Rabbi Kook – Perfection and Self-Development – Lesson 2

[Rabbi Michael Abraham] Last time we were in the passage from Rav Kook in Orot HaKodesh about perfection and self-perfection. I’ll just briefly remind you, because this is the context for what I want to continue with. He opens with a description of some more abstract reality that somehow peeks through the reality that we see. And after that he connects it to the progress of the world, or the world’s transcendence—that there is some force or something that keeps causing the world to rise all the time, some kind of engine that keeps making the world rise. He explains that for this reason the world was also created lacking, so that it would have somewhere to rise to. And this introduction is really a broader introduction that speaks in general about the relation between what we see and what lies behind what we see. And after that he goes into the question of perfection and self-perfection a bit more concretely, and he says that the Holy One, blessed be He, seemingly cannot perfect Himself, because He is perfect, because He is infinite. And since that is so, then in fact one of the perfections is missing from Him—self-perfection is one of the perfections—and the solution to that is that He created deficient creatures, or a deficient reality, whose progress or transcendence is the way He perfects Himself. Through it He perfects Himself, essentially. And if I connect this back to his introduction, then that basically means that somehow, through the dynamic of the world’s transcendence—kind of what you said when we came in—through the dynamic of the world’s transcendence, we are actually supposed to grasp or feel the engine of this whole thing, which is in fact the Holy One, blessed be He. Meaning, what stands at the basis of the world’s self-perfection. And indeed that self-perfection, or our spiritual and value-oriented work, is really a higher need. I spoke about service for a higher need—that we are actually doing this for Him, He needs us, because without it He is basically stuck, He cannot, He cannot perfect Himself. Another point that came up there is that without accepting this assumption, that there is something there behind reality—the existence of the infinite—then there is no direction, there is no meaning to any standard. Meaning, if I do not define some abstract point as being the peak of progress, of this transcendence, yes, this infinity that can no longer transcend further, then basically it is impossible to define what we see around us as transcendence at all. Because who says it isn’t decline? Meaning, who dictates the direction?

[Speaker B] Why do I need, let’s say objectively for the sake of argument—I can define up and down without saying that there is some specific point above.

[Rabbi Michael Abraham] No, you need something that gives the direction. You can’t define up.

[Speaker B] But the direction is from the point where I am right now, in a certain direction.

[Rabbi Michael Abraham] But what is up and what is down? You need to move away from the place where you are. What is up and what is down? There has to be something absolute at infinity that defines what the positive direction is, meaning defines the direction of the axis. Who says the axis is this way and not that way?

[Speaker B] I can also define it according to the place I started from.

[Rabbi Michael Abraham] No, you can’t define it that way, because from myself I go out both toward minus infinity and toward plus infinity. The question is where minus infinity is and where plus infinity is.

[Speaker B] Yes, I’m saying I can still define it by the surroundings here. I can say that my right side is minus.

[Rabbi Michael Abraham] You can define whatever you want, but there is symmetry between right and left. What breaks the symmetry? Why is one side really progress and the other side decline? In what real sense is there a non-arbitrary plus and minus here? So there has to be some external, objective standard against which things are measured.

[Speaker B] And that standard has to be a point at the end, meaning something…

[Rabbi Michael Abraham] No, it doesn’t have to be, but infinity here is just a metaphor. The point is that it’s an absolute point, not dependent on any perspective, something that defines direction absolutely and not relative to something else. He himself writes here that… He says: “And the more science becomes established on the foundation of development, so evolution comes closer to the clearer divine light and arrives at the loftier vision, by which one cannot judge all existence from the standpoint of its partial relation—that is, from the relation between one part of it and another—for its true values will not be found there. Rather, from its inner essential law, in the matter of the general relation of all of it and all its parts to divine perfection—that is the honored thing and the most fitting foundation upon which to establish the concept of existence as a whole.” Meaning, when you measure things one against the other, you never have a way of knowing what is more and what is less. By the way, that is exactly why a world without God—yes, after Nietzsche, when they said God is dead, as it were—the standards collapsed. Meaning, Nietzsche already writes this even before postmodernism came from potential into actuality. He says that where there is nothing absolute that determines the direction of the axes, you cannot determine which axis is positive and which is negative. What counts as progress and what counts as regression. And indeed, a world devoid of God is a world that reaches the conclusion that basically everything is the same. There is no progress and no decline, and everyone is equally right, everyone is equally true, everyone is equally beautiful—yes, in art—and everyone is equally moral. Meaning, you have no external standard. So once there is no external standard, the human being becomes his own standard, so he is always right. Meaning, if I am my own standard, then I am always right by definition. And if each person is his own standard, then each of us is by definition right. You need something that is not conditioned by anything else, and that is what gives things their standard.

[Speaker C] So that, that’s probably David…

[Speaker D] I wasn’t here at the beginning, but what I really heard—Haydo the musician explained this to me many years ago—that there was a revolution, meaning the human being, specifically the human being, was put at the center, and that was a rebellion against God. But in every field—for example in art it’s very obvious—there are no more rules, whereas once there were rules.

[Rabbi Michael Abraham] Right, exactly. Meaning, once—there can’t be rules if there’s nothing except me. A rule is something that is supposed to dictate things to me. But who is going to dictate things to me? There is no one outside me. Right. I can say what I think, someone else can say what he thinks, but who is going to determine which of us is right if there is nothing outside us that serves as the standard of things, and we’re only comparing one to the other? Comparisons of one to the other are simply meaningless; they won’t give you anything. And even when you do that, and you compare two people and say this one is better than that one, or two societies and say this one is better than that one, or this art is better than that art, in the background you are assuming some standard that is outside both of them. Because otherwise how can you determine who is to the right and who is to the left? Meaning, if you look this way you’ll see that he is on the right and the other is on the left, so what? So there is something very problematic in a world without God. People try to create alternative standards; they don’t really succeed. Meaning, the discourse continues, of course—to our good fortune, I think—there is some inertia from belief in God that keeps rolling into all sorts of ideas of atheistic morality, never mind—but what…

[Speaker E] Most people still believe.

[Rabbi Michael Abraham] Yes, but even a society that doesn’t—as a society—a society that doesn’t believe in God or doesn’t speak in that language still speaks in terms of moral values, with criticism, with everything, without feeling that in fact this has no basis. Just a second. Those who really take it all the way—for example the intellectual avant-garde, the people who are… meaning, it’s no wonder the masses remain basically okay as they always were, but among the intellectuals there is a big fracture. Because the intellectuals are really the ones who draw the conclusions. They say, wait a second, this really has no basis, we’re just talking nonsense. Meaning, there is a real break here. The fact that we don’t feel it is because, fortunately, there aren’t too many intellectuals. Meaning, if there were too many intellectuals today we would be in a very bad situation. I’m truly convinced of that. Meaning, if there were too many intellectuals in the world, the situation would be extremely grave.

[Speaker D] Intellectuals in the sense of being free in their thinking?

[Rabbi Michael Abraham] Yes, in the sense of people who do what they really think—meaning, who think about what they do and then do it, or act according to the principles they think through, and so on. That’s a very dangerous world—a world with lots of intellectuals.

[Speaker B] But moral behavior, as distinct from morality itself, couldn’t it also stem from some natural tendency?

[Rabbi Michael Abraham] It could, of course. But on the other hand, you have no ability whatsoever to criticize someone who doesn’t have that natural tendency.

[Speaker B] You can criticize him, but it could be that in practice that would hold up no less than…

[Rabbi Michael Abraham] Maybe yes and maybe no, maybe yes and maybe no. I’m not sure about that. That’s what they claim.

[Speaker C] I’m saying that at first it would hold up, but at some point it would start… Kant’s categorical imperative.

[Rabbi Michael Abraham] Kant’s categorical imperative comes out of the blue.

[Speaker D] But in some sense it’s not out of the blue, it’s not.

[Rabbi Michael Abraham] Yes, but he assumes the existence of a categorical imperative. There may be a categorical imperative, but why should I obey it? So Kant says, if you’re moral, then by definition you respond to the categorical imperative. Fine. And if I’m not moral, then I don’t respond. That turns it back into a definition. What you now need to explain to me is why I should really respond, not define whether I’m moral or not. Why should I respond? So Kant himself bases one of his proofs for the existence of God on morality, meaning because he too understands that obligation to the categorical imperative cannot arise without some external standard.

[Speaker B] The definition of… meaning I can understand that this requires an external standard. Is that external standard, by definition, God—God in the sense of the Creator of the world and its ruler?

[Rabbi Michael Abraham] No, it doesn’t have to be, but clearly God is such a thing. If someone believes in the existence of something else that didn’t create the world, or I don’t know, some other kind of thing—maybe yes—then that would be consistent, right? Not necessarily correct, but in that sense it would be okay. Meaning, you need some external standard. To identify it with the One who created the world or brought Israel out of Egypt—

[Speaker B] It doesn’t seem to me like an idea that doesn’t exist in the world. Meaning, an idea that doesn’t exist in the world? Meaning, the notion that there is an external standard?

[Rabbi Michael Abraham] There is discourse like that, but there isn’t really a philosophy behind it. What is an external standard? There are people who say that one must obey the moral laws of the world. What, is there some thing that obligates that? They say no, no, of course not, it’s simply by virtue of our being human, or things like that, which in my view are empty of content. You can say whatever you want; language can bear anything. But it’s empty of content. What does that mean, by virtue of our being human? I’m like this and you’re like that, and that’s all. Right now I’ll kill you because I don’t feel like listening to your categorical imperative.

[Speaker C] Huh? You see it in certain cultures—they kill over family honor, and it’s legitimate.

[Rabbi Michael Abraham] No, here among us they kill over Sabbath desecration. Yes, no—let’s say that guarantee that they won’t…

[Speaker C] Kill—they think they’re moral and they do it because that’s what they believe, that’s what they think is right, legitimate.

[Rabbi Michael Abraham] An excess of moral standard.

[Speaker C] That’s their moral standard.

[Rabbi Michael Abraham] On the contrary, there it can definitely fit with the existence of an external entity, and that external entity does not necessarily impose good—it imposes a standard.

[Speaker C] But to say that it doesn’t come from something external and to say that that’s just the way it is—you see, there’s one culture that behaves this way and another culture… and both think that…

[Rabbi Michael Abraham] I’m not—again, I’m not saying that the existence of an external standard guarantees a moral world. What I am saying is that without an external standard, it is unreasonable to expect a moral world. With an external standard, it depends who it is. There are people whose external, objective standard is a standard that tells them, I don’t know, to be wicked. That too is possible. Fine, that’s also a kind of standard, but it’s a problematic standard. But I’m saying that without a standard there was some illusion, or maybe there still is an illusion in the world, that you can create a world without a standard—a world without believing in something outside human beings. Every atheist will tell you this whole thing, how fanatical religious people are and how they don’t understand what they’re talking about—of course you can be moral without believing in God, what do you mean, what does it have to do with anything, and besides, God is not moral anyway, as is well known. So what now? None of them—at least I don’t see there any serious accounting for this issue. There isn’t any. You can’t give a serious accounting for it. It really doesn’t hold water. Today it’s not popular to say this—if there is no God, the place is desolate and they kill me there, yes, those moralist sayings—but conceptually that’s how it is. I agree, by the way, people often mix this up—we talked about it in one of the previous years—they often mix it up with evaluating reality. In reality it’s not true. In reality a society without God can certainly behave in a fully moral way, or well. Yes, but I’m talking about the concept of morality, or the demand to be moral—more on the theoretical plane than on the practical plane. And the reason for that is that, fortunately, there aren’t too many intellectuals, as I said before. If everyone were intellectuals, then everyone would understand that it’s nonsense. And in the end, it seems to me, they would not really behave morally—or they would retreat, they would really understand that there is a standard, and stop fooling themselves into thinking there is no standard and yet one still has to be moral. In my opinion, that’s just words.

[Speaker B] I think I’ve encountered a lot of people, at least in writing, who I assume are intellectuals in that sense, and they say outright: yes, I don’t think there is a standard, and I still keep certain moral rules, and that is what is good for me. But I’m aware that it’s completely…

[Rabbi Michael Abraham] And his neighbor won’t keep them.

[Speaker B] So I’ll be angry at him, but that’s just how you’re built.

[Rabbi Michael Abraham] So be angry. He’ll kill you, and then be angry.

[Speaker C] But they would say, what primitive people—of course they behave that way.

[Rabbi Michael Abraham] Fine. In short, these really are speculations about what would happen in a world made entirely of intellectuals.

[Speaker B] But I’m saying there are many people who say openly: I am committed to this, and it’s only…

[Rabbi Michael Abraham] They say that, although in my opinion what really drives them at bottom is precisely a religious worldview—they just deny it. In my opinion there is no such thing, I do not believe in the existence of morality without that. I don’t believe them. You can call it paternalism—I don’t believe them. What does that mean, I don’t believe them? It’s unconscious. It’s not that three people really believe but aren’t telling you. “The heart does not reveal itself to the mouth.” Meaning, they don’t reveal it even to themselves. That commitment to values clearly testifies to belief in an external standard. The fact that they deny it is fine—maybe it doesn’t suit them, or whatever, for all kinds of reasons. That’s more a matter of psychology than philosophy. Okay. I’m already sounding exactly like a moral supervisor, but I think it’s true, what can I do. I think it’s true. Sometimes it’s even funny.

[Speaker F] That’s not true, Rabbi, that’s not true, nobody fell asleep.

[Rabbi Michael Abraham] Well, maybe because of those comments. The moral supervisors don’t say, okay, now I’m going to talk like a moral supervisor. But at least maybe it wakes people up. Okay. So now I really want to move on and talk about the problem of perfection and self-perfection in a slightly more formal sense, and after that come back to these introductions, because these introductions are important—to close the circle back. So I want to do it through a discussion of Zeno’s arrow paradox. Zeno of Elea, one of the Greek philosophers, cast doubt on the existence of motion in the world. Don’t ask me what exactly that means—did he really think there is no motion in the world? I don’t know. Another one of those intellectuals who talk, and I’m sure they don’t believe what they’re saying, but I don’t know, never mind for now. That really belongs to psychology, not philosophy. But he raised several arguments against the existence of motion. One argument is Achilles and the tortoise, yes, which is well known and is of course just a mistake born of simple mathematical ignorance. It’s just a mistake. But there is another set of paradoxes that are somewhat connected to the concept of continuity. Meaning, Achilles and the tortoise is a paradox that is simply a mistake, because unlike, I think, maybe all the other paradoxes—I don’t remember them all, but the ones I do remember—it is not connected to the concept of continuity. It’s just a mistake. But with the concept of continuity there really is a nontrivial ambiguity, and that’s what I want to talk about. But Achilles and the tortoise is simply about summing an infinite series and arriving at a finite sum, meaning he just didn’t understand that such a thing can happen—that the sum of infinitely many terms can sum up, or converge.

[Speaker B] Can sum up—when I learned limits, it can get closer and closer until we define it as the sum.

[Rabbi Michael Abraham] Fine, you still won’t get past two. Never mind. For solving the paradox, that’s enough. Because what I’ll show you is that you’ll never manage to get past two, even if you take infinitely many steps. On the other hand, those infinitely many steps also won’t take you more than two seconds. So this whole description in infinitely many steps describes the first two seconds of the race. Okay? And in the first two seconds of the race Achilles indeed does not catch the tortoise.

[Speaker C] And in the end, after two seconds?

[Rabbi Michael Abraham] Yes, of course after two seconds you continue onward. Meaning, he didn’t understand that a description in infinitely many steps can still be a description of the first two seconds of the race, up to those first two seconds. Okay? That’s just a mathematical error. But the other paradoxes are subtler. They are subtler, and therefore—to my astonishment—they have not really been dealt with to this very day. At least not as far as I checked ten years ago. Meaning, there are all kinds of general statements like, okay, it’s calculus and derivatives and things of that sort, but in a more precise translation, or an attempt to see where exactly the paradox gets solved, I didn’t see anything like that. I once wrote an article about this, so I checked a bit what exists in this area, and to my astonishment I discovered that, it seems to me, no real solution had actually been proposed. And by the way, the connection to calculus is also quite new, and it’s strange that this didn’t receive serious treatment. All right, how does the arrow paradox go? It’s kind of the most absurd of them. So it goes like this: there are several formulations; maybe I’ll read two of them. One formulation appears in Bergmann’s Introduction to the Theory of Logic. It says: “The flying arrow, in one indivisible moment of its flight, both flies and does not fly at once. And both statements are true: it flies, because were it not flying at every moment, its motion would not be carried out; and it does not fly, because within one indivisible and non-extended moment of time, it cannot perform any motion. Therefore say: the arrow both flies and does not fly at once.” Right? At the same moment in time the arrow both flies and does not fly. Why? If it were not flying at that moment, then it would not be advancing, right? On the other hand, in an indivisible moment of time it cannot fly. So it turns out that simultaneously it both flies and does not fly. In another formulation of the paradox—philosophically somewhat different, no matter, even though it’s very similar—it basically says: at every moment you look at the arrow, it stands in a different place. Suppose you photograph it. And at every moment it stands in a different place. Meaning, at every moment, so long as the moment is different, it stands in a different place. Maybe slightly different, but different. So now the question is: when does it pass between the places? When does it change its place? If at every moment it stands in a different place, then when does it jump? How does this change happen? That is another way of presenting the issue, only the first one is a formulation that undermines the law of non-contradiction, while this one is more, I don’t know, kinematic—more in the world of physics than in the world of logic.

[Speaker C] But it seems to me something is missing here. Suppose you take an object in this place—actually a pole in this place—and I photograph it in cross-section, then each time I photograph it, it’s photographed in a different place, if I have a camera that sees only one plane. Okay. But each time it is in a different place, because each time you see a different point.

[Rabbi Michael Abraham] Meaning, each time you raise the camera.

[Speaker C] Yes, and if I raise the camera, because there’s another dimension here that my camera doesn’t see. So the same with the arrow—time is also a dimension. Right? When you look at it at a certain time, of course it is at a certain point and it doesn’t move.

[Rabbi Michael Abraham] The question is: if it doesn’t move, then how does it arrive at a different point in the next moment if it doesn’t move? What do you mean? How does it help that there is another dimension here? How does it get to a different point in the next moment if it doesn’t move at that point?

[Speaker C] So he says if at every moment it doesn’t move…

[Rabbi Michael Abraham] Yes, exactly, that’s the question.

[Speaker G] There’s some kind of digital description here.

[Rabbi Michael Abraham] Right, exactly. It really is a digital description. Right, that’s his problem. But you really have to put your finger on where the problem is in that digital description. What is wrong with digitizing continuity? What did you lose here? At most, okay, do a finer and finer digitization—that still isn’t continuity.

[Speaker C] In the end, digitization is not continuity.

[Rabbi Michael Abraham] Exactly. Now there is something in this transition—it is not simply making the digitization denser and denser and denser up to infinity. However much you do, it is still that. So what is there in continuity beyond very dense digitization? That is basically the question in more modern language. Once—I don’t remember whether I mentioned this here—I talked about it at Bar-Ilan. I tried to demonstrate it through something that bothered me a bit, some problem that once bothered me: the stair problem. I don’t remember if I mentioned it here. Think about stairs. Suppose we divide a coordinate system. Okay. There is a unit—ten meters on the x-axis, ten meters on the y-axis. Okay. Or twenty meters on the y-axis if you want; they don’t have to be equal. Fine? And now I build equal stairs so that every meter I go up, say, two meters. Okay? Stairs like that. Now of course I can go up one meter every half meter. That too will get me from here to there. That basically creates a right triangle, where the hypotenuse is actually stairs, right? Only now the stairs are each time smaller in width and height. Now I increase the number of stairs, meaning it is always ten divided by n—that’s the number of stairs—and n gets larger and larger. So I increase the number of stairs and decrease the height of each one in such a way that it still starts here and gets there. Okay? Now what is the length of the—this is fractals.

[Speaker B] What? That’s fractals.

[Rabbi Michael Abraham] No, it’s not fractals. No. In my opinion that isn’t a fractal. Here, when you now look at the length, I’m asking what the total length of this path is. The hypotenuse, right? The stepped hypotenuse.

[Speaker G] And that equals thirty.

[Rabbi Michael Abraham] Wait, what’s the value? No, before what it tends toward.

[Speaker C] It equals the base plus the height.

[Rabbi Michael Abraham] Ten plus twenty is thirty, right? Because if you sum all the horizontal dimensions, that gives you exactly the base. Sum all the vertical dimensions, and that gives you the height. Meaning, altogether the path you traveled is simply the base plus the second base, right? Or plus the height. Okay. And what path did we traverse in the hypotenuse?

[Speaker B] Or the path you traversed?

[Rabbi Michael Abraham] In the hypotenuse—but also in the upward and horizontal parts.

[Speaker B] What do you mean?

[Rabbi Michael Abraham] The length of the stair path.

[Speaker B] Of the stairs, not of the hypotenuse.

[Rabbi Michael Abraham] The stairs are the hypotenuse. Right now it’s not a hypotenuse, it’s stairs. The hypotenuse is stepped. It’s not a line. Fine. So its total length is thirty.

[Speaker D] Now the question is why that doesn’t converge to the length of the hypotenuse.

[Rabbi Michael Abraham] Exactly. Now the question is: you increase n to infinity.

[Speaker C] Because apparently a point has dimension.

[Rabbi Michael Abraham] Wait, before that. What is the limit? Apparently, for a point—that is, there’s something in the continuous line that is not the limit of the steps when n goes to infinity. The limit of the steps when n goes to infinity remains A plus B. That’s the limit. By the mathematical definition of a limit, the limit is A plus B. But when you get to an actual continuum, suddenly there’s some kind of phase transition: it suddenly becomes the square root of A squared plus B squared. Meaning, the hypotenuse cannot be obtained by increasing the number of steps. Taking n to infinity will not give you the diagonal. It’s very interesting; it’s a very nice demonstration of this point: even if you carry digitization all the way, you still don’t reach the continuum. Now, around this problem of the arrow paradox, people are constantly talking in terms of the continuum, the problems of the continuum. You know, there’s also, for example, the description of a continuous line as a collection of very, very densely packed points—that’s a description that leads to paradoxes. Mathematicians already realized this long ago. You can’t treat a line—even though there are points on the line, there’s no problem defining points on the line, of course—but you can’t be satisfied with saying that a line is simply a dense collection of points.

[Speaker B] A collection even if it’s an infinite collection?

[Rabbi Michael Abraham] Even if it’s an infinite collection, yes. A line is not an infinite collection of points. An infinite collection of points is, in terms of cardinals, say, maybe a countable infinity—meaning that each point is defined and you can count it. But here you move to a continuum, and a continuum is an infinity that is not countable. No—infinite, there are several levels of infinity. No, it’s not a number of points.

[Speaker D] A continuum is—

[Speaker B] Not a number of points, you—

[Rabbi Michael Abraham] You can’t count the points. In the real numbers, it’s an infinity that is not countable. An uncountable infinity is no longer a collection of points. That’s exactly the point. It’s not that there’s a larger infinite number of points; it’s simply not a number of points. The cardinal is not a counter; that is, it’s not a counting number, the cardinality of the continuum. And that means it’s not a collection of points. A collection of points is some number of points, even infinite—you can count them and reach infinity. But it’s still an infinity of points, and of course each one is also located to the right of another. And of course in a continuum we know that between any two there is another one. By the way, that’s true even for the rationals: there’s the density theorem, that between any two rational numbers there is another rational, and that is still a countable set. So here too you have a collection of many.

[Speaker C] The rationals are countable, but the reals are not countable.

[Rabbi Michael Abraham] But I’m saying: the rationals, even though they’re countable, have the density theorem—that between any two rationals there is another irrational between them.

[Speaker B] Because between every—

[Speaker C] Any interval contains infinitely many rational numbers.

[Rabbi Michael Abraham] Right, meaning it has to be that between any two—

[Speaker B] Wait, but all the reals, say—that’s an infinity that isn’t countable. Not countable. But it’s still somehow a collection of many numbers.

[Rabbi Michael Abraham] No, it’s a collection—I’m saying, what is a collection? Obviously it contains many numbers, but you can’t say that the continuum is only the collection of numbers. There’s some additional thing here, an extra element beyond the set of numbers. That is, a continuum is not a set of numbers.

[Speaker C] That’s not how you define a real number.

[Rabbi Michael Abraham] It’s a set of numbers with some additional property, some additional dimension, some extra thing you have to put into the definition in order to get a continuum. If you put in infinitely many numbers, including from zero to one, even infinitely many, you still haven’t gotten a continuum. Just arranging them on a line so that they’re points you can describe as markings on a line—that means there’s something here beyond that set of numbers. There’s something else. And therefore this paradox is basically mapped onto questions of the continuum, of how one relates to the continuum. Now the problem is that when we speak in this language about the solution to the paradox of the flying arrow, we say, fine, it’s calculus—that is, we solve it with calculus. So you solve it with calculus, but the same problem exists in calculus too. At most, what you do as a mathematician in calculus is define it in a way that isn’t exposed to these problems. Definitions, of course, can always be consistent; the only question is what connection they have to reality.

[Speaker C] And what you brought from the steps, what you brought from the steps really is the expression of the paradox, that it’s not exactly—

[Speaker B] Calculus.

[Rabbi Michael Abraham] They solve—

[Speaker C] It’s in calculus, but it doesn’t solve the problem in— it’s not calculus. It’s not calculus. It doesn’t solve the problem.

[Rabbi Michael Abraham] No, it does solve it. Once you’ve already passed to the continuum, then you have to speak in a different language, in the language of calculus. You can no longer sum an infinite length. It’s already something else. Which is exactly the same transition as between points and calculus. Only there you see it much more concretely. So that means that when we translate this into calculus, all we’ve really done is an English-English dictionary. Because the same problem exists in calculus too. So what did we solve here? We didn’t solve anything here. The problem still exists.

[Speaker G] Only in a triangle?

[Rabbi Michael Abraham] Fine. I’m asking: but in the triangle how did you get there, yes.

[Speaker G] No, I’m not solving it.

[Rabbi Michael Abraham] Why? On the grid, where is the— I can— I’m asking what the difference is between a triangle made of steps when n goes to infinity and a right triangle.

[Speaker G] Difference? It’s a big difference, because I can give you lots more approximations.

[Rabbi Michael Abraham] Not approximation. n goes to infinity—the limit.

[Speaker G] I’ll give you diagonals instead of a straight line. Right? Since then it’s easier to understand, because it’s the base plus the height. I can do it with diagonals, fine, slanted lines, such that when you take them to infinity they also merge with the line of the hypotenuse.

[Rabbi Michael Abraham] Obviously. That’s not a solution.

[Speaker G] What’s the problem? The problem is that it seems to you like just a conceptual jump because it looks connected. But if you magnify it each time, you’re really left with the same structure.

[Rabbi Michael Abraham] No, but again, because you’re looking at a given n. I’m already taking the—

[Speaker G] I’m already taking the limit.

[Rabbi Michael Abraham] But because in the end—

[Speaker G] When you go to the limit you lose that.

[Rabbi Michael Abraham] No, no! That’s what I’m saying. The limit of this process is still A plus B. It’s not the square root; it’s not Pythagoras. But there are other processes that converge to that same limit, and then in your understanding—

[Speaker G] Other processes—what do I care?

[Rabbi Michael Abraham] Because in the paradox you can always tell me: look, your description is wrong because there’s another description that leads me to the correct result. Fine—that’s the whole paradox. The paradox says: show me what’s wrong in this description. Don’t show me other descriptions that are correct.

[Speaker G] When you get to infinity, intuitively you lose it.

[Rabbi Michael Abraham] No, but that’s not true. That’s the point—I’m saying no. At first I really thought that the limit when n goes to infinity is the square root. But that’s not true. The limit when n goes to infinity remains A plus B. It doesn’t depend on n at all. So why should the limit be different? It’s A plus B.

[Speaker G] Certainly, but not the hypotenuse—no problem.

[Rabbi Michael Abraham] Right. And now I ask: what’s the difference between this definition of the limit and an ordinary hypotenuse? What’s the difference? Because it’s—

[Speaker G] I’m saying again: you’re just looking at it at a different resolution.

[Rabbi Michael Abraham] No, again—you’re looking at n going to infinity as some kind of very large n. It’s not very large n. It’s as large as you like. That’s different. You can’t look at it in terms of resolutions; it’s not a matter of resolution. There’s something essential here. Just to illustrate a bit in the language of measure theory, okay? Points are zero-dimensional. Right? That’s zero length. An infinitesimal is zero length and one-dimensional. It’s not that points have zero length—they don’t have length. But an infinitesimal is zero length, yet its dimension is one. Meaning there’s something in the continuum that differs in character, in essence, from points. It’s not just a collection of many points. We’ve moved dimensions; we’ve moved to one dimension. Okay? So therefore I’m saying that translating this into calculus doesn’t really solve the problem. It only says: right, this problem exists somewhere else too, not only for you. Fine, I understand—now solve the problem for me. It’s like—maybe there’s another way to look at it. Do you know Russell’s theory of types? In the introduction to Principia Mathematica. There’s a book by Russell and Whitehead, three big volumes, where they try to build all of mathematics on set theory.

[Speaker B] There’s some person who actually read it.

[Rabbi Michael Abraham] Nobody read it. I hope Russell and Whitehead read it. But I don’t even know how the work was divided there. You know, in physics there are the books by Landau and Lifshitz, and the legend says there isn’t a word there from Lifshitz, and not a single idea from Lifshitz, and not a word from Landau. Meaning, Landau had the ideas and Lifshitz wrote them. By the way, Lifshitz came here to the Technion; I once spoke with him. I came to him with some problem. His ending wasn’t all that persuasive, but Landau apparently still was— anyway, here too I don’t know how the division worked between Russell and Whitehead. But there too, in the introduction to the book, he proposes solutions to self-reference paradoxes. Yes, like the liar paradox, or like the barber who shaves everyone who doesn’t shave themselves. There’s a whole set of self-reference paradoxes. And the solution he proposes there is that he builds a hierarchy of statements, types, yes? Statements that belong to different types, and each statement, by its rule, can refer only to statements lower than it in the hierarchy. And he defines there a hierarchy of statements; it doesn’t matter now exactly how. Now once you do that, of course a statement cannot refer to itself, because by definition it belongs to a type that is not itself—it is of a kind of things to which it cannot refer. So you solved the problem. But is that a solution to the problem? Of course not. You simply forbid formulating it. You build a language in which it is forbidden to formulate the problem. That is not a solution.

[Speaker B] It would be a solution if you could explain, translate every idea into that language. But obviously you can’t.

[Rabbi Michael Abraham] Obviously you can’t, because there are self-referential statements that are not paradoxical. And those statements— obviously you can’t, because there are self-referential statements that are not paradoxical, and those statements you won’t be able to formulate in that language. Meaning, this is exactly the failure of the analytic approach in philosophy—of the analytic conception, not of the approach. The analytic approach, if it’s a method, no problem; as a method it’s an excellent method. It shows, it insists on precision, it defines things properly—an analytic method is good. But the analytic conception is the conception that says everything is only a matter of definition. Meaning, if I offer a consistent definition, I’ve solved all the problems; all problems are merely ambiguities of language. There are no real philosophical problems. That’s basically the conception here. But it isn’t true. It’s like saying: there’s a death penalty for anyone who dares formulate the problem, and that way I solved the problem. That’s how Stalin generally solved problems—he simply killed everyone who raised them. Okay? So Russell doesn’t kill, but he basically forbids formulating the problem in his language. In that sense, calculus doesn’t offer a solution to the problem; it only defines a language in which the problems don’t appear. That is, it defines things consistently so that the problems won’t appear, but that’s not really a solution to a philosophical problem. I ask myself: what’s happening here? How can it be that an arrow—after all, a point in time exists in calculus too. On a continuous axis the points exist; no one denies that there are points on the continuous axis. The claim is that a continuous axis is not only a collection of points; there’s some further component here beyond the collection of points. Fine, so there’s another component. But now I say: at this point in time the arrow is standing here, so how does it get there? When does it pass? Right, that question is an intuitive question, and calculus won’t help here. Calculus is only a formulation of a language free of these contradictions, in which these contradictions can’t be posed or formulated. It’s like type theory, but it doesn’t really offer a solution to the problem. Another direction that was proposed—okay, we’re done with mathematics, with calculus. Now we return; now we come to physics. Some people propose a solution—I don’t know if anyone really does; I think maybe I thought of it myself—to propose a solution to Zeno’s arrow paradox in terms of the uncertainty principle. The uncertainty principle in physics says that for a body whose position is known, you can’t speak about its velocity. That is, a body cannot have velocity and position simultaneously. If it has a defined position, then it has no velocity; if it has a defined velocity, then it has no position. And this is a big question—philosophers and physicists disagree about it. Today it’s commonly thought that they also simply cannot both exist. But that still hasn’t been proved; that is, perhaps people are looking—no, no, no, there’s a fascinating seam there between philosophy and physics. There are apparently experimental means that could test experimentally whether it simply cannot be, or only whether we cannot talk about it. And what I just said sounds almost like an oxymoron.

[Speaker E] A limitation in the precision of all measuring devices?

[Rabbi Michael Abraham] No, no, no—not a limitation of precision. It’s an actual property either of reality or of our way of looking at reality, but it’s not a limitation of precision. That’s not a precision limitation; that has been proved, that’s clear. Bell’s inequality—there are experiments, meaning by now there’s even proof of this. But the open question remains whether it’s something that stems from us or something in reality itself. Physicists think it’s something in reality itself, though it’s not entirely clear. Bohm and Aharonov tried—Bohm, for example, tried to speak about so-called hidden variables in quantum theory, which basically say that in reality itself things do work out; it’s just that there are variables that aren’t exposed to us, they’re hidden, and they are really what account for all the physical anomalies. Fine, but that’s another discussion. For our purposes, apparently here too there’s a solution. Because what am I saying, essentially? You ask me when the arrow moves if at every moment it stands at a different point. But according to the uncertainty principle, if it is at a certain point, you can’t speak about its velocity. So what does it mean, what is its velocity when it stands at that point? When it stands at that point, it has no velocity. So this too could be formulated in two ways—there are philosophical subtleties here. But one could present it as: the term you are speaking about is undefined. The velocity of the body at the point where it stands. That is simply a contradictory term; it is undefined. And one can talk about this more in the language of physics, it doesn’t matter now—you can’t speak about that velocity. The question is not one that can be answered.

[Speaker C] The velocity while it stands. Exactly.

[Rabbi Michael Abraham] Here now—this is a point where I think you’ve actually put your finger on the point that is correct. Meaning, it really is a solution to the problem and not merely a prohibition against formulating the problem. Because in the solution via the uncertainty principle, my feeling is the same as with calculus. There too it’s an English-English dictionary. You’re offering me a solution to one thing I don’t understand by means of another thing I don’t understand. Fine, so that other thing is like this one—that still doesn’t solve it. But the point is that in both places I now need to look for an explanation. So my feeling is that there isn’t really a solution to the problem here, only a demonstration that it exists in another context too. So I suggest maybe reversing the picture. I suggest explaining the uncertainty principle by means of a philosophical solution to the arrow problem, and not using uncertainty to explain the arrow. The other way around. Explain the arrow, and use that in order to understand the uncertainty principle, or a certain element of it. And what I basically want to say is really what Ilan just said. It seems to me that there is simply a conceptual confusion here. It is not directly connected to the concept of the continuum at all. Indirectly yes, but not directly. It seems to me that this mixes up two different statements and sees them as the same statement. And many philosophers who followed him fell into the same conceptual error. Not directly connected to the continuum. When I say that a body is at a certain place, and when I say that a body is standing at a certain place, that is not the same statement. Not the same statement. A body is at a certain place means it is at x equals two. I said nothing about its velocity. It is there. When I say a body is standing at x equals two, then I am saying that it is there and its velocity is zero. Right? Meaning, even for a moving body I can speak about its being in a place. Leave uncertainty aside for a moment. I can speak about its being in a place, and I ask myself: is it moving? The answer is yes. It is moving at an indivisible instant of time; it is moving. How can that be? Here I take one more step. What does moving mean? You can define moving as changing position. The body changes its position. Fine, it is obvious that this cannot be. A body cannot change its position in an indivisible instant of time. Okay? But I can speak of a moving body in the sense that the body has velocity. To say that a body has velocity is not the same thing as saying that the body changes its position. A body has velocity at an indivisible instant of time. The body does not change its position at an indivisible instant of time. That’s true. But it has velocity at that instant. And my claim is that when I say the body is moving, what that means is that it is at a different place at every moment of time—not that it is standing, but that it is there. In what sense is it not standing? It has velocity. It does not change place, but it has velocity at an indivisible instant of time. By the way, another common mistake, among physicists too—or mainly among physicists—is that a body cannot change place in an indivisible instant of time: that is not a physical limitation; it is a logical limitation. Physicists think this is a limitation because a body cannot move at infinite speed. Because in order to change position in one instant you basically need infinite speed. Zero time multiplied by infinite speed would give you some finite distance. But a body cannot move at infinite speed; that’s a result of relativity—you can’t exceed the speed of light. So it seems from this that it is a limitation of physics, not of logic, but that’s not true. A body moving at infinite speed is not the same thing as a body being in two places at the same time. That is a logical limitation. Again, the transition to the continuum. A body moving at infinite speed means a body whose speed is greater than any speed we know. But that is not a body that is in two different points at the same time. Because if it is in two different points at the same time, that is a logical problem. It’s not the same thing—there are two bodies. It’s not a problem of a speed bound; you don’t need relativity for that.

[Speaker C] The problem is logically prior before it’s a physical problem.

[Rabbi Michael Abraham] Exactly. The physical problem is a harder problem: you can’t exceed the speed of light. But saying that you can’t be in two different spatial points at the same moment is only saying that you are one entity and not two. Meaning, this is a limitation of logic, not of physics. Physics only says the speed of light is the maximum—that’s relativity. But the fact that both cannot be true at the same moment, in two different points at the same time—that’s a limitation of logic. It’s impossible. Okay?

[Speaker E] Isn’t infinite speed also a logical problem?

[Rabbi Michael Abraham] Infinite speed is a hard question, because again, you have to get into definitions—what is infinity?

[Speaker E] Like, doesn’t infinite speed also mean in zero time?

[Rabbi Michael Abraham] Concrete infinity, not potential infinity. Fine, the logical paradox there is the one of concrete infinity. Right. So indirectly it is somehow connected to these questions, but I’m saying here it’s very clear. You don’t need relativity; that’s not the point. Now what does this mean? What causes us to identify velocity with change of place? What causes us to identify velocity with change of place is, first, physics, and second, the character physics has taken on. Because when we define velocity, every physics student knows—how do we define velocity? Change of distance over time, that is. Difference between places, change of place or change of distance—sorry, movement of distance, change of place over a certain time. When we define velocity, we are basically taking a difference of places divided by a difference of times. If you were at x equals one and t equals two, and at x equals three and t equals five, then what is your velocity? The difference in places—which is three minus two, that’s one—the difference in places divided by the difference in times, five minus two, that’s three, so your velocity is one-third. Okay? One-third meter per second, or whatever your units are. Now what happens if the velocity is not constant? Meaning, the velocity changes. Then we have to do this over a very small interval, over an infinitesimal, let’s speak a bit crudely. Yes, over an interval as small as we like, and over a small interval you can always assume it’s a straight segment. Then the definition of velocity is still difference of places divided by difference of times, only now they’re very, very small. And now I’ll take it to the limit, never mind—but even in the limit I’m still speaking in terms of an interval that is as small as we like. Right? What does that mean? That in order to define velocity I need an interval. I cannot define velocity at a point in time. But notice: anyone who knows physics knows that when we do this, it’s really a derivative, the concept of a derivative. So when I have position as a function of time and I want to calculate velocity, I differentiate the position function with respect to time. The result gives me velocity at every point in time. True, I pass through an interval in order to define velocity, but that interval is only a computational device.

[Speaker B] Meaning, that’s assuming it’s a differentiable function.

[Rabbi Michael Abraham] Yes, of course, a differentiable function. So it’s only a computational device. After I used that computational device, my result is velocity at every point in time, period—an indivisible instant. There is velocity at that point. Okay? It’s just that in order to calculate it I need to look at an interval as small as we like around that point in time, but that is only a computational constraint. Now, what happens when we define velocity in this way through the derivative? We are actually using an operational definition, a definition of how to calculate, but that is not the definition of the concept. Those are two different things. When I ask how to calculate, that’s the definition. Is velocity that? I think not. Velocity is not that. Velocity is a potential for change of place. Velocity is not change of place. If a body has velocity, then in the end it will also change place. But velocity is not change of place; velocity is a potential for change of place.

[Speaker E] In the physical sense.

[Rabbi Michael Abraham] What? Yes, velocity takes time, but I’m saying: the fact that you have velocity means that you really have the potential to change place. And therefore that potential is a property that exists in the particle at a point in time, an indivisible instant of time. It has velocity. It’s not true that it is standing still. It has velocity. It cannot change place at a point in time, at an indivisible instant of time—correct, because that’s a logical problem. So what? But it has velocity at every point in time. And therefore the distinction between a moving arrow and a standing arrow is a distinction that can be made at a single indivisible point; you don’t need an interval for it. The question is whether it has velocity at that point or whether its velocity is zero. If it is standing, then it is in that place and its velocity is zero. If it is moving, it is in that place but its velocity is not zero. Now how will I calculate the velocity? Fine—for that I need to look at an interval. Why? Because I need to see how it changes place. That will give me an indication of how much velocity it has. Potential. Exactly. But that is only a way of calculating; it is not the definition. Those are two different things. Okay? Now if that is indeed so, then the paradox of the flying arrow is, of course, solved on its own. Here I don’t need—again, behind this of course sits the continuum—but I don’t need to know calculus. I could have given this answer in the 12th century too. I don’t need to know calculus or know how to calculate it in practice in order to say what I just said. After calculus people already made use of it and so on, so in a certain sense we actually lost the ability to understand this, because we are already captive in a language in which the problem does not appear. But this really is the philosophical explanation of the problem. Now if I broaden this a bit more, then I say as follows. We are actually built—and here I’m going back a bit to Rabbi Kook now from this formal analysis—why do we fall into this mistake in the first place? We identify velocity with change of place because we basically think of, or know, the world in camera mode. We are built like a camera. When we film a movie, how is a movie built? A collection of frames. Yes, a collection of frames, right, packed closely together. Why? Because our basic perception of the world is the perception of a camera, not of a camcorder. When we want to produce motion, all we need to do is— and therefore the problem of the flying arrow is born, because we think about motion in the language of a camera. Now let’s try to abstract away from our usual mode of thought or cognition. Suppose there were some alien creature that perceived the world the opposite way. It perceives the world fundamentally as a camcorder. For it, the camera would be something it doesn’t understand. It’s the integral over the filming; position is the integral over velocity. Right? So fundamentally it thinks in terms of velocity, not in terms of positions. Okay? So it would perceive motion as something simple. On the contrary—position, for it, would be some kind of thing that has to be calculated. It would not be something it grasps in itself; in order to understand position it would have to take an integral. Meaning, its basic perception is a perception of a camcorder. For it, films would not be built as a dense projection of frames. No, it would just directly see the film. On the contrary—in order to see a static image, it would need to perform some combination on the films. We, in order to see motion, have to do some combination on the still images, or as—

[Speaker C] As we said, freeze the picture.

[Rabbi Michael Abraham] Yes, it would need to freeze it, exactly. Now suppose such a creature exists—and theoretically such a creature could exist—with a form of cognition different from ours, and thinking different from ours, and perception different from ours. Then it could see velocity at a point in time. It wouldn’t need to go to an interval around that point and take a derivative.

[Speaker G] It would only see velocity.

[Rabbi Michael Abraham] Exactly. It would see the velocity; position would not be before it. It wouldn’t understand what position even means. Position would be some fiction it would somehow generate out of the film. It would need an integrator; we use a differentiator, it would use an integrator. Okay? So what comes out is that we are prisoners of a perception that is static in its essence, and even dynamics we analyze in static language, and we approximate it through static language or calculate it in static language. But all that is trying to capture something that is not of that kind. We are trying to capture velocity, which is a property of an object at a point in time. We have no way—this is not accessible to us; we don’t know what to do with such a thing. In order to calculate it in our language, in our static world, we do this trick of framing or taking a derivative. But that is only because we are built this way. Okay? Now notice something nice. I think for physicists this is really quite a remark, when I understood this. In physics, after discussing the uncertainty principle and saying that a body cannot have position and velocity simultaneously, then basically every pair of quantities that—

[Speaker C] Do not—

[Rabbi Michael Abraham] Commute with each other, exactly—like time and energy, position and momentum, and so on—there are also two world-pictures in which one can present reality: the position picture and the momentum picture. The position picture is when all the properties that interest us are functions of x, functions of position. The momentum picture is when all the properties that interest us are functions of momentum, of p or k, it doesn’t matter. Okay? Now these are two forms that are orthogonal; they don’t talk to one another. I can move from one to the other—it’s a Fourier transform, for those who know; that really already is technical—but these are two pictures that you can never inhabit both of. Either you look at the world through the glasses of position, and then everything is a function of x, or you look at the world through the glasses of momentum, and then everything is a function of p. You can’t wear both pairs of glasses at the same time. That’s the uncertainty principle. The uncertainty principle says you cannot wear both pairs of glasses at the same time. You cannot talk about the position of a thing and the velocity of a thing together at the same point in time. Now understand that this is exactly what I said earlier. When I speak in the glasses of position, what is that? Those are camera glasses. A camera with zero exposure time—real zero, not just very short. Actual zero, meaning a mathematical instant of time. Okay? In such glasses there is no velocity. You cannot speak about velocity; velocity does not exist at a point in time. Because those are the glasses—you’re looking in the position picture. When you look at the world in the position picture, the concept of velocity is an artificial concept. You need to look at the position at one moment, the position at another moment, take a difference, divide by time, take a derivative. That’s how you extract velocity. In the momentum picture, it’s not done that way at all. In the momentum picture you get the velocity directly. You don’t need to take a derivative. Now of course the momentum picture is therefore very hard; even physics students have a very hard time grasping this, because the momentum picture is always some vague thing they don’t understand. Why? Because that really isn’t the way we think. We think in the position picture. But the momentum picture is a picture—those who know this stuff know—it is exactly as legitimate. So for that creature there’s no problem; that’s the momentum picture, not the position picture. Then it would have to extract positions from some integral over the momenta. And what this basically means is that the uncertainty principle, or this duality of pictures, really is—yes, postmodernists love this kind of thing, that everyone has his own picture and the pictures don’t speak to one another and therefore you can’t argue, and all kinds of things like that, which of course is the opposite of the truth, but never mind—you can move from one picture to the other by a Fourier transform. So that means exactly that these are not two different pictures; they represent exactly the same reality. But it’s like Euclidean geometry and non-Euclidean geometry, or things of that kind, and people always take it in the direction of relativism, when in fact it means the opposite. But these two pictures basically emerge from Zeno’s arrow paradox. So now I explain the uncertainty principle through understanding the problem in Zeno’s arrow paradox. I do not use uncertainty to explain the arrow; I use the arrow to explain uncertainty. Now of course I’m not going to get h-bar out of this—for anyone who wants that—I can’t derive from this the quantitative measure of uncertainty. Physics determines that there is also a certain definite amount of uncertainty. Yes, meaning, I can only tell you that if you speak about exact position you won’t be able to speak about velocity; if you speak about exact velocity you won’t be able to speak about position. I can’t tell you what happens if I speak about an inexact but small position—how much uncertainty there will be in velocity. That I can’t derive from here. That’s a result that comes from measurement; h-bar is some number, it doesn’t matter. Okay, but the principle that there are two pictures that do not speak to one another.

[Speaker C] What, what? Which one? You do force with the derivative? What, what? Which one? If you have the new interpretation, then you can already with the derivative—

[Rabbi Michael Abraham] No, no, I can’t, I can’t—that’s what I’m saying. The uncertainty principle is made up of two things. First, you cannot speak about position and velocity together. That I can explain through the arrow principle in a completely simple way. That’s something we actually should have anticipated long before the 20th century. If people had sat with this paradox—if I had managed, of course I live after uncertainty—but if I had managed to write this article in the 16th century, then I could have derived the uncertainty principle in the 16th century, without any measurement.

[Speaker B] This part of the uncertainty principle.

[Rabbi Michael Abraham] This part, yes. Of course, not h-bar.

[Speaker B] Not only the number, but this law as a practical law—that in reality we are actually unable to measure velocity.

[Rabbi Michael Abraham] And we will never be able to know the exact position and velocity of the same thing simultaneously, at maximal precision.

[Speaker B] As a physicist, you should have been able to understand that in advance.

[Rabbi Michael Abraham] What do you mean, as a physicist? I’m claiming it’s not physical; it’s a logical law. That’s exactly what I’m saying. That’s precisely the point.

[Speaker B] If it’s a logical law, then these are two different concepts. But how does it follow from their being two different concepts that we won’t be able to measure them simultaneously?

[Rabbi Michael Abraham] Not that they are two different concepts, but rather that you can’t—you cannot speak in terms of this concept and that concept at once. They are simply two ways of looking that do not sit together. Like when we look at a moving body with a camera, there is always some blur trail. We can’t really see motion with a camera.

[Speaker B] Yes, but won’t we be able to see, regarding the same body, both this picture and that picture? Not together—fine, not together. At the same time.

[Rabbi Michael Abraham] We—

[Speaker B] We’re talking about together, at the same time. Maybe we won’t think about them together, but with respect to the same time it could have both this picture and that picture.

[Rabbi Michael Abraham] That’s why I say: our dimension, at least—I said this is a debate between physicists and philosophers—whether this is a claim about reality or a claim about our perception of reality. In our terms, we could bracket that, from our standpoint. Now, what this claim is basically saying is that looking—that is, I can look at reality through static glasses, and from that static look, from the trail—after all, the exposure time is not zero—so from the trail I understand that there is motion here. So I understand that I need to say: there is something behind what I see. You see how this comes back and flips things around? In other words, motion or progress always indicates that there is something behind the things that I can’t grasp through what I see, through the static gaze, through the state itself. Meaning, I can’t talk about velocity through changes of state; changes of state are a consequence of velocity, they are not velocity itself. But I do have some indication that there is something there behind it. He spoke about the true reality peeking through the cracks of what we see. That is exactly this point. In other words, there is something standing behind what we see. It’s clear to us that it is there, we have clear evidence, but we have no way to grasp it directly, and that is exactly what is responsible for change. There is some kind of—when we talk about elevation, Rabbi Kook talks about elevation, about perfection, about change—that means there has to be some engine, some something behind it, standing behind the change; that is velocity. Potential. The potential of change. Velocity is of course with respect to change of place, but every change has a derivative at its foundation. And we always grasp the derivative as a kind of fiction. What exists is the location; the derivative is only a useful definition, it doesn’t really exist. That’s not true. It’s only because our glasses are static glasses. The derivative exists to the same degree. And therefore behind the reality that we see sits something elusive like that, which we can try to calculate, reach indirectly, but we have no direct way to grasp it. If there were, for example theoretically, if there were a way to measure velocity—not in the form of differences in place divided by differences in time, but to measure velocity at a point—then, I claim, there would be no uncertainty principle, or at least—it could be, maybe I’m wrong—but this consideration should not yield the uncertainty principle in such a case. At one point I thought about the Doppler effect, for those who know. The Doppler effect is seemingly hitting a car at a single moment in time, a beam that hits the car at one moment in time and I know the speed of the car at one moment in time. The radar. What? Yes, the radar. And then apparently there could have been a situation there—and that’s not correct. When you examine at higher resolution what the Doppler effect is, it’s not one point.

[Speaker C] It can’t be one point. What?

[Rabbi Michael Abraham] It’s just a frequency shift.

[Speaker C] It has a frequency?

[Rabbi Michael Abraham] No, so I’m saying, on the theoretical level, if I had some Doppler like that that managed to touch the car at one mathematical point and know its speed, then maybe I would know speed and position simultaneously.

[Speaker C] And that wouldn’t be because of the consideration

[Rabbi Michael Abraham] that you mentioned. It could be that such a thing simply cannot exist at all. Yes, it could be that such a thing cannot exist at all. Fine, but I’m saying on the principled level, theoretically at least. Say that cameraman perhaps has that, though then of course he would have no ability to measure position. So that’s why I say: but in principle there could be an instrument—the instrument he would use in his world would be an instrument that can grasp velocity at a point. You could call it an ideal camera, not a camera of frames, but a camera that grasps velocity at a point. What?

[Speaker B] We also can’t grasp position by a measurement at one single point. Why not? How can I? If I make any measurement at all, it will be through light rays or rays of…

[Rabbi Michael Abraham] No, fine, obviously. But I’m saying that our way of thinking, our form of cognition, is static in its essence. There are physical constraints, fine, obviously. That’s also why there is no camera whose exposure time is one point in time, zero. It’s always very small, but not…

[Speaker B] No, but for the sake of argument, one photon—can it… can I measure position by sending one photon?

[Rabbi Michael Abraham] No. One photon, in principle, also has no position. An ideal photon, a photon with a defined frequency, is found throughout the whole world. It has no position, because it has defined momentum. There is no such thing as one photon in one place. When people talk about one photon in one place, that’s a mistake. What one can talk about is only—just now I was speaking with someone who works precisely on these points at Bar-Ilan—when they do experiments on one photon, but it is always a photon

[Speaker B] that is a collection of frequencies,

[Rabbi Michael Abraham] exactly, but that is a small collection of velocities. Meaning, it is never one fixed velocity. But why are you saying one velocity?

[Speaker B] Never mind, also a photon…

[Rabbi Michael Abraham] No, but then it is no longer a photon in the full mathematical sense; it is no longer a photon.

[Speaker B] Okay, so one quantum—what should we call it?

[Rabbi Michael Abraham] No, it’s not a quantum. I’m saying a quantum only of frequency, yes? It’s already not a quantum. There is a terrible conceptual confusion there. Meaning, most physicists fall into this. I was just talking with him, and it really became clear that a lot of things physicists say are simply misunderstandings. Meaning, they talk about a photon, but what they really mean is some wave packet like that. That’s not a photon; it’s something else. Fine, a side issue. There really is no genuine uncertainty principle like they teach in class. Meaning, in the lab it’s never really that. Okay, but to our matter: now if I return, yes, to Rabbi Kook’s perfection and perfecting, then basically the claim is that when we now—I’m coming back now—what solution did he propose? That the Holy One, blessed be He, is perfect, so He cannot have the perfection of becoming perfected. That perfection is missing for Him, right? So what does He do? He creates deficient beings, and they become perfected as it were on His behalf; meaning, through them He becomes perfected. What does that mean? Let’s now try to think about a certain implication of the picture I described. Suppose a body has velocity and it gets stuck in a wall. Now physicists don’t like this way of describing it; this is layman’s language, okay? But when it gets stuck in a wall, it has velocity and it cannot translate that into change of place; in other words, the wall does not let it advance. Physicists of course say that the moment it gets stuck in the wall the velocity is already zero and there is no… but leave that aside. I am now speaking explicitly in layman’s language because I want to solve the philosophical problem, not the physical one. So that means it has velocity when it gets stuck in the wall, and it loses it at the wall; it cannot advance. Now this will come out in the form of heat, impact, an explosion, whatever, the stuff they teach in high school. But it does not actualize in the form of a change of location. What does that mean? It does not mean that it had no velocity. For physicists it means it had no velocity, because by definition… calculate operationally and you’ll see it has no velocity; the velocity is zero when it’s at the wall. Fine? But in the way I’m talking now, it does have velocity; it’s just that this velocity is not actualized in the form of a change of place, but in another form—exchange of momentum with other bodies, or dispersing through heat, or all kinds of things of that sort. And therefore, what we really see here is that there can be a situation in which a body has velocity, a potential for change of place, but that potential is not actualized. Here we have an example of the distinction between velocity and change of place. A body can have velocity and there will be no change of place there, and then indeed they cannot calculate its velocity; for physicists its velocity is zero. But we will see it through the heat that comes out, or through all kinds of other things. Okay? Now with the Holy One, blessed be He, there is a very similar phenomenon. He has velocity. He has the potential for progress. The fact that He is infinite does not mean that He has no velocity. It only means that this velocity cannot be actualized in the form of improvement, of elevation. Reaching the end. Exactly. So that does not mean that He has no velocity; He does have the perfection of becoming perfected. It’s just that the Arizal—I brought him last time—says that the Holy One, blessed be He, created the world in order to bring His names from potential to actual. What does that mean? In order for this velocity to come out into actuality, for it to be realized, that cannot happen within Him. So the perfection exists within Him; He is completely perfect. But in order for that perfection to appear, there has to be someone through whom heat will disperse, through whom the momentum will pass to him. Meaning, that the potential of that velocity will produce change of place in someone else, in someone who has somewhere to change to. Therefore the Holy One, blessed be He, created someone lacking, so that the potential of change would be actualized in him. And therefore when Rabbi Kook says that this is actualized through us, the meaning is not that He lacks perfection in the simple sense, service for a higher need, what the medieval authorities say—service for a higher need, that He needs us. He does not need us. He has it; the velocity is in Him, not in us. The ability to change, to ascend—that is the engine Rabbi Kook is talking about. This engine that causes things to change, which he says is divinity, is perfection—it is what gives them the possibility of changing. What does that mean? This potential, this velocity, is also found in Him. There is no contradiction at all between His being infinite and perfect and His being able to have velocity. That is exactly Zeno’s arrow. You can be in one place with velocity. There is no change of place, but you do have velocity. Now this velocity—somehow it mattered to Him that it also come out into actuality. For that to come out into actuality, there have to be deficient beings, who can change place. Someone who has not yet reached the wall can still move forward. Someone who is already in the wall can no longer move forward.

[Speaker E] We’re the heat of the Holy One, blessed be He.

[Rabbi Michael Abraham] Exactly. Or we’re the—yes, exactly—or we’re the body through which this potential of velocity comes out. And then this casts a completely different light on everything Rabbi Kook says. Because it basically means that the Holy One, blessed be He, has this perfection, He has the velocity. The velocity is the perfection, not the change of place. Now we can also understand that improvement—I spoke about two concepts of improvement: reaching a better state, and the very progress toward that better state. A penitent versus a completely righteous person. So here now it is completely distinguished. Meaning, perfection is essentially the velocity, the potential for improvement, not the improvement. That is the perfection. Where do we see that a person has such a potential? When he really does improve. But that is only an indication. The perfection is that he has velocity. And that is what a penitent has and a completely righteous person does not have. Okay? So these really are two different concepts; one must not mix them up. It’s not a contradiction; it’s simply two concepts: velocity and change of place. Velocity can appear without change of place, and change of place without—no, there is no such thing as change of place without velocity. Velocity is the potential. Exactly. And that potential also exists in the Holy One, blessed be He. In that sense, this perfection exists in Him and there is no problem at all; it is not really a deficiency there. Once you understand that it is not the change of place, the change of place can also appear in us. But the velocity—there is no reason at all it could not also be in Him.

[Speaker C] A completely righteous person doesn’t have this potential

[Rabbi Michael Abraham]?