Faith, Doubt, and Certainty – Lesson 1
This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.
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Table of Contents
- The purpose of the lecture and the framework of faith, doubt, and truth
- Proof, “begging the question,” and the claim that every proof presupposes its conclusion
- Validity versus truth, and skepticism arising from criticism of the certainty of the premises
- The precision of mathematics as “uselessness”: the story of the hot-air balloon
- Axioms, the sum of the angles in a triangle, and the metaphor of the safe and the key
- Wittgenstein, psychometric sequences, and Chomsky: learning as dependent on prior patterns
- Mathematics is not science: the impossibility of refutation and the distinction between mathematics and physics
- Where information comes from: induction and analogy versus deduction, and the root of skepticism
- The history of Western thought and the analogy to growing up: from dogmatism to rationalism to postmodernism
- Mathematics as “adolescent” thinking, pluralism, and the contemporary crossroads
Summary
General Overview
The lecturer presents the question of certainty and truth as a fundamental problem of proof and logic, and argues that the answer to skepticism is, in a certain sense, “yes,” but the main point is to understand what such a yes actually means. He maintains that logical proof does not generate certainty or new information, but at most preserves certainty that already exists in the premises, and therefore a conclusion cannot be more certain than the premises on which it rests. He proposes viewing logic and mathematics as precise but “useless” tools in the sense that they do not add information, and contrasts them with the accumulation of human and scientific knowledge through non-certain tools like induction and analogy. He describes a historical and psychological process in which naive rationalism, which demands proofs for everything, collapses into skepticism and postmodernism, and he presents a crossroads at which one can either remain skeptical or give up the demand that only what is proven is acceptable.
The purpose of the lecture and the framework of faith, doubt, and truth
The lecturer says that the topic may seem non-Torah in nature, and therefore out of place in a yeshiva, but that it has aspects that can also be seen as Torah-related. He declares that he will focus on the conceptual-philosophical parts rather than the halakhic aspects and the learning-oriented, Talmudic dimensions. He frames the basic question as how one can expect certainty at all, how a person knows that he grasps the truth, and whether we are condemned to perpetual skepticism, emphasizing that the answer “yes” requires clarification of what it means.
Proof, “begging the question,” and the claim that every proof presupposes its conclusion
The lecturer gives the example of a “proof” that every Jew must wear a hat in order to illustrate the sense that something is fishy in an argument that assumes its own conclusion. He defines “begging the question” as assuming the conclusion implicitly or explicitly, and argues that such a proof has no value because it adds no information beyond what was already assumed. He then goes on to make the sharper claim that this problem is not an exceptional fallacy but a feature of every logical proof, because the conclusion is already contained in the premises, and therefore anyone who truly knows the premises already knows the conclusion.
Validity versus truth, and skepticism arising from criticism of the certainty of the premises
The lecturer analyzes the syllogism “All men are mortal; Socrates is a man; therefore Socrates is mortal,” and distinguishes between a valid argument and a true or certain conclusion. He attributes to John Stuart Mill the claim that the certainty of a conclusion rests on the certainty of the premises, and that if the premises are not certain, the conclusion cannot be more certain than they are. He says that proof does not create certainty but preserves certainty, and therefore the chain of justifications always stops at a point where there is no proof, from which follows a skepticism according to which there is no way to attain absolute certainty.
The precision of mathematics as “uselessness”: the story of the hot-air balloon
The lecturer brings the joke about the hot-air balloon in which the answer “above my field” is perfectly precise yet completely unhelpful, and explains that the two features depend on one another. He argues that absolute precision stems from the fact that the statement adds no information beyond what we already had, and therefore it is “useless” in the intellectual-logical sense. He concludes that precise analytic tools such as logic and mathematics do not add new information, and therefore are not “useful” in the sense of expanding information, but only in formulating necessary relations.
Axioms, the sum of the angles in a triangle, and the metaphor of the safe and the key
The lecturer argues that although mathematical conclusions are hidden in the axioms, a student who knows the axioms still does not know how to formulate, use, and consciously grasp statements such as “the sum of the angles in a triangle is 180 degrees.” He presents a distinction between “information” that exists in consciousness in a latent way and “knowledge” as the ability to bring it to the forefront of awareness and activate it. He compares this to a safe whose contents are already there, while the proof or the teacher is the key that enables one to open the safe and reveal what is already inside, so that mathematics is the understanding of the necessary derivation from the premises rather than the collection of new facts.
Wittgenstein, psychometric sequences, and Chomsky: learning as dependent on prior patterns
The lecturer attributes to Wittgenstein the claim that a sequence such as 1, 2, 4, 8 does not necessarily compel 16, but depends on the rule one assumes, and that one can construct an alternative coherent function that yields a different continuation such as “minus seven and a third.” From this he concludes that the psychometric exam tests conformity to accepted thought-patterns or “wiring,” rather than a single logical truth forced by the data. He adds that Chomsky argues in linguistics that a person is born with linguistic patterns, and extends this to mathematics and learning in general by saying that a teacher cannot insert new patterns but only awaken and uncover patterns that already exist, and that someone who lacks the internal structure will not be able to learn the field.
Mathematics is not science: the impossibility of refutation and the distinction between mathematics and physics
The lecturer argues that mathematics is not science because science relies on observation, generalization, and experiment that may refute, whereas mathematics operates deductively and therefore cannot turn out to be false if the proof is valid. He uses the example “two plus three equals five” to show that an experiment with apples would not lead us to abandon the mathematical rule, but at most to question the physical claim that adding apples in a basket is described by algebraic addition. He illustrates this also through physical forces, where the appropriate addition is vector addition, and emphasizes that the failure lies in applying mathematics to describe the world, not in mathematics itself, because the mathematical theorem deals with ideas rather than the world.
Where information comes from: induction and analogy versus deduction, and the root of skepticism
The lecturer asks how we know anything at all if proofs do not add information, and answers that deduction never accumulates information because the particular is already contained in the general. He divides tools of inference into deduction, induction, and analogy, and states that science and new knowledge rely on induction and analogy or direct observation, whereas logic and mathematics deal only with deduction. He presents skepticism as the undermining of induction and analogy while accepting only deduction, and formulates skepticism as an “overly rigid intellect” that is willing to accept only what has a proof.
The history of Western thought and the analogy to growing up: from dogmatism to rationalism to postmodernism
The lecturer describes three stages in Western thought: a dogmatic stage of myths and acceptance on the authority of tradition, a stage of rationalist critique that demands proofs, and a postmodern stage in the middle of the 20th century in which those same intellectual circles lose confidence in rationality and slide into nihilism, pluralism, and the breakdown of ideologies. He compares this to a person who moves from dogmatic childhood to rebellious adolescence that demands proof for everything, and then discovers that the ideal of accepting only what is proven is impossible because there are always basic assumptions that are themselves unproven. He presents two paths: to become a skeptic who accepts nothing, or to give up the assumption that only the proven is acceptable and recognize that one must also include induction and analogy among the tools of thought.
Mathematics as “adolescent” thinking, pluralism, and the contemporary crossroads
The lecturer argues that only in mathematics are there “little geniuses” who achieve greatness at a young age, and interprets this as mathematics being a mode of thought characteristic of the adolescent stage in which one demands proof for everything. He says that the mathematician deals only with derivation and therefore not with information itself but with the structure “if the premises are thus, then the conclusion is thus,” and from this follows a pluralism in which different positions are equivalent as long as the logical derivation is sound and the premises differ. He concludes by saying that modern society still stands around the crossroads between a skeptical-pluralistic path of equalizing positions and an alternative that gives up the exclusive demand for proof, and he stops at this point and says they will continue at twelve.
Full Transcript
[Rabbi Michael Abraham] Okay, hello everyone. Our general topic today is usually not—maybe not perceived as a Torah topic, and so sometimes there’s a feeling that it has no place in a yeshiva. But the truth is that it has aspects that can also quite easily be called Torah-related. Well, it has aspects that can be considered, in a more classical sense, Torah-related; I doubt I’ll get to those. But since our goal is defined as engaging with faith, doubt, and truth, I’m really going to focus on the more conceptual, philosophical parts, and not on the halakhic parts and the learning-oriented, Talmudic implications. The basic question that was raised, that I was asked to speak about, is how one can expect certainty at all. How can a person know that he is holding on to the truth? Are we condemned to perpetual skepticism that we can never escape? The truth is that I could have stopped and said that the answer is yes. And that’s true, and that’s basically the answer in short or in long, but it’s very important to me to clarify what that “yes” means, and that’s really what I’m going to devote myself to more than to the mere statement of yes. Okay, so maybe I’ll start with a bit of a characterization of the field called logic—or at least classical logic—and we’ll see what exactly it tells us. When we raise the question of doubt versus certainty, closely related to it is the question of proof versus lack of proof. Right, certainty is usually linked in our minds to the concept of proof. So let’s try to examine the concept of proof a little, and maybe from that we can derive some insights about what we can expect in the realm of certainty. There’s a logical fallacy illustrated by the famous proof that every Jew needs to walk around with a hat, and the proof goes something like this: it says, “And Abraham went there there,” and a Jew like him certainly didn’t go without a hat—also “there there”—and therefore anyone who walks in the ways of our forefather also needs to walk around with a hat. Which was to be proved. What exactly is wrong with this proof? What’s problematic about it? There is, of course, a clear feeling that there’s some kind of—I don’t even know whether it’s kosher—but it’s definitely fishy. What exactly is the problem we see in this kind of argument? So this is what’s usually called in logical contexts begging the question. Begging the question is basically assuming the conclusion we want to reach. We want to prove some claim, we build an argument that proves that claim, and that argument in fact either implicitly or explicitly assumes the very conclusion we want to reach. In this argument, for example, we say “And Abraham went”—a Jew like him surely didn’t go without a hat. Where does that prophetic insight come from? Apparently from the assumption that every Jew needs to walk around with a hat, otherwise why assume that a Jew like him didn’t go without a hat? So in fact we implicitly assumed the conclusion that every Jew needs to walk around with a hat, without saying so, of course, in order to still achieve some impact. But in fact we assumed it, and when we assume the conclusion we want to reach, that’s what’s called begging the question, and of course the proof—there’s no value to it in such a case, because we haven’t added any information that we didn’t already have when we assumed the premises. As soon as we assumed the premises, that information was already in our hands. And if it wasn’t, then it won’t be in our hands after the proof either. Because if we challenge the premise, then of course we also won’t have to accept the conclusion. That’s the problem with begging the question. But here’s the thing: the problem of begging the question isn’t really a fallacy at all. There is no such fallacy, begging the question. Every logical proof begs the question. Let’s take the classical logical proof.
[Speaker B] One second, I’m disconnecting from the speaker because I have to walk around. Tell me if you can’t hear me. That also—
[Rabbi Michael Abraham] You didn’t hear me like that either. All men are mortal—that’s the first premise. Socrates is a man—that’s the second premise. And the conclusion is: Socrates is mortal. That’s a logical proof, a logical argument, what’s called a classical syllogism. That’s always the example they give when they teach logic. Okay? So let’s examine this argument. We have two premises: all men are mortal, and Socrates is a man, and from them we derived the conclusion that Socrates is mortal. Okay? How do we know that all men are mortal? I don’t know. From observations, from hypotheses, whatever, right? So the first challenge—and this was already raised by a philosopher named John Stuart Mill—a challenge to deduction, or to necessary inference, logical necessary inference, says the following: you want to prove this conclusion, and once you have a proof for it then it’s certain. It’s proven. But any such certainty rests on two premises—in this case two premises, sometimes you need more. Fine? What is your certainty with regard to the two premises? How do you know this or that? About one of them, say I saw it. Let’s say seeing is not something we’re going to cast doubt on right now. But what about this one? That’s a hypothesis, right? I haven’t seen all human beings, and certainly not all the human beings I have seen have died. I hope, at least. There are wicked people who in their lifetime are called dead, but most of the people I’ve met are alive—all of you today, may you live to 120. So how do I know that all men are mortal? I make an induction, right? I met some people who eventually died. I assume that this is true of all people, based on the group I met, and therefore I draw the conclusion that all men are mortal. Right? So in fact I made some kind of generalization here. I took several basic facts and built from them a general principle by generalization. Now I have a general premise, a particular premise, and a conclusion. But what is this general premise itself based on? On generalization. But generalization is not a certain thing, right? So if the premise is not certain, then clearly the conclusion that rests on it is not certain either. Because this conclusion is, after all, merely derived from the two premises. And if the premises are not certain, then the conclusion cannot be more certain than what we have in the premises, right? Once I asked high school students: what is more correct in geometry, the axioms or the theorems? The theorems have proofs; the axioms do not. So the axioms are arbitrary and the theorems are certain, right? Obviously that’s not true. Why not? Because the whole point of proof is to perform a deduction of the conclusion from the premises. In other words, to draw the certainty that is in the premises and transfer it to the conclusion. A proof can never produce certainty; a proof can preserve certainty. In other words, it can take the certainty I have in the premises and transfer it to the conclusion derived from them, right? But if in the premises themselves I do not have full certainty, then the degree of certainty I can have in the conclusion is at most what I have in the premises. Never more. Right? So that means that even the conclusion of such an argument is not a certain conclusion. It does indeed necessarily follow from the premises, but it itself—the conclusion itself—is not certain. This is called a valid argument, because its conclusion necessarily follows from the premises. But the conclusion of a valid argument is not necessarily a true conclusion. In other words, it is never a certain conclusion, and it is not necessarily true. Sometimes it’s true and sometimes it isn’t. Okay? So what this basically means—this was Mill’s point—is that we have no way of arriving at certainty. This is another formulation of skepticism. Since even if we present a proof, the proof can preserve certainty but never produce certainty. If we do not have certainty in the premises, then we cannot be certain of the conclusion. But as for the premises—if we have no proof for them, and at some point this chain always stops—then at the point where the chain stops we have no certainty. And therefore we will never have certainty about the conclusion either. And so Mill argues that deduction is an illusion. There is no such thing as genuine deductive logical certainty. The certainty is only a certainty of derivation: the conclusion follows from the premises, but there is never certainty regarding the conclusion itself. Or, as I said earlier, we are forever condemned to live in doubt. We never really have certainty. But I don’t mean to discuss that here right now—maybe later. Rather, I want you to notice another point, which may be related, but it’s different. The sentence “All men are mortal”—let’s say I know it somehow, I don’t know, from Elijah’s revelation, I know it with certainty. Fine? Then the conclusion too is certain. Fine? In this case Mill will agree with me that the conclusion is certain. Fine? Elijah can be related to certainties, not to doubts. In any case, there’s another point here as well. This sentence, “All men are mortal,” contains within it the particular claim that Socrates is mortal. Right? One of the human beings referred to in the first premise is Socrates. So the moment I know the first premise, that all men are mortal, I in fact already know this claim about all human beings. Right? That each one of them is mortal: Reuven is mortal, Yaakov is mortal, our forefathers did not die, someone else… never mind. All kinds of people are mortal, and Socrates too, may he be set apart for long life, is also mortal. So what do we actually see here? That this claim is already present inside the premise. Or in other words, what is this called, as I said earlier? Begging the question. In other words, this argument in fact assumes what it wants to prove. This argument starts from the premise that all men are mortal and derives the supposedly new conclusion that Socrates is mortal. But the premise we started from already includes that conclusion as well. I don’t need the argument in order to accumulate the information contained in the conclusion. I already had that information when I set out. In other words, if with Abraham our forefather this is the fallacy of begging the question, then this argument too contains the fallacy of begging the question. Or, in even sharper words: every logical argument begs the question. Otherwise it wouldn’t be a logical argument.
[Speaker B] He said, “He made the assumption that Socrates is a man.”
[Rabbi Michael Abraham] Yes, yes, that completes it.
[Speaker B] No, I’m saying—where is the assumption that Socrates is a man?
[Rabbi Michael Abraham] Of course—that’s my second premise.
[Speaker B] But the first premise—
[Rabbi Michael Abraham] When I know that Socrates is a man, then now it is included in the first premise. Once I know both premises—leave it, even if I was never taught logic—if I know both premises, I already know the conclusion.
[Speaker B] The conclusion is there. I just attached the second premise. Okay?
[Rabbi Michael Abraham] So in fact the conclusion that comes out of this—this is also an interesting conclusion—is that every logical argument begs the question. Begging the question is not a fallacy. On the contrary. If an argument begs the question, that means its conclusion is certain. And vice versa: if an argument’s conclusion is certain, that means it begs the question. So there is no such thing as the fallacy of begging the question. And it’s like the story I brought at the beginning about the two wagons—with the hot-air balloon—where two people lose their way in a hot-air balloon, they don’t know where they are, they ask the person standing down below in the field, plowing his field, they ask him, “Tell us, where are we?” He thinks for a moment and says, “Above my field.” So one of the people in the balloon says to the other, “That’s definitely a mathematician down there.” Why a mathematician? Two reasons: first, what he says is absolutely precise, and second, it doesn’t help us at all. And they tell this story in every faculty of the exact sciences. Everybody wants to claim the crown, and in academia there’s this feeling that if you’re no use at all then you’re a real academic. So let me explain to you why—really, not as a joke. I’ll tell you why: because what does it mean, “doesn’t help us at all”? This joke is actually a very serious thing. Why really does the mathematician have these two characteristics? These two features always come together. They are not two independent features. Why really is he absolutely precise? Why really is that statement absolutely precise? Because it doesn’t help us at all. That’s why. Because in fact it adds no information beyond what we already had. That is what it means that it doesn’t help us at all, right? This argument too doesn’t add any information beyond what we already had. This argument too doesn’t help us at all. In other words, saying “it doesn’t help us at all” in this sense—there are also lazy people who are no use at all—but “no use at all” in this intellectual sense, in this logical sense, means absolutely precise. That’s what it means. They’re not two characteristics—it’s absolutely precise and also useless—it’s just saying the same thing in different words. It’s the same thing. Something that is completely useless to us—that is the meaning of being precise. So if that’s the case, it turns out that logic, or precise analytical mathematical tools, are tools that cannot help us at all in this sense.
[Speaker B] Yes. On the other hand, the sentence about Socrates does add information, because in fact I already knew the premise.
[Rabbi Michael Abraham] I knew both premises before I set out on the logical road.
[Speaker B] So the second claim says that Socrates is a man and therefore he is mortal.
[Rabbi Michael Abraham] Sorry—what is “the second argument”? The second claim. The argument is the whole structure. The claim—fine. Now once I have these two premises, I no longer need someone to derive the conclusion from them. I already know it. After all, everything is inside this. Right? Because someone already defined earlier the two claims. And if you know those two premises, you don’t need to study logic—you already know that Socrates is mortal. Right? It is already included in the premises. Not necessarily in one of them—you’re right, you need the second one too. So what do we do with this? Does mathematics really not help at all? Did anybody ever feel that way—I don’t want to insult anyone or fail to understand—but did anyone feel that studying geometry, for example, was of no use at all? In life? What am I talking about? “Of no use at all” in the academic sense, as I said earlier, is actually a compliment. The sense in which people subjectively feel that geometry is of no use at all—intuitively. So what is really happening there? What is the “usefulness,” in quotation marks, that we are talking about when we speak of mathematical tools or logical tools? Look, take the four axioms of plane geometry, Euclidean geometry. Every child in fourth grade knows them. You don’t need to study them; they’re obvious. Like, through any two points there passes one straight line, that two parallel lines do not meet, and so on and so forth. Every child in fourth grade knows these axioms. And still, if you ask him what the sum of the angles in a triangle is, he won’t know—unless he’s a very, very intelligent child. Why not? After all, the information about the sum of the angles in a triangle is in fact already included in the premises. Otherwise it wouldn’t be mathematics, right? If the proposition that the sum of the angles in a triangle is 180 degrees is a certain proposition, one that has a mathematical proof, then this basically means that the claim that in every triangle the sum of the angles is 180 degrees is already hidden somewhere in the premises, in the axioms, as they’re called in the context of geometry, right? So why does someone who knows the axioms not know the conclusion? He supposedly does know it. It’s inside there. He knows it in the same sense that someone who doesn’t know Socrates—if all men are mortal, then he knows that if there’s some Socrates and he is a man, then certainly he is mortal. Okay? So here too, in some sense that child really does know that the sum of the angles in a triangle is 180 degrees—in Euclidean space. So what doesn’t he know? Why does he need the teacher? Maybe if he’s a genius, he doesn’t. But usually he does. Why? Because not everything we know in the sense of—let’s call it information theory—the information that is stored within us, does not mean that we really know it. In other words, that information is stored within him, true. But if he understands those fundamental premises, then within those premises there is also, in some form, the information about the sum of the angles in a triangle. But that doesn’t mean he knows it in the sense of being able to formulate it, use it, understand it, have it before his eyes—that proposition that the sum of the angles in a triangle is 180 degrees. In other words, it has not emerged to the forefront of consciousness, or of thought, or awareness, and that is not a trivial operation. Even though the information is indeed, in some sense, already latent within him. There is some kind of locked safe here, and the teacher’s role is basically to help the student open it, to discover what is inside the safe. If it were not in the safe, that student would never understand what the teacher is teaching him. It seems to me that even if he did understand it, it would not be mathematics. Because mathematics means not only understanding that the sum of the angles in a triangle is 180 degrees, but also understanding that this necessarily follows from the axioms of geometry. Otherwise he didn’t study geometry, he studied architecture or—I don’t know—engineering, engineering. The sum of the angles in a triangle is 180 degrees. The geometer isn’t interested in what the sum of the angles in a triangle is; what interests him is what follows from these premises and what doesn’t follow from them. Right? And in this sense I mean that the student learned nothing from the teacher, cannot learn anything from the teacher, if it isn’t already there within him in some sense, in the safe. There’s a philosopher—he’s sometimes called half-Jewish, although it’s the wrong half—Wittgenstein, one of the major philosophers of the twentieth century. He argued the following. In psychometric math exams, right? Let’s take sequences. In psychometric exams they ask about sequences. One, two, four, eight—what’s the next number? Sixteen. Great, you got 800 on the psychometric. Now if I say, instead of sixteen, some alien comes and says, “What are you talking about? Here it’s minus seven and a third. Obviously. Don’t you see? It’s minus seven and a third. You can see it immediately.” What would you say? Zero on the psychometric, right? Send him to a hospital. But that’s not true. His answer is no less well founded than what you say. Why do you assume it’s sixteen? You assume it’s sixteen because you assume that the sequence goes like two to the power of n. n equals zero is one, n equals one is two, n equals two is four. So you have some mathematical expression such that if you increase n by one each time, it generates this sequence, right? But I’ll tell you that this sequence also has a mathematical expression that generates it. Not hard at all to produce one in principle, right? an plus bn squared plus cn cubed plus dn to the fourth plus e. Take that expression—doesn’t matter now, let’s not get into the math—just to show you, because this is really a very important point. There are five coefficients here, agreed? Fine. Now let’s generate from it any sequence you like. Okay? Let’s plug in n equals one and require that the result be one. Fine? So a plus b plus c plus d plus e equals one. Now let’s plug in n equals two. a plus 2b plus 4c plus 8d plus 16e will equal two. Plug in n equals three and require that it equal four. Five equations. We have five unknowns, a through e. Five equations with five unknowns generally have a solution. Unless the equations are dependent in some way—so add another term, then there’ll also be fn to the fifth. Fine? That’s all. And now look: this alien will come and tell you, “Here it’s obvious, it’s minus seven and a third, you can see it immediately.” And you’ll ask him, “Wait, what do you mean?” He’ll say, “What’s the problem? One point five plus minus one-third times n plus three i, a complex number, times n squared—and right away you see that when you plug in n equals zero you get one, when you plug in n equals one you get two, and at n equals five it gives minus seven and a third.” Any sequence you want, you can now generate in this way. So now Wittgenstein asks: why—who said he’s wrong? Is the one who decided it’s sixteen really the one who deserves the 800 on the psychometric? In fact the other guy is no less correct. And that’s true. So what does the psychometric test actually test? Does it test intelligence, or does it test our wiring in the brain? One person has this kind of wiring in his brain, built more or less like all of us, and gets sixteen. And another has different wiring in his brain and reaches an answer that is no less correct—he arrives at minus seven and a third. What’s the problem? Especially if, when you ask him, he gives the reasoning, then he deserves an 800 on the psychometric exactly as much as the one who says sixteen. If you ask him and he doesn’t give the reasoning, then I don’t know, maybe there’s room to hesitate. As it is, not everyone among us would immediately say what two to the power of n is; our brains are much simpler. But if he says it’s sixteen, he’ll get an 800, right? So the other one says minus seven and a third, and he can’t quite say exactly what complicated function lies behind it—but what difference does that make? That’s what his brain does, and that’s perfectly fine. So Wittgenstein says that all learning, at least mathematical learning—but not only mathematical learning, even mathematical learning—all our learning is based on certain patterns that are already within us. The teacher does not insert new patterns into us. It’s impossible. Chomsky says this in linguistics—that a person is born with certain linguistic patterns, because without that you can’t teach him to speak. A person is not born a tabula rasa. In mathematics too he is not born a tabula rasa. Because if he were born tabula rasa, he would remain tabula rasa. You couldn’t teach him. Because what would you do? You start teaching him—let’s now teach you the laws of addition. Fine? One plus two equals three. One plus three equals four. One plus four equals five. So now he knows—great, so what is one plus five? Minus seven and a third. That too is no problem; I can define the operation of addition in such a way that it gives minus seven and a third. There’s no problem defining the operation that way. Fine? So what will you do? You’ll start explaining to him the operation of addition. Addition means kind of taking this and then taking that—you’ll bring him one plus two, one plus three, okay, but eventually the examples will run out. Somewhere around 800 or 900, when you break down, the examples will end, and suddenly at 950 he’ll say minus seven and a third. You have no way to teach him. There is no way to teach him—unless he comes with another structure, one no less coherent than ours. We probably all have fairly similar structures in most cases, and that’s what the psychometric tests. But if someone comes with some other structure, he is no less intelligent, and there is no way to teach him unless you discover his structure and then do some kind of change of basis or something like that and try to teach him according to his own method. But otherwise, there is no way to teach him. This is a powerful illustration of what I said earlier. In logic too it works exactly like that, and in mathematics in general it works exactly like that. Anyone who doesn’t have the structure within him will never be able to learn mathematics. Mathematics—everything it does is help you open the safe that is already inside you, except that you don’t have the key. So the teacher helps you find the key—your key, not everyone necessarily has the same key—but he gives you the key in order to discover what is already inside you. But if it isn’t there, you will never learn mathematics. What does that actually mean? It basically means that mathematics or logic never renew information for us that was not already inside us. Right? Or to put it now in more everyday language: a proof can never lead us to a statement, to a conclusion, that we did not already know beforehand. That’s basically what follows from this. Okay? In other words, every conclusion that a proof leads us to is a conclusion that, in some sense, must already have been within us. If not, we won’t succeed in reaching it even with the help of a proof. The proof is just—if we follow it—the key. It helps us open the safe, analyze our premises, and discover how this conclusion is contained within them. That’s the key to the safe. That’s all. So indeed mathematics and logic and all things that are maximally precise, all things that are analytic in the philosophical sense that interests us here, are useless in this sense: they don’t add information to us. It is impossible to accumulate new information by means of logic or mathematics. There is no such thing. The accumulation of information, by definition, is never done by mathematics or logic. That’s why, for example, mathematics is not a science, contrary to what is commonly thought. Even though there were universities, and maybe still are, where it’s located in the faculty of natural sciences. But that’s for technical reasons or because of misunderstanding. Mathematics is not science. Science is something learned through observation and generalization, and therefore it can also turn out to be wrong; it can be refuted by experiment. But mathematics does not need observation, is not done by generalization, only by deductive means, and therefore it can never turn out to be incorrect. If the proof is correct, then it is correct. Maybe we missed something—but not that some new datum illuminates something we didn’t know before. Information will never be added to us and never be taken away from us. This connection always exists. You can assume that—if the assumption is unacceptable to you, then of course you won’t have to accept the conclusion. And mathematics can help you check the assumptions—that’s true, that’s its use in science—but that is not mathematics. Science uses mathematical tools, and precisely in this way: you formulate your hypothesis in such a way that we know from which assumptions it is derived. And now, if we test it in an experiment and discover that it is not correct, we’ll have to go back and check our assumptions. But mathematics is exactly the complementary field. It deals with everything science does not. Mathematics deals only with the connection between assumptions and conclusions, while science deals with the assumptions and the conclusions. The connection—it draws on mathematical tools, but that is not its domain. The connection between assumptions and conclusions is the domain of mathematics and logic, not of science. For our purposes here, logic and mathematics are the same thing. Okay, so now—yes?
[Speaker B] Maybe it’s a matter of language? Language and mathematics—these symbols, are they a matter of language?
[Rabbi Michael Abraham] It’s not new information. No, that information was already within you. That’s why I said—the question is what you mean when you say “I knew it.” That information was already within you the moment you understood the premises, the axioms. You couldn’t open the safe, you hadn’t found the proof, you hadn’t found the decoding, you didn’t know how to analyze the premises you were aware of in order to extract from them the information hidden within them. The teacher helped you with that. That’s why I compared it to some kind of safe that is already inside you, and he helps you open it. Okay, so now of course the million-dollar question arises: so how do we know anything at all? If proofs cannot add any information we didn’t already know beforehand, right? No logical or mathematical tool can help us know something we didn’t already know before using that tool. So how do we know anything at all?
[Speaker B] We learned from the gut. We learned from the gut.
[Rabbi Michael Abraham] Okay, a transplant.
[Speaker B] Ah, I have a question about language. Okay.
[Rabbi Michael Abraham] Once—yes, I understand.
[Speaker B] The moment the Rabbi wrote one plus two, one plus three, it seemed to me that someone had to explain to us what this symbol does. Once you explain it to the other person and he understands it, he also learns something.
[Rabbi Michael Abraham] He’ll tell you that when you explain the symbol to him—I said this afterward—when you explain to him what the symbol is, there too he uses examples. But he’ll understand the symbol itself according to his own method too. How will you explain to him what this symbol means? I’m the student now—explain to me what this symbol means.
[Speaker B] When I add the number that I have—
[Rabbi Michael Abraham] What does it mean “to add”? There—you’re now teaching me that word.
[Speaker B] Come on, let me explain. Maybe I can give an example from reality.
[Rabbi Michael Abraham] No, we’re talking about mathematics, not reality.
[Speaker B] Yes, but I’m using an example from reality. Just like the Rabbi uses the safe, I’m using an example of furniture and books.
[Rabbi Michael Abraham] That won’t work, for a reason I’m about to say. It can’t work. Maybe it will work, but it can’t work. I’ll explain in a moment. Maybe to sharpen it—I’ll say it now already. Earlier I said that mathematics is not science. You can’t refute it. Let’s take, for example, the following sophisticated mathematical law.
[Speaker B] The following sophisticated one.
[Rabbi Michael Abraham] Two plus three equals five. Fine? I claim that it is impossible to refute it. Can anyone refute what I’m saying? Can anyone suggest a way to refute it?
[Speaker B] Two plus three equals six? Eight?
[Rabbi Michael Abraham] No, I just wrote this now. Can we put it to a falsification test? I want to ask whether this claim is a scientific claim or a mathematical claim. Science is about whether you can propose an experiment that would refute it, right? So let’s try to do an experiment. What do you suggest? Someone here mentioned some experiment earlier. What experiment? Exactly: take two apples, take a big basket, put in two apples, here are another three apples, put them in too, and now count how many apples you have altogether. If you discover that it’s six, then the scientific claim has been refuted. And if you discover that it’s five, then it has been confirmed. Right? So there you go, it’s really a scientific claim, no? So to show why this is not a scientific claim: the fact that you laughed already means you understand that it isn’t a scientific claim. If you discovered that it was six, what would you say? Maybe you counted wrong, maybe someone threw in some orange you didn’t notice that suddenly became visible. I don’t know, maybe there was some hole in the basket or something. You would never give up the rule that two plus three equals five, right? It looks like induction. It looks like induction. No, no, induction can turn out to be wrong. You can see two horses and both were black, and infer inductively that all horses are black. Then tomorrow morning you’ll see a white horse. So what will you say? That this horse isn’t really white? That it’s an optical mistake? No. You’ll say the induction wasn’t successful.
[Speaker B] And if for five hundred years we only saw black horses? Then too. If for five hundred years we saw black horses, right? And then suddenly a white horse appeared, what happened?
[Rabbi Michael Abraham] Then it fell apart, fine, the theory was refuted. The induction was refuted.
[Speaker B] But not here.
[Rabbi Michael Abraham] Here too, five hundred thousand years, it doesn’t matter, whether a quarter of an hour or five hundred thousand years, it makes no difference — you will never give up on two plus three equals five.
[Speaker B] If they told you that tomorrow the sun won’t rise. Tell me that tomorrow the sun won’t rise.
[Rabbi Michael Abraham] That’s not stronger.
[Speaker B] In my opinion it is stronger.
[Rabbi Michael Abraham] In my opinion it is stronger. That the sun won’t rise tomorrow — that absolutely could happen. Suddenly the earth stops, a meteor hits it, I don’t know what; it could happen. Yes, could it happen? Yes, of course it could happen. With no error at all. We’d just see what happened to that star of minus seven and a third. But we usually think differently. We usually think differently, and I’ll explain what I mean. Let’s see — let me show you a refutation, I’ll show you a refutation of the rule that two plus three equals five. You take a body, apply a force on it northward of two newtons, and a force eastward of three newtons. What is the total force? Not five. All right? That’s the parallelogram rule, if you remember. Not five. The square root of three squared plus five squared. All right? Why? So there, we’ve refuted the rule that two plus three equals five.
[Speaker B] Because there’s a variable here — what we said is that this x is equal, we’re relating to it, we’re relating to it as two units plus three units. These are units, and we’re adding them. This isn’t units, right, these are units of force. It’s not — it’s not abstract. It’s like two temperature, two—
[Rabbi Michael Abraham] Two degrees plus three degrees is five degrees, even though—
[Speaker B] —that’s also something abstract.
[Rabbi Michael Abraham] One is something concrete and the other is something — degrees of temperature are abstract too. We don’t get four concrete objects. Right, but from the example of the oranges you can infer that.
[Speaker B] But the example of temperature was given because we want to see that two plus three is the same thing. Temperature too. But we’re not bringing that as proof that two plus three equals five. You’re claiming that when you add it like that, it’s not correct to use the rule of addition.
[Rabbi Michael Abraham] Is that what you’re telling me? No — not because it’s abstract or not abstract. Because the addition — these forces too, even if they’re abstract, if you add the two forces that way, then yes, you can add them, right? Two plus three in the same direction will indeed give you five. I don’t know. I think… I know, it’s five. It will definitely be five. What’s the problem with forces as opposed to oranges? Forces and oranges. If you add forces that way, it also gives five. The problem is something else: adding forces is not done by arithmetic addition, but by vector addition. Right? So what does that mean? What do we call it? Vector addition and not arithmetic addition. It’s a different kind of addition. It’s not ordinary arithmetic addition. Fine. It basically means that this is some force like this, roughly something like this, with length three, and this one with length two. Never mind. A different kind of addition, the kind used for forces. So apparently we refuted the rule that two plus three equals five, but in fact no. Obviously we are not going to give up two plus three equals five. What will we do? We will create a different mathematical addition that correctly describes the combining of forces. But we will not give up the arithmetic rule that two plus three equals five. Why? Because at most what this experiment refuted was not the mathematical statement that two plus three equals five, but the statement in physics — not in mathematics — saying that the addition of forces is described by algebraic addition. Something else. Two plus three equals five is an abstract statement; it doesn’t deal with the world at all. It deals with an idea. Numbers are an idea. Mathematics does not deal with the world. Science deals with the world. Now science sometimes uses mathematics as a language to describe what it says about the world. So if, for example, adding temperatures — say that two temperatures, two degrees plus three degrees together is five degrees — there the addition, the algebraic one, is correct, so I use the language of algebra to describe the adding of degrees. But notice: the claim that two degrees plus three degrees is five degrees is a claim in physics and not in mathematics. Two plus three equals five is a claim in mathematics, but two degrees plus three degrees equals five degrees is a claim in physics. For example, if it turns out that the claim is not correct with regard to degrees, I will not give up the mathematical claim that two plus three equals five. Rather, I will attack the physical statement saying that the addition of degrees is carried out by algebraic addition — that is what has been refuted. That is a statement in physics, okay? Meaning, what we can refute is only physics. Mathematics we cannot refute, ever. Therefore with the apples it’s the same thing: even if I find six apples here and I find no miracle and no problem and no malfunction and no cheating and so on, still I will not give it up. At most, in the extreme case, I will give up the physical statement saying that adding apples into a basket is carried out by the operation of algebraic addition. I’ll say that maybe there is some other kind of addition there. But I will not give up the algebraic rule that two plus three equals five, because it is not subject to empirical testing.
[Speaker B] You’re saying this is a specific claim and you won’t be able to refute it — you could say it only applies to these apples. Here it’s the question of generalization. But about these apples, I proved it — here it’s the question of generalization.
[Rabbi Michael Abraham] What about these apples? I proved it. That’s not enough for me — it’s also a claim. Why don’t you give it up? Why don’t you give up the rule? Why don’t you give it up? Because it just sounds obvious that it can’t be otherwise. There are things that… My feeling is that if you tell me you accept the rule that two plus three equals six, that’s changing the language of algebra. Fine, the basic assumptions are prior assumptions, or a prior assumption. You can say whatever you want, but you cannot operate outside that framework; you will never be able to give it up. All right, that’s the… you won’t be able to give it up. Okay, so that is basically the difference between mathematics and physics. So we were asking: how do we accumulate information? Mathematics, then, never adds information for us because it doesn’t deal with information at all. Mathematics deals with relations between items of information or between claims. All right? It never deals with information itself, with adding information. But a mathematical structure that derives conclusions from premises needs flesh to sink its teeth into; it needs premises. Where do those premises come from? Notice: the premises are the thing that contains all the information. Mathematics just takes it out, opens the safe and reveals what’s inside.
[Speaker B] But where did the safe come from?
[Rabbi Michael Abraham] Where does that information come from? Not through mathematical tools. Yes, something? I can’t hear. Where does mathematics begin? Where does it begin? It has all its rules and so on, the rules of mathematics — what… There are all kinds, depending on which field of mathematics. Two magnitudes equal to a third magnitude are equal to each other, all kinds of rules like that that are used in geometry in the middle without introducing them properly and in a different form.
[Speaker B] Obviously those too are the basic assumptions of mathematics. Yes, so—
[Rabbi Michael Abraham] Where does the safe come from? How do we know this information, where do we accumulate information from at all? Mathematics can at most — or proofs, or logic — can at most expose to us information that is already inside us. But how did it get there? How did we collect it? Where did we find it? Not by mathematical means, not by proofs, not by logical means. Notice, I’m not talking about faith at all; we’re talking about science, about scientific information — I haven’t even gotten to faith yet. Where do we accumulate scientific information from? I don’t know. In what way? The tools — usually when we divide the tools of inference, the accepted division is into three kinds: inference of a particular from a general rule, that is called deduction. A general rule from a particular, that is called induction or generalization. I saw one horse with four legs, I assume all horses have four legs. I saw one exhausted student, I assume all human beings are exhausted. And analogy — analogy is things on the same level, the same category, particular to particular or general to general. Logic and mathematics deal only with inferences of the first kind, deductive ones. From the general to the particular. But we have two other types of inference: analogy and induction. Those do not lie in the domain of certainty, of mathematics, of logic. Those are the tools of science. Those are the tools by which we accumulate information. As I said earlier, with deduction you do not accumulate information, because deduction goes from the general to the particular. What does that mean? It means that the information about the particular was already really known to us when we knew the information about the general. So no information is added in a deductive process, in deductive inference. Added information always comes either through analogy or through induction, or of course by direct observation. That is, when I see something, grasp something. But anything that is not observation, but rather an inference that adds information for us, is either analogy or induction. It is never deduction; it is not logic. At least not standard logic. So what is the validity of these tools, analogy and induction? The objections to them — skepticism, basically — are usually founded on pointing to this phenomenon: that all information we accumulate is accumulated by means that are not certain. Because any certain means never accumulates information, never adds information. But if we were born, say, tabula rasa, without any information, then all the information we accumulated throughout life was, by definition, not accumulated by logical means. So really it can’t be accepted because it has no proof. That is skepticism, perhaps in a more precise formulation. Basically skepticism undermines analogy and induction. It accepts only deduction as a legitimate inference, or only proof. Only things that have proof are we willing to accept, and that is deduction. Things based on analogy or induction are not acceptable. That is skepticism. And there is a very interesting process here. Usually skepticism is perceived as something irrational, far from reason, not trusting reason. In a certain sense it is almost the opposite. Skepticism is reason that is too rigid. It is a reason that is willing to accept only things that have proof. Only things that have proof. If they have no proof, it is not willing to accept them. So this is actually a person who does go with proofs, who is rational, a person who acts with his intellect — perhaps too much with his intellect.
[Speaker B] And the skeptic is not usually — what is the serious skeptic? Not the skeptic who throws out everything.
[Rabbi Michael Abraham] I’m now going to describe just briefly a historical process, in a very, very general outline, that will clarify the matter a little from another angle.
[Speaker B] In the history of Western thought, Western thought is more—
[Rabbi Michael Abraham] familiar also because that’s where it happened. In the East I don’t think there was such a process, at least not on a large scale. There was some developmental process that is known in three stages. The first stage was a dogmatic stage. People accepted things because that’s what they were told — that was the tribe, the parents, I don’t know who — just because; because everyone knows that’s how it is, and that’s that. That is the dogmatic stage. A world of myths and customs and so on. At a certain point, a rationalist critique began to emerge, saying: wait a second, so what if everyone thinks that? Who says it’s true? Do you have proof? In Greece you can define this in some way, though of course only in very rough lines. And then some rationalist critique arises, and slowly — in Greece myth and rationality lived together — but gradually the myth is pushed aside in quite a long process, and rationality remains, until the twentieth century. Then in the middle of the twentieth century, suddenly, a very strange process happens, one that historians of culture have spilled quite a bit of ink over: around the middle of the twentieth century, a bizarre reversal takes place. Those same intellectual groups that led rationality — those same groups, not others — took over and expelled it. Those very same groups, like in Daniel there with the four beasts: the fourth beast does not fight the third, it replaces it harmoniously; it seems that’s Greece and Rome. So here too something similar happens. Those very same groups suddenly become what’s called postmodern. They kind of smash the tools and lose all rationality. People rack their brains over how this happened: how modernity — modernity is faith in progress, in reason, in rationality — how did it suddenly turn into its opposite, and by those very same groups? It’s not as though some conqueror came from the East and took over and killed everybody; these same groups — it came from within. And suddenly it became a kind of total intellectual nihilism. Today there’s a little bit of waking up from it, but not all that much, and not enough. And this — the fact that there are no standards anymore, not in art, not in beliefs and opinions. They don’t even like beliefs and opinions anymore, not just standards. It’s sometimes called the collapse of ideologies. People basically no longer believe in anything. Everyone is as right as everyone else, as wise as everyone else, as intelligent as everyone else. Now there are multiple intelligences — everyone is intelligent. I’ll try to explain this process through the structure I described earlier. And I’ll do it by analogy to the maturation of a single person, a typical person. I even know one up close. In the first stage, the person is a dogmatic child: whatever the parents say, the teachers say, society says — it’s self-evident, it’s obvious, it’s true because it’s true and everyone knows it. In adolescence, let’s call it that — and it doesn’t have to come exactly with puberty or adolescence, though I assume there is some correlation — a rational rebellion begins. The rational rebellion basically says: who told you this is true? Prove it. Do you have proof? There’s this kind of optimism that says: I’m going to be a rational person, I’ll accept only things that have proof, I’m done with these dogmas I grew up on. That’s part of adolescent rebellion, its intellectual side; there are personal aspects too, but I’m not dealing with them right now. Then this adolescent matures, and at some point he discovers, to his amazement, that this illusion that he would be rational in the naive sense in which he thought of that concept — that he would accept only things that have proof — is an impossible ideal. There is no such thing. Why is it impossible? Because of what I said earlier: proof cannot produce information or produce certainty, it can only preserve it. But you have to begin with information or certainty that you already possess, and then the proof opens the safe for you and reveals more things about which you can also be certain. But if you don’t have the initial safe, then you have nothing. And that initial safe, as a rationalist, you cannot accept, because we have no proofs for it. It is simply something that exists within us. As was asked here earlier: who says? Maybe we were just born that way, some structure of our minds — who says it’s true? So a person who does not accept things that have no proof ultimately has to throw everything out. And therefore this adolescent really has two possible paths at the end of adolescence. At the end of adolescence he suddenly discovers that his hope of accepting only certain or proven things as legitimate is a false hope. There is no such thing; it’s impossible. Because everything you accept is based on premises. And those premises, if they have a proof, are based on prior premises. Wherever it stops, there lies the safe, and that you will not be able to prove.
[Speaker B] In another second I’d like to finish this picture.
[Rabbi Michael Abraham] So now our adolescent has two paths before him. The first path is to say: okay, so really I accept nothing. If nothing has proof, then I accept nothing. And then he becomes a skeptic. A clear postmodernist, the natural stance. A second option is to say: wait a second, who said only proven things are acceptable? That assumption I made as a teenager itself needs examining. Who said it’s true? You can give it up. Try doing this for fun — do you have proof for that, that only proven things are acceptable? So why do you accept that? Or maybe that’s just a little game. Fine. So the second option is basically to give up the adolescent’s basic assumption and say: no, I’m willing to accept things that are not proven. And then I continue with things I accept and things I do not accept, but I do not demand proofs for everything. Okay? Those are two paths of maturation. It seems to me that around the middle of the twentieth century we can more or less mark that crossroads which we reached in the crisis of the Enlightenment. The great hope of modernity — from the days of the Renaissance until the middle of the twentieth century — that great hope that we would be rational, we would work with reason, we would throw out all myths, all beliefs, because reason would replace everything — that thing broke in the middle of the twentieth century, and rightly so. It broke rightly because from the outset it was an illusion. And people did not understand that. A historical process takes longer than you initially thought. Until a society digests something like that, it can take hundreds of years. But in my opinion that is the process. And when you arrive at this break and say: wait a second, with reason you can’t do anything; with logic you cannot ground any information; with logic or with proofs or with mathematics — nothing. Not religion, not an alternative to religion, not science, nothing. Nothing at all. So you either remain a total intellectual nihilist, a skeptic. That’s one option. And indeed, many — and this is the break I described earlier, when modernist rationality suddenly turns, just like that, into postmodern skepticism. That is not surprising at all when you look at it from this angle. Fine, that is the natural conclusion of this naive rationalism that accepts only things that have proof. So when you suddenly discover there are no such things, then you accept nothing; you become a skeptic. But there is an opportunity to locate an alternative and say: wait, but who said that only things with proof are acceptable? Maybe this adolescent assumption should be reexamined and perhaps abandoned. Who says proof is the tool by which — and only by which — I can ground the claims I adopt? To claim that, as we saw earlier, no claim can really be grounded that way. So if I do in fact adopt any claims at all — and again, I’m still not talking about faith or anything, even in science, in any field — then I must give up the assumption that only what is proven is acceptable. Or in other words, I need to add to my toolbox, besides deduction, which was already there in the Enlightenment, also analogy and induction. And this is just an interesting phenomenon — I don’t know if any of you know one; I don’t — I don’t know any genius PhD at age eighteen in any field other than mathematics. Only in mathematics do you have these little geniuses who get a doctorate at age zero. In no other field does that happen. And I think the reason is that mathematics really is adolescent thinking. Meaning, a little child can excel at it; you don’t need life experience. It’s not the natural sciences. These are thinking tools that characterize the adolescent. The adolescent is a mathematician: he wants to prove everything, and he accepts only things that are proven. But what about science? What about information that we accumulate? We do receive information. We think there is correct information and incorrect information; there are correct beliefs and incorrect beliefs. In this rationalist mathematical world there is no room for that sort of thing. Only mathematics. He doesn’t know that the sum of the angles in a triangle is 180 degrees; he only knows that if you assume the premises and axioms of Euclidean geometry, then you must also assume or derive the conclusion that the sum of the angles in a triangle is 180 degrees. But ask him, what is the sum of the angles in a triangle? He’ll tell you: it depends on your assumptions. Meaning, a mathematician cannot know any information. All he can do is tell you: if the assumptions are such and such, then the conclusion is such and such. He deals only with implication. He does not deal with content, with the information itself. Who deals with information? Religion, for example — though not only religion. Many fields we deal with are about information. But one very prominent intellectual field is science, or the sciences. That is why these tools really develop at a later age. Once this develops — not only develops in the sense that we know how to make analogies and inductions, but that we have trust in the tools of analogy and induction — that is the point I’m speaking about now. Not only the ability to make analogies and inductions, but trust. Because the adolescent has no trust in those tools. Since they’re not proof, I’m not willing to accept their conclusions. I’m only a mathematician; only proven things. Until he understands that he’s at a dead end: he cannot accept anything. And then either he really remains with that assumption that only proven things are acceptable, and then he becomes a skeptic or a postmodernist, as we’d say today. Or he gives up that assumption and says: wait a second, this assumption itself also needs to be examined. No — I am willing to accept things that have no proof, just as I examine things that do have proof.
[Speaker B] This choice between those two options — because really, the choice to remain with the assumption that only what is proven is acceptable is basically an illusion. It comes out of despair.
[Rabbi Michael Abraham] Because if he remains fully consistent with himself and goes on to become one of the skeptics, that skepticism doesn’t understand doubt. In any case, in the end this crossroads really is a very important crossroads, and it seems to me that historically we are more or less circling around it today. I usually mark the nineteenth century there, but the real transition — which even today is still in full force — is the question of which of the two paths to take. The skeptical path, the pluralistic one, let’s call it now, in which all positions are equal? Where did this pluralism come from, this idea that all positions are equal? Because if you’re a mathematician, if you think that only implication has validity, but not premises and not conclusions — say there are premises x and from them comes conclusion y; say there are premises a and from them comes conclusion b — then both are equally correct, because the implication was done properly, so to speak. That is, we built our arguments using correct logic, only our premises are different. So what does that mean? That someone who is supposedly enlightened, yes, a naive rationalist, will automatically become a pluralist. That is, someone for whom all positions are equivalent, for whom there is supposedly no truth, no standards in art or in any other field where this appears. All right, I think we’ll stop here and continue at twelve.