Defining Concepts 2

Table of Contents

• [0:00] The distinction between a constitutive definition and a directive definition
• [2:07] The philosophy of soccer – what is a foul?
• [3:52] Definition in geometry – is there punishment?
• [6:51] Divorce and the halakhic / of Jewish law definition of the commandment of separation
• [10:31] Kiddushin and marriage – natural and legal stages
• [14:04] The get and its effect on the natural dissolution of the family
• [24:02] Punishment as an indicator of a legal system
• [26:14] Is it permitted to substitute a consecrated offering?
• [27:50] The discussion of if one acted, it is ineffective and what it is aiming at
• [29:00] Punishment for the sin of rebelling against the command
• [31:31] Definition versus claim – a philosophical perspective

Summary

General Overview

The text establishes a basic distinction between systems of laws and definitions that constitute the domain itself, and those that direct conduct within a domain that exists even without the laws, and it argues that confusion between these two types creates philosophical and halakhic / of Jewish law errors. It illustrates the distinction through chess versus soccer, through the commandment of tzitzit and divorce, and through the sugya of if one acted, it is ineffective and the world of punishment. It argues that in everyday and value-laden contexts, many definitions are actually claims about an existing idea, and therefore truth and falsehood apply to them and disagreement about definitions is meaningful, whereas mathematics fosters an image of definitions as arbitrary and constitutive, while “sweeping” the transition from intuition to the formal world into the definition itself.

Constitutive Definition and Directive Definition

The text distinguishes between a constitutive legal system that defines the domain such that deviating from the rules means leaving the domain, and a directive legal system in which one can remain within the domain even when deviating from the rules, only one is acting improperly. It presents chess as a constitutive system, because someone who acts against the rules is not a “lawbreaker” but simply is not playing chess, while it presents fields like traffic law as ones whose rules direct conduct within an already existing domain. It ties the distinction to the concept of an offense and suggests that the presence of punishment mechanisms and rules for violations sometimes serves as an indication that this is a directive rather than a constitutive system.

Fouls in Sports Games versus Constitutive Games

The text presents soccer as an example of a system in which fouls are part of the tactic, and the game continues while the rules themselves determine what happens to the one who committed the foul. It states that in such a situation, deviation from the rule does not remove one from the domain but activates additional rules within the same domain, and therefore the rules do not constitute “soccer” in the same way chess constitutes itself. It emphasizes that in chess “there is no such thing as a foul,” because a move contrary to the rules of the game creates a different game, not a foul within the game.

Halakhic / of Jewish law Examples: Tzitzit as a Directive System

The text cites the verse, “And you shall place on the corner a thread of tekhelet, and it shall be for you tzitzit,” and interprets “and it shall be for you tzitzit” as an instruction directing a certain use as a symbol, not as a constitutive definition that exhausts the concept of tzitzit. It argues that such a statement makes more sense if tzitzit is a broader concept of “symbol” that could have been expressed in different ways, and the Torah directs which symbol to choose. It cites Ibn Ezra, who interprets tzitzit as connected to “the lock of the head” and to “a banner to be raised,” that is, as something that marks and expresses, and from this concludes that Jewish law here functions as a regulatory system that directs rather than one that constitutes the very concept.

Sefer HaChinukh, Divorce, and the Existence of a Natural Layer alongside a Halakhic / of Jewish law Layer

The text cites Sefer HaChinukh in the portion of Ki Tetze, which describes divorce as a commandment and speaks of someone who divorced “not according to the rules” as one who has nullified a positive commandment and whose punishment is severe, and presents this as a sign that the laws of divorce are understood there as directive rather than constitutive. It proposes a model in which building a family unit takes place in two stages: kiddushin as a prior legal layer, and marriage as a natural bond of shared life. It explains, in the name of Maimonides at the opening of the Laws of Marriage, that before the giving of the Torah a person “would meet a woman in the marketplace” and bring her into his home. It argues that the Torah adds a layer rather than replacing the natural one, and therefore there can be a reality in which a natural bond is dissolved without dissolving the kiddushin—a situation in which the woman is “divorced” in one sense but “still forbidden forever” in another sense.

“Divorced in the Heart,” the Ketubah, and Financial Rights in a Natural Separation without a Get

The text describes a situation of “divorced in the heart,” in which a couple has stopped living together but the get has not yet been given, and explains that various obligations can lapse even without a get, while the prohibition of a married woman remains until the get is given. It quotes, “From the moment he set his mind to divorce her, he no longer has rights to her produce,” and adds that “from the moment he set his mind to divorce her, it is forbidden to maintain marital relations.” It notes that this was attributed to Tosafot in Bava Batra and to the Rashash, and that from that moment the husband no longer inherits her. It presents a distinction between monetary laws that depend on marriage and laws of prohibition that depend on kiddushin, and accordingly argues that the act of “and he shall send her from his house” belongs to the natural layer, while the get belongs to the legal layer.

A Possible Symmetry in the Woman’s Initiative to Dissolve the Relationship and the Claim “He Is Repulsive to Me”

The text suggests the possibility that on the natural plane the marital bond may be more symmetrical, so that even if the woman leaves and dissolves the natural plane, a positive obligation is created upon the husband to send her away and give a get in order “to permit the woman” and not leave her chained. It presents this as a “revolutionary” direction in relation to the accepted halakhic / of Jewish law conception, and sets it against the halakhic / of Jewish law complexity of the claim “he is repulsive to me” and the question whether that is grounds for coercing a get, while stating that “today we do not coerce a get over this.” It qualifies this by saying that part of the picture is his own construction based on indications from halakhic decisors, and that he does not have clear proofs for every case.

Punishment as an Indicator, and the Reservation about It

The text argues that punishment sometimes serves as a sign that the laws are directive rather than constitutive, because in a constitutive system deviation simply removes one from the domain and makes punishment unnecessary, but it adds that this indication is not binding. It suggests that even in a constitutive system there could be punishment if “there is an obligation to play,” so that the punishment would not be for a foul within the game but for failing to participate in the required game. It distinguishes between a real “punishment” and a borrowed use in Monopoly, where “remaining in place for one turn” is not a punishment for deviation but a game rule of a quasi-random nature.

The Sugya of Temurah: If one acted, it is ineffective and lashes for “because he violated the utterance of the Merciful One”

The text uses the dispute between Abaye and Rava at the beginning of tractate Temurah to illustrate the difference between a legal result and an offense as defiance of a command. It describes the prohibition of substitution and the verse, “Then both it and its substitute shall be holy,” as a sanction in which the attempt to transfer sanctity causes both animals to become holy, and presents the discussion whether, when the Torah forbids an act, it nevertheless takes effect (if one acted, it is effective) or does not take effect (if one acted, it is ineffective). It brings the question, “For what does he receive lashes?” according to if one acted, it is ineffective, and Abaye’s answer that he is flogged “because he violated the utterance of the Merciful One”—meaning for the attempt and the rebellion against the command, even if the act did not take effect—and points to the conceptual implications for the question of degrees of punishment if punishment is not result-based.

The Fundamental Claim: Many Definitions Are Claims, and Therefore There Is Truth and Falsehood about Definitions

The text opposes the view that a definition is necessarily arbitrary and constitutive and therefore cannot be “correct” or “incorrect,” and argues that in many cases a definition is regulatory and directs us toward something that exists outside it. It points to the phenomenon that people say things like “the definition of good” or “you are mistaken in the definition,” and concludes that this shows a definition functioning as a claim about an idea, not merely as a linguistic stipulation. It sets up the difference between mathematics, which sharply separates definitions from claims, and everyday thought, in which that distinction is misleading because the definition is a “hidden claim” about essence.

The Example of Democracy: A Dispute That Is Not Dictionary-Based

The text presents disputes about a “democratic state” as an example showing that the argument is not necessarily a dictionary dispute about how to use a word, but a dispute about a value and about what is included in the idea of democracy. It suggests that if the definition were constitutive, the disagreement could easily be resolved by splitting into different concepts and different names, but in practice the feeling is that there is one thing being argued about. It states that shifting to the definition of “the value of democracy” only moves one step backward and does not solve the fact that there is a real dispute here about essence.

Who Is a Jew: A Dispute That Does Not Boil Down to Rights over the Use of a Word

The text argues that the dispute over “who is a Jew” should not exist if definitions are constitutive, because one could simply “part as friends” on a semantic basis and give different names to different concepts. It describes the halakhic / of Jewish law definition—“someone born to a Jewish mother or who converted according to Jewish law”—as suffering from circularity that requires “initial conditions,” and suggests a wording that begins from Sarah, Rachel, and Leah and so on in order to close the recursion. It tells how he came to understand, through the storm surrounding Rabbi Shach’s “speech about the rabbits,” that the dispute touches “the deepest nerve of the soul,” because both sides assume that there is one idea of what a Jew is and struggle over who is its authentic continuation, and not because of sentimental attachment to the word alone.

Morality versus “Pusari”: The Impossibility of Parting Semantically

The text gives the example of an argument with “the Eskimos” about attitudes toward the elderly, in order to illustrate that a moral dispute does not break down into two separate concepts called morality and “pusari.” It argues that a moral dispute includes claims of right and wrong, righteous and wicked, and therefore presupposes that the definition is aiming at some truth rather than constituting something arbitrarily. It explains that the structure is similar to if one acted, it is ineffective in the sense of “there is an obligation to play,” so that one who deviates is not simply a player in a different game, but is perceived as acting improperly within a general obligation.

Conventionalism versus Essentialism, and Definition as Observation

The text links a constitutive definition to a conventionalist conception of concepts as products of agreement, and links a directive definition to an essentialist conception in which a concept has an essence that is exposed through observation and description. It argues that when sides are actually disputing, the very dispute is an indication that they assume there is “such a thing” in reality or in the realm of ideas, and the definition is trying to hit upon it. It clarifies that definitions too can be made up of “or” and “and,” and illustrates this through a halakhic / of Jewish law inquiry about monetary damages and the question why not define liability as a combination of ownership and negligence together.

Mathematics, Constitutive Definitions, and the Claim that the Mathematician “Sweeps Dust under the Rug”

The text argues that the public perception that a definition is a technical and constitutive matter is influenced by mathematics, in which definitions are perceived as arbitrary and not open to value-laden dispute. It describes an axiomatic system in which there are definitions, axioms, derived claims, and rules of inference, and argues that this model is useful in mathematics but misleading in life. It suggests that mathematics does not persuade us about the world but about “a Platonic world,” because the transition from intuitions about the world to formal definitions is not decided mathematically but is pushed into the definition.

The Example of Convexity: A Topological Theorem and a Proof Based on Changing the Definition

The text presents a theorem: the intersection of two convex shapes is convex, and emphasizes that despite how intuitive it seems, mathematicians demand a rigorous proof because of fear of pathological cases. It defines convex as follows: every two points in the shape are connected by a straight line segment that lies entirely within the shape, and then the proof becomes immediate, because every such segment lies in each of the shapes and therefore also in the intersection. It argues that this is “cheating” in the sense that the real difficulty is justifying that the formal definition really captures the intuitive concept in the world, and mathematics “sweeps” that difficulty into the definition and proceeds from there with absolute certainty.

The Attitude toward Mathematics: Cheating as the Art of Isolating Certainty

The text presents the mathematician as someone who gives up empirical questions in order to preserve the purity of certainty, leaving it to physicists and other scientists to determine whether the axioms and definitions fit the world. It argues that mathematical genius lies in the ability to perform a kind of sorting work that extracts the certain parts and intelligently pushes aside the empirical “dirt.” It concludes that the mathematical approach to definition as constitutive is extremely useful within mathematics, but when it seeps into everyday thinking it conceals the fact that many definitions are directive, dependent on ideas, and open to real dispute.

We’re dealing with definitions. Last time, a little bit, I don’t even know how to summarize what we did, because it was some kind of general look, with all sorts of aspects, at this issue of definitions. I want to take from that what I need for what comes next. Basically, first of all, I want to distinguish between a constitutive definition and a directive definition. In analytic philosophy they distinguish between directive and constitutive systems of rules, regulative and constitutive systems of rules. What does that mean? It’s like Newton—right, exactly. So what? Right, chess is basically a constitutive system of rules, because it’s a system that defines the domain. It isn’t imposed on the domain—they don’t place it on the domain—but rather it defines the domain. Someone who deviates from the rules of chess isn’t a criminal; he’s simply not playing chess, he’s playing something else. So that’s all—it’s a matter of definition, not something imposed on some other thing. By contrast, say, the laws of physics or traffic laws—some say, although there you can argue a bit—those are directive systems. And directive systems in the sense that, say, someone who deviates from them—it’s not that he’s no longer driving on the road; he’s driving on the road, just not properly. Okay? It’s a little hard to deviate from the laws of physics, so maybe that example doesn’t help so much here. But the claim is that the laws do not define the domain. Those laws—you’re still in the domain even if you deviate from them; it’s just that you deviated. There are rules for how to do the… how to conduct yourself in that domain. Once I was at a very interesting lecture on the philosophy of soccer. So the main question he dealt with—it was a kind of philosophy-on-the-bar thing, they did philosophy lectures in bars, actually nice, some friends invited us to Tel Aviv. So he gave a lecture there, and one of the central discussions they had there—it was two guys from the Open University—they discussed the question: what is a foul in soccer? You commit a foul—after all, today, and I know basketball better, I’m more connected to basketball—but you make a plan for when it’s right to commit a foul and when not. It’s not that you committed a foul, tried to get away with it, and got caught. You didn’t get caught—there are fouls you commit as part of your tactic. Meaning, there are situations where you need to stop him with a foul, and you pay the price. You commit a foul and pay the price. Here you came right in the middle, Menachem. Which soccer? We’re talking about philosophy of soccer now. What do you say? So what is this thing? Is that foul a deviation from the rules? So what—when you committed the foul, are you still playing soccer? The assumption is yes; obviously. Nobody says, okay, game over, now we’ve switched to playing—I don’t know what—something entirely different, another game. It’s soccer; you deviated from the rules, but the rules themselves also relate to deviations from the rules. Meaning, part of the rules of soccer is what happens to someone who committed a foul. So what does that mean? That this foul is a deviation from rules that do not constitute the domain. Because if they did constitute the domain, then the moment you deviated, you wouldn’t be playing soccer—you’d be playing something else. Meaning, there’s no point in punishing someone who deviates from the rules of geometry, right? Meaning, you assume that two parallel lines do meet. So will you get punished? No. You’re just not doing geometry—or at least not Euclidean geometry. Okay? You can write a doctorate on other algebras and even make a name for yourself. Right—but Euclidean geometry you’re not doing. So the difference—the point—is whether this system of rules constitutes the domain, so that if you deviate from it you’re simply outside the domain, or whether it merely directs the domain, but even if you don’t behave according to it, you’re still in the domain.

In roughly the same context, there are several halakhic / of Jewish law aspects in which one can ask a similar question: does Jewish law constitute this domain, or does it direct this domain? For example, the verses speak about the commandment of tzitzit. So it says, after describing what should be there, “and they shall place on the corner a thread of blue, and it shall be to you as tzitzit.” What does “and it shall be to you as tzitzit” mean? Simply speaking, what it means is that what I’ve described until now—that is what you should use as tzitzit, and not something else. Do you hear the tune? The tune is that it’s directive, not constitutive, right? Not every collection of strings is tzitzit? No—the opposite, the opposite. The meaning is that any collection of threads could have been tzitzit; I just want you to do it this way. Meaning, this is what shall be your tzitzit, not something else. Meaning, if the laws of how to make tzitzit were constitutive laws and not directive ones, then you couldn’t say a sentence like “and it shall be to you as tzitzit.” That’s the definition of tzitzit, obviously. You can’t command me: make the tzitzit this way and not otherwise. You need to tell me: make tzitzit. That’s the definition of tzitzit; something else is not tzitzit. When they say to me, “and it shall be to you as tzitzit,” it means that this construction—you should use this as tzitzit, not something else. Which implies that something else also could have been tzitzit, except that the Torah forbids it. The Torah says: not that one, do this one. In fact, for example, Ibn Ezra explains what tzitzit is. It’s like the forelock of the head, a banner to be raised—a kind of thing that marks you, something that highlights you or expresses you, some sort of symbol. So if that’s what tzitzit means, then clearly tzitzit could also have been other things. You could have chosen other symbols that would symbolize you, and you chose—or rather, you have to choose—this symbol specifically, because the Torah’s description directs the concept of symbol. You could have chosen a different symbol; even if you had chosen a different symbol, it still would have been a symbol. The Torah wanted this kind of symbol and not another. So here, basically, the assumption is that the system of rules is a directive system and not a constitutive one—regulative.

Maybe another example: there’s Sefer HaChinukh in the portion of Ki Teitzei. He talks about the important commandment of divorce—that it is a commandment for a man to divorce his wife with a bill of divorce. At the end of the commandment he says: “And one who violates this and divorces his wife not according to these rules has nullified this positive commandment, and his punishment is great.” So it’s a little surprising to read such a sentence when talking about divorce. Usually the common view regarding divorce is that divorce is a procedure; it isn’t a commandment. Meaning, if you did what in yeshiva language they call a din, that’s not a commandment. Meaning, divorce is a definition. Someone who wants to dismantle a family unit—this is how it’s done. If he doesn’t do it this way, has he committed an offense? No—he simply didn’t dismantle the unit. Meaning, the way to separate a couple is in this way. If you didn’t do this, you’re still married. Meaning, it’s not that you’re a criminal deserving punishment for deviating from the rules. You see, this is exactly the same question: are the laws of divorce constitutive or directive? Now, in HaChinukh it says that the laws of divorce are directive; they are not constitutive. You can divorce in another way too—just as you could have chosen different tzitzit, not the blue and white and the number of strings and everything the Torah defines. You could have done divorce in another way; the Torah says it has to be done this way, and if you did it differently, then you’re a criminal. Meaning, but you still did divorce—just differently—so you’re a criminal. And would the divorce take effect? In principle, yes. Maybe I’ll add one sentence just to complete the picture, although that’s not our topic—it’s only an example—but in principle yes. Meaning, if this were a constitutive system of rules, then if you didn’t do it this way, it’s not that you played the game incorrectly. Like chess—if you don’t play by the rules, you simply aren’t playing chess, you’re playing another game. There’s no reason to punish you for such a thing. Meaning, if someone wants to play a different game, that’s his right. Okay? So when the system of rules is constitutive, every deviation from it simply means that you’re not in the domain, because the domain is constituted by that system of rules. So punishment is the sign that it’s directive, basically. What? So punishment is the sign that it’s directive and not constitutive. Right, right. Because if it were defining the concept of divorce, then if you didn’t do it this way there simply is no divorce here. Why punish you? What offense did you commit? You just didn’t separate. You wanted to separate, you didn’t separate; you failed to separate. There are many commandments that look like procedural commandments; some really are, and in the commandment of divorce, at least according to HaChinukh, not so.

So here I’ll just add some comment on what Ido said earlier. Okay, so what does that mean? That it’s possible to divorce without a bill of divorce? Is she divorced? And the answer is yes. She is divorced, but she is still forbidden forever. What does that mean? When we build—yes—when you build a family unit, it is done in two stages. Right? First betrothal and then marriage. Maimonides opens the laws of marriage: before the giving of the Torah, a man would meet a woman in the marketplace, bring her into his home, and they would live together. That is called marriage—the natural situation. And then the Torah came and added a formal legal-halakhic layer. You first have to do betrothal before you do the regular, natural act, before you start living together. Before you take her in—betrothal. So betrothal is a preliminary act; taking her in is the natural act. So you need to do that first. But why is Maimonides telling me this historical story? A man would meet a woman in the marketplace and… okay, we got it. History—once it was like that. He could also tell us at the beginning of the laws of tzitzit: once a person would walk around with a four-cornered garment and didn’t need to put anything on it, and now he needs to put tzitzit on it because the Torah said to put tzitzit on it. He doesn’t write that. Why doesn’t he write that? It seems to me that what Maimonides wants to say here is that the natural concept of betrothal and divorce—that which existed before the Torah was given—is a concept that still exists today. The Torah came to add something, not to replace it. It says: in addition to this natural concept, I’m adding another layer, another story. A prior story, actually—it adds it as a preface. Before marriage, you need to do betrothal. So what does that mean? If, for example, we live together without betrothal, you can’t say we’re not a married couple. We are a couple. The word “married” may already be inaccurate. We are a couple. We just didn’t fulfill what the Torah wanted; we didn’t create the first legal layer of betrothal. But we are a couple. The concepts of common-law spouse, Noahide marriage, all kinds of things that have come up in recent years, when the religious court took jurisdiction over civil marriages, to define it as Noahide marriage—it seems to me that the concept behind this, in my opinion—they don’t define it this way, and I think they’re mistaken about that, because that’s the right way to define it. No—they claim that after a Noahide marriage you need a bill of divorce. I claim that you don’t need a bill of divorce. Noahide marriage means you are a couple. But that doesn’t mean… you’re not a couple in the full sense, because you didn’t do betrothal. The bill of divorce dissolves the betrothal, not the marriage. What dissolves the marriage? “And he shall send her from his house.” Meaning, as they used to do. When there was only marriage, they started living together, agreed, started living together, and they are a married couple. How is that dissolved? They agree to stop living together, he sends her out of his house, and they separate. But then is she permitted? If she was married only by marriage without betrothal, then there’s a question what her status is. There is no halakhic prohibition, no capital liability and all that. That all belongs to betrothal. Without betrothal, you don’t have that. But yes, she is forbidden just as a Noahide’s wife is forbidden to someone else. From the Noahide aspect, even a Jewish man’s wife has some Noahide aspect. Every Jew is also a little Noahide, and he has a second story in which he is also a Jew. Okay? So the Noahide layer remains. The Torah did not replace it; the Torah added another layer. And the idea is—this is what I’m saying—that therefore this is called directive and not constitutive in the sense that the Torah did not replace that layer with a second layer that defines. The Torah directs how to do marriage properly. It could have been done differently too. It doesn’t define the domain; it directs it. The Torah says: do it this way and not another way.

Now there are many laws that depend on marriage and not on betrothal. And those laws are basically grounded in the natural bond, not the legal bond. All the obligations of the ketubah and the like—those are obligations that depend on marriage, not on betrothal. Therefore, for example, what happens in a situation—I’m saying, when we build the marital unit we need to build it in two stages: first betrothal and then marriage. When we dismantle the marital unit, we also dismantle it in two stages. When you give a bill of divorce, it dissolves both stages together. That is how divorce is usually done. But what would happen, for example, if someone sends his wife out of his house and did not give her a bill of divorce? Then he has dismantled the family unit in its natural sense. They no longer live together, they are no longer a couple in the natural sense. But the legal dimension of betrothal still remains. That he has not yet dissolved. Because something that was built with legal tools, you dissolve with legal tools. That belongs to the legal sphere, not the natural sphere. The natural sphere is living together and stopping living together. The legal sphere tells me, directs me, how to do it. So it says: you begin living together with betrothal, and you stop living together with a bill of divorce. So what will happen with someone who, in the language of the later authorities (Acharonim), is called a “divorced-in-heart” woman? Someone you decided to divorce, but you haven’t yet given her the bill of divorce, you still haven’t done the… yes. So the obligations of the ketubah, at least according to certain views—on this I once wrote an article and showed this—they lapse. She is a married woman, she is forbidden forever, she still needs a bill of divorce. Without it she may not remarry. But the ketubah obligations lapse. Even Torah-level obligations, by the way. To inherit her, to become impure for her, various things like that, a husband’s inheritance—and a husband’s inheritance, is that a question even if he sent her out of his house? That’s exactly what we’re talking about, exactly that, only he can send her out of the house, yes. No, so why if—why does she not have a ketubah? Because she didn’t receive a bill of divorce. No, no—not that she doesn’t have a ketubah, but that the ketubah obligations, meaning maintenance and these things and conjugal obligations. From the moment he set his mind to divorce her—the Talmud says in tractate Gittin—from the moment he set his mind to divorce her, he no longer has rights to the produce. Her usufruct property, yes, property she brings with her, and the produce belongs to the husband, while the principal remains hers but the produce belongs to the husband so long as they are together, yes? The moment he set his mind to divorce her—he did not give a bill of divorce, they are still a married couple at the halakhic level—he no longer has rights to the produce. From the moment he set his mind to divorce her, marital relations are forbidden. That’s in the Talmud, that’s a Talmudic law. Rashash and Tosafot—there is a Tosafot in Bava Batra—say that he does not inherit her. From the moment he set his mind to divorce her, regardless of the bill of divorce—there’s no bill of divorce, they are a married couple. So are you leaving force to betrothal and the bill of divorce only for the purpose of permitting her to someone else, or not? All the other monetary laws—almost all of them—apply. More or less. So then betrothal has basically become something that concerns only the issue of prohibition. Right, right, right. That’s the halakhic aspect.

Why am I bringing this up? I started with HaChinukh, who says that someone who divorces his wife not according to the rules has nullified this positive commandment and his punishment is great. Now we can understand that. Why—what does it mean that he nullified this positive commandment? I said: if it constitutes the domain, then if you didn’t do it according to the rules, she simply isn’t divorced. What positive commandment did you nullify? It’s a procedure; there’s no… right, wrong. If you dismantle the family unit in the natural sense, meaning you decided not to live together, there is a commandment to release the woman. Meaning, to dissolve the betrothal as well. Don’t leave her chained, what today is called an agunah, or forbidden to you, forbidden to the world, tied to you. Yes—there is a commandment upon you to send her away, to give her her life. If you are no longer living together, if you decided not to be a couple in the Noahide sense, to stop living together, you don’t want to be a couple, then the Torah says what must be done. And one who did it without the bill of divorce nullified a positive commandment and his punishment is great. Obviously there is no commandment to divorce a woman. But if you are dismantling the marital unit, this is how it is done, and if you did not do it this way—you dismantled the marital unit but didn’t do it this way—then that is a nullification of a positive commandment. Yes. And the natural bond lapses the moment he sends her away. And if she left on her own? No—it could be that then too it lapses. The asymmetry between husband and wife—about that I don’t know, because all this is my own invention. I don’t know. But the asymmetry between husband and wife might belong only to the halakhic world. He performs the betrothal, he gives the divorce, the bill of divorce. Because after all, even in betrothal you also need the woman’s consent, right? You can’t betroth a woman without her consent. And in divorce it’s different. But maybe in the natural unit it’s all by consent. On the natural plane it could be that everything is by consent—I don’t know. Like among Noahides. What happens among Noahides? I assume that among Noahides too, so long as the Torah does not define that there has to be asymmetry—and if it defines it in the halakhic plane, then maybe it’s only there. So long as there is no other source, why assume there is no symmetry here? It may be completely symmetrical. And then, on the contrary, it becomes very interesting. Because according to this, and this is a not-simple halakhic question, according to this, what happens if the woman decides to leave? She doesn’t want to, she leaves. Then she has dismantled the natural plane. But now a positive commandment has arisen to divorce her. But divorce in the halakhic sense is only on the husband. The woman cannot divorce the husband. Giving a bill of divorce is only the husband giving to the woman. And then it comes out that, in effect, a kind of symmetry is created here. Not completely, but more symmetrical than the usual way people think. Since, basically, the woman too can compel a divorce. She just leaves the house and no longer wants to live with him. Once that is the case, now we, as a religious court, if he doesn’t divorce her, he has to divorce her. In principle, that is his obligation. If he doesn’t do it, we are supposed to see to it that he does it. Because now a positive commandment lies upon him to divorce her, once the marital unit has broken apart. And if she too can dissolve it, just like he can, then it’s dissolved. Fine—that actually, to modern ears, I think sounds better. This thing is somewhat more symmetrical than is commonly accepted.

There are several situations where he has a commandment to divorce her, and it depends specifically on her. Right. If she doesn’t act according to Jewish practice and things like that. She can, in quotation marks, compel him to give a divorce. “Doesn’t act according to Jewish practice” isn’t compelling him for her sake, but the opposite—it’s his duty, not for her. But she created the situation. She created the situation, but she committed offenses. That’s not a tool she can play with in order to force divorce. Here I’m saying there is a legitimate tool to force divorce. Not to be a criminal for that—a legitimate tool. She doesn’t want to live with him. That’s very revolutionary in the halakhic sense. Because in the halakhic sense, then she has no ketubah. So she won’t receive the ketubah—yes, okay, no, I’m not speaking in that sense; she’ll lose the ketubah or lose… offenses. She committed offenses—what do you mean? A woman doesn’t want to commit offenses. She’s God-fearing. Even if she leaves, she’s a rebellious wife. If she just got up, as you’re saying, and decided to leave. Fine—that’s exactly the question. So she creates the… yes, she obligates him to give her a bill of divorce. Right, that’s what I’m saying. But in the accepted halakhic conception, it’s not so simple. She says, “He is repulsive to me.” Yes—with claims like “he is repulsive to me,” “I don’t want to live with him.” Whether that is grounds for compelling a divorce or not grounds for compelling a divorce is an extremely complex question. Today they don’t compel a divorce on that basis. It’s a matter—an extremely complex question. Meaning, if you look at it through these lenses, you would have to compel a divorce in such a case, because basically, once… after all, the whole idea of betrothal and the bill of divorce only accompanies the natural process; it does not constitute. That’s why I entered into this example, and I’m getting too involved in it. Meaning, I brought it as an example of a situation in which Jewish law directs life and does not constitute it. In the yeshiva conception or the accepted halakhic conception, we see Jewish law as a constitutive system of rules, not a directive one. Meaning, when you divorce a woman in this way, then she is divorced, and if not, then simply nothing happened—you did nothing, that’s all. There is no such thing as divorcing a woman without giving a bill of divorce, because giving the bill of divorce defines what divorce is. What I’m saying now is not correct, according to that view. There is such a thing as divorcing a woman—stopping living together, meaning sending her away. Literally, yes, sending her away. The Torah tells us how to do it correctly. Meaning, the system of rules is regulative, not constitutive. It directs; it does not constitute. Okay?

But from the moment you say, suppose the woman left, the question—again I return—is what monetary powers or rights exist, because there it does constitute. If she is entitled to all the monetary rights she has—those monetary rights are all on the rabbinic plane; that is the ketubah. So we need to discuss on what these rights depend. Halakhically, that’s not a simple question. I’m saying again: this whole picture—I have indications for it from many halakhic decisors, early and late, but this is a picture I built. So I don’t know, I don’t have proofs; I can try to think what the logic says, but I have no clear proofs about what happens in every situation. I don’t know. I can tell you what I think should be, but it’s not—no, we’ll go with you. Okay, okay. Let’s see where we get. How should it be—meaning, the agreement of both spouses? If the woman leaves, what do they agree on in the rabbinical court? Ah, what do they do in practice in the rabbinical court. Fine, the question is whether I agree with them, but yes, you’re saying that’s what happens. Okay, fine.

In any event, this distinction between constitutive and directive basically also applies in the world of definitions. In the world of definitions too, one can relate to a definition as something constitutive or as something directive. These systems of rules too are basically some sort of definition. Someone who sees them as constitutive, as a constitutive system of rules, says: this is how the domain is defined. Say you play the game of chess—these and these are the rules. What does that mean? These rules are basically the definition of the game, right? That’s the definition of the game. If you play by other rules, you’re playing a different game. That’s the definition of the game. And one can see the rules as something directive, not something constitutive. Then what does that mean? In the ordinary view they would say, fine, then that’s not a definition. Definition is by essence something constitutive, not something directive. If it’s something directive, then that’s not the definition, because you can still play even if you deviate from it—you’re still playing. You’re just not playing by the rules. You committed a foul, like in soccer, where the system of rules is apparently more directive, unlike chess. In chess there is no such thing as a foul. There is no such concept. Think about it. There are games in which there is a foul, and games in which there is no foul. What is a foul? If someone playing chess moves a pawn like a knight, what do you do with such a person? The other player gets up and says, thank you very much, goodbye. If you want to play another game, fine, but I’m playing chess. Right? In soccer there’s a foul, they blow the whistle, there’s a penalty, I don’t know, you get the ball from the side. Or in basketball you get two free throws. Okay? Meaning, there are rules for what to do with fouls; that means it’s directive and not constitutive. And in a certain sense, as someone said before, punishment is often the indication whether this system is constitutive or directive. Because once the system is constitutive, the concept of punishment loses some of its meaning. What is the punishment? You’re simply not in the game. So maybe there is punishment for… In Jewish law, by the way, I think there are parts that are constitutive and not directive. “Whoever betroths does so according to the mind of the rabbis,” for example; or “if one did it, it is ineffective.” Who says that if someone says it is ineffective there is nevertheless punishment? So that’s what I’m saying. That’s a good example—the dispute of Abaye and Rava there—but it’s a good example. I’m saying it’s an indication, but it doesn’t have to be that way, because there can be punishment on someone who does not play. There is an obligation to play. The punishment is simply another rule, one of the rules. Ah? It’s like Monopoly, where they tell you to stay in one place for a turn. Punishment? That’s not punishment, that’s one of the rules. They call it punishment. There it really isn’t punishment. To stay in place for one turn is not punishment at all. A certain number came up, or you did something specific and they assigned you to a certain place, like prison, and tell you to pay the pot. But there it’s clear that this isn’t punishment in any connotation, because it’s not something given to you because you deviated from the rules; it’s a kind of lottery. You brought up the… yes. No, true, it’s not like in soccer. In soccer it’s within the rules: you committed a foul, you get… the other side gets the ball. But there it’s defined as a foul—you deviated from the rules. In Monopoly the concept is borrowed; it’s not punishment, but rather a bad draw—you drew “go to jail.” But not that you did something that deviated from the rules and they impose punishment on you. It’s not the same thing.

So true, in Jewish law there are punishments for many things. Some of these things maybe are not constitutive; some of these things are not directive but constitutive. As you brought up before regarding “if one did it, it is ineffective,” there are places where—that, by the way, is Rava’s argument against Abaye. Abaye says—what does he get lashes for? He gets lashes “because he violated the statement of the Merciful One,” because he tried. That’s the argument. After all, wait, so what does he get lashes for if… Maybe I need to explain what the topic of “if one did it, it is ineffective” is. Say someone takes an animal, okay? It is sacred, he consecrated an animal as a sacrifice, and now he wants to transfer the sanctity to another animal. Maimonides explains that the motivation is basically that he found a cheaper animal and wants to save money. So he wants to… and this is expensive, don’t make light of it. Bringing a sin offering is not simple—it’s no small financial investment. Right? So he says, come on, fine, this too is a sacrifice, what’s the problem? If I had bought this one from the start, it would have been okay, so I want to transfer, to exchange the sanctity from one to the other. So the Torah says at the end of the portion of Bechukotai: you do not exchange. Okay? It is forbidden to exchange. “And it and its substitute shall both be holy.” What is that? That is a sanction the Torah imposes: if you make a substitution, now both will be holy; you’ve lost them both. Okay? So on this the Talmud discusses, at the beginning of tractate Temurah, which deals with these matters of substitution. There is a dispute between Abaye and Rava on the question whether, if one did it, it is effective, or if one did it, it is ineffective. Meaning, what is the simple assumption in Jewish law? When the Torah says it is forbidden to exchange, if I violated it and did make a substitution, does it take effect? Or does the Torah’s saying that one may not exchange mean that it’s impossible? Meaning, when you do this substitution—it’s actually a good example—that it simply does not take effect; the Torah does not allow it, there is no such procedure. You see that this is exactly the same question. And indeed there—now that you mentioned it—Abaye and Rava are discussing exactly this question. The one who says “if one did it, it is ineffective,” the other asks him: wait, so what does he get lashes for? When you conceive of the system of rules as constitutive, then when he made the substitution and the Torah said not to exchange—but if that is constitutive, it means you failed. You simply did not substitute. It is impossible to substitute. Okay? So what are you getting lashes for? You’re supposed to get lashes for making the substitution. That is exactly what the Talmud says there. And then they innovate there that he gets lashes because “he violated the statement of the Merciful One.” He gets lashes because the Holy One, blessed be He, said not to exchange and he tried to exchange. Not because he exchanged—he didn’t exchange; he tried to exchange. And for that he gets the punishment. And that is a completely different conception of punishment. Then, for example, if someone tried to exchange, or tried, I don’t know, to do something else, to make a divorce not according to the rules or something like that—there is no reason to assume that the punishment would be different, right? Because we are not talking here about the gravity of the offense, because in any case you didn’t succeed. All that happened here is that you rebelled against the command of the Holy One, blessed be He. Rebelling against a command should carry punishment of the same intensity. What difference does it make whether you rebelled against this command or against that one? Why? Because the command defines; it is not severe or lenient. The command defines. If you are talking about the results of what you did, you say there are results that are severe and results that are lenient, so here there is a severe punishment and there a light punishment. But if the whole point is only that you get lashes because you violated the statement of the Merciful One, then you get lashes for disobeying. What difference does it make in what you disobeyed? That’s a big question—why there are differences in punishments according to this conception. Fine. That gets into the whole theory of punishment in Jewish law regarding this issue. But I’m saying: punishment in this conception of “if one did it, it is ineffective” is punishment of someone who does not play. There is an obligation to play. You truly are not playing. If you are not according to the rules, then you simply are not playing. But there is an obligation to play. If you don’t play, for that there is punishment. Not for deviating, for committing a foul as in soccer. The punishment is for not playing, because you need to play. Every Jew is obligated to keep the commandments; he is obligated to play this game.

And what about the fact that betrothal takes effect with those forbidden by prohibitions? Yes, so that is a question raised there. What happens if “if one did it, it is ineffective”? Then why does betrothal take effect with those forbidden by prohibitions? Fine, there are many things they ask there. There’s a question: what happens if you perform betrothal on the Sabbath? It is forbidden to betroth on the Sabbath. So is the woman betrothed or not? She is betrothed. There are so many… in order to find where they actually say “if one did it, it is ineffective,” you need tweezers. There is almost no area where it is really said. All the examples the Talmud brings there, at the beginning of tractate Temurah, are rejected, each for different reasons. Here “if one did it, it is ineffective” doesn’t apply, and here it doesn’t apply. In the end you are left with something that requires special theoretical constructions in order to implement it. But that’s not the point. The idea of “if one did it, it is ineffective” really is, as you said, this idea. But it is not said in most areas; each area is according to its own matter. What does “if one did it…” mean if I murdered someone? Can one say “if one did it, it is ineffective”? He is not murdered because the Torah forbids murder? He is murdered. Right? So would they say here that I get lashes because I violated the statement of the Merciful One, not for the result but only for the rebellion involved in such an act? No—obviously not, right? So that already narrows greatly the offenses to which this applies. It only applies to offenses that are not result-based but behavioral offenses—exactly—or perhaps legal results, where you can say that they do not take effect. Jewish law can determine that they do not take effect. Jewish law does not determine that someone is not dead. If he is dead, then he is dead; that is a biological fact. Jewish law cannot determine here the matter of “if one did it, it is ineffective.”

So apparently, this is how people usually relate to it: a definition is always something constitutive. Meaning, the moment I say “this is the definition,” if I deviate from it then I simply am not in the game. I want to argue otherwise. I argue that usually, in most cases—not always, there are always different examples—but in many cases a definition is something regulative. Meaning, the definition tries to capture something real, not to constitute the thing but to show how it is correct to describe it, how it is correct to behave—or in the case of a definition, how it is correct to describe it. Okay? A definition is basically a claim. What we are very unaccustomed to in philosophical thought and in mathematics is making very sharp distinctions between definitions and claims. Definitions are perceived as something arbitrary. Define it—if you defined it, fine, that’s how we defined it. Anyone who wants to define it differently, good for him; then he is talking about a different concept. It’s worthwhile to synchronize the system of concepts, not to use the same word for different concepts. But a definition is perceived as an act that is arbitrary. Once we agree, that is the definition. Regarding a definition, one cannot say true or false. What is a true definition and what is a false definition? If you defined it this way, that’s the definition. Anyone who wants to define it otherwise, good for him. That is a conception of definition as something constitutive. Definition is something constitutive. It creates the thing—so how could it be incorrect? What I created, I created. And what wasn’t—there is something I am trying to capture? When you tell me that I defined incorrectly, you are basically saying there is some correct definition and we are trying to uncover it—not create it, but uncover it. Then they tell me whether I succeeded in uncovering it or not, whether this is a true definition or an untrue definition. That is basically the assumption that definition is something directive and not constitutive, if I speak in terms of a true definition and an untrue definition, or a good definition and a bad definition. There is no such thing as a good definition and a bad definition according to the constitutive conception. What I defined is the thing. If you want to define it differently, then that will be a different thing; it won’t be this. What is a good definition and a bad one? You understand, there is a certain duality in our relation to definitions. On the one hand we perceive definition as something constitutive, and on the other hand we do sometimes say: you defined it well, or I think the definition is different, you’re a little mistaken here, the definition is a little different. But wait—truth and falsehood are not about definitions. Truth and falsehood are about claims. You make a claim, and the concepts you use should of course already be defined, because a claim is made up of concepts, so the concepts need to be defined, and now you can use them to make claims, or make claims about them. But the definition itself is not a claim. The definition itself is something prior to a claim. But when I say that a definition can be true or false, good or bad, correct or incorrect, that basically means that the definition too is a kind of claim. What does that mean? It means that there are at least some definitions—and I think most are like this—that are directive and not constitutive. They are definitions that, in a certain sense, make a claim. When I define what a democratic state is—we spoke about this last time—when I define what a democratic state is, someone may come and say: look, that definition is not successful. In my opinion a democratic state also needs separation of powers, or without separation of powers, it doesn’t matter, everyone according to his own view. Now what do you mean, “in your opinion”? Fine—define, I don’t know, a plutocratic state without separation of powers, or with it. Make two concepts. What does it mean to say, “No, in my opinion you are wrong”? How can one say such a thing? So there are those who want to say: you were mistaken about the convention. I only want to explain to you how people use the term. When people use the concept “democratic state,” they don’t use it in that meaning but in this meaning. So there still isn’t something true called a democratic state; it’s a definition. It’s a definition we agree upon, and you just didn’t properly grasp our agreement. That’s all; it’s not really something real. But many times that’s not so. Because if, say, several democratic states were to decide to give up separation of powers, already in some significant number, then you could no longer claim that people don’t define a democratic state that way. At that point you would have to begin arguing—not saying, that’s not how people generally define it, you’re just using the wrong dictionary. If the definition is constitutive, then the argument is a dictionary argument. You are basically calling this a democratic state—incorrectly; in the dictionary, that isn’t what answers to the name “democratic state.” So it’s just dictionary. But the feeling behind such arguments doesn’t rest in the realm of the dictionary; rather, we are really arguing about what a democratic state is. Is it fitting for a democratic state to be this way, or not fitting? Should this be included in the definitions or not included? But there is some value that you see before your eyes—you see some value before your eyes, and now you want to describe it. Okay, right, and therefore I’m saying. But if now we are arguing about the definition of a chair, then I say—it may be that this really is what people mean when they say chair. No problem. That’s why I said there may be both definitions of this kind and of that kind. What I’m claiming is that there are also definitions of this kind—definitions that are directive and not constitutive, definitions that try to capture something outside themselves and may succeed or fail in doing so—as opposed to constitutive definitions, which do not capture anything at all but create the thing they speak about.

So that is basically somewhat the same thing, because what do we want to say when people say “democratic state,” when they mean the value behind it—that is what they mean. So it’s also basically like chair. No, so I said—you can relate to it on that plane. I disagree. No, because it’s something real; there is a value behind it that I think is this way, and I can argue. The value of democracy? Yes. Fine, so leave “democratic state”; let’s define the value of democracy—what does it include, what does it require, what are its characteristics. You’re just stepping one stage back; it doesn’t… So those are constitutive and directive definitions. And if there are directive definitions, that means, first of all, that there can also be truth and falsehood regarding a definition, not only regarding claims. Which of course would make a mathematician tear out his hair upon hearing such a thing. What do you mean? That’s the definition. You want a different definition? No problem. Develop a theory based on those concepts, define them properly, call them by some name—not preferably by the name of definitions already in use—and do whatever you want, develop a new field in mathematics or do something else. A mathematician never argues about definitions—he is not supposed to argue about definitions, by the way. Definitions are constitutive for him.

So, first, there is truth and falsehood in directive definitions. Second, of course, there can be arguments. Once there is truth and falsehood, there can be arguments. You will claim that the definition is this way, and I will claim that the definition is that way. Maybe let’s take an example. But that’s actually not so much a definition—it’s a description, not a definition. Okay, I’m claiming that we are speaking about the concept of definition; I’m defining the concept of definition. So I’m saying that when I speak about defining something, sometimes at least, I am basically making a claim, describing something; I’m not defining. Not defining in the constitutive sense in which people usually speak. But still you defined two definitions. Fine, fine, I’m distinguishing between two things, but both really exist in our world, and often we do not distinguish between them. Therefore it is important to put a finger on the fact that there are two things here sheltered under this concept, definition.

Okay, maybe I’ll take an example. Suppose there is an argument about who is a Jew. Okay? There are those who say: one who was born to a Jewish mother or converted according to Jewish law. Of course that is a recursive definition, because what is a Jewish mother? If you don’t have the definition of what a Jew is, if you want to tell me what a Jew is, that won’t help, right? One who was born to a Jewish mother or converted according to Jewish law—that is a circular definition. You are using the concept Jew in defining the concept Jew. So it is clear that this needs to be completed, and the first Jewish mother was Sarah. You need initial conditions. Exactly. There are initial conditions and there is the dynamics. So… Also the Ishmaelites, not… what? Also the Ishmaelites… Fine, never mind—Rachel and Leah, then. Yes, no matter. But let’s start from some point where we place the edge, and then the definition is closed, no problem. If you say: one who was born to Rachel and Sarah—or Rachel and Leah—or converted according to Jewish law is a Jew, and a child born to a Jewish mother is also a Jew. Fine? Now there is no problem, because I defined the concept Jew without needing to use it, and from then on I can use it.

Fine, that’s a satisfactory definition. So how is it that there are disputes over who is a Jew? Or what conversion is. Yes, of course—it’s the same dispute reflected. There are those who hold that the definition of Jew is not halakhic. Okay, then don’t use the word Jew; call it Israeli, call it Yankeleh. You’re just using the same word. That’s not an argument. So let’s part as friends. You’ll call what you want to call Jew; I’ll call what I call Israeli—deliberately I chose the opposite of what we’re used to—and that’s it, and we’ll part as friends. Your definition is acceptable. Definitions are not argued over. No problem. I’m even prepared to define the concept Jew that way, and the concept Israeli I’ll define as I say. Let’s part on a semantic basis. I’ll take something more provocative. But I think there’s really no argument over the definition—the dispute is over the result, over what follows from it. No, there is an argument over the definition, because if they wouldn’t budget them, then they wouldn’t care at all whether they are Jewish. People tell me that a lot; I don’t agree. We’ll return to that in a moment.

Let’s look at something more provocative—a dispute over morality. Okay? We have an argument with the Eskimos. The Eskimos take the elderly out into the snow so they’ll freeze there, right? And okay, we don’t usually do that, for one reason or another. Sometimes we do things even worse to them, but that is perceived as not okay. Okay? You should put them in a nursing home, or care for them at home, or I don’t know what—everyone with his own solutions. Okay? Now, if I see the system of moral laws as a constitutive system, then this argument has no meaning at all. There is no argument. The Eskimo simply is not behaving according to the laws of morality but according to the laws of, I don’t know, schmorality—what difference does it make? It’s only a question of name, so why should I care? When I say that there is an argument between us, what does that mean? It means that the concept morality—the definitions I give it, its characteristics—do not constitute it. They are claims about it. I claim that a moral person should behave this way, and someone who deviates from that—back to the issue of punishment—is not moral. Not immoral in the definitional sense, that he is not defined as a moral person; he is not okay. He is not a player in the moral field, but he is not okay. The Eskimo would say: what’s the problem? I’m fine. I’m not playing soccer; I’m playing chess. What do you want from me? What—must one play soccer? The answer is yes, one must play soccer. Meaning, that’s exactly the point: one cannot part on a semantic basis. And therefore, indeed, when people conduct a moral argument, nobody proposes parting on a semantic basis. So one says, I say it is permissible to murder, to perform abortions, and the other says abortions are forbidden—fine, it doesn’t matter, under certain conditions, some moral dispute, I just took an example. Okay? So one says: fine, then you’re a moral person and I’m a schmoral person, and everything is fine, we have no dispute, we just need definitions. Would anyone say such a thing? This dispute is a heated one. It is not a dispute about definitions. There is right and wrong here, there is wicked and righteous here, there are arguments. What does that mean? It means that we are not talking here about definitions, at least not in the constitutive sense, right? We are not defining the concept moral this way. If it’s a definition, then why should it matter? We are making a claim that in order to be moral you have to behave this way; if you behave that way, you are not moral—not that you are not defined as moral, you are not moral. Okay? So that basically means that the definition here is directive and not constitutive.

Now I claim that regarding “who is a Jew” as well—which is less clear-cut—regarding “who is a Jew” too, the fact that we are arguing—this is where the penny dropped for me, by the way, this was the first time many years ago—the dispute over who is a Jew I never understood. What’s the problem? Let’s define two concepts. You’ll call it Israeli, he’ll call it Jew, doesn’t matter. As far as I’m concerned call it Australian. What difference does it make what you call it? Choose yourselves two words, hold a lottery over who gets which one. If you have sentimental attachment to the word Jew, fine, enjoy. Israeli, by the way, is the more original term for what we today call Jew, and so in my opinion even for religious people…

That too is a definition. One of the definitions, for example, that once existed, even legally, was that a Jew is whoever feels Jewish. Yes, that’s following… we spoke about it once. What? Not Israeli—Jew. That’s Brother Daniel. Yes, I mean, when did it come up more… with Brother Daniel, I think, and it also came up in Rabbi Shach’s speech, the “rabbit speech” of Rabbi Shach. Remember it? Rabbi Shach’s rabbit speech—there was actually a very interesting political crossroads there, the question whether they would join the government or not, party Gimmel, yes, Agudat Yisrael—would they join the government or not? And then everyone was waiting for Rabbi Shach’s speech in Yad Eliyahu, because there he was basically supposed to announce what they were doing, and the coalition depended on it. Okay? So naturally all the cameras in the world—and from abroad too, not only from Israel—everything was there. And then Rabbi Shach began to speak about eating rabbits on Yom Kippur and all kinds of things like that. He didn’t speak about the coalition or anything. He just said, in his own way, that the Labor Party all eat rabbits on Yom Kippur and the kibbutzniks and so on, and everyone understood the political meaning, but he didn’t speak politics; he rebuked the Jewish people: how can you eat rabbits on Yom Kippur? Okay? Now I sat there—not eating rabbits on Yom Kippur, rabbits and desecrating Yom Kippur and all that. So I sat at home, listened to the speech—who didn’t hear it? I didn’t go to Yad Eliyahu as all my friends did, but I listened at home on the radio—and I said to myself: wow, this man is an idiot. He has all the cameras and microphones in the world—raise some argument that really speaks to them. Why do you tell them, “In what sense are these kibbutzniks Jews?” That’s how he says it there, yes, as one could understand; that was Rabbi Shach’s Yiddish-style habit. What? Exactly what Rabbi Keshetiel said about the speech. Ah yes? Okay. So I’m saying that was my feeling when I heard it on the radio. Fine—you define them as non-Jews. They think Jew means something else, or they… what’s the problem? It amused me—all the guys I was with in the army telling me how smooth things were on Yom Kippur and how tasty the rabbit was on Yom Kippur and all kinds of things like that. So you tell them: you eat rabbits, on Yom Kippur, and all that—you’re not Jews. They’d die laughing. Come on, put forward some argument. You’ve got microphones; the whole world is listening to you now.

The next day I understood that a whole wave had begun… of repentance, really. A storm began, unbelievable. A storm began in the kibbutz movement journal, pages and pages, in the publication of the kibbutz movement, and some people began saying: wait, actually he’s right—in what sense are we Jews? What is Jewish about us? That we speak Hebrew? Fine, okay, Belgians speak Belgian—what difference does it make? Just a convention. So what, what is actually Jewish? In all our values we are no different from any gentile whatsoever. And that’s simply true. There is nothing Jewish in our Judaism, right? So suddenly people were kind of being shown a mirror, and a phenomenon was revealed that I had never imagined existed. It was an enormous surprise to me. People began examining themselves: wait, in what sense are we Jews? And the educators shouted at Rabbi Shach, of course, and got angry and so on. And that surprised me too. No less than the straightforward people, as it were, who actually started discussing—maybe he’s right, perhaps what he’s proposing is repentance. But those who rose up against him also surprised me. Why do you care? Fine, so he defines it this way. You know how he defines a Jew. Why do you care? He touched a nerve. Because they claim that what they are doing is what is called being Jewish. What do you mean? What you are doing is not Jewish at all. Serving in the army—and then Chaim Herzog, the president at the time, said: what do you mean? The hands of the kibbutz are calloused from labor and from patrols and the elite units. At that time it was still very clear there and… and how can he say they are not Jews? That too made me collapse with laughter, because what connection is there between patrol units and working in the kibbutz and Judaism? You can say they are good people, you can say they are useful citizens, you can say they are idealists—but what does that have to do with Judaism? What, in American special forces there aren’t good people who go and risk their lives for the establishment of America—or Kuwait, it doesn’t matter; the American army isn’t busy with America. But there is no less self-sacrifice and no less idealism, and in every place there are such people. And calloused from labor—I think we are not exactly the world champions at that. Others also do work. So what is Jewish about that? You can say it is good, but what is the connection to Jew? Maybe those are gentile-calloused hands from labor. Calloused, as opposed to Israeli nationality. So I’m saying, the feeling was that from his point of view this is what is called Jewish.

And then, again, I’m bringing this as a continuation of what I said before. Let’s part as friends: you call it Israeli, I’ll call it Jew. We’re just bickering over concepts—for what? After all, we’re not really arguing. Because then I was in a phase where I thought definitions were something constitutive. Fine, you define Jew this way and he… why define the same word differently? Just choose a different word. These are only definitions, what difference does it make? Why do you care? So I tell you that you’re not a Jew. Right, according to my definition you really aren’t a Jew, but according to your definition you are. So let’s agree between ourselves—you call it Israeli or Jew, and I’ll call it Israeli or Jew, and everything is fine. Then what are the fights about? The fights are because both sides feel that the definitions they are proposing are directive and not constitutive. The concept Jew is one concept and we are all talking about it. And we have an argument over how to define it. There is an argument over the definition, and it is a real argument. Because the definition here is making a claim. The definition here is not merely constitutive. The definition here is making a claim. You are making a claim: who is the authentic continuation of Abraham our forefather? Rabbi Shach claims it is he, and the kibbutzniks, say for the sake of discussion, or Chaim Herzog, claim it is they. By the way, I’m not sure who is closer. I have certain suspicions, which I will not expose here, but I’m not sure who is closer to being the continuation of Abraham our forefather. Truly, there are aspects here—it’s not such a trivial dispute, although it looks very, very simple. And neither side is willing to give up and let the other take the definition away from them. Yes, clearly. But I’m saying, when people, when I talked to people about this, they said: fine, it’s a dispute over the rights to use the word. People have sentimental attachment to the word Jew, that’s all. Obviously that is not true; it’s disingenuous.

When I push you into a corner—because after all you are a conventionalist—we spoke last class about how the constitutive definition is basically based on a conventionalist conception, one based on agreement. Because that basically means that concepts are the product of agreement; they are not the product of observation. Right? Essentialism is basically the conception that says a concept has an essence. And when we define a concept, we are performing an act of observation. I observe this concept in some sense—not with the eyes, but in some sense, okay?—and I describe what I see. The definition is the product of that observation. Therefore the definition makes a claim. The definition describes something; it does not constitute it out of nothing. It describes something that already exists before it. And when we argue over who is a Jew, basically we are assuming—and both sides assume this, and that’s what is nice here. Both sides assume it. Because many times there is one side for whom it is very easy to argue, because he says it’s all convention—so what, I define it differently. It is easy, when you are under attack, to get out of it that way. It’s all convention—I define it this way, you define it that way. But no—here, when you are conducting an argument, that is an indication that on one thing you both agree. That there is such a thing as Jew, this idea of Jew, and you have an argument over how to define it. And this is a real argument; you can’t part semantically. We cannot say: you call yours Jew, I’ll call mine Israeli. Because you are calling a concept that has no correlate in reality, and you call it Israeli. Why should I care what you call it? There is no such concept. And now what do you mean there is no such concept—I agreed and now there is, I defined it. No. A definition describes a concept that exists even before you defined it. You do not make a convention and produce a concept out of nothing. You are trying to hit upon what exists. If no such thing exists, then there isn’t. That concept does not exist. And if they reach agreement that both and both? What? I didn’t understand. Both this is Jewish and that is Jewish? No problem. You can define that. There are disjunctive definitions. Like an argument also about family, yes? Leave it. Or. “Both and both” means you need both this and that, or that this also… No, it’s not “both.” If when you do this you’re Jewish, and also… Or! Right. That could be. Fine, there are “or” definitions, yes? Either you are defined as this or as that. I once thought—and this also relates to our matter—it is somewhat related to the series on dichotomy we once did. The halakhic inquiry regarding damages caused by one’s property. The famous Even HaEzel, who formulates it this way, though even before him people discussed it: am I obligated to pay because my property caused damage—the very fact that my property caused damage obligates me to pay—or is what obligates me to pay negligence in guarding? I was negligent in guarding. Why did you leave this ox here? Why there in Menachem’s place? Possibly, but this is related to the issue of dichotomy that I did speak about here. That was the first or second series, I don’t remember. Maybe both and both? Yes. So there the standard yeshiva-style inquiry is dichotomous. You have to decide: either what creates the obligation to pay is negligence in guarding, or what creates the obligation to pay is the simple fact that your property caused damage. And I always wondered: why not the combination of both together? Either-or or also-and, yes? Either… what creates the obligation to pay is either that you were negligent in guarding or that your property caused damage. Why do you have to choose? Or perhaps what creates the obligation to pay is when both conditions are met—which is actually the truth. Both negligence in guarding and that it is your property. There is no value to a duty of guarding when it is someone else’s property. So it has to be your property, and there also has to be negligence in guarding. Why do you always assume it’s either this or this? But I have no problem with definitions that include within them conjunction as well as disjunction, yes? Or and and. That is included; it is part of the definition; no problem. And there is still room to ask whether that definition is a constitutive definition or a directive definition.

So the meaning of what I’m saying, basically, is that when people say “you don’t argue about definitions”—that isn’t true. You absolutely do argue about definitions. We are constantly arguing about definitions, and we are right to do so, because it is correct to argue about definitions; it isn’t a mistake. Because people conceive of definition incorrectly. Because a definition is… there are definitions that are directive and not constitutive.

What happens in mathematics? That’s what messed up our minds. Even people who don’t engage in mathematics, but the conception of mathematicians is what basically penetrated into the broader layers of the public, so that somehow people who are more or less intellectual kind of understand that definition is a technical matter. Meaning, in mathematics one is very used to defining the concept, and that’s the definition. You want to define it differently? No problem. It will of course have different properties, and act accordingly. It’s just advisable not to use the same word, as I said before, but semantic separation is called for in mathematics. Mathematicians are not supposed to argue over whether this definition is good or not good, or correct or incorrect. You define and see what follows. There are no ideologies there. You want to examine the properties of concepts? Fine—define the concept and derive from that the properties of the concept. Group, space, point, triangle—all kinds of such concepts, which if you define them properly you can derive from that definition all sorts of properties. Can anyone argue whether the definition is correct or not? Define it differently; deal with another concept. No problem—do whatever you want. Meaning, they conceive of it as constitutive. They conceive of it as constitutive. And I think that indeed in mathematics the difference between definitions and claims is very sharp. Mathematics is a cornerstone in this sense. Meaning, every axiomatic system has definitions of concepts, and the definitions claim nothing. They are definitions. And there are the claims. The claims are divided into two: axioms and derived claims, claims derived from the axioms. Okay? And besides that there are definitions too, and there are rules of inference, all sorts of things… all the elements of an axiomatic system are separated. In any event, I claim that this distinction does not really exist, at least not in everyday thinking. In mathematics it is useful; it is fruitful. But in everyday thinking it is misleading. Because what we call a definition is actually a kind of hidden claim, an implicit claim. We are making a claim about some idea that exists somewhere. The idea of Jewishness, yes? The Jew. We have a claim: this idea—what is its correct definition? Is it one born to a Jewish mother or one who converted according to Jewish law? Or is Judaism just commandments, say for someone who claims that. Is the question how one becomes Jewish, or the question what Jews are? Two different questions. Okay? And someone else will claim: no, serving in elite army units, I don’t know, working hard, contributing to the state, no matter, speaking Hebrew, reading Amos Oz—everyone with his own definitions for why he is Jewish. Fine? What? Sense of humor. Sense of humor is Jewish. No Jew… doesn’t matter. Everyone defines himself. Diaspora Jew. “Diaspora” is by Halfi. But let’s see—we’ve defined what is really gentile. A real gentile. A complete professor.

So mathematics here, in a certain sense, does us an injustice. But there is such a function of definition, and it is very important. I don’t think one should hate mathematics. Meaning, there is such a function of definition that is constitutive. Why is it useful to relate to a definition as constitutive? Because then basically we know what we are talking about, and from there on we only derive the consequences. I’ll now bring an example that may sharpen this a bit more. There’s no board here, so I’ll do it maybe on paper. Again, I don’t have… here, I’ll do it on this sheet. Look, maybe we once talked about this—I think in the year of logic, five or six years ago, I talked a little about it there, yes.

There is a theorem in mathematics, a very interesting theorem in topology, very interesting. We define—we speak about convex shapes. What is a convex shape? We know intuitively what a convex shape is. A convex shape is something whose belly points outward. Define “cowlick,” something everyday. What? Define “cowlick.” Try defining “cowlick.” Okay, there is some clear mathematical definition for this, okay. But people ask in daily life… Okay, something convex. Yes. In any event, say a circle is a convex shape, right? Because the belly always points outward. In contrast, a banana is not a convex shape, because this side of it has its belly outward, but that side of it has a concave belly, inward. Like a bowl. Yes, a bowl itself too—if you look at it from outside it is convex, not concave. Only if you look at it this way it is concave. So shapes—convex shapes are basically defined as those whose boundary, from all directions, points outward. From all directions. Meaning, concave shapes—for mathematicians, and for good reason—are shapes that are not convex. Meaning, it is enough that there be one part that is not convex for the shape to be considered concave. Okay? I don’t know, maybe this would have to be a mathematical pathology—surely mathematicians have such things—but a shape that is concave from all directions. Can you imagine such a thing? Only pathologies, surely some Koch snowflake or something. Although I think the Koch snowflake is not… I don’t know, actually. Maybe. Like a three-dimensional egg-like thing. Yes, so some crazy thing of mathematicians—only they can invent such a thing—but in principle there is no such thing. Therefore it is clear why they define it in the direction of convexity. The convex shape is basically the fundamental definition, and concave is everything that is not convex. Okay?

Now there is a theorem in mathematics—and in a moment you’ll see why I’m entering into this technicality. The theorem says that the intersection of any two convex shapes is also convex. Okay? For example, I make an intersection of this circle with an ellipse, which is also a convex shape. Fine? So intersection means the common area of both. Do you see it? Here. There is an ellipse and a circle, and there is an intersection between them—the common area, the points belonging… intersection for us is wings, body dismembered, spare part, something completed. For mathematicians a shape is a set of points. The circle is the set of all these points, not the circumference. The circumference doesn’t matter how, and the disk is exactly… no, no, I mean, the question is what is the area and what is the line. Circle and circumference. A circle is the line, right? The circumference is the area, yes. Anyway, I mean this: this is a circle and an ellipse. With ellipse, by the way, there aren’t two separate terms, so there’s no problem there. So I’m saying, there is the set of these points, which is the circle, and the set of these points is the ellipse. Fine? The common points—that is the intersection of two sets. Geometric intersection is basically an example of the intersection of two sets. There is a set of points and a set of points, and there are points that belong to both sets. That is the concept of intersection. Right? There is the set of Israelis and the set of people whose names begin with A. These are sets that have a common part: there are Israelis whose names begin with A, there are people whose names begin with A who are not Israelis, there are Israelis whose names do not begin with A. There are three categories, and there is an intersection of these two sets. Okay?

Now when I do… the intersection between two convex shapes—when I’m speaking of shapes, then the intersection too is a shape, some set of points that is some kind of shape. This shape will always be convex. If the two shapes being intersected are convex, then their intersection will also always be a convex shape. Now the layman, yes, the mathematical amateur—fine—says, obviously, it’s very intuitive, what? Of course that’s true. But for mathematicians nothing is obvious. Meaning, there are always pathological cases one can show, yes? Mathematicians spend an entire semester proving, in a coordinate system, say there is a graph like this, yes? This is a coordinate system, x-axis and y-axis, and the graph goes down here. So if the graph starts here and passes to there and is continuous, it must cross the x-axis. That’s a theorem on which they spend more or less a semester in mathematics. You understand? Why? Because there is sine x over x, and at the origin you can somehow—I don’t know—pathologies of mathematicians, doesn’t matter. So mathematicians—not of mathematics. Of mathematicians. It’s the same thing, because mathematics is a field that mathematicians constitute, not direct. In any event, this theorem seems very intuitive, but mathematicians, after all—for them what does intuitive mean? You have to prove it. How do you know there isn’t some pathological shape where it doesn’t hold? You have to prove it. Fine, so let’s try to think how one proves it. I’d give you this as a riddle: think how to prove it. You have all the tools. You don’t need any mathematics for it, and that’s what’s beautiful. It was in a book I found once, some used book in some esoteric field of topology. Someone apparently wrote a book no one ever read. I found it somewhere once; I was a high-school kid, I think. Anyway, I read it—there is no problem. A high school kid can read the whole book. You don’t need to know anything, really—just follow what he says, no prior knowledge needed. And this was one of the theorems he proved in that book using elementary-school tools. But when I give it to an ordinary person to prove—even an intelligent person—it will be very hard for him to prove it by mathematical standards. Meaning to prove it, not to say: look, because the boundary… no, to prove it. Because these intersections always—you know, the meeting points are always the problematic places. After all, every boundary of a convex shape is always part of another convex shape, so clearly it will be convex. But the meeting points always create the problem. Can one produce, at the place where the two shapes intersect, some point where maybe some feature arises that doesn’t preserve convexity? That needs to be proved. How do you do that? Has it ever happened to you to prove that not? Exaggeration. Yes.

So when I try to prove this, there in the book—it’s didactic—he tells you, try to prove it, and then you try, you don’t succeed, and then you read and see what the proof is. You see there are people smarter than you. An educational book. Humility, humility. Yes, no, really, it’s very beautiful. In any event, it is very hard to prove—try it—because there are all sorts of combinations, you need to think of all the possibilities, and how in general do you define it in mathematically binding terms? To go rigorously all the way, meaning consistently all the way, fully defined. Maybe by contradiction, actually? There? Assume it isn’t convex. Then show there is a contradiction in the assumption. Well, can you show it? I’m saying, wait—that region that turns out not to be convex would have to belong either to the chord line of one shape—no, obviously the problem is generated by the meeting point. That’s why I said that, right, that explains most of the points. But there are points where you move from the curve belonging to the circle to the curve belonging to the ellipse. In that transition maybe suddenly some concavity can arise? I can’t imagine such a thing, but you have to prove, not imagine. No, concavity belongs not to a point but to a set of points. But that point connects two subsets. Either it exists to the right of the intersection point, or it already exists at this intersection point, or in the middle relative to the two sets? It has to exist on a segment too. This property of concavity is a property that has to exist on some segment. It’s not a simple question, because there’s differential calculus—you know it doesn’t really belong to any one point, but it’s around the point, and there’s concavity at a point. The second derivative of the heart? I have two derivatives… think—sorry for the digression—but think about a positive or negative second derivative. After all, that’s basically what defines this in ordinary functions, right? The second derivative exists at a point. At that point, if you take the second derivative, it will come out negative.

Okay, now leave the… how do you prove it? You prove it simply by first defining what convex is. The mathematician constitutes; he must educate us too. When you begin thinking about something, you must define it. We spoke—I don’t remember if it was last year or two years ago, I don’t remember—about proofs for the existence of God, and I showed there that every such proof assumes a definition. A definition: who is God? Before you define, you can’t prove the existence of something you don’t know what it is. You need to define it. Very important education: you need to define things before you start talking about them. Okay, whether the definition is a convention or not a convention. Now here you need to define what convex is. Convex means with the belly outward. That is not a mathematical definition. That is a definition to explain what I mean. But there is a mathematical definition. What is the mathematical definition? Here’s a proposal. Convex means that every two points you take inside it—if you connect them with a straight line, the whole line will be inside the shape. That is the definition of convex. Definition—notice—it is not a theorem. We would see it as a theorem. There is a theorem: in a convex shape, yes—say, let’s take the circle. I take two points; you see this circle? There are two points here. I draw a straight line between them—I hope those farther away can somehow see. I draw a straight line between them. You see that the whole straight line lies within the circle. And it is obvious to you that this will happen between every two points in the circle, right? Obvious. Now what happens with a banana? There are points such that if we draw the straight line, it indeed will be inside. For example here, in the broad part of the banana, right? Take two points and connect them—it will be inside. But not every two points. There will be points—for example the two ends of the banana—that if you connect them with a straight line, part of the line will be outside the shape. That means the banana is concave. Now you understand the idea? It is obvious that this definition is correct. Okay, this definition is correct. Notice that, yes? In a moment we’ll get there. So here, we’ve defined it. Done. Now the proof is trivial. One line.

Fine, let’s take the common part. I need to prove that the common part is convex. What do I need to prove? That every two points in it, if connected by a line, then the line also lies in the shape, right? Because that’s the definition. Look. I drew these two shapes here. I take any two points. Connect them with a line. Fine? Here I made this little line. You see? Now I ask: is this line inside the ellipse? It must be inside the ellipse. Why? Because the ellipse is convex by definition. And these two points belong to the ellipse. Is this line inside the circle? By definition, because the circle too is a convex shape. So if you take two points inside it, the whole line lies within it. What comes out? That the whole line lies both in the ellipse and in the circle, meaning the line belongs also to the intersection—the intersection between the circle and the ellipse. That’s all. A completely simple proof; no mathematical knowledge required. What is required? To define properly. So first of all, look at the power of definition.

But now look where I tricked you. No—the whole of mathematics tricks us, not that I committed some special trick here. Mathematics is one big cheat. In fact just today I spoke about this in the Talmud class in the morning. I have some women mathematicians there too, doctoral students whom I teach Talmud, so I told them that mathematics is cheating, and in the end they agreed with me. I told them mathematics is like sweeping the dust under the rug instead of throwing it in the trash. Because think about it. I’m really interested in a claim about the world. In the world there are convex shapes. Intuitively I understand what a convex shape is. That’s why we said before that this is a good definition. Right? A beautiful definition. A correct definition. Because there is some concept of convexity that this definition does not constitute. I understand it even before I knew the definition. After someone defines it, suddenly—wow, you captured the concept wonderfully. So I understand the concept; beautiful definition. True, good, convenient, useful, fruitful definition—everything is correct. But you see that all these superlatives about the definition can only be said if I compare this definition against something and say: yes, it is successful. As opposed to another definition that would not be successful. If the definition is constitutive, then what is successful about it? What it defines is what it defines.

Now beyond that, if I’m interested in the world, then I ask myself whether in the world, if I cut paper shapes like this, yes, and always make the intersection, will it always indeed come out convex or not? Understand that from here you cannot know that. Why? Because now I ask: who says that this definition really captures the intuitive concept of a convex shape as I see it in the world? Do you have a mathematical proof of that? We all feel yes, of course. But if you want to be a committed mathematician, not one of those cheats who rely on inner feelings—okay, prove to me that everything I call a convex shape corresponds to this definition, and vice versa, everything that corresponds to this definition… you can’t prove such a thing. So what does the mathematician do? He says: we define convex shape. Define. That is a constitutive definition, a definition that constitutes. He is not trying to hit upon anything, because if he tries to hit upon something, he can’t prove that he hit it. It’s impossible to prove. So what? He defines this definition, and from there on the path is very simple. But the difficulty I was dealing with, when I wanted to understand something about the world—over that difficulty he skipped; he didn’t solve it. He hid it inside the definition. Meaning, in order to prove this about the world, I should actually have had to go through the whole route. I should have said: look, first I’ll prove to you that everything you intuitively call a convex shape is characterized by this definition. This definition captures it well, one-to-one and onto, yes? Every such thing will be such, and every… it will fit completely. Now I can do the proof. Then you’ve convinced me mathematically with absolute certainty that there is such a property in shapes in the world. Not about something mathematical. But mathematicians can’t convince me of anything about the world. They speak about their own fantasy world. It’s a Platonic world, not our world. The leap between our world and the Platonic world—that is the dust that the mathematician sweeps under the rug. The rug is the definition. The mathematician’s constitutive definition, the definition that constitutes the concept, is actually aimed at an everyday concept. But mathematicians don’t want to deal with the question of who says it is aimed correctly, because that’s an empirical question, a question for physicists or other lowly creatures. You moved from hot-air balloons to carpet theater. Yes, lowly creatures, not noble aristocratic mathematicians. The mathematician deals only with things that are absolute, certain, theoretical, fully defined, incapable of criticism, constituted within the definition. What? In mathematics too there are axioms and something intuitive, and from there mathematics begins. Intuitive, yes—but the mathematician is not interested in whether that intuition is correct or not. Whether it corresponds to something happening in the world or not. As far as he is concerned, the route—the mathematics begins from the axioms and onward. The axioms will be supplied by the physicist, or by someone else, I don’t know, any empirical scientist. Meaning, mathematicians, in order to preserve their purity, their precision, their inability to make mistakes, give up all the problematic parts with which the human being deals. After all, I want to understand properties of the world, not of Platonic worlds. And mathematics deals with Platonic worlds. It sweeps all the dust of the road of our complex and insoluble world under the rug of the definition.

Therefore I say that the relation to definition as constitutive, which comes from mathematics, is a cheat. The mathematician deals with convex shapes because he is also a human being, besides being a mathematician. And as a human being he understands what a convex shape is; he does not start from a definition. He understands what a convex shape is, and then he proposes a very specific definition for it, one that is beautiful, but of course not necessary, or not necessarily exact. It seems right to us. There too the mathematician says “it seems to me right.” He just says: okay, that’s not mathematics. Put that under the rug. Mathematics begins from the rug onward. Okay? I sweep all the dirt under the rug and start from there. Everything is precise. Mathematics is wonderful. It explains everything with absolute certainty, impossible to argue with, everything pure, everything clean—except for your basic concepts and your definitions, where all the dirt is hidden. That’s where all the difficulties are.

By the way, this is an extraordinary art. It doesn’t mean you don’t have to be a genius to be a good mathematician. On the contrary. This distinction, the ability to extract the precise parts, the certain parts, and to sweep the dust aside intelligently under the rug—you have to be a sophisticated housekeeper. You have to be a genius cheat. Yes, right. You have to be a real genius in order to be an intelligent cheat. Okay? And precisely the ability to isolate, out of everyday thinking that gets tangled up and doesn’t know—after all, someone who isn’t a mathematician wouldn’t do that. Although the mathematician cheated, still only he knows how to cheat. Meaning, he says: look, I’ll tell you where you get tangled. I won’t solve the tangle for you. I’ll only tell you where you get tangled. You get tangled in the transition between the intuitive concept of a convex shape and the mathematical definition of a convex shape. So I’ll propose a definition. Let’s agree, the two of us, that this transition sounds reasonable to us. Beyond reasonable I can’t tell you anything, because that’s not in mathematics’ department. But I suggest: let’s start with this. From there on I’ll lead you by the nose to the result. No problem. It’s absolute. The art of making this separation—this selecting labor, sorting out the exact mathematical part from the empirical, physical, or scientific dirt—that is the art of the mathematician. And the physicist also often does this. He builds a mathematical theory, and then you know that the basic concepts out of which the theory is built are the dirt of physics. From there on it’s equations. You need to solve equations. You’re the mathematician from there on. So that’s precisely “choosing the food from the waste.” Yes, exactly. That’s “choosing the food from the waste.” One of the chief acronyms of the rabbinical court. Good.