Study and Ruling – Lesson 37
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
- Planning the end of the series and the upcoming topics
- Giving a get as a model for casuistry and an attempt at a general theory
- Scientific abduction, simplicity of theory, and the failure of “patches” in Jewish law
- Definition by extension and definition by content as a key to understanding
- Why not settle for a list of cases: understanding, learning for its own sake, and ruling in new cases
- The failure of theory as a tool: Maimonides’ negative attributes and understanding through negation
- Intuition after analysis: da'at Torah as the product of a process and not of a formula
- Neural networks as a model for analytic Talmud study and learning from cases
- The network does not do abduction: success without a “general idea”
- Dynamic versus static: Zeno, speed, and the reason for the many cases in “and he gave”
- Text generators, AI, and halakhic ruling
- Closing the session and continuing the series
Summary
General Overview
The speaker plans to finish the series in two sessions and, in the current session, presents the tension between casuistry and an attempt at positivism in the study of Jewish law through the topic of giving a get, while comparing it to the scientific method of abduction. He argues that the later authorities’ attempts to distill a general definition of “and he gave” fail because the theory becomes as complicated as the list of cases itself, yet the work still builds an intuitive understanding “through negation,” similar to Maimonides’ negative attributes. He explains this process using the model of neural networks: the multitude of cases trains the inner system to recognize correctness even in new cases without being able to formulate a general principle, and he suggests a hypothesis that the conceptual difficulties in giving a get are connected to the fact that this is a dynamic concept that people are trying to capture in static language. Toward the end, there is a discussion about text generators and AI, about the possibility of reconstructing an author’s or halakhic decisor’s “mode of thought,” and about the feasibility of halakhic ruling through AI.
Planning the end of the series and the upcoming topics
The speaker says that there are two meetings left before going into the between-the-terms break on the Ninth of Av, and he wants to finish the series in them. He says that the current session will deal with ways of studying Jewish law through casuistry versus positivism, and that in the next session he will speak about conceptual reasoning in learning.
Giving a get as a model for casuistry and an attempt at a general theory
The speaker presents the verse “and he wrote her a bill of severance and gave it into her hand” as a verse that generates an unusual number of topics in tractate Gittin, and focuses the discussion on the word “and he gave” and the definition of valid giving. He describes how the Talmud builds the concept of giving through dozens of different cases of valid or invalid giving, such as a get tied with a string, giving through a courtyard, transferring ownership of a courtyard, and writing on things from which benefit is forbidden. He describes the move made by the later authorities, who try to extract from the examples a general definition of giving, and presents Ketzot and Kehillot Yaakov as trying to formulate general hypotheses and test them against the cases.
Scientific abduction, simplicity of theory, and the failure of “patches” in Jewish law
The speaker compares the attempt at conceptual definition in Talmudic analysis to scientific abduction, where one seeks a simple theory that explains a wide variety of facts, and brings as an example Newton’s law of gravitation, which unifies different phenomena under a single principle. He argues that when a theory requires many ad hoc corrections until its complexity resembles the number of cases it explains, there is no theoretical gain in it, and it resembles definition by extension alone. He describes how in Kehillot Yaakov every case “adds another hedge” until the theory becomes equivalent to the list of cases, and therefore, in scientific terms, he would throw such a theory in the trash.
Definition by extension and definition by content as a key to understanding
The speaker distinguishes between definition by extension, which lists cases, and definition by content, which formulates general characteristics, and illustrates this with the phrase “a democratic state.” He presents the ideal of analytic learning as moving from collecting facts to a content-based definition that explains the facts and makes it possible to understand the principle that generates the distinctions between the cases.
Why not settle for a list of cases: understanding, learning for its own sake, and ruling in new cases
The speaker answers the claim that one could settle for a list of cases with a check mark or an X next to each one, and offers two answers. He says there is value in learning as understanding and connecting to Torah, and not merely as storing information, and he cites Rabbi Israel Salanter’s interpretation of “expound and receive reward” regarding the wayward and rebellious son and the idolatrous city as a claim that learning itself is the goal and not only a means to practice. He adds that even for practical application it is impossible to list all cases, and therefore only understanding the principle makes it possible to rule in a new situation that does not appear in the Talmud.
The failure of theory as a tool: Maimonides’ negative attributes and understanding through negation
The speaker asks why the sages do not simply transmit a direct theory instead of cases, and answers that there is no simple theory that can be formulated in a way that stands up to all the facts. He brings an example of the exceptional rule in the first Mishnah of tractate Bava Kamma, “what is common to them is that they are your property and their safeguarding is upon you,” and shows how the rule breaks down in relation to a guardian, a robber, one who positions his fellow’s animal, and a pit in the public domain, after which one adds corrections like “considered as though it were his” or “the Torah placed them in his domain.” He suggests that the theoretical discussion is meant to produce understanding “through negation,” similar to Maimonides’ negative attributes: it is impossible to formulate what the thing is, but rejecting incorrect formulations eventually creates an unformulated understanding that is properly oriented.
Intuition after analysis: da'at Torah as the product of a process and not of a formula
The speaker argues that the analytic work happens along the way, in the formulation of theories and their collapse against cases, and not at the end in the form of a valid theory. He says that in the end he does not have a general formulation in hand, but there remains in him a calibrated intuition that allows him to decide, regarding a new case, whether this is valid giving or not, and he calls this “da'at Torah.” He says that sometimes one uses analogies to cases, but does not present a general thesis because there is no theory that explains everything.
Neural networks as a model for analytic Talmud study and learning from cases
The speaker argues that today this process can be understood with the help of neural networks, and presents a network as a collection of nodes and connections that change through training. He describes supervised learning, in which one gives examples classified as “correct / incorrect,” and the network adjusts the weights so that it will also correctly identify new examples it has not seen, with accuracy increasing as more examples are provided. He tells about an experiment in which a simple network predicted with about 70% success apparently random keystrokes of 0 and 1, and argues that this demonstrates how training on examples makes it possible to imitate decision patterns even without formulating principles.
The network does not do abduction: success without a “general idea”
The speaker emphasizes that a trained network does not contain “ideas” and cannot provide a theory that explains the examples, but only arranges internal parameters that produce correct answers with high probability. He compares this to classical programming, where the programmer has to put the abduction into the software through if-rules, and argues that neural networks bypass the need for abduction and therefore succeed in complicated problems that human conceptualization struggles with. He identifies in this a direct parallel to the study of giving a get: the multitude of cases “trains” the brain to give the answer the sages wanted even without the ability to define the concept in a general way.
Dynamic versus static: Zeno, speed, and the reason for the many cases in “and he gave”
The speaker offers a hypothesis as to why giving a get is “delicate” and requires a multitude of cases, and develops it through Zeno’s arrow paradox and the distinction between “being at a point” and “standing at a point.” He argues that the confusion comes from mixing up speed with change of place, and suggests that speed exists even at a point in time, although the way to calculate it requires a span of time. He argues that human thought tends to grasp dynamics through a sequence of static states, like frames in a film, and therefore has difficulty defining dynamic concepts directly. He suggests that giving a get is a dynamic act that people are trying to define through “before / after” states, and therefore the conceptualization fails and many examples are needed in order to create a direct understanding of the act, even though the formulation remains static.
Text generators, AI, and halakhic ruling
The speaker is asked about an experiment with a random text generator that produces passages in the style of Rabbi Kook, and he argues that if a reader extracts lofty ideas even from a text behind which there is no intention to convey a message, this undermines the correlation between what the author meant and what the reader derives from the text. He distinguishes between a random generator that imitates modes of expression and AI that imitates a mode of thought, and argues that if the AI truly reconstructed Rabbi Kook’s “brain,” it would be able to give answers even to problems about which Rabbi Kook never wrote. He connects this to projects of halakhic ruling through AI that is fed the Talmud and the halakhic decisors in order to infer Jewish law for new cases, and expresses the view that there is no principled reason not to rely on that, even arguing that AI is one hundred percent objective, while mentioning that he wrote a column about halakhic ruling by AI.
Closing the session and continuing the series
The speaker says that in the next session he will speak about conceptual reasoning and declares his wish to finish the series by the end of the term. He closes with farewell and the blessings “Good night” and “Shabbat shalom.”
Full Transcript
[Rabbi Michael Abraham] Let’s begin. We basically have two more meetings until the end of the term. What’s called, on the Ninth of Av we’ll go into the between-the-terms break. So I want to finish the series in these two meetings. So what I want to do today is talk a bit about methods, right, casuistry versus positivism in Jewish law, or in the study of Jewish law. And we’ll see — maybe I’ll have time for something else too — and next time I’ll talk a bit about conceptual reasoning, the meaning of conceptual reasoning in learning. And that will more or less wrap up my plan, so it works out nicely, and I’d like to finish it. Okay, so first of all I really want to talk about casuistry, and the topic through which maybe this can be demonstrated well is the topic of giving a get. When I look at these topics of giving a get, there’s something there that seems very problematic. Usually, right, it says — there is a verse: “and he wrote her a bill of severance and gave it into her hand.” It seems to me that this is the verse that has the most discussions in the Talmud of any verse in the Torah. Every word there is a whole collection of topics: “and he wrote,” “her,” “bill,” “of severance,” “and gave,” “into her hand.” Six words, and every one of those words is a whole collection of several packed topics. I don’t think there’s another example of a verse in the Torah that gets this kind of treatment. In any case, I want to focus a bit on the word “and gave.” And that basically means for me, right, “and he wrote” — then the question is how the writing has to be done; “her” — that means for her sake; “bill” — what the structure of the get has to be; “of severance” — that there has to be an actual severance here, with all sorts of laws of conditions; “and gave” — the question of what valid giving has to look like; and “into her hand” — the question of where it has to arrive. Each one of these is a whole set of topics. There are words among these words that span dozens of pages — one word.
[Speaker B] In Keritot? Tractate Keritot? In tractate Gittin, sorry. Tractate Gittin, yes.
[Rabbi Michael Abraham] Okay, fine. In any case, I want to focus a bit on “and gave.” When the Talmud wants to describe the concept of giving a get, what valid giving is, it brings a very large number of examples — dozens and dozens. Each example, right, asks whether the giving is valid or invalid: the get in her hand while the string is in his hand, right, some get tied with a thread, I’m holding the thread and giving the get to the woman; or he writes her the get and delivers it to her in her courtyard; or he places it in his courtyard and transfers the courtyard to her; he wrote it on things from which benefit is forbidden. Endless, endless topics trying to characterize the concept of giving. Now this whole thing is strange on its face, because when the later authorities try — it starts already with the medieval authorities (Rishonim), but the later authorities usually do this in a more systematic and focused way — the later authorities try, from this set of examples, to distill a definition of the concept of giving. What must a giving satisfy in order for it to count as valid giving? And it starts with Ketzot, and after that Kehillot Yaakov has two or three huge sections on this issue. And what characterizes this discussion is basically what they are trying to do: they are trying to take the examples that appear in the Talmud and extract rules from them, right, general definitions. What is giving? What must giving satisfy? In fact it’s very similar to scientific abduction. I take experiments that I’ve done, I have results from different experiments, and from that set of experiments I try to define a theory, a general theory — which is basically exactly the same thing as what they are trying to do here. The problem is: what exactly do we do in scientific abduction? Usually, the more the abduction leads to a simpler theory that explains as many facts or experimental results as possible, the more successful the theory is. In the end, our goal is to take all the facts — facts we have observed — and explain all of them by means of a simple theory. As simple as possible, with some number of principles — I don’t know, two, three, four — with a few basic concepts, and that’s it. Clearly, if the complexity of the theory is more or less like the number of findings it is supposed to explain, then we haven’t done anything. Right? Meaning, for example, let’s take the theory of gravitation, which is always my favorite example in these contexts. So Newton looks at a collection of observations or scientific findings. For example, tides, or the paths of the stars, or the falling of bodies to the earth, and so on. And he says: all of these phenomena, which on the face of it don’t even seem related, I can explain with one simple law, the law of gravitation, which says that when there are two masses at a certain distance, they exert a force on one another that is inversely proportional to the square of the distance. Okay? Meaning, the product of the masses divided by the square of the distance. And this can explain tides, it can explain all the paths of the stars, and of course it can also explain the falling of massive objects toward the earth. So that’s a wonderful theory, because one simple law explains a huge range of phenomena. All sorts of phenomena, endless phenomena. And some of them on the face of it don’t even seem connected to one another, which is even better. Because the more varied the phenomena are compared to the simple theory that explains them, the greater the theoretical gain. Right? The gain in theory is that instead of holding an infinite number of facts, I can speak about one simple theory that basically expresses all the facts. Meaning, all the facts are private cases of the general theory. And therefore, the more numerous and varied the facts that the theory explains, the more grounded, more useful, and therefore more acceptable the theory looks. In other words, the goal of scientific abduction — abduction is moving from the facts to the theory that explains them — it’s like induction, except induction moves to a general description of the facts, whereas abduction is an explanation of the set of facts; it’s not just a description of them. So scientific theory is the result of abduction, not of induction. So abduction basically aims to reduce the set of facts before me to the simplest possible description. That’s the goal. Clearly, if the complexity of the theory is more or less like the set of facts it explains, then we haven’t gained anything. Okay? That’s clear. So going back for a moment to the topic of giving a get, what I want to say is that in the topics of giving a get, the theory we arrive at is basically of a complexity more or less like the number of cases it is meant to describe. In other words, Kehillot Yaakov — I think that’s maybe the clearest example I know, almost the clearest. I said there are two sections there, maybe even three, very, very, very long and detailed and exhausting, dealing with this. And how is it structured? It starts with Ketzot. Let’s say Ketzot presents at least three or four examples. What? I can’t hear. Ketzot deals with… What? Shmuel needs to mute. The theory, yes — Ketzot talks about three or four examples. So let’s say, for example, at first glance he says: what is called giving a get? Performing an act of giving — taking the get from the husband’s hand and passing it to the woman’s hand. Okay? That’s the initial proposal for defining giving a get. But it doesn’t stand up to the test of the facts. For example, if I place the get in my courtyard and transfer the courtyard to the woman, that is also valid giving of a get. Now I didn’t give any get here in the hand-to-hand sense. So Ketzot says: fine, then I have another proposal. Maybe giving a get means transferring ownership of the get from the husband to the woman. Now how do you transfer ownership of the get? Either through courtyard acquisition — I transfer the courtyard to her and through that… actually, by the way, that’s not courtyard acquisition. Courtyard acquisition is when I place the get in her courtyard. To transfer the get to her together with my courtyard is acquisition together with land, not courtyard acquisition. So through acquisition together with land I transfer the courtyard to her, or in her courtyard she acquires the get by courtyard acquisition. And if I physically give her the get, that is basically transfer of ownership by pulling or lifting or whatever. So here we have an excellent proposal: giving a get means nothing other than transfer of ownership. Ketzot says that this too does not stand up to the test of the facts. Why not? Because the Talmud says that if one wrote it on things from which benefit is forbidden, it is valid. You can write the get on something from which benefit is forbidden, give her the get, and it is a valid get. Now he says there is a dispute among the medieval authorities (Rishonim), but at least according to some of them there is no ownership over things from which benefit is forbidden; you cannot transfer ownership of such a prohibited item. So if writing it on things from which benefit is forbidden is valid, that means transfer of ownership is not required for the giving to count as valid giving, because there is no transfer of ownership there. In short, you see how the discussion goes. The discussion basically says this: I have a hypothesis, I test it against the cases that appear in the Talmud. If it explains them, excellent — a great hypothesis. If I have a case it does not explain, then the hypothesis has fallen — like Popper, like a scientific theory against experiments, against empirical tests. It works in exactly the same way. Okay, so that’s how Ketzot proceeds, and somehow in the end he remains with some unclear definition. I don’t even remember exactly what his conclusion is, but that doesn’t matter. Right now I’m only trying to show how this works in principle. Now Kehillot Yaakov tries to continue in the path of Ketzot and do a broader inquiry, dealing with many more examples — dozens of examples, I think, or at any rate many — and he tries to arrive at some definition that will cover all the examples. Meaning, once you succeed in formulating a theory with which all the examples fit, or from which they are derived, then you have an excellent theory, exactly like in scientific abduction. Now it turns out that if you follow what Kehillot Yaakov does, every case that comes before him adds another patch to the theory. In other words, you have a case that doesn’t work — okay, so let’s round off some corner in the theory. Another case comes — again I round off another corner. Another case. In short, in the end the theory is made up of so many patches, so many stitched-together tears, that it is more or less equal to the number of cases in the Talmud. Or, in scientific terms, I would throw such a theory in the trash. It’s worthless. Because the theory is at a level of complexity that is more or less like the number of cases it explains, so what did you gain? A theory is supposed to be economical. It is supposed to explain all the cases through a few simple principles. But here, basically, every case adds another principle to the theory. If every case adds another principle to the theory, then what have you done? Just list all the cases and let that be your theory. You know, in logic they distinguish between definition by extension and definition by content. When I want to define a set, right, I can define, for example, the set of democratic states in two ways. I can simply list the democratic states — that’s called definition by extension. The extension of the concept “democratic state” is the collection of all democratic states — that’s definition by extension. Definition by content is a definition that says: all the states that have a regime with separation of powers, with elections and civil rights, whatever all the characteristics of a democratic state are. That’s definition by content. Now in principle these are supposed to be equivalent, right? Definition by extension is fine, meaning it’s a correct definition, but it doesn’t help us at all. Fine — you listed all the facts for me, so you gathered the information in one place. The goal of science is not to gather the information in one place, but also to try to understand it and explain it, and for that you actually need definition by content. For example, if I wanted to define the concept of a democratic state and let’s say I don’t have a definition, I try to look around, I look at which states people refer to as democratic. I make a list of democratic states — that’s definition by extension — but that of course hasn’t advanced me anywhere in understanding the concept of a democratic state. I look at that list and I try to distill from it what characterizes all the states on the list, and then I’ll arrive at several characteristics that will be the content of the concept of a democratic state. And then I’ll arrive at several characteristics that will be the content of the concept of a democratic state. And now I’ve done a theoretical act. In other words, collecting examples is not a theoretical act; it’s good preparation, but that’s the empirical part. But once I have the facts, I need to explain them. And the explanation comes from looking at the facts and trying to understand what they have in common, or what explains why all these facts behave in the same way. Okay? What are the characteristics that actually generate the common behavior. And that is really the theoretical work. So now, if I go back to giving a get, if I were to collect all the examples the Talmud brings — I said, dozens of examples of valid and invalid giving of a get — and I made an orderly list, I would have gathered all the information. I would have done the empirical work. But analytic learning — and this connects to the topic of our series — analytic learning is not satisfied with that. Maybe that’s Rabbi Ovadia’s dream, okay? Meaning, to gather all the facts and know all the facts, all the material, everything written in all the books. Okay? But in the yeshiva world, at least the Lithuanian world, though not only Lithuanian I think, the policy is different. Collecting the facts is perhaps a first step. It’s a means, not an end. It’s a means for trying to produce from those facts understanding, or to produce the theory that explains the facts. To do scientific work in that sense. So therefore, if I collected all the cases about which the Talmud says that the giving of the get is valid or invalid, that would be an initial basis for the work. But it is really only a means in order afterward to try to define the concept of giving a get. Once I arrive at such a definition that succeeds in explaining all the valid cases and also explaining why the other cases are invalid, then I know I have a good theory. Okay? That is really the goal here. Now you understand that what I’m doing is: I take a case, propose a hypothesis, right, like Ketzot did. The first hypothesis: giving simply means an act of giving, transferring the get from the husband’s hand to the woman’s hand. But I have examples that don’t fit that. Examples where there is no such giving and nevertheless it is a valid get. Or alternatively, if I had examples that did satisfy this condition but the get was nevertheless invalid. Fine? Examples of that type topple the theory. What do I do? I rescue it, or articulate it, as they say in philosophy of science. I basically add qualifications, sand down corners, add further definitions or further principles each time in order to cover more and more cases — whether the giving is valid or invalid, the theory has to explain both types of cases. Meaning, why in these cases the giving is valid, and why in those cases the giving is invalid. So every such case basically adds another principle to the theory. If so, then in the end the theory will be some structure that is a set of, as I said before, little tears patched together, which is not really a theory. Because it’s basically just like listing the whole set of facts. You haven’t gained anything. Only if you manage to produce a structure that is relatively simple compared to the number of facts do you have some theoretical gain, only then does the theory have meaning. And what happens in Kehillot Yaakov is that the structure he reaches in the end is made up of more or less a number of components or elements that is more or less like the number of cases he came to explain. Every case added another: yes, but only if such-and-such. Ah, and besides, only if such-and-such. And every case adds another qualification. So you understand that if the number of qualifications is like the number of cases, then all you’ve really done is list all the cases for me in a different language. You haven’t really done scientific work here. But that’s it — there’s no way, at least Kehillot Yaakov didn’t find one, and other later authorities don’t find one either — there is no way to produce a simple definition of the concept of giving. Okay? And maybe I’ll bring one more example related to something I may want to touch on later. When we do work in moral philosophy, ethics, right, in philosophy — how is meta-ethical work usually done? Work in philosophical ethics. You take a set of cases, try to judge them good or bad, moral or immoral, but that’s just collecting the findings, definition by extension. Now you have to try to define a moral principle, a moral principle that will explain the dimen— not the status. Right, the number of basic principles that spread it out. So what you want to do afterward is produce a moral theory that will give you a tool, or explain to you, why this set of cases is moral behavior and why that set of cases is immoral behavior. If you succeed in producing such a tool — for example, I aim for the greatest benefit for the maximum number of beings. That is, for example, a utilitarian theory of morality. Utilitarian not in the sense that I get something out of it. Utility here is altruistic. It is to bring the maximal benefit to all beings in the world, to maximize benefit. That is a very economical moral theory, because it tries to define with one principle something that will basically give you a tool, in every case, to decide what is moral and what is not moral. Never mind whether it actually does the job, but that’s the aspiration. That’s what they try to do. There could be another theory, it doesn’t matter — there are all kinds of theories in meta-ethics — but that is basically the work they try to do. But you understand that it makes no sense to give a theory that says: if you have a case like this, do this; if you have a case like this, do this; if you have a case like this, do this. Even though that theory also tells you what to do in every single case, it does not give you any understanding why in all these cases you have to do this or that. For understanding you need a theoretical explanation. You need to do abduction. Okay? It is not enough just to list the cases. So here too, in giving a get, what the later authorities are trying to do is produce some kind of theory, an abduction. Except they don’t succeed. They don’t succeed because the theory that results from this abduction is at a level of complexity like the number of cases it is supposed to explain. This is what in philosophy of science is called an ad hoc rescue of a theory. I have a theory, it doesn’t fit a case, okay, I’ll add some principle so it will fit. This is except for cases that happen at the North Pole. So no problem, my theory is wonderful, because the case I checked where it didn’t work happened to be at the North Pole. What? This theory is wonderful except in North Pole cases. You haven’t explained why, what is special about the North Pole. But you are indeed describing the facts correctly. Once you have an explanation, it will usually also be economical, and it will be an explanation with a small number of principles that explains many more cases. Therefore theories of this sort, with ad hoc additions that rescue all the cases and somehow fit them artificially into the theory, are not theories worth much. That’s not really theoretical work. And the question is, then, why is it really like this — that’s one thing — and second, if it is like this, then what is the point of doing all this work? “What is this service to you,” as was said.
[Speaker D] Exactly, Rabbi, that’s what I wanted to ask. Why do we need a theory at all? Why not just make a list of cases and next to each case put a check mark or an X — whether it works or doesn’t work?
[Rabbi Michael Abraham] You just asked two different questions. The second one I’ll answer now; the first one I’ll answer in a moment, because I also asked it. As for the second question, you’re basically saying: why not just make all the cases? For two reasons. The first reason is that I’m interested in understanding, not only in knowing all the cases in the Talmud. If my goal were to know everything written in the Talmud, then indeed there would be no need. If my goal is to understand, or to know in the sense of “and Adam knew Eve his wife,” right? Meaning, to know in the sense of connecting to it, understanding the meaning of the matter, not just holding the information. Because from my point of view, the cases of the Talmud are not even the relevant information. They are actually conveying to me the ideas those cases express. And my goal is to understand the ideas, not the cases. I just use the cases in order to extract the ideas from them, to distill them. That is the conception of the analytic scholars. The analytic scholars think that the goal of learning is to understand the theory behind the laws and behind the Talmud, not to know like a parrot how to recite all the laws in the Talmud.
[Speaker D] The question is whether you really need that. Do you need to know why? Do you need to know the meaning?
[Rabbi Michael Abraham] I think yes. You can say no, that’s fine — then we have a disagreement. So that’s one thing.
[Speaker D] But if you do need it, why do you need it? What I need to know is whether my act of giving counts for something or not.
[Rabbi Michael Abraham] Can you explain to me — what? Why do you need to know the cases?
[Speaker D] To know the Jewish law. But—
[Rabbi Michael Abraham] The cases are not the Jewish law. Why not? There are cases dealing with things that have nothing to do with practical Jewish law at all.
[Speaker D] Cases of giving a get that are not Jewish law?
[Rabbi Michael Abraham] Erasing Amalek, the wayward and rebellious son — there are lots of cases and lots of laws. Why do you need to know them?
[Speaker D] No, okay, I was talking about giving a get now.
[Rabbi Michael Abraham] No, I’m asking you about those. I know you were talking about giving a get. I’m asking about them — why do you need to know them? If you assume that the only reason to learn is to know what to do, I have a few difficulties with your position. I understand. This is what Rabbi Israel Salanter writes. The Talmud says that the wayward and rebellious son and the idolatrous city never existed, and why were they written? “Expound and receive reward.” So he says: what, I need the three verses about the wayward and rebellious son so that I’ll have something to learn? I’ve already finished the rest of the Torah? What kind of answer is “expound and receive reward”? He says we’re not reading the Talmud correctly, not understanding it correctly. The Talmud says that the passage of the wayward and rebellious son was written so that we understand the idea of “expound and receive reward.” The idea of “expound and receive reward” means that I do not learn in order to know what to do; I learn because learning itself is the goal. It is not a means to know what to do. And the passage of the wayward and rebellious son teaches me the idea of “expound and receive reward”; it was not written so that I should learn it and receive reward for it. And that’s what it teaches: that we do not learn in order to know what to do. Of course one has to do what one has learned — to learn in order to do — but it’s not that the goal of learning is action, that learning is a means and the goal is to know what to do. No, the goal is to understand Torah. That is the first answer. The second answer — and about this it’s harder to argue — says that even if you know all the cases in the Talmud, the next case that comes before you will not be identical to any of them. So how will you know what to do about it? How will you know how to rule on a case that does not appear in the Talmud?
[Speaker D] There are infinitely many cases. You can’t list all the cases, it’s not—
[Rabbi Michael Abraham] Limited?
[Speaker D] Exactly. What exactly?
[Rabbi Michael Abraham] You can’t cover all the cases, and even if there are many, the Talmud certainly didn’t cover all of them. The Talmud brings several dozen cases about giving a get, and that’s a lot, and in other topics it brings fewer. So what do I do with a case that doesn’t appear in the Talmud? If I understand what giving a get is from the cases that appear in the Talmud, I create an understanding of what giving a get is. Once I have understanding, I can also rule on a case that doesn’t appear in the Talmud. But if I had only counted the cases and memorized them, that wouldn’t help me if a different case came before me. In order to infer from them to another case, I need to understand the idea behind them, and then maybe I can apply that also to the new case. So there is practical benefit even in the sense of halakhic application. Even if I do accept your assumption that the goal of learning is practical halakhic application, there is still value in understanding — though here of course the conception is that understanding is a means and not an end. In the previous answer I gave, understanding was an end; it was not a means. So then I don’t need excuses.
[Speaker D] Fine, so really in the Rabbi’s first answer — is the Rabbi assuming that behind every case there is an idea that can be found?
[Rabbi Michael Abraham] Of course. What do you mean? How was the Jewish law determined behind the case? How did the sages determine the Jewish law in that case? Randomly? Did they draw lots? They understood what giving a get is, or tried to understand what giving a get is, and from that conception they said: this is valid giving and this is invalid giving.
[Speaker D] The cases reflect the—
[Rabbi Michael Abraham] The understandings they had, according to which they decided what is valid and what is not valid.
[Speaker D] I see. When the Rabbi says “idea,” he doesn’t mean some metaphysical idea or something?
[Rabbi Michael Abraham] No, no, no—I’m talking about the theory of Jewish law itself, what makes a bill of divorce valid. I understand, the metaphysical ideas are a separate question. That already goes into… what stands behind the rules of Jewish law? After you’ve conceptualized them, now you can ask a further question: what do they themselves actually reflect? There’s… you can go deeper and deeper below the surface on this issue. But fine, that’s already another set of discussions. In any case, that’s regarding the question of why we need to do the scientific work. But practically speaking, that’s what we’re trying to do. In yeshiva-style conceptual learning, that’s what we’re trying to do. And in my opinion, that’s also what a halakhic decisor needs to do, and we spoke about first-order halakhic ruling and so on. Now, the question I asked, and that’s the second question you asked, is: if we don’t arrive at the right theory, then what’s the point of the whole enterprise? If in the end, regarding the giving of a bill of divorce, we arrive at a theory that isn’t worth much, because the number of its components is more or less equal to the number of cases the theory explains, then what’s the point of the whole story?
So here my answer is that the goal of the enterprise is basically like Maimonides’ negative attributes. What do I mean? It may even give some meaning to Maimonides’ strange thesis about negative attributes. The claim is basically this. I want to understand what exactly… what is a valid giving of a bill of divorce? Now how can I do such a thing? If in fact I don’t have… I’ll put it differently, I’ll start differently. The Sages want to convey to me the principle, what the definition of a valid giving of a bill of divorce will be. Why do they do it through cases and not just give me directly… the result? They give me cases, and let’s say the Kehillot Yaakov managed to arrive at a good theory. Why didn’t the Sages just give the theory directly? Why did they give me cases and leave me afterward to distill some kind of theory from the cases? Let them just give me the theory directly. After all, in the end you want to convey the theory to me, right? If that’s the goal, if that’s basically what I need to know in the end. That’s the assumption. So give it to me. Why do you bring me a case and leave me to do the work?
The answer is, at least part of the answer, that there is no such theory. There’s no way to formulate that theory and pass it on to me. I think I gave you an example when we spoke about rules, about the meaning of rules in this series some time ago, when we spoke about why the Talmudic text doesn’t contain rules, the Mishnah and the Talmudic text and so on. So I gave an example there of an exceptional place where a rule does appear—the first Mishnah in tractate Bava Kamma. And the first Mishnah there says: “What is common to them is that they are your property and their supervision is upon you,” regarding the primary categories of damages, “that they are your property and their supervision is upon you, and when they cause damage, the one who caused the damage is obligated to pay compensation from the best of his land.” So there, it gave us the rule. Anything that is my property and whose supervision is upon me—if I fail in supervision, I have to pay from the best of my land. And if damage occurred, of course I have to pay. So they gave me the rule, right?
Excellent. But now I start checking that rule and I see that it doesn’t work. I’m not even going into the Talmudic text on page 6 that I brought, where you can see that it doesn’t—but even without that. Just, let’s try checking whether it works. Take, for example, a paid guardian, someone who received an animal to watch. If the animal went out and caused damage, and he was negligent in guarding it, he has to pay, right? But the animal is not his property. After all, “anything that is your property and whose supervision is upon you, you are liable to pay.” If the definition is that it has to be your property, this is not the guardian’s property; it is the depositor’s property. The depositor entrusted you with his property, so why are you liable to pay? So the medieval authorities (Rishonim) already discuss this, and they say that it is considered as if it were his with respect to the legal (in halakhic sense) of damages. You’re beginning to recognize the laths here? Meaning, I have a theory that says anything that is my property, I’m responsible for it—boom. But then I have several cases that are not my property, and nevertheless I’m responsible. Yes, but it’s considered as if it were my property for the purpose of damages. So practically speaking, you don’t really need “my property.” This is exactly like the move made by the Kehillot Yaakov.
After that, the same thing applies to a robber. A robber too—if the animal caused damage, he has to pay. Or someone who places another person’s animal onto another person’s standing grain—he also has to pay. There’s a dispute between Tosafot and Rashba there in the chapter HaKones as to whether this is under the category of a person who causes damage or property that causes damage. If it’s a person who causes damage, fine. But if it’s property that causes damage—I placed your animal there. Your animal caused the damage, not mine. And the Mishnah says it has to be my property in order for me to be liable, but it’s not my property. Yes, so it is considered as if it were my property for the purpose of damages. Again, the later authorities (Acharonim), the medieval authorities (Rishonim), discuss this.
In short, it doesn’t work. What about a pit in the public domain? A pit in the public domain is ownerless. So how am I liable to pay if I dug a pit in the public domain? So Rabbi Yehuda says that there are two things that the Torah treated as being in one’s possession in order to make him liable for damages: leavened food on Passover and a pit in the public domain. Again, something that isn’t really mine, it is not my property, and nevertheless if it causes damage I have to pay. So this rule doesn’t really work. This rule that anything that is my property is the definition of liability for damages? No, that is not the definition. Look how many exceptions I just listed for you. But that is the definition—that’s what appears in the Mishnah. What does that mean? Apparently the Mishnah really could not find a way to formulate a rule that would stand the test of all the facts. No—there just wasn’t a way to conceptualize it, or at least the Mishnah didn’t find such a way.
So what do you do? You give some approximate rule, in this case, even though we all need to understand that although the Mishnah gave the rule, clearly there are certain exceptions where there too he will be liable, because intuitively it’s obvious to me that there too he should be liable. Ah, it doesn’t fit the rule? Fine, there are a few exceptions; maybe I’ll add a few more laths. Does that mean that the fact that the thing is mine is irrelevant to liability for damages? It is very relevant. Obviously the basic liability for damages is when something of mine causes damage, right? Clearly that is the basic liability. Only what? Only that it isn’t exhaustive. That is the basic liability, but there are exceptions where that condition is not met and I still have to pay. So you can’t say that saying “it’s my property” is irrelevant, that they told me something absurd, something completely unrelated. It is related, but it is not exhaustive. There are all kinds of exceptions; you need to add qualifications, all kinds of sub-principles and so on, epicycles and differentials. Okay?
The same thing is what happens with the Kehillot Yaakov. So what does that mean, basically? On the one hand, my claim is that the Talmudic text uses cases because it had no way—there was no way—to formulate the theory directly for me, to convey the theory directly to me. There’s no way to formulate the theory simply. The Talmudic text preferred to convey the idea through many, many cases, from which I would build some kind of understanding of what the correct theory is, what is called a valid giving of a bill of divorce, more generally. But then what do Ketzot, the Kehillot Yaakov, and all the later authorities (Acharonim) do? What people always do in yeshivot—everyone does this. You try to build theories on the basis of the cases, and each time you throw out the theory because there is another case, and then you build a broader theory that tries to contain that case too, and then throw that out as well. You’re constantly doing this work, attempts to do abduction, and then discard the result. Another, slightly broader abduction, discard the result, and so on. A kind of process like that where in the end, as I said before, you’re left with nothing. So why was all this work done?
A very simple reason. Our way of generating a good understanding of what a giving of a bill of divorce is, is by rubbing up against the cases. Even if in the end my attempt to generate a theory from the cases fails, I at least understand by negation what is not a giving of a bill of divorce. And that is Maimonides’ negative attributes. When Maimonides talks about negative attributes with respect to the Holy One, blessed be He, he says: I cannot explain, I cannot understand what He is, but I can describe what He is not. And Maimonides’ claim is that by understanding what He is not, some kind of intuitive understanding also develops in me regarding what He is. It’s just that I don’t know how to formulate that. I only know how to formulate what He is not, but the formulation of what He is not—by my negating all kinds of things from Him—I succeed in generating some kind of positive understanding regarding the question of what He is. Because otherwise—I think this gives very, very good meaning to Maimonides, because on the face of it, when you read Maimonides, this is complete nonsense, the doctrine of negative attributes, just logical absurdity. But that—I spoke about in other contexts.
For our purposes, what I want to say is that the same thing happens in our analytic learning. In our analytic learning, basically, we have many cases. In the case of giving a bill of divorce, since it is much more delicate, the Sages needed many more cases in order to give me a good understanding of what the giving of a bill of divorce is. That understanding, in the end, has no formulation that describes it; I don’t know how to conceptualize it. But it exists within me. It exists within me in such a way that for any case that comes before me, I’ll have a good intuition as to whether this is a valid giving of a bill of divorce or not. How did I internalize that understanding of the giving of a bill of divorce? How did that happen? Through the kind of conceptual work I described earlier. I propose a hypothesis, and then a case contradicts it, so I throw out the hypothesis and try a more sophisticated hypothesis. Throw that out too; I understand that the more sophisticated hypothesis is also incorrect, so I try a third. Another case comes and I throw that one out as well. You see that every time I encountered another case, it sharpened some additional aspect for me. And somehow, even though the theory collapsed, what remained in me was an unformulated residue of understanding of what a valid giving of a bill of divorce is. I do this by peeling away all the shells.
Think of a tree trunk or a block of stone from which you are trying to produce a sculpture. You need basically—and several philosophers spoke about this—the sculptor did not create the sculpture. The sculpture exists; all he removes is what hides the sculpture from us. Meaning, he removes all the parts of the stone that interfere with him, and the sculpture remains. So who created the sculpture in the end? The sculpture was there; he just removed what prevented us from seeing it. Of course that’s not literally true, but I mean on the conceptual level, that’s basically what we also do in analytic learning. We remove all kinds of arguments that we negated, but through the negation of those arguments I understand what is not a giving of a bill of divorce, and indirectly an unformulated understanding is created in me of what is a giving of a bill of divorce. And therefore it is very important to do the whole process that the Kehillot Yaakov did, even though in the end we are left without a theory, or with a theory that is worth nothing. Because what matters is not the result; what matters is the path we took. The path we took built within us some kind of intuitive understanding. After I encountered so many cases, I can already understand intuitively, regarding any other case that comes before me, whether this will be a valid giving of a bill of divorce or not.
And that is basically the goal of the whole thing. The goal of the whole thing, in the end, is to generate within me the correct intuitive understanding. And the theories, in this particular case—there are places where the theory does succeed in explaining the cases—but in a case like this there is still value in building a theory. Theories and their negation, this kind of evolution of theories, because in the end after this whole process I remain with the unformulated understanding, but now it already exists inside me. I have some kind of good intuition. Any case that comes before me, I’ll be able to know whether it is a valid giving of a bill of divorce or not. And if they had tried to do this by trying to give me a definition, they would not have succeeded, because there is no definition that does the job. And the more delicate the matter is, of course, the more cases are needed in order to formulate it for me, or to generate the intuitive understanding in me. And since in the giving of a bill of divorce the case, the formulation, or the principle is delicate, you need many, many cases in order to help me internalize it at sufficiently fine resolution. That is why all those dozens of cases were needed.
[Speaker B] I have a question. I don’t understand why the Rabbi calls it intuitive understanding, because in the end, when all the cases you learned come up, you are then required afterward to do an analytical analysis, so in the end you give a halakhic answer as a result of analytical analysis, not as a result of intuition.
[Rabbi Michael Abraham] No, no—the claim is that I do the analytical work along the way. Analytical work is to formulate a theory, find cases that contradict it, topple it or refine it, find another case, again topple it, refine it—that’s analytical work. After I’ve finished this whole story, I’m left with nothing, with piles of ruins. I did the analysis and the patient died, as they say. Analysis is dissection. So analytical work is basically dissecting work. I did conceptual dissections, and in the end the patient died. No—he didn’t die. The structure died, but the structure is not the issue. The structure is the conceptualization of the issue. So I don’t have a conceptualization, but the issue itself has been internalized deep inside me, and now I have a good intuitive understanding of what is and what isn’t. Now when new cases come before me, in light of the whole path I took, I already have a calibrated intuition, so that I can say regarding each case whether it is a valid giving or not.
[Speaker B] Wait—and at that stage, when you have this calibrated intuition, are you also required here to do more analysis?
[Rabbi Michael Abraham] No, in order to give an answer. I don’t know how to formulate this understanding, but I can say regarding each case whether it is valid or not. I have Torah judgment. I have Torah judgment. From this point on I have no formulations, but I tell you that this is a valid giving. Sometimes I use cases to show you some analogy, but I’m not presenting you with a theory or a full thesis. I don’t have one. Okay.
Now, the nice point here is that today we have tools to understand this, which we didn’t have until twenty years ago. Today we have excellent tools for understanding this process. This is exactly what happens in a neural network. Exactly. And if we remember that in our brain we have a neural network, then it shouldn’t even surprise us. What happens in a neural network? For those who know it, it’s simple, but I’ll try to describe it simply for those who don’t know. A neural network is a kind of miracle that no one knows how to explain, but people know how to use it. Okay? Yes, like OpenAI, who created some kind of golem and had no idea what the power of the golem they had created would be. They themselves didn’t think those would be its capabilities. There’s no way to understand it. At least from what I read, even the experts. When I encountered this when I started my doctorate, people had begun working on neural networks. And someone kind of explained the idea to me a little, and we played with it a little, because it came from physics. We started playing with this thing and it was simply unbelievable. I had a friend who built me a program with a network of I don’t know what—I think ten sites. Yes, a neural network is always nodes connected to one another, or vertices, a kind of network of vertices connected to one another with links, with signals. Think of a fishing net. So each node is a site, a vertex, and between the vertices there are lines—not every one to every one, but some yes and some no. That’s called a network. A neural network. Fine?
Now this network basically doesn’t merely symbolize; rather, it is actually a description of how our brain works. In our brain there are neurons connected to one another by all kinds of connections. That’s how our brain works. Now it turns out—I’ll describe to you what happened there in that experiment—my friend built a network that had ten vertices, one-dimensional, the simplest possible. Ten vertices, nine bonds between them; that is, one is connected to two, two is connected to three, simply a row. No one is connected—it’s only nearest neighbors, as it’s called, meaning only close neighbors. Fine.
Now what happens in such a neural network is that I train the network—that’s what it’s called, training the network. I give it an example. For instance, I want the network to identify a particular person’s face. So I give the network various pictures. Some of the pictures are of that person, and some of the pictures are of other people. And I tell the network, when I give it a picture of that person, I tell it: this is a correct image. When I give it the other pictures, I tell it: this is not a correct image. Yes, this is supervised learning; there is also unsupervised learning, but right now I’m describing supervised learning in a simple way. So I give it the examples, and it organizes its internal weights, or what it infers from the examples it received—that’s called training the network. It changes its internal parameters accordingly so that they fit exactly the correct answers and yield the correct answer for them, and for the incorrect answers yield the answer no. So it organizes itself so that it fits all the examples on which I trained it.
It turns out that after it adjusts itself that way, when I bring it new examples that it hasn’t seen, it will identify them with a high probability. It will identify what is correct and what is not correct; it will give not-bad answers. And the more examples I train it on, the more reliable answers it will give me on new cases that it has not seen. The more examples there are, the greater its capacity, with finer resolution and higher reliability of identification.
What that fellow did for me was tell me: type zeros and ones completely at random. Whatever you want, okay? Zero zero one, one zero zero zero zero zero one zero, whatever you want. Okay? And the network has to predict for you what your next keystroke will be based on your keystrokes up to now. Now it turns out that even the simplest network—a very simple network, with ten bonds, with yes, you have to be careful about overfitting, absolutely there is overfitting there—the claim is that the network basically takes my previous keystrokes, sees, learns what the sequence was, whether after zero I type one, or I do zero twice and then one—it tries to learn the rationale of how I type, even though I think I’m typing something arbitrary. And I try to fool it, I try very hard to fool it, and in the end the network tries to predict what the next keystroke will be.
Now this drove me completely crazy. I tried to fool it with all my might and simply failed. It’s not that it predicted every keystroke, but with the simplest network I described to you—which really was a very trivial network in terms of its power—it managed to predict something like seventy percent of my keystrokes, when in principle it should predict fifty percent, right? That’s the random chance for a network that hasn’t been trained at all. Seventy percent systematically—that’s impressive. Its predictive ability rises, and the more we continue, its prediction will rise further, and if I had kept typing it would have managed to predict more than seventy percent. Meaning, the network learns me, and by learning me it manages to think like me, and basically tries to predict what I will say or do next. And this is basically a way in which I teach or train a network according to examples that I give it, and from that training it manages to generate, to perform this task also on examples it did not encounter.
I think that if you think for a moment about this process, even someone who doesn’t know the details of neural networks—it doesn’t matter, I’m not an expert either—but if you think about the process I described here, then you’ll see that it is exactly, exactly what I described when I spoke about analytic learning. Exactly the same thing. Meaning, our brain is basically some sort of neural network. It encounters cases—this is supervised learning. It encounters cases, and the Sages tell us: these cases are a valid giving of a bill of divorce, these cases are an invalid giving of a bill of divorce. I organize my neural network accordingly. After I’ve encountered enough cases—since this is delicate here, it is subtle—I need many cases in order to train, so they give me many dozens of cases regarding the giving of a bill of divorce. In other topics, where things are less subtle, they give fewer examples. And then after I’ve gone through all these examples and made all the distinctions, that is the way I trained my network, and now when a new case comes before me, there is a decent chance that I will give the answer that the Sages wanted me to give. Basically, I am training my internal neural network.
Now there is a very important point here, and it is a bit hard to grasp for someone who doesn’t know this professionally, but this is the fact. A very, very interesting fact. The trained network built the connections between its nodes in such a way that it fits all the examples up to now. But the network does not do any abduction. If you ask the network what is the idea behind the examples, it will look at you blankly. There are no ideas. The network doesn’t work with ideas. It’s all very technical. It organizes the various weights of the connections between the nodes in such a way that they fit the examples, and after it is trained, bring it a new example and it will give you the correct answer—or hopefully, with high probability it will give you the correct answer. But it cannot tell you what the general idea behind it really is.
I’ll perhaps try to formulate this; maybe I’ll compare it to something else. What happened before there were neural networks? Before there were neural networks, when I programmed software that would recognize faces or perform some action, I had to feed the abduction into the software. I had to build the theory of how to identify a face and then write a program—classic software, not neural-network software—if this then do that, if this then do that, simply according to the principles I discovered through my abduction, the scientific principles. I had to do abduction, and then I could program the software and it would do exactly what I do. It would simply apply the theory I found to the cases that come before it. And therefore it was impossible really to solve complicated problems with such software, because abduction has very limited ability; it won’t succeed in generating complex concepts, complex theories.
When they found this trick of a neural network—a trick of the Holy One, blessed be He, registered in His name; they just discovered it—when they found this trick of making software also based on a neural network, they bypassed the need for abduction. The programmer has no idea how faces are recognized. He doesn’t know. He did not put face-recognition techniques into the software. He only trained it, and now it knows how to do it through its internal parameters. But when you ask it, how do you do this work? Give me the theory. It doesn’t know how to give the theory. It’s irrelevant. It doesn’t work in terms of abduction at all.
And that is exactly—really exactly—I think there is no better explanation than neural networks for what I described to you earlier with the giving of a bill of divorce. I think that thirty years ago, or forty years ago, it would have been very hard for me to explain this thesis about the giving of a bill of divorce to people. I actually wrote it in an article I wrote when I was still doing my doctorate; that was really in the early years. And I didn’t use the concept of a neural network there. But today, when I look at it, I think that since in those years I had also become familiar with the concept of a neural network, it influenced me, though at the time I didn’t connect it. Today I understand that that article was simply an application of the concept of a neural network. It’s an article in the weekly page of Bar-Ilan, not some scientific article or anything; maybe I’ll send it to you at some point.
[Speaker C] Which means that the neural network also influences people in the form of a neural network. What?
[Rabbi Michael Abraham] I didn’t understand.
[Speaker C] That the way a neural network, yes,
[Rabbi Michael Abraham] This is exactly second-order learning. So in the end, the claim is that the purpose of analytical study is to organize my internal neural network so that in future cases it will give the correct answer. The catch is that when an AI does this, in a human being it passes through my cognition—not like in a computer, which is ultimately just hardware. In a person it goes through cognition. So how do I actually train my neural network? Through the analytical method I described earlier, the kind used by Kehillot Yaakov or Ketzot. I build a theory, test it empirically against facts, refine the theory, throw it out, refine again. Another example, a fitting example, a non-fitting example, building theories, discarding theories. In the end, it works through the analytical effort of trying to build theories, because we are not computers—we are human beings. And that’s why it’s built this way. But at the very end, we work like a computer. Meaning, ultimately we can function even without understanding—or not without understanding, but without being able to formulate what the definition of a valid giving of a bill of divorce is. I have no idea. But I do have a good intuition that tells me, for every case that comes before me, whether it counts or not. And that’s why I think the neural-network example really sheds wonderful light on this analytical process I tried to describe here. I really need to write this up sometime; I’ll try to do something with it.
Anyway, I want to add one more remark, since this took me a long time. I want to add one more remark. Why really is this case of giving a bill of divorce so delicate, so subtle, so hard to conceptualize? Why do we need so many cases in order to train our neural network there? So I have a certain hypothesis about this, and I want to make the following claim. This comes from another article I wrote—not from exactly the same period, roughly around then—an article in Iyyun on Zeno’s arrow. Maybe I’ll send links on WhatsApp. And there my claim was the following.
You know Zeno’s paradox of the flying arrow? Zeno raised paradoxes attacking the concept of motion. He showed that if we assume there is motion in the world, that leads us into contradictions, absurdities, paradoxes. And therefore his claim was that motion is a kind of illusion. There really is no motion in the world at all, as our teacher said. It’s all illusions.
Now one of the paradoxes he raises is the paradox of the arrow in flight. What does that paradox say? Look at a flying arrow. At every moment you see it, it is standing in a different place, right? Look: at this moment it’s here, at the next moment it’s here, at the next moment it’s here. Yes, we’re talking about an arrow that is flying. So at every single moment it is basically standing in a different place. So the question is: when does it move between the places? Every moment it stands here, and the next moment here, so when did it pass from here to here? In order to get from here to here it has to pass through. How—when does that happen if at every moment it is standing in another place? There are somewhat more sophisticated formulations of this. I’m putting it a bit simplistically, but that’s the formulation.
Now usually, at least from what I’ve seen, this paradox of the arrow—all of Zeno’s paradoxes in the end are solved mathematically. Achilles and the tortoise is solved by infinite series; I’ve spoken about that in other contexts. And this paradox of the arrow in flight is usually solved with infinitesimal calculus. What does that mean? The claim is that it’s actually incorrect to describe a continuous line as a collection of discrete points that are extremely, extremely close to one another or touching one another. That’s not a description that can really capture a continuous line. Therefore you need—there are continuity assumptions here, doesn’t matter—you need certain additional parameters here, parameters of density of this set of points, in order to generate the concept of continuity. Okay? And therefore basically—again in a very simplistic formulation, and those who understand should forgive me for the superficiality—basically the line is not made of points, I’ll put it that way, but of infinitesimals, which are basically differentials or infinitesimals, which are essentially infinitely tiny lines—tiny toward zero, not toward infinity. Meaning: as small as you like, but still lines.
What’s the difference between such a line and a point? Again, I’m saying all this in layman’s terms; mathematicians will be angry at me when they hear things like this. But what’s the difference between such a tiny line, such an infinitesimal, and a point? A point and an infinitesimal both have zero length—but for a point, it’s not that its length is zero; it has no length. That’s not the same thing as length zero. A point has no length. An infinitesimal has length, and its length is zero or tends to zero. Okay? For example, think about blind people—people who can’t see. Usually we picture to ourselves that what they “see” when they don’t see is that they constantly see black in front of their eyes, like when we close our eyes. But in fact a blind person is someone who sees nothing, not someone who sees black. Not seeing is not the same thing as seeing everything black. He doesn’t have sight. Even a black image is a kind of seeing something. He has no images before his eyes—not even that neutral black image. Do you follow? It’s a distinction that is a bit hard to grasp for a seeing person. But conceptually I think we understand it. Same thing here: an infinitesimal is a line. Its dimension is one, so it has length; it’s just that its length tends to zero. A point has dimension zero, not one, so it has no length. Not that it has zero length, but rather it has no length. Therefore a point and an infinitesimal are not the same thing, and the claim is that what makes up a continuous line is not points but infinitesimals. All this is really infinitesimal calculus for laymen, so to speak.
So the claim people usually make in order to solve the paradox of the flying arrow is that you cannot speak of an arrow being at a point in time or a point in space. Because there is no such thing as a point of time or space; rather time and space are built out of infinitesimals of time and space, not points. I was never satisfied with this solution, because it’s obvious that a line has points on it. It is true that you can’t describe the line as a collection of points that are very, very close to one another or touching one another, but it is not true that you can’t speak about a point in time or a point in space. Of course you can. Therefore I still ask: at every point in time, the arrow stands at a different point in space. That sentence can be said even in the world of infinitesimal calculus, and therefore in my view this mathematical solution proposed for the arrow paradox doesn’t really solve it. And as far as I saw, I never encountered another solution people proposed to the problem, so I wrote an article about it in Iyyun, and there I proposed a different solution.
The solution does not require infinitesimal calculus at all. On the contrary: after solving the problem on the philosophical level, I tried to offer explanations also for infinitesimal calculus. I didn’t use infinitesimal calculus to solve it; I used the solution in order to understand what infinitesimal calculus is, and also what relativity and quantum theory are, which has all sorts of interesting implications scientifically as well. And the claim basically says that the difference between—meaning, when I say that at every single moment the arrow stands at a different point, I am mistaken. What is correct to say is that at every single moment of time the arrow is located at a different point, not standing at a different point. What is the difference between the arrow being located at a point and the arrow standing at a point? The difference is that if the arrow stands at a point, that means it is there with zero velocity. In contrast, being located at a point says nothing about what the velocity of the arrow is. Even an arrow that is moving, whose velocity is not zero, at every moment is located at a different point.
Now when you ask me when it passes from point to point—that does not happen at another point in time. At that very point in time in which it is here, it also has the velocity that carries it to the next point. Because the arrow at every moment is located at a different point, but it has velocity. If I had said that it stands at a different point, then indeed the question would be how it gets to the next point, because standing means being there with zero velocity. But it does not stand at a different point; it is located at a different point. And that is a philosophical explanation which, in my opinion, simply solves the problem even without needing infinitesimal calculus.
Now to understand the root of the confusion—why people get so confused by this paradox and feel the need for infinitesimal calculus—I made the following claim. I said: when I look at the velocity of a body, people generally think that a body cannot have velocity at a point in time. In order to define velocity, you have to look at a certain interval of time, maybe a short interval, but an interval. It cannot be that a body has velocity at one point in time. Why? Because we understand the concept of velocity as follows: how much time will it take it to traverse a certain length or unit of length? So in order to compute velocity you have to look at a stretch of time, see how much distance it covered during that stretch of time, divide the stretch of distance by the stretch of time, and the result is the velocity. Okay, and without getting into derivatives and things like that. So the result is the velocity.
My claim was that this is the operational definition of velocity. That is the definition of how to calculate velocity. But it is not a definition of the concept of velocity. “Operational” means the way to compute it, but it is not the definition of the concept. The concept of velocity exists at a point in time. The way to compute the velocity requires me to wait through a stretch of time and see how much distance it covered so that I can divide the distance by the time. But the result in the end, as anyone with a little familiarity with infinitesimal calculus or basic mechanics knows, is that after I differentiate the position function, the result is the body’s velocity at a point in time. At every point in time the body has a velocity. What does that mean? It basically means that even though the calculation of velocity requires me to observe the body over some period of time—short, but still some period of time—the result of the calculation is a result about the velocity of the body at a particular point. And if you ask physicists, they’ll tell you this is a fiction. The velocity of the body means over a small interval around that point. But there is no velocity at the point. And I claim they are mistaken. There is velocity at a point. The way to calculate the velocity requires me to look at an interval. But that is only a way of calculating. In the final analysis the body has velocity even at a single discrete, defined point in time.
And therefore, when I say that the body is located at a certain point, that does not rule out the statement that it has velocity at that point. In order to see that this velocity is realized by a change of position, I have to wait a tiny bit of time. At a single point in time it will not change place. But at a point in time it has velocity. The way to verify this is to wait one more moment and then see that it really passes to another place, or that it changed place. That reveals to me—it is a sign, not a cause—that it had velocity at every point in time belonging to that interval. Is what I’m saying clear? In short, what I want to say is that there is a difference between velocity and change of place. Velocity is not change of place. Change of place is the way to calculate the body’s velocity. And change of place, of course, happens only over an interval of time. But velocity exists even at a point in time. And we have a tendency to mix up these two concepts: velocity and change of place. They are not the same thing.
My claim is that velocity is the potential a body has to change place. Therefore if we wait a bit and the potential passes from potentiality into actuality, then we will see that the body changes place. But that waiting, that need to wait a bit, is only so that it can reveal to me retroactively that it had velocity at every point in time. I won’t be able to notice that unless I wait a bit and see that it changes place. So the important point here is that there is a difference between velocity and change of place.
Now what is the root of the problem—and I expanded on this more there, and I’ll send that to you on WhatsApp—the claim is that the moment we try to define dynamic concepts like velocity, we usually do it through a relation between endpoint states. When I want to define the velocity of a body, I say where it is here, where it is here, how much time it took to get from here to here; I divide place by time. That is basically defining velocity through the two endpoint states. Each endpoint state is a position and a point in time; another position and another point in time. Difference in position divided by difference in time equals velocity. This operational definition is basically trying to define a dynamic concept by means of static states through which it passes—from this state to this state. Or to define velocity through positions. Positions are a static state, and velocity is a dynamic magnitude. But our mind works in a way that is static in principle. Therefore if we want to grasp dynamic magnitudes, we need to define them through static states that come before and after. From looking at the change in the static states, we can understand that there was something dynamic in between.
If you think about it—say, think about a movie. When we watch a movie, it is actually a huge number of frames shown to us at a very high speed, at a very, very high frequency. Frame after frame after frame at very high speed. But each such frame is a still image. It is a static picture. It’s just that in the next image the horse is already here, in the next image it’s here, and if you do it very fast and very densely, it looks to us like the natural galloping of a horse. But it’s just an illusion. What they project before us is a collection of static images. Why? Because all of our perception is static perception. And our static perception, when I want to perceive something dynamic, constructs it for me through the projection of different static states one after another, and my brain tries to reconstruct from that the dynamic understanding of the image before me. There is no way to convey to me directly the dynamic magnitude, the concept of velocity. And therefore we always tend to define velocity through differences of place and time. But that is only because of our limitation, because we think and perceive in a static way. Parenthetically, that’s also how I explained the uncertainty principle in quantum theory, which also has implications for debates about relativity theory. I’ll send you the article if anyone wants to read it.
But for our purposes, what I want to claim regarding the giving of a bill of divorce—and with this I’ll finish, because this is not really our topic here—in the giving of a bill of divorce, my suspicion is that what we are trying to define is actually an act of giving. You understand that this is a dynamic magnitude. How do we try to define it? Notice: all the definitions I suggested earlier, what were they really saying? They were definitions through static states of before and after. For example, transferring the bill of divorce from my hand to her hand is called giving. Do you understand what I did there? I described a dynamic magnitude, the act of giving, by means of the state before—the bill of divorce was in my hand—and then it passes over and is or comes to be in her hand. But that is defining the dynamic magnitude through a static state of before and after. It may be an operational definition, but it doesn’t really capture the dynamic concept itself.
The same thing with placing the bill of divorce in my courtyard and transferring ownership of it to her, or things of that kind—these are all definitions through static states, because that’s how our thought works. But there are also dynamic things in the world, and we have no direct way to grasp them. We try to reconstruct them through changes in static states.
For example, in that article I brought the question: why do the Sages say that a penitent is preferable to a completely righteous person? What does the penitent have? At most he reaches the perfect state and becomes completely righteous. What does a penitent have beyond a completely righteous person? The answer is: his movement toward the perfect state. His movement toward the perfect state does not have value only because in the end one arrives at the perfect state. If that were the case, then a penitent would at most be like a completely righteous person. When they say that a penitent is preferable to a completely righteous person, what they are saying is that the value of repentance is not only that in the end you are in a perfect static state, but that the movement toward the perfect state—and movement itself—has value. It is not merely instrumental, bringing me to a more complete state; rather, my very being in motion is something of value.
But even so, when I want to define the concept of repentance, how will I define it? Yesterday he was detested and abominable, as Maimonides says, and today he is beloved and close, and so on. Yesterday and today—a definition through static states. You were a sinner, you repented, you became righteous. But repentance is not exhausted by the question of what was before and what was after; in the transition something dynamic happens, and the consequence of that is the move from one static state to another. That is a consequence of the fact that there was something dynamic here. Repentance is something dynamic, and therefore it is very difficult to define the concept of repentance. The same is true of giving a bill of divorce. Giving a bill of divorce is an attempt to define an act, a dynamic act that changes a state. So I define it through the state before and after, but I won’t really succeed in capturing the point. It is hard to define a dynamic magnitude, and therefore they give me very, very many cases. For each one I try to fit some definition via changes in static states. In the end I will throw it all out, because there is no way to define it through changes in static states. But that is the way to train my neural network so that in the end it will also latch onto new situations with the correct answer.
That is a suggestion for why, in the giving of a bill of divorce—that is, my principle itself has nothing to do with everything I said now about the static and the dynamic. The principle I wanted to propose is the comparison to a neural network. What I proposed at the end is only a possible explanation for why, in the giving of a bill of divorce, so many case-possibilities are really required, so many examples. Because giving a bill of divorce is a dynamic magnitude, and it is hard or impossible to define it through static states. Therefore they use many, many, many examples, in the hope that in the end I will have some kind of immediate understanding of the dynamic act of giving a bill of divorce. I won’t be able to formulate it, because any formulation of mine will always work in static language, language of states. But through the states I will understand—just as through the frames in a movie I actually grasp the dynamics of what is happening on the screen. For me it is dynamics, even though what they are projecting for me is static states one after another. In my mind I produce from that an understanding of what dynamic thing is happening there. Okay? That is basically what happens in the study of giving a bill of divorce.
Okay, I’ll stop here. This took more time than I planned. Still, next time I want to finish already, because by the end of this period I want to finish this series. So next time I’ll speak about moral intuitions—not ethics, I’ll skip ethics already—I’ll speak about intuitions. Okay, comments or questions?
[Speaker D] Yes, Rabbi, when the Rabbi spoke about the experiment with the neurons, so if—
[Rabbi Michael Abraham] So in that sense broad familiarity is preferable—wait, wait, one second. Yes.
[Speaker D] It reminded me of a passage in the second book of the trilogy that I remembered I didn’t really understand, so maybe this is an opportunity to ask. There’s a passage where the Rabbi explains another experiment he did. The field of computing developed a lot, and people manage to create random text generators. Right? And there the Rabbi explains that basically, if you give a computer one text by Rav Kook, it can generate a few more like that, and if you give those to a reader of those texts, he won’t know how to identify which one really is by Rav Kook. In short, the Rabbi talks about this there at length, develops it. I didn’t really understand what point the Rabbi wanted to get to, because the Rabbi says that the conclusions are within us, and when we read the text we only express conclusions that already exist within us, right? Right? But I don’t understand why this experiment.
[Rabbi Michael Abraham] My assumption—and there I made the comment, maybe that’s what you mean—my assumption there in that discussion was that when the computer produces some random text that resembles Rav Kook, there is not really any message behind that text. Nobody thought to convey a message through it; rather it uses Rav Kook’s characteristic words and phrases and builds a passage out of them. Now the moment you relate to it, you’ll study it—if you think it’s really by Rav Kook, you’ll give a major lecture on it.
[Speaker D] No, but that’s it, I understand, but that’s exactly the point I don’t understand. It’s obvious that when I read a text by Rav Kook I relate to it as something special because it comes from Rav Kook. Now if you give me another text that is random, not by Rav Kook, and tell me that it’s not by Rav Kook, then I won’t relate to it that way, meaning—
[Rabbi Michael Abraham] No, but I’m telling you it is by Rav Kook. You’ll relate to it like Rav Kook, right? I’m lying to you. Fine? I tell you: this is by Rav Kook. Now you’ll give a major lecture on it and you’ll find lofty ideas there.
[Speaker D] If they lie to me and tell me it’s by Rav Kook, then maybe I’ll reach the same conclusion, but okay.
[Rabbi Michael Abraham] Exactly—but they lied to you. So what does that mean? That even in genuine passages of Rav Kook, everything you get out of it you get out of yourself, not from him. That’s the claim.
[Speaker D] No, but why? I actually don’t agree with that conclusion, because the very fact that I draw conclusions from Rav Kook is because I know it comes from him. I start from the assumption that Rav Kook wrote this, and that gives it value because he wrote it.
[Rabbi Michael Abraham] You give him credit that what he wrote is not nonsense, that there are ideas there. So now you try to understand what ideas stand behind it, and you find certain ideas that you derived from the text. Now if from a random text that I tell you is by Rav Kook, and from that too you extract those ideas, then who told you there is any connection between the ideas you derive and what Rav Kook wanted to convey to you? After all, here nobody wanted to convey anything to you, and you extracted lofty ideas—so maybe with Rav Kook too it’s like that. The correlation between what you derive from the text and what Rav Kook wanted to put into the text is zero.
Now I commented there—I thought maybe that’s what you wanted to note—that if this had not been a random text generator but rather artificial intelligence, then this would not have been an objection. Why? Because artificial intelligence really imitates Rav Kook’s way of thinking, and then when I derive ideas from it, those really are Rav Kook’s ideas. Meaning, he himself could have written that passage, because the artificial intelligence genuinely imitates him. What was happening there with the random text generator that I brought was a text generator that talks only about forms of expression. It’s not AI. It’s not an attempt to embed Rav Kook’s mode of thought. There still weren’t machines like that then—or at least they were in their infancy—so there it was simply generating combinations of words characteristic of Rav Kook, but there was no idea behind it. If today you gave OpenAI—there are already such things today—generators of Rav Kook texts produced by AI, my claim would not be correct. Because there, if you derived conclusions, it is definitely possible that those are conclusions that fit Rav Kook’s way of thinking.
[Speaker D] What do you mean? But wait, wait—that’s already a huge novelty. You mean the Rabbi is saying that if I now give a text to AI, no matter by which late rabbi, I can learn novel ideas from it from that rabbi? Oh! How? How can it be that AI comes up with new things for me?
[Rabbi Michael Abraham] Think in a very extreme way. AI still doesn’t know how to do this yet, but in a very extreme way, suppose I trained the AI on a huge, huge number of texts by that same rabbi, okay?
[Speaker D] And I—
[Rabbi Michael Abraham] I’m not talking about word combinations like the random generator. With AI I train it to be like Rav Kook. Suppose I trained it on many examples. In the end, the AI is built like Rav Kook’s brain; it duplicated Rav Kook’s brain. Okay? Now it encounters a new problem that Rav Kook didn’t get around to dealing with, okay? So Rav Kook’s brain handles this new problem, and that is what Rav Kook himself would have produced regarding that new problem. So there is no problem at all learning from here Rav Kook’s view on the new problem. Of course, if the AI were perfect and knew how to do this in the best possible way. For now it still doesn’t know, but it is already getting closer. And therefore today, if AI did this in that same experiment, what I said there would not be correct—and by the way I noted that there. I’m talking about random. A random generator is not the same thing. It’s not AI. It uses AI software in order to generate random text; it does not imitate the mode of thought, it imitates the mode of expression.
[Speaker D] Yes, yes. No, but actually this is a very interesting thing, meaning—
[Rabbi Michael Abraham] You can take it the way Kopel is working today on texts—on programs that will issue halakhic rulings. Give it a new case, and it will rule for you.
[Speaker D] Based on what?
[Rabbi Michael Abraham] On the Talmud, on halakhic decisors. Yes, on halakhic decisors. He feeds into it everything people study when they study Jewish law. He feeds that into the AI, so the AI also learns Jewish law. And after it finishes learning, now a new case is brought before it, and now it has to issue a halakhic ruling on that case.
[Speaker D] So that really touches on today’s lesson. Meaning, it relies on all the cases it learned, and they bring it a new case that doesn’t exist—
[Rabbi Michael Abraham] And it—
[Speaker D] draws the conclusions on its own.
[Rabbi Michael Abraham] Right, right. Those are projects that Kopel is working on now.
[Speaker D] Working on now. Would it be possible to rely on that according to Jewish law? Honestly, that sounds to me—it sounds more reasonable to rely on that than on a rabbi.
[Rabbi Michael Abraham] There is no reason not to rely on it. By the way, I wrote a column about this—issuing halakhic rulings by AI—where I spoke about, I had a series on the differences between AI and human thought, and among other things I discussed issuing halakhic rulings by AI. Look for it on the site; if you don’t find it, I’ll send it to you.
[Speaker D] Meaning, unlike a human being, AI is one hundred percent objective—meaning it won’t be influenced by anything.
[Rabbi Michael Abraham] That can be a disadvantage or an advantage, but yes, there can be differences. Yes.
[Speaker D] Wow, okay, thank you.
[Rabbi Michael Abraham] Okay then, goodbye, good night.
[Speaker D] Good night, Sabbath peace.
[Speaker B] Good night.