Conceptual Analysis – Lecture 7
This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.
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Table of Contents
- The domino puzzle and the chessboard
- Concepts, characteristics, and Kant
- Zeno’s paradoxes: motion as inconsistency in conceptualization
- The paradox of the flying arrow and the distinction between “being located” and “standing still”
- Velocity at a moment in time: quotient versus potential
- Perception, camera, film, and a picture of velocity
- Doppler and measuring velocity at a moment in time
- Definition by negation, logic, and Maimonides’ doctrine of negative attributes
Summary
General overview
The text argues that concepts are abstract entities that have existence, and that our understanding of them is the product of observation and cognition, not just thought, so definitions and properties are ways of representing a concept rather than the concept itself. The example of the chessboard and dominoes shows how colors that are not essentially relevant can help reveal an abstract mathematical property. Zeno’s paradoxes are used to show that the contradictions point to a failure in our language and conceptualization, not to the nonexistence of motion; in particular, the arrow paradox is resolved by a conceptual distinction between “being located” and “standing still.” From this, a philosophical conception of velocity is advanced, according to which velocity is a potential for changing position over time, and conceptual analysis is presented as a process of refinement through contradictions, including a parallel to Maimonides’ doctrine of negative attributes and to the image of the sculptor and carved relief.
The domino puzzle and the chessboard
The puzzle asks whether it is possible to cover an eight-by-eight chessboard from which the two squares at the ends of the main diagonal have been removed—say A1 and H8—using dominoes, each of which covers two squares. The claim is that this is impossible, because every domino always covers one black square and one white square, whereas removing two squares of the same color leaves unequal numbers of black and white squares, making such a covering impossible. The black-and-white coloring is not a property essentially relevant to the problem, but rather a representation that makes it possible to notice an abstract mathematical property of two classes of squares, a property that existed even on an uncolored board but is harder to formulate and detect.
Concepts, characteristics, and Kant
The text argues that the characteristics we ascribe to a concept are not the concept itself, but only characterize it, just as a collection of properties of a table is not the table itself; the table is the object that has those properties. There is an abstract entity of the concept that we cannot really speak about without beginning to characterize and define it, and the definitions are the way we grasp it. This claim is presented as an extension of what Kant showed regarding objects, and here the same move is claimed regarding concepts.
Zeno’s paradoxes: motion as inconsistency in conceptualization
Zeno presents paradoxes to show that the concept of motion contains internal contradictions, and this is usually presented as though he denied the existence of motion. The text questions this and suggests that the more plausible conclusion is that our conceptualization and description of motion are problematic and lead to contradictions, and that one should therefore look for a different conceptual framework for describing motion without contradictions. The paradox of Achilles and the tortoise receives a mathematical solution by means of the finite sum of an infinite convergent series, in which the time needed to catch the tortoise is one and one-ninth seconds, so dividing the time into infinitely many steps does not prevent the catching from actually occurring.
The paradox of the flying arrow and the distinction between “being located” and “standing still”
In the arrow paradox it is claimed that at every moment the arrow “stands still” in a particular place, and therefore the question arises: when does it move and pass between places? The text argues that the failure comes from confusing saying that the arrow is “located” in a place with saying that it “stands” there, because a body can be in a certain place while in motion without standing still. The proposed solution is that the arrow is in a different place at every point in time, but at no point in time does it stand still; it is always moving, and therefore there is no difficulty in saying that it moves at every moment while at every moment being somewhere. The text presents this as a conceptual analysis in which “being located” and “standing still” are of the same kind in the sense of location, but are different species in their relation to motion at that same point in time.
Velocity at a moment in time: quotient versus potential
The text argues that people have trouble accepting velocity at a moment in time because defining velocity as distance divided by time is an operational definition for calculation, not a definition of the concept itself. According to the computational definition, at a single instant there is no change of place, and therefore dividing distance by zero time yields the absurdity of infinite velocity, so it seems that there is no velocity at a point, only over a small time interval around the point. Against this, the text advances a philosophical position according to which a body does have velocity at a point in time, and velocity is the potential to change place over time, while the actual change of place is a result that becomes evident when one waits for a stretch of time. The example of a ball hitting a wall is used to illustrate that the potential for change of place can be present even when the change of place is not realized because of a limitation, and the potential can manifest itself in other ways such as heat, recoil, or exchange of momentum.
Perception, camera, film, and a picture of velocity
The text explains that the difficulty in seeing motion at a moment in time stems from a perceptual limitation: the visual system works like a camera that captures static positions, and motion is created for us by connecting many frames, as happens in film, cartoons, or the movement of a cursor that disappears and reappears in another place. An imaginary creature with a sense that registers velocities rather than positions would see velocity at a moment in time without waiting for a change of place, but would not know location. The text connects this as a hint to Heisenberg’s uncertainty principle in quantum theory and to two ways of looking, called the position picture and the momentum picture, in which choosing a description in terms of position rules out a grasp of velocities and vice versa.
Doppler and measuring velocity at a moment in time
The text brings the example of police radar based on Doppler technology, where a beam is sent at a moving car and the change in the return frequency is measured, from which the object’s velocity is calculated. The example is presented as an illustration of the principled possibility of detecting velocity through interaction at a moment in time, and not by way of a quotient of change of place over time, even though it is noted that the description is not entirely precise.
Definition by negation, logic, and Maimonides’ doctrine of negative attributes
The text argues that the cases of the chessboard and of Zeno’s paradoxes show how definitions of concepts sometimes work by way of negation: one proposes a definition, discovers a contradiction in it, and understands that the definition is a limited representation rather than the concept itself. Someone who identifies a concept with its definition mistakenly concludes that when the definition collapses the concept does not exist, whereas the correct conclusion is that the linguistic description or conceptual framework does not grasp the concept in its fullness. The text connects this to Maimonides’ doctrine of negative attributes and suggests that the negations do not leave a vacuum but rather refine perception and teach something positive at a more abstract level, just as negating “velocity is a quotient” leads to understanding velocity as the potential to change place. Logic is presented as a tool with a destructive role of locating contradictions and cleaning the picture of them, but this role is described as constructive destruction that ultimately leaves a refined concept free of contradictions, in the image of the sculptor who removes waste or of carved relief in which what is unnecessary is removed and what remains is the work itself.
Full Transcript
[Rabbi Michael Abraham] Last time I spoke about—those who remember—the domino and chessboard puzzle, and that was in the context of trying to present concepts. Right, I tried to show that concepts are basically some kind of entities, things that exist. That our understanding of concepts is the product of observation. We are basically looking at concepts with the mind’s eye, or some kind of cognition, and not just thinking. And we saw various implications—that one can argue about concepts, and one can change the definition of concepts, and all kinds of things like that. In the end I finished with that puzzle of the chessboard. Right, whether we can tile an eight-by-eight chessboard after cutting off the two squares at the ends of the main diagonal—say A1 and H8. We removed them, and the question was whether it is possible to cover the board with dominoes, each one covering two squares. So I said that because this comes in the context of chess it’s easier for us to think about it, and basically the claim is that it can’t be done. Because if you think about it you’ll see that every domino covers two squares of different colors. That is, one white and one black. Now when we removed the two extreme squares of the diagonal, we removed two squares of the same color. Because the diagonal is made up of squares of the same color. And since that’s the case, what remains is a board with an unequal number of white and black squares. And that means it can’t be covered with dominoes. Because a domino always covers one black and one white, and therefore however many dominoes we place, it will always have to be an equal number of black and white squares. That’s actually the proof that you can’t cover this board with thirty-one dominoes. And what I wanted to show through this is that really the color of the squares is not relevant to the solution at all. In other words, this property of the chessboard as I see it is not the point at all. If the board weren’t colored, you could ask exactly the same question and give exactly the same answer. It would just be harder for us to formulate the answer. In order to formulate the answer successfully, or to notice the answer, it’s more convenient for us to deal with a colored board. Colored like a chessboard, black and white. But in truth the color isn’t relevant. We added a feature here that isn’t really in the problem. If nobody had ever invented the chessboard, and we weren’t familiar with game boards like that—checkers or chess or something like that—nobody would think of starting to color a board in that way. And then it really would have been very hard to solve this puzzle, even though the color isn’t really relevant. What we see here is that many times the way a thing appears to our eyes reflects some property of the thing that is actually the essential point. After all, all the black squares—even on an uncolored board—all the black squares have some feature in common. And so do the white ones. It’s just that I don’t know exactly how to define that feature, and I wouldn’t have thought that that feature was important for solving this puzzle. The black-and-white coloring helped me focus my thinking on that point. So this is an example of how we present some abstract property of these squares by means of colors. But really it has nothing to do with colors. The colors are only a representation. The property itself of these squares is some mathematical property of theirs. That is, it has nothing to do with colors; the colors merely represent the existence of that abstract property. And in a more general context, what I basically want to argue is that with concepts we give these concepts various characteristics. We define a concept through one characteristic or another of the concept. And my claim was that these characteristics are not the concept. These characteristics characterize the concept. Just as the collection of a table’s properties is not the table. The table is that object that has those properties. So I claim that with concepts too it’s like that. In other words, there is some abstract thing that we can’t really talk about until we begin to characterize it, find its properties, define it, and through that we actually grasp it. Just as Kant showed regarding objects, I claimed regarding concepts. And maybe I want to bring one more example of this matter, and that will bring us back to another example of this issue—the way a thing appears to our eyes as against the thing itself. There is Zeno’s paradox called the paradox of the flying arrow. Right, Zeno presented a set of paradoxes whose purpose was to show that the concept of motion is a concept that contains internal contradictions. And so basically his claim—I don’t know exactly what his claim was—this is usually presented… sorry? Yes. Usually this claim is presented as though he denied the existence of motion. That in fact there is no such thing as motion. Okay? I’m not sure that’s really what Zeno claimed, but maybe I don’t know—I didn’t read his words in the original, of course. Not at all. But this whole set of paradoxes was intended to show that the concept of motion is a concept containing internal contradictions. And therefore such a thing can’t exist; there is no phenomenon of motion in the world. Now why do I somewhat doubt that Zeno brought these paradoxes in order to deny the existence of motion? Because it’s hard for me to believe that a sane person denies the existence of motions in the world. Rather, what could be the case? It could be that our conceptualization of the concept of motion contains internal contradictions. We don’t have a good description, a non-contradictory description, of the concept of motion. I think that may be the conclusion from Zeno’s paradoxes, and therefore maybe what needs to be done is not to deny the existence of motion, but to look for a different framework, some other conceptual framework within which it will be possible to describe concepts of motion without running into internal contradictions. Okay? So let’s take the example—one of the well-known paradoxes he brought is Achilles and the tortoise. Right, where Achilles races the tortoise—let’s say he runs ten times faster, okay? So he gives the tortoise a ten-meter head start. Then the gun goes off, they start the race, and say Achilles runs ten meters per second and the tortoise runs one meter per second. Okay? So after one second Achilles is at the point where the tortoise was at the beginning of the race. Meanwhile the tortoise has advanced one meter, because in one second the tortoise covers one meter. A pretty speedy tortoise, but okay, for our purposes that’s the tortoise. Now Achilles will also cover that one meter; it will take him a tenth of a second, but meanwhile the tortoise will already have advanced another ten centimeters. Achilles reaches those ten centimeters, and the tortoise has advanced another centimeter. Achilles reaches that centimeter, and the tortoise has advanced another millimeter, and so on. And therefore Zeno’s claim was that Achilles will in fact never catch the tortoise. Now our own eyes can see that when Achilles races the tortoise, he does catch it. So this thing is really a paradox; it’s not a proof that Achilles won’t catch the tortoise. So what does it show us? It shows us that our description of the motion of Achilles and the tortoise—the concept of motion—apparently our description of that concept is problematic; it’s a description that leads to contradictions. That’s the conclusion one might draw from Zeno’s paradoxes, not to deny the concept of motion. I simply don’t think that’s the conclusion that follows from them.
[Speaker C] Maybe Zeno just isn’t presenting the situation correctly? What? Maybe Zeno doesn’t describe the fact properly—once he gets to that meter in one leap he’ll pass the tortoise. What kind of presentation is that?
[Rabbi Michael Abraham] It doesn’t work with one leap. What’s “one leap”? Tell me how many seconds it takes him to do it. Give me the amount of time it takes him to do it. “One leap” solves nothing. Tell me how long it takes him, and every time you tell me at what time, I’ll show you that the tortoise is still ahead of him. So in fact he doesn’t catch it. You can’t solve it that way; it doesn’t work like that. The solution is simple, of course. Achilles and the tortoise—the mathematical solution is very simple. All the claim says is that if you look at the times it takes for each such step, so we said: the first step was one second, in which Achilles passed the first ten meters. After that he passed another meter; that took a tenth of a second. After that he passes another ten centimeters; that took another hundredth of a second. A thousandth, one ten-thousandth, and so on. So how much is that altogether? 1.111111… seconds. Right? Yes.
[Speaker C] There you go.
[Rabbi Michael Abraham] Exactly. During the first one and one-ninth seconds, Achilles really does not catch the tortoise. 1.111111… is one and one-ninth, right? So during the first one and one-ninth seconds Achilles really does not catch the tortoise. If you do the kinematic calculation, you’ll see that the time it takes Achilles to catch the tortoise is one and one-ninth seconds. The moment the time goes beyond one and one-ninth seconds, Achilles has caught the tortoise. All Zeno did was divide the first one and one-ninth seconds of the race into infinitely many steps. But that doesn’t mean we’ll never get there.
[Speaker B] An infinite series too can have a finite value.
[Rabbi Michael Abraham] Yes, it can converge to a finite sum. So the fact that you divide it into infinitely many steps doesn’t mean it will never happen. It will happen after one and one-ninth seconds. It’s just that you chose to divide those one and one-ninth seconds into infinitely many small intervals, yes, into a sum of infinitely many segments that keep getting smaller. Okay, divide it however you like, but after one and one-ninth seconds Achilles has caught the tortoise. And therefore here there is really no paradox at all. It’s just that, of course, one needed this modern concept of convergent infinite series—that is, that the sum of a series, even if it has infinitely many terms, can still be finite. Right, a series here—in this case a geometric series whose ratio is one tenth. Any geometric series whose ratio is less than one basically converges. Fine, so that’s the easy part. But the paradox I wanted to deal with here is the paradox of the flying arrow. I also wrote an article about it; I assume some of you have already heard this from me. Zeno makes the following claim. Look at an arrow flying through the air. At every moment I observe it, it is basically standing in a different place, right? At this moment when I observe it, it stands here. A second later, when I observe it, it stands here. Another second later it stands here. In fact at every moment it stands in a different place. The question is: at which moment does it pass from one place to another? Right, that’s one formulation of this paradox. There are various formulations. Basically this can’t happen, because at every moment it is there—at one moment it is here, at one moment it is here—so when did it pass in between? How did it pass from this place to this place? When?
[Speaker D] But what are you defining as a moment?
[Rabbi Michael Abraham] When is it in motion? Right, after all, at every moment it stands in a different place, so when is it in motion?
[Speaker D] What is a moment?
[Rabbi Michael Abraham] A moment is some period—a point in time.
[Speaker D] A point in time of what? What is it equal to?
[Rabbi Michael Abraham] Equal to nothing. A mathematical point.
[Speaker D] A point in time has to have some particular time value. A quarter of a second, half a second, a tenth of a second.
[Rabbi Michael Abraham] Yes. One second.
[Speaker D] No. In one second it already passed; it isn’t in one particular place.
[Rabbi Michael Abraham] x equals one. Not the whole second, but the endpoint of the interval. x equals one, t equals one.
[Speaker D] I don’t know what t is.
[Rabbi Michael Abraham] Time. t is time. Yes. Okay. I’m asking what happens at time t equals one—say I begin measuring time from a certain point. I say, okay, one second has passed, two seconds have passed, two and a half seconds have passed. Every such point is a point on the time axis. Think of the time axis as a continuous number line. Okay? So on that line there are points.
[Speaker D] Points where every point has some specific time value. Yes, right.
[Rabbi Michael Abraham] Now I’m asking about the point t = 1.35. Okay? That’s one mathematically well-defined point on the time axis. The arrow isn’t moving there, right? At that point it is in a certain place. It is standing in a certain place. At another point in time you’ll see it standing in another place. So the question is: when, then, does it move? At every point it’s standing in another place, so when does it pass between the points? Okay, that’s one formulation of this paradox. Now, to my surprise, when I started dealing with this, I saw that at least as far as I looked, there really wasn’t a solution to this paradox—up until I wrote the article, I didn’t find a solution to this paradox in the literature, until I wrote the article. What I did find was people who suggested solving it through infinitesimals in mathematics, infinitesimal calculus, and the claim basically is that on the time axis there are no points; the time axis is made up of arbitrarily small intervals. Anyone who knows a bit of differential and integral calculus knows that what’s being discussed there is infinitesimals, right, not points. Meaning, the time axis is not really made up of points but of intervals—but very, very, very small intervals. I’m really using very imprecise formulations here, but there’s no choice, I can’t get into the formalism here. So the claim—I saw various people claiming that in fact infinitesimals, right, the calculus Leibniz and Newton began, differential calculus—it actually solves the paradox of the arrow in flight. Because then we understand: there’s no such thing as an arrow standing at a point in time; rather there is always a small interval of time, and then within that interval—it’s an interval—so it can move; it’s not a point at which it is in one defined place. My problem with that solution is that it solves nothing, because all you’re telling me is that there is, at most, another description—a different description of motion in which maybe it will be impossible to formulate the paradox, but that doesn’t solve the paradox. I’ll ask the same question about infinitesimals too; I’ll ask exactly the same question. After all, there are points on the time axis. You can’t deny the fact that there are points on the time axis, there is a mathematically well-defined point in time. So now I ask: at every such point the arrow has some position. You can’t deny that either; in the infinitesimal description too that’s the case. And if so, then what have you solved? You showed me that you can wrap up or bypass the problem by means of some formulation that talks about intervals rather than points, so you don’t want to talk about the points and therefore there’s no problem. The fact that you don’t want to talk about it doesn’t mean there is no problem. So don’t talk about it—but I am talking about it. I think I mentioned—I don’t remember whether I mentioned—the introduction to Principia Mathematica by Russell and Whitehead. It’s some monumental book like that, three volumes, on mathematics, trying to ground all of mathematics in set theory. So in the introduction to that book, which almost nobody has read, including me—so don’t suspect me of having read it—it’s three such volumes that contain only formulas, there are almost no words, it’s all just formulas. I did read the introduction. In the introduction, Bertrand Russell proposes a solution to the paradoxes of self-reference. Right, like the liar paradox, or the barber paradox. Right, the barber who shaves all the people who do not shave themselves. There’s such a barber in Seville, who shaved all the people who do not shave themselves. And now the question is whether he shaves himself or not. If he shaves himself, then he belongs to the group of people whom he does not shave, so he does not shave himself. But if he does not shave himself, then he belongs to the group of people whom he does shave, so he does shave himself, and so on. So that is a set of paradoxes—the liar paradox, right, “This sentence is false,” about that sentence itself. These are sentences that contain self-reference, right, self-reference or self-indication. These are sentences that refer, among other things, to themselves as well. So Bertrand Russell proposed a solution to this whole collection of self-reference paradoxes by defining there a language, what is called type theory, right, the theory of types, which divides statements into a hierarchy of statements. Never mind, some hierarchy that he defined there, and his assumption is that every statement cannot refer to other statements that are on its own level, right, on its own level, but only to statements lower than it in the hierarchy. Now of course you won’t be able to formulate statements of self-reference anymore, because a statement cannot refer to any statement on its own level, and certainly not to itself, because it too is on its own level. Okay? Like my favorite saying that I am my own brother, because I and myself have the same parents, you know, and therefore we are brothers. A statement cannot refer to any other statement that is on its own level in the hierarchy, and certainly not to itself, which is also on that level. But this of course solves nothing. What he is really proposing is a language in which it will be impossible to raise the paradox. A language that forbids a statement from referring to certain statements, and in particular to itself; consequently, of course, you won’t be able to formulate the paradox in the language he proposes. Is that a solution to the paradox? It’s a Stalin-style solution. Meaning, whoever raises the paradox gets shot in the head, and then naturally nobody raises the question, and that’s how we solved all questions. It’s a prohibition against formulating the paradox; it’s not a solution to the paradox. I think that the solution I mentioned earlier too, right, by means of differential calculus, also is not really a solution, but rather a prohibition, or a language that forbids expressing it, but does not really explain what is wrong with this paradox. In order to explain what is wrong, you have to show me what is mistaken in my formulation, not offer me an alternative formulation in which it will be impossible to formulate the problem. Now my claim was that the arrow, contrary to the way Zeno formulated the paradox, at every point in time the arrow is not standing in another place, but rather is in another place. That is not the same thing. What is the difference between saying that something is in a certain place and saying that that thing is standing in that place? For example, a moving car at every moment is in a different place, but it is not standing in that place. Right? It’s traveling. So you can’t say that at every moment it is standing in another place. At every moment it is in another place, but it is in that place while moving. There is a difference between saying the object is standing in a certain place and saying the object is in a certain place. An object that is in motion still has a place; it is in some place, but it is in motion. Okay? When I pass through Tel Aviv on the train, even if the train didn’t stop, you can still say that when I am in Tel Aviv, I am in Tel Aviv. That doesn’t mean I am standing there; the train is moving, but I am in Tel Aviv. I’m not standing in Tel Aviv, but I am in Tel Aviv. Now notice: once you understand this, you don’t need any differential calculus, and this problem could have been solved two thousand years ago. The formulation of the paradox simply conflates being in a place with standing in that place. The claim is that the arrow is at every moment in a different place, but at none of the moments is it standing. It is always in motion; only while moving, at every point in time it is in a different place. That’s all. And now there is no question of when it moves. When does it move? At each and every moment it moves. That does not contradict the fact that at each and every moment it is also in some place, because being in a place does not contradict its moving. Standing does contradict its moving; being in a place does not. Okay? Now look at what I actually did here: I performed a conceptual analysis of what it means to stand in a place or to be in a place. I drew a distinction between two concepts. If you want to go back to the scheme I gave a few lessons ago, then basically I see that the concept “is in” and the concept “stands” belong in a certain sense to the same genus, but they are different species within that genus. Whether I am in a place or standing in a place, I have a certain location. That is the similarity between these two concepts, being in and standing. But there is also a difference between them. In one of them I am moving at that point in time, and in the other I am standing at that point in time. So that is if I connect this to the method of conceptual analysis I spoke about in the first lessons. But I want to go a bit deeper into the meaning of the distinction I made here. Look where this confusion actually comes from—and here we do in fact return, by the back door as it were, to infinitesimals—where does this confusion come from, or this assumption that a body cannot have velocity at a point in time? After all, that is what really confused Zeno. I want to argue that a body can have velocity at a point in time. The fact that it is at a certain place at a point in time does not mean it has no velocity at that same time. Alongside the fact that it is in that place, it also has some velocity; it is also moving. What makes it hard for people to understand this? Why did this complicate things so much for people? Because when people think about the definition of the concept of velocity—let’s move now from the more general concept of motion to the concept of velocity—how do we define velocity? We define velocity as the rate of change of position. That is basically the definition of velocity. Meaning, when someone moves at a speed of one hundred kilometers per hour, I am basically saying that as an hour passes, he covers a distance of one hundred kilometers, right? That is the meaning of the concept that I am traveling at one hundred kilometers per hour. So basically, if you think about velocity in that way, then velocity is some quotient of distance divided by time. Right? How much time it takes me to cover a certain distance. If one hundred kilometers takes me one hour, I divide one hundred kilometers by one—one hour—then the velocity is one hundred, measured in units of kilometers per hour. Okay? So basically velocity appears to us as a quotient, right? The quotient of distance divided by time basically gives me the velocity. Now if the velocity is not—this is for constant velocity. If the velocity is not constant, then it’s a derivative; never mind, that’s more complicated, but even there we are basically taking very small segments along the path and asking ourselves, on a very, very small segment, how long did it take the body to traverse it. And when we take that segment to be very, very small, we can define the velocity on that small segment. Okay? That’s what we do when velocity changes from moment to moment. When it is constant, I simply take the total distance the body traveled, divide it by the total time, and the result is the velocity. But if the velocity changes—the body accelerates, slows down, changes speed—then I need to move to varying velocities, and at every point I need to calculate the velocity on a small segment around that point. Okay? And as a result, people basically feel that the concept of velocity cannot exist at a mathematical point in time. It has to be some interval of time. Because at a point in time the body cannot traverse a distance. Right? At a point in time the body is in a well-defined place. Understand: if at one point in time the body were to traverse any distance, however small, what would the velocity of the body be in such a case? Infinite, right? Because the distance it traveled is, I don’t know, some distance, a millimeter, but the time it took to cover it is zero. So divide the distance by the time and you get infinity. We are talking about infinite velocity. Okay? Therefore there is no such thing—when you talk about a point in time, the body is in one place; it does not change place at a point in time. But it is still at velocity, in motion. And that is what is so confusing here. I am saying that on the one hand, at a point in time the body does not change place. But it still has velocity. But this is what people do not understand, because what do you mean—if it has velocity, that means it changes place over time. So what does it mean that at a point in time it does not change place but it has velocity? So if you ask physicists, they will tell you that velocity is basically a fiction, where what we really mean is the rate of change of position on a small interval around that point. There is no velocity at a point in time. There is velocity on a small interval around that point in time. Pick any point in time you want in order to calculate the velocity at that point. Look a little ahead and a little behind, at a small interval around that point. Make that interval very, very small—there are mathematical ways to do this precisely—but make that interval very small and calculate the ratio between the distance and the time it takes; that will be the velocity. Therefore velocity always exists only on an interval, not on a point. I just take a small interval around the point and say: that is what is called the velocity at the point. That is what physicists will answer you if you ask them this question. But if I think about it philosophically, I do not accept that answer. I claim that a body has velocity at a point in time. It is not a fiction. And the fact is that after doing the calculation—called a derivative, the derivative of position with respect to time gives me the velocity—after doing the calculation, whatever point in time you plug into that result, you will get a different velocity. Meaning that the body has a velocity tied to one specific and well-defined point in time. At every other point you choose, the body’s velocity will be different. Meaning that a body has velocity at a point in time. So the physicist can tell you: yes, but that is a fiction; it doesn’t really have velocity at a point in time, because the axis is not made up of points but of small intervals. Okay? We call it velocity at a point in time, but not really—in reality itself the body does not have velocity at a point in time. But here it already depends very much on the question of how you define the concept of velocity, and here once again we arrive at conceptual analysis. I want to argue that the definition of velocity as a quotient—a quotient of distance divided by time—is what is called an operational definition. A definition—how to calculate it—that is the way to calculate velocity; it is not the definition of the concept of velocity. When you ask me how to calculate velocity, I tell you: divide the distance by the time. That is the way to calculate velocity. And if I ask you what velocity is—here I want to make a philosophical claim. Velocity is not how much distance you cover in how much time. That is only the way to calculate velocity. Velocity is the potential to change place in time. The change itself is a result of the fact that the body has velocity. It is not the definition of the concept of velocity. The way to calculate velocity is the quotient of distance divided by time, but it is not correct that this quotient is the definition of the concept of velocity—that it is not. It is a definition of how to calculate; it is not a definition of what velocity is. Those are two different things. And this is a philosophical claim, not a physical one. I want to argue that in physics, if they ask you what the definition of velocity is, they will say: the derivative of position with respect to time. That is the definition of velocity in physics. But I say that on the philosophical level, I want to argue that this is only a definition of how to calculate it, whereas the definition of the concept of velocity itself is the potential to change place. That is the concept of velocity. The potential. A body that has velocity has the potential to change place. Now a body can have the potential to change place even at one isolated point in time. I look at the body at that point and I say: this is a body that has the potential to change place. How do I know? Wait a tenth of a second and you’ll see that it will be in another place. I will not know this by observing the body at that specific point in time. At that specific point in time it is in a defined place; it does not change place. It is impossible to change place at a point in time. But it is possible to have the potential to change place at a point in time. Okay? My potential to change place is determined by the velocity. Someone asked here about acceleration—I’m using very simplistic formulations. In fact you need all the derivatives, the whole Taylor series, not just the first derivative, but I’m not going into that here now. For our purposes we’ll make do with the formulation I’m giving here. So the claim is that the right way to look at velocity is as the potential for a change of place, and not how long it takes me to change place. The implication of the fact that I have velocity—that I have the potential for a change of place—will become clear if I wait a little in time; it will become the fact that the body will actually change place. That reveals to me that the body has some potential to change place. It will move from potential to actual if I wait a little more time. Meaning, if I summarize, my claim is that at a point in time a body cannot change place, but that does not mean that at a point in time the body has no velocity. At a point in time the body does have velocity. More than that I’ll say: at a point in time the body is moving, but it does not change place. It is moving—meaning it has velocity. It is in motion; it is not standing still. So that means the body is moving. A change of place cannot occur at a point in time. And in order to see this potential move from potential to actual, you need to wait some interval of time, and then we will see that the change of place also appears in actuality. Let’s formulate this again in a way that may annoy professional physicists, but I think it’s a good illustration. Take a body—a ball, okay? I throw it at the wall. Okay? Now when it reaches the wall, at the moment it meets the wall, it has a very high velocity, right? But the wall is standing there and does not let it continue forward. So the physicists say that at the moment it met the wall its velocity is zero. And I want to argue: no. At the moment it met the wall its velocity is one hundred. Okay? It’s just that the change of place, which is the result of this potential called velocity, cannot move from potential to actual. The velocity cannot be translated into a change of place, because the wall does not let the body continue forward. So then what happens? Heating—depending whether it is a plastic collision or an elastic one—the ball may move backward instead of continuing forward. Maybe it will stick to the wall, depending on whether it’s plastic or elastic. But it will come out in the form of heat, or in the form of backward velocity, or it will push the wall a little forward, exchange momentum, and so on and so on. The fact that the body has the potential to change place, that it has velocity, will not always be expressed in an actual change of place. You have the potential to change place, but the actual change of place will not always appear. If there is something that prevents it from appearing, then it won’t happen. So I had the potential to change place, but there is something that does not let me bring it from potential to actual. So it will come out in other forms: in the form of heat, in the form of backward velocity, and so on. And that is basically my claim: on the physical level I’m not arguing with the physical descriptions. On the philosophical level, I want to argue that when a body hits a wall, its velocity when it is adjacent to the wall is one hundred—one hundred kilometers per hour. Fine? But that is velocity in the sense of potential for a change of place. The change of place will not happen, and the body will not succeed in translating this potential, bringing it from potential to actual, and changing its place. There is, when we look at the…
[Speaker C] Sorry, how long does that last? It meets it and its velocity is one hundred—until when does it continue to be one hundred? Until when…
[Rabbi Michael Abraham] I’m talking about points in time. There’s a point in time of the hundredth, and there’s a point in time as if immediately after it, even though it’s not legitimate to speak that way continuously, because there is no point in time after an existing point, but as if it drops to zero velocity and comes out differently. Okay. Okay. I’ve said things here that make mathematicians’ and physicists’ hair stand on end when they hear them. But never mind, I’m trying to illustrate a philosophical point here. So what actually causes us to make this mistake? And here I come to why this example is important for us. When I look at a body and ask myself whether it is in motion or not, clearly in order to answer that I need to see whether it changes its place or doesn’t change its place. Right? Meaning, if it changes its place, then I understand that this body is in motion. But if I look at a body at one mathematical instant, at that instant it does not change its place. Okay? So if that’s the case, I won’t be able to know that the body is in motion. And therefore Zeno basically said, wait a second, at a point in time the body is not in motion. And now I correct that: the body is in motion. It’s just that I can’t see it, because the motion is not translated into change of place, since at a point in time a change of place is impossible. Which means that in fact we have some limitation, a limitation in that we are unable to see motion at a point in time. But that doesn’t mean there is no motion. There is motion at a point in time; we just can’t see it, because when we look at motion, we see it through changes of place. We cannot see the very fact that a body has velocity. We can only see the implication, that the body changes place; that teaches us that the body has velocity. What does this come from? It comes from the fact that our visual system operates like a camera. A camera captures the location of the object. How do we make a movie? A motion picture. Once they used to call a movie a motion picture. What is a motion picture? They would project images one after another at high speed and you would see an object moving. In fact, that’s what happens today too. Meaning, when we watch a certain film, something happening, what they are really projecting to us is very, very dense pictures one after another, only each time there is a small change in the object, and our brain connects the different frames and as far as we are concerned we see motion. But the truth is that this is an illusion. At every moment we see a static picture. In the film itself nothing moves. The film is, in principle, a collection of pictures, one picture after another after another, like they do in animated films. In cartoons, each time they draw one picture, then they draw a picture with a small movement and another picture, and project them in sequence, and it seems to us that the drawing is moving. Or like the cursor on a computer, the pointer on a computer. You press an arrow and you see the pointer moving, up, down, right, left. Of course it is not moving and not going anywhere. It simply turns off here and lights up here, turns off here and lights up here, turns off here and lights up here, but to our eye it looks as if it moves to the right. It doesn’t really move to the right. It turns off here and lights up a little to the right, turns off here and lights up. And lights up a little further to the right. Each time a different cursor is simply created in a place farther and farther to the right. But our brain looks at this and sees motion. It sees one cursor here that is simply moving to the right. It doesn’t understand that each time there is a different point here. The previous one went out and a new one lit up. That’s basically how we see films. Now this stems from the fact that our viewing system, our eyes, our vision, is a static system. It is a system that grasps location, or position, or state. And in order to produce motion I simply need to make many successive positions and project them quickly, and then the brain somehow completes it or turns it into some sort of sequence that is in motion. Think about some other imaginary creature, endowed with a different cognitive system from ours. It sees velocities, not positions. It sees motions. And when it looks at a body at a point in time, it sees that the body has velocity. I learn that a body has velocity through seeing it change place. But it doesn’t operate through place; it directly sees velocity itself. It doesn’t need to reconstruct from the fact that there was a change of place and arrive at the conclusion that the body had velocity; no, it directly sees the velocity. It has some kind of sense that detects velocities, not positions, the way our sense is built. Okay? Yes, exactly, that’s where I’m heading, to uncertainty. So the claim basically is that that imaginary creature I described here would see the fact that a body has velocity at a point in time. It would not need to wait for a stretch of time in order to see that the body has velocity. We need to wait because at a point in time we see the body located somewhere; we have no way of knowing whether it has velocity or doesn’t have velocity, whether it is merely there or standing still. Remember the distinction I made earlier. I don’t know whether it is there in the same place or standing still in the same place. And that is exactly the reason Zeno got mixed up here. Because he failed to distinguish between the concept of being located and the concept of standing still. And why? Because when, in our static sensory system, we look at a certain body at a point in time, we cannot know whether it is moving, whether it has velocity or doesn’t have velocity. For us, being there and standing still are perceived in the same way. To the eye they are perceived in the same way. In order to see the difference between being there and standing still, you need a camera exposure time, a certain amount of time, not one mathematical instant, but you need a small stretch, even a tiny one, but still a certain stretch of camera exposure time. Incidentally, when a camera photographs a moving object, then indeed it gets smeared a bit on the film. It depends on the exposure time relative to the speed of the body. If the exposure time is very short, then the body has to move very fast for us to see a trail in the picture, right? If the exposure time is long, then even a body moving at not such a high speed, you’ll see a trail in the picture. But in principle there is always a trail. It’s just that if the exposure time were one mathematical instant of time, you would see the body standing still. Otherwise a moving body will always be some kind of blur in the photograph. Okay. And why? Because our exposure time is not zero. We cannot open the camera for one point in time; rather it will be an interval, maybe small, but still an interval of time. All right? That is the limitation of cameras. And think about that person I described before, that creature I described before—I’ll now call it this way: our eyes are basically an ideal camera. Our eyes capture positions at a point in time. Except that they are open over a long period of time, but at each and every point they capture a position, the position at that point. Now think of another creature that instead of eyes has a video camera. Not a still camera, but a real video camera. Not an illusory video camera like ours, which actually projects static pictures to us quickly one after another, but rather for it there is an ideal video camera, a device that captures the very existence of motion in a body even at a point in time. That’s its device. Okay? That person would see a body in front of him at one point in time. He would open the video camera for one point in time. He would see that this body has velocity. He would see it even at one point in time. But he would have no idea where the body is located, about the body’s position, because his perceptual system does not see locations but velocities. Now these two forms of looking—and here this is really only a hint for… those in the know—these two forms of looking are actually what underlies Heisenberg’s uncertainty principle in quantum theory. This basically says that if I look with an ideal camera, I will not succeed in seeing velocities, only positions. If I look with an ideal video camera, I will succeed in seeing velocities, but I will have no idea about positions. Now we need to choose either we are in this mode or we are in that mode, but it is impossible to look at the object with a camera and a video camera at the same time. If my perception is currently in camera mode, I will see positions. If it is in video-camera mode, I will see velocities. I cannot see both. Okay, those who are in the know, those who once studied quantum theory, know that in fact at the basis of Heisenberg’s uncertainty principle there are really two pictures; this is called the position picture and the momentum picture. Momentum, for our purposes here, is velocity. Yes, the point is that there is one mode in which we describe all of reality in terms of position, and there is another completely different mode, not connected to that mode, in which we describe everything in terms of velocities. And the problem is that position and velocity are two forms of observation that are not—they are orthogonal, yes—they do not talk to one another. If you choose to look in terms of positions, you will not see velocities; if you choose to look in terms of velocities, you will not see positions. But these are two forms of observation, and these two forms of observation are actually the picture that emerges from an ideal camera and from an ideal video camera. Well, that’s enough remarks on quantum theory; I really drifted a bit there. But what I want for our purposes—the important point for our purposes—is that you should notice what I actually did here. I took—basically what I did was the same thing I did with the chessboard, in the chess-and-domino problem. Because what I was really trying to show is that we take the accepted definition of the concept of velocity and try to peel it away in order to understand what the concept itself is. And then suddenly we discover that the definition is perhaps some kind of representation of the concept, but it is not the concept itself. The concept of velocity exists at a point in time. The definition as the quotient of place divided by time cannot be applied at a point in time. Someone who grasps the concept of velocity as the quotient of place divided by time will not be able to accept the claim that a body has velocity at a point in time. He will have to see this as fiction; he will not be able to accept such a thing. And then indeed you get Zeno’s problems and the flying arrow and all the rest. But someone who understands that our definitions are only a certain way of describing the concept—it is not that the concept is the definition, the definition is merely a certain way of describing the concept—may understand that this way indeed leads me to problems, and then I understand that it has limited meaning and one must be careful in using it. And in fact the concept itself is a concept, it is something very elusive, very abstract, so I have no choice: I must deal with it through definitions and characteristics and properties, because I don’t know what to do with the concept itself, as we saw with Kant and with objects. But on the other hand one must always remember that yes, I have to use definitions and characteristics, but one must always remember that these are definitions and characteristics in which all kinds of problems may appear, because they are drawn from my world of thought. The concept itself must not be identified with the way I describe it. Just like with the chessboard: the fact that I described the black and white squares through their colors is incidental; I chose the language of colors to show that there are two kinds of squares here, but the truth is that they have some property that is not connected to colors; the colors are only a form of representing that property. So here too we see that in fact concepts are concepts that have some kind of existence as ideas, yes, some sort of independent existence in the world, and the definitions we try to attach to these concepts, the characteristics we try to attach to these concepts, are the result of observing the concept, and that observation can get us into trouble because the observing system, our system of thought, may not really be able to grasp the concept properly, and then we get entangled in contradictions or in one philosophical problem or another. So we learn from here another lesson, beyond the conceptual analysis itself of the concept of velocity that solved Zeno’s problem. I want to point out something else here: in fact, good conceptual analysis sometimes shows us where the definition we gave a concept fails, where it collapses. So what are we left with? We are left with some conception of the concept that we have no way really to define fully, or at least perhaps we have no way. The ordinary way of defining this concept suffers from internal contradictions; it gets us into trouble. So the conceptual analysis led us to contradictions within the description of the concept, but that does not mean the concept does not exist. As I said at the beginning with Zeno, Zeno’s conclusion is not that the concept of velocity or motion does not exist, but rather that the language in which we describe this concept is an unsuccessful language. It does not manage to capture this concept fully, and therefore we fall into contradictions. And we need to understand that what stands behind this is some abstract concept, for which one of the ways of capturing it is in times that are not a single point in time but intervals of time; then we can indeed use this language, seeing velocity as the quotient of place divided by time, and everything is fine, no problem at all. But on the philosophical level, when we speak about the paradox of the flying arrow, that is a philosophical paradox, not a mathematical and not a physical one; there we will need to give up the physical definitions. Incidentally, an interesting point again for those who know a bit: you know that the police radars that catch cars driving at high speed—when they check the speeds of cars, many of the radars work with Doppler technology. Doppler is basically this: you send a beam of light toward the moving car, and it turns out—there is what is called the Doppler effect—that if the car moves, the beam returns at a frequency slightly different from the frequency with which it arrived. And the difference between the frequencies is proportional to the speed of the object. The device basically does the calculation, and that is how it discovers the speed of the car. Now in the description I gave here—and again, it is not really accurate, but it is rather tricky to understand why it is not accurate, but never mind—in the description I gave here, notice that this description refers to velocity as a velocity that exists at a point in time. It is not a quotient. The velocity here is not a quotient of place divided by distance. The velocity here is basically determined by an impact at a point in time: the beam hits the car at a point in time and returns. And from the difference in frequencies I know the speed of the car. So there is basically a way here to detect the speed of a car at a point in time, and that is not done by a quotient. It does not have to change place for me to test the point in time. In principle it does, that’s the truth; this whole description is a bit deceptive, but in principle I think it illustrates well the meaning of this concept, velocity at a point in time. In principle, there may be a method of measurement that succeeds in giving me the fact that a body has velocity through an interaction with the body at a point in time, not over a small interval of my choosing but at a point. Okay? That may be a possible illustration of this issue. In any case, what we are really learning here is that both the chess example and the matter of velocity are in a certain sense definitions of a concept by way of negation. And this is a very important point also with respect to Maimonides’ theory of negative attributes, and all kinds of things of that sort. In all these cases, like the chessboard with the colors or velocity with Zeno’s paradoxes, we basically propose a definition of the concept, find a contradiction that follows from that definition, and then we understand, in some more abstract sense, what this concept actually is, like the concept of velocity. So we begin with the simple conception that velocity is a quotient, distance divided by time, and then Zeno comes and says, wait a second, if it’s a quotient then something here doesn’t make sense, because at a point in time the body doesn’t move, so when does it move, and all the paradoxes I discussed earlier. Then I say, fine, so there is no choice, one must somehow either qualify or give up this language in which we describe the concept of velocity or the concept of motion. Then what are we left with? In the simplistic conception, careless philosophers basically say: apparently velocity or motion does not exist, this concept does not exist, the phenomenon of motion in the world does not exist. But that is nonsense, of course. Why? Because they assume that the concept of motion just is its definitions. And the moment the definition collapses—if it is contradictory, such a definition cannot stand—then motion also does not exist in the world. But that is of course incorrect. Because the concept is not its definitions. The definitions are an attempt to describe the concept. If the definition collapses, the concept has not collapsed; rather what collapsed is the way we describe it. And from here we learned that this description of the concept is not a complete description of it. It taught us something more about the concept. Meaning, I understand the concept—now I understand the concept of velocity better, not less well, as a result of Zeno’s paradox. Because what I understand now is that the concept of velocity is a potential for change of place. If I wait a certain time, I will see the change of place, and I can calculate the velocity as the quotient of distance divided by time. But if I want to speak about velocity at a point in time, I have now learned from this analysis that I can speak about the velocity of a body at a point in time. One can speak about it at a point in time, except that I need to speak about the potential for change of place, not the change of place itself. So one can call this, in effect, a definition by way of negation. It is not change of place; it is what underlies change of place. But it is not really only negation, it also taught us something positive. Because I have to pass through change of place and then make an abstraction and say: no, velocity is not change of place, it is the potential to undergo change of place. And now I really understand what velocity is. So I peel away the layers, but it’s not that I’m left with nothing in the end. In the end I’m left with the abstract thing, but the layers helped me very, very much to understand it. Without them I wouldn’t have understood it. If I hadn’t understood what change of place is, I wouldn’t have been able to define velocity as the potential for change of place. So I have to pass through the simplistic definition, the inaccurate one, the one that does not stand up to the tests involving change of place, and then peel it away, understand that it leads to paradoxes and problems, and say: it is related to change of place, but it is not change of place; it is the potential to change place. You understand that here, in this case—and by the way it is often like this—conceptual analysis is very often built on finding contradictions within a definition. You give me the concept through one of its definitions, I find a contradiction within the definition, and that is basically the meaning of conceptual analysis. But then one needs to understand what one is left with. Because people often say that logic is mainly concerned with destruction, not construction. In logic it is very hard to build something. I can prove to you that there is a contradiction within your structure or your conceptual world, but with logic one cannot build a conceptual world. There is no logical way to build a conceptual world. You build a conceptual world however seems right to you, and afterward you can check using logic whether there is no contradiction in it. And therefore Aristotle already noticed this, and many after him as well, that logic has a destructive role, a role of tearing down and not of building. Of course this is constructive destruction, because you destroy the contradictory things in order to remain in the end with something that contains no contradictions. You clean your picture of contradictions by means of logic. But—and this is exactly what I was talking about here—after you clean the picture, it’s not that you are left with a vacuum. You are left with something. That something is the abstract concept when it is clean of all the contradictions and all the confusing languages that lead to those contradictions. So conceptual analysis, by means of contradiction, basically teaches me… yes, I agree with what Benny wrote here. It really is an interesting philosophical question; I have a lot to say about it, but we won’t get into it here. This discussion of the via negativa—think about this when Maimonides spoke about his theory of negative attributes. Maimonides basically says that when I say of the Holy One, blessed be He, that He is merciful, I really cannot say of Him that He is merciful. I can say of Him that He is not unmerciful, that He is not cruel. Because to say something positive about Him is really to grasp something in Him Himself, and that we do not have the ability to do. What we can do is understand what He is not. On the face of it, this claim is nonsense. Why? Because by exactly the same token I can say of the Holy One, blessed be He, that He is cruel, because I can say that it is not true that He is merciful. Just as it is not true that He is cruel, it is also not true that He is merciful. Because no positive attribute can be ascribed to Him. So just as you cannot ascribe to Him the positive attribute of being merciful, how do you solve this? It is not true that He is not merciful. But by the same token you also cannot ascribe—you can say in exactly the same sense—that He is cruel, because it is not true that He is not cruel. Because nothing positive about Him is true. Okay? So is saying that the Holy One, blessed be He, is merciful more correct than saying that He is cruel? In the simple reading of Maimonides’ theory of negative attributes, there is no difference at all. You can say He is merciful and you can say He is cruel, and you have said the same thing. Basically you have said that the concepts of cruelty or mercy are alien to Him; they do not apply to Him. You are only negating things of Him, you are not saying positive things about Him. That is the accepted view. And then it turns out that we are left with nothing. So in fact Maimonides is saying that one cannot even say of Him that He is merciful, even in the sense Maimonides is trying to present in this picture. Even to say, in the sense that He is merciful, meaning to say that it is not true that He is not merciful—even that, in fact, I cannot say. Because it is simply not true that anything applies to Him; you have said nothing by that. It is also not true that He is not cruel, and it is not true that He is not angry, and it is not true that He is angry—nothing is true of Him. So what, are you merely saying that He is detached from our world of concepts, and that’s it? What is the meaning of saying that He is merciful when you nevertheless try to say something about Him and treat these statements as truer than the opposite statements? I think, it seems to me, that what Maimonides means in the theory of negative attributes is the same conceptual analysis I did here. When you arrive at contradictions, you are basically negating the conceptual system you use with respect to the object—or with respect to the concept, in the case of velocity. But you are not left with a vacuum. After you negate the problematic description, you are left with something more abstract, which is actually a deeper understanding of the concept or of the object. You are left with something. Maimonides writes this too, incidentally: in the theory of negative attributes people think we are only negating. No—the negations also teach us something positive. Only it is not entirely clear what that means. What does it mean that they teach us something positive? So do they actually teach us that He really is merciful? Then after all you have said something positive about Him; you cannot escape that. These examples show us what is actually meant. Because when I negated from the concept of velocity the quotient of place divided by time, I did say something about the concept of velocity. It is true that it is not a quotient of place divided by time, but from this I learned that it is the potential for change of place in time. So I did learn something positive about it. Therefore the way of negative attributes is a way that teaches us something about the matter. Finding contradictions in things is not destructive; finding contradictions in things is constructive. It is the construction of an abstract concept from which we peel away the contradictory parts, the problematic parts, and we try to remain with something free of contradictions. Think like a sculptor, yes? This is an example—I think I saw it in Tziona Levi Otzberger or in Aharon Bart, I no longer remember, one of the two. A sculptor, for example. The sculptor does not make the sculpture, right? Take a block of wood or a piece of stone—what does the sculptor do in order to produce the form? He basically removes all the waste, and what remains is the sculpture, right? What is called in Jewish law engraving out the inside. Like when writing letters on the Sabbath or in a bill of divorce or whatever it may be, one can form them by engraving out the inside. What does that mean? Suppose the whole page is full of ink; you erase all the ink and leave only the shape of the letter. Fine? So in fact you did not create the letter, you only removed the waste, and the letter remained by itself. So in what sense did the sculptor create the sculpture? The answer is that he created it in a negative way. The moment you remove the waste, what remains is your creation. Because removing the waste is the way to create the sculpture, and what is produced in the end is the sculpture. That sculpture is your sculpture. It is not that the sculpture already existed and you merely removed what prevented us from seeing it. Yes, one could describe it that way in a simplistic manner: the sculpture already existed, you did not make the sculpture, you only removed the things that prevented us from seeing it. But that is of course nonsense. You created the sculpture by way of negation. You removed all the things that interfere, and in that way you created the sculpture. Negative analysis of concepts, or analysis of concepts through contradictions, basically does the same work. We try to refine the concept by cleaning it of various contradictions and of various descriptions that lead us to problems. Someone who thinks that after this cleaning we are left with nothing is the same person who identifies the concept with the language that describes it. Because he thinks concepts have no existence in themselves. Concepts are our creations. So in fact the definition is the concept, and if there is a contradiction in the definition then the concept has collapsed, so there is no concept. But I want to argue that no, concepts have an existence of their own; the definition is our attempt to grasp the concepts, understand them, define them. It has to go through a thorough laundering, cleaning, removal of contradictions and removal of inaccurate elements, so that in the end I remain with a good conception of the concept itself. Okay? Fine, I’ll stop here, and if anyone wants to ask or comment, now is the time. That’s it, okay. So goodbye.
[Speaker D] Thank you very much. Sabbath peace.
[Rabbi Michael Abraham] Thank you very much, goodbye. Sabbath peace.