Dispute and Truth – Lesson 17

Table of Contents

• Dispute, truth, and “these and those”
• You can’t “live with contradictions,” and the laws of logic as necessary
• Formalization, models, and the qualification to logic
• Levels of explanation, sufficient conditions, and breaking points
• A toolbox for resolving dichotomies and contradictions

Summary

General overview

The speaker connects dispute and truth through a systematic clarification of apparent contradictions, pluralism versus monism, and the possibility of understanding “these and those are the words of the living God” without giving up the law of non-contradiction. He argues that there is no real possibility of “living with contradictions,” and that the laws of logic are unlike the laws of physics, because they are necessary in every possible world and even with respect to the Holy One, blessed be He. At the same time, he qualifies this by saying that the contradiction is sometimes created by a mistake in formalization—that is, by an unsuccessful translation of rich content into logical-formal language. He offers a toolbox for resolving dichotomies: combining explanations that turn out to be sufficient only together, moving from binary categories to a continuum, and exposing a third possibility by attacking the assumption shared by both sides.

Dispute, truth, and “these and those”

The speaker presents two interpretations of “these and those are the words of the living God”: a monist interpretation, according to which there is one halakhic truth and only one opinion is actually correct, even though both opinions are legitimate in a tolerant sense; and a pluralist interpretation, according to which both opinions are right. He sharpens the point that on the factual plane, you cannot hold two contradictory truths, and that if there is a dispute about reality, then one side is mistaken. But he rejects the assumption that there are no mistakes, and allows us to say that there are factual disputes, while sometimes also showing that some of them are not really disputes about reality at all, but only appear that way. He concludes that if there is truth, then there is only one truth, and pluralism in the sense of “multiple contradictory truths” is equivalent to saying there is no truth, because the law of non-contradiction prevents a thing and its opposite from both being true at the same time.

You can’t “live with contradictions,” and the laws of logic as necessary

The speaker attacks the clichés about “the Torah is above reason” and “the unity of opposites,” as well as the approach that dismisses contradictions as something that shouldn’t bother us, and presents this as misunderstanding or intellectual laziness. He explains that the term “laws of logic” is misleading if we understand it like the laws of nature, because the laws of physics are learned from observation and could have been different in another imaginable world, whereas the laws of logic are not the result of observation and are valid in every possible world. He argues that even with respect to the Holy One, blessed be He, contradictions such as “a round triangle” cannot be applied, and that expressions of “cannot” here do not indicate weakness but meaninglessness. Therefore, the stone paradox (“a stone that He cannot lift”) is a meaningless concept, not a refutation of omnipotence. He cites the Maharal in the introduction to Gevurot Hashem about the superiority of the sage over the prophet, because truths such as “two plus two equals four” are true in all worlds and do not depend on observational apprehension.

Formalization, models, and the qualification to logic

The speaker presents formalization as the stage at which a problem from life or from science is given a mathematical/logical garment, and then the difficulty or paradox may indicate that the garment does not correctly represent the content, rather than that logic has “failed.” He illustrates this through Popper: “two plus three equals five” is not a scientific theory, because no experiment would cause us to give it up. He then shows, by means of vector forces, that arithmetic addition is not a suitable model for adding forces, even though mathematics itself is not thereby refuted. He brings in Vandevelde’s law in Belgium from the book of Chaim Perelman, and the judge’s ruling that allowed the sale of twenty liters of wine because the purpose of the law was to prevent someone from wasting his salary, not to prevent commerce. From this he concludes that even the a fortiori reasoning of “included within two hundred is one hundred” can fail when a formal model is not suited to the purposive meaning. He adds an example from topology—“the intersection of two convex shapes is convex”—to show that a mathematical proof depends on the definition that was chosen, and that there is no mathematical proof that the formal definition always matches everyday intuition. He argues that Schrödinger’s cat and the two-slit phenomenon are surprising phenomena, not paradoxes, and that the paradox is sometimes born from the model, not from reality.

Levels of explanation, sufficient conditions, and breaking points

The speaker first presents examples of parallel levels of explanation: repentance and leaving religion are explained on one side psychologically and on the other side philosophically; Newton’s falling apple receives a theological explanation alongside a gravitational one; and in Oscar Wilde’s “The Happy Prince,” the lead heart breaks both as a poetics of heartbreak and as a material-physical result of temperature. He identifies a parallel here to Pardes and the seventy faces of the Torah, but argues that the picture becomes more complicated, because an “explanation” is supposed to be a sufficient condition for what is being explained, and therefore two independent explanations, each claiming to suffice on its own, come into conflict. He concludes that usually not every one of the explanations is a full explanation, and that the correct explanation is their combination together as a sufficient condition, as in “he found a fly and did not object; he found a hair and did object” in Gittin. He explains that miracles are points of mismatch between a physical description and a theological description, where “there is no choice” but to break the laws of nature, and he calls this a “topological defect,” like the break in coordination between Cartesian and polar coordinates at the origin. He parallels this to interpretation by saying that a scriptural decree is a breaking point between the plain sense and the secret level, where the tools of the plain sense do not produce an adequate fit.

A toolbox for resolving dichotomies and contradictions

The speaker states that there is no possibility of a “jump into the vacuum,” and that dealing with a contradiction requires identifying what is not correct or what was formalized badly. He then proposes systematic mechanisms for reconciling opposing views. As a first mechanism, he suggests turning two competing explanations into a combined explanation, in which neither component by itself is a sufficient condition, as with the psychological-philosophical combination in repentance. As a second mechanism, he proposes a solution of the heap-paradox type by moving from binary categories to a continuum, so that the concept of a “heap” becomes graduated rather than simply “yes/no.” He applies this also to an argument against exams, which mistakenly assumes a world of pathologically lazy and industrious people, with no intermediate continuum. As a third mechanism, he proposes identifying a third possibility by attacking the assumption shared by both sides in the dichotomy, arguing that a contradiction arises only where there is an overlapping domain, and therefore one should look for the common element in order to break out of the framework. He illustrates this in the debate between creationism and evolution, where he proposes a third possibility in which there is no contradiction; and in the division between Religious Zionism and Haredi Judaism, where he cites the rabbi of Ponevezh, who said, “I am as Zionist as Ben-Gurion,” in order to formulate a position of “secular Zionism” in a religious person, without the religious-Zionist hyphen. He concludes with the argument against free will—“either there is a cause, and then there is determinism, or there is no cause, and then there is randomness”—and presents a third possibility in which there is no cause but there is purpose, so that the choice is causeless in the deterministic sense but directed toward an end. The failure, he says, lies in the formalization that identifies “no cause” with “randomness” and ignores the axis of purpose.

What I basically want to do today is talk a bit—since we’re talking about dispute and truth—I want to come back to the connection between them, between dispute and truth. And I want to deal a bit with contradictions, or apparent contradictions, apparently different positions, and the question whether there is really a dispute here, and what the truth is when there are different positions. We touched on this a bit in previous sessions, but I want to do it a little more systematically. So we talked about the topics of “these and those are the words of the living God,” and I presented two possible interpretations there. One interpretation, a monistic interpretation, says that there is one halakhic truth, but “these and those are the words of the living God” means that both opinions are legitimate in some sense. And one should relate tolerantly to each of the two opinions, but not because both are correct. Only one is correct—that’s monism. Pluralism says that both opinions are correct. Meaning, “these and those are the words of the living God”—both are correct. That’s the claim, and I discussed the implications of that throughout all the stages of that Talmudic passage: the divine voice, its conclusion, the Talmud’s explanation of the conclusion, and all those things. After that I spoke about “these and those” with regard to reality. I spoke about disputes about facts—how can there be a dispute about facts when, of course, the assumption is that in reality, at least, you can’t talk about two contradictory truths. Maybe on the legal plane you can say that the Holy One, blessed be He, wants it so that from His perspective both this and that are acceptable, even though they are opposed. Why? Because what really matters to Him is not what you do, but that you do it מתוך analysis within the halakhic-Torah system and so on. And if that is your conclusion, then from His perspective that’s fine. In other words, He has no goals in the content; His goals are in the method, you could call it that. Suppose so. I don’t accept that, as I said; I think that’s not true, and I also brought evidence for that. But at least there one can understand a pluralistic position. A pluralistic position basically says yes, there is more than one halakhic truth. From the perspective of the Holy One, blessed be He, it’s fine. With facts that’s impossible. With facts, if x happened, then “not x” is not true, and if x did not happen, then x is not true. So regarding reality, regarding facts, that can’t be. And then I said that this is really where that thesis comes from—that there cannot be disputes about facts. Because if there is a dispute about facts, that means one side is mistaken, and the assumption is that there can’t be mistakes. I said that assumption is not true. Of course there can be mistakes, and therefore there is also no reason not to say that there can be disputes about facts, although with some of the disputes I also showed that they are not really disputes about facts; it only looks that way. The bottom line basically says that obviously, if there is truth, then there is only one truth. There isn’t more than one truth. You can say there is no truth—that is, if you are a pluralist, then what you are really saying is not that there are many truths, but that there is no truth. Meaning, what is required of you is only some kind of methodology, but as far as the content is concerned there is no truth—it’s the law of non-contradiction. It can’t be that a thing and its opposite are both true simultaneously. I want to touch a little more on these topics, on this question—the question of pluralism, multiple truths, contradictions, whether one can live with contradictions. So maybe a brief introduction—actually two introductions. One introduction deals with the question whether contradictions are some kind of decision of ours. Meaning, is it possible to live with contradictory beliefs, or is it impossible? Meaning, there’s no such thing—contradictions have no place. Very often we hear various statements that the Holy One, blessed be He, is beyond reason, or the Torah is beyond reason, and where philosophy ends faith begins, or various slogans of that sort, and the assumption there is that logic is some kind of human form of thought of ours, but there could also be some other logic. What seems to us like a contradiction, maybe it isn’t a contradiction, or maybe one can live with it, and therefore the great alarm over contradictions is not justified. It may be that there are things that are contradictory—like the story they always tell about a student. A student comes to a rabbi and asks him some question, says, this Talmudic text needs serious analysis, such-and-such is contradictory. The rabbi says to him: look at Rabbi Akiva Eiger in Ketubot on page whatever, 26. He looks there and sees in the margin a note by Rabbi Akiva Eiger, and he doesn’t understand what it has to do with anything, so he goes back to his rabbi and says, well? I didn’t see any connection between what Rabbi Akiva Eiger says there and my question. So the rabbi says to him: did you see that Rabbi Akiva Eiger there comments on two consecutive Tosafot? On the first he remains with “it requires further analysis,” and then he moved on to study the second, right? In short, if you have a “requires further analysis,” you can move on and keep learning. In other words, the claim basically behind this is that one can live with contradictions. That it’s not something especially frightening. And certainly when we’re talking about faith in God and Torah and such lofty matters that are supposedly beyond human reason, and all kinds of things like that, then there is some tendency there to resolve contradictions with some statement like, yes, the unity of opposites, or all kinds of strange slogans of that sort, I don’t know. Yes, the unity of opposites is a Christian idea, as is known—a book by Nicholas of Cusa from the 15th century, I think. For some reason in recent generations, I think mainly since Hasidism, this somehow became very popular in the Jewish world too. The point I want to make clear is that it is simply not true. Meaning, you can’t talk about living with contradictions. What confuses people in this context is that we are used, at least in recent generations, to speaking about logic as the laws of logic. There are the laws of physics, there are the laws of biology, there are the laws of logic, the laws of parliament, whatever—various systems of laws. And the laws of logic too are a system of laws. So indeed, at least in the world of logic, people sometimes raise other logical systems, not the regular logical systems, but other logical systems. And I think this creates in people the sense that the system of logical laws is like the system of physical laws. Meaning, in our world the laws of physics are these, but there could be another world in which the system of physical laws would be different. And there, say, a body with mass would remain hanging in the air and not be pulled toward the earth, okay? Because the law of gravity doesn’t exist there. Or the opposite: it would actually fly upward because masses repel each other and don’t attract each other. Okay? Meaning, the laws of physics are not necessary things. What does “not necessary” mean? It means that I can imagine a world where they are different, where they are not true. Okay? In logic they present this in modal terms, as it’s called. The modal meaning basically says: when I want to speak about some proposition and say that it is necessary, what I really mean is that in every world I can imagine, that proposition is true. That’s what it means for it to be necessary. Because in our world the laws of physics are necessary—you can’t bypass the laws of physics—but we don’t call them necessary laws. Why? Because there could be another world I imagine in which the laws of physics are different. In our world it is necessary. Therefore, when you want to speak about necessity in the sense I’m speaking about here—logical necessity—then the laws of physics are not logically necessary. They are not logically necessary; rather, the laws of physics are scientifically necessary, let’s say, but not logically. And that means that in another world that I can imagine, the laws would be different; these laws would not be true. That is the modal meaning of necessity. So now, regarding the laws of physics—they are necessary in this world, but they are not modally necessary, they are not necessary in other worlds I can imagine; there there may be different laws of physics. Does the same hold for the laws of logic? Can I also imagine a world where contradictions do hold? Not like in our world. Or a world where there is also a third possibility, unlike the law of the excluded middle, where either x is true or not x is true—there is no third possibility. That is called the law of the excluded middle. No, maybe there is a world where there is also a third possibility: either x is true, or not x is true, or I don’t know, or a frog croaks. I don’t know—something third. Okay? So maybe the laws of logic too are like the laws of physics: it’s a system of laws. It’s a system of laws in the sense that here these really are the laws, but there could be another world I imagine where the laws are different, where there is another logic, just as there could be another physics. The claim I want to make is that that is not true. And therefore using the term “laws” for the laws of logic is misleading. Not recommended. You can say it’s semantics, do what you want, but it’s very misleading. Because they are not laws in the same sense that the laws of physics are laws. We call the laws of logic laws because we formulate them as a set of principles. But there is no possibility of different principles. It’s not that in another world I can imagine a world where the laws of physics are different, the laws of logic are different. What do you mean? In that world both x is true and not x is true? If x is true, then not x is not true. That is true everywhere. It is not a law of nature that just happens to be so in our world but could be otherwise in another world. No. It is true in every world whatsoever. Or in other words, if I learn the laws of physics from observation—and why do I learn them from observation? Because in another world they could have been different. How do I know these are the laws? I need to observe and see what the laws are in this world, right? Because without observing, how would I know? Maybe they’re like this, maybe they’re different. The laws of logic are not learned from observation. Why not? Because I don’t need to know the character of our world in order to say that these are the laws of logic. In another world too these would be the laws of logic. Or in other words, in order to know the laws of logic I do not need observation. Logic is not a science. Okay? Logic is a branch of mathematics or of philosophy; it is not a science. Invention—that already carries a connotation. So I’m saying, when you say it is an invention, you are already assuming something, and that is a problematic connotation. I do not agree that it is an invention, but I do think it does not arise from observation. But it is not an invention. Invention means it is our arbitrary decision; we could arbitrarily have decided otherwise. No, it is not arbitrary. It is necessarily true. Invention means it is not true. True means true but not necessary. But no, it is necessarily true. What does it mean that it is necessarily true? For example, the Maharal in the introduction to Gevurot Hashem—I think it’s there—has three introductions. In one of the introductions he writes there why a sage is preferable to a prophet. He says every prophet has some maximal range up to which he sees, yes, in Kabbalistic-world terminology, there are even descriptions of which world each prophet reached, how far his attainments reached. But the sage grasps all the worlds together. Because, say, two plus two equals four in all worlds. You don’t need to experience that; you don’t need to see it; it is not something visual. This idea that two plus two equals four is always true; it does not depend on this world or another world. So if it is true, it is true in every world. And in that sense a sage is preferable to a prophet. Because his concepts are true for every world. The prophet—think of a prophet as a scientist—understands the laws of nature of the different worlds. Up to the point that he understands. What he does not understand, he does not understand. But the sage does not need observation. Two plus two equals four regardless of observation. It is always true. So if it is always true, it will be true in all worlds. I do not need to have apprehension of them in order to know that two plus two equals four in all of them. Therefore a sage is preferable to a prophet. Meaning, there is something about logical, mathematical truths and the like, that makes them stronger than scientific truths. We may sometimes have less confidence in them. Philosophical statements—either they are true or they are not true—we may have less confidence in them than in scientific statements. Wait, sorry, one second. We have less confidence than in scientific statements, but on the other hand scientific statements are true after we have observed them, right? And in another world they could be different. But our a priori concepts are true in every world; they are not the result of observation. And in that sense they are stronger than scientific conclusions. Yes, mathematics is more true than science. Even though for science I have observational confirmation and for mathematics I do not. But fine—still it is more true because it does not need observational confirmation. Not because I don’t have observational confirmation; it does not need it. It is true regardless of observation, true in a way prior to observation, because it is a priori. So therefore this proposal to live with contradictions is a proposal that stems from misunderstanding. It is some sense that the system of logical laws is like the laws of physics or chemistry. It could be this way, it could be that way. So in another world the laws of logic are not valid, or are not relevant with respect to the Holy One, blessed be He, or with respect to spiritual worlds, or spiritual ideas. Not true. The laws of logic apply to everything. Meaning, if the Holy One, blessed be He, exists, then it is not true to say that He does not exist. If He supervises, then it is not true to say that He does not supervise, and vice versa. It doesn’t matter whether I’m talking about the Holy One, blessed be He, or about whoever you like. I cannot say that the Holy One, blessed be He, can create a shell that penetrates every wall and also a wall that stops every shell. Right? Self-contradictory. It’s either this or that or neither—but not both together. Even the Holy One, blessed be He, cannot. Why can’t He? Because deviating from the laws of logic is not even called an ability at all. There is no such thing. The Holy One, blessed be He, cannot make a square circle not because there is some defect in His omnipotence, some limitation in His powers, but because there is no such thing as a square circle. Explain to me—if you ask me whether the Holy One, blessed be He, can make a square circle, explain to me what you are asking, and then I can try to answer you. But what you are asking is meaningless. When I say that the Holy One, blessed be He, cannot make a square circle, again, the “cannot” here is an inaccurate expression. It is not that He cannot because He lacks sufficient power. He cannot because there is no such thing; it is simply undefined. Therefore there is no problem saying that the Holy One, blessed be He, cannot deviate—or cannot deviate—from logic. He cannot deviate from logic not because He is weak, because He lacks enough power, the way in physics, say, where He can perform miracles and deviate from the laws of physics. He cannot deviate from the laws of logic because deviating from the laws of logic is an oxymoron. It’s like a square circle. There is no such thing as deviating. There is nothing whatsoever outside the laws of logic. Everything—logic rules over everything. There is no such thing as outside the laws of logic. Okay? It’s like saying square circle. Yes, and this is the famous stone problem—whether the Holy One, blessed be He, can create a stone He cannot lift. Yes, so the proof is, either way, that He is not omnipotent. Right? What is the mistake in that argument? The paradox of omnipotence, yes. After all, it does not attack the Holy One, blessed be He; it attacks the concept of omnipotence. And the claim is that the concept of omnipotence leads to contradiction regardless of the Holy One, blessed be He; it has nothing to do with Him. But that is a mistake. It is not true that it leads to contradiction. Because when you discuss—say I believe that the Holy One, blessed be He, is omnipotent. Now someone comes and tries to attack me and show me that I am not right. Yes? So he basically has to take me and derive a contradiction from my assumptions. Okay? So he says to me, tell me, can the Holy One, blessed be He, create a stone He cannot lift? I didn’t understand that sentence. What do you mean? I assume He is omnipotent. What does it mean whether the omnipotent one can create a stone that the omnipotent one cannot lift? There is no such stone. What is this stone that He cannot lift? It’s like a square circle. I do not understand that sentence. That sentence is formulated in your conceptual world. In your conceptual world God is not omnipotent, so you can talk about a stone He cannot lift. But in my conceptual world, where I say He is omnipotent, you cannot ask me: tell me, can He create a stone that the omnipotent one cannot lift? That is like asking me whether He can create a square circle. If He is omnipotent, then there is no stone He cannot lift. Not just that none exists—there is no such concept as a stone He cannot lift; it is undefined. It is against the laws of logic, not against the laws of physics. A stone that the omnipotent one cannot lift is a meaningless notion. It is a meaningless concept, not a sentence. Okay? So it is that kind of question. It is like asking me whether He can make a square circle. That means that logic is the framework for all our thinking in every area. Not only in some areas or others—in every area. There is nothing whatsoever that departs outside logic. If you encounter a contradiction, one of its sides is not true. Don’t tell me stories that you’re leaping into the empty void and the unity of opposites and you’re some great believer and therefore not troubled by contradictions. You’re a great babbler. Nothing more than that—or a great lazy person. Meaning, someone who is not willing to invest thought, and therefore instead of trying to think why there is no contradiction here, or what the problem is, or which side to give up, he jumps into the empty void. I don’t know what he does there. If you say all kinds of words from Hasidism or Kabbalah in general, you sound terribly deep. So the conclusion is basically that the laws of logic are a rule from which one cannot deviate. All right? You can’t say, “I live with contradictions.” If there is a contradiction, one of the sides is not true. Okay? That is the first introduction. The second introduction somewhat qualifies the first. Usually when we carry out logical analysis of reality, we have a tendency not to notice that there is some stage along the way. In logic that stage is called formalization. Yes, form—giving form. When I formalize some scientific problem or a problem from life or whatever, I give it some mathematical or logical form, and then I use mathematical or logical tools to analyze it. Okay? Now very often, after I have formalized the problem, it becomes a problem in logic, and now I can use logical and mathematical tools to prove various things or to rule out various things, and so on. And then sometimes I can also arrive at paradoxes and all kinds of things like that, and then that has various implications. But very often, once I have arrived at a paradox, it means that the formulation, the logical garment I gave to the problem that interests me, is not correct. Meaning, it does not represent it correctly. And therefore, although you have a problem on the logical level, that does not reflect a problem in the content itself. It simply means that your translation into the language of logic was not a successful translation. Okay? Yes, I’ll bring some examples if you like. For example, I used to teach mechanics here in physics, and in the first lesson I asked them whether the law that two plus three equals five is a scientific law. According to Popper, a scientific law is a law that can be subjected to a test of falsification. Is that familiar? Meaning, Popper said that one cannot prove a scientific theory. One can only try to falsify it. For example, the theory that all ravens are black. Okay? How would you prove it? Observe all the ravens? You never know that you have observed all of them. Maybe there is one more, maybe there will be one more in the future—you never can. There is no way to prove a scientific theory. What one can do is try to falsify it. Take a raven and see. If it is black, very good. But if it is pink, then you have falsified the theory that all ravens are black. Meaning, one can falsify a theory; one cannot prove it. Okay? Popper goes one step further and says: if so, I have a criterion for what counts as a scientific theory. A scientific theory is a theory that can be subjected to a falsification test. For example, the theory that God exists is not scientific. That does not mean it is not true; it means it is not scientific. Why? Because you cannot propose an experiment to test it such that if it fails, that means there is no God, and if it succeeds, then maybe there is. You can’t prove it, but at least you haven’t falsified it. Okay? There is no such experiment. At least I don’t know of one. Okay? So that is a non-scientific theory. That doesn’t mean it isn’t true, by the way; it is non-scientific. Fine? So Popper basically claims that a scientific theory is a theory that can be subjected to a falsification test. For example, the theory that every object with mass floats in the air and does not fall to the ground is a scientific theory. A scientific theory that is not true. Because I can subject it to a falsification test. I’ll put the object here and see. If it falls to the ground, then the theory has been falsified. Meaning, this theory as a category is a scientific theory. In this case, a scientific theory that is not true. Fine? But it is a scientific theory, because it can stand a falsification test. One can test its predictions, do an experiment, and see whether it works or not, whether it fits or not. Now I asked them whether two plus three equals five is a scientific theory. Or in other words, do you have an experiment that could test this theory—falsify it or confirm it? There is no such experiment. What, does everyone agree? Wait, wait, wait, one second. Those are explanations afterward. First of all I’m asking: can anyone think of such an experiment? Notice—not an experiment that will falsify, but an experiment that can falsify. Even if you know in advance it won’t happen. So what? Also with gravity I know that won’t happen. Every object you put in the air will fall. I promise you. And still that is a scientific theory, because in principle if it did not fall, the theory would be falsified. So that is a theory that can be subjected to a falsification test. Fine? Here, I’ll propose an experiment to you. Right? Right. Take a plate, put two balls in it, take three more balls into the plate, count how many you have altogether. If it comes out five, excellent. If it comes out minus four, then you have falsified the theory that two plus three equals five. Right? So yes, it can be subjected to a falsification test. Now I’ll ask you a question. Suppose you put in two balls, added another three balls, and it came out one. You counted how many there are altogether—one. What is your conclusion? That you counted wrong, that there was an error in the experiment, that some balls disappeared, turned transparent, disappeared from the plate—I don’t know what. Even though you see no indication of that, right? But you will never conclude that two plus three equals one. Never. No experiment will ever bring you to that. You will always say there was an error, or that we missed something, I don’t know exactly what. What does that mean? It means that this experiment cannot really falsify the claim that two plus three equals five. Because if it fails, even then I will not give up the claim. I will say there was some problem in the experiment. Okay? That means that two plus three equals five is not a scientific theory. It does not stand a falsification test. It does not emerge from observation, it is not connected to observation, and it will not be falsified by observation. It is true because I understand the concepts two, three, and plus, and I understand that together they make five. Okay? That’s all. No, it is not a result of observation and not a result of anything. One can of course use observations to help show a person—take two plus three, see, altogether it comes to five. That is a didactic aid. But if it did not come to five, he would still think that two plus three equals five. Meaning, it is not really based on observation. Why am I saying this? Because, as I told you—remember, this was at the start of a mechanics course—I told them: look, I have an experiment that falsifies the claim that five plus five equals ten. Take a certain body, apply a force of five newtons northward and another force of five newtons eastward. What is the total force acting on the body? Square, square, root. Five times root two, right? Seven and something. Okay? Root fifty. Vector, vector, yes. So what does that mean? That five plus five does not equal ten; it equals seven and something. Here, I have falsified the scientific theory that five plus five equals ten. But when we do that experiment and see that this is not the total equivalent force, we do not conclude that five plus five does not equal ten. Rather, we understand that the mathematical theory appropriate for describing the addition of forces is not arithmetic, the addition of numbers, but vector calculation. Vector calculation already knows how to perform those computations. Meaning, if I reach the conclusion that some logical argument or some mathematical argument is not correct, I will not say mathematics made a mistake. I will say that this mathematical theory is apparently not suitable for describing the reality I am talking about. Or in other words, there was a problem in the formalization. There was a problem in translating the real problem into a mathematical problem. For example, let’s go back to forces. I want to know the total force acting on the body: there is five this way and five that way. My formalization is: okay, this is probably arithmetic addition. I need to do five plus five and see what comes out. It comes out ten. Even with a supercomputer you’ll find out it’s ten. Okay? So what does that mean? It means I formalized the problem of adding forces by means of arithmetic theory. One could actually say that physics is a model of arithmetic; in mathematical language that’s what it’s called. A model—meaning, it satisfies those rules of arithmetic. Well, no, that’s not true. But the claim that physics is a model of arithmetic is not a claim in mathematics; it is a claim in physics. And therefore, if I discovered that it doesn’t work, that is not a refutation of mathematics. It is a claim in physics, saying that the addition of forces is not well described by arithmetic addition, but requires vector calculus. Okay? But that’s physics, not mathematics. Meaning, although there is a mathematical theory here—five plus five equals ten—I did an experiment and reached the conclusion that it is not correct. What is my conclusion? That mathematics is incorrect? No. My conclusion is that my physical assumption is not correct, that the formalization I made was not correct. Okay? Yes, exactly. So the qualification I want—the second introduction I want to bring here, the qualification to the first introduction—is that although logic is always correct and one cannot live with contradictions, one must notice carefully that even if we have arrived at a contradiction, that does not necessarily mean that we have to give up something here. Sometimes our formalization doesn’t work. Let me maybe give you another example. There was a law in Belgium, by someone named Vandervelde. I read this in a book by Chaim Perelman. He was a law professor in Belgium, a Belgian Jew. And he says there was a law called the Vandervelde Law, which forbade selling more than two liters of wine to workers. Why? Because the legislator wanted them to bring their weekly salary home, and if they spent it all in the pub, then there would be no livelihood at home. So he says: up to two liters you may drink, but that’s it. More than two liters—it is forbidden to sell to you. Okay? Now a worker comes to the pub, I don’t know, and he wants to buy twenty liters of wine. The seller says to him, listen, impossible—one may not sell more than two liters. Yes, more than two liters is forbidden, but I want twenty, not two. Twenty. Two hundred minah? When I sell you twenty, I certainly sold you two; I just also sold you another eighteen. So what? But I still sold you two. This is what in Jewish law is called an a fortiori argument of “included within two hundred is one hundred.” Fine, they went to the judge—there are surprises. They went to the judge. And the judge says the buyer is right. The buyer is right. If he wants twenty liters, sell him twenty liters. Ah, what? What do you mean? If I sell him twenty liters then I sold him more than two, and the law forbids selling more than two liters. How can that be? The judge answers as follows: after all, the law is really intended so that you bring your weekly salary home, so that you don’t spend it on wine in the pub. Okay? But if someone wants to start investing in the wine business, buying quantities of wine, putting them in a warehouse and selling them—I don’t know, entering the wine business. There is freedom-of-occupation law. He is allowed to do business in that too, right? And someone who buys twenty liters is apparently buying it for commerce or I don’t know exactly what. Whatever the case, in any event it is not merely wasting one’s weekly salary but perhaps investing money. He wants to invest it in wine. What’s the problem? That the law does not forbid. So notice what came out here. There is here an a fortiori argument that even some of the rule-based authorities say cannot be challenged; it is an a fortiori argument of “included within two hundred is one hundred.” “Included within two hundred is one hundred” cannot be challenged—why? Because an ordinary a fortiori argument can be challenged when the supposedly more severe case is in some other aspect less severe than the source case. The relationship between them is not simple. Okay? But an a fortiori argument of “included within two hundred is one hundred” is not based on the second being more severe than the first; it is based on the second containing the first. The second contains the first itself, plus something. So here I do not say that because the source case is liable, all the more so the derived case is liable. Rather, the derived case is liable because the source case is within it. And when you buy ten liters, you certainly bought two liters. Not because ten liters is more than two and therefore more severe and surely the legislator also forbids that—no. Rather, he forbids the two liters that are within the ten. Meaning, he forbade two liters; within the ten there are two liters. Did you sell two liters? Then you violated the prohibition on selling two liters of wine. True, after that you also sold another eight. But those first two liters are forbidden to sell. And that is a much stronger a fortiori argument. Here it cannot be challenged. What possible challenge could there be? After all, you sold two liters. You did not do something more severe than two liters, where one could say more severe in this respect but not in that respect. But here no—you sold two liters. That is exactly what the legislator forbids. And it turns out that even an a fortiori argument of “included within two hundred is one hundred” can be mistaken—those rule-based authorities are mistaken. Even such an a fortiori argument can be challenged. Here, we saw it. By the way, there are halakhic examples of this too. An a fortiori argument of “included within two hundred is one hundred” can be challenged—what does that mean? After all, you cannot deny that within ten there is two. That ten is two plus eight. Right? So if ten is two plus eight, then if you sold ten liters, you sold two liters and after that you sold another eight liters, so with respect to the first two liters that you sold, you violated the prohibition on selling two liters of wine. What difference does it make that you sold another eight? There can be no challenge to that; it is logically necessary. Right? In logic. But you did not formalize correctly. You think that the way to interpret this law is arithmetically. If two is included within ten, then obviously if two is forbidden, ten too is forbidden. No—the problem is in your model. You formalized the problem badly. Yes, exactly. Meaning, you took the law in a formal, literal way. And if you think about what it means, you actually see that your formalization is not correct. And the formalization—you will not succeed in proving in any way that two is not included in ten. Of course two is included in ten; there is no way to refute that. So when I refute, what does it really mean? I did not refute logic. I refuted the connection between the logic and the reality it is trying to describe. And my claim is that this logical or arithmetic model does not constitute a good representation of the content I am trying to represent. That is the problem. Okay? So in other words, we have two introductions. One: there is no way to escape logic. The second: no problem we deal with is logic. We always get to a logical problem after we have done a formalization. But no problem is pure logic. After we formalize, then you have a problem in logic—but notice that in the background there is always the question of the formalization, whether you formalized correctly. Therefore, if you encounter a problem in logic, that does not necessarily mean that you have to throw something out. What? I don’t see why this is connected to the question of an interpretive limitation. That is the basis of the whole problem. There is a certain scale, and in some sense no—an interpretive limitation is situating something in certain circumstances; it is adding a datum. We are talking here about nighttime; we are talking here about spoons and sleeping; we are talking here about—this is adding a datum. Maybe—I don’t think it is the same phenomenon. It does not seem to me the same phenomenon. It is really a mistake to say that he violated the law. What? If the legislator had said it, fine—but the judge has to adhere to the law. His claim is that this is what the law says. Meaning, he claims that purposive interpretation is part of the tools for interpreting the law. He interprets the law as containing an exception. Logic itself is not an exception. Not an exception—the law itself, which says “do not sell more than two liters of wine,” that is not what the law says. The law says: do not invest your weekly salary in wine. That is what the law says. That’s it. Once you assume that, everything is fine. So once this is a judge and not a legislator, he cannot legislate; he only interprets. So if he interprets, that means that this is how he understands the law itself. So it means that this a fortiori argument is not correct. Of course it is logically correct; it is not legally correct, and logic does not provide a good formalization of the legal problem. Exactly like with vectors and forces. There too mathematics is not a good formalization for physics. This mathematics—arithmetic—is not a good formalization for physics. Therefore, although I encounter a problem, I am not bothered by it; the formalization is apparently not correct. Okay, so those are the two—those are the two introductions. I’ll give you an example, a very amusing example. Let’s see if I have a marker here somewhere. I once found this—I was once in Boston, I went into a used bookstore, and I found some really high-school-level book on some area of topology. I don’t remember what anymore. Sort of popular mathematics for high school. Let’s see if I have the—no, not here. All right. Okay, let’s do it orally. Look, there is a theorem in topology that says that the intersection of any two convex shapes is also a convex shape. Now, let’s speak not in precise language—of course I’m not a mathematician—but in ordinary everyday language, we know what a convex shape is. A convex shape is a shape with a belly outward, with all due respect to the mathematicians here. Okay? Like a ball. Yes, like a ball. As distinct from a bowl—the opposite of concave. Concave is like this, convex is like this, okay? Very coarse, of course. So now take two convex shapes, and let’s say a straight line is also convex for our purposes. Only concave is not convex, all right? Let’s say a triangle, for example, is a convex shape. Fine? Because there is no inward belly anywhere; at most there is a straight line or a pointy corner that is really concave, but still—but still—there is nowhere any concavity. Okay? Therefore it is a convex shape. A triangle or a square or whatever—anything like that is a convex shape. A crescent moon, for example—no. A banana shape—no, because one of the boundaries there is concave. Okay? Now, how does one prove that the intersection of any two convex shapes is convex? Because think of a triangle and a circle, yes? So here is the triangle and here is the circle, and there is some region here common to both. That region is also convex. And of course any intersection you like—keep intersecting as many shapes as you want—if they are all convex shapes, then the intersection is also convex. Now, I’m not drawing it here on the board, but if you try to think about it a little at home, it is not so trivial to prove. Yes, that’s where I’m going. But when you look at it, if you think about it in simple everyday terms—what is a convex shape?—it is very hard to prove such a thing, if at all. But let me show you a trick for proving it in one line. Because, as mathematicians do, first we need to define well for ourselves what a convex shape is. We have an intuition of what a convex shape is, but let’s try to give a definition. Mathematicians work with definitions. So the definition they propose for a convex shape is that for any two points inside the shape, if you connect them with a straight line, the whole line lies inside the shape. Take a circle, for example: any two points you take, connect them with a line, the whole line lies inside the circle. By contrast, think of a bowl, yes? A bowl—say this is its lower boundary—or a banana shape. So if I take a point here and a point here and draw a straight line, then you see that part of the line is not inside the shape. Because it is concave. Okay? So therefore, if that is the definition of a convex shape, then the proof is one line. Take two convex shapes—doing this orally is silly—but take two convex shapes, say these two circles, okay? Two convex shapes, two circles, and here in the middle is their shared region. And what I need to prove in order to prove that this shared region is convex is that any two points inside it, if I draw a line between them, the whole line lies inside this intersection, right? Now you understand that those two points belong both to this circle and to this circle, because they belong to the intersection. Whatever belongs to the intersection belongs to both this and that. Now, if those two points belong to this circle and it is a convex shape, then this whole line lies inside this circle, because by definition it is a convex shape. And the same goes for this shape too, right? So if the whole line lies both in this shape and in that shape, then it also lies in their intersection. Which is what we had to prove. Now that is very simple, right? So where was the problem before? That we didn’t define it. Right? We didn’t define the concept of a convex shape; we thought of it intuitively, and therefore it is very difficult to prove this claim. And the moment we made a precise definition, everything was fine. That is how the mathematicians would answer you—and not the physicist. And that is not true. There is a law of conservation of difficulty among physicists. The law of conservation of difficulty says this: you never succeed in bypassing difficulties. If you bypassed a difficulty, that means it is somewhere there; you will encounter it along the way, always. I tell you, I have all kinds of frustrations with these things. Every morning I had some new idea of how to get around all sorts of problems in my doctorate or something like that, and I started to progress and always met the difficulty—either from here or from there, it always appeared. Meaning, there is no way to get around it, okay? Here too, same thing. Did we really prove here that the intersection of any two convex shapes is a convex shape? The answer is no. No, we did not prove that. What we proved is that a convex shape in its mathematical definition satisfies this theorem. That is true. But who said that our intuitive definition of a convex shape is well described by the mathematical definition? Intuitively it sounds very right, but intuitions—we are mathematicians, and intuitions do not help mathematicians. Who said that the definition according to which if you connect any two points, the whole line lies inside the shape—that that is called a convex shape? It is supposed to fit perfectly with our intuitive understanding of the concept of a convex shape, right? If it really fits perfectly, then once we prove it in the mathematical world, it is also true in the everyday world. But who said there is such a correspondence? There is of course no mathematical proof of that, nor can there be. Mathematics starts from the definition. But who said that this definition corresponds to our intuition about convex shapes? There will always be some point that is concave, inside that point. You are offering me some other proof now, so leave it—probably you’re not right, but let’s leave it, it doesn’t matter, that’s not our issue here; I’m only bringing an example. What I want to say is that when mathematics defined the concept of a convex shape, it was basically offering me a model; I did a formalization here. I have an intuition of what a convex shape is. Mathematics clothed it in a precise definition, okay? And from there onward one can begin to work. The big question is whether the definition actually fits my intuitive concept of a convex shape. As long as I have not proved that, then this mathematics has not really proved to me that the intersection of any two convex shapes is convex. In other words, mathematics proved a mathematical theorem. But if I ask physically, in the world, whether the intersection of any two convex shapes is a convex shape—I don’t know. I don’t know, at least not with mathematical certainty, because I don’t know whether the mathematical definition describes my intuitions precisely, whether there won’t be some exception where in my intuition it counts as convex but mathematically it will not be convex, or vice versa, okay? Meaning, in order to prove a claim about my everyday concepts using mathematical tools, I need to believe that the mathematical formalization is indeed precise, but there is no proof of that. Because mathematics begins after the formalization; it begins from the definition. It cannot prove to me that the definition fits the everyday concept—that is not its business, and it does not even try to prove that. Okay? Here is another example of the fact that yes, in mathematics everything is always perfect, no mistakes, everything certain, no room for argument, everything is true—but none of our real-life problems is logic. Logic is always some kind of model. Or a formalization of the problem we are dealing with. And therefore, when we encounter a logical problem, very often we need to think backward and see: maybe we really proved something here, but maybe our formalization was simply not correct, and therefore we got into trouble. Okay? Schrödinger’s cat is not a paradox; it is a surprising natural phenomenon. What is paradoxical there? The cat is alive and dead. Why? Who said it isn’t? It is. There is a natural phenomenon there, there is. What is paradoxical about that? He produced the model that way. It is surprising, but what is paradoxical about it? Also the particle that goes through two slits—that is not paradoxical, only surprising. Fine. All right, so now I want to move on to what is called dissolving dichotomies or resolving contradictions. Look. I’ll maybe bring a few examples of apparent contradictions and how one can treat them and show that this is not a contradiction. Again, the option of living with the contradiction does not exist. To say that this is the unity of opposites and leap into the empty void—that is good for anyone who buys that nonsense. But if one really wants to handle contradictions, then one needs a certain toolbox that will help us handle contradictions, or different opinions, and see what the truth is in relation to the opinions. That is exactly the connection between dispute and truth. Look, for example, say someone became religious. Not in the old sense of repented, but in the modern sense of “became religious.” He did not regret his sin and did not change his worldview—those are totally different concepts. So his secular friends ask themselves: what crisis did he go through? What caused him—what does he need this nonsense for? Right? They look for the psychological explanation of the matter. His new religious friends: at last he understood the truth. He left falsehood and came to the truth. Meaning, the religious are philosophers and the secular are psychologists. Okay? Now someone leaves religion. So what do his religious friends say? He wanted to permit sexual prohibitions to himself, right? You are psychologists. And what do his new secular friends say? Well, at last he understood the nonsense he was living in and that one has to leave all that nonsense already. They are philosophers. Okay? Who is right? I think both are right. Every person who does some action can be explained on the psychological plane and can be explained on the philosophical plane. On the psychological plane, he broke up with his girlfriend, went through a crisis, and therefore became religious or left religion. On the philosophical plane, he had difficulties with faith, or difficulties with secularity, or whatever it may be, and he thought the alternative was more correct. Okay? You do not necessarily have to choose between those two explanations. On the contrary, usually a person has some philosophical justification for what he does, and along with that he also has psychological reasons for what he did. We are all flesh-and-blood creatures, and there are psychological things that cause us to act in one way or another. Meaning, here there is ostensibly an argument between two opposing sides, but actually both can be right. Both can be right. Another example of this matter. Newton is sitting under a tree and an orange falls on his head, heaven forbid. Newton asks himself: why do oranges fall on the heads of innocent Christians? And he discovers the law of gravity. The law of gravity pulled the orange and therefore it fell. So I ask myself: what was wrong with the theological answer that says he sinned yesterday and therefore the Holy One, blessed be He, dropped an orange on his head? Yesterday he did not turn the other cheek, and therefore an orange falls on his head as punishment. Good explanation. He was a devout Christian. So I assume he would have accepted such an explanation. Such an explanation. Well then, why do you need gravity explanations and all these things? Because here too we have parallel planes of explanation, exactly like in the previous example. On the scientific plane, that apple fell on him because the force of gravity overcame the force applied by the branch that held the apple. And therefore it fell. Okay? On the theological plane, it fell on him as punishment for a sin he committed yesterday. And therefore we are dealing not with two conflicting explanations, but with two explanations each of which belongs to a different plane. There are parallel planes of reference or planes of explanation. Okay? A third example: there is a very famous children’s book by Oscar Wilde called The Happy Prince. Know it? I think every child read it, or at least saw it. The Happy Prince is some prince who lived in a palace and of course had a happy life, and he did not know how much suffering there was outside. Oscar Wilde, yes—it’s a well-known tragicomic piece. And after he died they built a statue of him on some very high pedestal. The prince died, yes, on some very high pedestal, and from there he could see beyond the palace walls, because he was already so high, and suddenly he saw all the misery and everything. The statue, yes, saw all the misery and all those matters. He was entirely covered in gold and precious stones and a sword studded with diamonds and things like that. One day a swallow arrives there, whose companions had flown to cold Europe—sorry, whose companions had flown in the winter to Egypt, to warm places—but she was late, she stayed there and got stuck in cold Europe. So she began staying beneath the prince there; she lodged there beneath the prince in that statue, and he began sending her on errands to all kinds of miserable people around the city. Give them the precious stone from here and the gold leaf I have here and the diamond from my eye and my sword—in short, distribute all the gold and diamonds to all the wretched people of the city. At a certain point the cold intensifies, winter is already at its height, and there is a heartbreaking farewell scene. The swallow parts from the prince because she feels she is about to die. Not for nothing do they migrate to Egypt, because they cannot live in Europe in that cold. Then she says goodbye—she says, I can’t, I’m about to die, and so on—after having done all this charity and kindness and so forth, and she dies. And then the story goes like this: “Then there came a low cracking sound from inside the Happy Prince. His leaden heart had split clean in two. Ah, how bitterly cold it was.” Now the question is: why did his leaden heart split in two? Because it broke over the swallow’s tragic fate? Or because of material strength—simply the temperature dropped so much that the metal did not withstand the temperature and it cracked? Which explanation is more correct? Apparently both are correct. The scientific explanation is material strength and temperature, temperature resistance; and the poetic explanation, the human explanation, I don’t know what to call it, is that the leaden heart split because it could not bear the tragedy. Okay? So ostensibly we have a collection of parallel explanatory planes. You can see an example of this in Torah interpretation. In Torah interpretation we have peshat, remez, derash, and sod—plain meaning, hint, homiletic interpretation, and secret meaning—right? One can interpret the very same verse in several different ways. Why? If one of them is correct, then the others are not correct as different interpretations? No—“the Torah has seventy faces.” You can interpret the Torah in several ways, in plain meaning, hint, homily, and secret; these are parallel planes of reference. Okay? That is, you can interpret things or explain things on different planes in parallel, and the explanations do not necessarily compete with one another. But this whole story is more complicated than what I have described so far. Why? Because let us return to Newton and the apple. Okay? When we speak about a scientific explanation—the apple fell because of gravity. Okay? The assumption is, every explanation—not just a scientific explanation—an explanation is supposed to give me a sufficient condition for what happened. A necessary condition is not enough, and a necessary and sufficient condition is not required. It needs to be a sufficient condition in order to count as an explanation or cause. Yes? For example, the paper burned because I lit a match with fire near the paper. Okay? What does that mean? The match, the fire of the match, is an explanation for the paper’s burning. Why is it an explanation? Because obviously paper is… not from something else. Okay? Therefore it is not a necessary condition, but it is a sufficient condition. Meaning, if paper is in fire, it will burn. That is sufficient. It is not necessary. That is called an explanation. Meaning, an explanation has to be a sufficient condition for the thing explained. Okay? That is the definition. Now understand that if so, then the multiplication of explanations I proposed for Newton, for example, is very problematic. Because on the one hand you tell me that gravity caused this apple to fall on Newton. When you treat that scientific explanation as an explanation, what you are really saying is that if Newton were sitting there and the force were the same force and the branch had the same strength, then the apple would fall on him whether he sinned or did not sin. Because that is enough; it is a sufficient condition. So if the whole situation is like that, then the apple falls, period. So what room is left for God’s decision and for Newton’s sin? If Newton had not sinned, the apple would not have fallen. And alternatively, if you say the theological explanation is an explanation—that the Holy One, blessed be He, dropped it on him because he sinned—then that means it falls on his head even if gravity by itself could not overcome the support of the branch. Right? Otherwise it is not an explanation. An explanation needs to be a sufficient condition. That means that if you sinned, the Holy One, blessed be He, drops an apple on your head. It does not matter at that moment whether gravity is such or whether gravity is otherwise. That means that if I treat these two planes of explanation, each one of them, as an explanation, then something here cannot work. Because each one is a sufficient condition. There cannot be two independent sufficient conditions. Independent means that if one sufficient condition holds and the other does not, it still happens. Fine? That cannot work. If you tell me Newton needs to get an apple on the head because of the sin, then he gets an apple on the head because of the sin, even if gravity is not enough to bring the apple down. Because that is the explanation. It means that if he sinned, that is a sufficient condition for the apple to fall. Okay? And if the scientific explanation is an explanation, then that means gravity is a sufficient condition for the apple to fall, even if Newton had not sinned. So what does that mean? It basically means that these two explanations really do contradict one another. Unless you say that somehow there is always a fit. Whenever you sin, then gravity is such that it overcomes the branch, and vice versa. And if you do not sin, then there is no fit. But I see no logic in there being such a fit. I have free choice whether to sin or not to sin, while the strength of the branch and the weight of the apple are givens. They do not depend on my choice, so I do not think that such an option exists, this matching-up. Okay? In other words, what I want to say is this: even regarding the four explanatory levels I mentioned for verses—plain meaning, hint, homily, and secret—there too the picture is not so simple. Although according to Maimonides’ view, really no, only the plain meaning is the interpretation of the verse; the others are not interpretations. There is only one correct interpretation of a verse—Maimonides in the second root. Why? Because what is the assumption really? The assumption basically says that you can explain it, say, on the plane of secret meaning, and you can also explain it on the plane of plain meaning. And there is some kind of correlation such that everything that fits plain meaning also fits secret meaning. Somehow it all works out, so that you can basically explain everything on the plane of plain meaning, and you can explain it also on the plane of secret meaning, and both explanations hold up. But that means there must be some kind of correspondence between the principles on the plane of secret meaning and the principles on the plane of plain meaning. Or in other words, that it is all just a translation into another language of the same thing. It is not two explanations. It is the same thing explained in two different languages. If there is a complete correspondence between them, then it is like explaining it in Hebrew and explaining it in English. Those are not two explanations. They are the same thing explained in two different languages. But if they are really different explanations, then the correspondence between them is probably not complete. It’s like—think for example of a coordinate system. We can look at any point in a Cartesian system—its x and y—and I can look at it in a polar system—its r and theta, its distance from the origin and its angle relative to the x-axis. Okay? And there is a correspondence between those two forms of description. So they are just languages, ostensibly, right? But not entirely accurately. That correspondence breaks down at the origin. At the origin, r equals zero and theta can be anything. Okay? The line r equals zero and every theta. Okay? So that correspondence is not a perfect correspondence. And that creates topological defects and all kinds of things like that. Never mind, but I’m saying that when you create a correspondence that is not synchronized principle-for-principle, then you will have breaks. Or let’s return to Newton. Newton sat under the tree, an apple fell on him. There is a theological explanation: he sinned yesterday, therefore the Holy One, blessed be He, dropped it on his head; and there is a physical explanation: gravity overcame the force of the branch and therefore the apple fell. Now assuming there is some correspondence between these two things, so what—is this just two languages describing the same thing? I said before, that is not plausible. Because in the theological language there is choice and decisions of the Holy One, blessed be He, in response to human choice, whereas in the physical language everything is deterministic—whatever the circumstances dictate is the result. It does not depend on… It is very hard to believe that one can produce a correspondence between that language and this language. It is not two languages; they really are two different explanations. Therefore, for example, I think one indication of this is: why does the Holy One, blessed be He, need miracles? To perform miracles? What is the definition of a miracle? The definition of a miracle is basically a topological defect. What is a miracle? You have an explanation of something that happens in terms of the laws of nature, in terms of physics, okay? And you have theological explanations, God’s considerations as to why He wants something to happen this way or not happen this way. Okay? Usually perhaps this can fit together in one way or another, but there are places where God’s will is x, but the laws of nature, if left to themselves, would do y. In such a situation, the Holy One, blessed be He, has no choice. He must intervene in the laws of nature, freeze them, and dictate to the system—for example, to the Red Sea—to split. Even though the laws of nature do not allow such a thing, the Holy One, blessed be He, in His theological considerations decided that the sea should split, and therefore He has to break this correspondence that exists between the natural explanation and the theological explanation. Those points that we call miracles are the same points where there is a break or a mismatch between the theological explanation and the physical explanation. The physical explanation would lead to one result, but the theological plane wants another result. And because there is no correspondence, there is no choice but to break it, and therefore the laws of nature are broken. Why does the Holy One… what, is the Holy One, blessed be He, not omnipotent? What is the problem with creating the laws of nature in such a way that they always carry out what He wants carried out? The fact that He needs laws is ostensibly a defect in His omnipotence, right? Why would He need miracles? Miracles are for the weak. Someone who is truly omnipotent would build the laws in such a way that what happens will always be what ought to happen, even from the theological point of view. The answer is that this cannot be done. Even the Holy One, blessed be He, is subject to the laws of logic. It cannot be done. Why not? Because of the lack of correspondence I mentioned before—because one side depends on choice and response to choice, and the other is deterministic. You cannot create a rigid system whose results will always fit what the theological considerations say should happen. Since that is impossible, the Holy One, blessed be He, creates the system in a way that perhaps does this as well as possible, so that as few miracles as possible will be needed—but here and there He will still need miracles. There are topological defects at the origin where the correspondence does not exist, just as between Cartesian and polar systems. Okay? There are points where the correspondence between the physical description and the theological description breaks down; it does not exist because it is not a complete correspondence. By the way, what is a miracle called in the interpretive context of plain meaning, hint, homily, and secret meaning? It is called “a scriptural decree.” The topological defect in those problems is “a scriptural decree.” What is “a scriptural decree”? It is a principle that in the system of plain-meaning explanation one cannot understand, but in the system of secret meaning one can. Right? There really is some… So it is not that the law is simply like that for no reason. The law is like that because there is some rationale why it is so, say in the domain of secret meaning. Therefore people who use plain-meaning interpretive tools cannot understand it, so they call it “a scriptural decree.” “A scriptural decree” is the parallel of a miracle. A miracle is a break in the correspondence between the laws of nature and theological considerations; when that breaks, it is called a miracle. And “a scriptural decree” is the break in the correspondence between plain meaning and secret meaning. In those places where the correspondence breaks down, we call it “a scriptural decree.” Okay? In any case, for our purposes, this means that parallel explanatory planes are highly problematic. We are terribly used to this. In a certain sense we are very used to it, we relate to it with Olympian calm. That you can explain things on plane A, you can explain them on plane B, and the fact that he became religious because of psychology is an excellent explanation, and the fact that he became religious because he reached the conclusion that it is more correct is also an excellent explanation, and never the twain shall meet. This one is in psychology and this one is in philosophy. Not true. If it were really an explanation, then it would mean that the philosophical consideration would have led him there even had there been no psychological crisis. And if the psychological explanation is correct, then it means that if he broke up with his girlfriend he would have become religious even if it did not seem logical to him. Because a correct explanation means a sufficient condition. This only means that the psychological explanation and the philosophical explanation are not really explanations. Neither of them is a sufficient condition. The correct explanation is the combination of the two together. For a person to become religious, there has to be a psychological crisis that causes him to reconsider his path, and then, of course, he also has to reach the conclusion that this seems truer to him, and that is why he takes this particular step and does not go become a Shinto monk. Okay? Meaning, you need the combination of both things, the psychological and the philosophical together, and together they provide the sufficient condition. Meaning, the explanation is both together, not each one separately. Therefore the picture—because the intuitive feeling that there is a contradiction between the psychological perspective and the philosophical perspective—is correct. It is not a mistake, as I presented it earlier, as though there are parallel planes of explanation, so what’s the problem? This is psychology and that’s an explanation; this is philosophy and that’s an explanation. Not true. No, that is a mistake. The explanation is apparently the combination of the two together. There has to be a psychological crisis and philosophical deliberation; the two together can produce a change—becoming religious, leaving religion, whatever, a change in a person’s worldview. If one of them is missing, it probably won’t happen. Even if you think it is truer, why would you suddenly make a change and start reconsidering your path? You need some crisis to make you reconsider your path, for example. Okay, so the assumption that these two things are explanations means that both together are the explanation—not that each one separately is an explanation. That is only if you want to reconcile the two. Right. Or to say that one of them is correct and the other isn’t—obviously. I’m saying that if you want to live with both, then usually it does not work. Usually you have to say that this means that neither one by itself is an explanation; rather only the combination of the two together is the explanation. Neither one is a sufficient condition as long as I think the other is false. I’m saying: if you think only one of them is true, then there is no problem, because there is no problem to solve. I’m saying that in a place where I am convinced that both explanations are correct, then one has to notice carefully: if both explanations are correct, that means that neither one is really the explanation. Only the combination of the two is the explanation. Okay? So that is a first example of dissolving dichotomies. Someone offers you one explanation, someone else offers another explanation. The answer may be that these are parallel explanatory planes and really both are correct—like “he found a fly and did not object; he found a hair and did object,” what we saw in Gittin 6b. What does that mean? You need both the psychological aspect and the philosophical aspect. Neither by itself would have produced the result, but both together produce the result. It is exactly like the fly and the hair. Okay? So that is one mechanism for dissolving a dichotomy. Dichotomy means what? The fly and the hair. A hair. A strand. What the Talmud says—we talked about it, the Talmud in Gittin concerning “these and those are the words of the living God.” Never mind. In any case, that is one mechanism for dissolving contradictions. Another mechanism is the heap paradox. I mentioned it this morning, I think. The heap paradox basically says: one pebble is not a heap; adding one pebble to a collection that is not a heap will not change its status; but ten thousand pebbles are a heap. That’s it. Now those three assumptions together cannot all be true. They are inconsistent. Okay? Because if one pebble is not a heap and adding one pebble changes nothing, then two are not a heap either, then three are not a heap either, then four aren’t, then five aren’t, all the way up to ten thousand. That does not fit. Those three assumptions do not fit together. So what is the answer? The answer is that adding one pebble does change the situation—a little. What does that mean? It makes you a little more of a heap than you were before. Or in other words, the term “heap” is not binary. It is not either yes-heap or no-heap. There are different degrees of heap-ness between zero and one, and as you add more stones it becomes more and more of a heap. There is something that is a bit of a heap, something that is fairly heap-like, very heap-like, really a heap, completely a heap, and heap. Okay, I don’t know, I don’t have enough words—but between zero and one. Choose any number you like; that is the degree of heap-ness of the thing. This is logic—fuzzy logic, yes. It basically means that very often when we have a contradiction, the way to resolve it is to understand that we live in a world that is not dichotomous. And therefore, we do not have to choose either this option or that option; there are intermediate options. For example, someone wants to prove that there is no point in having exams. Why? Because diligent students study even without the exam; lazy students won’t study even with the exam. So either way there is no point in exams. Where is the mistake? The mistake is that the world is not divided into pathologically lazy people and pathologically diligent people. There is a whole continuum of levels of diligence or laziness. Now it is true that for the two extremes the exam is irrelevant. The pathologically diligent will study in any event, and the pathologically lazy will not study even with an exam, also in any event. So the exam won’t make any difference one way or the other. But someone who has a certain level of diligence—without an exam he would not study, but with an exam he would. So there are levels of diligence that are not zero and not one, but something in the middle. And with regard to them, there is indeed a point to having an exam. And so on—you can show in many dichotomies, many problems that arise because we treat the two sides as an exclusive dichotomy, mutually exclusive. It’s either this or this, but not both. But sometimes there can be intermediate states. That is the second mechanism—I’m doing it quickly because I have to finish. The third mechanism is to show that these two sides do not really contradict one another, because there is a third option. Usually it is built like this: either this or that; either way a certain result follows; and therefore the proof of that result is correct. No—there is also another possibility. What? Or both this and that. Or both this and that, or a third option. There is also a third option. I’ll bring a few examples so you can see what I mean. Let me give one example. Look—maybe before that, one more preliminary sentence. Whenever we speak about a contradiction between two things, it means they have something in common. There is no contradiction between, I don’t know, socialism and kindness. Or I don’t know—just a random example—socialism and cold. Cold, low temperature. There is no connection, right? These are two things that do not speak to one another. When we speak about opposites or a contradiction between two things, it means they belong to some shared area. They have something in common, and within that shared thing, one is this and the other is its opposite. Okay? Say salty is the opposite of sweet because both are tastes. But salty is not the opposite of triangle, because salty is a taste and triangle is a geometric shape. Okay? Meaning, you need to belong to the same category in order to be opposites. Fine? Now, between any two opposites or between any two people who disagree, they always agree on something. If they do not agree on something, there is no dispute. That is always the case. And therefore—notice why I’m saying this—because very often they present us with two possibilities and say: look, either you are here or you are there, there is no third possibility. Almost always there is a third possibility that we are not thinking about. I’ll give examples in a moment. And where is it located? Think about what is shared by these two dichotomous sides. You are either here or here. There is something they share. And then think: maybe one can disagree with that shared thing. And then I basically say: I am neither here nor there because I do not agree with the thing the two of you share. Let me give you an example. It is common to think that the disputes between creationists—believers in creation—and neo-Darwinians—believers in evolution. Okay? Usually it is thought that either you adopt the evolutionary explanation or God created the world. Now both sides in the argument agree on this. No—both sides in the argument. Both sides in the argument agree that you have to choose either this or that, that there is a contradiction between them. The neo-Darwinians agree that if there is God, then there is no evolution; if there is evolution, there is no God. The believers also agree to this. The difference between them is the question of which of the two contradictory sides they choose. The neo-Darwinians choose evolution; the creationists choose faith, creation. Okay? But both agree that there is a contradiction. But now here we have a third option. So there is no contradiction. Meaning, He built the laws within which evolution proceeds, I don’t know, something like that. I’m doing it very briefly just to illustrate the point. Another example: yes, we are used to the division between Religious Zionism and Haredism. Now ostensibly there is no third option. You are a religious person. If you are in favor of Zionism, you are a Religious Zionist. If you are against Zionism, you are Haredi. What else could there be? Either you are for Zionism or you are against Zionism. Hardal? Doesn’t matter, for this purpose it is the same thing. What is a Zionist Haredi? Hardal—well, same thing. So make up your mind. You are in favor of Zionism, then you are a Religious Zionist. Just word games. Ostensibly, do you understand? There is a dichotomous argument here. After all, the law of the excluded middle says either you are Zionist or you are not Zionist, right? There is no third possibility. And that’s it. Either Haredi or Religious Zionist—what else could there be? I’m asking what else could there be—not secular; secular is obvious. I mean within the religious world. What else could there be? The answer is that yes, there could be something else. For example, a person could be Zionist but his Zionism would not be religious. Like the rabbi—like the rabbi, the rabbi of Ponevezh would hang a flag on Independence Day in Ponevezh—he would hang a flag in Ponevezh on Independence Day, and he would not say Tachanun. But he also did not say Hallel. Okay? The students asked him: either way, if you are Zionist then say Hallel, not just omit Tachanun, and if you are Haredi then say Tachanun too. Meaning, either way? So he told them: I am Zionist like Ben-Gurion. He too does not say Tachanun and does not say Hallel. Now that’s a nice joke, but the truth is that it’s a serious answer. What he meant to tell them is that I am a secular Zionist. I am a religious person, but my Zionism is secular. What does that mean? That I can be a religious person and also a Zionist, but there is no hyphen between the religious and the Zionist. Meaning, I am both Zionist and religious, but my Zionism is not religious. So here you have a third option. What did I attack—what is the shared assumption of Haredim and Religious Zionists? I’ll tell you what is shared: that this whole business has to have religious significance. If positive, then you are Religious Zionist; if negative, then you are Haredi. He says no—this matter has no religious significance for me whatsoever. I support it the way a Norwegian supports Norway. That’s all; it is not connected to my religiosity. It has no religious significance at all. Do you understand that here I came out against the shared assumption of the two sides in the dichotomy? The principle that both of them agree to. And that is how a third option emerged for me. What is the difference between the Haredi you are talking about and the Zionist? In what sense is he Haredi? Because he wears a long coat? Don’t bother me with sociology. I’m speaking now about the ideological dimension. So he is Zionist, he is not Haredi. I’m trying to show you that whenever you attack the shared assumption, you suddenly discover that there is a third possibility, even though they present it to you as yes or no, what else could there be? Final example, and with this I’ll stop. There is a very well-known argument in favor of determinism or against free choice. The argument goes like this: suppose a person performs a certain act. Either he has a cause or he has no cause, right? There is no third possibility. If he has a cause—and we said a cause is a sufficient condition—then that means it is deterministic. Whatever comes out is the result of the cause. If there is no cause, then it is simply random. It is just something that happened on its own. Either way, this is not free choice. Free choice is not randomness and not determinism. Therefore there is no free choice. Which is what we had to prove. Now here too, when you look at it at first glance, the logic is pure logic; one cannot argue with it. Either yes or no—what else could there be? Either there is a cause or there is no cause. Is there another option? The answer is yes, there is another option. You identify “there is no cause” with randomness, but that is not correct. It may be that there is no cause, but the thing is not random; rather, it is the result of a choice. Both are things that happen without a cause. The difference is that what happens out of deliberation or choice is directed toward a purpose. And what happens just like that happens just like that—not from a cause and not toward a purpose and nothing. But in both cases there is no cause. Or in other words, you are looking at things only on the axis of cause. But there is another axis, the axis of purpose. So something that has no cause and no purpose is random. Something that has no cause but has a purpose is free choice. Something that has a cause, whether or not it also has a purpose, is determinism. So here we are. Here you have a third option. Even though from the outset they told you either there is a cause or there is no cause, ostensibly there is no way out. It is a compelling argument, logic. Yes? Remember the formalization—the problem of formalization. The logic is always compelling; there is never any mistake. It’s just that our whole problem is not really logic. Always check the formalizations. All right, I’ll stop here. Good luck on the exams, for those who have them. Goodbye.