Artificial Intelligence – Lesson 7 – Rabbi Michael Abraham

[Rabbi Michael Abraham] What is thought בכלל? What mechanisms of inference or thought do we know? In logic it’s customary to divide things into three forms of inference. One form is from the particular to the general; that’s called induction. Another form is from the general to the particular, deduction. And a third form is from one particular to another, or from one generalization to another on the same level, and that’s called analogy. And, simply speaking, those are basically the only three kinds of inference there are. What else could there be besides from particular to general, from general to particular, or from particular to particular and general to general? It doesn’t seem like there’s any other option. Actually, beyond those three people also tend to add what’s called abduction. Abduction is basically a move from particular examples to a theory. In other words, induction is a move from particular examples to a general law of which those examples are particular cases. Abduction is a move from particular examples to a theory from which the examples can be derived. They’re not particular cases of it, but the examples can be derived from the theory. Those are basically the forms of inference. What’s the difference between them? So logic, in the most basic sense, deals with deduction. Basically, the rules of inference from the general to the particular, or necessary rules of inference — that’s what’s called deduction. If all frogs are green and this thing is a frog, then this thing is green. That’s a deductive argument; the conclusion follows necessarily from the premises. In other words, it’s a necessary argument. Analogy would be something like: this thing is a frog and it’s green. That other thing is also a frog, so presumably it too is green. That’s analogy, right? I’m basically inferring from this frog to that frog. Here it’s obvious that there’s no certainty in this conclusion; it’s a speculative inference. The comparison may be right, and it may be wrong, but it’s not a certain inference. And induction is: this frog is green, and maybe this one is green too, so probably all frogs are green — that’s a generalization. And here too, this inference is not necessary, because maybe I generalized incorrectly. It could be that these two examples are green frogs, but maybe the other frogs, or some of the other frogs, are not green. So again, there’s an element of speculation here, and the inference is not necessary.

Now the question is what exactly the relationship is between these forms of inference. When I want to talk about analogy and induction, then usually we… let me put it this way: suppose I said, this frog is green, that one is also a frog, therefore that one is green too. What is that comparison based on? Why do I assume that if this frog is green, then probably that one is green too? After all, I don’t know anything about that object except that it’s a frog. And if I make the comparison between it and this frog, then I’d really be doing the same thing with any frog. Or in other words, implicitly I’m assuming that all frogs are green. Right? In other words, when I say this is a frog and it’s green, and that’s also a frog and it’s green, I’m really inferring this from the mere fact that it’s a frog — not from something specific to this animal or this object, but from the fact that it is a frog. So analogy from one frog to another actually assumes induction in the background. It assumes that the example I saw here is an example that holds for all things of the same kind, all objects of this type. So really there’s induction here. That’s on the one hand. On the other hand, when I see that this frog is green and infer from that that all frogs are green, how do I make that inference? Why assume from this frog to all frogs? Presumably, I say: this frog is green, so that one is probably green, and that one is probably green, and that one is probably green, and that one is probably green, and that one is probably green — so, apparently all frogs are green. Or in other words, when I do induction it’s basically—

[Speaker B] We can’t hear.

[Speaker C] Rabbi, we haven’t heard you for the last minute or so, maybe a bit less.

[Rabbi Michael Abraham] Wait, can you hear now? Can you hear me?

[Speaker D] Yes, yes, now—

[Speaker C] Yes.

[Rabbi Michael Abraham] Okay. What I’m saying is that when I do induction — this frog is green, therefore all frogs are green — induction, in practice, is based, it seems to me at least, on a great many analogies. This frog is green, so that one is probably green too, and that one is probably green too, and that one is probably green too, and so on, and therefore all frogs are green. Or in other words, when I do induction, it’s basically the sum of a lot of analogies. So it turns out that the difference between analogy and induction is not clearly something that really exists. In other words, at the foundation of an analogy there sits a hidden induction, but induction itself is based on a great many analogies. So it’s not entirely clear that we really have two different forms of inference here. That’s regarding analogy and induction.

Now let’s move for a moment to deduction. In deduction I say that I infer a conclusion from the general to the particular, because if all frogs are green and this thing is a frog, conclusion: this thing is green. Why is that really necessary? Why is it that if I accept the premises I have to accept the conclusion as well? Here’s the thought experiment I often use to illustrate this. Suppose some alien comes before you, from who-knows-what planet, some kind of Little Prince. He arrives here and lands, and you say to him, peace be upon you, Your Highness, welcome to planet Earth — you know, all human beings are mortal. Meaning, he doesn’t know human beings, it’s new to him, so I’m introducing him to the pleasant side of humanity, namely that they’re mortal, meaning there’s light at the end of the tunnel. That’s point one. Point two, I tell him: Socrates is a human being. Okay, the prince says, fine. And I say to him: so then you understand that Socrates is also mortal, meaning he’s destined to die? So the prince opens his eyes wide at me and says: what are you talking about? How did you infer that conclusion? Why do you think that’s true? I tell him: no, you didn’t understand. You agreed that all human beings are mortal. Yes, he says, if you say so, why shouldn’t I believe you? And then I say: and you agree that Socrates is a human being. Okay, you probably know him, so you probably know. Fine, so from here it follows that Socrates is mortal. No — why? How does that follow? I didn’t understand. I accept the two premises, but I’m not prepared to accept the conclusion. Why? Why do you think that someone who accepts the premises also has to accept the conclusion? Here’s a thought experiment to challenge us: what would we do in order to explain to this fellow that he must accept that Socrates is mortal? What would force him to accept that conclusion? The conclusion by itself he doesn’t need to accept. He needs to accept it only if he accepts the two premises — that all human beings are mortal and that Socrates is a human being. But assuming he accepts those two premises, he must also accept the conclusion. And the question is: fine, he must — but he doesn’t. He doesn’t accept the conclusion. The question is, what can I do to explain to him why he must accept it? What exactly compels him to accept the conclusion?

So the route I would try to take is this: I’d tell him, look, when I say to you that all human beings are mortal, what exactly did I tell you? I basically told you that Moshe is mortal, that David is mortal, that Yossi is mortal, that Muhammad is mortal, and so on — ten billion people, each one of them is mortal. The abbreviated linguistic way of saying that is to say that all human beings are mortal, but that’s only a shorthand. What I really told you here is ten billion statements, okay? This one is mortal, and this one is mortal, and this one is mortal. Now that’s the meaning of the statement “all human beings are mortal.” So you understand that if Socrates is a human being, then one of those ten billion statements is that Socrates is mortal, right? If you accept that all human beings are mortal, then you accept that David and Yosef and Muhammad and Socrates and so on — every human being, each one individually, is mortal. So one of the statements you already agreed with me on is that Socrates is mortal. So now how can you deny the conclusion when I tell you Socrates is mortal?

So what method did I use to prove this to him? What I basically said to him was: look, if you accepted the premises in their general form, then you already accepted the conclusion as well. The conclusion is already accepted by you. When I stated the conclusion, it wasn’t supposed to be something new from your perspective. Once you accept the premises, within them you’ve already accepted the conclusion. And therefore you have to accept the conclusion that Socrates is mortal, because it is contained within the premises you already accepted. It’s not something new. In other words, the conclusion contains no new information beyond the information found in the two premises that you accepted. And if you already accept that information, then you’ve accepted it — that’s it — so it’s true according to you as well. Therefore you have to accept the conclusion. In other words, what I’m really telling him is that the reason he has to accept the conclusion is because there is nothing new in the conclusion. The conclusion didn’t tell you anything, any information, beyond the information you accepted when you accepted the premises. The conclusion doesn’t contain additional information. Okay? Therefore you have to accept it.

[Speaker E] But that’s not deduction, by the way. What? What you’re saying now isn’t deduction.

[Rabbi Michael Abraham] Why not?

[Speaker E] No, that’s analogy. A somewhat different analogy, but I’ll explain what I mean. Usually deduction is taken to be a logical chain where one thing follows from another. For example, if I meet a person with a yellow mustache, okay? I don’t need any premise in order to understand that he smokes. That’s deduction. When I meet a person with dirty shoes—

[Rabbi Michael Abraham] It seems to me that we’re using completely different language. What you’re saying is absolutely not deduction, and what is usually called deduction is what I said. Deduction is defined as inference from the general to the particular. What you described is everyday conclusion-drawing. His mustache could be yellow for reasons other than smoking — because of something else, or someone else smoked next to him, or all kinds of other things. That may be a sensible conclusion, but it’s not necessary. In other words, someone who accepts the premise that the mustache is yellow is not forced to accept the conclusion that the person smokes. It may be likely that he smokes, but it isn’t necessary. Deduction is a necessary argument. It’s an argument whose conclusion follows necessarily from the premises. You cannot accept the premises and reject the conclusion. And that’s what happens when you infer from the general to the particular. If you know that all human beings are mortal — that’s the general rule — then in the particular case, Socrates, who is one of those human beings, is also mortal. In other words, if you accepted the premises, then you must accept the conclusion. That’s what is usually called deduction.

[Speaker E] I think here and there it’s analogy, because all human beings—

[Rabbi Michael Abraham] Let’s agree not to argue about words. I’m defining the terms now as I’m going to use them, okay? We can argue about the terms afterward. So the claim is that the necessity found in deduction lies in the fact that deductive inference does not add information for me. There is no information in the conclusion beyond what I had in the premises, beyond what is latent in the premises. Deductive inference, all it does is bring out information that was latent in the premises; it doesn’t add information beyond what was latent there. And that is exactly why this inference is a necessary inference. This inference is necessary because you already accepted the information; you can’t fail to accept the conclusion if it’s included within the information you already accepted. The fact that deduction is necessary lies in the fact that it does not introduce new information for me. That’s the reason. Can I take a question?

[Speaker F] What? Can I ask a question? Say, I’m trying to prove a theorem in geometry, and I start from Euclid’s most basic premises possible, and I get to some theorem about similar triangles and prove it. How do you want to tell me that no information was added in such a move?

[Rabbi Michael Abraham] Right, that’s what I said. Okay.

[Speaker F] So what did I get from that move, then?

[Rabbi Michael Abraham] What do you mean, “not information,” then? You extracted more and more information that you already had when you accepted the axioms. It’s not new information; it’s information you already had. Deduction, to my mind, is like a key to a safe. You have a locked safe full of lots of jewelry, diamonds, things like that, but you don’t have a key. If you don’t have a key, that doesn’t mean those things aren’t yours — they are yours. But they’re inaccessible. Now someone comes and gives you a key. So you open the safe and take the things out and can use them. Did you acquire new things? The answer is no. Those things were yours beforehand; they were just inaccessible to you. The key helped you access things that were yours all along. Deduction, or mathematics, or logic does exactly that. Basically, someone who holds the premises has within those premises all the information that follows from them, all the propositions that will come afterward — except that the information is locked in the safe, inaccessible to me. It’s hard for me to get those theorems out of the axioms, whether in triangles or whatever other theorem it may be. Learning geometry is basically extracting more and more information from the premises, not adding new information. And that’s why geometry is a branch of mathematics and not of science. Mathematics is something necessary, something that can’t be refuted; you can’t bring an experiment that will show it’s wrong. Why? Because if you accept the premises, the conclusion is already inside them, and therefore it can’t be that you accept the premises and not the conclusion.

By contrast, in analogy and induction these are two forms of inference in which the conclusion contains information beyond what was in the premises. Suppose I say: this frog is green — that’s my premise. Second premise: that one is also a frog. Now I say, fine, conclusion: that one is green too. The conclusion that that frog is green was not inside the premises. My premises dealt with this frog being green and that one also being a frog. That’s what I knew in the premises. Those two premises do not contain within them the information that that other frog is also green. That’s information beyond what was contained in the premises. Therefore the analogy has added information beyond what I had at the beginning, beyond what I had when I held the premises. And induction is of course the same. This frog is green, and that one is green too, so all frogs are green. You understand that my premises, which dealt with these two frogs being green, do not contain the information that all frogs are green. That is obviously additional information beyond what I had when I started. So induction too is a process, or an inference, that adds information beyond what I had initially. Both analogy and induction do that. Deduction only reveals information that was already in my possession.

Now there’s that joke about the hot-air balloon. I always use it in this context to illustrate the point. Two people lose their way in a hot-air balloon. They see below them a field where someone is plowing. One of them shouts down, tell us, where are we? They’ve lost their bearings, right? Where are we? The fellow looks up and says: above my field. So the guy in the balloon says to the other one with him: that fellow down there is definitely a mathematician. Why? First, what he says is absolutely certain. Second, it doesn’t help us at all. Those two aspects, or those two characteristics of the mathematician, are really two sides of the same coin. Because why is mathematics absolutely certain? Because it doesn’t help us at all. It is absolutely certain precisely because it doesn’t add any new information. Anything that adds new information cannot be certain. When you make an analogy — this frog is green, therefore that one is green too — that’s not certain. Maybe yes, maybe no. There’s a certain degree of uncertainty here. Why? Because it adds information beyond what I had before. Who says this added information is correct? There’s an element of speculation here. So inferences that add information are not certain inferences, not secure, not necessary. A necessary inference is one that does not add information.

So now I can divide fields of knowledge, let’s say, into fields that deal with adding information — what’s usually called science. Science deals with adding information, and therefore science mainly uses analogy and induction. Those are the tools by which we add information for ourselves. By contrast, mathematics and logic deal with deduction, and so they really do not add new information, but only extract more and more information — information out of the premises that I accepted from the outset. They simply add for me, or bring out for me, more and more information. I wouldn’t have believed there was so much information inside my safe, but it turns out there is. When you open it, you discover it contains far more information than you thought. But it’s all there. In other words, I owned that information even when the safe was closed. I just didn’t know it, or it wasn’t accessible to me. Okay? So the difference between mathematics and logic on the one hand and science on the other is that mathematics deals with extracting existing information, sharpening and extracting existing information. Science deals with adding information, accumulating information, and therefore science deals with analogies and uses analogies and inductions, whereas mathematics and logic use deductions. Okay, that’s basically the disciplinary division between them.

Now I want to say one more thing. Really, these three forms of inference are not actually three forms of inference. Let’s think again for a moment about the analogy from this frog to that frog: this frog is green, that one is also a frog, presumably it too is green. I already said earlier that at the basis of this analogy there is really a hidden induction. Basically I’m claiming that all frogs are green, and in particular that frog is also a frog, so it too is green. Okay? But really what I’m doing in the subtext is induction to all frogs. And if you look at it that way, you’ll see that the analogy I make between this frog and that frog is composed of two steps: one step of induction — this frog is green, therefore all frogs are green — and the second step is deduction: if all frogs are green, then in particular this frog is green. A move from the general to the particular. You see what happened here? In other words, I made an analogy from one particular to another, but how did I do it? In two steps: the first step was induction from the particular to the general, and the second is deduction from the general to this particular, which is included in it. So induction and deduction are simply stages in the performance of an analogy. When I make an analogy, it’s just in two stages: induction and then deduction. If I say Moshe is mortal, therefore Yosef is mortal, I’m really saying Moshe is mortal, therefore all human beings are mortal — that’s induction — and if all human beings are mortal, then in particular Yosef is mortal, and that’s deduction. Okay? So really when I add information, I’m using analogies. Analogies add information for me. This process of adding information is built out of induction — which is the stage where I accumulate the information — and deduction, which is the stage where I merely clarify something out of the information I accumulated. It’s not a process that adds information, but one that refines a particular piece of information that is already contained within the information I accumulated. Okay? So for example, when I discover a new scientific theory — a new scientific theory — then I’ve taken a step of induction, or more precisely abduction, but never mind, right now I’m not making that distinction. I take a step of induction; from several particulars I build a theory that deals with very many facts. Now I can take that theory and ask what will happen in this specific case — what the speed of the object will be under such-and-such conditions. Okay? So I use the theory, do the calculations, and discover what the speed of the object will be. What did I really do here? In creating the theory I took a step of induction — I added information. Once I have the theory, when I perform a calculation on a particular case on the basis of that theory, no information is added; I extracted existing information that was already latent within the theory. I just had to make an effort to extract it from there, but it was already there. No new information was added here beyond what the theory already contains. Okay? So in fact the essential scientific step is precisely the step of creating theories. Testing theories on this case or that is a deductive step; that, in effect, a mathematician could also do.

Now the point is that Popper — Karl Popper, a well-known philosopher of science from the twentieth century — defined a scientific theory as a theory that can be subjected to a test of falsification. Say you claim that all frogs have four legs. But, for example, the theory that God exists is a theory that cannot be falsified. Because I can’t think of an experiment I could perform whose result could refute the claim that God exists. In other words, that claim cannot be subjected to an empirical falsification test, and therefore according to Popper it is not a scientific claim. Popper argues that a scientific claim is one that can be subjected to an experiment of refutation. Okay? He also talks about the fact that theories cannot be proved, only refuted. Yes, the theory that all ravens are black — you can never prove it; you can never know whether you’ve seen all the ravens. Maybe there’s one you haven’t seen.

[Speaker E] What about evolution?

[Rabbi Michael Abraham] But wait — but you can refute it. Why? Because if you find one raven that’s pink, then you’ve refuted the theory that all ravens are black. So Karl Popper says that the definition of a scientific theory is not a theory that can be proved, but rather a theory that can be refuted. A theory that can be refuted is, in a certain sense, a theory that contains information, that adds information for me. Then I can test whether the information it added is correct. I’ll perform an experiment and check. But a theory that doesn’t add any information in the sense that I can’t measure it — that can’t be refuted. So it’s also not scientific. Because in order to refute it I need to take some bit of information, perform an experiment, and see whether that information is correct or not. But if it can’t be subjected to observational falsification, then that basically means that there is information in it — the statement that God exists is information — but it’s not information in the scientific sense, in the observational sense. It’s not information I can observe. Therefore, in a certain sense, you can say it doesn’t add information — not scientific information, in this case. So a scientific theory is a theory that can be refuted. By contrast, mathematics cannot be refuted. Yes, if I want to refute the proposition that five plus five equals ten, can anyone suggest an experiment that could refute that theory? Meaning, if it failed, it would turn out that the theory is wrong? There is no such experiment. When is the place to say why not? Yes, there is such an experiment. Take five nuts and put them onto a plate. Take another five nuts and put them onto the plate too, and count how many nuts you have altogether. If you get thirteen, then apparently five plus five is thirteen — in other words, you’ve refuted the claim that five plus five is ten. So here we have an experiment that can refute the claim that five plus five is ten. So apparently the theory that five plus five is ten is a scientific theory. But that’s not true, because even if I assume I do this experiment and I discover thirteen nuts on the plate, we would all agree that the conclusion would not be that five plus five isn’t ten. The conclusion would be that there was apparently some mistake in the experiment. I will never agree to draw from experiments like these the conclusion that five plus five does not equal ten. It simply won’t happen.

[Speaker G] More than that, it’s also true in ordinary experiments. What? It’s true in every experiment too — you don’t immediately reject the theory just because—

[Rabbi Michael Abraham] Yes, not immediately. But here it’s never.

[Speaker G] Maybe you’d test it a billion times and in the end say that five plus five isn’t ten — strange world, but…

[Rabbi Michael Abraham] So I’m betting no. It won’t happen. Not even a billion times. Yes, even ten billion times. And I’ll also explain why. Okay. I’ll maybe explain one more thing. I once made this point when I started teaching a recitation section in mechanics. I was at Bar-Ilan in my M.A., and I was teaching a mechanics course. So I asked the students there — after talking a bit about what a scientific theory is — whether in their view five plus five equals ten is a scientific thesis. Can it be refuted? Yes, and then I gave them the nuts example, or whatever example I used, but something like that, and I told them I didn’t think it could really be refuted. Then I said, look, I’ll give you an example. Suppose I take some object and apply to it a force of five newtons to the north. And now I apply another force of five newtons to the west. Now I ask, what is the total force acting on the object? The total force acting on the object is seven point something newtons, right? Not ten. Even though there’s a force of five plus a force of five, they’re not on the same axis, right? They’re not in the same direction. In the diagonal direction — never mind the details right now — but the result is seven point something. Five times the square root of two. So this experiment has apparently refuted the theory that five plus five equals ten. Here’s a force of five plus a force of five, and the result doesn’t come out ten. So what conclusion do we really draw after we see experiments like this? That adding forces is not described by arithmetic addition. That arithmetic addition is not the correct mathematical theory for describing the addition of forces; instead one should use vector addition. Vector mathematics. Okay, never mind the details, but it’s a different kind of mathematics.

[Speaker H] Is the theory that there is no God falsifiable?

[Rabbi Michael Abraham] No, that too is not. By the way, I’m not… If the theory that there is no God can be refuted, that means the theory that there is a God can be proved. So what happens with—

[Speaker E] The theory of evolution?

[Speaker H] Explain, if possible.

[Rabbi Michael Abraham] If you refuted the theory that there is no God, then you proved that there is. Okay. That’s it. You can’t prove that there is a God scientifically, and therefore obviously you can’t refute the theory that there isn’t one. And vice versa. You can’t refute the claim that there is a God because you can’t prove that there isn’t one. Okay? Clear. This whole discussion is not taking place in the scientific field. Yes, none of these claims stands the test of observational falsification.

[Speaker D] Sorry, Rabbi, why can’t you refute the claim that there is no God? If the Holy One, blessed be He, reveals Himself and performs miracles, then wouldn’t that refute the claim that there is no God?

[Rabbi Michael Abraham] Yes, but you… I’m talking about a planned experiment. A planned experiment means I can design an experiment and say, let’s go into the lab now and see what the results are. Obviously, if God decides to reveal Himself and the revelation is sufficiently unambiguous, then yes, all of us will be convinced. That’s not called a scientific refutation or a scientific proof. A scientific refutation or proof means performing an experiment, and the experiment must also be repeatable. Repeatable means that it can be repeated whenever someone wants and decides to; he can repeat the experiment and get the same result. Because a one-time experimental result is not considered a scientific result, a result that can’t be reproduced.

[Speaker E] So what about the theory of evolution, Rabbi? What about it? Isn’t it falsifiable? Like here too, there too. What—

[Rabbi Michael Abraham] What do you mean, not falsifiable?

[Speaker E] It’s a theory accepted today in science, right?

[Rabbi Michael Abraham] Right.

[Speaker E] So — can it be refuted?

[Rabbi Michael Abraham] Absolutely. Good.

[Speaker E] You can also fail to refute it because we live on assumptions.

[Rabbi Michael Abraham] What do you mean, fail to refute it? If one can perform an experiment that would refute it, then it’s a theory — a scientific theory. That doesn’t mean it will be refuted, and not every scientific theory is in fact refuted. Rather, every scientific theory is one that could be refuted, meaning there is an experiment such that if it fails, it would refute the theory.

[Speaker E] But you can’t even do an experiment.

[Rabbi Michael Abraham] Why can’t you do an experiment? They’ve done many experiments.

[Speaker E] Yes, but still they haven’t achieved anything in all those experiments. I’m talking about human evolution. Not… what about human evolution?

[Rabbi Michael Abraham] Evolution in general is a very well-founded scientific theory. There are very few scientific theories that are better founded than it.

[Speaker E] The theory of—

[Rabbi Michael Abraham] Evolution has stood up to many tests. It contains some components — I wrote about this and explained it in detail — it contains some components that are not falsifiable simply because they are logical tautologies. For example, that the survivor survives, what’s called the survival of the fittest, that is a tautological component within the theory. But that’s only one component. There is genetics, the formation of mutations, and all kinds of things like that. All in all, this is a fully scientific theory. Anyway, the claim is that a scientific theory is falsifiable, and that is tied umbilically to the fact that a scientific theory adds information for us. Because falsification means showing that part of that information is not correct. That’s what it means to refute. If the theory tells me that some fact of such-and-such kind ought to be true, then the experiment I perform to check whether that fact is true or not will be the refuting experiment for the theory. In other words, if the experiment fails, the theory is refuted; if it doesn’t fail, it isn’t refuted. By contrast, a logical argument or a mathematical proof cannot be refuted. It is not open to falsification testing. Five plus five equals ten, as I said earlier, is not refutable. Whatever you may say — if I put five nuts into a bowl… and another five nuts into a plate and count and get thirteen — and let’s even say I conclude there was no error in the experiment — the most I will conclude is that adding nuts into a bowl is not described by arithmetic addition. In other words, “five plus five equals ten,” that mathematical theory, is not the right theory for describing the addition of nuts into a bowl. That’s all. But at the mathematical level, five plus five equals ten is always true; it cannot be refuted.

[Speaker G] You tested infinitely many things, a billion things, and all of them happened in just that way — that five plus five equals ten.

[Rabbi Michael Abraham] And if not, the conclusion is that arithmetic theory is not suitable—

[Speaker G] —for describing those domains. You didn’t find any one where it did, say, not yet in the billion years that you were there, the Rabbi didn’t find—

[Rabbi Michael Abraham] All those domains are domains not described by arithmetic addition.

[Speaker G] But doesn’t the Rabbi feel that it’s obvious that that’s not what would happen? If human beings had counted—

[Rabbi Michael Abraham] A billion—

[Speaker G] things and it—

[Rabbi Michael Abraham] always came out ten, it’s obvious to me that no. Five plus five equals ten, period. No fact will move that.

[Speaker G] We saw it — that’s how we see it.

[Rabbi Michael Abraham] No, not because we saw it. Because we understand it.

[Speaker G] A child, after all, isn’t taught — a child isn’t taught a theory.

[Rabbi Michael Abraham] It’s a priori.

[Speaker G] A child isn’t taught a theory,

[Rabbi Michael Abraham] The child—

[Speaker G] doesn’t learn it a priori; he learns it—

[Rabbi Michael Abraham] until he—

[Speaker G] feels it very clearly, they keep bringing him back to the change.

[Rabbi Michael Abraham] You’re mixing two levels here, but you’re in good company. In the company of a philosopher named Saul Kripke, a brilliant Jewish American religious analytical philosopher, perhaps the most important philosopher of the twentieth century, or among the most important of the late twentieth century — he really made your claim. But unfortunately he was wrong, because there’s a difference. When I teach a child with the nuts and show him that five plus five equals ten, the nuts serve as a didactic device. In other words, he does not infer the conclusion that five plus five equals ten from the nut experiment; rather, the nut experiment helps him become aware of this a priori truth, that five plus five equals ten. But after he uses that experiment and repeats it in several contexts, he understands it as a truth not conditioned by those experiments. It is an a priori truth.

[Speaker G] And if you couldn’t teach the child — if you showed him five plus five and every time it came out thirteen in all things, across the whole range of reality?

[Rabbi Michael Abraham] Then the child would answer what I—

[Speaker G] answered earlier,

[Rabbi Michael Abraham] he would still know that five plus five equals ten. At most it would take me much longer to explain it to him.

[Speaker G] How could you explain it to him?

[Rabbi Michael Abraham] I can explain it to him the way I—

[Speaker G] How can I explain what right and left are? That’s truly a big question.

[Rabbi Michael Abraham] Exactly — and in the end we do manage to explain it, right?

[Speaker G] There are other hard questions like that, and the previous question wasn’t to the point.

[Rabbi Michael Abraham] But we do manage to explain it. They’re not hard questions.

[Speaker G] Not that you feel it beforehand; it’s ingrained in our feeling from the start.

[Rabbi Michael Abraham] Not in feeling — again—

[Speaker G] You come—

[Rabbi Michael Abraham] with feeling. It’s ingrained in us not in feeling, because we know that it’s true. It’s true a priori.

[Speaker G] Did you prove rationally that right is on the right and left is on the left?

[Rabbi Michael Abraham] I didn’t prove it rationally, but you asked how I can explain it, and here I’m showing you that it can be explained.

[Speaker G] Because I simply feel it.

[Rabbi Michael Abraham] Fine, right, because it’s an a priori truth. That’s exactly the point. That’s what I’m saying about mathematics too. Fine. In any case, the point is that science stands the test of falsification, whereas logic and mathematics do not stand the test of falsification. By the way, I’m saying something here that I’ll still have to cash the check on, but I’ll say it now because it touches our discussion: computer science, what’s called computer science, really doesn’t belong to the world of science at all; it’s a branch of mathematics. Computer science. They call it computer science, and that always bothered me. There are places that also speak of mathematical sciences, but that’s nonsense. No — mathematics and computer science are not branches of science; they are branches of mathematics. Up to the point when artificial intelligence appeared. Once artificial intelligence appeared, computer science came back into the bosom of science, because the knowledge of what artificial intelligence can and cannot do is not knowledge accumulated by proving theorems; it is knowledge accumulated by experiment. And knowledge accumulated by experiment is science, even though that science studies objects made by human beings and not natural objects. That doesn’t matter — it still studies them in an empirical, observational way. Here computer science detached itself from the world of mathematics. In other words, artificial intelligence does not belong to the mathematical side. There are also some mathematical theorems there that touch on the matter, but the field as a field really belongs to science. Classical computer science — Turing machines, computability, the classical computer, and all those things — I’ll still explain all that. For now I’m just putting it on the table; that check will be cashed later. That kind of computer science is a branch of mathematics, not of science. Because these are not things arrived at by empirical experiment, by empirical measurements, but by theorem-proving. Okay, so that’s regarding the three forms of inference.

Now notice what this really means. What this really means is that when we prove some proposition, a proof is a logical argument. When we prove some proposition, we’re really showing that it is contained within the premises, the premises on which the proof is based. Okay? The proof does not add new information for me; it reveals more and more information that is contained in the premises. So for example, let’s look at geometry, a field more or less familiar to everyone, I assume. Geometry is based on four axioms — Euclidean geometry, four or five, I don’t remember anymore, it doesn’t matter, how many?

[Speaker I] Four, and the fifth is disputed, and then you can derive the—

[Rabbi Michael Abraham] If it depends on the others.

[Speaker I] Yes, right.

[Rabbi Michael Abraham] Okay, okay, so there are four. The four axioms, and from them we can derive all sorts of theorems, all sorts of propositions. Okay? Now what I’m really claiming is that since this is a field belonging to mathematics, geometry, then the theorems are not new information added to me, but information extracted from the axioms. It’s true that on my own maybe I couldn’t open that safe; I need the help of the teacher. But the teacher merely gave me a key that helps me extract more and more information from the safe. And that’s why it belongs to the mathematical side and not the scientific side.

Now I’m going back to the hot-air balloon. The hot-air balloon story basically told us that the guy plowing the field down below is a mathematician. Why? Because what he said was absolutely certain but useless to us. And I said, what does it mean that it’s absolutely certain? It’s absolutely certain because it adds no information. I also knew that I was above him, so when he says “you are above my field,” he hasn’t added any information for me. Exactly for that reason, what he said is necessarily and certainly true, right? I also see that; it’s obvious. So the reason what he said is certain is that he added no information.

Now our feeling when we study geometry is that the learning does add knowledge. Because there are things I know after I’ve learned them that I didn’t know before I learned them. Think about a sixth-grade child. I think that child understands the four axioms of geometry quite well. He also knows that between two points there passes one straight line, and he knows that parallel lines don’t meet, and so on. And still, if you ask him what the sum of the angles in a triangle is, I assume he won’t know how to answer. After he studies geometry, he’ll be able to answer that it’s 180 degrees. So apparently this learning added new information for him, information he did not previously have. So I claim no — this learning revealed to him information he already had, but wasn’t aware of, couldn’t access, couldn’t get to. But really that information was latent in the knowledge he already had in sixth grade. It just takes skill and wisdom to extract that knowledge from those axioms he already knew in sixth grade. And that is what one learns in geometry. Therefore, unlike the fellow in the field, you can’t say that geometry didn’t help us at all — that it’s absolutely certain but doesn’t help us at all. It certainly does help us. It only “doesn’t help us at all” in the very specific sense that it doesn’t really add any new information. Yes, that’s a very specific meaning of the statement that it doesn’t help us at all. It doesn’t add new information, that’s all. But obviously it helps in the practical sense. Someone who gave me a key to open the safe helped me a great deal. He didn’t add a penny to my wealth. Everything that is mine now was mine before too. But clearly he helped me greatly; now I can also use it. It’s not enough that it was mine — now I can use it. The same in geometry. All the knowledge was already there even with the child in sixth grade, but he couldn’t use it because he wasn’t clever enough to extract, from the axioms he knew in sixth grade, the theorems he learned in tenth grade. Okay? So this is basically a parable for what I said before about handing over a key. It gives you the possibility of using the information, even though that information was already yours before.

Now I want to dwell on one more point — and again with another joke that surely all of you, or most of you, have already heard from me, I use it a lot. The joke about Abraham our forefather and the hat. How do we know that every Jew should walk around wearing a hat? There’s a simple proof: it says, “And Abraham went” — it says there, “and Abraham went.” And a Jew like him surely didn’t go around without a hat. So if Abraham went with a hat, then we, his faithful descendants who walk in his ways, also need to walk with a hat. Which is what we wanted to prove. You see? Every Jew must wear a hat. We proved it — a fine proof. So why is that considered a joke? Why does it provoke laughter? What’s wrong with that proof?

[Speaker J] The assumption.

[Rabbi Michael Abraham] What do you mean?

[Speaker E] It’s begging the question.

[Speaker J] You’re assuming that every Jew has to walk with a hat — certainly a Jew like Abraham.

[Rabbi Michael Abraham] It’s begging the question. Not just an assumption, but what is begging the question? Begging the question means, let’s say, I’m trying to prove some claim and I place that very claim itself as a premise at the base of the proof I’m using. If the conclusion is itself one of the premises on which I’m relying, that’s called begging the question. You can’t assume what you’re trying to prove. Okay? That’s usually seen as a fallacy. So I have a novelty for you: begging the question is not a fallacy. Begging the question is a synonym for a valid logical argument. Every valid logical argument begs the question. When I say all human beings are mortal, Socrates is a human being, conclusion: Socrates is mortal — I said earlier that if the Little Prince is unwilling to accept the conclusion, though he accepts the premises, how do I persuade him that he still must accept the conclusion? What I really tell him is that within the premise that all human beings are mortal — if you break it down into small change — one of those coins among the billion coins is that Socrates is mortal. So you accepted that Socrates is mortal. What did I really do here? I showed him that “Socrates is mortal” was one of the premises of the argument. Right? I showed him that it is contained within the premise of the argument. Or in other words, I showed him that this argument begs the question. That’s why you have to accept the conclusion — because it’s contained in the premises. So what am I really saying? That a logical proof works, that a valid logical argument is valid, and you have to accept the conclusion because the argument begs the question. The conclusion is already contained within the premises. And if you have validated the premises, then within them you have also validated the conclusion.

[Speaker E] So you can’t fail to accept the conclusion.

[Rabbi Michael Abraham] But there’s a difference here.

[Speaker E] There’s a difference between this and that. In the kind of begging the question with Abraham our forefather — why is it considered a joke? Because you’re basically assuming what you want to prove without any prior knowledge. There’s no knowledge. We don’t know anything about Abraham our forefather, or his hat, or this. In principle, with Socrates we’re talking about all human beings being mortal. That’s empirical. It can be demonstrated unequivocally.

[Rabbi Michael Abraham] No, no, no. You’re talking about something else. You’re talking about the question of how I know the premises. But that’s not an interesting question. I’m talking only about this: once I have accepted the premises, does that force me to accept the conclusion too? Okay? And once I’ve accepted the premises — and it makes no difference why — once I’ve accepted the premises, I must accept the conclusion. Why? Because the conclusion is contained within the premise.

[Speaker E] And what, you don’t have to accept the premise?

[Rabbi Michael Abraham] You don’t have to accept the premise. But if you accepted the premise, then I also have to accept the conclusion.

[Speaker E] No — with Socrates I agree with you completely, and that’s the example people bring for deduction. But there you can’t not accept it because it’s real.

[Rabbi Michael Abraham] No, what do you mean “it’s real”? Leave me alone with “real.” I’m now talking about the question after I’ve accepted the premises. I’m not talking about the question on what basis I accepted the premises. Not interesting. I’m talking to someone who already accepted the premises. Now I ask whether he also has to accept the conclusion, and why. The answer is: because the conclusion is included within the premises. I don’t care on what basis he accepted the premises. That’s not interesting. Once he accepted them, the discussion begins. Then the conclusion is inside the premises. In other words, when you accept the conclusion of a valid logical argument, you accept it because that argument begged the question. What you were asked to prove was already placed inside the premises.

[Speaker E] In everything except Abraham our forefather.

[Rabbi Michael Abraham] In every argument, including Abraham our forefather.

[Speaker E] No, I absolutely disagree with you. First of all, the premise — the premise is a false premise. Second, the premise doesn’t lead me to the conclusion, unlike Socrates, where it does lead me to the conclusion. Abraham our forefather doesn’t lead me to anything.

[Rabbi Michael Abraham] First of all, let’s remove from the stage the fact that the premise is false. Whether the premise is false is irrelevant, because we are talking with someone who adopted the premise, and therefore that isn’t interesting. The next step: now I’ll show you how the conclusion follows from the premises in the Abraham our forefather argument, okay? It says “And Abraham went” — first premise: it says “And Abraham went,” fine? Second premise: a Jew like him surely didn’t go around without a hat. Now, conclusion: Abraham our forefather went with a hat. Third premise: every Jew must walk in the ways of our old forefather. Conclusion: every Jew must wear a hat. Which is what we wanted to prove. That’s it — valid, and in the strictest way. Not first premise, not second premise — whether the premises are true or not is not interesting; you keep coming back to that, and it’s not interesting.

[Speaker E] Yes, but I want to build logic, I want a chain.

[Rabbi Michael Abraham] But I’m explaining again—

[Speaker E] With Socrates there’s a logical chain,

[Rabbi Michael Abraham] In all of them there’s a logical chain, the same logical chain. The logical chain is a chain leading from the premises to the conclusion. The question of what the premises are based on does not interest the logician one bit. It also doesn’t interest the logician whether the premises are true. That doesn’t interest him either. What interests him is: if the premises are true, do I also have to accept the conclusion? That is the logical chain under discussion. That’s all.

[Speaker E] Repeat the first sentence you said, by the way. Whether the premise what?

[Rabbi Michael Abraham] It doesn’t matter where the premise came from, and it doesn’t matter whether the premise is true. It doesn’t matter. As a logician, one thing interests me: if you accepted this premise, even if it’s false, I don’t care — you accepted it — now I ask whether you also have to accept the conclusion. That’s all. That is the concern of the logician.

[Speaker E] I didn’t understand where the connection is between the conclusion and the premise in this case.

[Rabbi Michael Abraham] In the case of Abraham our forefather? I explained. Fine, okay, I understood.

[Speaker E] You yourself see that as a logical chain without interruption?

[Rabbi Michael Abraham] Not that I see it that way — no one can see it otherwise. It’s a valid logical chain, period. You can’t argue with that. Except that it’s a trivial logical chain. Why? Because when I say that a person like him, a Jew like him, did not go around without a hat, what am I really saying? That every Jew must wear a hat, right? That’s just another way of saying the same thing. Or in other words, the conclusion that every Jew should wear a hat was smuggled in as one of the premises of the argument. So this argument begs the question. Fine — but every logical argument begs the question. So that’s not a defect. What is true, however, is that begging the question often turns the argument into a superfluous argument, not into an invalid one. The argument is perfectly valid; every valid argument begs the question. But it turns it into a pointless argument. Why is it pointless? Because if you want to persuade someone that every Jew should wear a hat, there is no point in building your argument on that very conclusion as one of the premises. Because if he doesn’t accept that conclusion, then what help is it to use it as a premise and prove the conclusion on that basis? After all, you’re trying to prove the conclusion to him. In order to prove the conclusion to him, you have to start from premises that he himself accepts. So if he disagrees with the conclusion, there’s no point in taking that conclusion and placing it as one of the premises. It’s simply a valueless argument. Not that the argument is not correct or not valid — it is valid in the strictest way. It’s just valueless; it’s not useful, because you won’t persuade anyone with such an argument. That’s all. But in truth that’s also true of Socrates. With Socrates too it’s the same thing. That is, if you accept that all human beings are mortal and that Socrates is a human being, then you are in effect accepting that Socrates is mortal. So all I’m really proving to you is something you already accepted along with the premises. That argument also begs the question. And therefore that argument too — if I don’t accept that Socrates is mortal, then either I won’t accept that Socrates is a human being or I won’t accept that all human beings are mortal. In other words, I’ll dispute one of the premises. Therefore there’s no point in proving to me that Socrates is mortal on the basis of those two premises, because if I don’t accept the conclusion then clearly there’s something in the premises I also won’t accept. So the argument is valueless. But it is a valid argument; the conclusion follows necessarily from the premises.

Therefore, what really matters to me from all this hair-splitting is to say what in philosophy is called the emptiness of the analytic. In analytic arguments — logical arguments — these are arguments that do not add information; they are empty. In other words, you will never persuade someone who was not already persuaded of the matter in advance. You won’t persuade him of that conclusion. So what are logical arguments for? What is logic used for if it really is superfluous? Because this basically means that logic is superfluous. Every time I prove something to you with logical tools, one of two things is true: either the conclusion was already accepted by you from the outset, in which case why did I do it? Or the conclusion was not accepted by you from the outset, in which case the argument also won’t manage to prove it to you. Because it turns out I’m begging the question. So where do we use logical arguments? The answer is, for example, going back to geometry: we use them where I know that I identify with the set of premises, but I don’t know how to get all the conclusions out of them. So I use logic in order to help a person extract the conclusions from the premises he has adopted. That’s all. But logic will never persuade someone of a position he does not agree with. Never. That cannot happen with logical tools. Because if it happened through logical tools, that means that somewhere in the premises that conclusion was already there. Which means he doesn’t accept the premises, so there’s no point in using that argument.

[Speaker E] Tell me, by the way — a person who knows geometry — no, not geometry, sorry. A person who knows mathematics: when you lay the triangle out before him, yes, what is the sum of all the angles — yes? Even if he doesn’t agree from the start, through calculation he still gets there. It is possible by the logical way, by reason, by this chain.

[Rabbi Michael Abraham] He doesn’t reach that sum by calculation; you reach that sum by proof. By a proof that rests on the premises. So if you don’t accept the premises, then there won’t be a proof.

[Speaker E] Fine, so I accepted it through a proof. When we talk about Socrates — yes? — in that case, even if I don’t accept the premises, I may fail to accept them, yes? But there is some information or science or various things that I supposedly… why do I accept a premise? Because I have some attainment from what lives around me, and I as a proposition—

[Rabbi Michael Abraham] Okay, what does that have to do with us?

[Speaker E] So I wanted to say that begging the question still — everything you said, yes? — begging the question in this case is still, really, a logical fallacy.

[Rabbi Michael Abraham] Again, for the third time, you’re making the same confusion. I’m not talking about the question of how one arrives at the premises. That is not the concern of logic.

[Speaker E] I just didn’t understand where the connection is between one, two, and three. Really?

[Rabbi Michael Abraham] The conclusion follows necessarily from the premises. The moment you challenge it, you are challenging the truth of the premises. But that isn’t interesting. I’m not dealing with the question whether the premises are true. I’m dealing with the question whether the conclusion follows necessarily from the premises.

[Speaker E] There can’t be logic in this case. Fine, let’s continue.

[Rabbi Michael Abraham] Fine. So the emptiness of the analytic is a very important point for our purposes, because what it really says is that mathematical calculations or mathematical and logical proofs cannot add new information for us. Reaching new information is not done with logical, analytic tools. Okay? That is basically the conclusion that comes out of this whole matter.

Now here I want to go one step further and speak for a moment about the scientific process. And what I basically want to say is this. There are reflections of David Hume, the British philosopher — Scottish, I think — in the eighteenth century, and he reflected on several basic scientific assumptions. In particular he reflected on causality and induction. In other words, the fact that something happened several times until now does not prove that it will continue to happen later on. If the sun rose every morning until today, that does not mean it will also rise tomorrow morning. That’s a reflection on scientific induction, yes? On our ability to generalize from cases we have seen to cases we have not yet seen. That is a reflection on induction. Then there is a reflection on causality. The reflection on causality basically says: we know that there is a principle of causality saying that every event has a cause. That is the principle of causality. And for example, when we look at a specific event — say I kick a ball and the ball flies — the kick is perceived by us as the cause of the ball’s flying. Right? The fact that I kicked it was the reason it flew. David Hume says: you have no observational way of knowing that. Why? Because what you see with your eyes is that there was a kick, and you see that immediately afterward the ball flew. You do another experiment and again you see the same thing: first there was a kick, and afterward the ball flew. But you do not see that the ball flew because of the kick. The “because” you do not see. And the causal principle, or the causal relation between the events, lies in the because. Because of the kick, therefore the ball flew. In order to say there is a causal relation between them, it’s not enough to say that first I kicked and then the ball flew. I could just as well say that first I kicked and afterward the President of the United States lost the election. So because I kicked, he lost the election? What’s the connection? There is temporal priority, but that doesn’t mean it happened because of that. Okay? When you talk about causality, you mean that event A not only precedes event B, but also causes event B. Event B happened because of it. David Hume says: there is no way to get that from observation. How do you see that the ball flew because of the kick? You see that there was a kick and you see that afterward the ball flew. The “because” is your interpretation.

Okay? In various podcasts recently I ended up talking about this issue of causality. So these doubts of David Hume, as Kant said, woke him from his dogmatic slumber. Suddenly he understood that there is a very, very basic problem in the justification or grounding of the scientific process. Because at least two of the cornerstones of the scientific process are induction and causality. And neither of them has an empirical basis. And the accepted conception in the scientific world is that science is based on observation. That’s what distinguishes it from mathematics or philosophy or whatever. Science is based on observations, on facts. And suddenly you discover that some of the cornerstones of the scientific process, the basic assumptions of the scientific process, are assumptions that are not derived from observation. They are not learned from observation. They are a priori assumptions, assumptions of reason, assumptions we come with from home. We did not learn them from experiment, from observation.

So Kant was deeply troubled by the problems Hume raised, and in order to deal with them — I’m doing this briefly just to complete the picture — in order to deal with them he proposed making two divisions of propositions. In language there are different kinds of propositions. We divide them along the logical axis and along the epistemic axis, the axis of cognition. Along the logical axis I divide between analytic propositions and synthetic propositions. What does that mean? An analytic proposition is one that analyzes the subject it deals with. Say “a ball is round” — that’s an analytic proposition, because being round is an essential property of a ball. From the very fact that it is a ball, I can derive that it is round. I don’t need anything beyond the information contained in the concept “ball” itself. So that is an analytic proposition. By contrast, when I say “this ball is heavy,” that proposition is already synthetic, not analytic. Why? Because the concept “ball” by itself does not contain the information that it is also heavy. There are balls that are not heavy. It is not true that every ball, as such, is heavy. So how do I claim that this ball is heavy? The claim is not analytic but synthetic. I make a synthesis of additional information together with the information latent in the definition of the concept “ball.” One has to make a synthesis with additional information. So that is the distinction among propositions on the logical axis: analytic and synthetic propositions. Analytic means they analyze, right?

There is another division on the epistemic axis, on the cognitive axis, and that is the division between a priori and a posteriori propositions. A posteriori propositions are propositions that arise from observation. Say I know that this table is brown — that is an a posteriori proposition. Why? How do I know it? Because I saw it. It is the result of observation. By contrast, the claim that two plus three equals five is an a priori proposition. Shmuel may not agree, but at least according to what I claimed earlier, two plus three equals five is an a priori proposition. It is not the result of observation. I know it prior to observation. So that is called an a priori proposition.

Now, if that’s the case, we really have two divisions of the world of propositions. A logical division into analytic and synthetic, and an epistemic division into a priori and a posteriori. Now since these are two divisions on different axes, we would expect there to be four categories of propositions. Analytic a priori propositions, analytic a posteriori propositions, synthetic a priori propositions, and synthetic a posteriori propositions. Right? Because there is no dependence between the divisions. One division is logical, one division is epistemic, and therefore there is no dependence between them. So each one can go with each one, and we would expect four types of propositions. But if you try to find an analytic a posteriori proposition, you won’t find one. Again, Shmuel won’t agree with this and Kripke also doesn’t agree with it, but at least according to what I described earlier, you won’t find such a proposition. An analytic proposition is a proposition that basically follows from analysis of the concepts involved in it. Such a proposition I know even without observation. Why? It’s enough to analyze the concept and understand that the proposition is true. I don’t need observation for that. Therefore an analytic proposition will not be a posteriori; it will be a priori.

The question is what happens with synthetic a priori propositions — can such a category exist? Prima facie, before Kant people thought not. Or in other words, before Kant people thought that these two divisions overlap. Every analytic proposition is a priori and every synthetic proposition is a posteriori. It’s the same thing. Let’s go back to the ball for a moment to understand why this makes sense. When I say that the ball is round, that’s an analytic proposition. It follows from the definition of the concept ball. That’s on the one hand. On the other hand, obviously I also don’t need observation; it’s enough that I understand what a ball is, and I can tell you that the ball is round. I don’t need to observe it to see that. So that proposition is analytic and therefore also a priori — no observation needed. What about “the ball is heavy”? “The ball is heavy” is a synthetic proposition, right? Because heaviness is not a property of a ball as such. So I need to add additional information in order to know that this ball is a heavy ball. It is not an analytic proposition; it is a synthetic proposition. But how do I know that information? Where do I know it from? If not from analyzing the concepts, then how do I know it? I know it by observation. I simply observe the ball, I weigh it, I see that it is heavy, and therefore I reach the conclusion that the ball is heavy. In other words, a synthetic proposition is one that requires the addition of more information beyond the information contained in the definitions of the concepts. And then I ask myself: where do I get that additional information from? Presumably from observation. Therefore a synthetic proposition is a posteriori. An analytic proposition is a priori. That was the accepted view until Kant.

Now Kant puts his finger on the following point. Consider the laws of nature. Think of some scientific law — Coulomb’s law, I don’t know, that two charges attract each other with a force according to some formula. Okay, so how would we classify that proposition, Coulomb’s law? Is it analytic, synthetic, a priori, a posteriori? First of all, it is not analytic but synthetic, right? Even if I know the meaning of the concept of charge, and of attracting, and of distance, and things like that, I still would not know that charges attract one another merely from conceptual analysis of the concept charge. Obviously I need observation for that. Right? So apparently it is an a posteriori and synthetic proposition. But Hume’s problem of induction says that observation doesn’t give me that. I see two charges, I see that they attract one another. I set up another two charges, and I see that they too attract one another. But how do I know that every pair of charges attracts one another? The problem of induction. How do I know it is always true? That is an assumption of reason. Or in other words, the proposition called Coulomb’s law is really an a priori law. It is not a posteriori, not just the result of observation. Observation is involved in it, but it is not based only on observation. In other words, one also needs a priori insights in order to arrive at it. And the same is true of all scientific laws, because all scientific laws are really generalizations. They deal with many, many cases, and we build them on the basis of a few isolated cases that we observed. We talked about this also in the series on probability. And therefore, whenever we arrive at the general law, we are always making a generalization. But generalization is Hume’s problem of induction. Generalization is our interpretation; it is an a priori matter. It has no observational empirical justification.

Therefore, all the laws of nature are laws that on the one hand are synthetic and on the other hand are a priori. They do not arise from observation alone, but they add information for me beyond what I had from observation itself. Now I ask: how can it be that information is added for me without observation? A synthetic a priori proposition — synthetic means it adds information, a priori means without observation. Or translated into philosophical language, I say: how is it possible for us to accumulate information not by means of observation? That is essentially Kant’s question of the synthetic a priori.

[Speaker K] No, but it’s a mixture — a priori you accept this that—

[Rabbi Michael Abraham] I said, there are components here that are observational, but they are not sufficient by themselves to reach the generalization. All they give me is the observations I actually observed — a few cases I observed. But when I talk about the law, the law jumps from those observations to the general law.

[Speaker K] So the generalization is a priori. So the generalization is a priori.

[Rabbi Michael Abraham] Right. And it’s a combination of two things. And the law itself — what there is in the law beyond the few observations I made — what there is in the law is everything beyond the observations. Because in the observations themselves I don’t need the law; I saw them. The law includes everything added for me beyond the observations — the induction, yes, which adds information for me. All of that addition is a priori. In other words, the scientific law is in essence an a priori proposition. But it is synthetic; it adds information about the world. It is not contained within… and that information is not contained within the premises from which I began. The premises from which I began are the observations I carried out.

[Speaker E] Wait — there is no analytic a posteriori? What? Analytic a posteriori doesn’t exist? No. Because the law of gravitation—

[Rabbi Michael Abraham] Again, according to me and according to Kant, it doesn’t exist. Ah.

[Speaker E] Because the law of gravitation, gravity as we call it, yes?

[Rabbi Michael Abraham] is synthetic a priori.

[Speaker E] Not analytic a posteriori. Why is it synthetic a priori and not a posteriori?

[Rabbi Michael Abraham] It adds information for me, so it’s synthetic.

[Speaker E] Certainly, unequivocally, yes?

[Rabbi Michael Abraham] It’s a priori because it is based on generalization and not on observation.

[Speaker E] Yes, but after we found it, all the gravity that we find supposedly for every—

[Rabbi Michael Abraham] The fact that we found it isn’t interesting. I’m asking about it itself, the proposition.

[Speaker E] No, in itself, yes.

[Rabbi Michael Abraham] That’s all. That proposition is synthetic a priori.

[Speaker E] As if the basic law — a basic law is a priori.

[Rabbi Michael Abraham] Right, synthetic a priori. Exactly. And Kant’s question is basically this: Hume, and Kant translates Hume’s question, is really asking how it can be that we accumulate information about the world without observation, in an a priori way. You understand that this really threatens the distinction I made earlier between science, which accumulates information and therefore is not done logically, and logic and mathematics, where I do not accumulate information and therefore I can do it a priori with logical tools, without observation and without anything. What Kant really showed us is that science too, which deals with the accumulation of information, is also a priori. It is possible to accumulate information in an a priori way. The distinction between logic and science is much more blurred after Kant than it was before him. Fine, I’ll stop here. It’s observation of the intellect. What?

[Speaker K] It’s observation of the intellect.

[Rabbi Michael Abraham] Yes, I’ll get to that — that’s partly where I’m heading. But that will be next time. Are there questions or comments? Rabbi.

[Speaker L] Yes. What’s the difference… you said that scientific induction is not necessary, meaning you can’t prove it, but what’s the difference between that and mathematical induction?

[Rabbi Michael Abraham] No — mathematical induction is deduction. Why? Unless you’re an intuitionist, but that’s a certain philosophy of mathematics that doesn’t recognize this, but that’s just hairsplitting. Mathematical induction — when you prove something by mathematical induction, what you’re really saying is this: let’s prove it for N equals one, and then I prove that if it is true for N equals K, it is also true for N equals K plus one. And from that I infer that it is true for every N. Right? Now you understand that this conclusion is not induction; it’s only called induction because of the form. It’s pure deduction. Because I proved it for N equals one, and if it’s true for one then it’s true for two; if it’s true for two then it’s true for three; if it’s true for three then it’s true for four — so for any number you give me, I have a deductive proof that it is true for that number too. But why—

[Speaker L] It’s really similar, no? It’s also… you have here a sequence of cases from which you infer the general case, but… not really.

[Rabbi Michael Abraham] Why? The big difference is that, for example, in scientific induction, if you look at — say you throw an object and it falls to earth, you throw another object and it too falls to earth. You say, okay, so all objects fall to earth. But you have no step that proves that if two objects fell then three will fall too, and if three then four will fall too. You don’t have that proof that carries you from N equals K to N equals K plus one. In scientific induction you skip that. You jump to every N whatever without having a proof for it. In mathematics you have a proof that if it is true for K, then it is true for K plus one. So if that’s the case, the move from one to two to three to four is a necessary move, a mathematical move.

[Speaker L] You’re saying that all the mathematical tools, all the operators and all those things that jump me from the K case to the general case, from K to K plus one, are assumptions we come with from home, and that—

[Rabbi Michael Abraham] No, that’s not right. On the contrary. If they were assumptions, then it would be like in science. I proved it. I proved that if it is true for K then it is true for K plus one.

[Speaker L] Yes, yes — I mean these are rules from which I derive, not cases from which I infer a rule.

[Rabbi Michael Abraham] Right. I proved the N equals three case; I didn’t just think it’s true also for N equals three. In scientific induction you don’t have this transition step that says if it’s true for one horse then it will also be true for the next horse. You assume it, but you don’t have a proof. In mathematics you have a proof of the transition step as necessary.

[Speaker L] Got it, thank you.

[Rabbi Michael Abraham] It’s only a formal similarity, so they call this too induction, but that’s nonsense.

[Speaker E] I remembered, by the way, regarding begging the question — which I didn’t want to raise in the middle of the lesson. I just didn’t understand why, when someone begs the question — say a person argues something against or in favor of some matter, and someone else comes and from the very thing he argues against he extracts proofs — yes? — then that is never accepted. But you’re saying it’s completely valid. I’ll give an example just for this: Yahya Qafih wrote a book, Milhamot Hashem, and Rabbi Yosef Tzuvairi with his pair of friends wrote a book Emunat Hashem. All the proofs against Rabbi Yahya Qafih in Milhamot Hashem were basically taken from the very book he was arguing against. So that wasn’t accepted at all. Why? Because how can you assume from the very thing being argued about?

[Rabbi Michael Abraham] I didn’t understand a word. Give me a concrete example. I don’t know what they did with it and all kinds of things — give me a concrete example of a concrete argument.

[Speaker E] For example, a concrete argument: I claim that Herzl Street in Lod is a street that goes in the shape of a peh. Then someone comes and says to me, listen, it says in the original blueprint that it goes in the shape of a peh, and the original blueprint doesn’t exist, and empirically it’s straight. But you want to prove to me that it’s a peh — but why would you prove it to me from that?

[Rabbi Michael Abraham] I proved to you that it’s a peh, and you didn’t prove to me that it’s not… what does that have to do with us? I’m really failing to understand.

[Speaker E] Like with Abraham our forefather — I’ll go back to Abraham our forefather.

[Rabbi Michael Abraham] Abraham our forefather is a fully valid logical argument.

[Speaker E] But where is the logic in his walking with head covering? I didn’t understand. Begging the question here is completely absurd.

[Rabbi Michael Abraham] I want you to understand something. Logic does not mean “common sense” in everyday language. Logic means an argument whose conclusion follows necessarily from its premises. That is called a logical argument, okay? Even if it doesn’t sound sensible to you at all, it is still called a logical argument. For example: all frogs have wings, this telephone is a frog, therefore this telephone has wings — that is a valid logical argument because its conclusion follows necessarily from its premises.

[Speaker E] Now I understand — you’re saying that logic is not common sense.

[Rabbi Michael Abraham] Not what in everyday language is called common sense, yes.

[Speaker E] Right, now I understand you. Because in terms of what seems sensible, it’s not accepted. In terms of logic — okay, fine, now I understand you.

[Rabbi Michael Abraham] Good.

[Speaker G] Can I ask the Rabbi something on another matter? Just like that, maybe — did the Rabbi comment on this issue of the Basic Law on Torah study that they’re trying to pass now in the Knesset? Someone asked me about it.

[Rabbi Michael Abraham] On the website? I don’t even know the details of the law yet, but it’s obvious to me that it’s nonsense even without knowing them. It’s part of the general corruption of those people. Is there really anything to discuss in that story?

[Speaker G] But no rabbi… there’s no — it’s passing, and the secular public thinks the religious public sounds to it like one melody…

[Rabbi Michael Abraham] I saw several rabbis who came out against this law. Again, I haven’t seen the details of the law at all, so I can’t speak about it specifically. I don’t need to leaf through it to understand what it is — I see who passed it and why they passed it, and I already understand what we’re dealing with. But yes, people definitely came out against it. It may be that their voice isn’t heard loudly enough, that I don’t know. It’s not—

[Speaker M] You can’t draw clear legal conclusions from this that it’s—

[Rabbi Michael Abraham] No. It’s an attempt to establish some future basis that will presumably justify granting privileges to Torah students because it is a value on which… as is well known, the world stands. Only Torah students, meaning hat-wearers who don’t enlist in the army — only they sustain the world, not just ordinary people who study Torah; that apparently doesn’t count.

[Speaker G] Rabbi, the simple solution, if this law passes and suppose as a substitute it also enables some exemption law from service because it’s a foundational value and then the High Court would have trouble touching it — the solution, in my opinion, would be to establish secular-Haredi yeshivot. They would study Torah in the loftiest way, to their taste. After all, Torah study isn’t really defined. The Rabbi devoted a whole course to that at Bar-Ilan, and maybe more, but they don’t have to agree with the Rabbi; they can disagree with him too. Then they can study Torah, remarkable Torah, and they do study, by the way. And they’ll call it Torah, and it is real Torah, and on that basis they can demand an exemption.

[Rabbi Michael Abraham] I think there’s a more immediate solution: simply refuse to enlist. That’s all. Exactly. Well, on that optimistic note, I think we can part.

[Speaker B] Thank you very much.

[Speaker J] Thank you very much.