2019-04-22 – Between Midrash and Logic – Lesson 17

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[Rabbi Michael Abraham] We go back from the table and find the solution for the operations. With the operations we can already try to identify things, because here we see, for example, that gamma appears in intercourse and money, but not in the other two. So it’s clear that this is pleasure, right? After that we saw that canopy and intercourse have beta. There’s something shared by canopy and intercourse that the other two don’t have, so canopy and intercourse are connection, right? That was beta. And alpha here was some parameter of intensity; we talked about that. That was—it usually works like this. In other words, the operations produce results, we make a diagram of the results, solve for the results, go back to the table and find the solution for the operations, and from the operations we begin the identification. After we identify all the parameters that appear in the operations, we go back—this is the end of the matter—we go back here, and then we say: for marriage, say, you need some dimension of connection. For betrothal you don’t need it, right? Beta is the connection. Okay? So marriage needs some dimension of connection, and so does a yevama, of course, because a yevama is a kind of marriage. That too needs some dimension of connection. So it doesn’t surprise me that beta appears here and here. Right? By contrast, gamma symbolizes pleasure. Pleasure is needed for redemption and for a yevama. All the others don’t need pleasure. Okay? Pleasure is needed for redemption and for a yevama? What pleasure? Yes. Redemption is—yes. To effect redemption and the yevama, you need something that involves pleasure: money for redemption and intercourse for the yevama. Money and intercourse are pleasure. Fine. Why is a yevama more intense than marriage? What? Than marriage—for a woman, in terms of the yevama. Because fewer operations work for a yevama than for marriage. Canopy does not effect a yevama. So because of that it’s more intense? So that makes it more intense, harder to effect. There are more zeros in the yevama column, because more operations fail to effect levirate marriage. Levirate marriage is only through intercourse, as we know. Okay? Betrothal is the easiest to effect. Marriage is already less so. Intercourse still works, but money doesn’t, for example, and things like that. Okay? Good, so that’s about identification.

After that I said that what we have here—or maybe I didn’t say it, I don’t remember anymore—what we have here is really an explicit description of a process that is usually thought of, I think—and I think we did talk about this, right?—as some kind of inspiration in the philosophy of science. We talked about the context of discovery and the context of justification. In science, usually when I find the theory, it’s some generalization that people treat as inspiration, as a flash in the night, I don’t know. I have no idea how one gets from the facts we measured to the theory. Because for any theory there are infinitely many generalizations; we talked about that in one of the first classes. I said that once I have some set of measurements, I can obviously fit them in infinitely many ways, right? Not דווקא in the form of a straight line. Okay? So in fact there is no direct way to get from the measurements to the theory. The line is the generalization, really—the general theory, right? The measurements are the points I explicitly encountered in the lab. Right? But the theory is my general hypothesis about the overall relationship between the two variables, here and here. Okay? There is no systematic way to get from the measurements to the theory. There are many possible generalizations here, and therefore the philosophy of science assumes there is some step here that is a leap. We have no logical way to describe it.

But already here we see that this isn’t entirely accurate. Right? After all, we choose the simplest one, and the criterion for what counts as simplest can definitely be a mathematical criterion. In this case, a linear line obviously has the fewest parameters. And the line is much, much simpler in many mathematical senses than other lines. Therefore there is also some logical dimension here, and in the most general sense that’s what I’m doing here. Because what we’re doing here—I said this technique has two meanings. First, it helps us fill in an empty square in the table, some fact that we don’t know. And on the other hand, it actually helps us find the theory behind the facts. In other words, to move from the points that I know—in our case, say, from the Torah—to my generalization, to the general law. And now I formulate the general law, say, like this: my general law says that in order to effect levirate marriage you need an operation of intensity two, with the capacity for connection and pleasure. Which of course takes me straight to intercourse, right? Because money has pleasure, but no connection. And therefore money does not effect levirate marriage.

So you see how I move from the facts—the facts are totally dry facts. Which action effects levirate marriage? Which one effects betrothal? Which one effects marriage? Redemption? These are all dry data. Out of the dry data, those are like the points I measured. These are the given things, these are the facts. The facts the Torah gives me. From the facts I try to build a theory. In that theory there may be theoretical conditions, and that’s a whole theoretical world. The theory of gravity contains masses and forces and fields and such-and-such effects and other effects—a whole world of things, none of which was directly observed by me. What I observed was only the phenomena themselves. There are objects that attract, or fall, or tides—those are the phenomena I saw with my eyes. Okay? Those are these points. The theory is to build some structure that explains the points I saw here. I basically did the same thing, except that we found some orderly, unambiguous logical schema that carries out this move from the facts to the theory.

That means we’re actually working here in two different directions. We start from some table that has data in it, say, and one item is missing. All the other data are known and one is missing. My initial goal is to fill in the missing datum, to know whether it’s a one or a zero. But along the way, in order to do that, I build a theory that explains all the data I have. Exactly like here. I ask myself: say this is the relationship between force and acceleration. I ask myself, if this is the force, what will the acceleration be? I only have these facts. So that’s basically like filling in an empty box, right? I’m looking for another fact that I don’t know, on the basis of the facts that I do know. What do I do? I build a theory on the basis of these facts. Now I say: if you apply the force, and this is the correct theory, then the acceleration is this. Okay? So it’s exactly the same process. In other words, the process is to take some set of facts, build a generalization on top of them, and derive additional facts from that generalization. That’s how I add information.

I started this year’s classes by saying that logic doesn’t help us describe the addition of information. We can’t use logic to describe adding information, because logic is always empty of information. All the validity of logic comes from the fact that the argument adds no information beyond what was already in the premises. Now here, this is a logic of adding information. It’s a logic of science, not of mathematics. And it’s a logic of how we produce more information on the basis of partial information we have in hand. We have these four measurements, and I ask what the result will be here when the force is such-and-such. So I’m trying to produce more information—and notice, I do it not by measurement, but by thought alone, which is absurd on its face. In other words, I take facts—four facts, that’s what I measured. Everything I did from this point on was pure thought. Nothing observational. That’s amazing, no?

And my claim is that my pure thought succeeds in reaching additional facts that exist in the world even though I didn’t observe them at all. And you also have a kind of verification that you’re right. Because if I understand, when you go back to the parameters and see, for example, that this is pleasure, and pleasure appears here and here, and from the Torah I know that there’s supposed to be pleasure here, that gives me—right. That’s the identification. I’m saying I’m right. Sure, but I mean even before the identification. Suppose I didn’t identify at all. I don’t know what alpha is, what beta is, what gamma and delta are. The whole thing works without identification; identification is a bonus. Identification is a bonus, but it does provide the seal, basically, right. But on the fundamental level, identification gives the generalization of all the information. In principle, identification is a bonus. No—for the generalization I don’t need the identification. If, for example, you ask me now—say there’s another action, another result, sorry, H, I don’t know, some other result—and I ask whether canopy would effect it or not effect it, I don’t need to identify alpha, beta, gamma, delta for that. I don’t need to. I’ll build the data and generate it. But identification is an added bonus.

Identification is what I said last time too: when the classic analytic scholars learn, they don’t deal with this kind of thing because they don’t have the tools to do it. But now we’ve basically gotten a tool that can help us understand what the Torah means when it says that money helps for this, intercourse helps for that. Those are all dry details. But I’m asking: what’s the idea behind it? The idea behind it is probably that this involves some kind of connection that includes pleasure; this one is pleasure and doesn’t require connection; that one is something else. In other words, each of these gives me one step deeper than what the ordinary analytic scholar can reach. This identification helps me understand the thinking behind Jewish law. There’s something here about understanding the conceptual dimension. After that I can begin asking why levirate marriage really involves connection—again, one more layer, one more level deeper. I can keep going and ask myself all sorts of interesting questions.

By the way, maybe I can even apply this very technique itself. After I’ve solved many, many problems, I can now make a table in which the microscopic parameters appear. And I’ll say: inside Y there appear alpha, beta, gamma, delta. Inside Y there appears alpha, beta does not appear, and so on. Then I say: and for this operation, what things will be required? Pleasure, connection, things of that sort. I can keep going, each time entering one level deeper, and understand why the phenomena are what I observe. And if the phenomena are the theory—that is, alpha, beta, gamma—then I’ll ask why about the theory, and then the theory will become the phenomenon, and I’ll find a second-order theory. Something deeper. So there’s something very, very powerful in this technique. It’s basically a move from facts to theory, something that is usually perceived as mere inspiration, not as something you can trace logically. Okay?

That’s exactly what I meant to say before: the leap that the philosophy of science says is made in theory—here, because of the identification—that gives you… The identification itself really is a kind of leap, because I have no systematic way to make the identification. I see that if gamma appears here and here, then gamma means pleasure. That sounds reasonable, but I have no mathematical way to do it. No, but isn’t there one of these parameters that I know for certain the Torah says, “this has pleasure,” “this has…”? No, where does the Torah say anything? I can understand what intercourse and money have in common. I ask myself; the Torah says nothing. Here it is abstract. There isn’t in the Talmud some… “Just as money has no…” No, I’m talking about the Talmud. And how does the Talmud know? I’m trying to imitate what the Talmud does—to imitate the Talmud. I’m asking how… I’m trying to imitate what the Sages did. For me, the Talmud is the Sages. About the Talmud I can say, if the Talmud says such a thing, maybe I can say it’s tradition—no, who says it’s tradition? They understood it, just as you’re trying to understand it.

I said that at the beginning of tractate Bava Kamma, for example, with the four primary categories of damages, the Talmud itself gives us the microscopic parameters there: “their common feature is causing damage,” “its way is to go and damage,” “another force is involved in it.” All these characteristics of the four primary categories of damages—the Talmud there explicitly gives us the alpha, beta, gamma, delta. Not the characteristics—okay, the laws that apply through horn, tooth, and foot, and so on. There they give us the solution directly, not the alpha, beta, gamma. Okay? But that’s rare. Usually it doesn’t happen. Usually we work with the laws. I talked about this in connection with the common denominator—that if the common denominator speaks about the microscopic parameters, it works differently than if it speaks about the halakhic outcomes, where one can refute it with the objection of “the strict side.” Okay?

Now, that is really the meaning of this model: there is here some systematic technique, an algorithm, for going from facts to theory, and I think that’s very significant. True, it doesn’t completely solve the inductive leap I talked about earlier, because identification, for example, is something we do by reasoning. I have no logical way to proceed with the identification. But it gives a very important instrument. Because I now see for each parameter in which halakhic actions it appears and in which it does not appear, and that really helps identify it. It helps, that’s all.

Now another important point. Look, actually—and this also touches on identification—when I look at this model, I could have built something else too. I could have said this is alpha, this is two alpha, this is three alpha. Okay? This is three alpha and… no, that didn’t come out right. This… yes, it could be two alpha, alpha-beta, and then Y would be three alpha-beta. What? No, here I can’t, because if this is two alpha and this is two alpha, there’s a relation between them. I need somehow to account for that. I’m looking for another solution to the same… let’s think for a second. Up above… let’s see here. Here alpha. Let’s make this alpha. Then here I’ll do alpha-beta, and here I’ll do two alpha. Here I’ll do two alpha-beta and gamma as before, because I have a constraint. And here it’s three alpha and beta. So why doesn’t that force what’s below? What? Because below there’s also gamma. Where? Why doesn’t Y force… no, erase that, that’s left over from before. Here it’s alpha-gamma… ah wait, no, that’s Y. Where was I? Here. Alpha-gamma as before. Alpha-gamma-delta. Okay? That’s also a solution, and it’s also minimal.

Now notice what comes out of this. I write this also… whether with filling zero or filling one—one, all one. Doesn’t matter. This is another solution for filling in one, and it’s also optimal. There’s no way to choose between them. That’s an important point, especially when we’re talking about identifying the parameters, as you can no doubt understand. Okay? That’s the black solution, which is an alternative solution. Fine. Now I ask myself what the new parameters are—call them alpha prime if you like, no matter, I marked them as black alpha and blue alpha. Fine? So now I have to identify the parameters מחדש; it’s not the same anymore. So of course gamma will still be pleasure, right? Here there’s a one and here there’s a one, and gamma is in both and not in the others. So gamma is pleasure, because that’s how we built it from the start. No problem. But now, for example, what will beta be? Look: here now there is no beta. Beta appears only in the two lower ones, in intercourse and document. In intercourse and document there is beta. Before, beta was connection, right? Intercourse and canopy. Now beta is intercourse and document. It has no simple meaning. What do intercourse and document have in common? I don’t know. It has no simple meaning, right? And delta, again, appears only in money, so maybe that’s value. That’s what we also talked about last time; that’s true here too, it hasn’t changed. In this case only beta changed its role. But in principle maybe there are other solutions where the parameters change entirely. And then notice: the identification changes, but nothing else changes beyond that. The predictions are the same predictions, everything will still be correct. There’s no problem with that. The only thing that changes here is the identification.

Now I ask myself: who is alpha, who is beta, who is gamma, who is delta? And of course I’ll get different answers. Now, before, for example, beta was in canopy and intercourse, so I said beta is connection. Now beta is in intercourse and document, so it can no longer be interpreted as connection. So what is it? I don’t know. On the other hand, who is the parameter responsible for connection? We don’t have one. There is no parameter here shared by intercourse and canopy. So what happened here? So I promise that the solution I’m… exactly. You said earlier that you need feedback, that this gives feedback confirming the solution. So you have to be careful with that. Because in fact there is a whole family of optimal solutions—a whole family of optimal solutions. Here I drew two of them; there may be more. One would have to prove theorems about it, but never mind. These are two solutions, for example. That’s good enough for us for the moment. The identification of the parameters will be completely different. Here, for example, I really don’t see a simple identification for beta. I don’t.

So what does that mean? In mathematical language, let’s call it this: we are rotating the solution in parameter space. Meaning that now the meaning of the parameters does not map simply onto connection, value, pleasure, and intensity—say, alpha. Rather, what will connection be? For example, connection is these two. What distinguishes these two? That in both of them, say, alpha is two. Fine? You see, that distinguishes only those two, right? So that means there is some dimension of connection in alpha that descended from beta. Okay? Alpha has acquired a different meaning, because beta also gets a different meaning. Everything is now switching meanings.

I’ll try to explain this a bit mathematically, but I think it clarifies the issue. Look. Suppose we have some coordinate system. Fine? And here is a point whose coordinates are these. Now I look at a different coordinate system. The coordinates of that point are now these, right? This is x-prime and this is y-prime. Right? Understood? Fine. There is some angle alpha between the coordinate systems. I rotate the coordinate system by alpha. Okay? There is a relation between the coordinates in the primed system and the coordinates in the unprimed system. Right? Say x-prime—which is this—you drop some line here and it gives me something. You can express it by x sine alpha plus y cosine alpha, something like that. Fine? And y-prime is something with a minus sign, or whatever, some other combination of them. Fine? So what does that mean? That x-prime equals A times x plus B times y, and y-prime equals C times x plus D times y. Right?

Now look. What is a rotation? A rotation means that my new x is a combination of the previous x and y, and vice versa. Right? That’s what happens here. Right? So what am I saying? Before, beta was pure connection. It was exactly arranged, right? It was exactly canopy and intercourse. Now what distinguishes canopy and intercourse? There is no single parameter that distinguishes canopy and intercourse. So that means that something from alpha plus something from beta together is really the connection—sorry, the connection. And pleasure minus value is really the new delta, for example, whereas before delta was just value. Okay? Therefore the simpler interpretation of the parameters comes from the blue solution. The black solution is really a rotation of the blue solution.

I’ll give you another example. People often make a comparison between Aristotle’s view of chemistry and the modern view of chemistry. In the modern view of chemistry there are of course many elements—the periodic table. For Aristotle there were fire, air, water, and earth. And then people say, “See? Aristotle was wrong, we’re right, and we’re smarter.” But that’s not precise. You can look at the Aristotelian description as a rotation—of course in a much more abstract sense—of our description. When you speak about earth, all you’re really saying is: it’s a combination of such-and-such a collection of elements with different weights, right? You can reduce Aristotle to us: A times x plus B times y plus C times z, and so on. Take all the elements and build earth out of them, something like that.

Now I could, for example, describe water. In the simple system, water is one of the elements. So the description of water would be 0, 0, 1, 0—the third element. Right? Are you with me? Fine? Meaning that water—to know what it is, you multiply by some vector of, say, fire, air, water, and earth. Fine? When I want to know what water is, water is that product, right? Zero times fire plus zero times air plus one times water plus zero times earth, which is water. Okay? Fire is this. Fire is this, right? Now what happens if I choose a different coordinate system? Not these, but—hydrogen, oxygen… fire minus air… no, no, let’s stay in the same neighborhood. Fire minus air is one component; two air plus three water; minus two water plus earth; and two earth minus fire. Whatever. Okay? It’s just a different coordinate system, and I can express water using that system too. It will be, I don’t know, let’s see if something works here, yes—it’ll be one and one. It’ll be this. Right? Look: one times this gives three water plus two air, and here it subtracts two water, right? So I’m left with one water. But I’m also left with two air, so maybe subtract this by two, and so on. You understand? I can build a different vector as the basis that presents the picture.

What is that? That’s exactly what I’m doing here. I’m simply saying that I’m not presenting the picture as a simple combination of value, connection, pleasure, and abstract intensity. Rather, it is pleasure minus value, and this is three value plus connection minus intensity, and so on. And that is the new system, and that is the black solution. Now of course, at the formal logical stage, it doesn’t matter which solution I choose. It’s not important. When I want to identify, the easiest way to identify the things is if I work in the diagonal system, in this simpler system. Fine? Then you might also find a system in which both are simple—two different interpretations. But if the interpretations are different and both are simple, then probably one can map one onto the other. So that would mean, for example, that connection is pleasure plus value. If indeed connection is intelligibly pleasure plus value, then yes, in one system connection will appear, and in another system pleasure and value will appear, and both will be simple systems. But they really will be equivalent in some way. Otherwise these are genuinely two competing explanations, and then I don’t know what to do. Right, no—I’m only saying it could happen. And if I have two competing explanations, then again, as I said, that stage is no longer logic. That stage is interpretation.

So I’m saying that in the end, very often the artistry still remains. You still have to be an artist; it’s not enough to be a mathematician. Because in the end, after you arrive at the solutions, you need to generate a whole set of solutions. Don’t choose the first solution you got. For each of the solutions, try to find an interpretation that fits with what appears where. And only the one that really settles into a simple interpretation—that’s the solution worth working with. Okay? Then I’m willing to identify parameters. Otherwise all the rest—and then I can of course move on.

Okay, now a few more remarks. What I said last time: this algorithm does not turn induction and analogy into deduction, as it might seem at first glance. Right? That question really bothered me after we finished this business. So what here is not deductive, really? There is a completely rigid mathematical technique for filling in the empty square, and basically that would mean that analogy and induction are really just kinds of deduction. How can there be a rigid algorithm for analogy and induction? But it still relies on the fact that you chose the simplest solution—basically the straight line that connects, and not the… Yes, but the criterion of the simplest solution is a mathematical criterion. I have a rigid logical way to describe the results of the analogy. Okay, but it’s not logical. What do you mean not logical? Mathematics and logic are the same thing as far as I’m concerned. But it’s something rigid, univocal. Analogy can’t have two answers; there is only one answer, and it is predetermined, and there is a rigid algorithm that gets us there. But what determines that in principle? But why is that principle logical? Can you explain it? What does explaining it have to do with anything? It’s not about explaining it. By “logical” I mean: usually when we think about analogy and induction, we think about things that don’t really have a necessarily true result. Another person might make the analogy a bit differently. The result of an analogy is not a necessary result, unlike a mathematical or logical calculation. But is induction absolute? No. Induction and analogy are both not absolute. On the contrary, deduction is absolute. Sorry—okay, so when you don’t find deduction, you want to turn this into deduction, and you’re saying it should be absolute. When you use deduction—here, it’s absolute. You ask me what the result in this table is—true or false? That’s also what logical calculus does.

What does logical calculus do? You build some compound statement and ask yourself whether it is true or false in light of certain premises, right? That’s what I did here. I have a collection of premises—that’s the table. In light of these premises, I have a rigid algorithm that tells me whether the result I’m interested in is true or false. So what’s the problem? The problem is that it doesn’t seem to me that analogies and inductions are supposed to be deductions. Science is not… science is not a deductive instrument. Science doesn’t use deductive tools. The path from facts to generalizations—that’s what we talked about before in the context of discovery, inspiration—that’s induction, right? Exactly. But here I do the induction using tools that give me a predetermined, unique result, by means of a mathematical device. Can a computer do analogies and inductions? Usually people think not. Well, what’s the problem? Give it the data, it’ll do whatever you want.

So that depends again on what you fed into it. Ah, so there are two things here. One of the respects in which analogy and induction are indeed still preserved here is that I choose what entries the table has—what actions and what results I put into the table. We talked about fine literature and jazz, right, that I wouldn’t learn one from the other. Abstract art and… yes, never mind. So the decision about what goes into the table is a decision I make intuitively, right? And that’s one aspect. But here we need to notice: even in deductive arguments—even in deductive arguments—who chooses the data? After all, a deductive argument also starts from premises and derives a conclusion from them. I chose specifically those two—why? Because they seemed relevant to the area I’m interested in. So what’s the problem? So here too I choose the premises. But once I’ve chosen the premises, the result that follows from them is unique. So I still haven’t explained where exactly this is not deduction. Do you see what I’m saying?

In fact this is still deduction, because even in deduction, even in pure logic or mathematics, you choose your premises. Obviously; a person always chooses the premises. What logic deals with—wait, wait—what logic deals with is inferring a conclusion from the premises that I chose. That’s what logic is about. Now that’s true here too. My axioms are the data in the table. But those aren’t axioms; those are things you choose because… Why? One, zero, one, one, zero, one, zero, zero, zero, one, zero, zero, zero, one, one, one. Those are my axioms, apart from that. Fine? And now I ask: I want to infer a conclusion from the axioms. So what exactly is this new fact? Why is it different? I choose premises, and the path from the premises to the conclusion is algorithmic. So the fact that I choose premises doesn’t make it non-deductive. In deduction too I choose premises. The question is how I proceed from premises to conclusion. And from premises to conclusion, here too, the path is unique and entirely mathematical. So again: where is the difference between deduction and analogy and induction?

Maybe in their distance? Deduction is still from particulars, and here it’s from the general… Just a second. Maybe from their distance in the cognitive place? You can give everything… you can derive anything from a starting point and build on that. The question is your starting point—say, in mathematics, that a point has no area and from that you build all of mathematics, or that an isosceles triangle is… Fine, you choose assumptions. But the question is at what level your assumption is. What do you mean, at what level? The assumptions there are those kinds of assumptions, and the assumptions here are different assumptions. Mathematics is the path from assumptions to conclusion; it is not the choosing of the assumptions. It is also… but maybe you can go back before… before that, and choose lower-level assumptions. Here too you can choose other assumptions, so what? Who told you otherwise? What characterizes mathematics and logic is that they do not deal with the question of how we choose premises. Rather, they are the tools by which we advance from premises to conclusion. And that’s what defines them as mathematics and logic: those tools are univocal. The analogy and induction are the premises we choose, and deduction is the tool afterward. I don’t see anything here.

What do you mean, analogy and induction? The analogy by which we infer analogically that there is a relation among money, canopy, intercourse, and document. They all… well, and you also infer analogically that there is a relation between “all human beings are mortal” and “Socrates is a human being.” Okay, and then you work with deduction. But there’s a difference. What difference? The difference is that deduction is the tool you use after you’ve chosen. But here too it’s exactly the same thing. Okay, I didn’t say… You choose a set of premises and use a mathematical tool to reach a conclusion. So what is the difference between deduction and this? I still don’t see any difference. Why should there be? I don’t know—the intuition, the initial intuition of anyone you ask, is that science is not mathematics. There is something there. Analogy and induction are things that are not univocal. For example, they are not necessarily correct. Deduction is necessarily correct. Wait, let’s define the terms. In deduction, necessarily, you are right. In analogy and induction, at least the feeling is that you are not necessarily right. Sure, because maybe you’re not right about the conditions—you choose here the parameters. But in deduction too. Again. No… suppose with money—who told you that exactly these parameters are the right ones? You choose money according to things you think, the way you understand money. You understand these parameters. No, these parameters are a result; you don’t choose them. They are a result of the model. It’s an algorithm. I don’t choose anything. There is a table.

How do we choose the… the things we put there, that we attribute to the table? Of course I have an algorithm that tells me how to build the solution for a given table. No—how do you build… I asked how you choose the entries of the matrix, the A, B, C, D. Yes. So what? That’s what I said earlier: I choose my premises, just like in ordinary deduction. I choose my premises, and now I move from the premises to the conclusion. The conclusion is the question mark there. I want to fill it in. And I see that the march from premises to conclusion is a unique mathematical march with one correct answer. So in what sense… then why isn’t this deduction, basically? There’s something very disturbing here on the philosophical level. I don’t know how to explain it mathematically, but in deduction you’ve already encompassed everything you want, and now mathematical rules make you find it in the same place. Whereas… No, I understand what he’s saying, but you’re repeating the problem; you’re not offering a solution. That’s my problem. My problem is: how can it be that in a process like this, where I start from a set of facts and look for a new fact that is not included in them, I succeed in doing it by a purely mathematical route? This is basically Kant’s problem of the synthetic a priori. How do I manage to know things about reality not through observation? These things do not come out of the facts I have. It’s some act of thought I perform. So why should this result be correct? How do I know it has any connection to those facts? Just like here, by the way—the same thing. How do I know that this thing will also correctly describe reality? It’s the result of some act of thought I performed when I drew the straight line. That’s exactly my difficulty.

I would have expected there to be something here that is not a rigid algorithm, because… I can say it’s a hypothesis. Fine, I understand that—a hypothesis. So give me some way to arrive at the correct hypothesis. But how can it be possible to generate hypotheses by a mathematical route? Meaning that basically you arrive at it with certainty. A hypothesis is usually seen as something that might be right, might be wrong; there are different ways to do it. How can there be a rigid mathematical algorithm that handles adding information? Mathematics does not deal with adding information. Neither mathematics nor logic. Scientific generalization adds information for us. And here suddenly I translate a process of scientific generalization into logical or mathematical tools. There is something very problematic here.

What? The generalization happened when we chose the entries for the matrix. After that… That’s true. That’s also true in deduction. Again I tell you: there too I basically did the work when I chose the two premises from which I derived the conclusion. I could have chosen infinitely many entirely different premises and wouldn’t have been able to get any conclusion from them. Okay, so that means that there too there is a process of choosing premises, so this still isn’t different from here. Okay.

I’ll tell you: we happened to hear a lecture at Tel Aviv University in computer science, and there someone—and now the penny dropped for me—someone there said that among AI people there is an ideological debate about what the role of artificial intelligence is. Is artificial intelligence supposed to arrive in the best possible way at the correct answer? Or is artificial intelligence supposed to arrive in the way most suited to what a human being does? Humanly. The answer closest to what a human being would do. And that was a very interesting remark for me, because I think it’s the solution to the problem here. This algorithm brings us, say, to the conclusion that here there is a one. That doesn’t mean the correct answer is one. It means that if a human being thinks about such a problem, he will arrive at the conclusion that it is one. But wherever human thought would err, this algorithm would err too. That is, it will never work better than we do. At most it will do perfect human work. At most. Fine? Meaning, at most it can successfully imitate the way a human being works. If a person has these data, he really will say that the result here is one. Fine?

If so, this algorithm imitates the way our mind works. In other words, it tracks the way our mind works. But that doesn’t mean—the mind can make mistakes. And here is the difference from mathematics. In mathematics the difference is not in the method. The method here too is an algorithm, and in mathematics too it is an algorithm. But in mathematics the result is necessarily correct, if the premises are correct, of course. And here, even if the premises are correct, the result may not be correct. But it matches the premises. No, it does not match the premises—no. It matches our way of thinking. That’s exactly the point—not the premises. In mathematics you can’t have that. If the premises are correct, the conclusion is necessarily correct. It cannot happen that the premises are correct and then you suddenly measure something and discover that the conclusion was wrong. But here that can happen. The fact is that one can refute an a fortiori inference, right? How can there be a refutation of an a fortiori inference? I made a logical inference, arrived at the answer that it is one, and then suddenly I find in the Torah that I was wrong. How can that be? With mathematics you will never find in the Torah that you were wrong. That’s necessarily correct; there is no such thing.

So what this algorithm is doing—and this too is very far-reaching—but it still isn’t deduction. It succeeds in imitating, in doing a very good simulation of how a person thinks about such problems. How we make generalizations, how we arrive at theories. But we as human beings can of course make mistakes. And wherever we make mistakes, this algorithm will make mistakes too. Or at least, let’s put it this way: wherever the ideal human being would make a mistake, it will make a mistake too. Meaning: it is the ideal human being, as far as I’m concerned. I can miss something in the inference if I don’t mechanize the process; one of the problems is that maybe I’ll make a mistake, I’ll get confused in something complicated. That’s the advantage of a mechanical process. Fine? So the computer will do it for me and reach the correct result—correct in the sense that it is the result I should have arrived at. But that doesn’t mean that this is really the result that will later be written in the Torah or measured as a fact in the lab.

A kind of Torah Turing test for this thing. Yes, yes, exactly. Basically there is here an algorithm—which I also think is very far-reaching—but it is not deduction. Meaning, it is not true that if all these premises are true and this algorithm gives me a one here, then for sure when I search the Torah I will find a one. That is simply not true. It could happen that I search the Torah and find a zero. What does that mean? That in my mode of thought I will not be able to reconstruct that result from the Torah. Because my method does not know how to get from these premises to the correct conclusion here. I have a bug in my way of thinking. It’s simply not true; a human being does not know how to get to that result.

Then the premises aren’t correct? What? No, no—the premises are correct. The problem is in the routes of inference, that’s what I’m saying. The premises are correct; I take them from the Torah. All the zeros and ones written in the table are data taken from the Torah. No—the routes of inference. No, all these premises are taken from the Torah. Zero, one, everything. Canopy, money effects canopy—I don’t know—money effects betrothal, marriage, everything I take from the Torah. Now I have canopy—does it effect betrothal or not? I don’t know; I didn’t find it in the Torah. So I calculate and it comes out that yes, it does, that it comes out as one. Fine? Now I go search in the Torah and suddenly I see that it’s not true. Here that can happen. In logic and mathematics it cannot happen. You will not find in the Torah that the Pythagorean theorem is false. There is no such thing. Things that are logical or mathematical necessarily follow from the premises. Here I can accept the premises and still discover that the conclusion of the inference is incorrect, and therefore this is not deduction.

Meaning, the method I’m using—we’ve managed, say, to produce artificial intelligence, that’s what we did here altogether: how a human being thinks, in a completely mechanical way. Now a computer can do the work a human being does. But that doesn’t mean the computer will always be right, because even the ideal human being will not always be right here. Unlike mathematics and logic, where if the premises are correct, then certainly the conclusion is correct too. So there’s the challenge between this and deduction.

About that dispute regarding AI—there is a side… What do you mean there is a side? It’s an ideological question. Someone else could come and say: I’m looking for an algorithm that… No, not that either. Artificial intelligence will do whatever you want. I’m saying what this algorithm does. This algorithm does not find the correct answer. It finds the answer that the ideal human being would find. That’s what it does. I know what I found. I’m not saying what one should find, what it is right to find, what can be found. I’m telling you what I found here. That’s all. It’s not an ideological statement; it just made something click for me. I’m not taking a position in those debates.

It’s also not really a debate—just a question. Obviously this problem is valid, and solving it would have value; and solving that would have value too. It’s just a matter of defining two sides and distinguishing between them; that’s the value of the debate. It’s not that there is really some actual disagreement here. Like disputes in hermeneutics, about interpretation. There are people who say you need to understand the author’s intention—that is the meaning of the book or the work. And there are people who say the meaning is the book itself. And there are people who say the meaning is simply what I get out of it. Right? So what kind of disagreement is there here? Whoever seeks the author’s intention seeks the author’s intention. Whoever seeks the meaning of the book seeks the meaning of the book. And whoever seeks what it says to him seeks what it says to him. So what are they arguing about? Tens and hundreds of articles and books—and I think thousands—grind away at this problem of what interpretation really is, when in fact there is no dispute at all. What is interpretation? It depends on what you are doing, that’s all. If you’re seeking the author’s intention, that’s what you’re seeking. If you’re seeking something else, then you’re seeking something else. What is proper? No—what do you mean, what is proper? What is proper is: do what interests you; that’s what’s proper.

I think the real debate behind these things is not what is proper, but what is possible. And that is a real debate. Because the deconstructionist interpreters, who say that interpretation is only what it does to me, are really claiming implicitly—and sometimes not even implicitly—that you will not be able to find any other interpretation. You have no way to reach the author’s interpretation, because you can’t extract from the book what the author meant. All you can know is what it does to you. What? And maybe with artificial intelligence it’s the same way. Meaning that the AI people who try to find the correct answer—the others are saying to them: you can try until tomorrow, but you’re the ones building the machine, so there won’t be anything in the machine beyond what you know how to do. Therefore, at most, you can build a machine that is an optimal human being. That’s all. You won’t be able to build a machine that is God. The one who builds the machine imposes a constraint on what he can expect from the machine. And maybe in that respect there really is a genuine disagreement. So I’m saying that is true here too.

Good. Maybe one more thing. What I want to show you now, following this, is a few examples of applications of the algorithm—entirely theoretical possible applications, I haven’t launched a startup yet. But I think my goal is more than the applications themselves; through them we can understand better what this algorithm does. Okay?

So look. One example: let’s say we have a collection of data about storms. Storms—typhoons. Tsunamis, typhoons, all kinds of things of that sort. And I have, I don’t know, some table. What, for typhoons that occurred in the Pacific Ocean? Yes, exactly. I have some table and I say, I don’t know, say Katrina, okay, tsunami in Japan, and so on. Fine? They always give these storms women’s names. Now here, say, core size, distance, speed, intensity, duration of survival, and so on. Fine? And let’s say, so we still stay within our framework, that we do this in binary form, zero and one. Although you see here that some of the data—distance, for example—is continuous, not zero-one. So let’s say the distance is above one hundred kilometers or below one hundred kilometers. We digitize the data. Okay? So we fill it in: one, zero, one, one, one, zero, zero, zero, zero, one, zero, one—I’m just filling in values, of course. Fine, say something like that. Fine? Now here we have a question mark. A question-mark one. Now I ask—not here, that’s not so interesting—say here. I ask: after how much time will the third storm stop? Now a storm has developed; I’ve already measured the core size, distance, speed, all kinds of things, and I ask: will its duration exceed a week? Fine? Again we digitize: above a week or below a week. Fine?

In principle I can apply this algorithm. There’s no reason not to do it. What I’m doing is just scientific generalization. And not only can I apply the algorithm—the algorithm will also give me the theory of storms. I’ll find here alpha, beta, gamma, delta, which will be the variables, the parameters. They’ll be the physical or meteorological parameters that affect all these physical properties of the storms. I’ll know that Katrina has two alpha plus beta, and this tsunami has two alpha plus three gamma plus beta, or something like that, and so on. Fine? And now I’ll be able, from that, to predict, say, how long the third storm will survive. That is exactly the process of scientific generalization. That’s how you build a scientific theory.

Now of course this is primitive because it’s only zero-one. But if it were continuous—and one can at least try to think, we haven’t done this yet, it’s on the agenda—to think about extending this algorithm to a case where the data are continuous, not zero-one. What, to look at it in terms of magnitude? Meaning, I have more of this and less of that… Meaning, without digitization. When you say it would be continuous, I don’t look at whether this is zero and this is one, but whether it’s, say, greater than that or… No, no—numbers! The size of the storm’s core is a kilometer, or three kilometers, or half a kilometer, or a hundred meters. How… exactly. So now I’m saying: I need to build a theory, or kinds of theories, that explain continuous results. Fine? So I’m saying that’s a future task. I don’t yet know exactly how to do it because we haven’t done it yet. But I don’t think it’s implausible to find such a theory, or such an algorithm, for continuous data.

I’m telling you in principle that this thing can be applied entirely apart from law, Jewish law, or anything else. All it means is: look, I have some horribly complicated problem here. It’s clear to me that there are all sorts of parameters inside the problem that determine its properties. Where did the storm originate, I don’t know, from what, where did it come from, at what angle? It doesn’t matter. These alpha-betas can be all kinds of things having to do with how the storm formed, the surrounding temperature, all sorts of things of that kind. Okay? But right now I know nothing about storms. I’m starting the science of storms from scratch; I know nothing about it. Fine? So no problem—I write down the data, I measure, just measure, like any scientist does. Fine? And now I’ll analyze the table: diagram, alphas, betas, gammas, deltas—I’ve found a theory of storms. Now of course it would be good to identify who alpha is, who beta is, who gamma is. Maybe one of them is ambient temperature, maybe one of them—I’d have to see. To say, whoever appears in Katrina and the tsunami, I’ll ask myself what Katrina and the tsunami had in common that the others didn’t. Exactly the process of identification. Then I’ll find what the parameters are that define the physics of storms.

Or another example. Someone who worked with us on this used it for an exercise. He took a table of computer monitors from the internet. He had all kinds of data, a collection of monitors from different manufacturers, and there were data points: resolution, response time, price, all kinds of things that at the time I didn’t know what they meant for monitors. Fine? He built a table. For one of the monitors, one datum was missing. They didn’t provide it; it just wasn’t there. In principle, from the table, it was possible to reconstruct it. Again, we digitized, of course. Response time is again something continuous. So you say: above one hundred milliseconds or below one hundred milliseconds, fine, just as an example. Then I turn that into zero or one. And of course after that you can cut more finely. Once you’ve determined that it’s above one hundred milliseconds, you can now ask another question: is it above two hundred? You can keep applying—this is one way of reaching a continuous theory. I take the digital theory and apply it several times in succession. Each time I partition the problem more and more until I can reach whatever precision I want in the answer. In principle that’s also a possible answer. Of course it’s not computationally efficient, but it is a possible answer.

And basically you can reconstruct data. What stands behind that? What stands behind it is the assumption that there are parameters characterizing these monitors that control, for example, how much resolution this one will have—what material it’s made of, for instance, or in which factory, or how much it cost. Cost, for example, can certainly allow better performance than something with lower cost. Fine? So there are all sorts of facts here. I can apply this to facts—that’s what I want to show you—not only to judgments, not only to laws or legal rules. It can also show us correlation and not only… Yes. Once I know that, say, resolution and price are both alpha, then I understand that resolution depends strongly on price, for example. Okay? Things of that sort. Or perhaps this depends more strongly on price than that one. Each probably depends somewhat on price, but which one depends more strongly? In other words, what I’m trying to convince you of is that there really is here a systematic way to produce scientific generalizations in any field whatsoever, with the basic assumption, of course, that there are some microscopic parameters—that is, that there is a scientific theory behind the events. These aren’t just things that happen for no reason. There are parameters behind the events that determine them. That is the assumption of all science: that they are what actually determine what will happen.

Can you prove that there is a scientific theory before you find it? No. We assume that there is a scientific theory. That’s an assumption. Not specifically about storms? No—generally. Science always assumes there is a theory, and once it assumes that, it says, fine, if there is, then this is the theory. But maybe there isn’t. Maybe it’s all accidental. Maybe there really are no rigid rules that govern the behavior of the world. Maybe there aren’t any at all. Maybe it’s just a collection of cases. That is an assumption. The assumption of science is that there are rigid rules here. I’m saying: if you assume that assumption, you can use this technique and derive a system of rules for any scientific problem. But the number of columns, for example in the case of storms, will always be greater—there are columns you won’t even be aware of. Fine. Of course you don’t always arrive at the correct answer. But that’s true in science too. Maybe you didn’t take other data into account. Science advances. I’m saying: at the first stage I take all the data that sound to me somehow connected. Then I build some theory. If it doesn’t work, one option is to understand that either some of the data I took are irrelevant, or additional data are missing and must also be added. Then I build another table until I get there. That’s how theory is refined; that’s how science always works. Okay? Good, let’s stop here.