Faith – Lesson 30

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[Rabbi Michael Abraham] Years don’t matter, and not only our own years but also other people’s—the collective experience of all of us. That’s not—okay, in another moment I’ll get to that in more detail. Right now I just want to draw your attention, maybe I already noted this, that when Kant asked his question, he asked not whether synthetic a priori judgments are possible, but how synthetic a priori judgments are possible. Meaning, he assumes they are possible and asks himself: but how can that be? You understand that in effect he put on the table the theological formulation—or the exposing formulation, as I called it—of the question. Because in principle he should have drawn the conclusion that Hume drew: if this thing is so complicated and has no justification, then apparently you really can’t believe it, and synthetic a priori judgments are not true or not reliable. But he doesn’t say that. He says: obviously they are reliable. Now the only question is how this can be justified. He is not asking whether it is true, but how it can be that it is true. In other words, he assumes that it is true and asks how this can be. That is exactly the theological formulation, what I called the exposing one, which basically says: I move from the conclusion to the premises, not from the premises to the conclusion. He asks the question in the direction I’m talking about here, not in the physico-theological direction.

Now let’s see how one answers this question. So, Kant basically did what— in order to answer this question of how synthetic a priori judgments are possible, you understand that now any answer to this question will knock down our proof, because if there is a good answer to this question, then there is no need to arrive at the existence of God. So let’s see what answers can be given to this question. Kant’s own answer was what he called the Copernican revolution. Yes, of course the original Copernican revolution was Copernicus’s, but Kant claimed that he was making a second Copernican revolution. And he says: what did Copernicus do? What was Copernicus’s revolution? Instead of looking at the planets around me, at the galaxies around me, from my own point of view, as though they all move around me, let’s look at it from the point of view of the sun. At the planets in our galaxy, yes, from the point of view of the sun. And then suddenly it became clear that everything falls into place amazingly well; everything becomes relatively simple circular or elliptical paths, instead of all the chaos that existed when we put the origin of the coordinate system on the earth. In other words, sometimes changing the point of view, or reversing the point of view—instead of looking from the earth toward the sun, let’s think as though we are looking from the sun toward the earth—okay, so changing the point of view suddenly clarifies the picture and removes many of the difficulties we had in that picture and many of the complexities we had in that picture.

Kant says something similar: let’s remove the difficulties by changing the point of view. What does that mean? We usually assume that we stand opposite the world, we look at the world, observe it, describe it. Science is basically our tool for describing how the world behaves. So there is us, opposite us there is the world, and we try to know the world. And about that David Hume asks—and rightly so—how do you know that you know the world out there correctly? Who told you that you’re hitting the target? The fit between the picture you have in your consciousness and what is happening in the world has no justification. Who said there is such a fit? But Kant says: let’s reverse the point of view. He wants to argue—this is Kant’s transcendental argument, or transcendental arguments—that science does not deal with the world itself. Science deals with the world as I perceive it. Right? When I speak about some scientific phenomenon, I am not speaking about how it happens in the world; I am speaking about how it happens in my consciousness, how I perceived it. That is the only thing I know. I cannot speak about the world in itself, as Kant says; I speak about the world as it is apprehended through my eyes.

Now this is the distinction he makes between noumena, which is the world itself, and phenomena, which is the world as I perceive it—the phenomenology of the world, how I perceive the world. And his claim is—and this is his Copernican revolution—that science does not deal with noumena; it deals with phenomena. It does not deal with the world in itself; we have no access to the world in itself. It deals with the world as we perceive it. Why is this important? Kant says: because if the laws of science really do not deal with phenomena in the world, but with phenomena in our consciousness, then it is less surprising that this works. It is less surprising that our consciousness, even if it arose by chance, still correctly describes what happens in the world, because it is not the world itself; it is the world as it is perceived in consciousness. And therefore, Kant says, our subjectivity exists not only in the perceiver but also in the perceived. In other words, the world that I, as a scientist, am trying to describe—the world too is in some sense inside me. Not only the conceptual and observational tools I use, which are my tools; the phenomena I deal with are also in effect inside me. And therefore, Kant says, what is so surprising about the fact that my tools work? My tools work on things that are inside my consciousness. So I am not asking how the fit between me and the world happens. It is not a fit between me and the world. It is a fit between me and myself, between the world as I perceive it.

Or if I translate Kant’s formulation a bit more, I would say this: the world itself undergoes some sort of filtering when I perceive it through my senses, through my intellect, and the picture that emerges after this filtering fits my categories of thought. Because it is not the world itself; it is the world after it has passed through the filtering of my categories of thought and observation. So therefore, Kant says, this can solve the question of why we assume our tools are reliable.

Now that is Kant’s answer. In my opinion—and in the opinion of many better and greater than me—that answer does not hold water. It’s a bad answer. Why? Let’s try to think this through. Suppose that in the world itself—or maybe I’ll put it this way, look. Let’s take an example, one that I think I’ve mentioned more than once: the example of colors, yes, Mary’s room. So I see something as green. The color green does not really exist in the world itself. The color green is a cognitive, conscious phenomenon. It exists only inside my consciousness. In the world itself there is apparently some crystalline structure such that an electromagnetic wave strikes it and only the green wavelength is reflected back to me, and therefore I see that object as green. But the color green exists only in my consciousness. The same is true of sounds. Sounds and sights and all sensory data really exist only in my consciousness; they do not exist in the world itself.

But of course that does not mean that everything I say is a statement about myself. Rather, I use my subjective, phenomenal language to describe the world itself. When I say that the object in front of me is green, I said something about the object, not about myself. I simply used a language that belongs to my consciousness—the language of colors. But what I really meant to say was: it has such-and-such a crystalline structure, light breaks on it in such-and-such a way, and it reflects the wavelength that I perceive as green. Yes? If a tree falls in the forest and no one is there, does it make a sound? The answer is of course no. It does not make a sound, because sound is something created only when an acoustic wave hits the eardrum. If there is an acoustic wave in the world, that does not make a sound. The sound is created in consciousness when I put my ear there and the acoustic wave hits the eardrum. Then, inside my consciousness, a sound is created. Therefore sounds and sights are cognitive phenomena, not phenomena in the world itself.

Now, by mistake there are commentators on Kant who think that Kant is really talking about subjective things and not about the world itself, but that is not correct. Kant is speaking in a subjective language in order to describe the external world, the objective world. When I say that something produces such-and-such a sound, or has such-and-such a color, and so on, that is just my language for describing what is happening out there. Therefore if one person says, “I see green,” and someone else says, “What are you talking about? It’s red,” then we have a disagreement. If that were merely a statement about me, there would be no disagreement at all. In my consciousness there is green; in his consciousness there is red. What’s the problem? The claim is about the world. What is the claim? That in the world itself, the object has some crystalline structure such that the color human beings perceive when they look at light reflected from that structure is red or green. There we have a dispute. So the dispute is formulated in my subjective language, but it is a dispute about something happening in the world. Therefore Kant is trying to explain how science describes the world itself, and yet why we may still trust my a priori categories when I use them to describe the world.

Now this is Kant’s problem—this is what he tried to say. But here is the problem. Let’s think about Newton’s second law. Newton’s second law establishes a relation between force and acceleration, right? Force equals mass times acceleration. So if I apply a force of four newtons to a body whose mass is one kilogram, then its acceleration will be four meters per second squared. Right? Four equals one times four; force equals mass times acceleration. Now the question is: what really is the body’s true acceleration in the world? Not how I perceive the body’s acceleration. So what do you want to tell me—that in the world the body’s acceleration is really two, but I see it as four? Or that bodies with an acceleration of two in such a situation—I just won’t see them? I will only see bodies that obey Newton’s second law? Somehow I filter the phenomena, and therefore my consciousness can perceive only phenomena that fit my categories of thought—that is, Newton’s second law? This is pure science fiction. If there is a body moving with an acceleration of two, there is no reason I shouldn’t see it, as long as the force and mass fit Newton’s second law. But if the force and mass do not fit Newton’s second law, then suddenly I just won’t see it? Why shouldn’t I see a body moving with acceleration two, when under other circumstances I do see a body moving with acceleration two?

You understand—anything is possible. It could be that there is some deceiving demon of Descartes here, that someone is deceiving me, each time showing me things, hiding things from me, making a fool of me. But that claim is no more plausible than the claim that the senses are reliable. I don’t see what we gained with this claim compared to the difficulty we faced before. This claim is completely speculative and has no basis. So why should I adopt it in order to explain other things that also have no basis? What did I gain? Some demons driving me crazy and arranging the observations I see to fit exactly my categories of thought.

In other words, I’m saying this: even if I accept that everything I say about the world is said in my own subjective cognitive language, there is still a difference between cognition and thought. My cognitive constraints are not my conceptual constraints. My cognitive constraints determine what I will see and what I will not see. My conceptual constraints determine what the properties of what I see will be—like Newton’s second law. There is no connection between them. I can perceive something, from the standpoint of the constraints of perception, that does not satisfy the laws of my conceptual constraints; I will simply be surprised, and I will see that my conceptual laws are wrong. But there is no reason I should not see it. Therefore the cognitive constraints—no, we have no indication that they overlap with the conceptual constraints, or vice versa. But if there is no overlap, or no basis for assuming there is overlap, then the question remains exactly as it was. So Kant did not offer any solution to the difficulty he raised.

Now I want to widen the frame a bit and say this. There is a certain feeling that accompanies engagement in philosophy, that philosophy has countless systems, and since it is not science you cannot test it in a laboratory and decide which system is correct, so basically there is no point in dealing with it. Fine—there is this school of thought, that school of thought, everyone invents his own gut-level speculations, and in the end you can’t make progress. That is a very strong feeling many people have about philosophy. I think the question we are dealing with here is an excellent example of why that feeling is mistaken.

The question I asked here is basically the question of the fit between the intellect and the world. Why should what my intellect tells me also really fit what is happening in the world itself? Now, several answers to this question were proposed throughout history—I don’t know how many, eight, ten answers in the history of philosophy. These answers are surveyed by Hugo Bergmann in his book Introduction to the Theory of Knowledge, epistemology. Chapter nine is called “The Rationality of the World”—rationality in the sense of the world’s fitting our rational thought. And he presents this as a foundational problem of philosophical epistemology. Yes, the most fundamental problem of epistemology is why there is epistemology at all. In other words: how can we trust our epistemology? And he goes through a set of answers that were given throughout the history of philosophy and rejects them one after another, including the Kantian answer.

Now the question is: so what is left? If no answer works out, what remains? In my opinion, the only thing left here is belief in God. In other words, belief in God basically says: since I assume that God coordinates between the structure of my thinking and what happens in the world, He gave me tools that are reliable. They did not arise by chance—if we go back to the Scotland train—my sight, my intellect, my senses did not arise by chance; they were created by an agent who made them in a reliable way, so that I can accept their outputs as a reliable description of the world. So that can explain why I assume there is a fit between these two things.

How do I know there is a God? It is an immediate feeling, intuition, whatever you want to call it. You can say maybe I’m mistaken there—but the type of proof I’m speaking about here, the theological proof, basically says: maybe I am mistaken and maybe not, but if I trust the senses, then I assume there is a God. It could be that I’m mistaken, that there is no God, and that one should not trust the senses. But if I do trust the senses, then I am implicitly assuming that there is a God. Yes, remember? This is the hypothetical character of this kind of argument. It is hypothetical: if you trust the senses, then there is a God. You can always say, yes, I do not trust the senses, and fine, then you do not need to believe in God. But if you trust the senses, then you are in effect assuming that there is a God.

[Speaker B] Wait, wait—but that depends on the person. Someone else will say the probability that there is a God is so small that it’s insignificant. Right. Another example: in the past we talked about the multiverse approach. One person says there are many universes and so on. Another says the probability of a multiverse is so tiny that it has to be God. And I read books where they bring both sides: they say the probability that there is a God is zero, therefore I believe in the multiverse. Everyone believes what he wants.

[Rabbi Michael Abraham] We dealt with the question of the multiverse when we discussed the physico-theological proof, and I think I answered that there. In our context it comes up from the opposite direction, which I’ll get to in a moment. In our direction—but I want to sharpen the point, which is why I gave that whole methodological introduction to this kind of argument, because it is a new kind of argument, different, the reverse of what is customary in philosophy. I do not want to prove that there is a God. You can always say to me: look, I do not trust my senses, and therefore I am exempt from the conclusion that there is a God. What I want to say is only this: if you trust your senses, then you are not exempt from this conclusion. Then you are assuming that there is a God. You can say: look, I don’t trust my senses, and then no—then there is no God. I am speaking only to people who do trust their senses, their a priori intuitions, and everything else I spoke about before. And therefore of course, whoever does not accept the premise also does not have to accept the conclusion—that is true of every argument.

But there is just one thing you cannot do. You cannot say that the stone inscription was created by chance and then start packing your suitcase. That is impossible. And that is what I also want to claim here. So I leave two options: either to think, I do not trust the laws of science, my observations, my a priori insights, and then I also do not need to assume that there is a God; or, second option, I do trust them, and then of course I am necessarily, implicitly, believing in God. But there is no option of trusting science and the senses and my rational thinking, and still assuming that there is no God. That option does not exist. You have to choose one of the first two options.

Now anyone who is honest with himself and says, “I do trust my senses and science and my thinking”—he can always tell me that he doesn’t believe, but let him say that to himself, not to me. If he himself really does not believe that, then fine. But if he does believe that, then let him know, when he stands in front of the mirror, that he is in fact believing in God. That is the claim.

Now when you go through all the answers that were given throughout the history of philosophy to this question, Hugo Bergmann rejected all of them. Now look, he didn’t do this—true, toward the end of his life he moved closer to Judaism, and I think he even observed some commandments a little. He was Jewish—meaning, I don’t mean he converted, of course—but he became closer to Jewish religiosity. But he wrote this book independently of that; it is a philosophical introduction to epistemology, meaning an introduction to the theory of knowledge. He surveys the answers to this question and rejects all of them.

Now look at two beautiful sentences he writes. Especially when someone says something in passing—that is always the most beautiful thing. Look here: “However, if all these attempts to explain our trust in the epistemological tool, in the epistemic tool, have failed, then Hume’s claim remains in full force. The principle of the regularity of nature, the supreme principle of science in general and of natural science in particular, this principle, which underlies every scientific inquiry, remains without proof.” Okay? It remains without proof. And then what? There is no choice except—look what he writes, this is the conclusion: “If we do not wish to cut the Gordian knot of proving the rationality of the world by means of a metaphysical assumption, such as God’s guarantee”—that’s Descartes—“or Leibniz’s pre-established harmony, or the action of the intellect on pre-conscious sensation, subjective idealism”—that is, saying that everything is a statement about us—“then there is no other way left to us except to see the rationality of the world as a guiding, regulative assumption of science’s path.” Or in other words: one cannot believe the senses. This is a methodological assumption; it is not a statement about the world, what I called actualism.

Now, whoever is not an actualist—well, then he is an actualist. Whoever is not an actualist, meaning someone who trusts his senses and the principle of causality and induction and scientific assumptions, and therefore also scientific laws, does not see these merely as methodological assumptions—“I have no idea whether they are true or not”—but says: this also seems to me probably true. Not certain, but probably true. Then he cannot see this as merely a methodological assumption. So what is left? A metaphysical assumption. Right? A metaphysical assumption such as God’s guarantee. That is the only thing left.

You see, precisely he—someone who in the end rejects this conclusion—actually puts the question in its full force after surveying all the possible answers in the history of philosophy that were given to this question. He rejected all of them, and then he says: okay, so really one cannot see the laws of science as claims about the world itself. They are methodological organizations we make within ourselves. They are not claims about the world. So I ask: okay, but someone who nevertheless assumes that these are claims about the world, who trusts the laws of science—and after all we see that they work in the end—must assume that there is nevertheless some reason, some basis for this trust that we place in the scientific, observational, and conceptual tools. And he says: the only thing left is a metaphysical assumption such as God’s guarantee. In other words, even though in passing he does not want to enter into the issue of God’s guarantee, he himself says that this is the only possible way to believe in the laws of science in an informativist way and not in an actualist way. That is basically his claim.

And if I come back to our discussion, then apparently there really are many approaches in philosophy, as I said—eight, I don’t know, ten answers that were proposed to this question throughout the history of philosophy. But look, it is possible to reach conclusions philosophically. It is not true that everything remains open. All these arguments can be rejected. I explained the Kantian one; the others are even weaker than it. All of them are rejected, and we are left with a question you cannot escape—a philosophical question you cannot escape. We have only one of two paths, just as Hugo Bergmann himself, as an expert in this field of philosophy, says. You have two paths; you have to choose between them. Either these are merely methodological assumptions—science is not making claims about the world, science is making claims about us, what I called actualism—or science is making claims about the world, informativism, but then you need God’s guarantee. Because without that there is no justification for it. That’s it—that is the choice. Now you can choose this or that, but you cannot say: I do not see these as methodological assumptions, I trust the laws of science, but I am an atheist. That does not work. It is not consistent.

[Speaker B] But people like Stephen Hawking are not consistent, obviously. Okay, but he didn’t believe in God, he was—

[Rabbi Michael Abraham] He thought he didn’t believe in God. He thought he didn’t believe in God. I claim—

[Speaker B] I—

[Rabbi Michael Abraham] I claim that people of that type, who see science as a tool that talks about the world and not as a methodological assumption, are in fact covert believers in God, or they are inconsistent. One of the two. Now, if you want the charity principle, then let us judge him favorably and say that he is really a covert believer in God and not that he is inconsistent. But that is what I claim about every— and all in all it seems to me that almost every reasonable person does not see science as merely methodological assumptions. He believes in science as something that describes the world. And therefore, in my view, every reasonable person should conclude that he in fact believes in God, even if he is not aware of it. So he is an unconscious believer, but he still believes in God.

[Speaker B] There will be Stephen Hawkings who will speak against this.

[Rabbi Michael Abraham] Okay, because he wasn’t aware of it. He thought he didn’t believe in God, but he was mistaken. That’s the meaning of this argument. What this argument basically shows a person is: listen, you’re not being consistent. If you think that inscription made of stones came about by chance, but then you start packing your suitcase and getting ready to get off the train, you’re not being consistent. So one of two things must be true: either implicitly—maybe you’re not aware of it, but implicitly—you believe that this inscription was arranged by someone, or you really are inconsistent, and then stop packing your suitcase. You don’t need to pack your suitcase; there’s no basis for it. That’s really the claim. It’s a cruel choice, but that’s the claim, that’s the choice facing people. Now notice something—maybe one more comment. There’s another book by Hugo Bergmann, and to me it’s an interpretive puzzle. One day I’ll have to ask people who know him and his thought a bit better. He has another book called Thinkers of the Generation. And that book gives an overview of the thought of modern thinkers—thinkers of the early twentieth century, the first half of the twentieth century, the thinkers of the generation he was talking about. Now, in the introduction to that book, he goes back to this question, and there he says that God is the ultimate and only necessary explanation. Now, when I looked a little at when the books were written, Thinkers of the Generation came out long before Introduction to Epistemology. So that’s a bit strange. In Introduction to Epistemology he says: if we don’t want the guarantee of God, then there’s no choice but to assume that these are methodological claims; the laws of nature are methodological assumptions about the world. But in Thinkers of the Generation he says: no, no, there is a God, and therefore we can accept this as claims about the world itself. And I know that actually fits the later Hugo Bergmann, not the earlier one. Because later on he returned to believing in God and even to observing commandments. So this is an interesting interpretive question about the stages Hugo Bergmann himself went through. I don’t know the answer. In any case, though, in the introduction to Thinkers of the Generation you can also find the conclusion as I’m presenting it here. So I want to explain what exactly we found here. The claim of the physico-theological proof is an argument that says the following: if you use the tools of rational thought, you must arrive at the conclusion that there is a God. In other words, the tools of rational thought lead you to the conclusion that there is a God, because the world is complex, a complex thing does not come about by chance, and therefore there is a God. I assume rational thought and arrive at the conclusion that there is a God. Here I’m saying that I’m taking the opposite path. I’m saying: if there were no God, I could not assume rational thought. If I assume rational thought, only God is what makes it possible. In other words, God is the basis of rational thought; rational thought is not the basis of belief in God. These are two opposite ways of looking at the relationship between rational thinking and God. What I really want to claim here is that without belief in God there is no rationality. Not that rationality generates it or yields it—yes, one can infer the existence of God from it—but God is the condition for it. Without belief in God, you cannot use the tools of rational thought. Someone who does not believe in God is not rational. He is not irrational because belief in God is true—that’s the physico-theological proof, which says that if you don’t believe in God then you’re throwing away a strong philosophical argument, so you’re simply not thinking rationally. Here I want to argue something much deeper: if you don’t believe in God, you have no justification for using the tools of rational thought, because that is the only justification for them. And therefore it’s very strange that an image developed in the world that… rationality is the antithesis of belief in God. Belief in God is supposedly for the non-rational people, and rational thought is atheistic, yes, it belongs to those who don’t believe in God. That’s ridiculous in every sense I can think of. It’s exactly the opposite. These are people consoling themselves with fools’ comfort. That is, I think that without belief in God there is no basis whatsoever for the trust we place in rational thought. There is no basis and no justification for using the tools of rational thought. And I’ll remind you, perhaps, of what I told you in the methodological introduction. Here we’re actually arriving at that claim. Let me share the screen with you for a moment. I’ll remind you—I already did this once in the introduction. Now I want to show you what that introduction was for. Here I’m closing the circle. The claim is that if I have an assumption like this, that if A then B, then based on it I can build two kinds of arguments. Right? I can assume: if A then B—that’s the first assumption. Second assumption: A. Conclusion: B. Right? If they give me chocolate, then I’ll feel good. They gave me chocolate; conclusion: I feel good. Okay? So that’s one direction. I can also use that same assumption itself, A implies B, but the second assumption will be not-B, and the conclusion will be not-A. That too is a valid argument. If they give me chocolate, I’ll feel good. If I don’t feel good, apparently they didn’t give me chocolate. Because if they had, I would feel good. Right? These are two arguments, both logically valid, and both are built on the hypothetical assumption that A implies B. What we’re really saying here is exactly that point. I’m saying that from rational thought it follows that God basically—let’s say that A is there is no God. B is rational thought and science are not reliable. All right? Now I’m saying: if there is no God, then there is no basis for rational thought, so rational thought is not reliable. Now one can say, one can take it in the other direction and say okay, so if not-B—and whoever thinks that rational thought and science are reliable—then he must arrive at the conclusion of not-A. In other words, it is not true that there is no God; rather, there is a God. The assumption is that without God there is no basis for rational thought. But I can use that in the reverse direction, logically, and say: someone who believes in rational thought infers the conclusion—so he can infer the conclusion that he probably also believes in God. Okay? And that is a logically valid argument; it’s not wishful thinking. It’s a logically valid argument. Of course, you can assume that you don’t accept it—that is, it’s not true, you believe in B, rational thought really isn’t reliable in your eyes. Fine, if you believe in B then I didn’t say anything. But if rational thought is reliable in your eyes, scientific thought, then you necessarily believe in God. Therefore this line of argument is basically a route that goes backward, as it were, from the conclusion to the assumption. It’s not really like that. It goes from the consequent of the implication to the antecedent of the implication. But there are still assumptions here and a conclusion that follows from those assumptions. It’s a logically valid argument; there’s no trickery here. Okay? So that is the structure of this kind of argument. Now I just want to point out one more aspect, another aspect of this argument before I move to all the objections that came up here—evolution and learning from experience. We’ll probably do that next time; I’m sorry I’m dragging this out, but I’m trying to do it systematically. I want to talk a bit about a fascinating phenomenon that many philosophers and mathematicians have noted—the amazement in the face of the phenomenon that the book of the universe, yes, Galileo’s statement, that the book of the universe is written in the language of mathematics. Meaning that we can use mathematical language to describe what happens in the world. So Galileo was amazed by this: how does that happen, how can that be? I want to connect that too to the move I’m making here. Mathematics basically describes the forms of our thinking, how we infer conclusions from assumptions. Mathematics claims nothing about the assumptions and nothing about the conclusions, only about the relation between the assumptions and the conclusions—the mathematical logic; at the moment I’m not distinguishing between them. But it turns out that these ways of thinking, which after all deal only with ourselves, not with the world—at least according to the common conception of mathematics; mathematics does not deal with the world, it deals with the world of ideas, if you like, or with us, but not with the objective external world—it turns out that this thing is extremely effective when we come to describe what happens in the world. And the question is, how does that happen? I think this is just another aspect of the same question I asked here: how can it be that our a priori tools of thought fit, are coordinated with, what happens in the world itself? That is really the proof from epistemology. And I want to argue that this means that these tools of thought—something or someone implanted in them some kind of fit with what happens in the world; otherwise this should not have happened. Otherwise I could amuse myself with mathematics and with what happens inside me endlessly, but it should not have turned out to be such a powerful tool for describing what happens in the world. And it keeps turning out to be such a tool. Now I want to sharpen this claim a bit more. Look, in principle, usually when I hear the standard formulations of this puzzle, I don’t understand them. It simply doesn’t seem puzzling to me at all. Why? Because mathematics deals with the relations between concepts and ideas. Concepts of one sort or another have properties of one sort or another, so from this it must also follow that they have properties of another sort. And there are relations between ideas and between claims that mathematics deals with. Now if those relations are necessary relations, then it’s no wonder those relations also hold in the world. Yes, if velocity is the derivative of position with respect to time, then position is the integral of velocity with respect to time. It’s not surprising that if this holds, then it holds; it has to be that way—that’s the fundamental theorem of calculus, right? That integration is the inverse operation of differentiation. So this mathematical discovery, the fact that it serves me in the world, is not surprising at all. It has to be so. The fact that in the world too every triangle has three angles doesn’t surprise me. That’s the property of a triangle, that it has three angles. There’s nothing here that should be surprising. What is surprising—and this is what I want to sharpen a bit more—is that mathematical theories developed without any connection to what happens in the world are sometimes found, after a very long time, to be fruitful and useful in describing the world itself. For example, when I talk about geometry, I’m not surprised, because geometry was clearly developed מתוך looking at the world. You look at the world, and you look at the properties of things, and then you build a mathematical theory called geometry—or geometries, there are many. Okay? But it begins from some sort of observation of the world. Say, non-Euclidean geometry is already less directly an observation of the world, but let’s say it’s an extension of Euclidean geometry. But I don’t know—group theory, or complex numbers, or things of that sort—these are things that people created seemingly out of their wild imagination. They are not the result of observing the world, or at least not a straightforward result of observing the world. And it turns out that after you create these concepts and these claims, they have a very fruitful, complex, interesting content that ultimately, amazingly enough, comes back and serves us when we try to describe the world, even though it was developed with no connection whatsoever to what happens in the world. That is surprising. The basic mathematical relations—if the axioms hold, then the theorems or propositions also hold—well, of course that’s clear, because it has to be that way; that’s exactly what mathematics shows us. That isn’t surprising. If our world has Euclidean geometry, then the sum of the angles in a triangle in our world is one hundred and eighty degrees. That doesn’t surprise me at all; it has to be that way. In Euclidean geometry the sum of the angles is one hundred and eighty. What does surprise me is that a theory—the theory of complex numbers, developed on the basis of some generalization unrelated to the world, unrelated in any way—and it turns out that it is extremely useful in describing the world. Or structures like group theory, Hilbert spaces, fields, rings, vector spaces, all sorts of things of this kind, all kinds of strange mathematical creatures that in the end turn out to be amazing tools for describing the world itself. That is surprising. Because it means that something in our thinking really is coordinated with what happens, coordinated with what happens in the world itself. Now the claim—I’ll perhaps clarify this through something I think I already said once. When I started at university, at the beginning of my master’s degree in physics, they had me teach a mechanics course. So I asked the students there whether, in their opinion, the statement two plus three equals five is a falsifiable statement, a claim that can be refuted. Could they suggest an experiment that would refute this theory, this claim? So after I shook them up a little, I managed to get out of them what I wanted. Yes, it’s possible. Take a big basket, put two apples into it, add another three apples, and count how many you have altogether. If altogether you don’t have five but seven or two, then you’ve refuted the claim that two plus three equals five. So ostensibly, the claim that two plus three equals five is a claim in physics. But of course that’s not true. I’ll ask even on the practical level: suppose I put in two, added three, counted, and got four—there are only four apples in the basket. Would anyone think to say, ah okay, so two plus three really equals four, not five? Never in a million years. We would say there was an error in the experiment, we got confused, one apple disappeared, we didn’t count correctly the apples we put in. We would never give up the claim that two plus three equals five. Why not? Because two plus three equals five is not a claim about the world. It is a claim about the concepts two, three, equals, and plus. And the relation among them dictates that two plus three equals five; it is a necessary relation. It is not open to refutation. If I were to discover consistently that adding apples into a basket, two and three, gives four, the conclusion I would draw is that adding apples into a basket is not described by arithmetic addition. Arithmetic addition is not the theory for which this is a model. That is the conclusion I would draw. Why did I say this at the beginning of a mechanics course? Because I told them: look, let me refute for you the claim that five plus five equals ten. Let’s take a body and apply to it a force of five newtons northward. Now let’s apply another force of five newtons eastward. What is the total force acting on the body? Ostensibly, mathematically, five plus five equals ten. The answer, of course, is no—not true: five root two, seven-point-something. Why? Because those forces are perpendicular forces. Right? So the resultant force is something lying diagonally between the perpendicular forces, yes, the parallelogram rule, and that diagonal is five root two. So does five plus five equal seven-point-something? Have we refuted the mathematical proposition that five plus five equals ten? No. We have refuted the physical claim that the addition of forces is described by arithmetic addition. That’s not true. The addition of forces requires vector addition, not arithmetic addition. But that is a claim in physics, not in mathematics. We will not succeed in refuting the claims of vector calculus, nor the claims of arithmetic, in the laboratory. There is no chance. What we can do in the laboratory is decide which mathematical theory is suitable for describing the physical situation I am dealing with. That decision is a physicist’s decision, not a mathematician’s. And that can be refuted. If you think this can be described by arithmetic addition, I can refute that experimentally and show you that it’s not true; vector addition is needed, because that is a claim in physics. Mathematics cannot be refuted. Why? Because it does not deal with the world. Mathematics does not deal with the world; it deals with the relation between ideas. The world may be a model of some mathematical theory. In the language of model theory used by mathematicians, they say that any reality that satisfies the assumptions of a mathematical theory is called a model of that theory. And there can be many models of the same theory. For example, when I talk about vector calculus, then forces in mechanics are a model for vector calculus. Forces add to one another according to vector calculus. But accelerations and velocities also add to one another according to vector calculus. Moments, and whatever you like. There are many vectors. Each of these is a model of the theory of vector calculus. In other words, it turns out that in our world there are many parts or phenomena that constitute models of abstract mathematical theories. That is the miracle of mathematics. And the question of how this happens is really just another aspect of the same question I spoke about earlier: how can it be that our thinking is reliable, that it fits what happens in the world, and that we can use it to decipher what happens in the world. Okay, I won’t go into this further because these are already details. So okay, we’ll stop here, and I promise, without making a vow, next time I’ll get to the objections and the responses to them. What has come up here more than once, and each time I postponed it—I simply wanted to build the structure systematically. Next time we’ll talk about the objections: evolution, experience, all sorts of things of that kind.

[Speaker D] Okay, any questions? You—

[Rabbi Michael Abraham] Can you hear?

[Speaker D] Will there also be the proof from morality?

[Rabbi Michael Abraham] I can barely hear, it’s very faint.

[Speaker D] The proof from morality, like also…

[Rabbi Michael Abraham] Yes, yes, that’s the next talk. Meaning, after we finish this talk, that’s the next stage.

[Speaker C] Okay, so Sabbath peace.

[Rabbi Michael Abraham] Sabbath peace, goodbye.