection between Borges and The Name of the Rose—what’s his name, the Italian? I’m blanking on the name. Doesn’t matter. The author of The Name of the Rose, I’m forgetting his name right now. Umberto Eco, exactly. Also a very interesting creature, a very interesting Renaissance man. In any case, Borges’s story speaks about some planet, Tlön—that’s what it’s called. And on that planet there are idealist inhabitants. What does that mean, idealists? Inhabitants who do not believe in the existence of an objective world. Yes, Umberto Eco, exactly. They do not believe in the existence of an objective world, of objective objects. They perceive everything as some collection of my subjective perceptions. You do not exist, my computer does not exist; they are only images inside my imagination. Whether I exist—that already brings us to Descartes’ cogito. In any case, that is the idealist or solipsist view or something of that kind. Now he gives this phenomenon a fascinating literary description. He says that in the southern hemisphere of Tlön—he of course has all kinds of encyclopedias and hidden scrolls through which we can learn about the culture that existed on the planet Tlön, all of it his inventions of course, it’s just his imagination running wild—so he found some manuscript from which it emerges that in the southern hemisphere of Tlön there was no language that used nouns at all. So instead of saying, “The moon rose above the river,” they said, “It mooned above the ongoing flowing.” There are only verbs, no nouns. Why? Because in an idealist world there are no objects; objects are only our perceptions. All I can say is that I see mooning above ongoing flowing. “River” is already some static name we give, an object, of which the ongoing flow is a property. Okay? Or “mooning” is a process or an occurrence that happens to the object moon. But there are no objects in an idealist world. So if we want to present this in language, then the language will be made only of verbs, with no nouns. When we speak in terms of nouns and verbs, our language reflects—almost without our noticing it—our philosophical conception, namely that we are not idealists. That we understand there are objects, and the properties are their properties. But in Tlön that is not so. Now he says that in the northern hemisphere, this is even more brilliant. In the northern hemisphere there is a fascinating twist. There they do have nouns in the language, but the nouns in the northern hemisphere are any collection of characteristics you want. Usually when we define a noun, say this phone—we say it is rectangular, made of plastic, you can call with it, you can talk, you can watch YouTube, okay, doesn’t matter, all kinds of properties of this device. It costs such-and-such, it is built this-and-that way, and so on. A collection of properties. That collection of properties is the phone. And since in the idealist world there are no phones—there are only our perceptions of phones, or only the properties of phones—then in that world the objects are not only phone, book, lamp, computer, table, the things we have, but what? Any collection of properties you like is a noun. For example, the cry of a bird in the distance together with the color of the cloud that passed over my head at seven in the morning—that is called “cry-cloud.” A cry with a cloud, yes? That is a noun. Why not? Any collection of properties can be a noun. Now why is that brilliant? Because he puts his finger on a very beautiful illustration of the absurdity in idealism. In other words, what you really want to say is that this object, the phone, there is nothing in it different from “cry-cloud.” This is a collection of properties and that is a collection of properties. So I can define this as an object and that as an object too. Any collection of properties. The height of so-and-so together with the voice frequency of someone else—that too will be some noun, let’s call it “world-release.” Okay? The names don’t matter; choose whatever names you like. So in an idealist world there is no difference between these strange objects and what we are used to as objects. Why? Because they do not believe that behind the phone there is an object either. It’s a collection of properties. Well then, if a collection of properties is what is called an object, then any collection of properties can be an object. What is special about the phone or the table or the lamp compared to “cry-cloud”? What will the realist, let’s call him that—the parallel to the Platonist—the realist answer to this? He will say: what do you mean? A phone is something that exists, and it has a collection of properties, and that collection of properties is anchored as one distinct group because they are properties of the same object. By contrast, “cry-cloud” is two properties of two different objects. So you cannot use the collection of properties—he was mistaken—to define a noun. What is the difference between the realist perspective and the idealist or solipsist one? The difference is the question whether you understand that behind the collection of properties there is an object that has the properties, or whether the collection of properties is the object. Okay, that is basically the claim. Borges wants to illustrate for us the absurdity in the idealist view, so he says: look, if you think all objects are merely collections of properties that you perceive, then if a collection of properties is what I call an object, every collection of properties should be an object. And there are really two kinds of answers to this. An Aristotelian answer and a Platonic answer. The Aristotelian answer says: yes, but it is more convenient for me to speak of the phone as an object and not of the cry-cloud. It is more fruitful, more useful, it helps me understand and use things better, so I use this division, this definition of objects. But you’re right, in principle there is no difference. That could be an object just like this. The Platonist will say: no, absolutely not. I see that here there is an object that possesses this collection of properties, and therefore this collection of properties can be defined as a group of properties constituting an object, because they characterize a certain object. By contrast, cry-cloud is different properties of different objects, and therefore there is no reason to bundle such a collection of properties and call it a noun. It simply does not exist. So it all begins with the question whether you understand that the object is something that exists in reality, or whether the object is just our abstraction, our collection of properties. You see the connection to the Aristotelian-Platonic dispute. Let me sharpen it a bit more, and with this we’ll finish. What I want to say is that the Aristotelian and Platonic dispute is basically the same dispute that Borges describes between idealism and realism, but this time regarding concepts, not regarding objects. Borges describes there how I perceive objects—phone, lamp. Is it a collection of properties, or is there an object that has the properties? That is a dispute between realism and idealism. The dispute between Plato and Aristotle is about concepts. What is horseness? What is democracy? It’s not about the objects, not about the horse. The first dispute I described is a dispute about the horse—whether the horse exists, or whether it’s just a collection of properties that I call a horse. Plato and Aristotle argue about horseness. Plato says horseness is something that exists, and therefore the collection of the horse’s properties—having four legs, a tail, galloping, being rideable, the properties that distinguish a horse—there is reason to define a concept for them. Why? Because the idea exists, and this collection of properties is the collection of properties of that idea, and therefore it is a natural definition. By contrast, a definition of animals in New Zealand—that has no idea behind it. Therefore you have gathered things that have no connection between them, and there is no reason to define this as a species or kind of its own in the taxonomy of animals, in our zoology. It’s not that there is no point in using that coordinate system; it is not correct to use that coordinate system. It is not natural to use that coordinate system because it does not exist, it does not describe something that really exists. That is precisely the difference between Plato and Aristotle. You see that it is exactly the same argument we saw earlier with regard to objects, now appearing between Plato and Aristotle with regard to concepts. Plato understands that the concept of democracy is not elections every few years, separation of powers, human rights, and I don’t know what else you’d define there, some set of characteristics. Why not define countries that grow peaches, have at least four million inhabitants, and lie on the coast as “flopobatic states”? Why not define such a thing? So Aristotle will tell us: because it is not useful, it is not fruitful. Not because it is not correct, not because there is a difference. You can define it if you want, but it is not useful. Why is it not useful? Because my mind is built in a certain way, and these definitions are useful to me and those are not. But he does not claim that it is not correct in some sense, or less correct. Plato will say: there is no idea behind that; it is a collection of properties like cry-cloud. It is a collection of properties that do not come together into one thing. What unifies the collection of properties of a democratic state such that it is worth giving it a name? What unifies them is the concept of democracy. These properties characterize the concept of democracy. The direction is from the concept to the properties, not from the properties to the concept. The collection of properties is not the concept; rather, there is a concept, and when I look at it, I see that it has several essential properties, and those are its properties. A completely different perspective—the Platonic and Aristotelian perspectives. And that parallels the distinction between realism and idealism with respect to objects, to things. Well, I’ll stop here because I have to leave, which is why I moved today earlier, and I apologize that I also won’t stay here for questions if there are any—you can send them by email or… Well then, have a peaceful Sabbath.
[Speaker B] Thank you very much. Have a peaceful Sabbath.