Torah and Science, Lesson 2

Table of Contents

• The youth’s dilemma and Kant’s synthetic a priori
• Cognition versus thinking and scientific generalization
• Uncertainty, control mechanisms, and the scientific method
• A response to the claim “you just changed the name” and the practical difference as against skepticism
• Kant, broad empiricism, and intuition
• Morality, relativism, and the objectivity of cognition
• Torah, Jewish law, and moral boundaries versus halakhic boundaries
• Non-certain seeing and dependence on the observer’s system
• Deduction, induction, analogy, and the uncertainty principle between information and certainty
• Mill’s critique of deduction and non-Euclidean geometries
• The claim that there is only analogy
• Mathematics, Popper, and combining forces
• “One may not derive punishments from legal inference,” a fortiori reasoning, and the rule that two hundred includes one hundred
• The example of the Vandervelde law and wine
• Summary of the introduction: uncertainty, probability, and reasoning as Torah-level

Summary

General Overview

The text presents a late-adolescent dilemma: whether to accept only claims that are proven and certain, or also claims that are probable even if unproven. It places that dilemma within Kant’s problem of the synthetic a priori, about the possibility of making general claims about the world without direct observation. The argument is that the move from observed particulars to a law of nature or a theory cannot be mere internal “thinking,” but rather some kind of “cognition” or “seeing with the mind’s eye” that makes induction possible—while being careful to note that this is not sensory certainty and is open to error, and therefore requires control mechanisms in the form of the scientific method. The same logic is then applied to morality: if morality is only a product of an internal structure, it has no binding validity, so an objective dimension of cognition is needed to explain the sense of obligation without turning it into absolute certainty. The text goes on to argue that the distinction between certain deduction and non-certain induction and analogy does not hold once mathematics and logic are applied to reality. No claim about the world is certain, but uncertainty is not the same as “false,” because there is a level of “probable” that carries weight. In the end, the claim is that science is a kind of reasoning, and reasoning has the status of Torah-level authority; therefore the question of Torah and science cannot simply be dismissed because science is uncertain, but must be handled carefully, while distinguishing between scientific theory and meta-scientific interpretation.

The Youth’s Dilemma and Kant’s Synthetic A Priori

The dilemma is whether to stick with the adolescent view that only something proven and fixed is acceptable, or to agree to accept also things that are probable even if unproven. Kant’s problem of the synthetic a priori is presented as the question of how one can learn or make claims about the world without observation, since a law of nature is a general principle and not a direct product of observing specific events. The argument is that one cannot empirically deny the contribution of scientific theory to our understanding of the world, and so the question remains: how is a claim about the world formed without direct observation of the general rule?

Cognition versus Thinking and Scientific Generalization

The text argues that understanding generalization as an act of thinking creates a problem, because if thinking is a function of subjective structure, there is no reason to assume correspondence between its conclusions and the world. The proposed solution is that generalization is “cognition” and not “thinking,” and that in the course of generalization a person “sees” with the mind’s eye the general law through the phenomena, similar to what Maimonides describes at the beginning of The Guide of the Perplexed. The principle of simplicity and Occam’s razor are presented as descriptions of how this cognition operates, not as a logical justification that produces certainty.

Uncertainty, Control Mechanisms, and the Scientific Method

The text warns that the term “seeing” may mislead, because sensory seeing is perceived as highly certain—“hearing should not be greater than seeing”—whereas this non-sensory seeing is not certain and can involve mistakes. The scientific method is presented as an attempt to create control mechanisms for processes that are not secure, by testing predictions, comparing them to measurements, and precisely defining types of generalization and simplicity. Logic and mathematics do not need that kind of control, because a valid proof is enough to establish the truth of a proposition, whereas in science there is a constant need for control in order to avoid slipping back into dogmatism.

A Response to the Claim “You Just Changed the Name” and the Practical Difference as Against Skepticism

An objection is raised that the move from “thinking” to “cognition” is just a change of terminology, and the text replies that the gain lies in converting a principled contradiction into a lack of information. Pure thinking “cannot” add information about the world, whereas with respect to cognition one can say that we do not understand the mechanism, but we do see that it works in practice. The claim is that the move from facts to theory does not happen only within thought, but out of interaction with the world, even if we do not know where that interaction takes place. The practical upshot is that one can trust the conclusions and not slide into skepticism, because skepticism follows only if this is an internal thought-process that cannot guarantee any connection to the world.

Kant, Broad Empiricism, and Intuition

The text accuses Kant of “throwing out the baby with the bathwater” by translating the propositions of science into propositions about perceptions rather than about the world, thereby giving up the possibility of saying anything about the world itself. The proposed solution is said to be empiricism in a broad, non-sensory sense, because cognition is not sensory observation, but it is still a way in which the world “sends messages” to a cognitive process that is not understood. This cognition is also identified as intuition, and induction itself is defined as a cognitive process rather than a thought procedure.

Morality, Relativism, and the Objectivity of Cognition

The text brings moral principles as another example of the same logic. If morality is merely a product of how we are built or how we think, then it has no objectively binding validity and collapses into relativity and relativism. The adolescent question is asked: “Who says?” And it is argued that there is no observational answer to the question “Why be moral?” Evolutionary explanations only sharpen the problem, because they present morality as a contingent byproduct. The claim is that anyone who holds that morality is binding owes an explanation of why his own structure binds more than someone else’s structure. The only possible solution is that morality derives from something “out there,” and it is said that this “ought somehow to be connected to the Holy One, blessed be He”—not as a scriptural decree, but because “I see that it is correct that murder is forbidden.”

Torah, Jewish Law, and Moral Boundaries versus Halakhic Boundaries

The text distinguishes between a moral command and a halakhic definition, and argues that the halakhic definition of murder does not always overlap with the moral definition of murder, and that is why the command “You shall not murder” had to be written. Examples are given of cases in which Jewish law grants validity or establishes boundaries that do not overlap with morality—for example, punishment based on a halakhic transgression with prior warning, not simply on the basis of a “moral transgression.” An example is brought of an oath made in writing or by a child: in halakhic terms that is not an oath, while in moral terms it is an oath. The Ri Migash, Ibn Ezra, and the first lecture on oaths are mentioned.

Non-Certain Seeing and Dependence on the Observer’s System

The text gives two reasons why objective cognition does not create certainty. First, we are dealing with a scale of probability rather than a binary, and the scientific method helps sort things according to degree of probability. Second, seeing depends on the circumstances and on the observer’s system, as in the examples of red or green cellophane, and in examples of observational instruments like a microscope. Therefore one gets different pictures of one truth, not “multiple truths.” The claim is that our current measuring tools may lead to conclusions such as evolution or the age of the universe, and there may be reason to return to those questions later when applying the principles discussed here.

Deduction, Induction, Analogy, and the Uncertainty Principle Between Information and Certainty

Traditional logic divides inference into three types: deduction from the general to the particular, induction from the particular to the general, and analogy from one particular to another. It is argued that deduction is certain because it adds no information beyond the premises, whereas induction and analogy are not certain because they do add information. From this comes a kind of uncertainty principle between the level of certainty and the amount of information. Deduction is associated with logic and mathematics, while induction and analogy are associated with science, and it is noted that mathematical induction is actually a form of deduction.

Mill’s Critique of Deduction and Non-Euclidean Geometries

John Stuart Mill is presented as criticizing deduction as a fiction, because the deductive premises themselves rest on analogy or induction, and so deduction always merely “continues” a non-deductive move. Non-Euclidean geometries are presented not as arbitrary relativism, but as different descriptions of different realities depending on the metric of the space. In each space there is one correct geometry, and the difficulty is identifying which space one is in. The text argues that the premises of mathematics come from outside mathematics—from science, or from a choice to deal with premises “in Platonic worlds.”

The Claim That There Is Only Analogy

The text suggests that in practice there are not three forms of inference but only one: analogy. Analogy is made up of two steps: induction, which rises to the general, and deduction, which descends from it to a new particular; together they yield the comparison between particulars. The claim is that this is “a triangle with its tail in its mouth,” where each time one isolates a different side, but the move is really one move.

Mathematics, Popper, and Combining Forces

A question is raised whether “two plus three equals five” is a scientific claim in Popper’s sense, and an experiment with apples is presented and rejected, because even if the counting results were to deviate, no one would ever conclude that the arithmetic proposition had been refuted; rather, one would say there was a mistake in the experiment. A mechanics example is then brought: two forces of ten newtons in perpendicular directions do not yield twenty but ten times the square root of two. The conclusion is not that “ten plus ten equals twenty” has been refuted, but that the physical assumption was refuted—that ordinary algebraic addition appropriately describes the addition of forces. What is required is vector addition. From this the text argues that when mathematics or logic are asserted about reality, a non-mathematical assumption is added regarding the applicability of the tool to reality, and that assumption is what stands trial.

“One May Not Derive Punishments from Legal Inference,” A Fortiori Reasoning, and the Rule That Two Hundred Includes One Hundred

The text brings the rule “one may not derive punishments from legal inference” and Tosafot’s discussion of the Mishnah in tractate Bava Kamma as against the Mekhilta on “If a man opens a pit, or if a man digs a pit,” along with Tosafot’s claim that the Babylonian Talmud disagrees with the Mekhilta. The Maharsha in his second edition is presented as explaining that in an a fortiori argument of the form “when two hundred is included, one hundred is certainly included,” there is no refutation, and therefore there one may “derive punishment from legal inference,” because digging includes opening. The text responds that even there there is a refutation, in the form of the Kesef Mishneh’s reasoning: perhaps the more severe case requires a greater punishment, and therefore one cannot impose the punishment of the lighter case. From this the argument is made that when a deductive inference is applied to the world, it loses certainty, and there may always be some non-logical supplement that undermines the move.

The Example of the Vandervelde Law and Wine

A case is brought of a Belgian law called the Vandervelde law, which forbade selling two liters of wine in order to prevent workers from spending a weekly wage in the tavern. A worker who bought five liters claimed he had not violated the law, while the prosecutor argued that “when two hundred is included, one hundred is certainly included”—five liters includes two and more. The judge ruled in favor of the worker based on purposive interpretation, distinguishing between a purchase out of weekly wages and a purchase out of savings. The point of the example is to show that even an inference that appears logically compelled is not necessarily decisive when applied to legal reality.

Summary of the Introduction: Uncertainty, Probability, and Reasoning as Torah-Level

The text concludes that science is not certain and that no claim about the world is certain, but says this can be taken in two directions, and rejects the direction that shuts down questions of Torah and science by saying, “Science was mistaken because it isn’t certain.” The jump from “not certain” to “not true” is said to be an invalid jump, because there is a level of “probable,” and “reasoning” means probability, not certainty. Rabbi Shimon Shkop is cited as saying that even Torah is observed on the basis of reasoning, and the idea is brought that reasoning has the status of Torah-level authority—for example, the obligation to recite a blessing before eating food in tractate Berakhot is treated as reasoning, because “one must thank the Holy One, blessed be He.” From this the text argues that science is a kind of reasoning, and in that sense “it has a status like a verse.” Therefore, a contradiction between Torah and science is a contradiction that must be dealt with, while still leaving room for scientific error—especially in conclusions that are interpretive or meta-scientific interpretations, such as inferring “there is a God or there is no God” from evolution. That is not a direct scientific claim, even if it is a second-order interpretation of the findings.

Last time I talked about this dilemma that shows up at the end of adolescence or on the threshold of adulthood. Should we remain with the young person’s view that only something proven is acceptable, only something certain is acceptable, or should we also be willing to accept things that are reasonable even if they are not proven? And really, the expression of this dilemma is found in Kant’s problem of the synthetic a priori, which is basically a general label for the question of how we manage to learn or make claims about the world without observation. When we say or claim something, or arrive at the conclusion that there is some law of nature, a law of nature is a general principle; it is never the result of observation, because observation is always of specific events. And yet we claim that there is a law of nature, and that claim is a claim about the world. And I talked about how it seems to me that the second possibility can even be ruled out empirically—the claim that scientific theory contributes nothing about the world. So what we are left with here is still something unresolved. In other words: how can it be that we make a claim about the world without observation? The conclusion was—and I think I already said this—the conclusion was that apparently there is some kind of capacity here connected to cognition and not to thinking. In other words, the dilemma starts from the fact that we understand our act of generalization, our move from the facts we observed to the theory, as an act of thinking. That is, we have observations in which we saw certain events, and from there on thinking begins. Then we say: thinking is a function of our structure, of how we think. What does that have to do with what is happening in the world itself? Why should we assume there is a correlation between the conclusions of our thinking and what is happening in the world itself? And the possible answer to that—and it seems to me the only possible answer—is that there is some kind of cognition here and this is not thinking. When we generalize, we are actually looking—not with our physical eyes but with the eyes of the intellect, what Maimonides calls at the beginning of Guide of the Perplexed. We have some ability to see the general law through the specific phenomena. We observe the specific phenomena with our eyes, but through those phenomena we can see—see in a different sense than sensory sight—the correct generalization. The principle of simplicity, Ockham’s razor, what people often bring up, is really only a description of how that cognition works. Now, as I think I said, this sounds a bit mystical, and on what is it based, all of a sudden? Where in the brain is this department located that is responsible for seeing things not through the senses? I hope you don’t suspect me of mysticism—and if you do, then you don’t know me—but still, it seems to me that it is hard to avoid this. In other words, rationality lies here; this is not mysticism. When we arrive at generalizations, and the fact is that generalizations are facts, that means we have another certain faculty that perhaps we are not aware of or do not know how to point to exactly where it is in the brain, but we have such a faculty. And this faculty is apparently some kind of cognition and not thinking. In other words, the answer to the young person’s question—when the young person asks the adult: how do you know the claims you make about the world? After all, you didn’t see it; you have no proof, so how do you know?—the only possible answer is: yes, I did see, in a certain sense. In other words, I really do see this thing in the world. I see with the eyes of my intellect that this is how it is. Now, one has to be careful with this term “seeing,” because it can mislead. In sensory sight we have complete trust. In other words, what we saw is there. “Hearing should not be greater than seeing”—even the Sages say that sight is the highest certainty. I do not mean to claim that this non-sensory seeing is certain to the same degree. Not at all. And when I call it seeing, I do not mean that it is certain like sensory sight or that error is impossible here. There are those who will take this in the direction of: fine, it is above reason, above cognition, and therefore it is certain and cannot be challenged. Exactly the opposite. Precisely because this is such an amorphous and undefined thing, it is entirely possible that we make mistakes there, and very often in our generalizations we do make mistakes. Therefore it has to be done in some controlled way. And the controlled way to make these generalizations and these processes of “observation,” in quotation marks, is basically, to a large extent, what one might call the scientific method. The scientific method tries to create control mechanisms to help us in those intellectual procedures that are not certain. Logic does not need control mechanisms. In other words, what is logically certain is certain; I don’t need any control for that. If I have a proof for a theorem in mathematics—no one, I’ll get to this in a moment—no one needs to observe anything in order to confirm it. If the proof works, the theorem is true. But in science we constantly need controls in observation, checking predictions, comparing them against what we measure, on all kinds of generalizations of this sort or that sort, defining carefully which generalization is simpler than another. The whole attempt to define a more precise method is ultimately just an attempt to describe that form in which we “observe” reality, in quotation marks, and to do it in a controlled way so as not to slide back into the dogmatic stage, the stage in which we accepted things simply because they seemed right to us. We are trying to do it in a more controlled way. Yes. What did you do here? You said that you have a problem understanding how thought, which is subjective, can arrive at conclusions on the basis of observation. Then you said: I don’t know how to answer that, so I won’t say this is thought, I’ll say this is cognition, which is also completely subjective. So I changed the name—what did I gain? So I’ll tell you what I gained. You’re right on the principled level, but I’ll tell you what I gained. With thought, my problem with the fact that thought makes claims about the world is not that I don’t know how it does it, but that I know it cannot do it. It is impossible. In other words, if it is something that happens inside me, how can it be related to what happens in the world? And this cognition or sight also—regarding cognition, I can say: look, I don’t know how this sense is located, I have no idea how it works. But if it exists—by the way, I also don’t know how my sight works. But the fact is that it works. So once it works, I can claim that there is such a faculty even though I don’t understand it. That is much better than saying: here is something impossible, but the fact is that it works. If it is impossible, then it is impossible. You changed the name and solved it that way? No, it’s not a name. It’s not a name. My claim is that I really am creating an interaction with the world. This is not wordplay. My claim is that when I make a scientific generalization, the process of moving from the facts to the theory is not a purely intellectual transition. It does not happen only in my thinking. It also happens out of interaction with the world. Now, I haven’t put my finger on that interaction; I do not know where it happens. But the fact is that it happens, because in fact I arrive at generalizations that work. Thought could not have given me that; a priori it cannot give me that. In other words, it is impossible. Why should something that occurs inside me correspond to what is happening out there? Or certainly, how do I know that? In other words, even if by chance it is true that what fits within me fits what is happening in the world, how can I know that? All my means of control are also found only within me. What practical difference does all this make—thought or cognition? You just classify it as cognition and say it has consequences. What practical difference does it make? The practical difference is whether I can believe it. When a skeptic comes along—when a skeptic comes and says: look, this is exactly all the claims that were raised against the rationalists. And those claims are correct. On the other hand, Hume raises the same claims against the empiricists, so what do you do? The problem is a philosophical problem. But in practice we do the same thing all the time. And these are not philosophical questions. Wait, one second. So I ask myself: can I really believe my intellectual conclusions, or am I deceiving myself and should I be skeptical? So the answer I give is: no, you need not be skeptical. You should be skeptical if you agreed that this is a process of thought. But since I say this is a kind of cognition—what, do I know all my brain functions or intellectual functions? I do not. There are many other things I do not know. Even thought—you don’t know thought either? Thought too? Of course. But the character of thought I do know. The character of thought—therefore I say there it is not a question of not knowing. If I simply did not know something, that would not be a problem. But there, essentially, even if there is something I do not know that is a thought process, by virtue of its being thought, it cannot correspond to what is happening in the world. So it is not a matter of missing information—of lacking more information, and if we learn more then we’ll understand. Here I transfer it into a formulation or a description in which information is missing. Fine. Information is missing. We have a lot of missing information. That is not essentially different. No, but Kant tries to throw out the baby with the bathwater. He doesn’t try to, but that is what he does. He throws out the baby with the bathwater, because Kant basically says that the correspondence is not to the world but to the world as I perceive it. Then in effect it is a correspondence between cognition and cognition, not between the world and my thought. And indeed he argues: I cannot say anything about the world, because he accepts this claim that my thought—well, the world owes nothing to my thought. And therefore he translates all scientific propositions into propositions that do not speak about the world but about my perceptions of the world. So that is throwing out the baby with the bathwater. But why isn’t it, in practice, that once you turned it into cognition and you say there is some cognition inside, you are in effect relying on observations? Basically you’ve become an empiricist. Correct—an empiricist in a broader sense, not a sensory one. I do not know another solution to this problem. Right? If I really assume that thought cannot do the job, there is no choice—cognition does it. There is no other solution. But didn’t we say that observations cannot teach us anything? Sensory observations. Sensory observations. And now you call it cognition, and now you say that this can teach us something? Because it is not sensory observation; it is another cognition. But this cognition is a direct result of observations. Without observations you would not arrive at cognition. No, it is not a result. If it were a result, it would be thought, not cognition. My claim is that beyond the cognition by which I observe specific events, there is another element that is also cognitive, not intellectual. And that is an added something that knows how to do this induction and generalize. Yes, induction is not a process of thought—that is my claim. Induction is a cognitive process. In other words, beyond observing the specific facts, the process by which I move to the general theory is also a cognitive process. Is this not what is called intuition, this cognition? Yes, it is intuition, right. You can call it intuition, yes. You are basically saying that the world succeeds in sending messages to some place where this cognitive process takes place, a process we do not understand, while thought itself does not see these messages? Correct. That is why it seems to us that this is thought and not cognition, because we do not see that there is actually some interaction here with the world. But it seems to me that the facts say that this is cognition and not thought. Therefore I say: I have no interest in developing mysticism for no reason. I think there is simply no other way out. You cannot explain otherwise how to accept synthetic a priori claims, how to accept general claims about the world. And what is the difference between thought and cognition? It is such a blurry boundary that it is a little hard to understand. I think these are two well-defined concepts. Thought is something that happens inside me without interaction with the world. An intellectual process that ignores reality. Exactly. Such a process cannot add—I spoke about this, that it adds information. Pure thought cannot add information. Information is stored in observations. Thought can at most uncover information already present in the observations. But to move from the details to the general law is to add information. And thought cannot do such a thing—cannot add information about the world. I’ll speak more about that. I just want to show one or two more aspects so we understand the significance of the matter. Very often we do indeed reach the conclusion that certain things we think are the result of thought, and because of that they receive the same criticism of subjectivity—that it is not binding, and so on. For example, moral principles. A completely different context, but I only want to show the logic. I think we all share some basic moral principles. On the other hand, many times the questions arise—and again, the young person’s question when he comes to the adult and says, wait, who says? Why should one be moral? Do you have a proof for that? Maybe morality says the opposite? There are lots of such questions, and when you hear them, you have nothing to say. What are you going to answer to the question of why be moral? Because one should be a decent human being. Okay, and why be a decent human being? That is just repeating the same thing in different words. How will you answer that? What lies behind the dilemma? What lies behind the dilemma, and from there people arrive at moral relativism and so forth—what lies behind it is exactly the same problem. Since the claim is that morality is a product of the way we think, of how we are constructed, then why does it have validity? In other words, someone else is constructed differently, so for him morality is something else. It is exactly the same question. If something is a product of the way I am built, of how I think, then it cannot have objective validity. It cannot be binding—not even for me, not just for others. Because I too am built in a way that wants to speak slander. So should that also be done? In other words, not everything that is built into me is therefore binding. So how did I decide that this built-in thing is not binding and that built-in thing is binding? That means I have another criterion besides the fact that it is built into me. The fact that it is built into me is not enough. So what is enough? Why? Why is it binding? In other words, there is something beyond the fact that it is simply in me. The fact that it is in me—so what? There are others in whom it is not found. Can I make claims against them? And indeed as a result of this, because there is no answer to this question—there is no answer to this question—there really are people who will say: fine, moral relativism, the same old thing. Okay, I cannot convince you, but I will kill you so that you won’t kill me, and I will put you in prison so that you won’t cause harm. I have no good answer for why I ask you to behave morally, why I judge you when you do not behave morally. But the reason I have no good answer is because I am looking for the answer under the wrong streetlamp. I am looking for the answer in the observational world. And in the observational world there really is no answer. What, can one observe morality and see what correct morality is and what incorrect morality is? And precisely because of this people argue: look, there are moral disagreements and so on, because it is not the result of observation. But morality comes from somewhere. Exactly. Where does it come from? I do not know. But the same question—no, when a child grows up, I teach him what is good and what is not good. Okay. Then he will ask you: wait, but who says—this is a person’s subconscious. I tell him: listen, this is how one should behave. No, the subconscious is still inside us. So you tell him that this is how one should behave, and he says: thank you very much, but I think one should behave the opposite way. What will you say to him? Why can’t he tell me that one should behave the opposite way when he arrives at independent reason? Right, right, that is what I said—adolescent rebellion. So now what do you answer him? There is nothing to answer him. After all, you do not know how to answer him; you cannot point and say: here, don’t you see? It cannot be seen. But I think that even though it cannot be seen, the young person is not right. This thing is not the result of thought that takes place within us, because if it were, then he would be right. It is not that I do not know how to answer him; it cannot be answered. There is no answer to that. And if you are scientific, then you will say, as people do now, that from the standpoint of evolution there are reasons why we behave as we do. Right. And that only strengthens the question. What? That if evolution created certain patterns of behavior, then that exactly means they were simply produced by chance. So why does that bind me? Right. So I was created evolutionarily differently and I want to murder. Why is it binding? If you are a scientist, why do you think it needs to be binding? It isn’t binding. No, no, no, no problem—that is exactly what I am saying. Anyone who says that morality really is not binding—I haven’t said anything against him; that is what he says. But those who think morality is something binding and who argue that one should criticize another, condemn or praise moral or immoral behavior, they are supposed to give themselves, and others, some account of that. The fact that you are built one way and the other person is built another way—so what? Why is the way you are built worth more than the way he is built? In other words, moral relativism comes from the same place as epistemic relativism, cognitive relativism. In other words, where does it come from? It comes from the fact that you are convinced this is not the result of observing something, right, but rather: this is how I am built, this is how I think. Fine, this is how I think, very good—and he thinks the opposite. So wait, I want to understand: I understand that you are trying to say this comes from the cognitive side, just as there. Now I want to understand: this cognitive thing that you define—is it objective? Yes. Is it fixed? It is objective. Yes. So if it is objective, we all morally should have thought the same way. No. Let me explain now why not, for two reasons. For two reasons. That really is the next question. Because that is always the dilemma. If you say—if you say there is some such Idea of the good in the Platonic world of Ideas which I observe, and through that I understand what it is to be good and what it is not to be good—that is basically the only possible answer. I still don’t understand what is good and what is not good. Scientifically—no, no, if you accept some religion or some authority, fine. But if you don’t accept that, then the question is not relevant. Why? You are simply assuming the very objection I raised earlier, and I disagree with you. Why? What do you mean? A person who—where does the concept of good and not-good come from? Ah, that is exactly what I am asking—where does that concept come from? Because after all, if it is embedded within me, then it has no validity at all. So where does it come from? There has to be something out there from which it comes. Now that something out there—I also agree that it seems to me it somehow ought to be related to the Holy One, blessed be He, somehow, fine? But not because of scriptural decree. No, not because of scriptural decree. Absolutely not. What is scriptural decree? Because I see that murder is wrong. But why can I impose my morality on someone else? Why can you impose it? I do not believe I need to be moral. Fine, then you need treatment. No, no—sorry that it sounds terribly communist, but… but that is reality. In other words, someone who truly and sincerely does not believe this—I do not think he can be judged. Judged in the moral sense. I will judge his actions. His actions I tell you are immoral. Why do you judge his actions? Because you need to behave the way I tell you. Right—the way morality says, not the way I say. But how do you know that this is morality? Because I saw that this is morality, because I saw. How did you see? But no, I’ll ask you—someone who is unfit to stand trial. Right, exactly. Someone trying to walk through a wall… But you won’t murder someone because it is written. Not because it is written—because murder is immoral. That is exactly the point. No, not because it is written, because murder is immoral. In other words, again I say: this is my position. Someone else can come and disagree. But I say that if your position is that morality has something binding in it, something objective, then you are in exactly the same problem. Exactly the same problem. But if the Torah says, “You shall not murder,” for example, which is something supposedly absolute, and now we are calling it a cognition that comes even before that—how does that work? Why does the Torah need to put it into the Ten Commandments? That is already another question, and the Talmud says: why were obvious rational principles written? And sometimes it says: if you want, say it comes from a verse, and if you want, say it comes from reason. In other words, it is not such a big deal. Beyond that, I think the halakhic definition of murder does not always overlap with the moral definition of murder. Therefore the fact that “You shall not murder” is a moral command does not mean that its definition overlaps with the halakhic definition of “You shall not murder.” And for that reason it needed to be written. And similarly with “You shall not steal,” and so on. I once brought this in the first class on oaths: there is an oath made not by explicit verbal formulation, not by speech, but by writing, say, or an oath by a child or something like that, where in halakhic definitions that is not yet an oath, but in moral definitions it is an oath. That is the Ri Migash we brought there, and Ibn Ezra and others. So why is it written, if the moral command already exists? Because the halakhic boundary, first of all, comes to give force to the moral definition sometimes, and beyond that it also gives certain halakhic boundaries that do not necessarily overlap with the moral boundaries. For example punishment—lashes are not given on the basis of a moral offense; they are given on the basis of a halakhic offense. You need prior warning in order to punish. Okay. If you say that this is at a lower level than the senses in terms of certainty, then where do we get the power to kill people on the basis of such a thing? Right, so in a moment I’ll say that, and then we’ll see, because I was in the middle of saying it. I said there were two reasons that still mean that even though this is the result of some kind of cognition, it is not certain. By the way, notice that everything I’m saying now is also true of laws of nature. I am bringing morality as another example, but it is the same thing. You could ask exactly the same thing about laws of nature: how do you know this law is the correct one? It is the result of a generalization. It is just something built in. Now the new radical critique really does say this about scientific thinking. Newtonian mechanics is only a male construct, or whatever. In other words, it has no—what? “Who in His goodness renews every day, continually, the work of creation”—every day is a new miracle. Yes, okay, every day is a new miracle, but in the same form and with the same forces. But that is a different discussion. I’m saying there is a force that each day there is a new miracle. Yes. In any case, there are two important points to notice here. One of them I said earlier. One of them is that just because I say this is sight does not mean it is certain. Exactly as a law of nature is a kind of cognition and I still do not think it is certain. The level of certainty, of course, is also not always the same. There are things that are clearer and things that are less clear. By the way, the scientific method is supposed to help us classify things according to their degree of probability. Yes, it is not just probable or improbable; it is not binary—probable or improbable. There are different levels of probability. You know, there are all kinds of such concepts in Jewish law, where very often the binary treatment of them works against us. For example, there are places where we see that doing something in an unusual manner is permitted. So everyone asks: what do you mean? Doing something in an unusual manner is rabbinically prohibited—so how can they say that since it is not being done in the normal way it is permitted? The answer is: what is “an unusual manner”? It is a continuous scale. So there are certain levels of unusualness that are rabbinically prohibited, and at a greater level of unusualness it is completely permitted because then it is no longer the thing at all—it is no longer the act itself. So there are all sorts of concepts like that. The same thing when I talk about sight. This “sight” also does not mean I have full certainty here like in sensory sight, and even in sensory sight I do not think the certainty is absolutely full. Fine? So that is why I say: the fact that this is sight is only a philosophical solution to why I am allowed to believe it. But the question is at what level I believe it—that is an entirely different question. And it is not something so obvious, as a fact that scientific generalizations are disputed and are sometimes corrected because we are not right about them—and that is perfectly fine. This is a weaker mechanism than the mechanism of sight. That is one thing. The second thing, and it is also connected to Kant, is that very often when I see something—and in this matter Kant was certainly right—first, it depends on circumstances, and second, it depends on the observing system. If I look at the world through red cellophane, I will see it as red. If I look at it through green cellophane, I will see it as green. So what is the truth? Is it red or green? Are there many truths? No, there is only one. It is not a multiplicity of truths; there is one. Every observing system will see that one truth differently. It is not this and not that because you are using something to see it as red. Exactly. But what if there is a situation where without the cellophane you would not see anything at all? Because you have no ability to see it. Things you have no ability to see. You look through a microscope and then you see them; someone else might look through another instrument and see differently, because it also depends on the instrument. Yes, that is exactly the difference. There are things we cannot see with our eyes at all, and the cellophane is the only means through which we can see them in the first place. In such situations, different pictures can absolutely arise. Therefore those different pictures do not mean that there is a multiplicity of truths or that there is not one truth; it only means that we see it from different angles or in different ways, and therefore we can arrive at different conclusions. So, for example, evolution or the claim that the world has existed for a million years—is that because we currently have means to measure it that way, and it may not be correct? Maybe. When I get to those problems and we try to apply what we discussed here, we’ll come back to that. I don’t want to jump ahead. You said it is a weaker mechanism than sensory sight, but to me it looks much stronger. It seems to me something that, for example, is terribly hard to change; it is much easier to influence a person by a logical proof and change his thinking than to change his cognition. I would even say that someone who became religious probably had a strong, deep inner cognition opened up for him, which had been somewhat dormant and suddenly opened. But cognition seems to me much stronger than sensory sight. No, I do not think so. The degree of certainty I have in the specific thing I observed—say, I saw that this fell to the ground when I let go of it, right?—my degree of certainty that it falls to the ground is much stronger than my degree of certainty in the general law that bodies with mass attract one another. Usually, it seems to me, from what we see—but again, fine, everyone with his own definitions; I am not making a claim for my own side, I am just reporting. That is how it seems to me, but I don’t know, someone else can think differently. Fine, let’s move on a bit. Here I want now to move a bit further and define these patterns of thought a little more precisely. Look, there are three types of inference that are distinguished in traditional logic. There is deduction, induction, and analogy. Deduction is movement from the general to the particular; induction is movement from the particular to the general; and analogy is from one particular to another particular or from one general to another general—in other words, between two things on the same level. Fine? So if I say all human beings are mortal and Socrates is a human being, then I infer that Socrates is mortal. I begin with a general proposition, all human beings are mortal, and infer a conclusion about a particular case within it. That is deduction. Induction is when I say Socrates is mortal, Moshe is mortal, so apparently all human beings are mortal. I begin from particulars and move to the general. Analogy is: Socrates is mortal, so apparently Moshe is also mortal. There is simply a similarity between these two things. That is analogy. What is basically the difference between them in terms of certainty? It is perfectly clear. Deduction is absolutely certain, right? You cannot argue with the conclusion of a deduction. But analogy and induction, yes. Analogy and induction are not certain. You can make a mistake in a generalization or in a comparison. It could be that I made an incorrect analogy—that can happen. Okay? Now where does that come from? It comes from what I already talked about with the hot-air balloon. It comes from the fact that deduction does not add any information beyond what is in the premises, and therefore it is certain. Because if you accept the premises, then within them you have already accepted the conclusion as well. If you say that all human beings are mortal, then you have already accepted that Socrates in particular is mortal. So the conclusion is completely certain because it does not add information. And here we see the other side of the coin: the reason analogy and induction are not certain is because they add information. An analogy says, for example, if Socrates is mortal then Moshe is mortal. The premise is that Socrates is mortal; the information that Moshe is mortal is not included in the premise—it is additional information. So by means of analogy I add information. The same is certainly true of generalization, which adds even more information, right? A generalization speaks about a whole group, so it adds information. Once things add information they are no longer certain. This is the uncertainty principle between certainty and the amount of information. Now the disciplinary division is this: deduction belongs to the domain of logic and mathematics. They deal with propositions that do not add any information and are certain—therefore mathematics. Okay, so it is always right and does us no good, like we said there. Because it adds nothing—it does us no good in the sense that it adds no information. Yes. Induction also—what? They also use induction in mathematics. No, mathematical induction is something else. Mathematical induction is in principle deduction, unless you are an intuitionist. But that is just philosophy—it is deduction. It is called induction because it also looks like moving from the particular to the general, but after all you prove that whatever is true for n is true for n plus one, so it is a proof. So analogy and induction, both of them are not certain because they add information. These are the scientific rules. Scientific rules are rules of analogy and induction, not deduction. Therefore the distinction among these three ways of inference is really a disciplinary distinction. In other words, deduction belongs to logic and mathematics. That is over there. That branch never adds any information, and therefore it is safe. It takes no risk, adds no information, so it is safe, and you are always right. No more certain than the premises from which it began? Yes, I’m saying: the derivation is certain. I spoke about that. The derivation of the conclusion from the premises is certain, not the conclusion. That is exactly the point. Gödel’s incompleteness theorem says that there are always not enough premises to reach… I’ll explain that perhaps later. Gödel—in a moment. And analogy and induction belong to the domain of science. In the domain of science, we are trying to accumulate information—that is our goal, after all. Fine? Now once we accumulate information, doubt always follows in our footsteps. It can never be certain, because these are procedures that add information. Now in truth there is a critique of deduction by Mill, John Stuart Mill. He basically argues that deduction is a fiction. There is no such thing as deduction. What does that mean? Exactly this claim. Suppose I say that every human being is mortal, Socrates is a human being, therefore Socrates is mortal. Is the statement that Socrates is mortal certain? Obviously not. Right? It is based on the premises. Statistically. No, not even statistically—simply not certain. Leave aside now what it is. Fine? It is not certain. It is based on the premises. So Mill asks: and how do you know the premises? Through analogy or induction, right? So deduction always really presupposes analogy and induction within it. In other words, mathematics can never do anything if you do not have some observation or scientific generalization at the beginning. After that, with mathematics you can analyze the results of those observations and accumulate information. Or you can analyze premises that have nothing to do with observation and reach conclusions that have nothing to do with observation—you can sail around in Platonic worlds. That is perfectly fine. From this you get contradictory geometries. Actually no. There precisely I do not accept that interpretation. Non-Euclidean geometries. No, I do not accept that interpretation. These are different premises. They are not different premises but different observations. But you turn the observation into a premise. Fine, so that is what I said earlier. I say: people often bring non-Euclidean geometries as an example of intellectual relativism. It is exactly the opposite. Non-Euclidean geometries—after all, every geometry lives within a space with a certain metric. Right? The metric induces a geometry. In other words, if you are on the surface of a sphere, there is one geometry. If you are on a plane, then there is another geometry. Right? So what does that resemble—a multiplicity of truths? Not at all. It means that on the plane there is one correct geometry, and on the surface of a sphere there is another single correct geometry. That is all. Obviously every reality has a different description. Does that say anything about relativism? Nothing. In other words, the question is in what space you live. Now I do not always know in what space I live. Sometimes the geometry will be an indication for knowing what space I live in. Fine, but still I say that this is not a function of positing premises in an arbitrary sense. You can posit such premises and reach such conclusions, posit such premises… People often bring non-Euclidean geometry as the basis for a relativist philosophy: that you can basically assume any premise you like and reach whatever conclusions you wish. That is incorrect. Non-Euclidean geometry says that for every—it does not say something about mathematics, it says that mathematics is relative to the premises with which it began. Exactly. That is what I said earlier: that mathematics can take—and where do the premises come from? From outside mathematics. So that is what I said. In other words, in the end the premises have to come either from science, or from some thought of my own, or I simply posit them because it seems interesting to see what follows from them. No problem. But those premises are the basis for all the conclusions that follow from them. Therefore Mill’s critique of deduction essentially says that deduction is a fiction. There is no such thing as a genuinely deductive thought process, because deduction is always a continuation of some induction or analogy that was there at the beginning. And if I put it in a more pictorial way, I would say the following. Well, maybe even before the picture, one second. What is the relation between… yes. Regarding Mill’s comment, I don’t know where you want to take it, but if I put ten black balls into a box, and I take out one ball and say that this ball, by deduction, is black, there is no premise here… Of course there is—the premise is that you put ten black balls into the box. That is not a logical premise; it is an empirical premise. You know that you put ten black balls there. It is neither induction nor analogy. What? It is observation, never mind—fine, it is observation. Fine. But it is grounded in itself. Right, right, but science never works that way. In the scientific context that is not relevant. In a specific everyday context… But deduction has its own… It is like my looking at this wall. I see that there is a wall here. That too is not analogy or induction or anything—I know there is a wall here. That is not deduction. Right, right, that is what I am saying. In itself there is no… the question is what it grounds. Sure, so that is exactly what I am saying. Obviously in non-scientific contexts you can speak about plain observations of specific things and know them without analogies or inductions or anything like that, simply because you saw them. You can even regard sight itself as some kind of generalization—the assumption that sight works. Fine, I am willing to accept that. It is not important for me for the purpose of the discussion. Now what is the relation between analogy and induction? That too is interesting. What do you think? Which is stronger? Induction. Induction is stronger. Do you agree? Analogy is more limited. And therefore? And therefore it is stronger. Seemingly analogy is stronger than induction. Remember the uncertainty principle between the level of certainty and the amount of information? Now in induction I add a lot of information; all it takes is one raven not being black for my induction to collapse, so it is very risky. But there are more examples in order to get to— No, with induction there is no rule about how many examples there are, and with analogy there is not either. I can make an analogy on the basis of ten examples. If I take ten examples and say that the eleventh is also like that, that is an analogy. There is no rule how many initial examples are needed. But in use, induction is stronger, yes—the more examples you have. No, but more examples does not mean induction. More examples can occur in both analogy and induction. There is no rule how many examples are required. For me, both induction and analogy can be based on one example. Of course then it is weaker—both in analogy and in induction. So seemingly analogy—I don’t know if it is stronger, but it is less weak. It is less weak because it takes less risk. It says something only about one single element. I say: Socrates is mortal, therefore Moshe is mortal. About Moshe, either I am right or I am not right. But if I say that all human beings are mortal, with every single one it may be that I missed something; that is a greater risk. Again, the same uncertainty principle. The more information I added, the more my level of certainty drops. On the other hand, if you think about it, that is not quite accurate. But you say that in induction you say there is some rule, some reason that they are all like that; in analogy you say… Ah, so now I am getting to that. In truth there is also another side to the coin. When I make an analogy, say, if Socrates is mortal then Moshe is mortal, why am I really comparing them? Because of induction. Obviously. After all, in the background there sits some hidden induction that says apparently human beings are mortal. So notice what we get here. What we get here is that contrary to what people usually think, there are not three ways of thinking; there is only one. There is only one—only analogy. Analogy. Only analogy. We perform analogy in two steps. We begin with induction and finish with deduction. Then we say like this: Socrates is mortal; we do induction— all human beings are mortal; and then we descend by deduction—if so, then Moshe too is mortal. Right? The first step is induction, the second is deduction, and the sum of the two is simply analogy—an analogy between Moshe and Socrates. Do you understand? So in fact there are not really three ways of thinking or three ways of inference. It is only a matter of isolating them; it is like a triangle, and each time we look at a different side of it, but this triangle has its tail in its mouth. In other words, it is not really three things. Now I want to continue a bit and see what that means. Look—and here I want to say something about mathematics and logic, or about their relation to reality. When I taught mechanics at the university, I began with a question. I asked the students whether they could subject the proposition two plus three equals five to a falsification test. In other words, is this a scientific claim? According to Popper—I mentioned this already—according to Popper, a scientific claim is a claim that can be falsified, that one can propose an experiment that would falsify it. Okay? Now the question is: is the proposition two plus three equals five a scientific claim? What do you say? No. So let me propose an experiment and tell me what you think of it. It’s a matter of definition. Okay, let me propose an experiment and then we’ll come back to that. Suppose I have a bowl and I put two apples into it, and after that I take another three apples and put them in as well. Now I count how many I have altogether. If it comes out five, then I have confirmed the proposition two plus three equals five. If it comes out four, or seventeen, or minus two, then I have falsified that proposition. What’s wrong with that? That it won’t come out that way. That it won’t come out that way—never mind. A scientific proposition does not have to be falsified; it only has to be falsifiable. Two plus three. Fine, so it turns the proposition two plus three equals five into a scientific proposition. When you say I put in two, I put in three—who defined that this is two, who defined that this is three? Everything is based on that same arithmetic of two plus three. So basically you are not putting in two, you are not putting in three, if you are saying that this puts it into question. Ah, I agree—but I would put even more emphasis on the “plus” than on the two and the three. We will soon see. Fine, I’m willing to accept that too. In other words, let us do a thought experiment. You said it would not be falsified, but let us try. What could happen? We put in two apples, then add another three apples, count them, and wow, it comes out seven. It comes out seven. What is the conclusion? Will anyone infer from this that two plus three does not equal five? Or that you don’t know how to count, or there was an error in the experiment. Never. Even if you repeat this a hundred times, one after another, and it comes out seven, it is obvious that there is some error in the experiment, right? We would never infer. That is an indication that two plus three equals five is not a proposition in physics. It is not falsifiable. Now why did I bring this at the beginning of a mechanics course? Because the next example is less trivial. Take a body and apply forces to it. Fine? There is a force of ten newtons northward and another force of ten newtons eastward. What is the resultant force? Ten root two, right? Fourteen point something. Not twenty. So have we falsified the proposition that ten plus ten equals twenty? No. But here we do think it is correct. There is no error in the experiment. So what is the conclusion? The conclusion is that adding forces is not described by algebraic addition. Right? So notice what that really means. When we assumed the hypothesis that was later falsified—that ten plus ten here would give twenty—we did not assume only the arithmetic proposition ten plus ten equals twenty. We also assumed here a proposition in physics: a proposition that says that the physical process we are dealing with is described by arithmetic. In other words, that arithmetic addition is a good description of the addition of forces. What we falsified was that. We did not falsify ten plus ten equals twenty. We falsified the claim that algebraic addition is the correct mathematical form for handling the addition of forces. Incorrect; you need vector addition. Okay. What does that actually mean? It is a parable. What this parable really says is that what we deal with in life makes the division I mentioned earlier lose its meaning altogether. The division between deduction and analogy and induction, or between science and logic. Because once logic or mathematics tries to say something about life, it is no longer mathematics. If someone draws a triangle here and discovers that the sum of its angles is not one hundred eighty degrees, would anyone do anything to Euclid because of that? No. The conclusion would be that our world does not have Euclidean geometry but a somewhat different geometry. That is all. In other words, our assumption when we use geometry, which is a mathematical field, and use it on the world, contains implicitly another assumption besides the mathematical assumptions: the assumption that this mathematics is the right instrument to handle this reality. That is a physics assumption. And as a physics assumption, it stands before the test of falsification—either it is correct or it is not. So very often we have the illusion that logic gives us certain results—again, the derivation of the conclusion from the premises. That is true perhaps in the world of Ideas. In our world, never. There is a very interesting implication of this for the rule that punishment is not inferred by logical reasoning. You know there is a rule in the Talmud that one does not punish on the basis of something learned by a kal va-chomer, an a fortiori argument. Okay. Now the first Mishnah in tractate Bava Kamma brings—yes, the four primary categories of damages. “This is not like that, and that is not like this”—the pit, the fire, the grazing animal, and the burning. So “this is not like that” means that they make all kinds of distinctions there: you cannot learn this from that, and that from this, and from both of them the third. Everything is fine; everything is needed—you cannot learn one from the other. Tosafot asks there: what do you mean, and if it could be learned—after all, punishment is not inferred by logical reasoning? So if I learned, say, horn damage from ox and pit, fine, then what—would you obligate payment? But after all, punishment is not inferred by logical reasoning. Why is that punishment? What? Is that punishment? So Tosafot assumes yes, and it is really an interesting question why he treats it as punishment, but fine, never mind—it is still an interesting question why this is punishment. Right, in the simple sense it is monetary liability, so it is a long discussion, I won’t enter it here. But Tosafot assumes yes, this is punishment—or at least that the rule that punishment is not inferred by logical reasoning also applies to such a thing, even if it is not exactly punishment. And Tosafot says—he brings a Mekhilta, and in the Mekhilta it says: “When a man opens a pit” or “when a man digs a pit”—if one is liable for opening it, then for digging it all the more so. So why does Scripture write digging? It would have been enough to write opening. This teaches us that punishment is not inferred by logical reasoning—or that monetary liability is not inferred by logical reasoning. Fine? So Tosafot says: even in monetary matters—exactly because of your comment—Tosafot makes an effort to prove that even in monetary matters punishment is not inferred by logical reasoning. So how then does that work in the Mishnah there in Bava Kamma? Tosafot says that the Babylonian Talmud disagrees with the Mekhilta on this point. And the proof is that on page 49 they derive something else from “when a man opens a pit” or “when a man digs a pit”; the Babylonian Talmud does not accept this. Maharsha asks there, in his later edition and on that passage on page 49, Maharsha points out that why should the Babylonian Talmud disagree with it? Not necessarily because the Babylonian Talmud disagrees with the claim that monetary liability is not inferred by logical reasoning, but because in this particular kal va-chomer it is clear that one does infer liability by logical reasoning. Because this is a kal va-chomer of what in the literature of the rules is called “if two hundred is included, certainly one hundred is included.” What does that mean? What is the relation between opening and digging? When I dig the pit, clearly opening is included in that. I just did more things besides opening. Right? That is called “if two hundred is included, certainly one hundred is included.” In other words, the severity of digging compared to opening is not because digging is more severe than opening—sorry, because digging is more severe than opening, but only because within digging there is also opening. Besides that there are some additional things. But opening is there. So if you are liable for opening, then it cannot be that if you did a few more things you suddenly are not liable, because opening is there. In an ordinary kal va-chomer you are dealing with two different things. “If her father had but spit in her face, would she not be ashamed seven days? Then from the Holy One, blessed be He, surely she should be for fourteen days,” as the Talmud in Bava Kamma says. That is a kal va-chomer that is not of the “if two hundred is included, certainly one hundred is included” sort. Rather, the assumption is that the Holy One, blessed be He, is a more serious matter than if her father is angry with her. So if from her father it is seven days, then from the Holy One, blessed be He, it is fourteen days. Fine? That is an ordinary kal va-chomer, not one of “if two hundred is included, certainly one hundred is included.” And here someone could come with some refutation, or say it may not be true, and in fact perhaps her father is still more severe than the Holy One, blessed be He, because you did not grasp it correctly; there is another side to the coin. But when there is a kal va-chomer of “if two hundred is included, certainly one hundred is included,” like the relation between opening and digging, there can be no refutation. There is no refutation, because within digging there is also opening. Or in other words, this rule that punishment is not inferred by logical reasoning—I think I mentioned it the first time—there are, as is commonly said, three explanations for it. One explanation is that maybe there is a refutation to the kal va-chomer, and therefore one does not punish by logical reasoning. By the way, that is Maharsha’s explanation, and it will be important later. And it can be refuted, because say one person began digging the pit and you only deepened it. No, then they are already two different people. But the second person did not open the pit; he only dug the pit. So here I have digging. He is probably also called one who opened it. Why? But he did not open it. Why? Because before that it was not a pit. Why? Because digging the first centimeter is not digging a pit. Digging the last centimeter is… not opening. No, it is not opening. You did not open a pit. Let’s see the pit—there is no pit. Only after he created a pit here, he created an open pit here, then that means he both dug and opened. Okay, so now another person comes and deepens it further than the open pit—does this second person… That is a different question; the Talmud discusses that. The tenth handbreadth—the tenth handbreadth, from the standpoint of digging, he does indeed dig a pit. No, not clear. Because there is already a pit here, so perhaps he is not even digging a pit; the first one dug this pit. But that is another discussion. Look, one explanation is Maharsha’s explanation, that perhaps there is a refutation, and therefore one does not punish. You cannot be certain; maybe you will find a refutation of the kal va-chomer, and therefore you do not punish. A second explanation appears in Kesef Mishneh and in several places: it may be that the punishment for the lighter case is not sufficient for the more severe one. Right? Therefore you cannot take the punishment for the lighter case and apply it to the more severe one. What about a more severe prohibition? Yes. And the third explanation is that there is some derivation from the verse “the daughter of his father or the daughter of his mother,” and therefore punishment is not inferred by logical reasoning. And it seems to me I already noted this in the first lesson, I no longer remember; if not, I’ll note it now: I never understood this matter. What does it mean to say there are three explanations? It is two explanations and a source. It is not three explanations. Scriptural decree is not an explanation. There is a verse that teaches it, and there are two explanations of the meaning of what we learned from the verse. We have some notion that if there is—it is not an alternative explanation. Fine. In any case, these are the three… Now in Maharsha’s own terms, how would we formulate his claim here? He basically says: look, why does a kal va-chomer not generate punishment—why is punishment not inferred by logical reasoning? Because maybe there is a refutation. But a kal va-chomer of “if two hundred is included, certainly one hundred is included” cannot be refuted. This is deduction, in my earlier language. It is “if two hundred is included, certainly one hundred is included,” Socrates is one of all human beings, right? This is deduction; deduction has no refutation. Therefore, in opening and digging, certainly one is punished by logical reasoning. Therefore the Babylonian Talmud does not accept that from here one learns the rule that punishment is not inferred by logical reasoning. If for opening he is liable, then for digging all the more so. The relation between opening and digging is one of “if two hundred is included, certainly one hundred is included,” right? So that means that there cannot be any refutation of this kal va-chomer. What is that expression? It means that within two hundred there is one hundred. Fine. So now that means there cannot be a refutation of this kal va-chomer. How could you prove that opening is not inside digging? Every digging includes opening as well. If the claim is that this is more severe than that, you can bring another aspect in which that other thing is actually more severe. But here it is not a question of one being more severe than the other. This contains that plus something more. There cannot be a refutation here. This is exactly deduction. Deduction means that when I say all human beings are mortal, I have said that Socrates is mortal and more than that. So how could there be a refutation of such a thing? Okay? Therefore he says there cannot be a refutation. Therefore the Babylonian Talmud does not derive from here as the Mekhilta does, but the Mekhilta does derive from here that punishment is not inferred by logical reasoning even in such a kal va-chomer. Fine, so you might say this is because perhaps they do not follow Maharsha, that the reason one does not punish by logical reasoning is concern for a refutation. But I do not think so. This is not really a kal va-chomer. What? It is another kind of kal va-chomer. Right, a kal va-chomer of “if two hundred is included, certainly one hundred is included”—that is what it is called in the general literature. But that is not correct; there is a refutation to it. A kal va-chomer of “if two hundred is included, certainly one hundred is included” does have a refutation. What is the refutation? The refutation is the reasoning of Kesef Mishneh—that is the refutation. What are you telling me? You are telling me: look, if for opening he is liable, then for digging he is certainly liable, because opening is included in digging. Then Kesef Mishneh says: just a second, but digging is more severe. It may be that this requires a greater punishment, in which case it is not correct to impose the punishment of opening. What is that if not a refutation of the kal va-chomer? After all, it is a refutation of the kal va-chomer. This reasoning of Kesef Mishneh, which is not based on saying there is no refutation, is itself a refutation. Right? And what does that really mean? It means that when you take an inference that is completely deductive, that in the world of Ideas cannot have a refutation, once it becomes a proposition about the world it has ceased to be mathematics. There can always be a refutation. I will give you another example from the legal world. I saw this in Chaim Perelman, a philosopher of law who deals with legal rhetoric and so on, from Belgium. In any case, he brings a law—there was a law in Belgium called the Vandervelde Law. I don’t know if I’m pronouncing the… Vandervelde. Okay? This law basically said that one may not sell… one may not sell two bottles of wine at once. Fine? Is that because of pairs? What? Is that because of pairs? Or two liters of wine. Never mind. Two liters of wine, yes. I don’t know if one can switch to another scale; then it won’t be two, it will be pi. Pi liters. Fine. In any case, there was concern… so because… in other words, they prohibited selling two liters of wine. Why? Because there was concern that laborers returning home with their weekly wages would spend it all in the tavern and not bring the money home. So the government there in that area prohibited it. Now a laborer came and bought five liters. So he ended up in court. So two is prohibited but five is not? “If two hundred is included, certainly one hundred is included” should say yes. He said: two is prohibited, but I did not buy two—I bought five. So the prosecutor says to him: exactly, “if two hundred is included, certainly one hundred is included.” Five means you bought two, and besides that you bought another three. Do you want to say that because you bought another three you will be exempt for the two you bought? After all, you also bought two. This is “if two hundred is included, certainly one hundred is included.” And the judge decided he was right. Who? The laborer. The one who bought the five. Why? Why is he right? There is logic to it. Because there is also another possibility. But what he was basically arguing was that the prohibition against buying two liters of wine was meant to ensure that you would not spend your wages in the tavern when you return from work. Five liters are not bought out of your weekly wage. If you want to invest your savings in wine, what can I tell you—not to invest your savings and open a wine business? Who ever told you such a thing? We only want your wages to go home, to reach the bank. Once you take it out of the bank, do with it what you want. Now you can accept this or not accept it, but this is another example that there is here a kal va-chomer of “if two hundred is included, certainly one hundred is included,” and when you come to reality it does not necessarily work. In other words, logic, which people always see as… this is true in the world of Ideas. But when we apply it to our world, even logic is not certain. Nothing is certain, because whenever you apply it to the world there is always some further assumption that this logic is applicable to the part of the world with which I am dealing—just like with the vectors we saw, just like with the wine—that the mathematics according to which five is greater than two is applicable to the legal question whether one may buy five liters of wine. It is the same logic. Therefore whenever we deal with reality—and as I showed earlier also in Jewish law, in matters between people and in vows—logic is not certain; logic loses its certainty. Because there is always some non-logical appendage or some scientific appendage that you assume. You assume that this mathematics can be applied to this reality or to this legal question. And that assumption is already no longer mathematics. That assumption can definitely be attacked. Okay, so basically I’ll just summarize, because I see that the time has already come. Here I have more or less finished the introduction. I tried to explain what the relation is between logic and certainty, or between deduction and analogy and induction. And I will now state my goal, because I said it once already but I want us to keep our heads with the overall line of the argument. The fact that science is not certain, as we have now seen, and that no claim about the world is certain, is a fact that can be taken in two directions. The first, obvious direction says: fine, if it is not certain then everything is okay, we can close up shop and go home. There are no Torah-and-science problems; everything is fine, because every time there is a problem—what problem? Science made a mistake; after all, it is not surely right. Right? In that way we solve the problem in principle. Now I do not want to go in that direction, as I said earlier, and therefore I prefaced all of this to say that true, science is not certain, but nothing in life is certain. I mentioned Rabbi Shimon Shkop the first time, who says that yes, we also observe the Torah because of our own reasoning, so why should we not observe our reasoning? Right? Therefore the same applies here. We have a reasoned judgment that this science, although not certain, is reasonable. And when I spoke about reasoned judgment, and I return to the question you asked at the beginning, when I spoke about reasoned judgment and said that reasoned judgment has the status of Torah-level authority, and we brought various proofs for that—what I now need to claim is that science is a kind of reasoned judgment. In other words, because it is not certain, there was room to say: fine, if it is not certain then there is no problem, so who says it is right? Everything I tried to do now was to show the mistaken jump from “not certain” to “incorrect” or “doubtful.” There is also “reasonable” in the middle—do not forget that. And when we speak about sevara, reasoned judgment, and probability—of course they are from the same root—when we speak about sevara we are speaking about exactly this. Sevara does not mean certainty. Sevara means that it is reasonable. And that has the status of Torah-level authority. It is sevara that has that status, not certainty, and not error. The obligation to bless before eating that they brought in tractate Berakhot—that is not an obligation based on a sevara whose foundation is certainty of the “if two hundred is included, certainly one hundred is included” kind. There the sevara is simply that one should thank the Holy One, blessed be He, and not eat without a blessing. That is all. A very sensible sevara, reasonable, entirely correct—but it sounds reasonable. That, that thing, has the status of Torah-level authority. And therefore I do not see how one can argue, or use all the tools that people usually use in these debates—look, science is not certain, fine, so if there is a contradiction that means science made a mistake here in this matter. It does not work that way. Because once this scientific sevara—I didn’t get to define the boundary, now I suddenly remember, but fine—once this sevara gives me plausibility, that is enough for it to carry force like a Torah-level principle. Therefore the question does indeed exist—the question of Torah and science. Once I have a scientific result, despite all the problems we have discussed until now, namely that it is not certain, it is at least a sevara. And I do not know who can say that this is not even in the category of sevara. It seems to me that this is usually regarded as very strong sevarot. Fine? So it is at least a sevara. In that sense it has the status of a verse. It has the status of a verse. So now if we have a contradiction, that is a contradiction that has to be dealt with. Fine? On the other hand, of course, one has to know that this is really true—that is, science may indeed be mistaken. That is an option. It does not mean I will use that sweeping claim all the time whenever there is some problem: no problem, it is not necessary, done. But yes, that is an option, especially when we are dealing with conclusions that are some kind of interpretation of the science and not the scientific theory itself. Although the scientific theory too is a kind of interpretation, as we saw. But there is another level of interpretation—meta-scientific interpretation. That is already not the scientific theory but something beneath it. One indication, for example, is that it cannot be subjected to empirical testing. A scientific theory can be subjected to empirical testing even though it is the result of generalization, but a meta-scientific interpretation—for example, that from evolution we arrive at the conclusion that there is a God or that there is no God. Neither that there is nor that there is not. That is not a claim that belongs to the scientific world, although I do claim—and later I will speak about this—that it is a second-order interpretation of the scientific findings. In that sense, yes, it is true. But here one has to walk a tightrope. On the one hand, science is not God—that is, there can be mistakes. On the other hand, I am not willing to use that fact to throw all the questions into the trash, because sevara has the status of Torah-level authority. So with that, I hope, we can already begin to work. Whoever is not on the email list and wants to be…