Doubt and Statistics – Lecture 18

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[Rabbi Michael Abraham] I’ll say it again: if this were poison, then I would behave the same way in both a majority that is present before us and a majority that is not present before us. But I would do it, in the case of a majority that is present before us, out of ignorance; and in the case of a majority that is not present before us, out of scientific-statistical understanding. So there is still a difference. If someone comes and asks me, “Tell me, is it really ninety percent kosher?” he won’t ask me whether I would drink the poison or not; he’ll ask me whether it’s ninety percent. I’m supposed to tell him: I have no idea. I truly can’t know. If I get to heaven and they judge me there for the statement of whether it’s ninety percent or not, I won’t say it’s ninety percent. If I have to decide whether to drink, I’ll assume it’s ninety percent—just as an assumption born of ignorance, because I have no better option. Better that cup than that cup. And that’s just my assumption. This is a real difference; it’s not just word games. Statisticians tell you that when there is a distribution, you can’t talk about probabilities. Yes, I’ll maybe give you a place where I once ran into this, when I spoke about the fine-tuning question. Right—the probability that the values of the constants in physics would be exactly what they are, because the precision of the values of the constants is very important for the functioning of the world. And if there were even a small change in the values of the constants, the world would look completely different. Maybe there would be no biochemistry, no biology, no life, nothing. This whole story is very sensitive to the values of the constants. So the question is: what is the probability that the values of the constants would be exactly as they are? So let’s say, for simplicity, that we have only one constant—just one physical constant—and it has to have an exact value. “Exact” meaning a real number between zero and one, let’s say, just simply, a real number between zero and one, a point on the interval from zero to one. An exact value. For the sake of the discussion. Okay? Now there are infinitely many such possibilities between zero and one. Okay? What is the probability that the value would come out exactly as it is now? Zero. But of course every draw you make will produce some number. So a number will always come out, even though the probability of getting that result is zero. So how does it happen? If the probability of getting it is zero, how does it happen? The answer is that you can’t define such a draw. In statistics you can’t define a draw that produces a real number from the interval zero to one. It is not defined at the statistical level. Because at the statistical level, when you try to define the distribution—right, what is the probability of each of the numbers—you get zero for each of the numbers. So try to calculate: what is the probability that the number one-third will come out? Exactly one-third. Or one-half, or one-quarter, whatever. The result is zero. You can’t say anything about such a thing. Nothing. Even though the result, when you supposedly make a draw—it won’t really be a draw. Because you’ll choose some point somehow, but it won’t be a draw. Because a draw is always subject to an orderly mathematical statistical description, with distributions and all that. Fine—but if you ask me what the probability is that such a number will come out, I say zero. Even though you can’t really speak here about probabilities. Understand: if the probability of each number is zero, then the total collection of all the probabilities does not add up to one. After all, the condition for a distribution function is that its integral is one. Or that the sum of the probabilities of all the events is one. In this case it isn’t. Every event has probability zero. So you can’t sum all the events to one. In order to sum them you have to do an integral, and then not talk about a point but about a small interval, as small as you like, around the point, and then it starts to be defined. You can’t talk about a point. At a point it is not defined. And nevertheless, when I ask myself the philosophical question, not the statistical one—what is the probability that the values of the constants are exactly as they are? What is the probability that one-third would come out? The probability is zero. Or in other words, it is impossible that this came out just like that without a guiding hand. But when I formulate this to a statistician—and I’ve already tried to do this—he tells me: you’re talking nonsense, I don’t even understand the sentences you’re saying. There is no such draw; it is undefined. I say okay, such a draw is undefined, but philosophically I can still ask that question. Is it reasonable that the values of the constants came out exactly this way by chance? Can that be? At the statistical level, the question is undefined. You cannot define a distribution. But at the philosophical level, it is certainly a question that can be asked. I would answer it: the probability is zero, and if that’s what came out, that is a sign that there is a guiding hand. That is the fine-tuning argument. But again, I’m not going into that here; I’m only bringing it as an example of a question that is not defined at the statistical level and yet, at the philosophical level, I can ask it and draw conclusions from the answer. I think the probability that such a value of the constants would come out is negligible—it is actually zero—and therefore there is probably a guiding hand here. Even though what I just said cannot be defined at the statistical level. But philosophically, what I’m saying makes sense. Maybe you can speak about plausibility and not probability. Plausibility is a philosophical concept, not a mathematical one. Okay, I basically want to say the same thing here: when I ask what the probability is that this piece of meat is kosher, whoever answers ninety percent is mistaken. Simply mistaken. It’s not correct. You have no information, you can’t define a distribution function. All you can say is that the plausibility is ninety percent, not the probability. And therefore most people will still assume that it’s ninety percent, but that is not really the result of a statistical calculation; a statistician would not accept such a thing. A statistician would say: there is no answer. You wouldn’t get an answer out of him even with a tractor. He would refuse to answer such a question. It’s a mathematically undefined question. And that is exactly the point. So the Torah tells us: if you don’t have statistical tools to handle this, then I tell you—count sides. But all that is only in the case of something separated, not something fixed. And there is something here, I think, that is illuminating, at this seam between legal explanations, or scriptural decrees as they are called, and statistical explanations. There is something here that contains both an element of this and an element of that. In the background there is a very deep statistical insight. Following that, there is a legal distinction here. Okay. Well, I hope I managed to clarify that, because there is something here that I think is really, really deep. It seems to me so. Now, I’m not sure this explanation can explain all the passages in the Talmud and in the medieval authorities and later authorities that use fixed and separated. And as I’ve said in the past, and I’m saying again now, apparently the distinction between fixed and separated began from some kind of reasoning. And I tend to think that this is the reasoning I explained here. But after we already established the distinction between something fixed in its place and something that separated, from that point on it undergoes formalization. And now people don’t keep going back all the time to the philosophical or mathematical roots of this distinction. From now on we have a rule intended for use by halakhic decisors. They are not supposed to understand statistics or philosophy. And the halakhic decisors tell them this: if the thing is in its place, then it’s fifty-fifty; if the thing separated, then we follow the majority. Not in every situation, I think, will you succeed in explaining it the way I explained it here. It seems to me that in not every situation will this work. For example, there are places where the group is in a moving state and not a stationary one. A collection of mixed things, let’s say kosher and non-kosher animals that got mixed together, and we don’t know which is non-kosher and which is not, and they are in motion. All of them, the whole mixture. So now, if I choose one of them, that is called separated and not fixed. Why should that matter? The mixture happens to be on a railroad car, and now I enter the mixture and choose a piece of meat or choose an animal. The logic is the logic of fixed, not of separated. But they discuss it as separated. Why? So I think there is some process of formalization here, and that makes sense. We don’t expect every halakhic decisor to be a statistician or a philosopher or whatever, and to understand the idea and analyze every situation statistically or philosophically. They did it once, they gave us the distinction between a majority present before us and a majority not present before us, fixed and separated within a majority present before us. From there on it passes into legal thinking. We don’t go to experts in statistics every single time all over again. And therefore there will be cases where this logic may not work the way it is explained here. But in the basic cases that appear in the Talmud, I think it works excellently. And it explains the Talmud wonderfully—the cases it deals with and the distinctions it makes and why the verse appears for this and not for that—it explains the Talmud’s distinctions very well. After that it already becomes, certainly among the medieval authorities and later authorities who take it completely formally, and maybe even in the Talmud itself, a kind of formalization, and we already use this distinction without going back each time to its roots. As Maimonides says, the Torah works according to the majority. Every Jewish law fits most situations. There will be situations where it doesn’t work, but we operate with the basic rule. We do not require a person to keep returning to first principles and seeing whether the rule applies in this case or not in that case. But there is one point I’ll say nonetheless: when the whole group is in a moving state, and there I said that they apply the principle of separated and not fixed, even though the logic I’m talking about is the logic of fixed—one could formulate it in a way such that maybe the logic would also be the logic of separated. What does it mean that the whole group moved? Think about pieces of meat from the shop. If all the pieces were lost, all of them were lost and all of them fell in the same place. Now every piece that I pick up has gone through a process of separation. Right, they all separated together with it, but in fact it separated from the original place, and another one separated from the original place, and another one separated from the original place. So why should I care that they all separated? As long as I have a piece in my hand that went through a process of separation, I can apply to it this calculation of the sides, to tilt the decision. Therefore, when I speak about a mixture that is entirely mobile, it may be that there is nonetheless in it the logic of separated and not of fixed, because I am really seeing each one of the elements in the mixture as though it separated—just that they all separate together. They are not in their place. And if that is so, then it is indeed correct to apply the process of counting sides that speaks of separated. And again, this is already a discussion that, let’s say, I don’t know how to make a clear statement about it without the Talmud—whether on my own I would have classified it as fixed or separated. I tend to think I would have classified it as fixed. But the Talmud classifies it as separated, and so do the medieval authorities and later authorities. That doesn’t bother me. First, because it has undergone formalization and now we no longer go back to the original logic; second, because maybe even this case itself can be understood with the original logic. There are simply many things here, all of which separated. Fine—but still, each and every one of them separated. And about each and every one of them I can ask: from which original group did it belong? The fact that those groups are no longer in the original place—so what? When it separated, I ask from which group it came. I don’t care that afterwards the railroad car burned up. Why should I care that the railroad car burned up? Or that all the other pieces separated. The question I am asking about the piece I am holding in my hand is still the same question: from which original group did it come? Yes, you can formulate that question as a question of separated and not of fixed. That doesn’t bother me. I think this explanation is really—I don’t know—the best one I know for this matter of fixed and separated, majority present and not present before us.

[Speaker C] Could the Rabbi explain for a moment the logic of the other side? If all the pieces separated at once, what is the logic of applying fixed and not separated?

[Rabbi Michael Abraham] Because basically I’m saying: there is a mixture here; nobody separated. The whole room is simply moving on wheels. So what? Think about a group of people standing in a room and I throw a stone into it. But that room is a railroad car, a moving car. I would not attach any difference here between fixed and separated. But I’m saying that in a somewhat formal way, after the formalization we make of the distinction between fixed and separated, I say okay, the car is moving, meaning all the people are moving. If each one were moving separately I would see him as separated from the original place. And now they all separate together. So I still see it as separated. And I’ll say again: I would not have said this on my own purely from reasoning. But after the Talmud says it, and after this distinction between separated and fixed has already undergone formalization, it doesn’t bother me. Okay? Now there is another interesting point here. An additional explanation for the law of fixed, which may be similar but is not identical. And this is an explanation Oren proposed—Oren, the site editor—in a comment, actually in a question, and also in a comment to one of the columns that we’ll discuss in a moment. But in a question he asked on the website he proposed an explanation for the law of fixed, and in my view it’s a nice explanation. We’ll see implications later. Good, here it is. Yes, he thought of an explanation for fixed, and he says this: when a person—or even an animal like a mouse—chooses an item of a certain kind from a mixture of items, the probability that he chose an item of a certain kind depends not on the relative quantity of items of that kind, but on the chooser’s preference considerations. Right? So for example, if a person throws a stone at a group of people in which there is a majority of Jews and a minority of gentiles, the probability that he intended a gentile and not a Jew is fifty percent. Why? The fact that there are nine Jews and one gentile there doesn’t matter, because the question is what he intended. If he intended to kill a Jew, then he intended a Jew; if he intended a gentile, he intended a gentile. After all, our whole question here is about intent, remember, not about whom it hit. Whom it hit in the end—it hit a Jew, I know. The whole question is what he intended from the start. So here it depends on his intentions. If he wants to kill a Jew, he intended a Jew; if he wants a gentile, a gentile. This is not some random choice where we wait to see on whom the stone falls. My question is whom he intended. So you understand that this is really a case of fixed. Because it’s either that he intends a Jew or he intends a gentile. It is the result of a human choice or human intention. It is not some random process choosing a person from among ten people. Right? In the same way, if a mouse takes bread from nine piles of matzah and one pile of leavened food—a passage in tractate Pesachim—there is a fifty-fifty chance which kind of bread he will choose. Why? Because it depends on his preferences, not on whether the majority or minority are the piles of matzah. Right? Because really, it depends on the mouse’s taste. Maybe he likes leavened bread, maybe he likes matzah. But who said he chooses randomly among the ten piles of bread? It depends on his preferences. And since I have no information about his preferences, then I also don’t know, so it’s fifty-fifty. The determination that it is fifty-fifty is not a statistical determination; it is a determination of ignorance. I don’t know his preferences, but there are no statistics here, so it’s fifty-fifty. And that is very similar to what I said earlier. In a moment I’ll explain the difference. He says the same thing also with the nine shops. If a person entered a shop and took meat from there, and afterwards forgot which shop he entered, the question is which shop he intended to enter when he went to buy meat. And that is a question that has two possible answers: to enter a kosher shop or a non-kosher shop. With equal probabilities. Of course that’s not correct. It’s not probabilities because it’s preference. Preference is not a lottery. But the more correct formulation is a formulation of ignorance, of a default. I say: I have no information, I don’t know the distribution of the person’s preferences. Not a distribution—rather, what he prefers: does he prefer this or prefer that. That has one answer. Okay, but I don’t know it. Since I don’t know it, it’s like a coin where I don’t know whether it is fair or not—what would I bet? Fifty-fifty. But out of ignorance, because I don’t know. And also when it depends on a person’s preferences, or a mouse’s preferences, or things of that sort, then basically I am supposed to treat it as fifty-fifty because the distribution according to the number of possibilities is not a good distribution. Right? This is not a random process; it is a process that depends on preferences. If it’s a person then certainly, but even a mouse’s preferences are preferences. He chooses bread he likes over bread he doesn’t like. He doesn’t choose by randomly falling onto some pile and eating the pile he fell on, right? The mouse has some kind of taste. Who said whether he likes this kind of bread more or that kind more? Why assume it follows those nine shops? Now notice: here it’s much stronger than what I said earlier. If I now have to ask statistically—if I’m betting with you on what the probability is that the piece the mouse took is leavened food versus kosher—here it could be that I would even bet fifty-fifty. Why? Because truly, if the mouse likes leavened food, then he’ll take leavened food. Why do I care that there is only one pile of leavened food and nine of kosher, of matzah? He’ll go to the leavened food. He’ll choose that piece from among the ten, the pile that contains leavened food, because that is what he likes. Remember Kopel’s example? Kopel’s example was: we have a box of balls, right? And now I lift a ball by hand, and I ask: what is the probability that it is black or red? Then I’ll say: that depends on the number of balls in the urn. And if I haven’t lifted the ball at all yet, then he says you can’t ask that question. Right? That was his explanation for the law of fixed. I’ll ask a third question now about that case: what happens if I put in my hand but not randomly—I choose. Now there are one hundred balls, ninety-nine red and one blue. I chose a ball, but I looked, saw the colors, and chose a ball. Now I ask you: what is the probability that I chose the blue one? Fifty-fifty.

[Speaker C] But that’s only in choosing, not in picking.

[Rabbi Michael Abraham] Yes yes, of course. I’m talking about choosing. Only when a person chooses, or a mouse—it doesn’t matter—not in the sense of choosing and picking as free choice. Even the mouse, for me, is choosing, not picking. Just as in discussions of free choice, for a mouse that would be picking. But here I’m talking about the mouse’s preferences, what it likes to eat. And since I don’t know what it likes to eat, I have no information, so I assume there is a fifty percent chance that it likes leavened food and a fifty percent chance that it likes kosher food, matzah. Okay? So if it likes leavened food, it will choose the leavened piece even though it is one out of a hundred. Because that is what it will choose; that is what it likes. It has no problem choosing. Therefore there I would even bet fifty. It’s not just that I say, in the absence of information and out of ignorance, I choose fifty. Oren proposes that this is not ignorance; it is actual statistics. Now this distinction is very nice. I mean, it’s very nice and very correct. Because when a person enters a shop to take a piece of meat, after all he has a tendency—he enters kosher shops or non-kosher shops. Now he doesn’t remember which shop he entered, and he has some preferences. Even if he wasn’t aware of them, he has habits as a result of previous entries into shops. Everything basically depends on his basic preferences, even if he does it unconsciously, or at least it could. Therefore, in such a situation it is definitely possible that even the statistics will be fifty-fifty. Not just that out of ignorance, since there are two possibilities, I say fifty-fifty. The statistics truly would be fifty-fifty. That is a much stronger explanation than the one I proposed earlier. Now, it is very similar to what I said earlier up to halfway. Earlier I said: there are no statistics here. And because of that, out of ignorance I will assume it is fifty-fifty. Oren wants to argue: first of all, there are no statistics here, like I said. But he wants to say: if there are no statistics here, I have an active calculation that does the work in place of statistics. Not that since there are two possibilities and I am ignorant, I assume fifty-fifty. No. Rather, there are two possibilities and I really bet that it is fifty-fifty. But it’s not all that far from what I said. Think about the mouse, for example. The mouse, after all, depends on its preferences. Now I have no information about its preferences. Maybe its preference is leavened food, maybe kosher food. So there is still ignorance here. It is a fifty-fifty that is the result of ignorance. But it is ignorance such that naturally the bet is fifty-fifty. In the case of nine kosher shops and one non-kosher one, what I said earlier was: since I don’t know the distribution, it’s fifty-fifty out of ignorance. What Oren claims is: no. If you choose a shop, then true, it’s ignorance because I don’t know your preferences. But it is ignorance that truly leads to fifty-fifty. Because either your preferences are this or your preferences are that. Not because there are two possibilities—even though it is nine against one, it’s fifty-fifty. Do you understand? It leads much more strongly to fifty-fifty than my explanation does. My explanation only says: I have nothing else to assume, so I assume fifty-fifty. Even though common sense says it’s ninety-ten. Plausibility, not probability. He says: since this is plausibility and not probability, Jewish law tells me fifty-fifty. Oren says no, no, no—it is a probability of fifty-fifty. That is already a genuinely probabilistic explanation.

[Speaker D] If a person enters a shop—if a person enters a shop and he didn’t know, because when he entered he didn’t know whether it was kosher or invalid—

[Rabbi Michael Abraham] That’s the discussion I continued with. That’s the discussion; that’s what I asked him in that very question that I’ve now raised for you. Look at the continuation of his discussion. That’s what I asked him, and then he proposed some interpretive narrowing, that the Talmud is speaking only of a situation where he didn’t remember.

[Speaker D] That narrows the whole discussion tremendously. Because for example, with the balls, if a person doesn’t know the color, then again here too you don’t have that issue. Right. And also with a mouse, it may be that leavened food and matzah are the same to it. Right, right.

[Rabbi Michael Abraham] It narrows the discussion, and indeed I would say that in other cases that won’t be the law. They’ll simply make interpretive narrowings in the Talmud. That’s okay.

[Speaker D] Yes, but then the Talmud would have to say “if they fell, then permit it in its place”—that’s really quite an interpretive narrowing. It’s not some rare thing.

[Rabbi Michael Abraham] Fine. Look what Oren brings here from the Jerusalem Talmud, you see? It’s in the continuation of that same thread. After I asked him exactly this question, he says: look, Jerusalem Talmud Shekalim. “Nine shops sell properly slaughtered meat and one sells carrion meat; if they became confused for him, he must be concerned. But with something found, we follow the majority.” Right, this is separated and fixed in the language of the Jerusalem Talmud. What does “they became confused for him” mean? It means he really entered a shop and knew where he was entering, but afterwards the shops became confused for him—he no longer remembers whether he entered this shop or that shop. If so, then when he entered it was not a lottery. It was a choice; only now he doesn’t remember what the choice was. That’s what he argued. And if so, then it’s not even an interpretive narrowing; rather, the Jerusalem Talmud truly is speaking about the case Oren proposed. I told him I’m not sure that’s the meaning of the expression “they became confused.” “They became confused” could mean that already when he approached, he didn’t notice that he needed to choose a kosher shop, and he doesn’t remember whether this is a kosher shop or a non-kosher one. And I even brought him an example of using the phrase “they became confused” in that sense, in such a context. I’m only bringing this as an illustration of the possibility that even this interpretive narrowing is not so far-fetched. It may really be what the Jerusalem Talmud means. Therefore, in my view, this difficulty is not so terrible. I also asked that very question. It’s not so terrible in my view. And this is an explanation in my direction, but in the end it arrives at the calculation of fifty-fifty in a more convincing way than my ignorance-based fifty-fifty. Okay, so these are the two explanations for the law of fixed. Now I want to move on to another passage that also requires fixed and separated, and I think that there perhaps one can see the difference between my explanation and Oren’s explanation. It all started with a question I was also asked on the website. What he asked was about a passage in tractate Ketubot on page 9, the “open entrance” passage, as it is called. Right, we’re talking about a husband who came to his wife and found that she was not a virgin. Then he came to the religious court, and the practical difference concerns the marriage settlement: whether she gets one maneh or two hundred. In the marriage settlement, a virgin gets two hundred and a non-virgin gets one maneh. So now the question is whether she is a virgin or a non-virgin. Now, that is one question concerning the marriage settlement. The second question is whether she is forbidden to him, because if he found her to be a non-virgin, then she had already had relations with someone else. If she had relations with someone else when she was a married woman, then she becomes forbidden to both the husband and the man involved. On the other hand, there is the possibility that she had relations even before the betrothal, in which case she does not become forbidden. They might say maybe it was a mistaken transaction—he thought she was a virgin and so on—that’s the question of mistaken transaction. He is not making the mistaken transaction claim; he is claiming that she is forbidden. Now the question is whether she is forbidden to him or not. Okay? So the Talmud there on page 9 says there is a doubt here. A doubt whether she committed adultery under him, after the betrothal, and a doubt whether she committed adultery before the betrothal. If it is with a priest, then it doesn’t matter whether it was under compulsion or willingly; there are many details here. I’m not getting into the details right now, but the Talmud presents it as a doubt: did she commit adultery before the betrothal or after the betrothal? This fellow on the website asks: let’s say this woman got married or was betrothed at age twenty-two, and betrothal usually lasted twelve months in their time. At age twenty-three the husband had relations with her, and then he found that she was not a virgin. Okay? So she had had relations.

[Speaker C] The question is whether she had relations after the betrothal or before it. He says: why don’t we follow the majority?

[Rabbi Michael Abraham] Twenty-two years—let’s say from age three she is already fit for intercourse, yes? Her virginity no longer returns, so from age three until age twenty-two, that’s nineteen years. From age twenty-two until age twenty-three, from the moment she became betrothed, that’s one year. So if I ask when she had intercourse, why is this an even doubt? Let’s go after the majority: there are nineteen years against one year. Most likely she had intercourse before the betrothal. Why don’t we go after the majority? That was the question. Now, at first glance, I answered him that applying statistics here is incorrect. Also Oren’s explanation—even though I didn’t mention Oren’s explanation here, Oren in a talkback to that column came back and repeated his explanation, and that’s why I remember it now too, because it’s a relatively recent column, column 509. So my claim was that it is wrong to use statistical tools here. After all, we are talking about a human decision to have sexual relations. The question of when they had those relations is a question of choice—when they decided to do it. It’s not that he happened to sleep with her by accident with his eyes closed and she with her eyes closed, eyes that hadn’t seen the calendar, yes, that don’t know what date it is, after the betrothal or before the betrothal, and the question is which moment in time we landed on. They decide whether to have relations or not, and they decide whether before the betrothal or after the betrothal—it depends on when it happened. But there is no room here at all to use statistical distributions; it’s irrelevant. Therefore it seems to me that this question, in terms of simple statistical thinking, doesn’t even get off the ground. Now of course the question still remains: then why fifty-fifty? Here I say: it’s fifty-fifty because of ignorance. What does that mean? Since I have no statistical way to deal with it, then it’s fifty-fifty. But that fifty-fifty of ignorance also requires explanation, whether it is fixed or separated. Because even if there is ignorance, still if it is separated then we follow the majority; there is a scriptural decree, “follow the majority.” Only if it is fixed is it treated as half-and-half. Meaning, the fact that I showed this is not statistics is still not enough. Because even things that are not statistics still go after the majority, because there is “follow the majority.” We’ll see that in a moment. But the claim basically is—yes, for example when I ask, I told him there, when I give a person a die and ask him to choose a face, like the balls in the urn that I mentioned earlier, I give a person a die and I say: choose a face. He sees everything; he has to choose. And I ask myself what the chance is that he’ll choose five. I have no idea. I have no way to answer that, right? Because it’s not a lottery; he chooses five because he likes five. It’s not that there is anything random here. It could be that I would still say it’s one-sixth because I have no information about his preferences. So I would say, I’d assume all his preferences are equal, but you understand that statistics simply doesn’t belong here at all. It’s irrelevant to statistics. There is no random process here, no lottery here; it’s a choice. Yes, the question is how the taste was formed. Why does he like five? Maybe that’s a random process, and here maybe statistics can be used. There are people who like five, people who like two, one, six, I don’t know. If that is distributed uniformly, I don’t know exactly how, then maybe one can use statistics about how his taste was formed, and not about his specific choice of some face on this particular die, but rather how he developed a taste for liking the number five. And that’s a question—that maybe really is statistical. Fine, let’s leave that. In any case, my claim is that the moment we are dealing with human choice, then you can’t apply statistics, because the whole question is what they want, and what they want is not random, so there is nothing statistical here. That’s how I initially tried to answer this question, and you understand that this is very similar to the way Oren explained fixed or separated, for our purposes of course. One second—but if there is a sample of a hundred people who chose such a distribution of numbers, isn’t that statistical? Again? If there is a sample of a thousand people who chose one distribution or another, one face or another. What does that have to do with statistics? So basically I have statistical information that there are preferences that tend in certain directions. Each one of them chose—again, there is nothing random here. Each one chose deterministically. What does that have to do with distributions? By chance or not by chance you got such a datum, but these are all deterministic choices; there are no distributions, no lotteries, nothing. Each person chose what he chose. Something will always come out, right? After each person chooses deterministically, if you count a hundred people, you’ll always get something—twenty, seventy-thirty, I don’t know, forty-sixty, whatever you want. Something will always come out. The question is whether that something represents a random distribution. Seemingly not. Everyone here is making a deterministic choice; whatever result came out came out; something always has to come out. Again, unless you go back to the question of why they are constituted with that taste, that one likes five and that one likes two and that one likes six, and then maybe statistics can be discussed. This relates to the question of why statistics works in psychology. When people in psychology behave in a certain way, those are usually decisions. So how do we use statistical tools to analyze behaviors in psychology? I talked about that in The Science of Freedom. And the claim is probably that the distributions of people arise from their preferences, and the preferences are formed randomly. But if a person really chooses—not the result of a preference but a choice—here again picking versus choosing comes in, then really statistics can’t be used; it has no meaning. Okay, in any case, there is another claim that can come up here. After all, if this couple—in other words, the adulterer and the adulteress—had to choose, there is even logic to say they chose before the betrothal, because then she was not forbidden. Why assume they chose after the betrothal? Here the Talmud itself says: “stolen waters are sweet.” Meaning, there are people who prefer to do it when it is forbidden. When it is forbidden it has a different taste, okay? And beyond that, there is also the question whether they had an opportunity to do it earlier. When did they meet? When did they fall in love? When did the opportunity arise? Their choice also depends on that. It’s not that a person sits at home and decides whether to do it with the girl when she is sixteen, or wait until she is betrothed at twenty-two and do it then. That’s not how it works. The question is when they met, when they had the opportunity, when they wanted to—it’s something that is not uniformly distributed over the years, but on the other hand it also doesn’t let them do it when she is permitted or when she is forbidden. If they met only after she was already betrothed, then they did it after the betrothal even though it was forbidden, because they had no other option—that’s when they met. So therefore, in short, in the end there is some whole system of considerations here saying that it is wrong to use statistics regarding this question. Even though there are nineteen years against one year, statistics cannot be used. That is basically the claim. Now the problem that came up—I later noted an additional problem. There is another problem, and there it will already be harder to apply the explanation I gave here. The Talmud in Kiddushin 79 discusses a mikveh that we know was measured and found valid, with forty se’ah—let’s say on Sunday at noon. Okay? Now a person immersed in it on Monday morning. On Thursday morning I measured the mikveh and it turned out that it already had only thirty-nine se’ah; the mikveh is invalid. Meaning, someone immersing in it is not purified. What happens in such a case? So there is a discussion here; seemingly these are laws of doubt. The question is whether we go after the earlier presumption or after the current presumption. In the simple conception, we go after the earlier presumption and the mikveh is considered valid unless it is proven deficient. And that was proven only on Thursday morning, so the assumption is that until Thursday morning it was valid, and that’s it. Okay? But the same question we asked earlier can actually be asked here too. Why don’t we check where the immersion was on the time interval between the time we knew it was valid and the time we know it is deficient? In the description I gave, half a day or three-quarters of a day passed from when we knew it was valid until the immersion. From the immersion until now, three days have passed. Let’s go after the majority. Most likely the mikveh became deficient over the last three days, so the assumption is that he immersed in a full mikveh, a valid mikveh. Why do we assume this is a doubt and need the current and earlier presumptions? Let’s go after the majority of moments in time, like we said regarding intercourse—that one goes after the majority of moments in time, nineteen years against one year. Here too we can go after the majority of moments in time. Why don’t we go after the majority? Now actually this is a somewhat tricky point. How many se’ah were in the mikveh on Sunday at noon? Let’s say exactly forty se’ah. On Thursday I find thirty-nine. So if my assumption is that someone drew out the extra se’ah, simply took water from the mikveh, then what I said before is correct. But if I assume it was evaporation, then obviously evaporation happens gradually. Slowly the mikveh decreases. Now if from Sunday at noon until Thursday morning one se’ah evaporated, then by Monday morning ten percent of a se’ah, fifteen percent of a se’ah, I don’t know, something like that had evaporated. Okay? That is enough to bring it below forty se’ah, so the mikveh was invalid—say if it had exactly forty se’ah. So I can actually do a somewhat more precise calculation and just try to estimate along the time axis when it crossed below forty se’ah. Assuming that evaporation happens continuously—which of course is itself a question: when was there strong sun, weak sun, at night there isn’t any, all kinds of things like that. Of course one can complicate this more and more, and maybe that very complexity itself says: forget it, we don’t get into this; every such thing is considered by us an even doubt, because these calculations won’t get us anywhere. There is a certain logic in saying that Jewish law works this way. You can’t operate according to all sorts of calculations and models and so on. I once remembered the Mishnah in tractate Mikvaot. There was a Jew, a school principal in Kiryat Yam, who went to do a doctorate in mathematics with Hershkowitz at the Technion—Daniel Hershkowitz. And he did a doctorate on mathematical problems in Jewish law, meaning Jewish law and mathematics. So he came to me for some meeting to ask me whether I knew problems like that. And one of the problems I gave him was a problem from the Mishnah in Mikvaot. What happens? There is a Mishnah in Mikvaot that says there is a water channel in which water flows, let’s say at a constant rate for purposes of discussion. And this water is drawn water. Now this channel reaches a mikveh that has forty se’ah of non-drawn water. It passes through the mikveh and continues onward. The mikveh is in the middle. All the while, drawn water enters the mikveh and water leaves, but of course it mixes inside the mikveh. And what leaves is not exactly what entered. Slowly the non-drawn water of the mikveh will leave, and at some point there will be at least three log of drawn water there. In the end, after a long time, the overwhelming majority will be drawn water. The question is how long one may still immerse in the mikveh. That is what the Mishnah asks there, and the medieval authorities (Rishonim) there make calculations upon calculations and reach wonderful numbers. So I told him: solve the problem with calculus. Solve it with calculus and check whether the medieval authorities (Rishonim) are right or not. To my surprise he says to me, listen, at the Technion they can’t solve the problem, the mathematicians at the Technion. I didn’t understand that. In any event, he went to mathematicians—that was the problem. Physicists solve it with their eyes closed. Because when you build a mathematical model of reality, then mathematicians are, I think, less practiced—some mathematicians at least—less practiced at attaching a mathematical model to reality. And I did it with Leibniz, with differentials, the dy/dx, in that way you can show it very quickly. I solved it and we reached some result under certain assumptions. And what were the assumptions? And those assumptions, by the way, are written in the medieval authorities (Rishonim). Those assumptions say that every drop of drawn water that enters the mikveh mixes uniformly, and therefore what exits the mikveh exits in the same mixture as exists in the mikveh as a whole. Which of course is not exactly true. That is an assumption of an adiabatic process in thermodynamics or fluid mechanics. And then basically the claim is that every part that enters the mikveh mixes equally, and what exits is distributed between drawn and non-drawn according to the general mixture in the whole mikveh. That is the claim. Under that assumption, this is a first-order differential equation; it can be solved fairly easily, and one arrives at some answer that is different by thousands of percent, sometimes, from the results of the medieval authorities (Rishonim). Thousands of percent—meaning there is no connection. It’s not an approximation of ten percent here or five percent there. There is simply no connection at all. They made some wild assumptions there because they didn’t have calculus. So it’s not even close to the correct answers. Now, of course, it’s also clear that my answer wasn’t correct either. Because it’s not true that it mixes immediately and uniformly, every drop that enters, and then what exits reflects the overall mixture in the mikveh. If you want to make an accurate calculation, there is really no way to do it. You can try to do some computer simulation, and even that I think is impossible. It’s a simulation that’s impossible to do accurately. And therefore, basically, my calculation too is not correct. I assumed certain assumptions of uniform mixing. That is of course much better than their assumptions, but still they are assumptions—assumptions that are not truly realistic. So in such a situation there is logic to say that when they ask me for practical Jewish law, the ruling is not like me but דווקא like those medieval authorities (Rishonim). Even though they are wrong by thousands of percent, because I too am wrong by thousands of percent, maybe in certain circumstances. It depends—if it’s a slow rate, when the rate is very, very slow, then they are right and I am right too. When the rate is fast, their error starts to accumulate, but when the rate is fast my model too is no longer so good. Because if the rate is slow, then you can assume that the mixing happens uniformly throughout the whole pit. When it’s fast, then there isn’t enough time to mix before it already exits. Therefore the calculation is so complicated that there is definitely room to say that we are talking about halakhic decisors. Halakhic decisors—remember what I said at the beginning of the lesson? Halakhic decisors are not supposed to be mathematicians or philosophers, and therefore halakhic decisors are basically supposed to do the calculation in the simplest formal mathematical way there is, without understanding mathematics and without going back to the foundations of the calculation. You establish some calculation, and that is the calculation that will be valid on the halakhic level. Even my calculation, which is seemingly more precise, is also not correct. So what’s the point? Basically, you just need to establish some formal rule, and that’s that. The medieval authorities (Rishonim) make a simpler rule than mine, and therefore there is no problem—we go with what they say. So something like that I want to say here too. When you start making crazy calculations about when they met and whether they were permitted and what they preferred and all kinds of things of that sort, then even if you reach some more realistic model, it will never really be the correct calculation. And if this mikveh is now evaporating uniformly over the whole week, then I can determine when on the time axis it dropped below—but that’s not true. At night it doesn’t evaporate; on hot days it evaporates more; at noon it evaporates more than in the morning, more than in the afternoon. You can’t really make this calculation. And therefore there is room to say that since the exact calculation is so complicated that it cannot be done, one makes the simplest calculation. And what is the simplest calculation in this case? A fifty-fifty doubt. This very much resembles the logic of fifty-fifty out of ignorance, not fifty-fifty out of calculation. And this is fifty-fifty out of ignorance, because fifty-fifty out of calculation, what Oren suggested for example, in this case won’t work. Because in this case Oren’s explanation won’t work, since think about the intercourse—we’ll speak not about the mikveh for the moment. We’ll talk first about the intercourse. The people are not in a situation where they can decide whether he will have intercourse with her when she is permitted or whether he will have intercourse with her when she is forbidden. And then he makes decisions and the question is what his taste is. That’s fifty-fifty, Oren claimed. But that’s not true; it depends on when they met, when they wanted to, when they fell in love, when they had the opportunity—it depends on very many things. These things are admittedly not random, you can’t apply statistics to them, but obviously this is not Oren’s fifty-fifty. Is your preference to do it when she is permitted, or your preference to do it when she is forbidden? It has nothing to do with my preferences; it depends on lots of other things. Therefore here I would not bet on fifty-fifty, but I would still say that on the level of decision out of ignorance, yes, I would say fifty-fifty. Because it’s ignorance. Okay? What happens regarding the mikveh? Regarding the mikveh, this of course does not fit what Oren said. There there is no decision by a mouse or by a person deciding when to diminish the mikveh—unless we assume someone drew water from the mikveh. But if it was evaporation, just some natural phenomenon, then there really isn’t the option of saying what Oren said. On the other hand, yes, on the other hand here basically I would—so what would I apply? I would apply probability out of ignorance. And once I am ignorant, I have no information, what should I do? Assume fifty-fifty. And no—we already remember that not so. Because when there is ignorance and I have no information, the Torah still tells me there is an available majority. Now the question is whether this is separated or fixed. If it is separated, I go after the majority; if it is fixed, I do not go after the majority. And therefore now we are at the question: okay, we understood that Oren’s explanation doesn’t work, but mine does. The question is whether according to my explanation, the majority both in the mikveh case and in the open entrance case—whether that is called fixed or called separated according to my explanation. So I’ll leave that for you to think about; next time we’ll continue with this matter. Sorry for the quarter-hour delay of the lesson, I just had a yahrzeit today, so I couldn’t get here on time. Okay, that’s it. Sabbath peace. If anyone wants to comment or ask, or run to see the NBA, to see whether we won, lost, who against whom, who against whom—I know. Fine. According to Rabbi Shimon Shkop’s explanation of the sides, that means that in a religious court, in a doubt of majority in wisdom or majority in number, in practice we follow the majority in number, if we count sides. Correct. That is exactly parallel to the question of many pieces in one store. But that is the Jewish law. Yes, the question is then what is the reasoning of Beit Shammai, who hold majority in wisdom? Interesting question. It could be that they would also hold regarding stores that you go after the majority of pieces. There are halakhic decisors who say that. Anyone else? Okay, have a peaceful Sabbath. Peaceful Sabbath. Goodbye.