Giving a Get and Quantum Theory
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
- Opening and dedication of the lecture
- The difficulty in defining the giving of a bill of divorce in the tractate Gittin discussions
- Giving as a process rather than a state
- The failure of simple definitions and the complications of abstraction
- Zeno’s arrow paradox and the problem of motion at an instant
- “And at midnight” and the death of the firstborn as a seam-instant
- The uncertainty principle and quantum theory as a framework of two descriptive languages
- Velocity: operational definition versus essential definition, and instantaneous velocity
- Velocity as the potential for change of place, and its separation from actual change of place
- A camera, an ideal movie camera, and the structure of consciousness as seeing states
- Implications: repentance, perfection and perfecting, and process as having intrinsic value
- Collectives, fluidity, and the judgment of the community in Maimonides
- Returning to the giving of a bill of divorce and the Sages’ method of examples
- A paradigm for Torah study and halakhic intuition
Summary
General overview
The speaker presents Rabbi Michael Abraham as a multidisciplinary scholar engaged in Torah, philosophy, and physics, and dedicates the lecture to the merit of Rabbi Shugerman, who is undergoing medical treatment. Rabbi Abraham opens with the difficulty that arises in the tractate Gittin discussions in defining “and he placed it in her hand,” and argues that the Talmud chooses a method of many examples because giving a bill of divorce is a process that is hard to define sharply through a theoretical rule. He develops a distinction between states and processes through Zeno’s arrow paradox, through the difference between an operational definition and an essential definition, and through the uncertainty principle in quantum theory, and concludes that genuine halakhic understanding is formed through intuition built from cases, not from a list of simple rules.
Opening and dedication of the lecture
The speaker warmly welcomes Rabbi Michael Abraham and introduces him as someone who teaches Torah at what is called the Advanced Institute at Bar-Ilan University, the kollel of Bar-Ilan, and as a rosh metivta at the hesder yeshiva in Yeruham. He mentions the “Middah Tovah” publications on the thirteen hermeneutical principles, and the books “Two Carts and a Hot Air Balloon” as an original blend of science, Torah, and philosophy, and adds that Rabbi Abraham holds a doctorate in physics. He says that Rabbi Shugerman is in the United States undergoing difficult medical treatment, that the public is praying, with God’s help, that it turn out well and that he have a full recovery, and that the lecture is dedicated to his merit and recovery.
The difficulty in defining the giving of a bill of divorce in the tractate Gittin discussions
Rabbi Abraham recounts that the topic arose for him while studying passages in tractate Gittin that try to define what “giving” means in a bill of divorce from the verse, “And he wrote for her a scroll of severance and placed it in her hand.” He describes how the Talmud brings dozens of complicated examples, such as writing the bill of divorce on a slave and giving her the slave, transferring ownership of a courtyard, throwing it into her courtyard, benefit prohibitions, and cases where the bill of divorce is in his hand and the string in her hand or vice versa, and he feels that this method is cumbersome compared to simply giving a definition. He notes that later authorities such as Kehillot Yaakov tried to distill a general principle, but in the end arrived at a “definition” that has nearly as many parts as there are examples, because each formulation runs into a different case and requires further revision.
Giving as a process rather than a state
Rabbi Abraham argues that giving a bill of divorce is an action or a process, not a state, and therefore the mere fact that the bill of divorce is in the woman’s hand is not enough for the divorce to take effect. He says the question is “how did it get into her hand,” and illustrates that if the woman takes the bill of divorce for herself, or if it reaches her “by way of the birds of the sky,” there is no divorce, because an act of giving is required. He argues that actions are harder to characterize than states, and therefore one tries to describe the action through pairs of states, before and after, and from the many cases to develop an intuition for what valid “giving” is.
The failure of simple definitions and the complications of abstraction
Rabbi Abraham presents attempts to define giving as physical handing over or as transfer of ownership, and explains that each of these fails the test of the examples in the Talmud, such as a bill of divorce placed in a courtyard transferred to the woman, or examples involving benefit prohibitions brought by Ketzot. He describes how, when one cannot delimit the action through simple endpoint states, there arises a need for an abstract definition, and then the Sages give a kind of “sense of smell” through cases: “this is a good giving” as against “this is not a good giving.”
Zeno’s arrow paradox and the problem of motion at an instant
Rabbi Abraham says he was reminded of Zeno’s paradox of the arrow in flight, in which the arrow appears to be “standing” at every moment at a different point, and the question becomes, “when is it moving?” He presents a formulation according to which motion requires a stretch of time and not an “indivisible instant of time,” and asks: if there is no motion at any single instant, then how does motion accumulate? He notes a common modern solution based on infinitesimal calculus and the conception of a continuum that is not a collection of discrete points, but he qualifies this and says there is such a thing as an instant of time, and that the mathematical mode of description does not necessarily dictate the actual structure of time.
“And at midnight” and the death of the firstborn as a seam-instant
Rabbi Abraham raises a question about “and at midnight,” which he had initially attributed to the Rabbi of Brisk, but later realized was probably from his student, the author of Ohel Yehoshua / Emek Yehoshua, where it is argued that there is no “instant” of midnight, only a seam between the first half of the night and the second. He quotes the answer that attaches that seam to the seam of death itself, because “there is no instant of death,” only a transition from life to death, so “they were alive until midnight and they were dead from midnight.” He says this expresses a view that an instant is not something that really exists, but adds that he thinks this is “not correct.”
The uncertainty principle and quantum theory as a framework of two descriptive languages
Rabbi Abraham turns to “quantum theory” and presents the uncertainty principle, according to which one cannot speak of or measure position and velocity at the same instant; if there is position, there is no velocity, and vice versa. He rejects using this principle as a direct solution to Zeno’s paradox as an “English-English dictionary,” and prefers first to understand why there is no problem with the arrow itself. He explains the uncertainty principle as describing two “pictures” or descriptive languages that are not translatable into each other: the position picture, in which velocity is not defined directly but calculated indirectly, and the momentum / velocity picture, in which position is not defined directly but calculated indirectly.
Velocity: operational definition versus essential definition, and instantaneous velocity
Rabbi Abraham distinguishes between “how velocity is calculated” and “what velocity is,” and argues that the ratio of distance to time is only an operational definition. He explains that in variable velocity one uses “instantaneous velocity,” calculated through time intervals that get smaller and smaller around a point up to a limit, that is, a derivative, but argues that this is a computational tool and not proof that velocity does not exist at a point in time. He suggests that velocity is a property that characterizes a point in time, whereas change of place is an indication that becomes visible over an interval of time.
Velocity as the potential for change of place, and its separation from actual change of place
Rabbi Abraham argues that velocity and change of place are not the same thing, and that velocity is “the potential for change of place.” He illustrates this with a body thrown at a wall that arrives with velocity but the wall does not let it “bring that potential into actuality,” and with a ball colliding with another ball and transferring momentum so that the first one’s change of place becomes smaller. He uses this to answer Zeno by saying that the arrow “is moving at exactly the same moment that it is also standing” in the sense of “being present,” because it can be in a particular place with nonzero velocity, whereas a question like “at what moment does it change place from A to B” is meaningless, because change of place requires two places and not a single instant.
A camera, an ideal movie camera, and the structure of consciousness as seeing states
Rabbi Abraham explains that an ideal camera with zero exposure time would show the arrow standing still in every photograph, and therefore “you can’t see motion with a camera,” because it captures positions. He proposes the concept of an “ideal movie camera” that would present velocity without place, and explains that it is hard to imagine because “our mind works like a camera and not like a movie camera”; that is, human perception grasps states and infers process only from their change. He parallels this to quantum theory, where the position picture and the velocity picture are separate languages, and concludes that the arrow paradox arises from conflating “the object is at a place” with “the object is at rest in a place.”
Implications: repentance, perfection and perfecting, and process as having intrinsic value
Rabbi Abraham raises the question “who is greater, a penitent or a completely righteous person,” and explains that the advantage of a penitent comes from the fact that repentance is a process and not only a final state, and therefore its value is not exhausted by arriving at the state of a completely righteous person. He defines “that the same sin comes to his hand and he does not repeat it” as an operational definition for measurement and not an essential definition, and adds that there may be a “penitent” whose process of progress does not show up as visible improvement in his state because of circumstances, “like a wall that stops him.” He brings Rabbi Kook in Orot HaKodesh on “the problem of perfection and perfecting” and suggests that the question arises from confusing process with change of states, so that perfecting is not necessarily “becoming better” but rather a quality of potential change, while change of states does not apply to the Holy One, blessed be He.
Collectives, fluidity, and the judgment of the community in Maimonides
Rabbi Abraham raises the claim that enterprises are better off being in processes of change even apart from the result, and infers from this that a process is not merely an abstraction if it has properties of its own. He compares this to the debate over the existence of collectives and argues that a collective is not a fiction, because there are properties that characterize the aggregate and not the individuals, such as the fluidity of water, which is not a property of a single molecule. He cites Maimonides on judgment on Rosh Hashanah and during the Ten Days of Repentance, where the Holy One, blessed be He, judges the individual, the city, the state, and the whole world, and explains that after the judgment of individuals there is still meaning in judging the city because “the collective is an entity that stands before the Holy One, blessed be He, for judgment.”
Returning to the giving of a bill of divorce and the Sages’ method of examples
Rabbi Abraham concludes that the difficulty in defining “the giving of a bill of divorce” stems from the fact that it is a process, while the human mind grasps states and therefore struggles to describe a process directly. He says that the Sages give “many, many examples” in order to build an “intuitive picture that cannot be formulated as rules” of giving, and that the Kehillot Yaakov’s work of negation and repeated revision is a kind of “theory of negative attributes” that teaches “what is not the giving of a bill of divorce.” He adds that just as in mathematics velocity is defined through place because there is no other direct route, so too here, but in the giving of a bill of divorce the endpoint states are not always the necessary result of a single process, and therefore there is no sharp rule that stitches all the states together.
A paradigm for Torah study and halakhic intuition
Rabbi Abraham argues that this is a paradigm for analytical Torah study, because many final formulations “do not grasp the bull by its horns,” and yeshiva-style conceptual distinctions produce poles whereas the truth is a complex combination of them. He mentions the Chazon Ish’s “fifth section of the Shulchan Arukh” and defines it as the halakhic intuition of decisors and Torah scholars who understand the process behind the states. He concludes by claiming that someone who knows sources and facts by heart can still be “a donkey carrying books,” and he closes with a prayer for the full recovery of Rabbi Shugerman.
Full Transcript
[Speaker A] Good evening. I welcome with blessing and joy Rabbi Michael Abraham. Just to let the public get to know him a little—I assume not everyone is familiar with the Rabbi—when we spoke, I told him that he is multidisciplinary. First of all, he currently teaches Torah in what is called the Advanced Institute at Bar-Ilan University, the Bar-Ilan kollel, and also serves as a rosh metivta in the hesder yeshiva in Yeruham. Beyond that, anyone familiar with the publications of Middah Shavah—what’s it called? Middah Tovah—which came out first as pamphlets and later as books on the thirteen hermeneutical principles, can appreciate the depth from there. And beyond that, he is also a philosopher. Again, anyone who knows Two Carts and a Hot Air Balloon, which is the first book in the quartet—it hasn’t all come out yet, right? How many have come out? Two have come out. Yes, a series of four books is supposed to come out, and for anyone who has read it and knows it, it is a very original blend of science, Torah, and philosophy. You can certainly appreciate this special creation, which is really unfamiliar in our circles, and it demands that you keep your head in the game and understand deeply what is being said there. And some of those ideas we’ll hear today. That’s one background point. And the Rabbi also has a doctorate, I think in physics—physics. So as I said, there really is a multidisciplinary figure here. And I told him that today we are dealing with Torah and science, so that’s one option, and there are other options too—we kept our options open. In any case, what I would still like to add is that at this hour, Rabbi Shugerman is in the United States undergoing difficult medical treatment, and we have prayed and are praying, with God’s help, that it turn out well and that he have a full recovery. And we dedicate this lecture, just like the previous ones, to his merit, to his recovery, that with God’s help he return speedily healthy and whole. So please.
[Rabbi Michael Abraham] Good evening. Actually, the topic I want to talk about today came up for me when I once studied a passage in tractate Gittin—or really several passages in Gittin—that dealt with the attempt to define what it means to give a bill of divorce. As is well known, the Torah says, “And he wrote for her a scroll of severance and placed it in her hand.” In order to sever the marital bond, one has to write a bill of divorce and deliver it into the woman’s hand; an act of giving is required for the divorce to take effect. When the Talmud tries to define this concept of giving—what exactly does the Torah mean when it says “and placed it in her hand”—something happens there that is actually rather awkward when you look back after going through the passages. The Talmud brings dozens of examples in several places, throughout tractate Gittin and a bit elsewhere too, lots of examples trying to characterize exactly how this business of delivering the bill of divorce, of giving the bill of divorce, has to be done. And somehow the feeling is that instead of multiplying and burdening us with so many examples, the Talmud ought to tell us the definition—just define what has to happen for this to count. Why bring examples, all kinds of terribly complicated examples: he wrote the bill of divorce on a slave and gave her the slave; he put the bill of divorce in his courtyard and transferred ownership of the courtyard to her; or alternatively he threw the bill of divorce into her courtyard; he wrote it on something from which benefit is prohibited; the bill of divorce in his hand and a string in her hand, and vice versa, the opposite way around; the bill of divorce in her hand and the string in his hand—various such examples that seem pretty crazy. I mean, it’s very hard to understand whether anyone really thought such a case would ever actually happen. But that in itself doesn’t bother us; there are even more imaginary things in the Talmud—wheat descending in clouds and other such discussions. But still, somehow the feeling is that this looks a bit… a bit cumbersome as a way of defining a concept for which they could just offer us the simple definition. And indeed, if you look at the later authorities on these passages—for example Kehillot Yaakov—I think he has two sections on this, maybe three, I don’t remember, long and detailed, where he tries to grind through all these examples and extract from them some definition that could serve us practically. If something happens in some form different from all the cases described in the Talmud, how will we decide whether there was a valid giving here or not? So you have to distill from the cases in the Talmud some more general principle. In the end, the conclusion of the author of Kehillot Yaakov there—and I think also of other later authorities who deal with this topic—is that he arrives at some definition whose number of stages is roughly the same as the number of cases in the Talmud. In other words, it seems that in the end he didn’t really succeed. He tries one definition—let’s say this is the definition—and then case number seventeen doesn’t fit, so we revise it a bit, and then case number three wakes up, and then we revise it a bit more. In the end, the number of revisions is like the number of examples, so it’s not clear how much we’ve really progressed with this rule, with this theoretical principle we tried to distill from the cases. And then I asked myself why this happens, and why specifically here. It happens elsewhere too, but here I think this is a very striking example of this cumbersomeness that in the end apparently we just can’t get around. Even when you try to find some sharp theoretical definition, you don’t really succeed. You arrive at something very cumbersome, and that’s probably why the Talmud chose specifically to show it by means of examples rather than to propose the theoretical definition, because a theoretical definition is probably not really possible all the way through. Why does this happen? So in order to try to understand that, I’ll give a kind of introduction. I’m basically following, more or less, the associations I had then, though of course I hope in a somewhat more orderly way, and afterward I’ll come back to this puzzle. Basically the first clue to the matter is that the concept of giving is a concept that describes an action or a process, not a state. In other words, you can talk about states and you can talk about actions. Giving a bill of divorce is an action. The fact that the bill of divorce is in the woman’s hand is not enough for her to be divorced. It’s not enough just that it is found in her hand. Something has to move us from the first state to the second state. That action is what contains the divorce, that is what causes the divorce. So really what we are trying to characterize here is not a state but a process, some action. And that already seems to me to be the first hint of the difficulty, because characterizing states is relatively easy—not always, but relatively easy. But characterizing actions is much more complicated. Of course, very simple actions I can define—physical giving, taking something and giving it to someone else. In that sense, if that were the definition, it would be simple. But you can see from the Talmud that this is not the definition. That is one possible way of doing it, but there are many other possibilities too. And because of that, we don’t really have a way to get directly at the action in order to define it, and so we try to describe the action or extract some insight about it through pairs of states—a state before and a state after. We describe many pairs of states, and the Sages tried to convey to the learner some kind of feeling from all these states so that he would try to develop an intuition for what exactly this action called giving really is. We have no way to get directly to the action; we have to characterize it through its endpoints, the state before and the state after. The problem is that characterizing it through states is very problematic. For example, Ketzot discusses this, and many other later authorities do as well. Some try to define that perhaps what is required is giving—physical giving. That of course does not stand up to the examples in the Talmud. Someone who puts a bill of divorce in a certain courtyard and transfers ownership of the courtyard to the woman—that too effects divorce, even though there was no physical act of handing over the bill of divorce itself. Another possibility comes up: maybe it is a transfer of ownership. In other words, it was mine before, I perform an act that transfers the bill of divorce to the woman, and now the bill of divorce becomes the woman’s. But that too does not stand up to the facts. Ketzot brings examples of prohibited benefit and various cases showing that this too is not the basic definition of the concept of giving a bill of divorce. And then things begin to get tangled, because we are no longer able to draw two simple endpoint states that delimit this action and define it. Rather, it is something more abstract. And because it is something more abstract, we try to give—or the Sages try to give us—a kind of sense of smell. In other words: look, this is a good giving, this is not a good giving. This one is good, this one is good, this one is good, that one is not good. From that, to try to develop some intuition indirectly: how exactly are we supposed to decide which giving is valid and which is not. Well then, if indeed the problem is a problem of trying to define a process—and that’s exactly the problem, because it’s impossible, or very difficult, to define it except through endpoint states—then here I really want to broaden the discussion a bit and deal in general with states and processes.
[Speaker C] So what’s the Jewish law—if the bill of divorce is in the woman’s hand, then she is divorced?
[Rabbi Michael Abraham] Of course not. The question is how it got into her hand. In other words, if she takes it and now it is in her hand, she is not divorced. There has to be an act of giving. In other words, if holding…
[Speaker C] Does the fact that the bill of divorce is in the woman’s hand produce “the bill of divorce is in her hand” on its own?
[Rabbi Michael Abraham] If a woman takes the bill of divorce for herself, she is certainly not divorced. If the bill of divorce reaches her hand by way of the birds of the sky, she is not divorced. There has to be an act of giving. So I want to widen the discussion a bit in order to better understand this difficulty in defining actions, or the relation between the action and the states between which the action moves us—the state before and the state after. And afterward to return to a few implications. This introduction will be less Torah-oriented, at least in the conventional sense, and afterward I’ll come back to implications that, at least in part, touch on what seem to me important aspects both in study, in thought, and maybe also in Jewish law regarding the giving of a bill of divorce. Good. The next stage on this track, on this intuitive or associative track, suddenly brought back to me the memory of a paradox raised by Zeno. Zeno, a Greek philosopher, tried to undermine the concept of motion, and in order to do that he raised several paradoxes that show that the concept of motion is inconsistent, meaning that you can’t really maintain it. There are several better-known paradoxes—Achilles and the tortoise, everyone knows that—but that’s not the paradox I mean. There are other things that may be a little more paradoxical. One of the paradoxes that survived for quite a long time, and in my opinion more than people think, is the paradox of the arrow in flight. Zeno’s arrow paradox works like this. You look at an arrow that is flying. If we look at the arrow during its flight, we see it in every moment, basically, located at a different point. So Zeno asks: at every moment the arrow is really standing at a different point, so when is it moving? In other words, when does it pass from point to point? At every moment we look, the arrow is standing—it’s just standing at a different point. Is there some moment at which we see it changing its place, moving from one point to the next point? That’s in a more popular formulation; there are slightly more precise formulations, but that doesn’t matter for our purposes. Or another formulation: such an arrow has to move and stand still at the same indivisible instant of time. Why? Because in an indivisible instant of time, you can’t move. Motion requires some span of time; you can’t move in a single instant. On the other hand, if in no single instant of time there is any motion, then how does motion occur at all? In the end, time is only the collection of these single instants. So how does it occur in the end? Rather, it must be that at every moment some motion takes place, right? Otherwise it couldn’t accumulate over an interval either. So it comes out that the arrow is both moving and not moving simultaneously in the same indivisible instant of time. Another formulation of what is, I think, not exactly equivalent, but similar. Now, the modern intuition after infinitesimal calculus says—and I think this held sway for quite a while, too long in my view—that there is a very simple solution: the view of the time axis as if it were composed of discrete points, individual points, very very densely packed next to each other—that is simply the wrong way to look at it. Infinitesimal calculus tells us that the continuous axis is really composed of segments as small as you like, what are called infinitesimals there, and not of isolated individual points. And therefore, you can’t really speak about the state of the arrow at a point, at a discrete instant of time, because there is no such thing as a discrete instant of time. This reminds me of a question I once saw—I thought it was from the Rabbi of Brisk, but not long ago I realized it was probably from one of his students, the author of Ohel Yehoshua—is that his name? Emek Yehoshua, something like that. He asks a question about “and at midnight,” right? At midnight the Holy One, blessed be He, killed the firstborn, and one can’t calibrate midnight exactly, and there are contradictions between verses and Talmudic passages on this matter. He says he doesn’t understand at all how it can be that the death of the firstborn took place exactly at midnight, because there is no such instant as midnight. Midnight is the instant—but it isn’t an instant at all. In other words, there is half a night until midnight, and the second half starts from midnight, but there is no such thing as midnight. Midnight is only the seam between the first part and the second part. So how can something happen in something that has no existence? In a time that has no existence? There is no such thing as an instant. And there this is not merely hairsplitting, because there they speak about around midnight or exactly midnight, so this issue really has a place there. The interesting answer he gives is: correct, but death itself is exactly the same way. When the firstborn died, they were alive, and at some stage they became dead. Right? There is no instant of death. Death is simply the seam that passes between the stage at which they were alive and the stage at which they were dead. So just stick one onto the other. They were alive until midnight, and they were dead from midnight. But there is no such instant as midnight, and there is also no such instant in which death occurs. Again, a kind of conception that says an instant is not really something that exists, though of course not for mathematical reasons. And I think this is also incorrect, but… Fine. So what bothers me about this explanation, which hangs the whole thing on infinitesimal calculus, is two things. First: there is such a thing as an instant of time. I see no obstacle at all to thinking that there is an instant of time. True, how exactly one builds a continuous axis out of such instants is a more complicated question. It is not simple just to compress points and put them next to one another; you need to add a few more elements to the story. But in the end, it seems to me that the simple intuition is that there is no reason in the world to assume that there is no such thing as an instant of time. There is. It may be very hard to grasp, impossible to compress, but I see no reason at all to say there is no such thing as an instant of time. More than that: even the assumption that says that if in infinitesimal calculus we use units that are tiny segments—that is only the convenient mathematical mode of description. But that does not mean that in reality, in the actual structure of the world, the time axis as it really exists is not composed of points. There is no reason in the world to assume that. Besides, I think there is a much simpler solution anyway, so there’s no reason to say that. So the next stage in this line of thought: let’s leave the arrow in flight for a moment. We have already left two things. There is the problem of the bill of divorce, and there is the problem of the arrow in flight. Now stage three—I said in the title: quantum theory. In quantum theory there is a principle called the uncertainty principle, and that principle basically says that you cannot speak of or measure—or there simply is not for the particle—a position and a velocity at the same instant. In other words, if it has a position then it doesn’t have a velocity; if it has a velocity then it doesn’t have a position. This is one of the marvelous features of quantum theory—marvelous in both senses. If you specify the position exactly, then the velocity is not known. It has a velocity—
[Speaker D] But it isn’t known.
[Rabbi Michael Abraham] That’s one formulation. Bell’s inequality. And it’s not completely simple that this is so, but I don’t want to get into those subtleties. In any case, ostensibly, when you look at this statement without even getting into its roots and why it is true and so on, the first conclusion you could draw from it is that the very concept of the speed of a particle, or an arrow, or a body at a given moment in time is simply an empty concept. It’s empty because if I say that at a given moment in time, or in a given place, then that means it has position and speed simultaneously, right? Basically, what was Zeno’s paradox? He’s really telling us: after all, at every moment the arrow is in a different place; at every instant of time it is in a different place, and then of course the question arises: when does it move, or how does it move, and so on. So first of all, it could be that the initial conclusion that comes out of the uncertainty principle is that this statement, the speed of an arrow at a given place, at a given point in space or time, is really an expression with no content, no meaning. Because if there is position, there is no speed; if there is speed, there is no position. But that sounds more like wordplay, and not only because I never liked an English-English dictionary. An English-English dictionary is always an explanation of one word you don’t understand by means of ten others that you understand even less. We like an English-Hebrew dictionary. An English-Hebrew dictionary means that through the explanation we try to understand more, not just point to something else we also don’t understand and thereby solve everything. Meaning, once we understand quantum theory, maybe from within quantum theory we’ll be able to explain this. But to explain one paradox by pointing out that there’s another paradox too doesn’t help us all that much. So it seems to me that this plane אולי not be very productive. What we might do instead is actually reverse the direction of thought, meaning to go from the arrow to quantum theory, not from quantum theory to the arrow. What do I mean? We’ll try to understand why, in fact, there is no problem with the arrow, that Zeno’s problem doesn’t exist. Maybe that will shed some light on quantum theory, because using quantum theory, as I said before, to explain the arrow is an English-English dictionary. Fine. Blessed are You, Lord our God, King of the universe, by whose word all came to be.
So for that we need to understand a bit the relation between processes and states. Now what I’m trying to do is understand, or solve, or decipher Zeno’s paradox. Roll the carpet backward. Meaning, after we understand this, we’ll return to the uncertainty principle; after we understand this, we’ll go back again to the questions about giving a get, and maybe to a few more implications. So when we speak about some dynamic process, we’re speaking, for example, about speed. Let’s take the concept of speed as an example. Speed is basically a change of place along the axis of time. As I mentioned earlier when I described the paradox, the initial feeling is that the concept of speed simply cannot exist at a discrete instant of time. Speed needs some interval of time, because at a single point, how could it be that a body has speed at one discrete moment of time? But that is apparently not true. In physics, or in mechanics, when speed is defined, it is defined as the ratio of distance to time. Meaning, say I traveled one hundred kilometers and it took me an hour, so the speed is one hundred divided by one, one hundred kilometers per hour. In other words, speed is defined as the ratio of the distance I traveled divided by the time it took me to travel it. From that definition there really does arise a certain feeling that at a discrete moment in time, speed cannot be defined, because speed has to show itself in some change of place, and then I divide that by how much time it took, and the result is the speed over the interval I looked at. But speed at a single moment is maybe something that doesn’t exist at all. But the truth is that this is not correct. It’s not correct because we need to distinguish here between two meanings of the concept of definition. When I spoke earlier about the definition of speed and said that it is the ratio between distance, distance divided by the time it took to traverse that distance, what I actually gave there was an operational definition. Meaning: how you calculate speed. But when the question is asked, what is speed, that doesn’t necessarily overlap with the question how do you calculate speed. In other words, what is speed is a completely different question.
For example, if there is some body that is not moving at a constant speed but at some speed that changes over time, then it’s very hard to define—really it’s impossible to define—its speed in terms of the distance it traveled divided by the time it took. So what do they do there? There they usually define in mechanics instantaneous speed. What is instantaneous speed? Maybe I’ll use this sharpener tool. Instantaneous speed: say there is some body moving like this—this describes its position as a function of time. All right? If the speed is constant, then of course the graph looks like a straight line. If the line isn’t straight, then the speed isn’t constant. Why? Because in equal intervals of time it doesn’t traverse equal distances, right? So the speed changes. What do they do in mechanics? They define instantaneous speed. Say, what is the speed here? At this point, which is different from the speed here and different from the speed here. At every point there is a different speed. But how do you calculate such a speed? After all, speed at a point isn’t something we know how to calculate. So for that they invented infinitesimal calculus and they tell us: take some interval around this point, see how long it is in terms of the time axis, how long this process takes, and what distance we traveled from the first point to the last point—that is, what distance it covered. Divide the distance it covered by the time it took, and that is the… and take a smaller and smaller interval around the point you’re interested in. The result that it eventually approaches, that is basically the speed at that moment in time. That’s what is called a derivative in mathematics. Never mind, it’s just to understand the idea.
Now this mode of calculation still tells us that, true, in the end we are speaking about speed at a single moment in time, but that’s really a fiction. We are really speaking about speed over a small interval of time around that point, not about the speed at that point in time. But that is true only with regard to the operational definition. This definition is a definition of how to calculate speed. But when I ask myself what speed is, that’s a completely different question. And the answer to it cannot be given—does not necessarily have to be given—in terms of distance divided by time. Distance divided by time is the way to calculate it. But the question is what it is. That is only the practical implication of it. But the question is what is it, what is it itself. Maybe you’re already starting a little to see the relation to the first question of giving a get versus edge cases. And then we can go back and say that there is no reason in the world to assume that speed exists only over an interval of time. Speed, as this sort of calculation gives us, exists at every point in time. And every point you place, every different point in time you place, will in principle give you a different result for the speed. Meaning, every moment in time has its own speed, which is a different speed. That strongly suggests that it probably characterizes the point and not the interval around it. Going to the surrounding interval that becomes smaller and smaller is only a way of calculating it. We have no other way to calculate it. But in fact the result that is obtained at the end stands on its own. And that is what is actually called speed. The method of calculation is only how to arrive at that quantity. But in the end speed itself—the definition of what speed is—is not the operational way of calculating it. So what is it? Speed is simply the potential—maybe before that I’ll define—there is a difference between speed and change of place. Ostensibly, at first glance they look equivalent. A body that has speed is a body that changes its place over time, right? A body that has no speed, meaning a body that is standing still, does not change its place over time. What I said earlier, when I distinguished between an operational definition and an essential definition, what I’m really saying now is that we have to distinguish between the concept of speed and the concept of change of place. They are not the same thing. What is the relation between these two concepts? Very often—and in a moment we’ll see not always—they appear together. But that doesn’t mean they’re the same thing. It only means they appear together. What is the relation between them? The relation between them is probably, or at least I propose, that speed is the potential for change of place. Meaning, if a body has speed at a certain moment, that means that… in the next moment it will be in a different place. Meaning, speed is not the change of place itself. Speed can also exist at a single moment in time. The implication of speed—how do we see that a body has speed? We have to observe it over an interval of time to see that it really moves, that it changes place. Then we know, we have an indication that the body has speed. But it had speed at every moment. We can’t see that with our eyes; we need to watch it over a certain stretch of time and from that infer that it had speed. Meaning, we need the change of place as an indication that the body has speed. By the way, there are indications of speed that don’t require an interval of time. The Doppler effect, for those who know it, measures speed without needing an interval of time.
So what are we really saying here? Maybe I’ll bring two examples to sharpen this further. Say a certain body has speed and I throw it at the wall. What happens the moment it touches the wall? The moment it touches the wall it arrives with some speed, which can be quite high, but the wall doesn’t let it move forward. Right? For physicists—any physicist here, his hair is probably standing on end when he hears me talking right now—but I think these are still things that can be said. When the body reaches the wall, basically it arrives with some speed, perhaps very high, and the wall does not let it bring that from potentiality into actuality. Meaning, it has speed, but the change of place will not appear. All right? I’m trying to illustrate here why I claim that speed and change of place are not the same thing. Change of place is a result of the fact that the body has speed, but it is a result that does not necessarily always appear. For example, in this case the body has speed, and the wall does not let it actualize that speed—that is, it does not let it change place—and then of course its speed drops to zero, as every physicist will say. What do you mean? It has no speed; its speed will be zero. It will be zero after the wall does not let it go forward. Or another example: if a body, a ball, collides with another ball. All right? Then clearly it transfers to the other ball what is called momentum. It transfers to it some capacity to change place that it did not have before. But it pays for that. It itself will now change place less. Meaning, it had a high speed, but the result of that speed is not expressed only in change of place. Part of it went toward handing over that potential to the body it collided with. All right? These are examples only to illustrate the point I’m trying to make here: that speed is basically the potential for change of place. It is not change of place itself. Usually when a body has speed we will also observe that it changes place. But there are cases like the ones I just described—cases where there is speed but there will be no change of place. Sometimes it can come out in the form of heat; it doesn’t matter, in all kinds of other forms, but not in the form of change of place. Because then the speed is—
[Speaker C] The definition of force.
[Rabbi Michael Abraham] What? In physics, no, that is not the definition of force; that’s what’s called momentum. Momentum is basically the speed. Speed is a different definition. No, but I think that in the ordinary definition of speed, even in the operational definition of speed, it seems to me that this is the result. Meaning, when the body got here, when it got here it had a very high speed. That’s clear. And it stopped, right? No, so there was a certain moment—the moment at which the speed stopped—there was a certain moment at which it had a very high speed and did not manage to realize it through change of place. All right? After that, of course, it immediately dropped to zero, that’s true. But I’m talking about that very point when it happened. Why am I saying this, really? Let’s go back, let’s return for a moment now to Zeno’s paradox. I’m beginning to roll the carpet backward. The claim I really want to make is that with Zeno’s arrow—the question was: if at every moment it is standing in a different place, then when does it move? And the answer is: at the very moment at which it stands, it also moves. Let’s ignore quantum theory for the moment; we’ll come back to it in a second. What does that mean? This naive intuition that says that a moving body, at every moment, is standing in a different place, and then I look for the moment at which it passes from point to point—this is an intuition that says a body cannot both move and stand still at the same moment. So if it is standing at this moment, apparently it moves at another moment, and I can’t find when that other moment is, because at every other moment too it is again standing still, just in a different place. But that is not true. This body has speed even at the point where it stands—that is, not stands, is located. When a body moves, let’s put it this way, the arrow goes like this, and say now I catch it at this place, in another second it will be a bit farther. A body that is standing still is a body with zero speed. A body that is in a certain place can be in that place with a speed that is not zero. It is there, and it has speed. So when I ask the question when does it move, that is an incorrect understanding of the concept of motion or the concept of speed. It moves at exactly the same moment that it also stands still. Meaning, it has speed at the very moment at which it stands still. It also has speed. Speed is a quantity that exists at a moment in time, not over an interval. The potential for change of place.
[Speaker C] Fine, so it has speed.
[Rabbi Michael Abraham] When does it change the motion? When, when does it change the motion? All the time it changes the motion. It doesn’t change the motion—you said the place. Sorry, yes. Change happens in that same place? It’s not in the same place; it’s in this place, and a moment later it’s in this place, and a moment later it’s in this place. To say—look, to say when does the body change place is exactly what is confusing here. Because to say when does the body change place is really meaningless. What do you mean, when does it change place? Change of place requires reference to two places. When does it change place from A to B? You understand that the answer cannot be one single moment. It cannot at the same moment be both at A and at B. But when I ask when does it have speed, there is no problem at all: both at A and at B it has speed, albeit different speeds.
[Speaker C] Speed gives that meaning, but you said—you defined change of place as—
[Rabbi Michael Abraham] Not speed.
[Speaker C] Right. So the question is: does change of place happen at some point?
[Rabbi Michael Abraham] Certainly. No, it does happen, yes, over an interval. The claim—the claim that a change of place cannot occur over a stretch of time if it does not happen at an instant—that claim is actually the wrong one; it is Zeno’s mistaken assumption. Why? Because what happens at an instant—there is something happening even at an instant. Only that something is not a change of place, but the existence of velocity. And that exists even at a single instant. Meaning, the body has velocity, some potential for a change of place. That velocity characterizes it at every single moment of time. And the fact that it has velocity is expressed in this: over a stretch of time it will change place. To ask at which instant of time it changes place—that is simply a meaningless question; it is not a paradox. It is like asking whether virtue is triangular or square. What is the answer to that? There is no answer; the question is meaningless. What does that mean? These are not paradoxes, not things that have meaning and I just do not know which is true. Here I am simply using a meaningless concept. A meaningless concept of course will have no sense at all. A body has velocity at a point in time, but a body cannot change place at a point in time. That certainly it cannot do, and there is no problem with that either—it does not need to do it. It changes place over an interval of time. What happens at a point in time is that it has the potential for a change—for a change of place. And that is what is called velocity. If I continue maybe one more step further, let us go back to that arrow and try actually to talk about the camera I mentioned earlier. I am photographing the arrow now along its flight. I photograph it at a certain instant, I see it here. I photograph it at the next instant, I see it here. I can of course photograph very, very densely, and then perhaps if I run the pictures quickly I will see the illusion as though there is in fact a process here, a continuous process. That is how cameras usually work. A movie camera is that kind of camera, one that samples this at high speed and transmits it, and we basically see some sort of motion here. At one instant, in one frame, at a given moment—take an ideal camera, a camera whose exposure time is a discrete point in time, if such a thing is possible. All right? That camera will of course show the arrow completely standing still. Meaning, it will photograph the arrow at every moment you want and show it standing in a different place. Does that mean the arrow is not moving at the moment it photographed it? No. So how—where, then? So when do you see the motion? The answer is that motion cannot be seen with a camera. It is simply not the right instrument for seeing it. A camera captures positions. A camera can capture the position of the arrow at every point in time. A camera cannot capture the motion. But that does not mean there is no motion. It does not mean that at that moment the body has no velocity. It does; one simply has to put on the right glasses. With glasses shaped like a camera, we succeed in seeing only positions. Now when Zeno claims that at every single instant the arrow stands in a different place, he is basically assuming a camera-like perspective. He says: let us photograph the arrow at every instant with some ideal, very, very fast camera, and we will see that at each instant the arrow stands in a different place. So when does it move? I do not see it moving. Of course you do not see it—put on different glasses, and with those glasses you will see it moving at every instant while you also see it being in a different place, and not standing still but being in a different place. Let us call those other glasses, say, a movie camera. But not a movie camera like the ones we know today. A movie camera as we know it today is a transmission of static pictures, only at very high speed. Then we see some illusion of motion. That is the movie camera we know today. But let us speak about an abstract, ideal device. I do not know how to build it, but we can think about it—and that is an ideal movie camera. This is a movie camera whose exposure time is also zero, and when it films the body at a certain point, it sees only its velocity; it does not see its place. A different device from a camera. All right? About that, everything we said about a camera would be true, only with the roles of place and velocity reversed. Meaning, we would see only the velocity; we would have no idea what the place is. We would only see the motion within it; it would not matter where exactly it is. Of course this is very abstract, because I do not know how to build such a thing, but it seems to me one can definitely define this sort of abstract notion. And why is it really hard for us to understand—and here we enter the root of the matter—why is it really hard for me to describe to you what this is, not only to you but also to myself, not only to you, what this ideal movie camera is? Because our mind works like a camera and not like a movie camera. Our thinking, our perception, works like a camera. Meaning, we are used to grasping states. We see the situation as it is before us. Our eyes are a kind of camera, a camera that takes pictures at every moment. Fine, but what we see are states and not processes. How do we understand that there is a process? If the state changed—meaning, if at one moment I saw one state and at the next moment I see another state—I infer from that that there is some process here. I never see the process. I have no instrument with which I know how to see the process. We do not have such an instrument. Our consciousness, our awareness, basically works in the form of a camera. Therefore I also do not know how to build, or even to describe, that abstract device called an ideal movie camera, because it is a device that does not work with our kind of mind. It would have to be some other creature that looks this way. Now, this is just an anecdote—I say this parenthetically—if I go back to quantum theory by way of rolling up the rug, then in light of this distinction one can also understand a bit of the uncertainty principle in quantum theory, which seems very mystical. In quantum theory, the uncertainty principle describes these properties of the particle—position and velocity—as expressions of two descriptive languages, two different languages of reference. The language of position and the language of momentum or velocity. The world, the position picture or the velocity picture. There is a picture in which we speak in terms of positions. In that picture, velocity does not exist at all, is not defined, does not take part in the game. We can calculate it if we calculate the position at one time and the position at another time afterwards, take the difference in positions, divide by the difference in times, and discover what the velocity is. We have no way to speak directly about velocity in that language. There is another language, which does not speak with this one at all—there is not even a way to translate. There are some mathematical notions, but there is no way to translate between these two pictures. And the second picture is the momentum picture or the velocity picture. In this picture one speaks only in terms of velocities. And in this picture positions do not participate at all; there is no way to define them at all. Only indirectly. If I know an initial position and I measure the velocity, I can calculate what the position will be later. But that is a calculation. I cannot observe it, I cannot speak about it, I cannot define it in this language of the velocity-world, the velocity picture. These two things, which are basically the more mathematical expression of the uncertainty principle, are not, it seems to me—or at the principled level, at least—not much more than the expressions of the movie camera and the camera. The camera I spoke about earlier is the prism, or those are the glasses, through which we look and receive a picture in terms of place. In terms of state. Place in relation to velocity, but any state as against a process can be spoken of in the same way. We receive the picture in terms of states, states that are each time a different state. In the picture of velocity, that is basically what I earlier called the ideal movie camera, and in the ideal movie camera we receive only—we look only at processes without the states between which those processes carry us. And these two pictures do not speak to one another. It is no wonder this gets complicated, and it seems to me this is also, to some extent, the basis of the uncertainty principle in quantum theory. It removes at least some of the fog, not all of it. All right, so basically what we are saying here is that the arrow paradox is solved not in connection with differential calculus or infinitesimal calculus, but simply because there is a confusion of concepts here. And it has nothing at all to do with differential calculus. So—a confusion of concepts. One must not confuse the statement “the object is in a certain place” with the statement “the object stands” or “rests” in a certain place. Those are two completely different things. It can be in a place even when it has velocity. When it is at rest, that means it is there and its velocity is zero. Those are two different things. When I say that the body is, I always have to say it is in some place, right? When I say that the body is at rest, I can stop there—I do not need to say in what place. “Is” has no meaning if you do not tell me the place. So it is not the same—it is not the same concept. All right, so now let us really move one step further and ask ourselves what this says about the question—or perhaps before what this says about the question of the bill of divorce, I will bring a few implications that arise from this distinction between states and processes. What I am basically claiming is the following; I will just summarize. The basic process that I described is a process we call velocity. Velocity is a kind of process. What does that mean? The state is the position, and the state changing means that behind this there stands a process. A process is a change of states. All right? A process takes me from state A to state B, and perhaps to many other states as well. But there are always, let us say, in a segment of a process, endpoint states: a state at the beginning and a state at the end, and the process takes me from the state at the beginning to the state at the end. A process does not have to be only velocity. In velocity, the state is position and the process is velocity. But there are other processes too. Processes of someone repenting—then his spiritual state is supposed to improve, right? That too is a kind of process that can be described through a change of states. He has a certain spiritual state, he repented, now he has a higher spiritual state. And there too we can ask ourselves: what is repentance? Is repentance improving our spiritual state, or is repentance performing the process? What do you say? The Talmud talks about this, does it not? Who is greater, a penitent or a complete righteous person? There is a dispute in the Talmud, but somehow it is accepted that the one who is greater is the penitent. Now tell me, how can it be that a penitent is greater than a complete righteous person? At most, if he repented perfectly, he reached the perfect state of the complete righteous person. How can one say that a penitent is greater than a complete righteous person? What does the penitent have beyond the fact that he was simply worse for part of the time and in the end arrived at being a complete righteous person? So in the end perhaps his state becomes equal—but how can it be greater? Right, so the claim is that being a penitent does not mean moving to a higher state; that is an indication that you are a penitent. Being a penitent means that you are in a process of advancement. You are in a process of advancement. How do you see that? Exactly as with velocity: by seeing—or you yourself see, it does not matter whether externally or internally—you see that you are constantly rising to a better spiritual state. So that means that you are a penitent. So here we have, for example, an instance: if we say that a penitent is greater than a complete righteous person, then what are we actually saying in this language? What we are saying in this language is that the essence of repentance is the process, and it is not just a means to reach a better state. If it were only a means to reach a better spiritual state, then a penitent would not be greater than a complete righteous person. Rather, clearly there is value in the very fact that you are in a process, and not only in the fact that as a result of the process you are also in a better spiritual state. One second—who spoke there? Nice comment. A penitent is one to whom the same sin comes, the same situation, and he does not repeat it. That is the indication that he is a penitent. That is the process… No, that is the way to measure that he was a penitent. But that is not what a penitent is; that is an operational definition. It is a definition of how you know he is a penitent. Put him into the same situation and see that he is in a better state. So there—you have seen that he advanced. But that is only the way you know he is a penitent; but who is a penitent? That is not the correct answer to that question. That is an entirely different question. You are asking for an essential definition, not an operational one. A penitent is someone who has advanced. And in fact it is not always possible to measure that. As we also saw regarding velocity and change of place, velocity does not always actualize itself as a change of place. For example, there can be a penitent who is in more difficult circumstances and he did not advance, but he also did not deteriorate. So you would not measure it—it is like a wall that stops him, yes, in the terms of velocity. You could not measure it, but he is still a penitent; he advanced in the processual sense. He is in a process of repentance, even though it did not actualize itself as a better spiritual state. There is perhaps another example: Rabbi Kook in Orot HaKodesh speaks about a philosophical problem called the problem of perfection and self-perfecting. Meaning, the Holy One, blessed be He, is supposed to possess all human perfections. Rabbi Kook says: one of the human perfections is to perfect oneself. Meaning, the very fact that I improve myself—that is one of the human perfections. How can that perfection exist in the Holy One, blessed be He? Can the Holy One, blessed be He, improve Himself? He cannot, simply speaking, right? Why can He not? Because first of all He is perfect; He cannot be better than He is, right? And in general change does not apply to Him either. Whether that is the same thing or not, it seems to me, is equivalent to the question we are asking here, but let us say these are at least two indications. So the Holy One, blessed be He, cannot perfect Himself. It follows, then, that actually He cannot be perfect in that way, because something is missing: one of the perfections is missing from Him, namely that He can perfect Himself. Meaning, that perfection is not found in Him, right? I am presenting this, of course, somewhat absurdly. But what he means is to ask the question: if the Holy One, blessed be He, cannot perfect Himself, then one of the human perfections is not found in the Holy One, blessed be He—how can that be? But according to the way we are explaining now—I am not one hundred percent sure this is his intention, because the language there, I am not sufficiently skilled in understanding his wording either, so I do not know whether this is his intention or not; take a look afterwards, I am not sure—I think what can be explained here is exactly what I said earlier: that this self-perfecting, this human advantage of being one who perfects himself, does not mean becoming better. That is an implication of the fact that you are self-perfecting. That implication we will not find in the Holy One, blessed be He. He cannot be better than He is; He is perfect. But the potential for change, that quality by virtue of which change is produced, certainly exists in the Holy One, blessed be He. If it were not in the Holy One, blessed be He, then it could not exist in us either. Meaning, that potential certainly exists in the Holy One, blessed be He. Of course one must not confuse the process with the change of states. Meaning, a change of states will not appear in Him. But this capacity—to repent, to perfect oneself—this capacity certainly has its source in the Holy One, blessed be He. So therefore this whole business is actually built on a mistake, this whole question, because once again there is a confusion between the change of states and the process, and it is not correct to identify the two. A question: meaning that if we see someone advancing, then the Holy One, blessed be He, does see? When you see his spiritual state with ordinary glasses, then look for the special glasses and ask whether he is advancing. We cannot even see him with the ordinary glasses, so certainly not with glasses whose appearance we do not even know. But basically that also means that even if we say the process does not actualize itself? No, it does not actualize itself; there is no change. By the way, it seems to me that Rabbi Kook writes there—part of the formulations there—that the self-perfecting of the Holy One, blessed be He, is done through us. Meaning, the fact that we become more perfect—that is the self-perfecting of the Holy One, blessed be He. Meaning, it is like a body, say, that collides with another body: it itself stops, it has velocity—how does that actualize itself? This body no longer changes its position, it stops, but it passes on to another body the possibility of changing position, yes? In the analogy we made earlier. So it seems to me that this definitely also fits with what we were discussing. All right, there are several more examples, even in non-Torah contexts. For example, we say—I once saw someone say—that it is very good for factories to be in processes of change. Again, irrespective of whether they become more efficient for the purposes for which they operate. The very fact that you are in a state of change is itself good for a factory; it airs things out, it creates some ventilation in what is happening there. Now this claim basically says once again that the process itself is important as a process, regardless of what the result of the process will be. The very fact that a process is taking place is the important thing. Now one must understand that if the process were only some abstraction of ours—because many times people think that in fact velocity is our abstraction, not something that really exists. We define a body that changes positions and call that thing “the body has velocity.” But if that were really so, it could not be that there are properties that are properties of velocity and not of positions. All the properties would have to be properties of the positions. Velocity would only be something standing behind the things. But if I have properties that are properties of velocity itself, and not of the change of place that follows from it, then that means velocity too is something that really exists. All right? That is obvious. Similar problems often arise regarding the existence of collectives. Yes? Today too there is a very common position that collectives are basically a fiction. What exists are the individuals, and the nation or a community or something like that—those are definitions for convenience, for day-to-day functioning, but they do not really exist, they are not some kind of entity that truly exists. But that too is not correct, for exactly the same reason: because there are certain properties that characterize the collective and do not characterize the individuals that compose it, and if so, it cannot be that there are properties of something that does not exist. Meaning, if it has a property, apparently it exists; otherwise, whose property is it? An example: water is ultimately just a collection of molecules, and water as such is liquid, but none of the molecules that make up the water is liquid. Liquidity is a property of the aggregate; it is not a property of the individual particle that composes it. So that means there is something of which liquidity is the property, and that is not the water molecules. So what is it? It is apparently some collective entity. Maimonides says that the Holy One, blessed be He, judges during the Ten Days of Repentance and on Rosh Hashanah. He judges the individual person, He judges the city, He judges the state, He judges the whole world. And each one—the intermediate, the righteous, and the wicked—is judged on its own. And the obvious question, of course, is: how can that be? After He judged all the individuals, what is left? What is there to judge in the city after He judged all the individuals? The answer is: that is not correct. The city too has to be judged. There is something beyond the behavior of each and every individual, and that too the Holy One, blessed be He, judges. The collective is an entity that stands before the Holy One, blessed be He, in judgment. All right, we are already approaching the end, and I just want to finish with a few remarks. So, returning to the giving of a bill of divorce—that was only the point of departure, that was the point of departure. The difficulty the Sages faced in describing to us what a valid giving of a bill of divorce is, as I said earlier, is probably the result of the fact that giving a bill of divorce is a process. And when we try to define a process, we have no way to do it, because our mind works like a camera and not like a movie camera. We know states. The bill of divorce is in her hand—that I know, I see it. The bill of divorce is in his hand—I see that too. If you tell me first in his hand and afterwards in her hand, excellent, I understand. But the change whose eventual implication is that the bill of divorce was here and passed there—the process itself I have no way of describing. Why? Because my mind works like a camera and not like a movie camera. Even the intellectual mind, not only consciousness; our intellectual grasp also works like a camera. And therefore the only way really to do this is to give us many, many examples, each one of which is basically a definition through endpoint states. It tells us what was before, what was afterwards, what one sees with one’s eyes. And through these endpoint states we are supposed to try to create some concept that somehow stitches together all these examples, and that will basically be the definition—and it is not the definition because it cannot be formulated—but it will basically be the correct conception of giving a bill of divorce. Whoever studied these examples well, compared them, thought about them, analyzed them, understood the meaning of each one and the difference between one and another, forms within himself some intuitive picture, not formulable as rules, of what giving a bill of divorce really is. The whole process in Kehillot Yaakov that I described earlier—where he basically brought one more example and then rejected the previous definition, revised it, then another example and another revision, another example and another revision—that is basically the doctrine of negative attributes. Meaning, he teaches us what giving a bill of divorce is not. After we have negated everything, what remains is giving a bill of divorce. We do not know how to describe it in words. We have no clear rules for how to describe it. Just as in mathematics, when we talk about velocity, we define it through place: place minus place divided by time minus time. Because we have no way to define velocity except through the states. In a place where the endpoint states are necessary, where they are a necessary result of the process, there is no problem. That definition is fine; it is indirect, true, but it works. Because every time there is such a process, if I was in such a state, afterwards I will be in that state. But giving a bill of divorce is probably not such a thing. Giving a bill of divorce is a more abstract process, and many times it can connect many pairs of states, and in each of those pairs there stands behind it a process whose idea is the same idea. But there is apparently no simple way to describe it, so the Sages give it to us through many, many examples in order to give us some sense, some nose, to try nevertheless to understand how this thing stands, how it works. And perhaps one last sentence: it seems to me that this is a paradigm for Torah study in general. Very often our feeling in learning is that the final formulation we arrive at after studying the topic analytically does not seize the bull by the horns. Meaning, fine, I have ruled out many things, I already know that it does not work with this and that it is not this formulation and not that formulation. All the yeshiva-style conceptual investigations, for example: if an ox that causes damage is liable because it is his property doing damage, or because of negligence in guarding it—that is obviously neither this nor that. It is some combination of those two things together in proportions that are very hard to define. One has to begin with the two poles, but afterwards begin to connect them and see that actually the combination of both of them is what defines liability for damages caused by one’s property. So this illusion that says states capture the real thing is a somewhat dangerous illusion. Many times this is called “the fifth section of the Shulchan Arukh,” yes, the Chazon Ish called it that, or the halakhic intuition that halakhic decisors or Torah scholars have. And basically what that means is this: even someone who knows all the facts correctly and knows all the halakhic sources by heart can still, in principle, be a donkey carrying books. And it is not advisable to ask him Jewish law questions. There has to be something where you understand how this business really works, not through the described states. Even if you know all the states, you do not understand the process, you do not understand the ideas standing behind it. We know—I think almost everyone knows—examples of people who can know all of Mishnah Berurah by heart; I would not ask them Jewish law questions. So that is it. Truly, may there be a complete recovery for Rabbi Shugerman.