2019-04-22 – Between Midrash and Logic – Lesson 16

Table of Contents

• Concluding the topic / passage: common denominator, refutation, and a microscopic constraint
• The final table and the preferred solution
• Identifying microscopic parameters מתוך the actions
• Alpha as abstract power and the formality of a document
• The model as a two-way tool: inference and theory
• An analogy to the scientific process and theoretical entities
• The context of discovery and the context of justification, and mechanizing generalization
• Implications for dispute, exegetical derivations, and halakhic ruling
• Benefit as a refutation versus other parameters, and continuing next lecture

Summary

General Overview

The text concludes the topic of betrothal using a tabular-algorithmic model of a common denominator, in which refutations are not just “additional columns” but can also appear as microscopic constraints on hidden parameters. After solving the final table and presenting vectors for the acts of money, canopy, intercourse, and document, the speaker moves to the model’s philosophical meaning: he suggests that the model is not just a tool for filling in missing slots, but also a mechanism for building a theory that explains halakhic facts, similar to the scientific process. From there he tries to identify the microscopic parameters as conceptual features, such as benefit, value, and connection, and expands the discussion into philosophy of science, the context of discovery and the context of justification, and the possibility that fully mechanizing generalization might reduce the space for inspiration and even for dispute.

Concluding the topic / passage: common denominator, refutation, and a microscopic constraint

The common denominator is built on a learning table in which money and intercourse teach us about canopy, where each of the teaching cases on its own is refuted, but together they operate without adding new data. The refutation, “there is benefit in money and intercourse, but there is no benefit in canopy,” is first understood as an additional column that breaks the inference, but afterward it is defined as a microscopic parameter gamma that exists in money and intercourse and does not exist in canopy. The constraint that gamma must appear in both teaching cases and not in the conclusion breaks the preference for “one filling” because it requires adding another parameter, so the dimensionality of the solution rises and equalizes solutions that had previously been preferred by the topology of the diagram. The constraint functions in the same way as adding a column, because solutions are compared through the cost of the required parameters versus changes of direction, and once every filling has a drawback they become equivalent and a refutation is created.

The final table and the preferred solution

The speaker presents a final table after correcting the refutations and notes that gimel is divorce and against her will, and from that he derives diagrams for filling in zero and filling in one. He determines that the preferred solution is filling in one and writes a parametric solution that produces four dimensions through alpha, beta, gamma, and delta. He sums up the topic by representing the acts as vectors: money is one, zero, one, one; canopy is one, one, zero, zero; intercourse is two, one, one, zero; and document is three, zero, zero, zero. He defines arriving at the solution as an abstract decoding of the components within each act and of what is required in order to apply each result.

Identifying microscopic parameters from the actions

The speaker defines gamma as benefit because it was introduced as a constraint based on the fact that in money and intercourse there is benefit, while in canopy and document there is none. He argues that in order to identify the other parameters one has to begin with the actions and not with the results, because money, document, intercourse, and canopy are familiar acts whose components can be analyzed, whereas results like betrothal, marriage, and redemption are categories that Jewish law defines. He suggests identifying delta as “pure value” because it appears only in money, and from that concludes that redemption requires pure value, whereas betrothal does not require value, because intercourse can effect betrothal without value. He suggests identifying beta as “connection” because it appears in canopy and intercourse but not in money and document, and connects this with the understanding that marriage requires a connecting component, whereas betrothal is a formal act that does not require connection.

Alpha as abstract power and the formality of a document

The speaker has trouble giving alpha a sharp identification, but proposes that alpha is an abstract, “uncolored” power that can be given color through qualitative parameters like beta and gamma, so that alpha functions as a general quantity of ability to effect legal statuses or legal results. He explains that a single parameter can increase in intensity because it is the common axis that makes it possible to infer conclusions from the table at all. He notes that the greatest intensity of alpha appears in a document, and suggests that a document is a purely formal act that has no “color” of its own, so it is natural to identify it with a pure parameter of legal force, whereas intercourse and canopy contain substantive components such as connection.

The model as a two-way tool: inference and theory

The speaker presents the algorithmic device as a tool that works in two directions: it makes it possible to fill in a missing slot and decide between zero and one for purposes of inference, and it also makes it possible to build a theory that explains the halakhic information and to decode “what the Torah requires” in order to effect betrothal, marriage, and redemption. He argues that no one “in the world of Talmudic learning or Jewish law” has formulated a conceptual rule for what is required in order to effect marriage, and the model makes it possible to move from familiar acts to identifying parameters, and from there back to the results in order to understand the structure of the halakhic requirements. He defines the process as progress to a meta-halakhic level that explains why certain acts effect certain results and others do not.

An analogy to the scientific process and theoretical entities

The speaker compares the process to science: just as a collection of facts like tides, falling bodies, and the paths of stars leads to a theory of gravitation that includes unobservable theoretical entities, so too written halakhic facts lead to a theory that includes theoretical entities like alpha, beta, gamma, and delta. He demonstrates a “scientific” common denominator using a book, a ball, and a marker that fall, and suggests that the microscopic parameter there is mass, and that different intensities of alpha can represent different masses. He argues that a common denominator is exactly scientific generalization, and that there is no extra move here beyond the process of building a unified explanation for the facts.

The context of discovery and the context of justification, and mechanizing generalization

The speaker brings in the distinction in philosophy of science between the context of discovery, which is considered creative and even “prophetic” and is not of interest to philosophy, and the context of justification, in which a theory is tested by means of predictions. He argues that his algorithm provides a mechanical form for explaining part of the context of discovery, because it shows a systematic way to move from facts to theory through a rigid logical process. He acknowledges that a painful issue still remains: choosing the facts relevant to the table. But he raises the possibility that even this could in principle be mechanized by means of an enormous table of all the data, so that solving the matrix would mechanically filter out irrelevant columns and yield the correct result. He argues that this possibility rests on the assumption of a one-to-one relation between a cause and a halakhic result, so that it cannot be that the same Jewish law has two alternative causes.

Implications for dispute, exegetical derivations, and halakhic ruling

The speaker suggests that if there really is one complete solution, then on the theoretical level there should not be dispute in logical studies, and dispute arises because people are working with partial tables. He distinguishes between a common denominator and a verbal analogy, and cites the dispute in tractate Sukkah between Rashi and Tosafot regarding the other hermeneutical principles, whether they are like an a fortiori argument or like a verbal analogy, and argues that the approach saying everything is like a verbal analogy is isolated and implausible. He says that halakhic ruling is not done in a completely mechanical way even if one could produce a mechanical “book of laws,” because applying Jewish law to a case requires analyzing the situation. He answers the question, “why don’t we administer punishments based on logical inference,” by proposing the reasoning of “perhaps there is a refutation,” and emphasizes that the possibility of error always exists, but someone who always worries about error will never do anything. He argues that there are no principled limitations on using the rules of interpretation, so long as one knows how to use them responsibly.

Benefit as a refutation versus other parameters, and continuing next lecture

The speaker is asked why the Talmud introduces benefit rather than value, and he answers that benefit was introduced because it produces a refutation in this specific topic, whereas other parameters like beta and delta do not produce a refutation at any stage of the argument. He notes that elsewhere the Talmud might have exposed a different parameter, and that here identifying gamma was easy for him because the Talmud itself supplied the microscopic datum. He concludes by saying that there is another point he wanted to discuss but there will not be time, and he says he will continue with it next time.

What I want to do today—we’ll see how far we get—is finish the topic of kiddushin. Really, we already laid the foundations last time; I just want to finish presenting the continuation of the topic properly this time, not the way I did it last time. The Rabbi asked whether what happened earlier, during the break, was part of what you said would also be useful? A doubtful matter. I’m relying a bit on what we discussed. After I finish the topic, I want to jump ahead a little and discuss the meaning of this model in general. What is it actually saying? What is its more philosophical meaning? What is its more general significance? And then in the coming sessions maybe we’ll come back a bit to a few more anecdotes or a few other applications of this model. I’m jumping ahead; it really would have been better to do this at the end, but I want to use this topic itself, and since we’re finishing this topic here, I want to show through this topic what the meaning of this model really is. So look. Last time we talked about refutations on the microscopic plane. And the claim was that usually we present the refutation as another column or another row, whose data don’t line up according to the expected hierarchy, and therefore it refutes the inference. But there we saw—and I demonstrated this with the derivation by common denominator—that there are situations in which the refutation is expressed through a microscopic constraint. Right, maybe I’ll write it again on the board to remind you what we’re talking about. The table was basically this table. Okay, this is just a common denominator, right? Just a simple common-denominator table, not the larger one yet, where money and intercourse teach us about chuppah. Right? So money teaches us about chuppah by an a fortiori argument; that’s this quartet. Opposite this quartet there’s a refutation from redemption, because money requires redemption and chuppah does not require redemption. Then we said intercourse teaches us about chuppah by an av-based derivation. Here this is comparison, not an a fortiori argument. And that av-based derivation also has a refutation: intercourse acquires a yevamah; here there’s a refutation—these two cells are a refutation of the av-based derivation. And the claim is that what the common denominator does now is connect all the data; it doesn’t add any additional data. Once we use the two teaching cases together, with that same table itself, the result here will be one, even though with each one separately it gets refuted. Each one separately gets refuted, and even so the whole common denominator works, and the various kinds of common denominator that we saw are one-one, zero-zero, or zero and one. Now the next stage that appeared in the topic, and I presented it at first as another column, was that money and intercourse involve pleasure, while chuppah does not involve pleasure. At first I presented that as another column, right? A column like this. That there is pleasure in intercourse and in money, and there is no pleasure in chuppah. Intuitively we understand why this is a refutation. It’s a refutation because those two share a common feature that this one does not share. Therefore I can’t know whether what is operative here with respect to betrothal is the feature common to all three, or rather this feature that exists only in those two; and if I assume that this too operates with respect to betrothal, then in betrothal too the result here could be zero. And so that is a refutation. But in the further analysis we saw that pleasure should not be written as another column that adds another datum or another result. A microscopic parameter. Exactly—pleasure is actually a kind of microscopic parameter, and therefore I’m claiming there has to be some parameter—let’s call it gamma—that exists here and here, but not here. And this one doesn’t have gamma. What I’m going to do now is take this common-denominator table, solve it as one normally does, but with a constraint that one of the microscopic parameters must be present in this and this, but not in this. Once I solve it with the constraint, it should break the common denominator. But if I solved the common denominator without the constraint, we would get that filling in one is the preferable filling. That’s the meaning of common denominator, and we did that, we saw it. Now the claim is that if there’s a constraint, the constraint will show us that this filling is not preferable. Okay? That’s how this is supposed to work. So let’s do it as an exercise. So look: the diagrams I’m drawing are basically common-denominator diagrams, there’s no difference, right? The table is a common-denominator table. The constraint comes afterward when I look for the solution. Okay? But the diagrams are common-denominator diagrams. So here there’s Y, here I, and here P. That’s filling in one. Okay? Exactly what I did last time. Now what does it mean that this is basically a common-denominator diagram with filling one? When I talk about filling in zero, then the diagram is this one. Here it’s zero, here it’s I, and here it’s P. I’m just copying the common-denominator diagram; there’s no need to do anything extra, just copy it. Where will the difference be? The difference will be when I come to solve it. Okay? When I come to solve it, unlike what I did before, here I have to solve it under a constraint. Okay? Let’s say what we had before we can already remember: common denominator—say if there were no constraints, then obviously this is less good because it has two direction changes. In every other respect it’s the same. Okay? It has two direction changes, and therefore filling in one is preferable. Now, in order for that to break, I need the constraint to give some advantage to this diagram. The constraint doesn’t change the topology of the diagram. Direction changes won’t change, the number of points won’t change, and connectivity won’t change, right? The only thing this constraint can affect here is that apparently the number of parameters required to solve this under the constraint is larger than the number of parameters required to solve this under the constraint. That’s the only possibility. The only possible solution path, really. Right? So if we actually manage to show that the constraint leads me here to three parameters instead of the two I had before, then we’ve shown that the two fillings are equivalent, and therefore this is a refutation. So the constraint will do the same job as adding a column. Okay? So let’s see how this works. There’s a somewhat more complicated problem here, because to solve this with a constraint I have to do it like this—look. I start from alpha. Now I say as follows: I want there to be a gamma parameter that exists in money and in intercourse, right? Let’s see in whom else that parameter must be found. That is, clearly—wait—clearly Y and P must contain gamma, right? Agreed? Because if B and M manage to contain Y and P, okay? But H doesn’t manage to contain either Y or P, so that means this parameter gamma is what is responsible for the difference, for the refutations. The refutations are exactly the differences, right? So this is a guess, because I don’t yet have a fully ordered algorithm. On a computer you can do it, but I don’t have a fully ordered algorithm for doing it. So I make an intelligent guess. I say: clearly Y and P need the gamma parameter that I’m looking for. Okay? So I start from there. Now here I write alpha, because that’s always how one starts. Okay? What happens here? So here I’ll put alpha and also some theta; I don’t yet know what it is. It’s hidden as far as I’m concerned. Okay? Here there is gamma and theta and also alpha, right? For this to hold. Gamma must be there, but there must also be something else. Okay? What happens here? Here there is gamma, but clearly there also has to be alpha, right? In order for it to contain it. But if there is gamma and alpha here, notice that some relation is created here, right? Therefore there also has to be some omega here, right? Where omega and theta are variables. What do I know about omega? Those are what I need to determine now, right? Basically that’s it. I’ve satisfied the constraint. That solves the problem. Now I just have to find omega and theta. Okay? What must omega and theta satisfy? First of all, they have to be nonzero. Right? There has to be something there, omega and theta. Also, omega cannot equal theta, because otherwise I once again identify these two, right? So omega is also different from theta. Okay. What else do I know? That there’s no hierarchy between them. What? That there’s also no hierarchy between them. We don’t know which is greater than the other, because then there would be some relation between them, or they would be independent. Yes, right. Okay, right. And I also need them to be independent—theta has to be different from gamma. Theta has to be different from gamma. If theta were equal to gamma—if theta were equal to gamma—we’d have a relation between them. So first of all there would be a relation like this, right? If theta equaled gamma, I could also derive that from the constraint. Because if theta equaled gamma then M would probably have managed to do it, those experiments. So theta is different from gamma. Now under those constraints, what can actually come out is that omega equals gamma and theta equals alpha. Theta can be alpha, so why should I add another parameter, right? I don’t add another parameter if I don’t need to. So theta equals alpha, and then here I basically get two alphas, and here it’s gamma and also two alphas, right? And I identify theta with alpha. Two alphas, two alphas, gamma and two alphas, right? And here I have omega, gamma, and alpha, where omega can’t be theta, it can’t be alpha. Okay, so let’s say omega is gamma. I don’t want to add parameters; I want the simplest model, so I try to make do here with two parameters, alpha and gamma. Right? So if I have only two parameters, alpha and gamma, and theta can’t be gamma, then I say theta must be alpha, right? What else is left? Unless I’m forced to add another beta here. But if not, then I say theta is alpha. Once theta is alpha, then I have here alpha, two alphas, gamma and two alphas, and here I have omega, gamma, and alpha—where omega is two gammas, where omega can’t be alpha of course because it can’t equal theta. It might perhaps equal gamma, and then here there would be two gammas and also alpha, but that isn’t legal. Why? Because I raised the degree in both alpha and gamma; it has to be either this or that. Therefore I have no choice but to define this as beta, and then an additional parameter gets added here. And there you see how this constraint works. There—a third parameter has been added because of the constraint. Without the constraint, that wouldn’t have happened. What happens here in this solution? You do exactly the same exercise, but it’s much simpler. Gamma to two gammas. You see? That solves everything, it satisfies the condition that there be gamma here and gamma here, and there’s no need to add another parameter beta. Right? What does that mean, basically? That the constraint caused the dimension here to rise to three, to go from two to three, and that is exactly the refutation. Because here there is dimension three and here there are two direction changes, so in that case each of the fillings has a drawback, and therefore they are equivalent; therefore it is a refutation. Okay, so that is basically the proof that if I impose a constraint on the solutions, it has the same effect as adding a column. This too is a refutation. Okay? What we need to do now—again, we could keep grinding through this, but there’s no point, it’s unnecessary. You do it in exactly the same way, only the constraints start getting more complicated, and of course it would take us much more time. At the bottom line, we eventually return to the final diagram of that whole table without the H; we erase the H and cast the H as a constraint. You just have to remember that the pleasure-constraint—when I say there’s a constraint of pleasure, what does that mean? That money and intercourse have pleasure, right? But in the final table there’s also a document. Right? And in a document there is no pleasure either. So you have to remember that the constraint now is on money and intercourse as opposed to chuppah and document. And everything else as usual. We solve the problem with that constraint; everything works. Okay? I’m not going to do all that on the board; maybe I’ll just write the last one because I’ll want to use it. So let’s do it like this, look. Let’s write the last one. Wait. Y, G is divorce and against her will. So what I have here is zero, zero, zero, one. Zero, zero, one, one. This is the final table after everything, after I already corrected the refutation; this is the end of the passage. Okay? So that’s the table. And now that this is the table, we draw the diagrams. Here the diagram will no longer be one of the diagrams we did in the past, unlike the common denominator, because here when you remove the H table you’re not left with something we already encountered before. So we have to do it anew, but it’s not too complicated; we’ve already done things like this. So now we have the following: in the zero filling, it looks like this. Let’s write G goes into K, goes into I, into Y, to where? And here there is P. Okay? That’s in the zero filling. And in the one filling, I have here G goes into K, goes into I, Y also goes in here, this goes into M, N goes into I, and P also goes into I. Okay? Basically the arrow from N to I disappears depending on whether the filling is zero or one. Let’s see here the list: zero, one, zero, zero, one, one. Okay, that’s basically the… fine, now I’m not going to do theta and omega and all the constraints again; I’ll write the solution directly. So look, since I’m also already telling you now that the preferred solution is one, I’ll do only one as well, because I already know this is the correct solution; it is preferable to G, to the zero filling. Okay? So I write the solution for one, and it goes like this: alpha, two alphas, three alphas. Here there is alpha and also beta, and here there are two alphas and also beta and gamma. And here there is alpha, gamma, delta—that’s already four dimensions, dimension four. Okay? That’s the solution that comes out here. Now there is also the other solution, but it’s not important, since this one is preferable. So for me this is the summary of the topic. Now why am I doing this? Because notice what comes out now. Basically now I’ve arrived at the final conclusion of the topic, I took into account all the data this topic raises, and I proposed the theory that explains all these data. And this theory basically tells me—maybe I’ll also add below: money is this vector: one, zero, one, one. Chuppah is one, one, zero, zero. Intercourse is two, one, one, zero. And document is three, zero, zero, zero. Okay? That is basically the solution to this topic. What do I now have in hand? Now I want to start discussing the question: what does what I’ve just found actually mean? So look: what this thing basically says—I’ll erase this for a moment, it’s not important, it’s already less… it’s the less good solution, so I’ll erase it. What this thing basically says is that after I take into account all the data in the problem, I have managed to decode in some abstract way what the components are in each of the acts, and what is required in order to generate each of the results, right? So for example, in money and intercourse there is pleasure; in H and money and intercourse there is pleasure. Okay, and in these two there isn’t. So which parameter here is the pleasure parameter? The third one. Right, because it’s the parameter that exists only in those two and not in these two. So gamma is basically pleasure, right? Now here of course that’s no great feat. Why? Because I put it in by hand. I determined gamma to be pleasure; I directly wrote gamma here and gamma here, and then started running. So by hand I inserted gamma as pleasure. That’s not such a feat; the passage itself told me that here and here there is pleasure. But notice: this actually gives us a very interesting tool for trying to decode the others as well. Because now I can try to ask myself, independently of the passage: which is the parameter that appears identically in money and chuppah, twice in intercourse, and three times in document? Or which is the parameter that exists in chuppah and intercourse but does not exist in money and document? Or which is the parameter that exists only in money? And that will actually give me a clue for identifying the parameters alpha, beta, gamma, delta, right? Here it came out mathematically. I know from these data that there exist some microscopic parameters alpha, beta, gamma, delta. I know there have to be four of them. And I also know, within each act, who is found there and at what intensity, and I also know within which result each of those parameters is required and at what intensity. Now where should I start? Obviously I should start—and that’s why I wrote it—from the acts. Why? Because if I ask myself what the meaning of those parameters is, if I do it on this side, I won’t get anywhere. If I know about some parameter that it is required to generate betrothal and marriage, but it doesn’t help for redemption, what does that tell me about it? Do I know anything about it? I know nothing. Because these are halakhic acts. I don’t know what Jewish law requires to generate betrothal or marriage or redemption or whatever, right? But money, document, intercourse, and chuppah are acts not defined by Jewish law; these are acts I know. I can see what there is in these acts. And therefore it is very important to finish this solution with a solution for the acts, and then when I begin, when I want to identify the parameters—who is alpha, who is beta, gamma, or delta—I go through the acts. And after I identify who the parameters are, I go over here and I say: if so, then I have also decoded what the Torah requires in order to generate betrothal or redemption or marriage. Do you understand? So in fact I have taken a step here that no one in the world has ever taken. No one in the world has ever written down in general what is required to generate marriage. Not which act—the acts are Jewish law—but what is the idea? What is common to these acts? Why do these generate marriage and others do not? Or what is common to marriage and redemption? Why do both work in the same way, at least partially, and another thing does not? No one even asks questions like that in the world of learning or in the halakhic world. Why? Because we don’t have the tools. It sounds a bit like speculative preaching, but here we have a completely ordered tool for cracking the meta-halakhic ideas. I can now understand why marriage—what marriage requires in order to be possible, what betrothal requires, what redemption requires, and all of them. Now I have a tool to advance on the theoretical, analytical level and understand what Jewish law wants. So look, let’s do an example exercise, okay? Let’s check. I now want to know what delta is. Okay? I want to identify who delta is. What do you say? What is delta? An intelligent guess. Mathematics gives me only parameters; I don’t know what they are. I only know there are four such independent things. But I don’t know who they are. Now I ask: who is delta? It exists only in money. What does that mean? A parameter of acquisition? Maybe I would say value? Physical, maybe? Yes, value. Basically that is what distinguishes money. Money is pure value, pure worth. Every other object is an object that may also have value in it, but it is not pure value. So delta is pure worth, let’s call it that. Okay? Basically once I identify delta this way, I say: delta is pure worth. After that I’ll see where delta is needed. For example, in order to do redemption, there has to be worth. There is no redemption without worth. Someone who has no worth will not succeed in doing redemption. That’s exactly the point, right? But by contrast, I don’t know, betrothal can also be done without worth. Maybe things with worth can also do it, but worth is not required. What is the proof? Because there are things that have no worth and they still succeed in effecting betrothal, like intercourse. Right? So that means that what is required for redemption is pure worth. That is the meaning of redemption. Okay. Now I continue. What is gamma? Gamma I already know, right? Gamma is pleasure. I fixed that in advance. So I’ve marked that too, and indeed it exists in money and intercourse, but not in chuppah or document. Okay, I move on. What is beta? It is found in chuppah and intercourse. Let’s try to think what chuppah and intercourse have in common that document and money do not. Betrothal? Chuppah? Right. I might even call it union. Right? Chuppah is bringing the woman together, entering together with the woman into the chuppah. Being together in one place. Intercourse is also a kind of physical union. Right? Money and document are formal acts. They don’t create any union. Maybe their result is some kind of union, but in themselves they have no meaning of union. So I would identify beta as union. Again, this is speculation. Someone may come and propose another interpretation, but this is certainly a reasonable interpretation. Okay, now when I ask myself what creates marriage, notice this: union and alpha. There has to be union here. We haven’t yet identified alpha, but there must be union here in order for marriage to come into being. Something that does not contain an element of joining cannot create marriage. Okay? By contrast, betrothal doesn’t need union. If we go back to all the yeshiva-style conceptual investigations, what is the relation between betrothal and marriage, I think that’s exactly what they say there. Betrothal is some sort of formal act that perhaps prohibits her to the world, but it does not yet create her connection to me. When does she become my own flesh, “his flesh” means his wife? When does she become part of me? The union is created only in marriage. Then there are the mutual obligations of the ketubah and everything else. Right? And she is permitted in intercourse only rabbinically. She wasn’t permitted earlier either, but rabbinically intercourse is permitted only after marriage because that is the stage of union. So we are not surprised to see that marriage requires union even though betrothal does not. Okay? Now of course the hard question here is: what is alpha? It could be will. Alpha is present in every one of these acts, but at different levels. In intercourse and in document it is strongest. In money and chuppah it is weaker. I don’t have a good answer for that, but I’ll remind you of something from one of the exercises I did last time. You remember those examples? Yes. And what did we have there? We had two parameters there that color, right? Beta and also alpha, and this is beta and also two alphas, and so on. And here—wait, I’ll do it the other way around. And here there was, say, gamma and two alphas, gamma and alpha, and here, say, alpha. Okay? So what does that really mean? It basically means—or let’s call it, you know what, for good order here let’s make this two and this one one, okay? What does this really mean? It means that alpha represents the abstract, uncolored intensity, which can be colored by beta and can be colored by gamma. The question is how much of it there is; that is represented by what intensity of alpha there is here. And basically alpha is the parameter that says that all these things succeed in generating some sort of legal effect or in performing substantive halakhic acts. What? Intensity of legal effect, obligation or effect, to generate a legal effect or something of that sort, on the completely abstract level that is shared by all of these. Meaning, not something specific, not union. Union will come with beta. Meaning that when we have beta and also alpha, that means there is union here with an intensity of three alphas, or with an intensity of two alphas, or something like that. Exactly like there. So alpha is basically the abstract quantity attached to those three qualities. Here there are different qualities, and here this is basically the quantity parameter. Now it doesn’t surprise me to find that there is one parameter that runs through everything, because from the outset I defined it that way. That’s also why I liked the fact that only one parameter can rise in its level, because that is the parameter that represents the axis along which everything is conducted, because if there is no axis shared by the whole thing then no conclusions can be drawn from this table at all. You can’t connect jazz with fine literature, right? These are things that don’t talk to each other. They have to belong to some shared abstract family. Alpha represents that. So these are acts that have the capacity to generate halakhic effects or legal results, and alpha is the intensity. Now I don’t know why it comes out this way, but this is how it comes out: the greatest intensity of alpha is דווקא in the document. Intercourse has less, and money and chuppah have even less. Because the document is basically pure—here document represents, let’s say, divorce—it’s pure. It has no other property; it only has the ability to produce—yes. With a document I can really understand it better, and with the others it’s a little less clear. A document really is a purely formal halakhic act. So I would expect what characterizes it to be a pure parameter. It has nothing else; it is an arbitrary definition. We could have defined, instead of a document, standing on one leg—after all, it’s just an act. By contrast, intercourse is union; you can’t define whatever you want. What joins, joins; what doesn’t, doesn’t. And therefore a document is a formal act. But there are formal boundaries to it. But there are formal boundaries that apparently may be smaller in quantity than a document, and therefore it has only that. Those are the formal boundaries you have. What in what? I’m saying maybe alpha represents a quantity of formal boundaries, and then it also works out—three, two, one and… No, the power to generate legal results or legal outcomes—that, in my opinion, is alpha. The power to generate legal outcomes. The document is strongest here because basically the document simply is that. Exactly—it is simply that. That’s what it has; it has nothing besides that. It is not colored by any color. With a document you can do whatever one does—you can make contracts, you can do kiddushin, you can do divorce. Whatever you write in the document is what you did with it. Meaning it has no color of its own. So with a document it actually doesn’t surprise me that it really attaches to alpha. I understand less with regard to intercourse, chuppah, and money. Okay? So I’m only giving an example here; another possibility is certainly possible. In a moment we’ll see why it also doesn’t bother me all that much if I don’t find the meaning. I’ll explain that in a moment. In any case, notice what’s happening here. We begin with a collection of halakhic data. We have a collection—you remember the microscopic data, like in this and this there is year, or this is year with respect to damage, beginning of action or end of damage—those are all microscopic data. They don’t appear here in the table. What appears in the table are halakhic facts. I now take the known halakhic facts, propose a theoretical explanation for them, and find a theory. That theory explains to me both what exists in each of the acts and what is required in each of the results—which of course is the more important thing. Since the acts are familiar to me, I use the information I have about them in order to identify the parameters. After I identify the parameters, I return to the halakhic results and I’ve managed to crack them. Meaning this device, this algorithmic device that we defined here, works in two different—perhaps opposite—directions. In one direction, it helps me define what the correct result is in the cell: is it one or zero? It is simply a device for making an inference. Okay? In that sense, it doesn’t matter to me to identify alpha, beta, gamma, and delta—who cares. The main thing is that there are such things; I do the calculation and I reach the conclusion of what the right answer is. Okay? But here there is an additional advantage. Because behind it—as we saw also with an a fortiori argument—there is a hidden theory. So now I can also learn in the opposite direction. Not only do I know how to explain what the correct filling is. Even if I already knew all the fillings, it would still be worthwhile to build the theory, because then it has analytical value. I want to understand what stands behind the ideas of the Torah. Why is marriage done by this and this and this and not by this and this and this? Why is betrothal done by all of them? Why and so on. So if I want to understand what stands behind the Torah’s determinations, I use this same device, but my goal is not to know what to fill in here; rather my goal is to understand the acts and the results in theoretical terms. When we explain in a scientific theory, what are we doing? Exactly the same thing. What are we doing? We see a collection of facts. We see tides, we see objects falling to earth, and we see the trajectories of stars in the sky. Those are facts. Now I take these facts and try to find a theoretical explanation. That theoretical explanation, in this case, is gravitation. The theoretical explanation contains entities… what? It takes twenty-five years to find such theories. Doesn’t matter, but the point is still that I’m looking for some theoretical explanation for a collection of facts. The facts I see with my eyes. Okay? Now I’m looking for a theoretical explanation. That theoretical explanation is built from all kinds of concepts and theoretical entities that nobody has ever seen. No one has ever seen the force of gravity. It’s impossible. You can’t see it, right? It is basically an entity that I invented within the framework of the theoretical explanation. So I take—yes, doesn’t matter, yes, right—every field, every force, every entity, the quantum wave function, everything, no matter. Every theoretical entity is basically not something I observed. I observe facts. In the case of physics, scientific facts. In order to explain them I build some theory that explains those facts, and that theory contains theoretical entities and theoretical concepts and rules and principles and relations between them. That is exactly what I’m doing here. I take the halakhic facts, and this I know—it is written in the Torah—facts. This does redemption, this does not do redemption, this does, everything is written in the Torah. Now I ask myself: what is the theory that explains these facts? That theory contains theoretical entities. These. These are theoretical entities; nobody sees them in the Torah, as it were—you won’t see them in the Torah. They’re behind it; they are part of the generalization I make in order to explain the facts in the Torah. So what I’m doing here is simply a scientific procedure. I take the facts written in the Torah and propose a theory that explains them. Okay? So this algorithm basically gives us two things: first, it gives us the possibility of filling in a missing cell, adding information—we began with the fact that this is an algorithm whose goal is a logic of adding information, right? And the second meaning of this device is that it enables me to understand the information; it offers me a theory that explains the information. Now I’ll say more than that. In philosophy of science they usually distinguish between the context of discovery and the context of justification. When we observe a collection of facts, they can be generalized in many ways, and really no generalization emerges trivially from the facts; it requires some measure of creativity, inspiration. Okay? A person suddenly jumps up and says there is a law of gravitation and it is responsible for the tides, for bodies falling to earth, and for the trajectories of stars—phenomena that seem obviously unrelated to one another. Before someone thought there was gravity in the world, no one would have dreamed there was any connection between those phenomena. Okay? But someone succeeded in inventing a brilliant idea that ties all these phenomena into one explanation. How does he do that? There’s no explanation for it. In fact, the standard philosophy of science treats that as a kind of prophecy. Philosophy of science doesn’t deal with the context of discovery. The context of discovery could come to you through Elijah revealing himself, or from your grandmother appearing to you in a dream—it doesn’t interest anyone. What we want now is exactly—or an apple falling on your head—we simply want, after you bring us the theory your grandmother told you in the night, we’ll test it in the lab; that’s the context of justification. And we’ll try to find predictions. For example, if this theory says that here there will be a one—what is our context? I go to the Torah and I look; maybe I’ll find that it really is one, and that will confirm my theory. For example, if I don’t have it in the Torah, then I don’t have it; so I tell myself that it’s one, but that is, for instance, one possible way of confirming it. One example would be to say: let’s now take one missing datum, throw it out. Okay? And now I ask myself: I will activate my theory, I will fill in that cell; if I get zero, then that exactly matches what is written in the Torah, and I’ve confirmed my theory. If not, I need to revise my theory. Okay? So this is basically a process that is entirely parallel—one moment—entirely parallel to the scientific process. But the great advantage here, and the important value of it, is that here we are actually proposing an entirely mechanical way of explaining the context of discovery. I’m showing you how we get from facts to theory, which is usually perceived as mysticism in philosophy of science. It’s mysticism—how can you get from fact to a theory that generalizes it? But here we have a completely systematic way to show it: we have facts, and I can produce from them a theory through an entirely mechanical logical process. The difference is clear between this and what you said before about the stars and so on, that here you basically have a list of the relevant facts. There too you somehow need to connect the data, but here the list is already connected. Here I need to connect—there are many facts in the Torah, and I didn’t put all of them into this table. I chose the data that seemed relevant to me. In other words, the page of Talmud chose. The page of Talmud chose, or I would choose if I make use of this, it doesn’t matter. But yes, a person does this; it’s not that the Torah itself gives no indications of what is relevant and what is not relevant. But wasn’t their claim that there was some structure to discover that? Were they also bringing data and each time trying to play with them? It’s not clear—which Amoraim are you talking about? Yes, the Amoraim. Certainly. So what I’m doing—not the Amoraim. No, in the end it’s impossible—what’s the assumption? Suppose this model were accepted in all the institutions in the world, then from today on people would have a logical method for getting from facts to theory. What is considered mysticism, actually inspiration—the thing philosophy doesn’t touch at all—I’m offering you a rigid logical algorithm that performs the operation. Your algorithm—but clearly, first, that’s under the assumption that we know the parameters more or less with which we’re supposed to play. No, I don’t know anything; the parameters emerge from the calculation. No, the facts. The facts from which I begin to generate the parameters. Right, still. No, you are right; that’s the sore point here. The sore point is that you have to decide which facts are relevant, whom you put into the table. There are infinitely many data written in the Torah. Right, but the mystical element from philosophy of science, in the context of discovery, still exists. Here it has been translated into how you choose relevant acts and results from which you can infer the conclusion that interests you. What—what is relevant? Which of them is connected? Jazz and fine literature, for example, won’t be connected, so I don’t put them in the table. Maybe even alpha is still mystical. What? Our alpha is still mystical. Fine, okay, so in this case because I didn’t manage to identify it, or perhaps I really do think maybe that is some sort of inspiration. And the question is what is the relation between defining the microscopic parameter and understanding the theory. Meaning, when I say there is gravity, what does that mean? It means that I know how to explain some sort of… the microscopic parameter in gravity is mass. I’ll show you, for example, a common denominator done in science. Look, I see—I think we talked about this once, yes? I take a book, I let go of it, it falls to the earth, right? Now I ask whether this marker will also fall. I don’t know—what proves it from the book, since it’s rectangular? Fine. A ball will prove it. I let go of the ball, it falls to earth. But what proves it from the ball, since it’s round? This pencil is not… this marker is not round. And the law returns: what is true of a book is not true of a ball, and what is true of a ball is not true of a book. The common denominator of the two is that they both have mass, and this marker also has mass, and if they fall to earth then this marker also falls to earth. Okay, all of them have equal alpha, all of them have equal gamma, and here once again relevance will come in exactly—but I’m saying that scientific generalization is exactly a common denominator. The common denominator we do here is exactly scientific generalization; there is nothing beyond that. Okay? It’s simply a generalization in which we take facts and find a theory. Now who is the microscopic parameter? If I analyze this scientific common denominator and look for a microscopic parameter, what will I discover? My alpha will be mass. Having mass. And perhaps, for example, the ball that falls faster will be two alphas, because it has a greater mass than the marker. Okay? So there are two alphas, three alphas, four alphas—that means mass can also take different values. Okay? In fact this thing is exactly the process of scientific generalization. And the nice idea here is that you… that one succeeds in locating the process that has always been considered mysticism—the process of scientific generalization. All of Hume’s questions—he says, what, what, how can you rely on these generalizations? These generalizations look completely arbitrary. Here there is a systematic logic of accumulating information on the one hand—accumulating information meaning filling an empty cell, that is accumulating information—and on the other hand a logic of the process of generalization. The process of generalization is no longer mysticism; there is a systematic way to do it. The mysticism can still lie in choosing the facts. It definitely begins there, yes, and that part definitely remains not… outside logic. Yes, what remains outside remains outside, there’s nothing to do. Yes. I have a question a little less related to the logical issue and more to the Talmud itself. Why does the Talmud introduce things like pleasure but not the issue of worth or similar things? Because it wouldn’t be a refutation. It introduced it as a refutation. When you want to learn from intercourse and money to chuppah, intercourse and money both have pleasure. All the other parameters except gamma—that is, alpha, beta, and delta—you won’t succeed in generating a microscopic refutation from them; at no stage will they constitute a refutation. You won’t be able to say: what about chuppah and intercourse, which have union. That wouldn’t refute any of the stages in the passage. And that is because of the fact that they oppose the… No, they identified only pleasure because that was simply the refutation they had. It fit in for them as a refutation. Because pleasure lines up here exactly against money and intercourse, so it constitutes a refutation of the common denominator that appears here. So they write: what about pleasure, since they contain pleasure. Somewhere else they might have uncovered beta rather than gamma. Okay? It’s an accident, a fortunate one for me because it helped me identify one of the parameters. But in principle it could be that none of them would be identified. If we work only on the plane of halakhic results and not on the microscopic plane, then I’ll have to do all the identification myself. Here, in this case, the Talmud has already done one of the identifications for me. Okay? So is our logic really an intermediate stage between the context of discovery and the context of justification? Their connection? Part of the context of discovery. Part of the context of discovery basically becomes something rigidly logical here. Now I’ll tell you more than that—I think, and of course this is a hypothesis, I also said this at the Nitzotzot conference—I think, or at least I have some feeling like this, that if we take… it may be that one need not leave anything at all to inspiration. That is, it may be that there is a completely mechanical way to make all these generalizations, to build all these theories. How? If I now construct a table theoretically, with all the data there are in the Torah—everything, infinite data, doesn’t matter, a billion columns, a billion, exactly. And I now build it, and there’s one datum that is missing for us. And I say: I don’t know what is relevant to what and what is not, I write them all down. Because I don’t know what is relevant and what is not, I write down everything. If I’m right that there is some rigid, coherent structure here, then maybe I won’t have to guess anything. It will give the correct result. And of course this is completely hypothetical, because who can write down all the data? And if you write down all the data, who can solve it? The tables—you see what is needed here for a table of four by what, six, with a constraint. So for tables of, I don’t know, a hundred by a thousand—which is nothing compared to the totality of the data in Jewish law—that’s something impossible to solve. But on the principled level I’m saying: it is quite possible, I have some suspicion that in fact the entire part of the context of discovery can be mechanized. Meaning there is no part that is inspiration. The part of inspiration stems only from the fact that I want to deal with a low-rank matrix, a relatively small matrix. Okay? And then I want to filter out many rows and columns that seem irrelevant. That’s where inspiration enters. But if I don’t make that clarification, then that is basically what will happen in a completely mechanical process as well. What will basically happen is that I will simply filter mechanically rather than inspirationally all the columns, everything. You’ll see that there are parameters here that are not relevant for generating constraints. Fine, but you still won’t know—you’ll solve the whole matrix. In the end we’ll get to the point where, what? In the end we’ll get to the point that the remaining columns really are not relevant. If you omit them, it won’t change the result. Yes, right. And that of course requires theorems here—what does it mean for columns to be independent—and it is apparently connected somehow to connectivity of the graphs as well. If they’re not connected, that means there’s probably no influence there between the two parts of the graph. In any case, what this algorithm basically gives us is both the ability to accumulate information and the ability to build the scientific theory. If I’m right in my hypothesis, then maybe no inspiration is needed here at all. Take all the data to the very end, work like an idiot, take all the data, whatever you want, solve the whole matrix; if some datum is missing, what comes out will be the right answer. Because it may be that in one number there are more direction changes and in the other there are more microscopic parameters, and you won’t know how to calculate it. Then no—then it’s a refutation. And what does a refutation mean? That zero or one, you can’t fill it in. That there is no true filling, or that I can’t? That there is no true filling—that if you take the totality of the results, it means there is no true filling. Where does that intuition come from for me? Because we talked about it when we talked about the common denominator. In the common denominator there was always the possibility—if I remind you, the common denominator was really learning from two teaching cases to a third thing. And I said that it could be that the X in this one and the Y in that one cause the law, and in this one there is neither the X nor the Y. But on the other hand, I preferred to say that the parameter common to both of them, which exists here too, is what creates the result. Exactly. What does that really mean? My assumption is that every law has only one factor. It is not possible for there to be two different factors for the same law, either X or Y. If the same law exists here in the two teaching cases, that means there is some factor, the common denominator, some factor that generates that law. That turns all learning into… One second, one second, I just want to finish. What this basically means is that if there really is such a one-to-one correlational relation between the factors and the legal results, then I really think no inspiration is needed here. Because then it means that if I take all the data I have in the Torah, everything, and solve it straightforwardly, like some supercomputer, okay? I solve all of it, I’ll get to the right result because there is no degree of freedom. Meaning there is one solution. Every result has one factor; if you found that factor, that is the factor, there is no other. Okay? So I have some suspicion that it is quite possible that even this element remains—for example, in the physical context, if I claim that the Holy One, blessed be He, created the world at the same level of constraint with which He wrote the Torah. The Sages do make some comparison between them—the ten utterances and the Ten Commandments. The ten utterances are the foundations of the world, what the physical world stands on, and the Ten Commandments are the foundations of the Torah. And the assumption is basically that the structure of these two things is similar in some sense; it is built on a similar logic. There is some analogy. An analogy between how the Torah is built and how the world is built, yes? “He looked into the Torah and created the world,” as people always say in this context, and so on. So in that context too, if I’m really right here with regard to the Torah—and common denominator also works in science, as we saw before—then that means that in science too no inspiration is needed. Give me the collection of all scientific facts in the world, write them into a table—yes, all the facts, theoretically of course—all the scientific facts, write them into one big table, solve that table, and out will come biology, chemistry, physics, whatever you want. Everything will come out. Huh? That’s deterministic in some way. No, not deterministic. The process of scientific understanding is deterministic, not the world. The world may be non-deterministic. Doesn’t that make all halakhic ruling completely mechanical? Meaning if I could do that, if there were such a thing, there would no longer be any dimension of choice in halakhic ruling. Right. Halakhic ruling—I’m not sure, because in halakhic ruling there is something beyond merely accumulating information. You also have to analyze it, you have to apply it, and it may be that the case before you requires some analysis, and that analysis is not connected to the table here. The table here only tells you what all the laws are that the Torah gives me. But the laws still have to be applied to the case. That introduces another certain parameter for us, meaning something in the style of… No, he says: here in this case what is needed is union, or this is similar to redemption, or this is similar to that, and then I—okay, and now I will use those data. Here it only gives me all the laws the Torah provides. But from the halakhic book to a halakhic ruling on a concrete case—that still requires additional involvement of a decisor. Meaning some kind of analysis of the situation. Could it be that there shouldn’t be dispute about a common denominator? What? Could it be that there shouldn’t be dispute. Right, on the theoretical level there shouldn’t be dispute if I’m right. The dispute is created because people take only partial tables, not common denominator—common denominator, not verbal analogy. With logical derivations, only because people take sub-tables, partial tables. Otherwise there really shouldn’t be dispute. It seems to me that that’s what we’re not allowed to learn—meaning common denominator from… Huh? Common denominator? No, verbal analogy. Common denominator one may. It is permitted to derive by common denominator even in matters one did not receive from one’s teacher? Yes. Only verbal analogy one can use only if one received it from one’s teacher. The other eleven of the thirteen hermeneutical principles—that is a dispute between Rashi and Tosafot in tractate Sukkah, whether they are like a fortiori or like verbal analogy. But most views hold that they are like a fortiori. There, I no longer remember whether it’s Rashi or Tosafot, one of them says it’s all actually like verbal analogy. But that is an isolated view and it is not reasonable. Also, even in verbal analogy itself there is human involvement. It’s not true that it comes to us straight from Sinai the way we learn it; the medieval authorities already discuss this, Nachmanides and others. The question is whether this is something that was always known in advance, as if what they specifically… No, not known in advance; they created these laws. They created these laws, they didn’t know them beforehand. No, the point is that they created them, but only they were allowed to create them because they had the knowledge… If they have the knowledge and we don’t, then it’s simply irresponsible. It could be that a certain common denominator is true, but it isn’t true with respect to the Torah. So I’m saying: if we don’t have the knowledge to do this, then it won’t be responsible to do it—not because it’s forbidden or permitted. Once I’m persuaded that I have these tools, that I have the skill, that I understand how they work, then I’m allowed to use them. But you can’t do—you can’t make one huge giant table like this and learn on your own. What do you mean, can’t? I said on the principled level—let’s talk about the Sages now, not me. About the Sages. In principle they could have made a huge table with all the data of the Torah and extracted what they wanted. Now, do I also know how to do that or not? Today I tend to think I do know how to do that. And the question is whether what they did was right. What they did we believe is right. Why? Why? Because they thought it was right, and I also think what I’m doing is right. So what? There is always the possibility that maybe you’re mistaken. Why do we not impose punishments based on logical derivation? For most of the medieval authorities that refers to a fortiori; according to Maimonides it applies to all the hermeneutical principles. Why do we not impose punishments? So there are later authorities who say: perhaps there is a refutation. Perhaps there is a refutation, right? The Amoraim did this, so how can there be a refutation? Even if the Amoraim did it, there may still be a refutation, but there’s nothing to do—these are our tools and we use them as best we can. There is always the possibility that maybe we are wrong. But someone who is always afraid of making a mistake will never do anything at all. If that is true, it is true of all science as well. Right, I think that’s true, and we also talked about this in the first classes—how many restrictions there really are on someone who wants to derive by the thirteen hermeneutical principles. I don’t think there are such restrictions; there is no restriction. One does not need ordination, one does not need a Sanhedrin, one does not need anything. These are rules of interpretation that we received, and we are allowed to use them. As long as we know. If we don’t know, then it is just irresponsible. We have to see that we really know how it works. Okay, another point I wanted to discuss here—I see I won’t have time already. Fine, we’ll continue with it next time.