Definition of Terms 7

Table of Contents

• Definitions as observation, and the claim that they can be right or wrong
• Kant and the distinction between types of proofs for the existence of God
• Mathematics, science, and geometry as examples of the question of “information from a definition”
• Rationalism versus empiricism, and the claim that there is no pure empiricism
• Aristotle, the empiricist revolt, and the danger of “logical” inference
• Hume, the problem of induction and causality, and Kant as qualifiers of empiricism
• Examples from physics: combining forces, vectors, and equations as the carriers of information
• Descartes’ cogito as a proof that it is possible to infer a fact from thought
• Anselm’s ontological proof: proving existence from a definition
• Observation with the mind’s eye as an answer to the claim of “information from a definition,” and doubts about the cogito

Summary

General Overview

The text brings the discussion of definitions to a close and sharpens the point that a definition is not always an arbitrary constituting act, but is sometimes the product of directed observation. Because of that, it also has a dimension of being right or wrong. The central claim is that the question whether one can extract information about the world from definitions depends on the clash between empiricism and rationalism, and on how one understands what “observation” is and what place the intellect and intuition have in it. Kant sharply rejects the possibility of deriving facts from a definition alone, but the text suggests that if a definition is grounded in observation through the mind’s eye, it can contain information that had not been noticed before conceptual processing. Along the way, the discussion draws connections between proofs for the existence of God, the relation between mathematics and physics, criticism of Aristotelian rationalism, the difficulties of “pure” empiricism in Hume, and historical examples such as Descartes’ cogito and Anselm’s ontological proof.

Definitions as observation, and the claim that they can be right or wrong

The text states that an interesting definition is usually an aimed-at definition derived from observing an idea, rather than a constituting definition that establishes an arbitrary use of a concept. It presents a position one sometimes encounters in conversation with someone who claims that no definition is constituting at all, but that all of them are aimed-at, because the very act of defining assumes that there is “such a thing” being understood. The text formulates the question whether definitions contain information—that is, whether one can derive facts about the world from a definition—and presents this as an apparently internal tension if a definition is seen as a detached construction. It argues that when a definition passes through the “crucible of definition,” it allows us to distinguish information that had been experienced but not noticed beforehand, and therefore a definition can carry knowledge that comes out of observation rather than arbitrariness.

Kant and the distinction between types of proofs for the existence of God

The text presents Kant’s division of proofs for the existence of God into the ontological proof, the cosmological proof, and the physico-theological proof. It explains that the ontological proof relies on conceptual analysis without observations, and tries to begin with a definition and end with a fact about the world, whereas the other two proofs begin with an observational fact and from there move toward a theological explanation. The text formulates Kant’s attack as a principled claim that one cannot begin with a definition and end with a fact, because if facts could be extracted from definitions, then one could simply “define whatever one wants” and generate facts with no connection to the world. In response, it argues that if a definition is the result of looking at the world by means of intuition and the mind’s eye, then there is nothing surprising about being able to extract information from it, because it already satisfies the empiricist demand that learning comes from observation.

Mathematics, science, and geometry as examples of the question of “information from a definition”

The text argues that mathematics does not begin from observation, and therefore by itself says nothing about the world, whereas science begins from observation even if it does not end there. It presents the theorem about the sum of the angles in a triangle as something derived from axioms and definitions, and stresses that this does not mean that this is how things must be in the world, but only in a world that satisfies those assumptions. The text states that in order to apply mathematics to the world, one needs an observation confirming that the relevant axioms hold in our space, and that even this observation is not “with the eyes” but with the mind’s eye and an understanding of the context. It adds that no one goes off to infinity in order to verify that parallel lines never meet; rather, one infers that from the structure of the assumptions. So the connection between mathematics and the world depends on an observational transition that makes application possible, and not on mathematics itself.

Rationalism versus empiricism, and the claim that there is no pure empiricism

The text defines empiricism as the view that information about the world is learned from observations and that thought merely processes it, and rationalism as the view that the intellect is a tool for knowing the world and that processes of thought can yield knowledge about reality. It gives a logical example in the style of “all human beings are mortal” in order to argue that logical inference does not create new information, but only unfolds what is already implicit in the premises, and empiricism demands that the validity of the premises come from observation. The text adopts a position according to which it agrees with empiricism, but broadens the concept of observation to include intuition as well, formulating this as an “expanded empiricism” in which the intellect is not a source ex nihilo but a form of non-sensory cognition. It argues that the empiricist solution is not sufficient, because scientific generalizations add something beyond what has actually been observed, and therefore a consistent empiricism is forced to give up laws of nature in their strong sense and settle for fitting the data observed so far.

Aristotle, the empiricist revolt, and the danger of “logical” inference

The text brings Aristotle as an example of rationalism, in which the “logical” assumption that bodies fall according to their weight became an entire physics, even though it turned out to be mistaken and could have been refuted by a simple observation. It states that modern science was born from an empiricist revolt in the 15th–16th centuries that rejected the claim that orderly thought alone is enough to describe the world. The text describes the Middle Ages as continuing Aristotelian rationalism, in which science was considered a branch of philosophy and theology was speculative, and it attributes to Maimonides a move of “converting” Aristotelian physics in chapters of Jewish law in the Laws of the Foundations of the Torah. It links the breakthrough of science with the destabilization of religious beliefs, and argues that scientific thinking carries a skeptical element that does not accept things merely because they seem logical.

Hume, the problem of induction and causality, and Kant as qualifiers of empiricism

The text explains that Hume points out that science in practice contains assumptions that are not directly observed, in particular that inductive generalization and causality go beyond what can actually be seen. It states that no one has ever seen a causal relation itself, but only a sequence of events, and therefore “because” is not exposed to observation. The text presents the conclusion that Hume and Kant tried to qualify empiricism and say that the scientific method is not pure observation, and that to this day the philosophy of science mainly reformulates those same problems. It presents the principle that “from definitions one cannot learn anything about the world” as the essence of the empiricist revolt, and then raises the difficulty that science itself rests on rationalist components.

Examples from physics: combining forces, vectors, and equations as the carriers of information

The text tells about teaching physics to biology students at Bar-Ilan and about the question whether “two plus three equals five” can be refuted, in order to show that mathematics does not make claims about the world; rather, physics assumes that a certain mathematics fits a real aspect of reality. It uses the example of combining forces to illustrate that ten plus ten is not “twenty” in a vector context, and concludes that what has been refuted is the physical assumption that arithmetic addition describes forces, not any law in mathematics. The text presents Einstein as someone who challenged the application of Euclidean geometry to the world, and argues that he did not “refute” geometry but the physical assumption that it describes actual space. It explains that in physics the information is located in the equation and in the axioms as a description of the world, and that once those are assumed, most of the work is mathematical until an experimental mismatch appears and sends the physicist back to repair the theory.

Descartes’ cogito as a proof that it is possible to infer a fact from thought

The text presents Descartes as a rationalist operating at the height of the empiricist revolt, looking for an absolutely certain anchor point that does not depend on sensory observation, which can be mistaken. It argues that the novelty in the cogito is not the trivial transition from “I think” to “I exist,” but the claim that “I think” is a premise that cannot be undermined. The text explains that Descartes grounds this by saying that doubting is itself a thought, and therefore any attempt to challenge “I think” leads to contradiction and makes the claim necessary at that very moment. It emphasizes that the cogito proves a point-like existence at the time the argument is being performed and does not prove the existence of others, and it describes Descartes’ goal as “getting a foot in the door,” showing a counterexample to the claim that thought cannot give information about reality.

Anselm’s ontological proof: proving existence from a definition

The text presents Anselm of Canterbury’s ontological proof (12th century) as a striking example of rationalism in which one infers existence in the world from a definition alone. It quotes the opening of the Proslogion as a prayer, and dismisses the criticism about “dishonesty” by saying that the argument itself is what matters, and that even in science, as with Andrew Wiles and Fermat’s conjecture, prior belief does not invalidate a proof. The text formulates Anselm’s definition of God as “that than which nothing greater can be conceived,” and emphasizes that this is a definition for the sake of the argument, not a full theological account. It describes the course of the proof: even the “fool” who denies God understands the concept, and so it exists in the mind; but if it exists only in the mind, one can conceive of something greater—namely something that also exists in reality—and therefore a contradiction follows with the definition. Anselm’s conclusion is that God cannot exist only in the mind, but exists.

Observation with the mind’s eye as an answer to the claim of “information from a definition,” and doubts about the cogito

The text returns to the claim that if one understands rationalism not as a detached thinking process but as observation with the mind’s eye, then both the cogito and ontological moves are not “information out of nothing,” but products of non-sensory cognition. It brings a possible criticism through Ron Aharoni (“The Cat That Isn’t There”) about a confusion between subject and object when the “I” observes the “self,” and argues that this may create circles and paradoxes. The text raises the question whether thought is allowed to formulate logic about itself and examine its tools by means of those same tools, and shows that rejecting this on grounds of “inconsistency” itself relies on logic. It concludes by suggesting that without some kind of observation, one cannot really get from the purity of a definition to a claim about the world, and therefore rationalist moves should be seen as intuitive observation rather than as a complete detachment from reality.

Okay, today is kind of a wrap-up—I’ll finish the topic of definitions. I honestly don’t even remember exactly where we were holding and what, but it seems to me that more or less I’ve already covered what I wanted. One more topic, and then we need to move on to the next subject. If anyone has requests, or I don’t know, suggestions or something like that, then gladly. Think about it as we go, maybe something interesting will come up for you. Okay. So if something occurs to someone by the end of the lesson—email, whatever, wherever—then we’ll decide what to do. So when we talked about definitions, especially in the first few lessons, I spoke a bit about a definition as a kind of observation. A directive definition and not a constitutive definition. And basically the claim was that there is also right and wrong within a definition. A definition is not just: fine, if I defined it then it’s my right to define whatever I want. Well, that’s also true—there are definitions that are constitutive definitions—but not all definitions are like that, and usually the interesting definitions are not like that. And I recently met someone who wants to claim that no definition is like that. Meaning, all definitions are directive, because otherwise why are you defining this concept at all? Clearly you understand that there is such a thing. Well, in any case, what I want to touch on today is really another meaning of that same issue, and that is the question whether definitions contain information. Meaning: can I derive information about the world from a definition? At first glance that sounds self-contradictory. Kant writes quite sharply that this is impossible, because a definition comes from me, I make some construction of a concept, and I can define whatever I want however I want. How can it be that I derive from a definition a fact about the world? Not derive from a definition—there’s some fact you have an intuition about, but you want to define it properly. No, so I’m saying it’s the opposite. That’s why I said this connects to the question of observation that gives definitions, or definitions as directive definitions. If we understand that the definition is the product of observing some idea—never mind, not necessarily with the eyes, we talked about that—then that’s something else, because observation can discern facts, and very often the definition contains information within it that the definition helped you notice. Meaning, you didn’t notice that information at all until you passed it through the crucible of definition. You experienced it, you encountered it in some way, but you didn’t notice it until you conceptualized it and defined it. Basically, the question whether there is information inside a definition is tied at the navel to the question of rationalism versus empiricism. Maybe before I get to rationalism versus empiricism, I’ll talk a bit about—I’ll start from the end so it will be clearer why I’m moving to that. Kant divides the proofs for the existence of God into three types. There is the ontological proof, the cosmological proof, and the physico-theological proof. The ontological proof is a proof based on purely conceptual analysis, without observations. You take a concept, define it, analyze it, and from that prove that there is God. A kind of hocus-pocus thing. The cosmological and physico-theological proofs are proofs that begin with some observation, with some fact I learned from observation. The cosmological proof starts from the fact that there is something—a universe, a world, something, whatever, without getting into the question of what exactly it is that exists—and then the claim is that if something exists, then things, at least of the kind we know, are things that must have someone who created them; they do not create themselves, and therefore there is God. I’m saying this only schematically. And the physico-theological proof has a similar structure, but it assumes something additional about the object you think exists, the one you observed—that it is complex, that it is designed, coordinated, all kinds of things like that. And then they say: look, a complex thing doesn’t create itself, so that’s a sign that there is God, someone who assembled it. So those last two kinds are kinds that begin with observation, with some fact I learned from observation, and then the claim that from this one can prove that there is God—that isn’t so terrible, because from observation you learn something about the world, so fine, there’s no principled problem in learning something about the world by means of observation. On the contrary, that’s empiricism: you look, and from that you learn what there is in the world. But the first type of proof—the ontological proof and things of that sort—tries to prove that there is God from a definition. From a definition. What does that actually mean? It means that I begin with a definition and end with a fact. I end with something that makes a claim about the world. But I didn’t add anything to the definition except the definition. Meaning, it’s just a definition. So how can it be that a definition contains information? That’s basically the point. How can it be that hidden inside a definition there is information about the world? Hidden inside it there is information about the concept I defined—that’s fine. But I defined it. It could also be fictional. I have no way to know whether there is anything corresponding to it in the world unless I made an observation. Meaning, I observed the world. If not, then it remains in the realm of definitions, and there you can do whatever you want, but it has nothing to do with the world. We talked about observing yourself—is that observation? Descartes—Descartes is an example of that. Descartes is an example of that. But he started—that too is a kind of observation. I’ll get to Descartes in just a moment. So basically Kant’s main attack is: how can that be? This kind of proof is impossible a priori. It cannot be. Because it cannot be that you begin with a definition and end with a fact. It just can’t happen. I can define whatever I want. If one can derive facts from definitions, then I’ll define whatever I want and derive whatever facts I want. What does that have to do with the world? What does the world owe me? It was here before me, as Mark Twain said. Meaning: how can one take definitions and derive information from them? So I already want to answer this right away, before we get more deeply into the matter: if indeed the definition is the product of observation and not of a construction made inside my thought in some autonomous way detached from the world—a kind of observation with my intuition, what is called the eyes of the intellect, and from that I derive the definition—then if the definition is grounded in observation, then it should no longer surprise us that I can extract information from it. That satisfies all the empiricist requirements: you observe the world, and from that you learn facts about the world. Because Kant’s claim really assumes that a definition is a detached construction. Meaning, you define whatever you want, a purely mental act, not connected to the world itself. Then he asks: so how do you derive information from it? But if the definition is really the result of some kind of looking at the world, then it can contain information about the world. So there’s no principled problem here. Like we also talked about, I think, regarding the difference between mathematics and science. In science, you observe the world, and yes, you make generalizations and so on. There are many problems with that simplistic description, but still, in science, basically, you begin with observation. True, observation alone is not enough, but you begin with observation. And in mathematics, apparently it does not begin with observation, and because of that it also says nothing about the world. Because it’s only definitions. And therefore this debate is really the same debate as whether mathematics can say something about the world. And in the simple sense, mathematics cannot say anything about the world. Mathematics is a collection of definitions, and you analyze them and find relations between them, but all of this is done inside yourself. These are constructions inside your head. It has nothing to do with the world. Afterwards, by means of observation, you may see that those constructions you built in thought fit what happens in the world, and then claim that what happens in the world can be described by that mathematical theory. But that is already the result of observation. You saw in the world that indeed—say geometry, yes?—I know that the sum of the angles in a triangle is 180 degrees. That comes out of four axioms and definitions. A set of definitions, axioms, and derivation rules, and from that it follows that the sum of the angles in a triangle is 180 degrees. Does that mean that in the world, the sum of the angles in triangles will be 180 degrees? The answer is no, certainly not. What that means is that if I define and assume such-and-such, then the result is that if there is a triangle in that reality, then it will have 180 degrees. Can I know from this that in the world, when I draw a triangle on a sheet of paper, the sum of its angles will be 180? Certainly not. All I can say is that if I observe the world and see that the axioms of geometry hold in the world—that between two points there passes one straight line, that two parallel lines do not meet, and all these axioms that I did not assume about the world, but axioms I assume about some Platonic world—and what is that? In what sense do I see? That’s a question, and in this context we talked about this issue. When I see—but it’s not seeing with the eyes, because with the eyes you can never see it—you see it with the eyes of the intellect, right? And that’s what I’m talking about here. So if I reached the observational conclusion that between two points in our world, in our space, there passes one straight line, and that two parallels do not meet, and that all the axioms of geometry hold, then because of mathematics I can also conclude that the sum of the angles in our world, in our space, in a triangle, will be 180 degrees. That, yes. But again, I had to have observation along the way in order to state this as a claim about the world. Before the observation, I can say that the sum of the angles in a triangle is 180 degrees, but in some Platonic triangle in a world that satisfies the assumptions I posit in mathematics, in geometry. If I want to state that about the world, then I need some observation. I need to look at the world and see that these axioms do indeed hold. To the extent one can see that—and we talked about this, that one sees it with the eyes of the intellect, not with ordinary eyes. No one walked all the way to infinity with two parallels and saw that they really do not meet. You understand from the context that they do not meet. So the claim that one can derive information from a definition is basically parallel to the claim that one can derive information from mathematics. Because from purely mental procedures, without observation—that is exactly rationalism. Right? Rationalism, as distinct from empiricism, is the view that I can learn about the world through mental processes. Aristotle, for example, was indeed a rationalist, and all medieval thought, let’s say, was rationalist following Aristotle. Not necessarily rational, but rationalist. Rationalist meaning that it sees the intellect as a means of knowing the world, through thought, not only through perception. And in the fifteenth-sixteenth century suddenly there was a kind of empiricist revolt. Meaning, there was some revolt saying: what do you mean? So what if you think that way? Who says that really describes the world? You need observation in order to learn about the world. And that is how modern science was born, basically—or empiricism in philosophy, parallel to the birth of modern science in scientific research. So this question—whether definitions contain information—is really the question of whether there is room for rationalism, or whether empiricism is right. Empiricism—just a second—empiricism basically wants to claim that there is no information in definitions. Definitions are mental procedures, and therefore you cannot extract information from a mental procedure. Information is extracted from observation. A mental procedure can process that information, but not produce new information. You can understand what that information means, what its implications are, what its meanings are, but it cannot produce information. A mental process does not create something from nothing. A mental process—we talked about logical arguments, which are like mathematics, right?—all human beings are mortal, Socrates is a human being, therefore Socrates is mortal. So that mental process does not produce new information. That Socrates is mortal was already implicit in the assumption that all human beings are mortal. If the assumption is indeed true, then the conclusion is also true. But who told you the assumption is true? Only if you reached the conclusion that in our world this assumption is true can you infer that Socrates is also mortal. If not, then you’re talking about a very hypothetical Socrates. Okay? And therefore, basically, thought—according to the empiricist view—is not a tool for knowing the world, but a tool for analyzing pieces of knowledge that come to us through observation. But to know the world, you need observation. Okay? And rationalism says no: the intellect is a means of knowing the world. And what I said when we talked about intuition—what I claimed was that basically I agree with empiricism, only I claim that what is called rationalism is not a purely mental process, but rather some kind of observation of the world, though not with the eyes, not with the senses, but observation by means of intuition. Through the eyes of the intellect, in the language of Maimonides. Okay? So therefore, basically, what we have here is that rationalism is expanded empiricism. It’s empiricism that doesn’t use only the senses, but also what is usually called intellect, though one could call it some kind of non-sensory cognition. Okay? Yes, one has to be careful with that. Yes. It’s not—it can be said in broad terms: you can define things; if you define things about the world in your intellect, then because your intellect is part of the world, those definitions are influenced by how the world is structured and why, and by means of that same intellect one can define things and its conclusions will be true in the world. Why should they be true in the world? Only because those logical inferences you make are based on information you have in your intellect from the world, so then you—“From the world” is already observation. No, one hundred percent, but the brain itself is part of the world. And with that same brain, if you want, one can define things that do not exist in the world, and then indeed the conclusions are not necessary and have no standing. But if you infer conclusions about things in the world, then it’s fairly compelling that this should also be true of the world. It’s not detached. Definitions—if you saw, I mean, the human brain—I don’t know—a computer produces computer results, and a machine produces machine results. Okay, and the human intellect, because it is part of the world, produces results that are part of the world. But a computer is part of the world too—what do you mean? No, one hundred percent. I’m only saying because the intellect—I don’t know—there’s a feeling that things in the world, the conclusions the intellect reaches, because it is built in a very specific way, then they should also be in the world. Therefore… That inference itself is the result of observation in a very broad sense. True, yes. Fine, so you’re smuggling observation in through the back door in any case, because this inference—who told you there is such a correspondence between the intellect and the world? Rather, you are assuming some assumption that is probably the result of observation. Fine, so you introduced observation from there. And also that—but it seems to me that beyond that you are making many mistakes regarding the intellect. True. For example, Aristotle—you mentioned Aristotle earlier. Aristotle, by means of the intellect, decided that all bodies fall to the earth in proportion to their weight, to their mass. Okay? That a heavy body falls faster than a light body. Fine? And this is a conclusion of the intellect that only sensory observation can refute. True. Okay, and there’s no escaping that. Obviously, if one attacks even that, then there’s no way out. But the fact that Aristotle thought this turned out to be mistaken, and the whole world followed him in this matter—as I said, you don’t need a particle accelerator to do the experiment. Aristotle could have done this experiment a thousand years before Aristotle. Take two stones, one heavy and one light, throw them from the second floor or from some mountain or wherever, and see which one falls faster. You don’t need sophisticated technology here. No one thought to do it because it seemed terribly logical, so why check? What’s the issue? If it’s logical, then it’s probably true. That is the danger in rationalism. But the solution—we talked about this—that the empiricist solution doesn’t really… I think we talked about it. The empiricist solution does not really provide an adequate answer because… it too is not free of mistakes. Not only is it not free of mistakes, but it assumes, it contains within it, a rationalist component. There is no pure empiricism. It is not true that we can learn from observation alone without involving our own mental processes that add information—indeed add information. Every scientific generalization is like that. When I make a scientific generalization, I take several particular cases and make some generalization and establish a law of nature. Okay? The law of nature contains more information than the number of specific facts I observed. In the facts I observed, I have information about those particular cases. It’s very simple, what… Many laws of physics, for example, use Euclidean geometry, right? Euclidean geometry is ultimately the fruit of the intellect. No, obviously. But if you can derive the four axioms from observation, that’s enough. But you can’t derive them from observation. Fine, so then that’s something else: then the observation is intuition. So the laws of nature use—that’s an example—that the laws of nature use things that come from the intellect. I see that these two lines are not approaching each other, say, over a meter, and I say: “Okay, if that’s so, then at infinity too they won’t meet.” That’s a kind of generalization. I’m basically saying they don’t meet, and then I say: “Okay, if so, then in our world geometry really is Euclidean.” Fine, if so, then all the laws of Euclidean geometry also apply to the world, and now we can talk about the world. Okay? And indeed Einstein challenged that. He said our world is not Euclidean. There is curvature of space, at least on large scales; there is curvature of space because of masses and all sorts of things like that, and our world is not Euclidean. So even that observation turned out to be an imprecise observation. Okay? It’s a good approximation, but it’s not an exact observation. Okay, so basically… Yes, maybe I’ll just bring the example I already gave, which is that when I got to Bar-Ilan, they assigned me to teach physics tutorials for biologists. Second year, not first year. In the first year they stuck me in the lab, to my sorrow. But I came from engineering, and they said, “You’re an engineer, teach lab.” I ran away from engineering because I didn’t know—I have two left hands, I don’t know what to do with instruments. I came to physics and they put me in the lab. Never mind. In any case, after a year they understood I had nothing to look for there, so I taught mechanics for biologists. When I opened the first tutorial I gave, I asked them whether the statement “two plus three equals five” can be refuted. Is it scientific? So I led them to yes—you know, students are passive—but I led them to yes, because what’s the problem? Take two apples, put them into a basket, add three more apples, count how many you have altogether. If it comes out… five, then you confirmed it; if it comes out six, then you refuted it. Meaning, it’s a scientific theory. “Two plus three equals five” can be tested by Popper’s criterion; it can be subjected to falsification, so it’s a scientific theory. And then I told them: what happens if you find six there? Six apples? If you found six apples, has the theory been refuted? From now on, will two plus three equal six for you? Be honest—obviously not, right? You’ll say there was an error in the experiment. Right? You will never conclude that two plus three equals six. You won’t give that up on the basis of experiment. Why not? Because “two plus three equals five” is not a claim in physics, it’s a claim in mathematics. It says nothing about the world. When you look at the world and see that it holds there, then you can take all the results of algebra, of arithmetic, and apply them to the world. But arithmetic is not conditional on observation. If you observed the world and got six, then you would conclude: okay, arithmetic says two plus three is still five, it’s just that this does not describe adding apples into a basket. Adding apples into a basket apparently requires some other mathematics to describe what is happening there. But you didn’t break mathematics, because mathematics makes no claims about the world. The assumption that mathematics says something about the world is an assumption in physics, not in mathematics. But even that no one—no one will ever say. What? That if he got six he would say: wait, two plus three is five, and with apples it’s six—no one would say that. But I’ll show you a place where they do say it; that’s why it started in a mechanics course. Yes, adding forces. Fine? Take some body. There is a force of ten newtons northward and a force of ten newtons eastward, and I ask what the total force is. Ten plus ten is twenty, no? So twenty. Not at all—fourteen point something. Yes, you know this? A force this way and a force that way, and the resultant force is fourteen point something on the diagonal, okay? Toward the northeast. So wait—ten plus ten equals fourteen point something? Ten plus ten is twenty, so have we refuted the law in mathematics, in algebra, in arithmetic, that ten plus ten equals twenty? The answer is no! We have refuted the claim in physics that says that adding forces is described by arithmetic addition. Wrong. It is described by vector addition, not by arithmetic addition. Meaning, the assumption that arithmetic addition is applicable in physics is an assumption in physics, not in mathematics. Mathematics says nothing about the world. Physics assumes that a certain mathematical theory fits a certain piece of reality, that one can use it to treat that aspect of reality. But that is an assumption in physics, and as an assumption in physics, of course it can be refuted. It stands up to tests of falsification. If we perform an experiment and it doesn’t work, then one must give it up. Here, for example, Einstein refuted the geometry. But of course he did not refute Euclidean geometry. He refuted the assumption that Euclidean geometry is the instrument that describes the geometry of the world. But that really is an assumption in physics, not an assumption in mathematics. He did not refute the proposition that the sum of the angles in a triangle is 180 degrees; that remains and is always true. What he claimed is that in our world this will not hold. Why will it not hold? Because in our world the assumptions of Euclidean geometry do not hold. Our world is curved. Parallels do meet in our world. Okay? Therefore you cannot take the results from geometry and apply them to the world. That is what Einstein claimed. He did not refute geometry, because geometry cannot be refuted. It is not open to refutation; it makes no claim about the world. So what stands behind all these things? What is this categorical difference between mathematics and physics? Mathematics is definitions. And definitions do not contain information. Information in the sense of claims about the world. They say nothing about the world. It cannot be that the axioms of geometry hold but the sum of the angles will not be 180 degrees. That cannot happen. Which means that if the axioms hold, then the sum of the angles is also 180 degrees. It’s tautological. Yes, exactly. Because the relation “if… then…”—that is logical inference. Right? “If… then…” is logic, not physics. Physics remains in the axioms. There is a book by Tarski, one of the greatest logicians, a positivist, early twentieth century, where he builds axiomatic systems for biology, for physics, for relativity theory, for various scientific fields. I have it at home—I once found it in some used-book store. He builds axiomatic systems for biology, for physics, relativity theory, for all kinds of scientific domains. And his ambition in doing this—I think he didn’t succeed, but his ambition in doing it—what does it really mean? And once again this connects to the domain of definitions we talked about with convexity. Remember that example of convex shapes? He is basically taking all the observational information and packing it into the definitions. And once you give definitions and axioms, and once you pack everything into the definitions and axioms, from there on everything is mathematics. That is basically what happens in an equation in physics. Or a physical law that is an equation, like Newton’s second law, or in the world of quantum theory, Schrödinger’s equation or something like that. Most physicists in the world, when they deal with a problem in quantum theory, say, take Schrödinger’s equation and solve that specific problem according to Schrödinger’s equation and find what is supposed to happen. So they’re mathematicians, not physicists. They are mathematicians if that is all they do—take an equation and solve it, plug in initial conditions or boundary conditions and solve it, that’s all. They’re engineers. You could basically say that about every… No, the engineer builds the system that he solves; the physicist solves ostensibly given systems, natural systems. Fine, but that difference is less important for our purposes here. So this aspect of work in physics is really only mathematics—solve Schrödinger’s equation and you’ll see what comes out. It’s like someone who analyzes forces by vector calculus; he too is only a mathematician. He simply applies vector calculus and finds what comes out. Where does the physics sit? Where does the information sit? The information sits in the axioms. The information sits in the equation itself. The thought that this equation really describes the world. The assumption that this equation describes the world is really the product of all physical research until now. Once I’ve assumed that, from then on I’m a mathematician. But all of physics sits inside that, inside the assumption that this equation describes the world. That is where all the physics sits; the entire physical factual content sits there, inside that law, inside that theory. From there on it’s all just… Now of course, if you discover—you solved Schrödinger’s equation and you discover that in the laboratory it doesn’t work, as usually happens—if you repeat it enough times and can no longer say the mistake was in the experiment, and there are other indications and never mind, all sorts of conditions need to be met—then you go back and say, okay, then apparently the equation is wrong. Here you are already a physicist. For the mathematician that would never happen. The mathematician solves the problem—that’s the solution, that’s all, there’s no… he finished the job in that context. The physicist did not finish… He works like a mathematician, but he doesn’t finish the work with the mathematics. Afterwards he also has to check whether it really works. And if it doesn’t work, go back and correct the equation. Say: apparently the equation is not the right equation. Not that it isn’t a correct equation—there’s no such thing as an incorrect equation. An equation is an equation. It is “incorrect” in the sense that this equation is not the correct description of what happens in the world—which is a claim in physics, not in mathematics. Okay? So this is really the question whether mental constructions contain information, whether definitions contain information. And this is the confrontation between rationalism and empiricism. Historically speaking, the thought of the world since Aristotle—let’s say from Aristotle as a model of the first systematic thought, never mind, the Greek philosophers were not only Aristotle—but from the point they began to think systematically about the world, the overwhelmingly dominant outlook was a rationalist outlook. Meaning, once you think in an orderly way and have reasonable assumptions and draw conclusions from them, then that is probably also what will happen in the world. Therefore, for them, science was a branch of philosophy. Philosophy was everything; within it was science, everything. Today philosophy is a separate branch. And I said—we once talked about this—that I don’t think that is so justified, by the way. But that’s how people think today. Philosophy is not committed to what the assumptions are; it only says that if the assumptions are such-and-such then that is the conclusion, like logic, say. That’s the somewhat thin conception of philosophy. And science has to check what is true, which assumptions are true. Meaning, you do not just start with assumptions and check what follows from them. So for Aristotle, as with what he assumed about mass—that bodies fall to the earth at a speed proportional to their weight because that seemed terribly logical to him—he drew his conclusions and built an entire physics, all of which is nonsense in terms of its description of the world. It may be internally consistent, but it does not describe the world correctly. Because for him science was a branch of philosophy, so he had a reasonable assumption and went with it—no problem, it didn’t have to stand the test of the facts. And so this continued through the Middle Ages. The outlook was rationalist, and therefore philosophy and theology were indeed very speculative. They invented there everything we were educated on: that the Holy One, blessed be He, has no body and no bodily form, and that He is infinite, and this whole great theology that was basically created in the Middle Ages—by the way, not so much by Jews, mainly borrowed from Islam and Christianity—and with the gracious help of Maimonides and the Jewish philosophers of the Middle Ages, it was Judaized and became principles of faith. Exactly as Aristotle was Christianized by the Church and became Christian principles of faith, and whoever denied Aristotle was a heretic. So Maimonides also did something a bit like that. The first four chapters of the Laws of the Foundations of the Torah are basically Aristotelian physics, and that became a sort of four chapters in a book of Jewish law. So he made a very similar move, and all this stemmed from the idea that if the intellect says so, then that is probably how it is. But it never occurred to him that even if it is logical, maybe that is not what happens in the world. It seems terribly logical to you—fine. In your mental constructions, you build all sorts of constructions about the Holy One, blessed be He, or about theology or what He does or what He should do, and from that you infer what happens in the world. Therefore it is no accident that the denial of religious beliefs came together with the breakthrough of science. And to this day I think that among scientists the percentage of believers is much lower than in the general population. Scientific thinking, in its essence, has something heretical about it—not that I’m condemning it right now, I’m describing it—because it is basically unwilling to accept things just because they are logical. Believing in God is logical: it solves some slot for me; the world is terribly complex, I need someone who made it, otherwise I’m left with some riddle—how can such a thing be? So if such a being is needed, then He exists. That is classic rationalist thinking. Yes—if such a thing is needed, then it exists. Then comes a scientist, who is a skeptic, and says: wait, who said so? Check and see. So what if it seems terribly logical to you? Maybe complexities arise by themselves? In that sense this is scientific thinking—at least the original science, which thought it was really pure observation. Until Hume somewhat reset the scientists and showed them that within the scientific method too there are lots of rationalist components. You also assume all sorts of assumptions that are not the result of observation—causality and induction and all of Hume’s problems. But in the early scientific thinking of the sixteenth and seventeenth centuries, when they thought we would cling only to observation and learn from that, Bacon’s logic, Francis Bacon’s, was the scientific logic—the logic of induction. That was a conception that came to replace rationalism. And its infrastructure was that from definitions one cannot learn anything about the world. From definitions and from my assumptions I cannot learn anything about the world, because these are processes that characterize how I am built—what does that have to do with what happens in the world? To know what happens in the world you need observation. Definitions cannot contain information. That was the essence of the empiricist revolt. Okay? Then in the eighteenth century Hume comes along and says: friends, if you want to ground science—which by then had already advanced quite a bit, in the eighteenth century there really was science, it wasn’t entirely empty—Hume looks and says: friends, the science we’re talking about is not the empiricism you’re talking about. Meaning, it contains many assumptions and hypotheses that come from us, not from observation of the world. And we need to give them up, and then science becomes very thin. Science does not claim there is a law of gravitation; it only says that all the data we have observed so far fit a situation in which there is a law of gravitation. But I do not claim that there is a law of gravitation in the world, because I have not seen it. If I want to be a consistent empiricist, then I have to stick only to what I have seen. And every scientific generalization breaks beyond what I have seen. Okay? Every principle of causality breaks beyond what I have seen. No one has ever seen a causal relation between things. You cannot see a causal relation. You can see that this happens and then that happens. How do you see that it happened because of it? The “because” is not exposed to observation. Okay? So at a certain stage, after Hume—and then Kant—they tried to qualify empiricism and say: within empiricism there are some rationalist components. We do not learn only from observation; it is naïve to think that all our claims about the world are the result of pure observation. Fine. To this day people are still tangled up in this issue—yes or no, and all kinds of definitions here and definitions there—but that’s basically everything that happens there. In my view nothing else happens there in philosophy of science. Meaning, they’re just different ways of formulating these problems. Now, in this historical process there are several very interesting points that bring out this issue of whether definitions contain information. The first point is the ontological proof. As I said earlier, that’s the name Kant gave it. Anselm, who created it, did not call it the ontological proof. Never mind—in Kant’s classification that’s what it’s called. Kant is eighteenth century and Anselm is, I think, twelfth century, the Archbishop of Canterbury, Anselm of Canterbury. He went through many upheavals, never mind, was thrown to France and returned to England and suffered quite a lot. In any case, what’s the point there? Basically he tried to prove God’s existence from a definition—which is a very blunt expression of rationalism. It basically says: I’m now going to define a concept, and I’ll show you from the definition that it exists. Now to say that it exists—that is already moving from the level of definition, where define whatever you want, to a claim about the world. Which of course, from empiricist glasses, is the mother of all sins. Meaning, it cannot be that you define a definition and from that infer the conclusion that the thing also exists. There is no such thing, it cannot be such a thing. What can happen is that you define something and afterward realize that it exists. You saw that it exists—fine. But you cannot infer from the definition that it exists; you need observation for that. It cannot be. Now Anselm did this move—I said in the twelfth century—and this is still deep inside the rationalist outlook. In the sixteenth century there was another interesting station on this axis of the struggle between rationalism and empiricism, and that is Descartes’s principle, Descartes’s cogito: “I think, therefore I am,” sixteenth-seventeenth century perhaps, early seventeenth, where Descartes basically tried to prove the existence of the person himself, and afterwards also the existence of God, by rationalist means—that is, from definitions without observation. Even though Descartes was also a scientist, not only a philosopher—also a mathematician, also a scientist—he was a polymath. But he grasped that science too needs to begin from some rationalist basis, which Hume later developed more fully, much later. Therefore he searched for some anchor point: how can I know something about the world with certainty? Now observation never gives me certainty. Observation—maybe you were mistaken, maybe this, maybe that, maybe you missed something. He was looking for something that would be based on a logical argument. A logical argument is the prototype of certainty. If you have a logical argument proving something, then it is secure, then it is certain. He was looking for some kind of Archimedean point on which one could build all philosophy and all science, and it would be secure. Otherwise we just keep spinning like a chicken—basically that is what he was looking for. Then he said: how can I find such a thing? So he began his methodological move—yes, he casts doubt on everything and so on—he began his methodological move and arrived at a logical argument whose conclusion is that he exists: I think, therefore I am. Now people don’t really understand the meaning of Descartes’s argument. He wasn’t just someone who doubted whether he existed and then finally calmed down—yes yes okay, I exist, we can relax. That’s not the problem that troubled him, whether he existed or not. The problem that troubled him was the struggle between rationalism and empiricism. Descartes is the beginning of the seventeenth century, when the empiricist revolt was already at its height. The claim was that by rational or rationalist tools you cannot arrive at claims about reality. Thought is not a tool for knowing reality. And this outraged Descartes, because he was a rationalist. He tried to show that there are things about reality that I can derive from pure thought. So he found “I think, therefore I am,” his own existence, which he had never doubted. That was not the point he lacked—whether I exist or not. He wondered to himself whether he existed or not—obviously he existed; he describes it. This is not a real doubt; it is methodological doubt. It is a methodological doubt whose purpose is to show that there is indeed a way to derive information about the world from a definition or from an assumption, from a mental process. In what sense does he exist? In what sense? Maybe it’s—we’ll talk about that. We exist because we’re here, or maybe everyone is just dreaming and it only seems to you, I don’t know. It seems to you? Then no—the only question is: to whom does it seem? Who is this “you”? I don’t know. Fine, in a moment—I’ll spell it out a bit more. I’m just placing it in context. In this context, the claim “I think” is really observation. Right. Observation of your own brain, observation of your own consciousness. Seemingly. And then that’s my next step. So this is the Cartesian project—yes, Descartes’ project; Cartesius is Descartes’s Latin name, that’s why Cartesian coordinate systems are named after him, because Descartes invented them. So Descartes’s project was basically to prove something on the basis of definition alone. Now the question is whether he succeeded. It’s a very non-simple question. I tend to think not, but I’m not entirely sure; it’s very tricky. But look, seemingly Ido is right. What do you mean, this is nonsense—obviously he assumed some assumption that is the result of observation. After all, he says “I think.” How do you know that you think? That’s observation. You observe yourself. People asked here earlier whether I can observe myself; I said I’d get to that. I observe myself, I reach the conclusion that I think, and I infer from this the conclusion that if I think, then there is someone who thinks; therefore I exist. He does an “either way.” Wait, that “either way” is in a moment—I’ll explain that. Right, you’re correct, but I just want to do it in stages. Because the observation they’re talking about is sensory observation, and this already is not… Never mind, never mind—even if it’s not sensory observation, there is still observation here. So this is the observation the Rabbi is talking about. Wait, hold on, step by step. That is exactly the two… Okay, one more second. So there is observation here, and therefore what is this pretension of thinking that there is basically a rationalist procedure here for learning about the world from thought? Not true. You begin from observation. Let’s say, in Kant’s classification, this does not parallel the ontological proof, which is a proof from conceptual analysis, but rather the cosmological or physico-theological proof. You observe something and then infer your conclusion about the world. But that’s not correct. That’s an incorrect understanding of the cogito. Why? Because by the same token he could have said “I walk, therefore I am.” Right? If I walk, then clearly someone who walks exists, otherwise who is walking? What is special about “I think”? “I sit, therefore I am,” “I am sad, therefore I am.” Sure, if something is said about myself, then obviously there is someone of whom these things are said, and I therefore also exist. So what kind of idiotic argument is this? Meaning: fine, okay, you assume that you exist, so you exist. Okay. That’s not the argument. It’s specifically “I think.” “I walk, therefore I am” is not interesting. “I think, therefore I am” is interesting. Why? Because Descartes’s whole argument was not at all to prove that if I think then I exist. That’s trivial. If I think then I exist, and if I walk then I exist, if I sit I exist, if I am sad I exist—it’s trivial. That is not Descartes’s innovation. Descartes’s innovation was: I want an assumption—“I think,” or “I walk,” or something like that—that cannot be challenged. If I can arrive at something that I am doing and that cannot be doubted, then now if I infer from it that I exist, that too cannot be doubted. After all, if I say “I walk, therefore I am,” someone will come and say: “Who said you’re walking? Thanks a lot. If I don’t know that I exist, I also don’t know that I walk. You want to prove to me that I exist from the fact that I walk? I don’t know if I walk; that too is doubtful to me.” He looked for an assumption that cannot be challenged. And he claimed that the assumption “I think” is an assumption that cannot be challenged. That’s Descartes’s claim, not “I think, therefore I am.” “I think, therefore I am” is already the conclusion; it’s like solving Schrödinger’s equation. The clever part is finding Schrödinger’s equation, not solving it. Formulating the equation, discovering that this is the right equation. In other words, Descartes’s innovation is not “I think, therefore I am,” but rather “I think is necessary.” The claim that “I think” is a necessary proposition. Now, if you infer from the fact that I think that I exist, then it also follows that the fact that I exist is necessary. The move from “I think” to “I am” is a trivial move; that is not Descartes’s innovation. That is obvious. Also if I walk then I am. Fine, if I say something about myself then I have assumed that I exist, okay, that’s not the innovation. Descartes’s innovation is that “I think” is a claim that cannot be challenged, and therefore if I think then I also exist, and “I exist” too is a truth that cannot be challenged. And here he arrived at the certain truth. That is still different from rational thinking. “I dream” is also… Rational thinking, apparently, could be transferred to a set of rules and let a computer generate the rational inferences. Why? It does not depend on the existence of a person, rational thinking. It is something abstract that stands on its own. No, that too isn’t right. Even “I think, therefore I am” can be done by a computer. But “I think” is still something subjective. Because it is the result of observation. A computer doesn’t think in my place, and then it won’t mean that I exist. No, because it wouldn’t think. A computer doesn’t think. Someone who doesn’t think cannot go through that process. Not the rationality involved in… No, someone who doesn’t think cannot go through such a process, because the computer is not a thinking being, so it will never give itself an account of “I think, therefore I am.” I also cannot, for example, do this regarding your thoughts, because just as I doubt whether you exist, I also doubt whether you think. I can do it only regarding myself. Therefore only a human being, or only a creature that has thought, can go through Descartes’s process. Only such a being can actually arrive at the foundation. It doesn’t mean that we’re always awake all the time, we always… No, now I still haven’t stated his argument. There is breathing, we also have a pulse… No no, not that—maybe not, I doubt it. You think… Exactly, that’s the point. You can doubt everything. He has the evil deceiver—I just haven’t yet stated the argument. How did Descartes arrive at the conclusion that the fact that he thinks is necessary? How did he establish that? Because if I were doubting—if I cast doubt on the fact that I think—that too is a thought. The doubting itself is also a thought. I think that doubt. Right now I have a doubt whether perhaps I do not think. That itself is a thought. Right? Which means one cannot doubt the fact that one thinks. I think. “I think” is a necessary claim. A claim whose opposite is incoherent. It is a proof by contradiction. If you assume the opposite, you run into a contradiction. If you assume that you doubt whether you think, that itself is a thought, so you still think. Fine? Therefore this is a claim that cannot be doubted. Or in other words, all these funny people who say: wait, maybe my thinking is imaginary? Maybe I myself am imaginary? Imagined by whom? Who imagines me? I never understood these skeptical wonderings. Maybe I’m only imagining that I exist. Who is the imaginer? The imaginer is me. I exist. What do you want? The claims of the evil demon, yes—Descartes’s deceiving demon. The fact that when we sleep we do not think and nevertheless exist even during that time. Why do we not think? Of course we think. We don’t have the feeling, the certainty. No, but we have mental processes like thought, dreams, whatever. They tell you there are eight hours of sleep; you don’t feel it the way Descartes’s arguments work. What—you experience your dreams in sleep? Say at a moment when you’re not dreaming. Let’s say at a moment when you’re not dreaming, okay? I woke up from sleep suddenly. If you didn’t wake up from sleep, then you don’t remember the dream. Also during sleep you don’t remember what you thought when you were awake. That’s one from the twentieth century, but let’s speak about the experience as Descartes experienced sleep when he was awake. When he was awake. He experienced sleep while awake, you’re asking? When he woke up. When he woke up. Fine, because when he woke up he did not know that sleeping Descartes existed. Sleeping Descartes knew that sleeping Descartes existed, but he didn’t know that awake Descartes existed, because sleeping Descartes didn’t remember that awake Descartes thinks thoughts; he only dreams dreams. Not relevant. You can say that Descartes as a function of time exists at this moment in which I am speaking now. Fine, correct. Exactly. He proved his existence at that moment when he made the cogito argument. At that moment. At every other moment, even a waking moment, no—you need to do the cogito anew. And the existence of someone else also can’t be proven, because I don’t know—maybe you don’t exist. No problem; Descartes did not have the goal of proving that I really exist, that’s why I told you. His goal was only to get a foot in the door. Meaning, to try to show that one can derive some fact from definitions, from assumptions. The fact of my existence at that particular moment when I make the cogito argument—that’s all. And did he continue from there, or did that only give him the ability to say, I can think thoughts that will already be about the world? He did continue from there. Perhaps that’s the only thought one can say about the world? Right. He did continue from there. He claimed that from this base one can also infer by logic that there is God. So he does continue further. His continuations, in my opinion, are much weaker—much weaker. By the way, the ontological proof too—the attempt to prove God’s existence. Meaning, if I exist then there is God because someone created me. And he has several proofs for the existence of God: an ontological proof, a cosmological proof, because if I exist then how do I exist? Someone created me, therefore God also exists. Then if God exists, then what He tells me is true. So there is science and there is—from this he derived everything. And after that, of course, there are observations. He can’t derive the details of science from this. He only derives the framework. Now that there is science and I have tools to know the world, etc., now I can take those tools, observe, and build science. Now, now empiricism comes in—he was also a scientist. But he needed the conceptual framework within which science operates to be grounded with certainty. He was unwilling to accept anything that would not be certain. So that’s the point people perhaps don’t really understand in Descartes’s cogito argument—in this argument. Descartes’s project was to show that thought is not an empty thing. That a definition or an assumption is something that contains information. And the cogito argument tries to show that regarding a certain piece of information that he chose—perhaps a banal, trivial piece of information, never mind—that I exist at this moment I conducted the project. What is missing compared to the rationalists? He shows possibility. He shows that such a thing can happen—that you infer a factual conclusion from thought. Because the rationalists claim, not merely by accident, that this isn’t true and that isn’t true; they say a priori that it cannot be. Thought cannot give you information. And Descartes wanted to break that. So he says one counterexample is enough to break it. I’m showing you possibility. Not certain, because this is a very special example that proves only itself. To go on to say that I can, from thought alone, say things about what is outside me, not only about the thought itself—that is already a further stage, a very big leap from there. That’s a big question. It could be that this itself is a refutation of the cogito itself. How can one refute the cogito? What? How can one refute the cogito? Well, that’s a discussion I’d need to do separately. I’d also need to prepare it, because it’s subtle. I need to think about it and get ready. I didn’t intend to do that now. If you want, we can play with it a bit now. The point is that I once talked about Ron Aharoni’s book on a cat that isn’t there. So its focus—and the cogito perhaps. The central example, but very beautifully he shows it with many other philosophical issues. He claims this is all of philosophy: that when I speak about myself—“Only about myself did I know,” as the poet says—when I speak about myself, the “I” and the “self” are two objects. There is the subject and there is the object. When I observe myself, it is very important to separate those two functions, because when I connect them I arrive at all sorts of circles that lead me to paradox. And if I speak in the language of “I observe myself,” I need to treat it as though the “self,” in quotation marks, is some other object. I can observe myself just as I observe anyone else, okay? And when one is not careful about maintaining that distinction, one says “I see myself” and also joins the “I” and the “self” and identifies them—which is exactly what the cogito does. It basically identifies them. “I think, therefore I am,” because if I cast doubt, that means I don’t think. The one casting doubt is the subject, and the conclusion “you don’t think” is the object; the object does not think. Okay? The cogito is just a classic example of that. So does the subject exist? No, because when you say the subject exists you have already said something in which the subject is an object. The subject is a subject, fine—that’s a tautology: a subject is a subject. When you say it exists, then you’ve already said something else. You’ve said there is such a thing, not only a consciousness that grasps, but also something grasped in consciousness, do you understand? Then, in his view, the leap is not legitimate. I’m not sure about this, by the way—I’m saying what he says. That is really the discussion around which the question revolves: did the cogito really do the job or not? The question is whether I’m allowed to make this leap, this identification between the subject as it itself serves as the object upon which it itself observes. After all… What is logic? Logic is that our thought discusses how we think, right? We formulate logic. What does it mean that we formulate logic? We are basically looking at ourselves and checking how we think. Who is checking this? That checker also uses logic, right? After all, my thought is what checks this. Is that legitimate? Are you allowed to do such a thing? Of course—why not? What’s the problem? What does “legitimate” mean? How can thought… As long as you have no thought, by what tools would you check how thought works? You use those same tools that you discover by using the tools. After all, I’m not talking about knowing, I’m talking about logic. Logic is to discover how thought works. But I’m saying: as long as you haven’t justified how thought works or discovered how thought works, by what tools are you using to discover that itself? I’m not justifying it; I’m discovering it. Logic is forced upon me. I don’t create it. I observe myself and discover that these are the laws… You’re not observing… Again, if you’re speaking of intuitive observation, then fine—it’s the same event as the cogito. But the conception was—the conception of those who raise this question—no, I don’t observe myself. I think about how logic can be. That’s not observation; that’s a mental process. So you’re making a logical refutation that it’s impossible to prove logic. The refutation you’re talking about regarding the observation of logic is itself also a logical refutation. What do you mean? You came and said that you define the definition by means of the definition you’re defining. So you are making a logical refutation which is itself something, so you cannot use that either. No, I can, because according to your view you use logic, but logic forbids such a thing. I don’t know what logic is; I know nothing. But I can show that your method is inconsistent. Exactly—you are inconsistent. That’s what I’m saying. So what? Such an argument too—to show that I am inconsistent and show inconsistency—is logical. Meaning, to reject something because of inconsistency is also logic. Okay, so according to your own view you can’t show according to my view. Fine. On the other hand, there is still some kind of problem here. True, you can’t stop it. Not a problem with anything. True, that’s the claim—that in the end a skeptic can’t say anything, so just be quiet. That’s the conclusion of discussions of this kind. But I’m saying that the cogito argument was basically his project. Now, the even stronger and earlier example, from the twelfth century, is the ontological proof. The ontological proof for the existence of God is also basically a proof—and there it’s absolutely obvious, there you don’t need all that I explained now—it’s right there on the table. Meaning, he takes a definition and derives from it the conclusion that God exists. Let me just give you a taste of this argument. It took a lot of fire, and in my view—I spoke about this in previous lessons—in my view this is truly a masterpiece, one of the masterpiece texts in the history of philosophy. Every sentence, every sentence is precise. Right now on my site there’s a notebook dealing with this, the first one, yes. So he says this: “Therefore, my God, You who grant understanding to faith, grant me, as much as You see fit, to understand that You are as we believe, and that You are what we believe.” This is an opening that is a prayer, yes—a prayer before you begin proving God’s existence, you pray to Him. So everyone mocked Anselm for that: to whom are you praying? You haven’t even proved He exists yet. Meaning, you are not really honest. You assume He exists even before you’ve proved Him. You didn’t come to the matter as a blank page. In my opinion this criticism is mistaken, first of all, but it’s also irrelevant. What do you care if I’m not honest? Check my argument. It’s like Andrew Wiles, yes, who proved Fermat’s Last Theorem. Before he invested many years in it, took enormous risks in terms of his academic career, and sat for years on this thing, investing years until he arrived at the proof—fine, I ask: when he approached this matter, did he believe that Fermat’s Last Theorem was true? Or was he a blank page? Either yes or no. It seems to me it’s obviously yes. No question about it. Someone who invests his whole career, risks his whole academic career on this thing, and years of work—it cannot be that he didn’t believe it was true. So what? What do I care whether he believed it or didn’t believe it? Check the proof. If the proof works, then it’s perfectly fine, and if there is a flaw in the proof, then not. What do you care what his point of departure was, what his motivation was? Exactly the same thing. So everyone mocks Anselm and applauds Andrew Wiles. What’s the difference? There too he didn’t have to be—No, but they checked the proof. They didn’t mock him, saying “you’re not honest,” and throw him out. Everyone understands that if someone presents a proof, you check whether the proof is valid or not. What do you care whether he is honest or not? And he is honest; it’s also not true that he’s dishonest. But even if he were dishonest, what do you care? Check his argument. He presents an argument—check the logic, that’s all. Besides, that criticism is also simply wrong. It is not true that he’s dishonest. It’s not dishonesty. He believes in God, and he says “grant understanding to faith.” He wants to ground his faith so that it becomes something that is part of understanding. He doesn’t say he doesn’t believe, or that his faith depends on getting a proof. He believes even without it. So what’s the problem? We believe many things for which we have no proof. Every one of us, including the greatest atheists. They believe there is no God; they have no proof that there is no God. Fine, they believe there is no God. Also, beyond that, when he wrote this, he had already finished the—he was already after the proof. No, it’s as though he’s documenting: I got up in the morning, brushed my teeth, prayed, said to the Holy One, blessed be He, “grant understanding to faith,” and now I approached the proof, and He granted it. It’s as though this were a chronological description of what he went through that morning. Of course not. This is a literary construction of the text. Therefore I say more than that: since this is a literary construction of the text and not a mere description, he doesn’t write here that he brushed his teeth. He only writes that he prayed. Why? Because that’s not—he also didn’t pray; that’s not the point. The text is built in such a way that it opens with a prayer because he wants to tell you exactly this: that I start from a point of faith, and there is no problem with that, contrary to what everyone would laugh at centuries later. There is no problem at all in the fact that I started from a point of faith; there is nothing inconsistent about it. Fine, that’s another matter. Then he says this: now he begins the—yes, so this is the prayer. Next sentence: “And indeed we believe that You are something than which nothing greater can be conceived.” Notice: “we believe.” What are you jumping to? Wait. First let’s see who exists, and then explain to me who He is. You first explain to me who He is and then you’ll prove it? No—again, wrong. In order to prove the existence of God, one must define who this God is whose existence we are proving. Right? We talked about this in the year when we discussed proofs for God’s existence—I don’t remember when that was, maybe a year or two ago, a year, I don’t remember. I said that every proof for God’s existence assumes some definition of God. With Anselm it’s right on the table. Because he was such a good philosopher, far better than all those who laugh at him, he knew that you have to begin the process with a definition. Here he defines it. He says “we believe”—yes, “grant us to understand that You are as we believe, and that You are what we believe.” And now he explains what it is that we believe, and then he will prove that indeed He is as we believe. How could he say it more clearly? No, he will prove part of what they believe about Him. What? That doesn’t prove everything he believes about Him. Why not? He proves what he believes. He says: “And indeed we believe that You are something than which nothing greater can be conceived.” That is the definition of God. I will prove to you that such an object exists. That is what I believe, and the faith—he’s not bringing in the whole Trinity, that He is good… That is exactly the point. He defines God here for the purpose of the argument; he does not define the Christian God. He defines God for the purpose of the argument. That is exactly the point. That is why this is such a precise text philosophically. It opens with prayer on purpose, and afterwards he brings the definition of what God is, and now he shows that from this definition he can prove that God exists. And this is the peak of—here it is, rationalism in its purest form. He takes the definition, and from the definition he shows that God must exist. And that means there is information inside definitions. Therefore the ontological proof is, in essence, rationalism: there is information inside definitions. So it says this—what is the proof? This is just to give you a taste of it. “Or is there no such nature? For the fool has said in his heart: ‘There is no God.’” Maybe there really isn’t. We believe this—that God is a being than which nothing greater can be conceived. That is the definition. Now he poses the question. Notice, every stage: prayer, next sentence is definition, next sentence is the question under discussion, the research question. Is this belief true or not? This is what we believe—that’s a fact. Now the question is whether it’s true or not; that is the research question. “For the fool has said in his heart: ‘There is no God.’” There is a thesis and an antithesis. Now I want to examine two hypotheses; I want to check which one is true. So one has to set them up against each other. Either there is no God, or there is God—two hypotheses. That is the research question. That is the next sentence. “But surely,” and now the argument begins. Seemingly it is very structured, but he uses these very things in the argument. It’s simply a wonderful text. It’s really built magnificently. “But surely this same fool, when he hears what I am saying—namely, something than which nothing greater can be conceived—understands what he hears.” Because otherwise he couldn’t say that this thing does not exist if he doesn’t understand it. If he says something about it—whether he says it exists or says it does not exist—he has to understand what the concept is that he is talking about. So he too understands the concept “something than which nothing greater can be conceived.” Up to this point we haven’t said anything, right? This is still only definitions and arguments; there are no observations yet, right? Everything is definitions. This concept reminds one of the concept of infinity. Now for mathematicians infinity doesn’t exist; it’s a limit, it’s really something you only approach. Anselm would say that infinity does exist. As some sort of idea, some ideal, yes, in the world of ideas—but yes. Otherwise we couldn’t think about it. But that’s another matter. Because “the Holy One, blessed be He, is greater than anything that can be conceived” is basically to say infinity. Indeed, in the language of medieval scholasticism that is the meaning of saying that the Holy One, blessed be He, is infinite. Mathematicians today could say: the Holy One, blessed be He, is greater than anything we know. That’s a more cautious statement, defining the same thing more cautiously. They do not say He is infinite in a positive sense, but rather that He is not finite. Also via negative attributes, in a certain sense—yes. It says what He is not, not what He is. So he says: then the fool understands what he hears. And then he says: “And what he understands is in his mind.” That’s a definition. What you understand is in your mind. “Even if he does not understand that what he understands exists in reality.” I didn’t say it exists in the world, but it exists in his mind. This is a bit like the cogito, yes? Meaning, if he understands the concept “something than which nothing greater can be conceived,” then this concept can dwell in his mind, yes. “For it is one thing for a thing to exist in the mind, and another thing to understand that the thing exists.” Two different things. “For truly when a painter thinks beforehand what he is going to make, that thing exists in his mind,” even though the painting does not yet exist, still it exists in his mind. “But he does not yet think of it as actually existing, because he has not yet made it.” Notice, he says—this is a very subtle point and many of his critics missed it. He doesn’t say here: but still the painting doesn’t yet exist. He says: the painting isn’t even yet in my mind as an existing thing. I do not think it as an existing thing—not that it doesn’t yet exist. Anselm distinguishes between two forms of thought, not between something that exists and something that is thought, but between two kinds of things that are thought. There are things thought that are in thought as an idea, and there are things in thought as an existing thing, where my thought grasps the fact that the thing exists. That happens only when the painting really exists, right? Only then do I grasp it as an existing object. But before the painting exists, the painter merely thinks it as existing? Yes—then it would be an illusion. But I mean, regarding the painting, he says that when the painting exists, the claim is not only that the painting exists, but that when I think it, I think it as an existing thing. When the painter planned it in advance before it existed, the painter held the idea of the painting, but he did not grasp the painting as an existing thing. Those are two forms of thought. Those are two objects in thought. He is not comparing the thing that exists with the thing in thought, but comparing two objects in thought. There is an object in thought that is the idea in thought, and there is the object that is in thought as an existing thing in thought. Okay? It’s a comparison between two objects in thought. Many objections to this proof collapse because of the distinctions I am making now, but I can’t go into them; I’m just trying to show you the pattern of thought. “But after he has painted it, that thing is both in his mind and he understands that it exists in reality.” Not that after that it exists in reality. He argues that his understanding of the thing is of a different kind. What now exists in his head is not what existed in his head before the thing existed. Because now what exists in his head is a grasp of an existing object, not only a plan for a hypothetical painting. Okay? “For he has made it.” Therefore, the fool too must be convinced that at least in his mind there is something than which nothing greater can be conceived. Right? That too the fool must agree with. He only says that in the world God does not exist, but in the mind God exists—that thing than which nothing greater can be conceived. Right? What are you doing? If so, then because he understands this when he hears it, and what is understood is in the mind. “But that than which nothing greater can be conceived cannot exist in the mind alone.” And here comes the punch. So you agree it exists in the mind, but that cannot be, because if it exists in the mind, then I say: after all, that than which nothing greater can be conceived—and it can also be conceived as existing, right? Even if it doesn’t exist, no matter, you’re an atheist, but it can be conceived as existing in the mind. And that is greater than a God who does not exist but exists in the mind as an idea. So that means that a God who does not exist, who exists in the mind, is not the greatest being that can be conceived, because I can conceive of a God who is infinite and who also exists. So that means I’ve reached a contradiction, because before I thought I was holding in my mind the object than which nothing greater can be conceived, but I showed that in fact there is an object greater than it that I can conceive, so that wasn’t the right object; I’ve reached a contradiction. Therefore the assumption that God does not exist leads to a contradiction, therefore God exists. Fine. Now there are many objections, never mind. Whoever wants can read. I really tried in some detail to analyze it in the first notebook on the site. But what I tried to show is how Anselm took a definition—“that than which nothing greater can be conceived.” That is a definition. Notice, it is even formulated in a modern cautious way. He doesn’t say “an infinite being.” He says “a being than which nothing greater can be conceived.” He formulates it in negative language, just like the potential infinity of modern mathematical thought. Okay? He doesn’t say “an infinite being.” He says “such that nothing greater can be conceived.” Fine? “As great as we like,” in modern language. Okay? And he says: if so, then from that itself, without any observation, I can prove that such a being must also exist, and therefore it exists. So this is a classic rationalist move, because I basically assume something and from that also prove a claim about the world—that such an object also exists in the world. Like with physics, that vector calculus also describes the world, or geometry also describes the world. This is a classic rationalist move. And this probably means that somewhere along the way—and here a much more detailed analysis is needed—but somewhere along the way something entered in; he assumed an assumption about the world. This does not really happen from a definition alone. And I claim that this is some sort of observation through the eyes of the intellect. There is some observation here, and in the cogito too there is some observation. True, it’s not observation through the senses, but it is observation through the eyes of the intellect. Because otherwise one really could not have reached that conclusion. But if we accept rationalism as something that is not only definition but observation, and here I…