Dispute and Truth – Lesson 1

We’re dealing with the topic of dispute and truth, philosophy in general and Jewish law in particular, and today I want to begin with a somewhat more philosophical introduction, to talk a bit about these concepts of dispute and truth, because in the end we’ll need the meanings of these concepts in order to deal with halakhic topics too, in halakhic contexts. So I’m going to talk, that is, about dispute, I’m going to talk about truth, and about the connection between them. Those are three different subjects. I’ll start maybe with a philosophical introduction. There’s a famous joke, right, in the yeshivot: how do we know every Jew has to wear a hat? It says, “And Abraham went” — there, there. Okay? A Jew like him obviously didn’t go without a hat, right? So if Abraham went with a hat, then we, his descendants and faithful students, have to walk in his path, so we too need to wear a hat. Which is what had to be proved. What do you say about that argument? What? Good joke. Who said Abraham wore a hat? A Jew like him didn’t go without a hat — I said that, didn’t I? Right, but the argument doesn’t prove anything. Right, meaning the problem — why is it a joke? Because this argument begs the question. What does it mean to beg the question? You want to prove that every Jew has to wear a hat. But if you dig a little into the assumptions of the argument, you discover that the conclusion we wanted to prove was actually hidden there inside the assumptions. Because when I say that a Jew like him didn’t go without a hat, what am I really saying? That a Jew has to go with a hat — in other words, a Jew doesn’t go without a hat. So the very principle I wanted to prove already appeared in the assumptions. That’s what in logic is called the fallacy of begging the question. I’m trying to prove something, and I assume the very thing I’m trying to prove. Okay, that’s a fallacy. Except that it isn’t a fallacy. You laughed for nothing. Every logical argument begs the question. Let’s look at an example of a classic logical argument that’s already coming out of everybody’s ears, anyone who’s studied a little philosophy and logic. The claim — the mode of proof says you begged the question, basically, as if that’s a method of proof. We’ll talk about it, okay. No, what I’m saying is that every time, every single time, you prove something, the very proof means the desired conclusion is inside the assumptions. Ah, okay. So I agree — let me spell it out now. Look. Suppose I say: all human beings are mortal, first assumption. Socrates is a human being — the major premise is the first one, the minor premise is the second one, the particular premise, right, about Socrates. Conclusion: Socrates is mortal. Which is what had to be proved. That’s a valid logical argument; it’s the classic example of a logical argument. When people study logic, this is always the argument they use to illustrate it, though of course it’s just a pattern that can be filled in a million other ways. Now this argument — imagine some alien landing by you from Mars. You say to him, welcome to planet Earth; not the friendliest place, but what can you do, not all of us end up in the right places. Now here on Earth all human beings are mortal. Just so you know: there are human beings here, all of them are mortal. He says, fine, very nice, happy to hear it — or not happy to hear it. Then you say to him: yes, and there’s one important scholar here called Socrates, and he too is a human being. He says, what? Interesting, probably worth meeting. Then you tell him, you know, that means you should hurry up if you want to meet him, because he’s mortal. Meaning he might die on you. Better hurry. He asks: why? No, I understood that all human beings are mortal, I understood that Socrates is a human being. Information, okay. But how do you get a conclusion from that? The conclusion doesn’t seem right to me. When you try to convince this person of that — and either he needs to be hospitalized or mend his ways — how do you do it? How do you approach this alien and tell him: look, if you accept the two premises, you have to accept the conclusion. And he says no: he accepts the two premises and doesn’t accept the conclusion. What can you explain to him? Any suggestions? Socrates belongs to the group of human beings, and the group of human beings is a group of mortals. The question is whether that explanation helps, because all you’ve done is repeat the argument, okay? But he doesn’t accept the argument. Okay, suppose so — then what? That’s a different starting assumption from ours, okay. So you couldn’t convince him. You’re basically saying you can’t explain it to him. Okay. So if someone really doesn’t accept that, then from your point of view he isn’t wrong — he’s just starting from a different point. And you’re saying that someone making a logical mistake isn’t mistaken? I disagree. On that point I think you’re mistaken, if you’ll allow me to say so. Look, the point is this. When I see such a person, what I really need to explain to him is this. Look, when I say all human beings are mortal, what have I really said? Let’s make a list of human beings. Yaakov is mortal, Socrates is mortal, Moishele is mortal, Muhammad is mortal, David is mortal, right, all of them. Instead of saying ten billion sentences about every person in the universe, I say in general: all human beings are mortal. Okay? It’s simply shorthand for saying ten billion statements, right? Now let’s break down that shorthand into small change, let’s write it out in full. What we’re really writing is: Yaakov is mortal, David is mortal, Moishe is mortal, Socrates is mortal, and so on, right? That’s what we said. It’s a translation; it’s just language. Right, there’s no argument here — just unpacking the claim. To say Socrates is on that list. Right. I need the argument, the second premise, the minor premise, in order to say that Socrates is on the list. You’re probably a mathematician, so that’s true, yes — and I have the second premise. Okay. You’ll soon see that it isn’t a compliment. So once he’s on that list, then if so, you’ve already accepted that Socrates is mortal. So what do you mean, afterward, that you don’t accept that Socrates is mortal? If you accepted that all human beings are mortal, then in particular you also accepted that Socrates is mortal, from the second premise that Socrates is a human being. So in effect you accept that claim, and then it’s just an internal contradiction. You can tell me: I both accept that Socrates is mortal and don’t accept that Socrates is mortal. Fine, maybe there are axioms that don’t exist, or universals… What’s the axiom of logic? Can you say a sentence and its opposite at the same time? Leave aside the complicated thing. In the initial description we said something complicated. Can you say, all right, maybe this logical structure — maybe you don’t accept this logical structure that every X is Y and A is X, therefore A is Y. Okay? Maybe you don’t accept that pattern. Forget it — let’s break it down into tiny pieces. Do you accept that if you say Socrates is mortal, you can’t simultaneously say Socrates is not mortal? Either you think this or you think that, but you can’t say both things; otherwise you’ve said nothing. Once you unpack it, you arrive at a logical contradiction that is totally frontal. It’s not something that needs some sophisticated logical analysis to expose. You’re saying a thing and its opposite, so that’s meaningless — then you’re saying nothing. That’s what I would try to explain to that person, and if it doesn’t work then yes, hospitalization. But that’s what I would try to explain to him, okay? Now what does this actually mean? That this argument begs the question. Right? Because what does that mean? That the conclusion, Socrates is mortal, is actually one of the premises I relied on. When I said all human beings are mortal, let’s break that into ten billion premises, and I use the second premise that Socrates is a human being, okay, so one of them is that Socrates is mortal. So I assumed that Socrates is mortal and arrived at the conclusion that Socrates is mortal — very surprising. So it begs the question. In fact, I’ll tell you more than that: why, when there is a valid logical argument whose conclusion necessarily follows from the premises, why must I accept the conclusion? Because you accepted the premises. Because it’s inside the premises. Meaning if you accepted the premises, then included within them, embedded in them, is also the conclusion. If you accepted the premises, then implicitly you already accepted the conclusion. It’s not that you now need to accept the conclusion — you already did. Or else don’t accept the premises and then you won’t have to accept the conclusion. If you accepted the premises, then hidden within them is already the fact that you accepted the conclusion too. Therefore, the validity of an argument is nothing other than its begging the question. A valid logical argument, whose conclusion necessarily follows from the premises, is basically a synonym for saying: a logical argument that begs the question. That’s it. So whoever wants to say that all logical arguments… But if you understand that logical arguments are not fallacious, then that means begging the question is not a fallacy. Why, in the case of Abraham our patriarch, is it a joke? Because in Abraham’s case the conclusion is one of the premises explicitly: every Jew has to wear a hat. In the case of Socrates, I need some combination of the two premises together in order to show that the conclusion is really contained in them. It’s a slightly more complex structure, but it’s there. It’s still there. So is it funny or not funny? I said before that it’s a good joke. If it’s funny or not funny, it’s not because it begs the question, but because it does so trivially, banally, bluntly. So it’s a joke. But other logical arguments don’t make us laugh even though they also beg the question, because there you need a bit of gymnastics to extract the conclusion from the premises. But after we’ve done the gymnastics, we discover that the conclusion really is inside the premises. Okay? Take geometry, for example. Okay? Everyone studies geometry in high school. So we start from four axioms and arrive at different conclusions — theorems, right, propositions. Okay? So you can’t say that someone who knows the axioms also knows the theorems. Let’s say you take a sixth-grade kid. A sixth-grade kid with a bit of intelligence knows all the axioms of Euclidean geometry. Ask him how many straight lines pass through two points, and he’ll tell you one. A sixth-grade kid understands that. Okay? If two parallel lines ever meet if we keep extending them farther and farther, he’ll tell you no. And so on. The axioms of geometry are something every sixth-grade kid understands, at least once you ask him. He won’t always gather the four axioms and make a structure out of them. But if… What? Yes, there’s one that’s a bit more problematic, yes, never mind. Either dependent on the others, or conversely redundant. No, no — either a novelty you can assume, or other things. Redundant. That was the parallel postulate… Redundant. Yes. So the claim is that someone who knows the axioms does not necessarily know that the sum of the angles in a triangle is one hundred eighty degrees. Right? Ask a sixth-grade kid what the sum of the angles in a triangle is, and he won’t tell you one hundred eighty degrees, unless he’s really a genius. Usually it won’t happen. But to know the axioms you don’t need to be a genius; every child knows that. What does that mean? After all, once the teacher proves it to him in class, he’ll see that the conclusion was really inside the premises. But by himself he wouldn’t have managed to get there. So it is begging the question, because otherwise it wouldn’t be a proof. Mathematics and logic — yes, proof always begs the question. But this is non-trivial begging of the question, and that’s why we don’t laugh; usually we cry when learning geometry. Okay? But that’s only not because it doesn’t beg the question — it does beg the question. It’s because the begging there is not banal, not trivial like with Abraham and the hat. That’s all. But there is no problem with begging the question as such. Right, there’s another joke that comes up here. Do you know the one about the two people who lost their way in a hot air balloon? They have no idea where they are, they’ve been drifting around the globe for days, no clue where they’re located. They see someone plowing a field below, so one of them shouts down: hey, where are we? The fellow below says: above my field. Then the guy in the balloon says to his friend: that fellow down there is definitely a mathematician. This is apropos mathematics, right? Why? Two characteristics. First, what he says is absolutely certain and completely precise. And second, it doesn’t help us at all. Now that joke is cute, but again, like begging the question, it’s completely true. Mathematics by definition is completely precise and also doesn’t help us at all. Doesn’t help us at all not in the practical sense, but in the sense that it doesn’t add information for me. The mathematical move from the axioms to the conclusion doesn’t add information that wasn’t already inside the axioms. In that sense it adds nothing. The difference between mathematics and science is that science adds information for me. Science, from the word to know — right? It adds information. Mathematics doesn’t add information. Mathematics reveals more and more information that in fact, when I held the axioms, was already in my possession; I just didn’t always have a way to become aware of it — that is, to understand that it was there, that it was true. Right, like an onion — or if you want, like a safe. Okay? A kind of safe where everything inside it is yours. But if you have no way to open the safe, then it’s yours theoretically, but you have no access to it. Same thing here: someone who holds the axioms of geometry hypothetically, potentially, knows that the sum of the angles in a triangle is one hundred eighty degrees. He has all the necessary information. And no additional information is needed in order to know that. What he needs is the skill to open the safe, to analyze the information he has and extract that conclusion from it. That’s what the teacher does. The math teacher doesn’t add information to you. The math teacher helps you understand, or quarry out of yourself, more and more information that was already there. Unlike a science teacher, who does add information to you. Okay? So the guy in the field down there is a mathematician in the sense that what he says is completely exact and certain. But precisely because it’s certain, it doesn’t help us at all. It doesn’t help us in the sense that something certain never adds anything beyond what I already knew, at the principled level. Practically yes. Practically, sometimes I won’t be able to derive the conclusions myself because the analysis is too complicated. But it’s still analysis. Okay? Analysis such that once you understand it, you can’t disagree with it. In science, you can disagree. Either it’s true or it’s not true. It’s not necessary. Okay? And in fact this says that the domains of mathematics and logic are empty domains. Empty in the sense — this is what in philosophy is called the emptiness of the analytic. The emptiness of the analytic means that when you have a logical or mathematical argument — for this discussion it’s the same thing — the conclusion adds nothing beyond what is already in the premises. In that sense it’s empty. Meaning the argument didn’t add anything. If you don’t accept the premises, you won’t have to accept the conclusion. If you do accept the premises, then you already know the conclusion, so in principle you don’t need the argument. Of course in practice you do need it, because you’re not always skilled enough to reach the conclusion, like in geometry. But at the principled level the conclusion is already in your hands. Right? Or in other words, this basically means the following. In logic they divide arguments into three types: analogy, induction, and deduction. Analogy is a comparison we make from one particular to another particular. Let’s say, I don’t know, this table is brown, that table is also a table, this thing is also a table, so it too is brown. That’s an analogy. Incidentally, probably not an analogy that works for all tables, never mind, but it’s a type of argument called analogy. Okay? What is induction? If this table is brown, then all the tables in the world are brown. That’s a move from the particular, or from a few particulars, to the general rule. As opposed to mathematical induction, which is actually deduction. Right? Mathematical induction is just a homonym; it’s really deduction. Unless you’re an intuitionist. But normal human beings understand that it’s deduction. And necessary. When you prove something by induction in mathematics, that’s a proof in every sense, and the result necessarily follows from the premises. It’s not — it’s actually deduction. It’s called induction because it resembles the move from particular cases to a general law. But real induction is scientific induction. When we see several massive objects falling to the earth and from that infer that all massive objects fall to the earth. That’s a generalization, that’s induction. Okay? So that’s a move from particulars or a particular to the general. There’s analogy, which is a move from one particular to another particular. What’s the difference between analogy and induction? After all, you spoke about the same thing. Analogy is from one particular to one particular, while induction is from one particular to a whole group. Which means induction is the sum of many analogies. We’ll deal with that in a moment. So that’s analogy and induction. What is deduction? Deduction is a move from the general to the particular. If all human beings are mortal, then Socrates, who is one of them, is also mortal. You understand that this is the opposite direction from induction, right? Induction goes from the particular to the general that includes it, and deduction goes from the general to a particular contained within it. Okay? Now of these three forms of inference, only deductive inference is necessary. Why? Right? From the general to the particular. From the particular to the general or from one particular to another — those are inferences, we use them, but they can be mistaken. It’s not necessary. We’re not certain to be right. Either the analogy we made is incorrect — here, this table is brown, that table is also brown, but there may be another table that isn’t brown. It’s not certain that the analogy is correct. But we do use analogy, meaning it is an argument. It’s not absurd, it’s not some fallacious form of argument. We use analogy. Okay? So he says that analogy and induction are arguments whose conclusion does not necessarily follow from the premises. Deduction does. And notice: even in deduction, it’s not that the conclusion itself is necessary. The conclusion necessarily follows from the premises, but anyone who doesn’t accept the premises doesn’t need to accept the conclusion either. You can’t say that the conclusion, by itself, is necessary. What is necessary is the derivation of the conclusion from the premises. Okay? In analogy and induction even that is not true. The derivation of the conclusion from the premises is not necessary. The premises may be true while the conclusion is false. Okay? What is the difference between deduc… why is deduction necessary while analogy and induction are not? Incidentally, analogy and induction are the tools used in science. Deduction is mathematics and logic. It’s the same division. The division between mathematics and logic on the one hand and science on the other is the division between deduction and analogy and induction. Why — why in mathematics do people sometimes use a logic of inductive thinking? But the logic of inductive thinking is deduction. It’s a mathematical description of inductive thinking. In any case, the difference between deduction and induction and analogy is that deduction, as we saw earlier, doesn’t add information for me. If all human beings are mortal, then obviously Socrates, who is one of them, is mortal. There is nothing in the conclusion beyond the information I had when I held the premise. Right? The move from premise to conclusion added no information. In analogy and induction, the inferences or arguments do add information. I say: if this table is brown, then that table is also brown. That adds information. Assuming this table is brown and that one is also a table — those are my two premises. The conclusion is that that one too is brown. That is not contained inside the premises, right? There’s information here that was added beyond what I had when I held the premises. And the same goes for induction, of course. Okay? So the difference between analogy and induction on the one hand and deduction on the other is whether it adds information for me. Now, in principle, everything that adds information for me is not certain. A certain thing is only something that does not add information, as we saw before — logic and deduction. Or in other words, there’s something we can call the philosophical or logical uncertainty principle. The uncertainty principle says that the product of uncertainty in position and in momentum or velocity is constant. So if I have great uncertainty in position, I have small uncertainty in momentum, and vice versa. Okay? Velocity and position don’t go together — knowledge of velocity and knowledge of position. Likewise, the product of the degree of certainty and the amount of information added is constant. Meaning: something that is completely certain, about which I have no uncertainty at all, adds no information. Okay? If my certainty is full, absolute, then the amount of information is zero. But in theory and in computational power… What? No, no, not without calculations. I’m speaking now at the logical level, the relation between propositions. It doesn’t matter right now how I arrive at it; maybe we’ll discuss that later. Analogy and induction are inferences that add information for me. Since that’s so, they are not certain. There is always a kind of trade-off, right — if you want certainty, you have to pay for it in terms of information, and vice versa. If you want information, you have to pay for it in the currency of certainty. Wait, so deduction is certain? No, that’s an assumption. That — that’s the assumption, yes. So if that won’t be true, then it won’t be certain either; I don’t know. Anyway, the claim is that there’s a kind of trade-off between the level of certainty and the amount of information this inference adds. If the level of certainty is full, absolute, then the amount it adds — the amount of information it adds — is zero. Something that adds infinite information has a level of certainty of zero. Right, to know everything — there’s no chance. Meaning, you can’t have certainty about anything if you want to say it about everything. If you want to say it about a very specific group of things, fine, then maybe that can be said, perhaps with certainty, perhaps with a higher probability. There’s this kind of trade-off. And once you look at it this way, incidentally, I return to your question: what’s the difference between analogy and induction? Let me ask you: which is stronger, analogy or induction? Induction is stronger. Why? Maybe weaker? What is induction? A collection of… First of all, according to what I just said, look: which one adds more information for us? Induction, right? Because it speaks about a general law, while analogy moves from one particular to another. That means induction is less certain. When you say something more speculative, you’re making a claim about each and every particular; any one of them that proves you wrong knocks down your induction. An analogy can be refuted only if you prove that this specific item you were talking about is false. If you prove it about another item, you’ve done nothing. Therefore, an inductive argument is a more speculative argument, less certain, because it adds more information for me. Again the same trade-off. In one-particular analogy, it adds very little information, so it can be at a higher level of certainty. The big problem that arises here is of course what the logical relation is between these two things, not the amount of information. Because induction is nothing but a collection of analogies. Right? That’s what you said earlier. When I say this table is, I don’t know, brown; a certain donkey has four legs, so all donkeys have four legs. That’s induction, right? What am I really doing? I’m making an analogy between this donkey and that donkey and that donkey and that donkey. The collection of those analogies is really induction. Right, but induction is a collection of analogies and therefore it’s more speculative. Any one of the analogies that falls knocks down the induction, but it doesn’t knock down the other analogies. But there’s another side to the coin, and this is the trickier side. Namely, that every analogy implicitly contains an induction. Think about it: when I say this table is brown, and that one is also a table, so probably that one is also brown — why? Why, if they’re both tables, do I assume that if this one is brown then that one is also brown? Because all tables are brown. I’m really saying that brownness is a property of a table as such. All tables are brown. And therefore, in particular, that table is probably brown too. Meaning when I make an analogy, I usually assume an induction implicitly. What? Exactly. I’m really assuming implicitly the induction that all tables are brown and applying that to that particular table. So the relation between analogy and induction is trickier than it seems. Analogy in itself clearly contains less information. But the logical route to analogy generally presupposes an implicit induction. Fine. Now more than that. If you look at it this way, then what comes out is this: in fact there aren’t three modes of inference. There’s one. Only analogy. Analogy from particular to particular. Except — how do I make the analogy from this table to that table? I take this table and make an induction from it to all tables being brown. And now I make a deduction: if all tables are brown, then in particular that table is brown. In fact, the systematic way to make an analogy is to do an induction and then descend from the particular to the general, and then descend from the general to another particular through deduction. No, what I’m saying is: I’m really always making analogies, only the way to make analogies is through inductions and deductions. It doesn’t matter — it’s a circle. You can describe it however you like, but it’s really one circle. There aren’t three distinct modes of inference here. It’s one circle. There’s, for example, John Stuart Mill’s challenge to deduction. John Stuart Mill — philosopher, right, from the nineteenth century, I think. The nineteenth. He challenged deduction and said: we think the conclusion of deduction is necessary, but it really isn’t necessary. Because how do you know that Socrates is mortal? You assume that all human beings are mortal and that Socrates is a human being. How do you know that all human beings are mortal? Where do you know that from? Very interesting. I want to know whether Socrates really is mortal. Derivability is nice, but you can’t buy groceries with derivability. I want to know whether Socrates is mortal. To know that, I need to know that all human beings are mortal. How do I know that? From induction. Simply from the fact that the people I’ve seen have died — those whom I’ve seen so far have at least died — and I infer that all human beings are mortal. Therefore it’s true of all human beings. So I do an induction, and once I have the major premise, which is the product of induction, from there onward I proceed safely: then of course Socrates is mortal. But behind your confidence in the conclusion of deduction, you’re really hiding an induction. So don’t tell me deduction is certain. Which is really to say what I said before: when you do induction, deduction is sitting there in the background. And therefore, if you look at it this way, when I arrive at the conclusion that Socrates is mortal, I’m really starting from examples of people I’ve seen. I’ve really made an analogy from the cases I saw to Socrates. That’s analogy. Only the way of the analogy is first to do an induction, arrive at a general statement, and then come down by deduction to the other particular conclusion. But in essence I made an analogy. And when I want to know facts about the world, facts about the world are always particular facts. Or at least — what do I mean, always? Facts are by their nature particular facts. The general laws only gather together collections of particular facts. Fine, but in the end, I don’t just want to know: if human beings are mortal then Socrates is human. That interests logicians. I want to know the world; I need information. Is Socrates mortal or not? To know that I also need to know the premises. It’s not enough for me to say that if the premises are true then the conclusion is true. “If, then” is a nice amusement, but if I want information then “if, then” is not enough — I need to know the “if,” the premises I start from. Okay? Right, think of someone who comes to a mathematician and asks him: tell me, what is the sum of the angles in a triangle? A responsible mathematician ought to answer: I have no idea. It depends on your assumptions. Under Euclidean assumptions, one hundred eighty degrees; under other assumptions, minus i to the power of i pi. No — i pi is also real. Pi, fine. The sum of the angles is pi, in degrees not in radians. Any sentence here can be made true; that’s the problem with mathematicians. So okay, how do I get there? Minus pi, in short. Fine, how do I get there? I build some non-Euclidean space, never mind exactly what, and I can arrange such a space so that the sum of the angles is whatever you want, made to order. No problem arranging that. Okay? Therefore a mathematician cannot tell me what the sum of the angles in a triangle is. He is responsible for the “if, then.” If your assumptions are such-and-such, leave it to me, I’ll calculate for you the sum of the angles in the triangle. But the assumptions are your responsibility, not mine. You need to tell me what your assumptions are, and then I as mathematician will tell you what the conclusion is. But I can’t give you information about the world. A responsible mathematician can’t give any information about the world. He’s only responsible for the “if, then.” So in fact you see that all the information we have about the world depends on some assumptions with which we enter the game. From there onward, we can use logic and mathematics to derive conclusions, but in order to take those conclusions as facts I need to begin from some assumptions. And those assumptions I don’t get from a mathematician, or in my capacity as a mathematician, but from — I don’t know — observation, induction, whatever you want, as a scientist. Okay? If I want to know what the sum of the angles in a triangle is in the world, that’s a scientific question, not a mathematical one. Right? Once, when I came here to study physics for a master’s degree, they assigned me to teach mechanics. So I asked the students there at the beginning, in the first tutorial: is the statement two plus three equals five a scientific statement? According to Popper, you know, a scientific claim is one that can be falsified. Something that can be subjected to a test of refutation. Okay? Meaning, if you can propose an experiment with one possible result that confirms and another possible result that refutes your claim, then it’s a scientific claim. A claim that cannot be put to a falsification test in an experiment is not a scientific claim. Seemingly, two plus three equals five can be tested. Take two balls, right, put two balls into a basket, take another three balls and put them into the basket too, count how many balls you have in total. If you get minus pi, then apparently two plus three does not equal five. But it’s a definition that two plus three equals five. What do you mean a definition? I know what two is, I know what three is — add them, you got minus pi. No, I’m counting. I’m counting. What? I’m counting. They’re not equal, not equal. I’m counting how many balls there are — they won’t be there. I’m counting how many balls there are. One basketball, one soccer ball, and one ping-pong ball. Fine, but the issue here is this. Suppose you counted. You put in two balls and then three balls, counted, and got minus pi. Okay? What would your conclusion be? That two plus three does not equal five? There was an error in the experiment, right? Yes — or someone doesn’t know how to count, or there was an error in the experiment. Right? Why? Because you will never get an experiment whose outcome is that two plus three does not equal five. It can’t be. Why not? Because two plus three certainly equals five — that is our a priori assumption. And the whole question is what happened here physically: did the factual experiment realize the arithmetic theory that two plus three equals five or not? Why did I bring this up at the beginning of a mechanics course? I told them: look, here’s an experiment that refutes the claim that five plus five equals ten. Take a body and apply a force of five newtons northward. Okay? And another force of five newtons eastward. What is the total force acting on the body? Five root two, right? Seven point something. Not ten. Right? Something intuitive; you don’t need to study physics for this. Two such forces acting on the body will move it in that direction. Right? By a force given by the Pythagorean theorem, five root two. Fine, but that isn’t ten. So did we refute the claim that five plus five equals ten when we performed such an experiment? Fine, five plus five doesn’t equal ten. Five plus five equals ten isn’t about forces, it’s about numbers. So it’s not on the same axis and not on that axis either. Numbers are all on a straight line. Again you’re making a physical model out of it; that’s not the right way to look at it. The device is right here in front of me, Menachem said here, not five and five and not ten and no such thing. Menachem said here: here’s five and here’s five and it turned right and didn’t get to ten. And now there’s an average. And to say the formula of the correct result — if he thinks the result is incorrect, that’s his mistake; look, the instrument shows me this. Fine. Incidentally, this is like pilots in vertigo. When a pilot gets into vertigo, that’s the training he gets: to drum into his bones that even when you’re convinced the instruments are wrong, go with the instruments. And that’s very hard when you’re sure they’re wrong, but if you don’t do it you’ll probably crash. Usually the instrument is right. Fine, never mind. For our purposes: what am I trying to say? Why is this a claim that can’t be falsified? Because the claim two plus three equals five is not a claim in physics; it’s a claim in mathematics. A mathematical claim is a claim about numbers, not about forces, not about balls, and not about anything else. When I want to apply that arithmetic claim to balls or to forces or to whatever, at that point I am already making an assumption in physics: that this piece of physics is a model for that mathematical theory. That arithmetic is a mathematical doctrine, okay? And now I say: addition of forces is a model — yes, one can use arithmetic to describe what happens when adding forces in physics, or adding balls to a basket, which is also, say, in physics. Okay? That’s an assumption in physics, not a mathematical assumption. And if it turns out that two plus three equals minus pi, and let’s say there was no error in the experiment, we haven’t refuted the proposition two plus three equals five in arithmetic; we have refuted the assumption that arithmetic indeed describes this physical situation. And that is an assumption in physics. An assumption in physics is by its nature falsifiable. Mathematics is not falsifiable — in the mind or in the Platonic realm, depending on whether you’re a Platonist or not. But it doesn’t deal with the world. Okay? And therefore it cannot be falsified. Mathematics does not make claims about the world. As far as the world is concerned, mathematics is only “if, then.” It can state “if, then” about the world. But no non-hypothetical claim, no absolute claim — not an if-then claim — about the world can come from mathematics. Mathematics is responsible only for the “if, then.” If I really want to know what is happening in the world, I first need a physicist to feed me the “if,” and then the mathematician will extract from that the “then.” “From strong came forth sweetness,” as it says. So the claim, in practice, is that logic and mathematics cannot teach me anything; they cannot add information for me. Information is added for me by scientific tools. And science deals with claims about the world. Mathematics does not deal with claims about the world. I had a big argument about this with some mathematician from the Technion, and I think he’s wrong. Mathematics does not deal with the world. His view is more complex — he’s a smart guy — but philosophically I disagree with him. Anyway, here we have Ron Aharoni, a mathematician, who has a book called The Cat That Isn’t There, based on a saying of William James. He says: what is philosophy? Philosophy is a blind man looking in a dark room for a black cat that isn’t there. That’s his description of philosophy. And Ron Aharoni wrote a book claiming very forcefully that there is no such thing as philosophy; philosophy is a collection of fallacies. There’s no such thing as philosophy. Philosophy by definition — the moment a problem is a philosophical problem, it is a fallacy. There are only problems in science. And mathematics is — it is basically science, just not claims about the world; that’s his position. Fine, I have two columns on my site about this. All of philosophy? All of philosophy. Yes. In ethics, for example, he would say you’re not asking what is right to do; you’re asking if you do this, will it do good, will it bring benefit to people, or cause harm to people. That’s a scientific question. Whether it’s right or not right — “good” or “not good” are just words. Now you decide whether you want to do good to people or do bad to people. If it’s about the soul, then go to psychology. That’s not mathematics, it’s psychology. Fine, but those are still factual claims about the world; that’s not philosophy. Philosophy, in his view, is an empty domain. Now he is wrong, but it’s not simple to show exactly where he is wrong. Anyway, the meaning, the conclusion, of what I’m trying to say is that when I speak about some claim and ask whether it is true, you cannot answer that question in terms of mathematics and logic. Mathematics and logic are not the right tools to handle that kind of question. At most, mathematics and logic can tell me: assuming these claims are true, then this conclusion is also true, or this claim is also true. But you always have to begin from some set of claims whose truth is accepted by you, and then start with mathematics and logic. Okay? That is basically the conclusion up to this point. Except that then the question always comes up: all right, so if that’s the case, then there is no way to accept any claim at all. Because the premises — I have no proof for them, so why should I accept them? And anything I do have a proof for is based on premises. And if I can’t accept the premises, then I can’t accept the conclusion either, because it’s based on them. I once asked students: in geometry, which is more correct, the theorems or the axioms? The theorems have proofs and the axioms don’t. They’re both equally correct. What does it mean that the theorems have proofs? It means reducing them to the axioms, right? That’s called proof. You can’t say the theorems are truer than the axioms. Proof means grounding the theorem on the axioms. That’s what proving means. But if you don’t accept the axioms, then you won’t accept the conclusion either. Or at least you won’t have to accept the conclusion. It may be that the axioms are false and the conclusion true, but not necessarily. In other words, what does this actually mean? It means that the concept of truth really belongs to the semantic field of science, not to the semantic field of mathematics. Mathematics and logic do not deal with truth — so what do they deal with? They deal with validity. The validity of an argument is the question whether the conclusion necessarily follows from the premises. Truth is the question whether a certain claim is true or false. Truth I check, say, against observation of the world. If the claim is “now it is daytime,” in order to know whether that is true, I look outside and see that now it is daytime. So comparing the claim with the state of affairs in the world that the claim describes — that is the tool by which I test whether the claim is true or not. Testing the validity of an argument is done with logical tools, not with tools of observation and comparison. Whether the conclusion necessarily follows from the premises or not — that is determined using logical tools. You don’t need observation for that. And if we have an argument that is valid and its premises are true? Then the conclusion is also true. Correct. The only connection between the question of validity and the question of truth is that connection: if there is a valid argument whose premise is true, then the conclusion is necessarily also true. Therefore logic is an interesting thing, apart from being an amusement. Logic is interesting because it helps me connect true claims to other true claims. It saves me work. I don’t need to test those claims too if they are derived from other claims that I’ve already tested. Okay? That’s all — it saves me work. But logic in itself is merely an intellectual amusement; it teaches me nothing. Meaning, if someone does not accept the truth of some premises, then mathematics and logic will not help him in any way. He won’t be able to do anything with them except amuse himself that if this, then that, and if that one, then that one. Beyond that — nothing. Now I want to show you where this logical-philosophical introduction takes us. There is a certain way of looking at things. The first stage is childhood, then adolescence, then adulthood. Right? Those are the three stages. And now, of course, I’m not talking about psychology — psychology doesn’t interest me. I’m talking about philosophical maturation, not psychological maturation. Okay? There are other aspects of growing up; not interested. I’m talking now about maturation on the philosophical axis. What does that mean? When a small child wants to know something, he asks his parents and they tell him. He asks: will the sun rise tomorrow morning? They tell him: yes, it will rise. Okay, understood. So the child accepts what the parents, or teachers, or adults tell him. We can call this the dogmatic stage. Okay? The stage in which if an authority figure tells you something, you accept it; you don’t ask questions. At some point the child grows a bit, and the point arrives when he passes from childhood to adolescence — again, only on the intellectual axis, I’m not entering psychology. So on that axis, this point comes when the child basically asks himself: wait a second, why? Why should I accept what they say? Right, the teenager’s protest against his parent — “who says? Prove it.” Right, that’s a very typical adolescent protest. What are they really assuming when they ask a question like that? The fact that you said it — so you said it. So what? Who said it’s true? Prove it. Why should I accept something you don’t prove to me? Okay, that’s adolescent rebellion. Adolescent rebellion continues until the teenager grows up and reaches the next point on the axis where he passes from adolescent to adult. What happens there? Suddenly he discovers that what he was really claiming was: I’ll accept only things that are proved to me, right? If it’s not proved, I don’t accept. He is basically assuming an identity between acceptability and proof — not provability, but actual proof. Okay? Meaning only a proven thing is acceptable. That is the adolescent’s assumption: prove it to me and I’ll accept it; don’t prove it, go home. When he matures, he suddenly understands that there is nothing that can be proved. The truths of life — nothing can be proved. Why? Because every proof is based on premises. There is no proof without premises. And if I don’t accept the premises because there is no proof for them, then proof won’t help me either. Like in geometry: if I don’t accept the axioms, then I won’t accept the claims proved on their basis either. Which means that if you carry your adolescence to its logical end — yes, with the assumption that you don’t accept things unless you have proof for them — then you cannot accept anything. No acceptability, no truth, nothing. You can adopt nothing as acceptable. Okay, that is basically the upshot. Now what do you do at this stage? The next stage is adulthood. So we passed adolescence; there was a transition point from childhood to adolescence. Another transition point takes us from adolescence to adulthood. How does one grow up? There are three ways to grow up. Lucky you — women are less philosophical. Whoever gets married stops dealing with these questions. At least for a while, until grandchildren start arriving, and then it’s fine, meaning there’s time and you can work on these things. There are three ways to grow up. How do I know there are only three and no more? Because what creates the tangle in the transition from adolescence to adulthood is the combination of two assumptions. Assumption one: only what is proved is acceptable. Right? That’s the adolescent’s assumption. Assumption two: there is nothing whatsoever that can really be proved unconditionally. If there is a proof, then the premises won’t be proved — right? You can’t prove something unconditionally. There were heroic attempts of that sort in the history of philosophy — what Kant called ontological arguments. These are arguments that try to prove a conclusion without premises, out of itself. Meaning the claim is that if you don’t adopt this conclusion, you land in a contradiction. For example, Anselm’s ontological argument for the existence of God, or Descartes’ cogito, “I think, therefore I am.” All of these are arguments Kant called ontological arguments. These are arguments that are supposed to lead you to a conclusion without assuming any premise. No premises needed. Logic itself yields a factual claim about the world, which is hocus-pocus. Kant did not accept this. He said it can’t be. Logic cannot give you facts, because logic is empty, as we said earlier. And indeed, when you examine these arguments, you see that they are all mistaken. Meaning there is no possibility of proving anything without assuming premises. But we won’t get into that here. In principle there is no such thing as a proof that is not based on premises. And once it is based on premises, then the question arises regarding the premises too: why should I accept them? I have no proof for them. If I do have a proof, then that proof has premises. In the end, I will arrive at some claims that I will have to accept without proof. Therefore the tangle of maturation consists of two assumptions. One assumption — this has an assumption — the first assumption is that only a proved thing is acceptable. The second assumption is that nothing is proved. There are three ways to grow out of that. Nothing is acceptable. Therefore the adolescent won’t accept anything; nothing is acceptable. No, that’s the adult. The adolescent rebels because he still lives in the illusion that he’ll manage to prove things. At some point he understands that nothing can be proved, and that’s the point of growing up — or of falling apart. Then you have three possibilities. With two assumptions, there are three possibilities, right — that’s combinatorics. Either you adopt both assumptions, and then what happens? I become a skeptic. Right? If only what is proved is acceptable and nothing is proved, conclusion: nothing is acceptable except this statement itself. Only this itself is acceptable: that nothing is acceptable. And that’s one form of maturation — adopting both assumptions. The combination of the two assumptions leads me to skepticism. Okay. Second possibility: give up assumption B. It is not true that nothing is acceptable — that nothing is proved, sorry. There are things that can be proved. And I’m not already talking about ontological arguments in philosophy because those are mistakes. Ah, then someone can come and say: I have some source of information not through logical argument, but someone who is all-knowing, and whatever he says is certainly true even without proof. Whether that’s the Holy One, blessed be He, or al-Baghdadi from ISIS, or your rebbe, or I don’t care who — some kind of all-knowing being whose words are definitely true. Then I say: there are things I can be certain about and for me they are acceptable. This is someone who accepts the assumption that only what is certain is acceptable, but does not accept the assumption that nothing is certain. He says certainty can come not from logic but from these or other transcendent sources. Okay? Other sources, not philosophical-logical or observational sources. Fine? I call that fundamentalism. Fundamentalism is an approach that adopts claims in such a way that it is unwilling to submit them to the test of critical thought. It’s certainly true, period. It cannot be false. I’m not willing to think about it twice. Now fundamentalism, incidentally, we often identify with cruelty, ISIS, all those types — but fundamentalism in its philosophical definition, as I’m proposing it here, can also go in the opposite direction. Mother Teresa is also a fundamentalist. For her, helping every living creature is not something subjected to critical thought. She is devoted to it, attached to it, she would give her life for it. She’s a fundamentalist, and in this case she’s a harmless fundamentalist. But philosophically she too is a fundamentalist. So fundamentalism is the willingness to accept claims without any — that is, you accept claims without being willing to subject them to critical thought. That is called fundamentalism, and it is one alternative way of growing up, as opposed to skeptical adulthood. That’s another way to mature. The third way is of course to give up the first assumption. The fundamentalist gave up the second assumption, that nothing is certain. But one can also give up the first assumption: that only a proved thing is acceptable. Who says that only something for which I have proof can I accept? What if it isn’t certain, but it makes sense? My common sense tells me it’s true. Why shouldn’t I accept such a thing? Why this assumption that I accept only what is proved? Why? Who said so? Not at all. I accept things that sound reasonable to me. They are not certain, true; I have no certainty about them. I’m not claiming those things are certain — no. But the fact that something isn’t certain doesn’t mean it isn’t acceptable. Acceptability and certainty — or truth and certainty — are two completely different things. There is a difference between saying that something is true and saying that it is necessarily true. In logic these are even two different branches of logic. Logic deals with what is true, and modal logic deals with what is necessarily true. Not deontic logic — that slipped out — never mind. Right, true and necessarily true are two different predicates in logic. So the claim is that one can give up — of course one can give up both assumptions, but that’s unnecessary, because giving up one is enough to mature. Therefore I speak of three possibilities and not four. In principle, combinatorics says there are four possibilities: adopt both assumptions, give up one, give up the other, give up both. Okay? But giving up both is unnecessary, because giving up one is enough to get out of the tangle. So it doesn’t create an additional option. What? What does this actually mean? It means that from the point of maturation, when you leave the point of adolescence and become an adult, you have three options and no others. Either to be a skeptic, or to be a fundamentalist, or to give up certainty but not give up acceptability. Meaning: to adopt claims, to say they are acceptable to you even though they are not certain. Fine, that’s the road; there’s no other possibility. Those are the three options. Can you be a skeptic at a high level? Call it skepticism if you like, but no, I don’t think so. In my view skepticism means saying: I know nothing in any way whatsoever. Every thing is as true as its opposite. This isn’t skepticism; it’s only lack of certainty. I think it’s true but I’m not sure. Human beings can make mistakes. I’m not sure. That doesn’t mean I don’t think it’s true. Those are different things. Sometimes people call that skepticism, but in my opinion that term is confusing and not very good. There is a difference between skepticism and the synthetic position. You can also be skeptical and still accept something as accepted, depending on the cases. Fine, true. You’re right that a person can treat certain arguments skeptically and others in what we’ll call this synthetic way. I call the third one synthetic, never mind why; it’s a Kantian term. Fine, but if I’m dealing with a particular argument, there are three possibilities. Okay? Now, in the history of our civilization too, you can distinguish these three stages. The first stage we’ll call the dogmatic stage, the stage of childhood. Pagan periods, where if the sorcerer said that in order to bring rain you need to dance around the fire and chant some mantra or another, then that’s what people did and assumed it was probably true; they didn’t think whether it was true or not. Then comes the rebellion of the Enlightenment. Wait — just because the sorcerer said it, who says it’s true? Maybe it isn’t true. You can connect this to ancient Greece, which revolts against religious traditions, against various things — yes, logical thinking, scientific thinking that begins to take shape in ancient Greece. The Sophists? Exactly. That’s the beginning of the age of Enlightenment on the historical axis, not the personal-biographical axis. Earlier I described the personal biography of an individual and how he matures; now I’m talking about the structure of our history. So that’s the beginning of the Enlightenment age. This age of Enlightenment, in my estimation, ends in the middle of the twentieth century. In the middle of the twentieth century there’s a very, very big crisis. The first half of the twentieth century is dominated by a philosophy called positivism. Positivism is the view that we accept only things for which we have a precise definition and proof — things that are certain. That’s really the height of the Enlightenment. Okay? Smart people, but thinking like little children. So that’s the point — and some tie it to the Holocaust, never mind — for one historical reason or another, it broke. It broke. Why? It broke because people realized that you can’t reach anything by proof. And for many people this led to a tremendous crisis around the middle of the twentieth century. Because people had spoken about the science of law and the science of morality and science — that is, science would lead us to all truths and solve all the universe’s problems. Today we are much more sober. We understand that science solves no real problem. It gives us knowledge in many domains, but it solves none of the real problems we wrestle with. And that created a very, very big crisis approximately in the middle of the twentieth century. Of course I’m generalizing very broadly, but roughly speaking it’s around the middle of the twentieth century. And from there emerged three channels of maturation, and all three can be seen in the world. One channel is postmodernism, which replaced modernism. Modernism reached its peak in positivism. There is truth, science will lead us to it — that is modernism, whose peak, whose most extreme form, is logical positivism. Against it, people may be surprised, but suddenly fundamentalism sprouted, which we still encounter today. Okay? Why? Because there is a kind of need to restore certainty. Exactly. Meaning, the positivists in the end remained with the adolescent’s two assumptions and reached the conclusion that there is no truth, and skepticism, and everyone with his own narrative. Right? What is logic? If all you accept is logic, then all you accept is the fact that the conclusion follows from the premises. You are not willing to claim that the conclusion is true. There you have it in its pure form. Everyone has his own narrative, everyone his own foundational assumptions, and therefore his conclusions too are as true as the other’s, because he is consistent with his assumptions. That is basically skeptical, postmodern maturation, which began to flourish in the middle of the twentieth century. There were buds earlier, but that’s when it crystallized. The reaction to that was fundamentalism. Because fundamentalism says: wait, I can’t prove anything, and with science I can’t get anywhere, but I’m not willing to give up certainty. So apparently the Sages, religion, or traditional thoughts of one kind or another — and there emerged another type of maturation: fundamentalism. And there people can slaughter others in the name of al-Baghdadi’s pure truth, or things of that kind, because they’re unwilling to submit the truths they advocate to critical examination. Incidentally, we too have fundamentalists, not only Muslims. Thank God they murder less. They do other problematic things, but they murder less. But this kind of fundamentalism exists among us as well. On the contrary, many identify religious faith with fundamentalism. Religious faith, in its essence, does not stand the test of critical thought, according to this view. And therefore, in many people’s eyes, religious faith is almost synonymous with fundamentalism. I disagree with that, but that’s how many people think. And the third maturation is the synthetic maturation, which says: okay, I can’t arrive at certainty, because I do not accept fundamentalism; everything can be submitted to critical thought. Proof won’t bring me certainty because there are always foundational assumptions about which I cannot say anything certain. But since I can’t reach certainty, I’m willing to lower the bar and say: okay, then I’ll settle for less than certainty. If it makes sense, if it’s common sense, all right, that too is acceptable to me. I give up the assumption that only what is proved is acceptable. I am willing to accept things that are not certain. That is synthetic maturation. Okay? Now, since this map is not clear to many people, they get very confused, and many times when they in fact hold a synthetic view, they think they are postmodern. Because they call this too skepticism, as you said. They say, well, if there’s no proof for it, then it’s not certain that it’s true. But they still fight for what they think. Which means they aren’t really skeptics. They’re just mistaken in thinking that if I’m not certain, it means it’s no truer than its opposite. No. It could be eighty percent, too — not everything is fifty-fifty. Sixty, seventy, I don’t know. Not every doubt is fifty percent. So what happens in the struggle taking place in the world today — suddenly you see how fundamentalism and postmodernism form a coalition. Incidentally, this is really these very days; I didn’t even think it would connect so directly to these very days. Right, the American universities, which are basically the purest expression of infantile postmodernism, are creating a coalition with infantile and more dangerous fundamentalism. Now, what is this coalition based on? On the fact that both sit on the same assumption: that only something absolute is acceptable. They disagree about the second assumption, and therefore they have no way to deal with each other. Postmodernism has no way to deal with fundamentalism, because it too agrees that if something is certain, then it is acceptable. It only says: I don’t believe in your God, so from my point of view I don’t see why this is true. If you believe, then you’re justified in murdering, and you’re justified in… yes, everyone has his own method. So there is no way to deal with fundamentalism from a postmodern point of departure. The only approach that can deal with it is the synthetic one, which says: true, I’m not certain, but I still think you are wrong. And if you are wrong and harmful, then I will fight against you in the name of my truth, which is not certain. But I do not identify truth with certainty. So what if you’re certain? I’m not certain. I think you’re certain only because you’re deluding yourself. And that doesn’t mean I don’t have a truth of my own, that I won’t fight for my truth and against your truth, and say you are wrong and I am right — which today is almost a dirty word, to say such a thing. They immediately tell you: you’re being paternalistic; you think you’re right and he’s wrong. A kind of stupidity I simply can’t understand. If I think X and someone else thinks not-X, then by definition I think he is mistaken. So saying he’s mistaken is paternalism? Saying he’s mistaken is simply logic. If I say he is certainly mistaken, that’s something else — that’s just arrogance. Who told you? Maybe you’re wrong. I can’t be certain I’m right. But if I adopt position X, even not with certainty, then obviously anyone who says not-X is, from my perspective, mistaken. That’s simple logic, not paternalism. So in fact, I think that in our world today we really see postmodern helplessness in the face of fundamentalism. And today, because of this, there is a kind of sobering up in the world. There is a sobering up whose basis lies in the philosophical analysis I’ve laid out for you here, even though it looks like politics and propaganda and things wrapped in a huge amount of noise. But when you analyze a complicated problem, even in science, you always have to understand its pure philosophical root. After that it gets dressed in lots and lots of noise. But when you look at the root, things become clear. And then people understand where one can really get to, how far one can get with this postmodern narrative approach that supposedly contains and allows everyone to be right, everyone equally intelligent, everyone equally right, and the wolf shall dwell with the lamb. And as Ben-Gurion said: if the wolf is going to dwell with the lamb, I want to make sure I’m the wolf. Even when the prophets’ vision is fulfilled that the wolf shall dwell with the lamb, I’ll make sure I’m the wolf in that picture. That’s what they forgot to make sure of. They want the wolf to dwell with the lamb, but they forgot that the wolf doesn’t want to dwell with the lamb; only the lamb wants to dwell with the wolf. The wolf will devour him. Okay? Therefore postmodernism has no way to deal with the fundamentalist outlook, because it recognizes the fundamentalist outlook as legitimate. There’s also the example that postmodernism itself is dogmatic. Yes, yes, obviously. It’s built in. That already belongs more to the realm of psychology. It belongs to the realm of psychology: human beings cannot exist without absolute truths, so the absence of absolute truth itself becomes an absolute truth. And those are psychological difficulties; for that, I think, one needs medication. There’s nothing else to do with it. They don’t take it, right — they don’t take it, that’s the problem. Anyway, my claim is that… and from here we begin to touch on the question of dispute and truth. Or maybe before that, before I get there. Let’s go back for a moment to the personal, biographical axis, not the historical one. The historical axis is civilization as a whole; the biographical one is about a private individual maturing from child to adolescent to adult. When the adolescent asks his father, “who says? prove it,” what does the father answer? After all, he has no proof. Fine, but it makes sense. True, I have no proof, but it’s true; accept it. I have experience, I don’t know — it’s true. Okay? Accept it. Does the child accept that? The adolescent? Certainly not. Everyone shaves on his own beard; no one shaves on someone else’s beard. Until he experiences the crisis himself and reaches the point of maturity, he won’t emerge from his adolescence. Therefore even if his father tells him till tomorrow that there is no proof but that’s how it is, it makes sense, it’s my common sense — he’s not interested. Why? Because remember: what has the adolescent already gone through? What stages has he already passed through? The stage of childhood. So when he sees his father, he identifies him with the child he himself once was. Ah, you accept things without proof? Yes, I used to be like that too. When I was a small and foolish child, I also accepted things without proof. But now I’m already smart; I’ll accept only things for which I have proof. He identifies the father’s third stage — synthetic adulthood — with the first stage, because only that first stage he himself has already passed through; he doesn’t yet know the third stage. In order to understand that there is a third stage and that it differs from the first, one has to go through it. You don’t understand it intellectually; you have to experience it yourself, go through the crisis, come out of it somehow, and then, if you choose to mature synthetically, suddenly discover that it’s different. But — and this is the sting in the whole move — why really is the child not right? What is the difference between the first and third stages? After all, the child too accepts things without proof, and in the third, synthetic stage you also decide to accept things without proof. So isn’t that dogmatism? What exactly is the difference? After you understood from experience. Fine, but after I understood, in the end I went back to being dogmatic. Experience — from practical experience, as if a cognitive process, you go through some cognitive process in which you… Fine, so I went through it. Okay, and now what? So in the end I reached the dogmatic stage. After a process. Okay, but that process is a circle. I came back to the same point from which I started. No. Ah — two differences, and they are connected. First difference: for the child, this is absolute truth; he doesn’t understand that he is accepting something that is not certain. The synthetic adult understands that it isn’t certain. He went through a process, encountered questions, and understood that he has no proofs to solve those questions. Therefore for him it is not absolute. He decides to accept it even though it is not absolute. That is one difference. The second difference, connected to the first: when the adult… he will also test himself, and he may come to the conclusion that he really was wrong, because it isn’t certain. Cross-check it with other places, examine the implications, look at the various contexts, examine its consistency with other assumptions of yours. You may conclude that your conclusion was not correct. Right — connected to the fact that it wasn’t certain. You are willing to test that claim because you know it is not certain. And here there is a difference from the child. The child accepts it because father said so. The adult accepts it because he thinks it’s true. But alongside that, he always preserves some level of healthy skepticism. He always says: look, but it isn’t certain. It needs checking. It may be true, it may be false. There are questions here. That’s the difference. Now, how does he check? If the checking is the difference between child and adult, what is this checking? Notice: this checking cannot be proof. Because proof — we already agreed — doesn’t exist, right? So what is the nature of this checking that the adult performs? How do you approach certainty? How — what — how? It isn’t logic, so what is it? How do you do it? How do you approach certainty? Look, there’s a field that has a bad reputation. It’s called rhetoric. Rhetoric. It’s the art of presenting arguments. In the United States, at least it used to be, I don’t know if it still is, a required subject in high school. Too bad it isn’t like that here. Now many people have a tendency to view rhetoric negatively. Rhetoric is basically treated as synonymous with demagoguery: how do I mislead you and present arguments in a way that confuses you and forces you to accept them even though they’re false? Without arguments. And that’s not correct. Demagoguery is misleading rhetoric. But there is also non-misleading rhetoric. There are arguments that aren’t logic, but they still help me see that a certain claim makes sense — that it’s reasonable. And therefore I accept it. And this is an art; it’s not simple to do rhetoric. It’s an art closer to the craft of a writer than to that of a logician. Because in the end there are two ways to relate to the question of how I adopt foundational assumptions. After all, that’s the difference, right? The adult adopts foundational assumptions without proof because it makes sense, and from there onward logic adds the conclusions. But how does he adopt the foundational assumptions? By intuition, I’ll call it. From his intuition. Now what is intuition? That’s a very interesting question. There are people who think intuition is really logical thinking, just unconscious. It happens inside us; we skip the steps, but basically it’s like autopilot. Right? When I drive a car and don’t notice what I’m doing at all, but I arrive at the place and signal and turn correctly and everything is fine and nothing happens to me on the way. Because I really do everything — I’m just not conscious of it. And there are people who think intuition is basically logical thinking on autopilot. You’re simply not aware of all the calculations you’re making because you’re doing them in your unconscious. And you’re doing them. I think that’s wrong. Why is it wrong? Because if it were a calculation, then once again the calculation would have foundational assumptions. A calculation is a logical argument. Where did those foundational assumptions come from? Also from a calculation? How does it begin? Therefore I say — and there is much to elaborate here, I don’t want to go into it because it would bring us into Kant’s philosophy — but I want to claim that intuition is a cognitive faculty we possess. We have an ability to know truths about the world in a kind of immediate way. Without experience? No, without sensory experience. Not through the senses. It’s a kind of seeing — I’m inclined to call it, in Maimonides’ language, “with the eyes of the intellect.” Right, but this thing is grounded in the ability to know; it isn’t only experience. Could one call it understanding one thing from another? Call it that if you like. I see — say, I see the principle of causality. How do I know the principle of causality? David Hume already taught us that causality is not the result of observation. You cannot derive it from observation. You never see that event A is the cause of event B. All you ever see is that after event A, event B occurs. But that A is its cause, that it produced it — that you cannot see by any means of observation. So how do you know it? Some people say: fine, it’s only a definition; it isn’t really causal. It’s convenient for us to speak that way. And I want to claim: no, I know it because I see it — not with my eyes, not with one of the senses. I have some observational faculty about the world called intuition. No, that doesn’t mean “because”; it means it comes together. Right. So the “because” here is a result of my assumption that there is a principle of causality and that there are causal relations between events. But I bring that assumption with me from home. It’s not the result of observation. So who says it’s true? My intuition says it’s true. I simply see in the world — “see” not with the eyes but with the eyes of the intellect — that this is so. This is the capacity to see general truths directly. That’s how we arrive at laws of nature, that’s how we arrive at all sorts of things — I won’t now go into issues in the philosophy of science. What? No, I’m not saying experience plays no role. I’m saying experience does not do the job by itself. I’m saying it’s not an analytic capacity and it’s not just experience either. What? Are we born with intuition? What? Was I what is called a tabula rasa? Was I, say, a blank slate. Okay, right. The way to refine that intuition — correct. It can be developed through experience, but it is a faculty built into us and not acquired from experience. The ability to draw conclusions is a faculty implanted in us. It is not the result of experience. Experience relies on the ability to draw conclusions from facts. But where does that ability come from? It itself is an intellectual faculty. Incidentally, this faculty I call faith. Faith not necessarily in God. Intuition is synonymous with faith. But when you say you believe in God, people think that means some subjective feeling — I feel there is a God. That’s a feeling, okay, but who says there is a God just because you feel it? So what if you feel it? Everyone feels what he feels according to how he’s built. I claim that when I say I believe in God, I mean I have an intuition that God exists. That is a factual claim. It is not a report on an emotional state. It is a factual claim. Just a second. But that factual claim is not necessarily based on a logical argument. And even if it is, the logical argument has foundational assumptions, and in any case it starts somewhere. And where do I get those foundational assumptions? From this intellectual or cognitive faculty that I call faith or intuition, which in my eyes are synonyms. That’s why when people say faith is an alternative to logic, that’s true, but it’s often presented incorrectly. Because faith is not something that belongs only to the religious world. In science too there are beliefs. The law of causality is a belief. Therefore faith is not an alternative to rational thought. Faith is flesh of the flesh of rational thought. Without faith there is no rational thought. Faith gives me the foundational assumptions, and logic derives from them the conclusion. Therefore in every field in which we employ rational thought, we begin with some set of beliefs. Belief in God is belief in God — one particular belief. We have beliefs in other things too. Isn’t an axiom identical with a belief? An axiom is not identical with a belief. If you treat the axiom as a true claim, then that is a belief. Someone else can say: an axiom is arbitrary; I use it in order to play a game. But I don’t believe it is true. When I state a certain axiom I’m not making a claim about the world. Okay? So if we’re speaking about intuition, and the intuitive faculty can lead to God in this way, then what would an atheist who is very educated say? What would someone who doesn’t know relativity say? An educated person. An educated person who doesn’t know relativity. What would he say? He needs to learn relativity. Right? The fact that you have intuition within you doesn’t mean that now you automatically know how to use it, or that you’ve already extracted everything you can from this tool — just like the safe I mentioned at the beginning. Someone who knows the assumptions of geometry can derive from them all the theorems, but he won’t derive all the theorems without the teacher’s help. That is true of science, true of faith, true of everything. Okay? Fine, we’ll stop here. It’s time for Mincha. Will this class go up on YouTube or something? Yes. All the classes are recorded and uploaded to Moodle. What’s it called? Dispute and Truth. But it will also appear on the course Moodle site and on my site under the series Dispute and Truth. Yes. Thank you very much.