Artificial Intelligence – Lecture 5 – Rabbi Michael Abraham

All right,

[Rabbi Michael Abraham] Last time we dealt a bit with the question of what intelligence is, what thinking is. I tried to demonstrate the problematic nature of the common views in the world of artificial intelligence, which define the intelligence of machines, animals, and all sorts of things of that kind, when in fact a very significant component is missing from that definition of intelligence—namely, the component of judgment and awareness. And my claim was that two things are required. That is, besides computational ability, which is a technical matter, two main things are required before I’d be willing to speak of the entity in question as an intelligent creature. The first thing required is that it exercise judgment, meaning that this not be a mechanical calculation. And the second thing required is that it have awareness of this matter, meaning that it do so consciously. I said that if it performs some non-deterministic calculation unconsciously—somehow something non-deterministic happens inside its brain without it being aware of it—I also wouldn’t call that intelligence. That is, it’s just something happening inside me; so what if it’s non-deterministic? But it isn’t me. The self is the conscious self. I am not acting here as an agent. I gave the example of Libet’s experiments, where I think you can see quite clearly the fact that unconscious freedom is not freedom. That is, you freely choose when to press the button, but if that free choice is made unconsciously somehow within your brain processes—assuming there are non-deterministic processes there—that still doesn’t make you the one who made that decision. And therefore I think that both in the realm of values, where I talk about choice, and in the intellectual realm, where I talk about judgment, freedom and awareness are required. Without those two things, you do not choose values and you do not exercise intelligence. That is the claim. And therefore machines, animals, all sorts of things, water too, right? I said that according to that logic you could also define water as having intelligence, inanimate things, and so on, cannot be defined as intelligent entities. That was essentially the claim.

[Speaker B] What is awareness?

[Rabbi Michael Abraham] Awareness is reflection. That is, you look into these processes. You observe them. It’s not that they’re just happening somewhere and nobody notices.

[Speaker B] In what sense is AI not aware?

[Rabbi Michael Abraham] If you think it’s aware, then it’s aware. I tend to think not. That’s another discussion. But assuming it is not aware, I’m saying that you can’t speak of it as an intelligent creature. You can argue about whether it’s aware or not; I think it isn’t, but—

[Speaker C] But that’s the heart of our discussion, the heart of our discussion—

[Rabbi Michael Abraham] No, that’s not the heart of our discussion. I don’t know—mine, at least, it isn’t.

[Speaker C] I’m just trying to understand: the heart of our discussion is this awareness. If, say, we could think that the computer… I and the Rabbi know what awareness is—we may not know how to define it, but we know what it is. If we could claim that a computer has awareness too, then we could close the course. I mean, obviously that’s the question.

[Rabbi Michael Abraham] What’s funny about “close the course”? Then we’d reach the conclusion that a computer has thought and intelligence. I’m not closing the course with any conclusion—neither the conclusion that it has awareness nor the conclusion that it does not. I’m trying to understand these concepts and the relations between them, and afterward everyone can draw whatever conclusions they want.

[Speaker C] Sure, I’m saying—if we thought that a computer has awareness, our intuition would be that it does in fact feel—

[Rabbi Michael Abraham] Then it would be intelligent…

[Speaker C] So what would it still lack in order to be a human being?

[Rabbi Michael Abraham] Nothing. Then it would be an intelligent creature and everything would be fine. I don’t understand—fine, so…

[Speaker C] Then it would be a person.

[Rabbi Michael Abraham] Right, okay, so what’s the question?

[Speaker C] So what remains to ask? If it’s a person, just in the form of a computer—

[Rabbi Michael Abraham] Then here we’ve created a person again… If something in what I’m saying seems unnecessary to you, then tell me. But as a general declaration that the course is unnecessary—I don’t… what am I supposed to do with that? I’m saying that whatever conclusion you draw, draw it at the end however you draw it. What I’m talking about seems to me meaningful; I don’t know—everyone can decide for himself.

[Speaker C] Heaven forbid, Rabbi, Rabbi, Rabbi, that’s not what I meant at all, heaven forbid. I’m only saying: because the Rabbi says the word “awareness” and then moves on as if it’s clear to us and we already know how to define that this is this and that is that, when that’s the focal point of everything.

[Rabbi Michael Abraham] No, no, no, I didn’t move on. Sorry. It’s just a misunderstanding of the flow of the discussion. Right now I’m defining concepts. I haven’t said anything about any conclusion regarding computers or anything else. What I said on the conceptual level is that the concept of intelligence requires both awareness and free judgment. That’s all. Afterward, when we apply this to computers and all sorts of other things, we can reach conclusions or not reach conclusions. Right now I’m defining concepts, that’s all. Now, of course, while defining the concepts I’m using examples that to me may seem obvious, but I have no problem with that—I’m not making a claim about the examples. That is, if you want to claim that water has awareness and intelligence, fine. I tend to think not, but that’s not my claim; that’s not what I’m dealing with right now. What I’m saying is that assuming we take water in the usual sense as inanimate, as something with no awareness and no judgment and so on, then water is not an intelligent creature. That’s all I want to claim—only this hypothetical “if… then…” argument. Now you decide: if you think it has awareness or judgment, then fine, water is an intelligent creature too. So right now I’m just putting the concepts, the tools, on the board. I’m only trying to define them. Okay, now, in order to sharpen this point further, I want to get into the question of semantics and syntax. That is really a question that touches the core of our discussions. Let’s go back for a moment to the Chinese Room. We return to the Chinese Room. In the Chinese Room there’s a person sitting there who doesn’t know Chinese, but in terms of input and output he functions like a person who does know Chinese. How does he do that? He performs some mechanical manipulations. Right? He basically receives input written in Chinese and somehow matches it with output that is also written in Chinese. He understands neither the input nor the output, but through endless time and swapping signs and all that, he manages to match an output to every input. Okay? Now, the activity that this person is doing—I’ll call it a syntactic activity. What does that mean? Semantics and syntax: semantics is meaning, let’s say, and syntax is structure, structure or form. And you can relate to thought processes purely on the formal level, and you can relate to them on the level of meaning. Right? When we formalize a claim in logic, for example, we can talk about a logical formula: P and Q imply Z. Okay? Now I can discuss this on the purely formal level—that’s the formula—and work on the formula while completely ignoring what P stands for, what Q stands for, and what Z stands for. I simply manipulate symbols. More than that, I can even ignore the meaning of “imply.” I can ignore that too. I say: P and Q imply Z—that’s a formula. And now I can say: from this formula one can derive another formula, without saying what P is, what Q is, what “and” is, what “imply” is, what Z is—I say nothing. I’m just performing manipulations on symbols. That is a syntactic action. Okay? I’m basically talking only about the formal dimension. When I think about the meanings, then I can say: P is clouds, and Q is rain, and clouds and rain imply cold; Z is the cold. So now I can think about the meaning of the statement and decide whether it’s true or not. Here I’m dealing with semantics. Okay? Semantics is the meaning of what’s written here, and syntax is the structure or external form of what’s written here. Now, we have two ways of reaching conclusions from what’s written here. We can reach conclusions while completely ignoring the meanings of the things involved—that’s the whole point of formal logic. And we can reach conclusions through thinking about the meanings, about the concepts, understanding what is happening here, and inferring a conclusion from that understanding. These are two parallel channels of operation: one on the syntactic plane, one on the semantic plane, and there are correspondences between them. In logic there are even theorems—the theorems of soundness and completeness and so on. Those are theorems that establish a full correspondence between semantics and syntax. That is, you can work semantically and you can work syntactically and you’ll get the same results.

Now let me perhaps bring an example that sharpens this point on several levels. Look at this table here, right? This is a problem from Hofstadter’s book. In Hofstadter’s book, Gödel, Escher, Bach, he talks about the foundations of computability and logic and how to relate to axiomatic systems and the like. Among other things, he brings a puzzle there that he calls the MU puzzle. Okay? Now, the puzzle is built as follows. We have an alphabet of three letters: M, I, and U. Okay? That’s our alphabet. A language whose alphabet consists of three letters. Now various combinations of the letters create words, right? MIU, MMI, UUIM, or any combination of M, I, and U creates words. Okay? Now we want to define what the legal words in the language are. Okay? So we have four rules. Rule one: add U to the end of any string ending in I. That is, anything that ends with I—you can add U after it. Here the “to” got reversed because English and Hebrew always play tricks. MI becomes MIU. Why? Because it ends with I. Anything that ends with I—you can add U after it. Okay? The second rule: if some string begins with M, everything after it can be doubled. For example, this begins with M, and after it there is IU, so you can double IU and then it becomes IUIU. That’s the second rule. The third rule: any triple I can be replaced by U. Okay? And the fourth rule is: if there are two U’s, you can delete them. They cancel out. Okay? Here is the formal formulation, here is a demonstration, and here is the verbal formulation. Okay? There are four rules in the language.

Now what’s the idea? The idea is that we start with the word MI. It is given that this is a legal word in the language. From here on, how do I know whether some word is legal or not? I check whether it can be reached by means of these four rules, starting from MI. For example, the word MIU is legal, right? Because you start with MI and according to the first rule you can add U. The word MIUIU is also legal, because if that word is legal then according to the second rule you can get MIUIU. Okay? Now the word MIUIUIUIU is also legal, right? You can double all four of those strings according to the same rule. If I reach three U’s I can delete them; if I reach three I’s I can delete them and replace them with U, and so on. That’s how you generate different words in the language. Those are the legal words in the language—that’s how the language is defined. Okay? Now the question is: is MU a legal word in the language? That’s the puzzle. What we have here are basically called grammar rules—four grammar rules. Of course, here the grammar rules determine what a legal word is, not what a legal sentence is as we usually think of grammar rules, but never mind. This language is made up of strings of letters that we call words. The alphabet is just three letters: M, I, and U—that’s it. Meaning, if you put in Z or X, that’s not a legal word. The word must be made up only of M, I, and U or combinations of them. But even there, not every combination is legal; only a combination such that if you start from MI and apply these rules in any order and any number of times, and you can get to it—that makes it a legal word in the language. Okay? Now the question is: is MU a legal word?

This puzzle is a very, very non-simple puzzle. If you try to solve it, if you think about it, you’ll see that it’s not at all easy to solve. If I gave a computer this puzzle to solve, what it would basically have to do is simply apply all the rules infinitely many times to MI and, I don’t know, make a list of all the words that come out. In principle, there are infinitely many such words. Right? How do I know? Because there are rules there that can be repeated infinitely many times, and you don’t return to the same chain, to the same string. Okay? Therefore there are basically infinitely many words in this language. But the way to know whether a word is legal in the language or illegal in the language is basically to see whether you can build it from MI by applying these rules. In some order—say rule two, then rule one, then rule four, then rule four again, then rule one and rule three, and so on. It doesn’t matter. In whatever way. And then see whether you can reach that word or you cannot reach that word. Now the problem is a very, very hard problem, because if I gave it to a computer, there’s no systematic way to solve it. Why? Because suppose I ask you, for example, about MU. Try to think how you could know whether by means of these rules you can get from MI to MU. I simply have to try applying all the rules back and forth—infinitely many possibilities. Who knows? Maybe there’s some possibility I missed that in the end will bring me to MU. And there are infinitely many possibilities, infinitely many words. We really have no way to write down explicitly all the possible words in this language, because we simply have to generate them, and that requires applying this generator infinitely many times. And therefore we have no closed way of describing this language, this collection of legal words in the language. The method Hofstadter, who wrote the book, chose—right?—the method he chose was by means of these four rules. That’s all. You’ve got four rules, and that’s a compact way to describe the language. Right, it describes the language, but that method does not give you a test for a given word. Indeed every word you generate that way is a legal word, but if a word is given and you want to ask whether it is legal—that’s the reverse problem, the inverse problem—there’s no way to solve that. How would you solve it? You’d have to generate all infinitely many words and see whether it appears in the list. There’s no way to do that.

So now, there isn’t even a way to check, say, “write for me all the legal words of length three in the language.” I don’t know. How can I know? I have to apply these rules many, many times, and then some of them cancel and some lengthen, so there are many, many ways to reach a string of three letters. And I don’t know which strings do. Basically there are three to the third power—there are twenty-seven possibilities, right? For three-letter strings when the alphabet has three letters: MMM, MMI, MIM, IMM, and so on. Then with U. So there are twenty-seven combinations of three letters. I have no way of knowing whether all twenty-seven are legal or not, because I have no systematic way to generate them. If these rules were rules in ascending order—that is, they generated all the words of length two, and then all the words of length three, and then all the words of length four—the problem would be solved. Because I could get to all the words of length three, and then I wouldn’t need to continue, because once I know all the words of length three, I check whether this word is legal or not. I go through the twenty-seven possibilities and see what’s there and what isn’t. Okay? But the rules here are such that some expand and some contract. You can delete U’s, you can delete I’s, you can add. So sometimes you can arrive at a three-letter word by a very, very long route—reach a hundred letters and then start shrinking and come back to three letters. And therefore there is no systematic way to check whether a certain word is legal.

Now this is a very common phenomenon in logic: the forward definition is straightforward, but the backward definition cannot be carried out. The backward check cannot be carried out. Think, for example, of any area in mathematics. Every area in mathematics is basically exactly this kind of area. Because if I give you, say, a theorem in geometry and ask whether the theorem is true, there is no systematic way to check it. Either you think of a proof or no proof comes to mind. Maybe yes, maybe no. But there is no systematic way to do it. On the other hand, there is a systematic way to generate true claims in geometry. Just apply the geometric inference rules to the axioms, and whatever comes out there is a true claim in geometry. Meaning, the forward direction can be done; the backward direction—that is, given a claim, check whether it is true—cannot be done. The inverse problem is much more complicated than the forward problem. Okay? Yes, of course this is connected to Turing machines and the halting problem, and we’ll still talk about these things, but right now I want to talk about semantics and syntax.

In short, that’s the puzzle. Now first of all Hofstadter does trick number one. He says: let’s now translate these letters into digits. What does that mean, translate? We’ll build another system that is completely equivalent to this one. M will be 3, I will be 1—just look, it’s simply similar, I resembles 1—and U will be 0. Okay? So this is basically a numerical system made up of 3, 1, and 0. Fine? Now each of the rules you saw earlier—let’s look at them again. Look, do you see these rules? For example, how would you formulate this rule? You see, this rule basically says that anything ending in I can have U added to it. Right? Or in other words, any number ending in 1 can be multiplied by 10. Do you agree that that is the meaning of the rule? Right? Think, for example, about MI. What is MI? In the numerical language, MI is 31. Right? Now they tell me that anything ending in I, like 31, you may add U to. Now U is 0. Adding 0 means multiplying by 10. And then we get 310. Okay? Likewise, when we want, say, “anything that begins with M can have what follows doubled,” what does that mean? In this case, for example, I double… first of all, anything beginning with M—you add to it… that is, suppose I have the first string MX, and from MX I create MXX. What am I doing? I am multiplying MX by ten and then adding X again afterward, right? That’s basically what I’m doing. I take, say, 31 and want to turn it into 311. Thirty-one times ten is 310, plus one is 311. So anything beginning with 3, I’m allowed to multiply by ten and add the units—or not all the units, everything after the 3. Okay? Now actually it’s not always multiplying by 10, it’s multiplying by… it depends on what this X is. If it’s one digit, then by 10; if it’s two digits, then by 100. Anyway, the principle should be clear. Now, for example here, if I have three digits in the middle, 111, then I can define mathematically how I do it, so I would, say, subtract 01110 from it. Okay? I can subtract 01110, and essentially that’s what I’ll get. I’ll get XY—say X00Y or something like that. Meaning, I can translate each such rule into an arithmetic rule, a rule of multiplication, addition, subtraction, and so on. Okay?

Now the system with the numbers is completely identical to the typographic system, to the system of letters. There is no difference between them.

[Speaker E] Isn’t that like a Boolean system?

[Rabbi Michael Abraham] No. Because—

[Speaker E] There too it’s representations of one-zero, zero-one.

[Rabbi Michael Abraham] Boolean uses two digits; here there are three.

[Speaker E] Yes, but roughly it works the same way, just with two there and three here.

[Rabbi Michael Abraham] “Roughly” in the sense that instead of ten digits there are only three, but it has no connection to Boolean—it’s not… and the rules here aren’t algebraic rules either. Meaning, this is not like Boolean algebra. Boolean algebra has algebraic rules; these are not algebraic rules—just rules, who knows where they came from. Notice, the point is that if I had given you these rules in their mathematical presentation, it would have been terribly complicated from your point of view, right? Because when I give it to you typographically, you immediately see: here it says III, erase it. If it ends in U you can add another U. If it ends in I you can add a U, and so on. So when I look at this as letters, it’s very easy for me to understand what these rules mean. If I had given you the mathematical formula for these rules, it wouldn’t have meant anything to you. That is, if something begins with 3, then you can multiply it by… by 100 and add the remainder after it once again. That tells us nothing. But understand that for a computer, in some senses, the numerical presentation is actually easier. Because it’s trained to multiply and subtract and divide and add, and those are the natural operations for it. It doesn’t care whether there is any meaning behind these rules. It has four rules and it will apply them; it will work with them. We, as human beings, try to understand the meaning behind the rules. So for us as human beings, it is much easier to handle this system when it is a typographic system—that is, a system built of letters and you look at the chain as a chain of symbols, not as the mathematical form of multiplications and subtractions and so on. It’s much easier for us to grasp it that way—at least for me, but I think for everyone. Okay? For a computer it wouldn’t be easier, but it doesn’t matter, because basically whatever the computer proves about the numbers will also be true about the letters and vice versa. Right? That’s obvious. They’re equivalent; these are two completely equivalent systems. Okay? The difference is that here I work with it on the typographic level: I delete three I’s and replace them with a U. The computer will subtract 01110 from the number, and the result of the subtraction will be the result from its point of view. Meaning, it is performing a completely different operation from the one I perform. But it doesn’t matter. The results we arrive at will be exactly the same results, except that mine will be written in letters and its will be written in digits. Right? Meaning, it’s the same thing, just two different ways of looking at it.

Now look: in some respects there is an advantage to the numerical presentation. As I said, for a computer it’s easy to do arithmetic operations. It’s harder for it to work with strings like this—or at least less natural. You can also give it instructions for what to do with strings, but the natural operations of a computer are arithmetic operations. Okay? A computer, basically, is a Boolean device—that Boolean point came up earlier; we’ll talk about what a computer is. But the basic thing is arithmetic operations; that’s the basic thing in a computer. When you do operations on strings in a computer, you have to build a programming language that does it, and what it basically does is the reverse translation of what I did here: from mathematics to letters, not from letters to mathematics. But it doesn’t matter. In principle these two systems are equivalent: what you prove here will also be true there and vice versa. Okay?

Now what is interesting here is that in a certain sense the mathematical operations actually do have meaning, unlike the typographic ones, which lack meaning in that sense. I understand mathematical operations: here I am doing subtraction, multiplication, addition—these are familiar operations. Here what am I doing? Deleting three I’s, adding a U—this tells me nothing; it does not fit into a conceptual world already familiar to me. In the mathematical operations, yes, I know what’s happening here. I don’t know why one subtracts exactly 111, but I do know what subtraction is, what addition is, what multiplication is, meaning it fits into some body of knowledge I already have. So in that sense, if I were looking for meaning, for semantics, and not just bare form, I might think that the context of the numbers is actually more connected to semantics than the context of the letters. Okay?

But I want to move to an even deeper plane. Let’s solve the puzzle. And after that let’s think about what exactly we did here, or how we managed to make the leap that solved the puzzle. So I’m sharing the rules again, and look how we solve the puzzle easily. We start with MI. These are the four rules, and the puzzle is whether MU is a legal word in the language. Now let’s count the number of I’s in every legal word. Okay? So look. We started with one I, right? This is MI, the one and only axiom—not the first. So we have one I. Now which rules can change the number of I’s? Let’s see. The fourth rule, for example, doesn’t touch the number of I’s, right? Agreed? The fourth rule doesn’t touch the number of I’s; the number of I’s will not change after I apply the fourth rule. In the third rule, the number of I’s can increase by three, right? Instead of U I can replace it by three I’s. Okay so the number of I’s in the result—the opposite. What?

[Speaker D] Yes, Rabbi, it’s the other way around.

[Rabbi Michael Abraham] The other way—from three I’s to U. Right. Again, the Hebrew here is confusing me; it removes three I’s. Okay? So I can—or it doesn’t matter for the number of I’s; it decreases it by three. What happens here? Say in the first one, what happens?

[Speaker D] Also, also doubles. If it had an I then it would double it.

[Rabbi Michael Abraham] The first one does nothing. The first. The second. The first adds U, right? Whatever ends with I, you can add U. It doesn’t change the number of I’s. So the first and fourth don’t change the number of I’s. Only the second and third. The third removes three I’s, and the second doubles the number of I’s. Agreed? If there are I’s at all, their number is doubled. Agreed? Yes. Now look, here’s a theorem about this system: the number of I’s in legal words cannot be divisible by three. There will never be a number of I’s divisible by three. Because—

[Speaker D] Then we’d turn them into U.

[Rabbi Michael Abraham] I didn’t understand?

[Speaker D] Then we’d turn them into U. If there are three I’s, they get turned into U. So then…

[Rabbi Michael Abraham] We turn them into U, so what?

[Speaker D] Then there won’t be… then there won’t be I’s, so it won’t be able—

[Rabbi Michael Abraham] To be divisible by three.

[Speaker D] Fine, that’s what I’m saying.

[Rabbi Michael Abraham] No, that doesn’t prove anything. So what if you turn every three I’s into U? Then you lowered the number of I’s by three.

[Speaker D] And therefore you won’t be able to divide… because there won’t be a case where it divides by three. Because if you can divide by three that means there are three U’s—and if there are three I’s, and if there are three I’s then they need to become U, so there won’t be three.

[Rabbi Michael Abraham] They don’t need to become U. They can become U.

[Speaker D] Oh, it’s not mandatory? Because it says “replace.”

[Rabbi Michael Abraham] No. Every word is the result of applying a rule. Apply it, produce another word. Don’t apply it, and you stay with the previous word. Right. It’s not that you must apply it. If you had to apply the rules, then what would “must” even mean? Then what would the alphabet be? There wouldn’t be any legal words here at all. I don’t understand.

[Speaker D] If you’re saying that if there are three I’s I need to replace them with U—

[Rabbi Michael Abraham] So what?

[Speaker D] Do you have to do that? Is it mandatory?

[Rabbi Michael Abraham] It can’t be that anything here is mandatory. Why? Because if all these rules were positive obligations, then there is no legal word you could show me. The word MI is legal, right? Now I ask: is MIU legal? Yes. No, of course not! Because then you’d have to apply the second rule, and then you’d have to double it. And to that too you’d have to apply that rule, and then you’d have to apply… you’d never stop. Obviously, every word is a word you can generate, and you can continue on and generate another word. But you can also not continue. You can apply all the rules in any order you want; every result will be a legal word in the language. But you can apply them—you don’t have to apply them. Okay.

Now I’m saying: so only rule two and rule three change the number of I’s, right? Now notice: the number of I’s at the beginning is one, right? MI—that’s the first legal word. Rule two doubles the number of I’s. What does that mean? That it can turn them into powers of two, right? Start with one, double the number of I’s—that’s two; double it—that’s four; eight, and so on. It will never be divisible by three, right? A power of two is never divisible by three. Okay? Now if that’s so, then clearly rule three won’t change anything. Because rule three only removes three I’s. But if you can’t get to a number divisible by three, then the number isn’t divisible by three; remove three I’s and it still won’t be divisible by three. Right? Seven is not divisible by three. Subtract three and you get four. Four also isn’t divisible by three. When you remove three I’s, then if at the start you had a number of I’s divisible by three, then afterward too it would be divisible by three. If at the start it was not divisible by three, then afterward too it would not be divisible by three. Meaning, it doesn’t change the property of divisibility by three, right? Rule three doesn’t change the property of divisibility by three.

[Speaker G] And rule two can’t—

[Rabbi Michael Abraham] Ever bring me to a word whose number—

[Speaker G] Of I’s is divisible by three.

[Rabbi Michael Abraham] Friends, please mute yourselves. You can listen to the news, but on mute. Okay. So what this basically means is that rule two gets me powers of two for the number of I’s, and rule three can reduce by packets of three I’s each time. Meaning, the number of I’s in a word will never be divisible by three. Now in MU, how many I’s are there? In MU? Zero. Zero is divisible by three. Therefore MU is not a legal word in the language.

[Speaker D] Rabbi, we can’t see the… the…

[Rabbi Michael Abraham] In the end, here is the proof—not too complicated a proof—that MU is not a legal word in the language.

[Speaker D] So how does it even appear there in the fourth rule, Rabbi—like, they turn that into MU?

[Rabbi Michael Abraham] No, that’s only an example. It’s not that whatever appears there is necessarily a legal word; it’s just illustrating the idea. Okay? If you arrive at MUUU, then you can turn it into… wait, again, I always get confused with the Hebrew—you can turn it into MU, yes. Okay? But apparently MUUU also isn’t a legal word in the language, because if it were a legal word in the language then MU would also be a legal word in the language.

[Speaker E] You just can’t get to it. Right.

[Rabbi Michael Abraham] So that is the proof that the word MU is not legal. Now let’s try for a moment to understand: what did we do here? To me this is a very important example. What did we do here? Do you understand that what we did here was something fundamentally creative, right? Who on earth made Hofstadter think of looking specifically for this property—whether the number of I’s in the word is divisible by three? I am basically proving a theorem: in all legal words in the language, the number of I’s cannot be divisible by three. That is a theorem. I just proved that theorem to you. Now this theorem rules out a huge number of possible words. Every word whose number of I’s is divisible by three—you can erase it, it’s not legal. This theorem proves lots and lots and lots of claims, but of course I could have thought of other theorems. Fine? I could think of other theorems about the number of U’s, about the difference between the number of U’s and the number of I’s, I don’t know what—depending on what I manage to extract from these rules. If I can extract some interesting theorem about that property, there are countless properties I could think about. The length of the word is a property. Maybe there’s a restriction on the length of the word—it can’t be a power of two. Just for example, I don’t know. We might think: can we reach a word of length eight? Or of length a power of two? Maybe not, because you keep doubling and adding, subtracting three—you’ll never reach a length that is a power of two. I’m just throwing that out, right? So I could have imagined infinitely many properties with respect to which I would search for interesting theorems, right? I have no way of knowing which property will prove fruitful. Meaning, which property I’ll be able to prove some interesting theorem about, one that would even help me solve the puzzle. But even aside from the puzzle, I now want to understand the system. So I can prove all kinds of theorems about this system. There are many theorems one can prove about this system. Not all of them will help me solve the MU puzzle, but they’re theorems that will tell me about many other words whether they are legal or not legal. Each theorem will determine some property of legal words or illegal words. Okay?

Now how did it occur to Hofstadter to choose specifically the property of divisibility by three of the number of I’s? Why not divisibility by five of the number of U’s? Or why not divisibility by four of the difference between the square of the number of U’s and the cube of the number of I’s?

[Speaker D] Or do it the other way around.

[Rabbi Michael Abraham] Yes, obviously. He constructed the puzzle that way, I assume. But I’m saying: suppose someone came along and solved it—not the person who created the puzzle. He solved it and that was his idea. That is usually how a mathematical flash of insight works. That is, you suddenly have some idea—you know where to look. An idea is not yet a proof. A theorem in mathematics requires a proof. But in order to understand what you want to prove, you need some kind of idea. The idea tells me that this theorem is probably true. Now let’s look for a proof and see whether that’s really so. So I have some initial intuition, a creative one, that says to me: “Look, this sounds like an interesting theorem; it’s worth checking whether I have a proof for it or not.” So there is some creative dimension here, and only after that comes the mathematics. That is, first you need to think that divisibility by three of the number of I’s is an interesting property, and then we start working like mathematicians: let’s prove that in legal words in the language the number of I’s is not divisible by three. That is mathematical work. Okay? I can do that systematically. But how can I get to the idea that proving the existence of this property will be the thing that gives me the solution to the puzzle? That this will be the interesting thing? That this will be something I can prove at all, even aside from the puzzle? There is no way to know that. And this is plainly a creative step.

Now how does one get there? I think that if, say, a computer—or anyone looking at the numerical formulation of the puzzle—had to solve it, it would have no chance. No chance. Try to formulate this property, that the number of I’s in the string is divisible by three—try to formulate that in arithmetic terms. The number of ones, right, in this number—I have a number with 3,1,0,1,1,1,3,1,3,3,3,0,0,1, right, a sequence of 3’s, 1’s, and 0’s in different forms, in different orders. Now I say: I have a property. What is the property? If we count all the ones in this number, we need to check whether they are divisible by three. Why on earth would we think of such a thing? Even if you had the arithmetic rules—here you can multiply by ten, here you can subtract one hundred and eleven, and so on—you still wouldn’t think of it. Because the thought comes specifically from looking at the typographic presentation, the presentation in letters. When you look at these rules in terms of letters—I add these letters, remove those letters—you can actually come up with this creative idea: let’s check divisibility by three of the number of I’s. That is an idea that can arise in the mind of a creative person when you look at these rules. But if you look at the arithmetic rules, you have no chance. Rabbi?

[Speaker D] Yes. It sounds to me maybe because when I translate it into arithmetic it’s a kind of game, because the fact that when I multiply by ten I get another zero on the side—I understand that—but really I added a zero on the right side. In the original system I’m not really multiplying by ten. There’s no such thing really.

[Rabbi Michael Abraham] The two forms are equivalent. You can look at it this way and that way too.

[Speaker D] No, formally I’ll get to the same thing, but I’m not really doing the same thing.

[Rabbi Michael Abraham] No, you’re mistaken—there is no “really” here. Think: if I came and presented you with this puzzle now, and I said nothing to you about M, I, or U, but gave you the four arithmetic rules: you have a number system with three digits—3, 1, and 0. You can build numbers. The legal numbers are only those built from these four arithmetic rules. That’s it—that’s the puzzle. Now tell me whether the number 30 is legal.

[Speaker D] So what I’m saying is that it would only look the same. The number 30 would be legal, say—it would look legal similarly to the formal system, but it’s not really the same thing.

[Rabbi Michael Abraham] It’s not really—

[Speaker D] It added that… it only looks that way from the outside.

[Rabbi Michael Abraham] You’re mistaken, you’re mistaken. There is no such thing as “really.” What does “really” mean? I could ask you this puzzle in arithmetic—that’s just a puzzle one can ask in arithmetic, that’s all. I asked you that puzzle. Fine? That’s it.

[Speaker E] No, you can’t really solve it in arithmetic, right?

[Rabbi Michael Abraham] The fact that here I chose—wait—the fact that here I chose to present the puzzle first through the letters and only afterward through the numerical translation, that’s just my choice. I could have asked you the puzzle in terms of numbers, and only afterward translated it into letters. That’s just an arbitrary choice. So what does that tell us? It tells us that we have many different ways of dealing with the same problem itself, and these are simply different presentations of the same problem, different languages for handling the same problem. But it turns out that in certain languages it is easier for us to arrive at a creative idea that will solve the problem. If the solution were systematic—meaning, if creativity weren’t required, if there were a systematic way to get there—you could do it both in the language of numbers and in the language of letters. But when the solution is creative, then the whole question is what gives you the idea. What gives you the idea may be that in this language you’ll get the idea, and in that language you won’t.

[Speaker C] Doesn’t that prove that what matters is how you experience the thing? You experience it differently when it’s in a certain language.

[Rabbi Michael Abraham] As far as creativity goes, of course, clearly yes. As far as our systematic operations go, it doesn’t really matter. Do it this way, do it that way—it’s completely equivalent. So if here, if we were talking about operations—if I needed now to generate, or they told me, “generate a thousand legal words in the language,” say that were the exam question you got—there would be no difference if you worked arithmetically or typographically. You apply the rules a thousand times and you get a thousand words. What difference does it make whether you did it this way or that way? Because that is a mechanical, deterministic operation that you can carry out systematically.

[Speaker C] No, but maybe one can infer from the creativity that even when creativity is not required, you are still experiencing things. It’s not that logic now is not… It doesn’t matter whether I experience things, but—

[Rabbi Michael Abraham] I don’t need those experiences in order to do it.

[Speaker C] No, you don’t need them, but we’re trying to learn what happens when we understand the rules of logic and mathematics.

[Rabbi Michael Abraham] Maybe I’m not trying to learn that. I’m asking a very simple question: it doesn’t matter right now whether I experience things or don’t experience things. What I’m showing here is that a solution like this cannot be done systematically, and because it cannot be done systematically, the language in which it is presented matters. A language that gives me some meaning for the things will help me arrive at the creative solution. A language that works in a completely mechanical way, like the arithmetic language—there it will be much harder for me to come up with the creative solution, because I’m functioning like the person in the Chinese Room. I’m swapping letters around; I don’t know what it means. Think of a person who speaks Chinese. A person who speaks Chinese, if asked this question, can answer it with some brilliant solution in the form of a very creative and interesting poem. The person in the Chinese Room probably won’t be able to do that. He’ll give a relevant answer, but that answer will be a mechanical one. It won’t be… he’s operating at the syntactic level; he’s not operating מתוך understanding of what he’s doing. The creative dimension isn’t there. It’s a completely mechanical dimension. And therefore I’m saying that what lies behind the Chinese Room and all the things we’ve been talking about is basically the distinction between semantics and syntax. That is, when you talk about the meaning of things, you are doing something completely different from mechanical activity.

Or in other words—I’ll already jump ahead a little and say this: if a computer had to solve this problem, what it would do is run through the rules and try to generate lots of words, and maybe by chance one of the words would be MU. And of course this is a problem that never halts, in the sense of a Turing machine—a problem for which there is no guarantee that you’ll ever get the answer, because in this case, for example, the word MU is not in the language. A computer will not be able to arrive at an answer of that kind. It just won’t. Not in any amount of time. Even in infinite time it won’t. Why? Because all the checks it performs will always yield more and more words, and none of them will be MU, because we know that MU is not a legal word. So it never knows—maybe later it will appear. But it never appears, because it isn’t legal. We know it isn’t legal. So I’m already telling you now that a machine or computer or, if you like, a Turing machine that tries to solve this problem will not halt.

[Speaker H] But AI today can solve it. What? AI today, say all the—

[Rabbi Michael Abraham] Wait, wait, wait—you’re jumping ahead.

[Speaker H] No, you said a classical computer program can’t solve—

[Rabbi Michael Abraham] I’m jumping—you’re jumping ahead. Yes, yes—no, no, no, you’re jumping ahead. When I get to artificial intelligence I’ll talk about exactly that. That’s why I’m going step by step. So the claim I want to make is that the very same problem can be solved in a creative way that is much easier than a mechanical way, which may be impossible or very difficult. And that is the difference between semantics and syntax. Let’s say: a person thinks about this problem; the machine calculates this problem. Do you understand the difference? When I say that I think about this problem, I am talking about the meanings that arise in my mind when I deal with it. I am not talking about the mechanical manipulations I do, but about what sits in my mind, what leads me to do those mechanical manipulations. When I do the calculation, like water solving the Navier-Stokes equations, then when I do the calculation I am not thinking; I am simply doing the calculation, doing the mechanical operations. It may be that I arrive at the same result and it may be that I don’t, but that is entirely on the syntactic plane. I am not thinking anything and still I am doing the operations. Or I am thinking and doing either the same operations or other operations. It is a completely different mode of activity. And the question whether everything that can be done in this mode can also be done in that mode—that is the central question in our subject. In the MU example, for instance, definitely not. No, in a deterministic machine you won’t succeed in solving this puzzle; the puzzle is unsolvable. This is not NP-complete, right? It’s not that it takes exponential time. No—infinite time. It cannot be solved; the problem doesn’t halt. Okay? Whereas with a creative solution I reached it after a proof of five or six lines.

Now that means there is a certain advantage to semantics over syntactic operation. Or in other words, what I want to argue is that intelligence may be connected—we’ll have to examine this—but I’m just giving a hint for what’s ahead—to the question of how far you can depart from mechanical activity. You exercise intelligence when you don’t do it mechanically, but instead ask yourself, “Let’s think creatively—what properties might help me?” Divisibility of the number of I’s by three. Boom! I made an infinite leap at the level of thought; mechanically I would never have gotten there in my life. And I solved the problem.

Let me give you another example of a problem. Look, there is a famous puzzle. Okay? You have an eight-by-eight chessboard. A chessboard—an eight-by-eight board, don’t get hung up on the chess. Okay? Now you have thirty-two domino tiles. Okay? Thirty-one, sorry, thirty-one domino tiles. Thirty-one domino tiles cover how much? Sixty-two squares. Each domino covers two squares, right? It’s that rectangular piece covering two board-squares. Okay? Thirty-one dominoes that cover sixty-two squares. Can you cover the board in such a way that the two corner squares on a diagonal are left empty? Right? All the rest—that’s sixty-two squares—and I have thirty-one dominoes, which is exactly the area of sixty-two squares, right? Now I ask whether you can cover the chessboard with thirty-one dominoes while leaving the two corner squares on a diagonal empty.

Let’s think about it for a moment. One possibility is simply to try all the existing possibilities. That is a very large number, but finite, right? Unlike MI, unlike MU. Here it’s a finite number. Therefore, for example, a computer could solve this problem. Why? Because there are only finitely many coverings, and it would check whether there is such a covering or not. It would just go through all of them. Unlike the previous problem, this problem is finite. And since it is finite, it may take a very, very long time to solve, but it will halt in the end; after enough time it will halt. So this problem is mechanically solvable. But since it’s so long, a human being won’t be able to check all those possibilities. A human being needs to use creative thinking here in order to solve the problem. So I ask whether you can use creative thinking to reach the solution.

[Speaker F] The corner squares are the same color.

[Rabbi Michael Abraham] Right, and therefore? The board in principle isn’t colored. It’s just eight by eight, right? No, it’s—

[Speaker E] It has nothing to do with color.

[Rabbi Michael Abraham] Black and white, what?

[Speaker E] It has nothing to do with color.

[Rabbi Michael Abraham] It’s a question of covering a board. The board could be an all-white board, like a sheet of graph paper, right, with squares.

[Speaker H] No, it does—it does matter if they’re the same color, because then every domino covers exactly two—like one color and the other color, right? You can’t leave two of the same color.

[Rabbi Michael Abraham] So the color itself doesn’t matter, because in principle it isn’t part of the problem. The problem is about an uncolored board. In principle you can solve it on the uncolored board. The creative step—and again, it’s very easy for us because we’re used to chessboards, so just because we’re used to chessboards we immediately get the association: let’s color the squares black and white. Right? But in principle I could have given you the problem on a white board.

[Speaker E] But that—you can’t, no.

[Rabbi Michael Abraham] Okay, someone already said that you can’t. I’m explaining. Now what am I actually saying? I’m basically saying this: let’s color the squares black and white. Now there is no principled reason to do that. It’s a creative idea. I could have colored them two blacks and a white, I could have left them uncolored, I could have done many, many possibilities—infinitely many. But I decided to color them black and white because I know the game of chess or checkers. Okay? So it just gave me a creative idea: let’s color the board. How did that help? Look—the solution is now immediate. Because the two corner squares on a diagonal are both the same color, right? The whole diagonal is the same color.

[Speaker E] Yes, so you get thirty-two instead of—

[Rabbi Michael Abraham] No, no, you get sixty-two squares and thirty-one dominoes covering them. In terms of area there’s no problem. The area works out.

[Speaker E] No, it sort of has to go out from the border of the square to the same color, sort of. I didn’t understand. I mean because the domino covers in the end it always comes down by color like two-two-two, and one always remains with one outside at the end. I mean I put it the other way.

[Rabbi Michael Abraham] Fine, I can place it lengthwise. What do you mean it has to go outside?

[Speaker E] Okay, so it’s the same thing, but regarding—not the same thing.

[Rabbi Michael Abraham] No, continue lengthwise, go all the way—who said not? So like this—

[Speaker E] It comes out, because it comes out thirty black and thirty-two white.

[Rabbi Michael Abraham] No, that’s something else. Now you’re making a different claim. That’s what someone else claimed earlier—I don’t remember who it was. That’s correct. After I colored the board—and that’s the creative step here—it becomes very easy to see that you can’t do it. Why? Because each domino covers one white square and one black square, right? It must; that’s always the structure of the board. Do you understand that if I cover sixty-two squares with thirty-one dominoes, then those sixty-two squares must consist of thirty-one white and thirty-one black, since each domino covers one white and one black? But if the two corner squares remain empty and they are both white, then that means I covered thirty-two black and thirty white. You cannot do that with thirty-one dominoes.

[Speaker E] The other way around, the other way around—thirty black, thirty-two white, because the two corners are white.

[Rabbi Michael Abraham] Therefore it’s thirty white and thirty-two black. Fine, it doesn’t matter—the idea is clear. So the point is that once I made the creative step and colored the squares on the board, I could immediately solve the problem. Do you understand that for a computer it would take a very, very long time? It would have to go over all the possibilities. Now in this case, because the problem is finite, it would eventually get to the solution. It would discover that it’s impossible—all the coverings, none of them look like that—and therefore it would answer me: no, it’s impossible. But that would take a huge amount of time and many trials, a great deal of computational effort. And I, with a creative idea, could immediately solve it with a two-line argument. Again, exactly the same thing: seemingly semantics and syntax work the same way. But when we get to creative matters, in creative matters semantics has an advantage over syntax. Someone who thinks only in syntax—that makes creative thinking very difficult. For mechanical thinking it doesn’t matter whether you apply it to semantics or to syntax. If it’s systematic, then you can do it on the numbers, on the letters—what difference does it make? It’s the same thing. But that’s only if you have a systematic way to do it. If you need a creative way, then it is better to work in semantics rather than syntax.

Let me give you another example. I actually spoke about this this morning in a class. There is a theorem in topology. In topology one distinguishes between convex and concave shapes. Convex shapes are shapes with the belly outward—a ball, right, whose whole boundary bulges outward. Concave shapes are, for example, a banana, or a crescent moon. Okay? Because a crescent has a part that curves inward, not outward. In general, a convex shape is one all of whose boundaries are convex. If there is a part of the boundary that is not convex, that’s called a concave shape. Fine? It doesn’t have to be all concave—there’s no such thing as entirely concave; that’s pathological. But I’m saying: a convex shape is one where everything is convex; a concave shape is one where not everything is convex. Okay? That is the definition, fine?

Now there is a theorem in topology that says that the intersection of any two convex shapes is also convex. It creates a convex shape. Fine? So for example, I—can you see the board?

[Speaker D] Not yet. Can’t see? Now.

[Rabbi Michael Abraham] Here, look. I take… this, say—well, think of it as a triangle, it’s a bit hard for me to draw with the mouse—this is one convex shape, this is a second convex shape, this is a circle, right? Just to be clear. The intersection between them is this. Do you see this shape? The claim is that if this is a convex shape and this too is a convex shape, then their intersection is also a convex shape. That’s a theorem in topology. By the way, the intersection of any number of convex shapes, but if we prove it for two then it’s true for any number. So one has to prove that the intersection of two convex shapes is also convex. Or thickness. Okay, suppose this is a circle for the sake of discussion. Now this is the intersection. Fine? This shape is the intersection. Fine? Now the intersection of two convex shapes is also convex. How do you prove such a thing? A theorem in mathematics, in topology.

So look, here is the creative idea that will help us. It’s hard to prove; I tried for a long time and didn’t succeed. The idea is this. First of all, in stage one, like good mathematicians, we need to define the concept of a convex shape. Think about this circle, okay? What does it mean to say that it is a convex shape? That it bulges outward at every point. But “bulges” is not language mathematicians use. Mathematicians want to give a sharp definition of what a convex shape is. So the definition is as follows: take two points inside the shape. See them? Now connect them with a line. Okay, I have two points that I connected with a line. If the two points belong to the shape, then the entire line connecting them is also inside the shape. That means the shape is convex. That is the definition of a convex shape. Let’s try to see the idea behind it. Look, for example, at a concave shape—okay? This crescent, this bowl shape. By the way, yes.

[Speaker E] We’re sort of looking at the belly from above, right? From outside. From outside, so that means every cut will cut it into two concave shapes?

[Rabbi Michael Abraham] An intersection is one shape—it’s the common part of the two shapes.

[Speaker E] That’s what I mean. And if I look at it from above, in topology you always look at the surface, how it folds, so it comes out that every convex shape in an intersection, right? I mean if you take it, can I draw here? No, no. In a case where I make a banana and cut and make two points, one at one end and one at the other—

[Rabbi Michael Abraham] Two points, as I just did. You connect them with a line?

[Speaker E] Right, yes. That line? Then I always get the belly.

[Rabbi Michael Abraham] Exactly. That’s why this definition—you see that it goes outside the shape?

[Speaker E] Ah, exactly what I drew.

[Rabbi Michael Abraham] Okay, so therefore it is concave, right? Right. Meaning, a convex shape is a shape such that for any two points inside it, the line connecting them belongs to the shape. A concave shape is a shape for which there exists a pair of points—not every pair, because these two here and here, for example, the whole line connecting them would be inside the shape—but there exists a pair of points such that the line connecting them goes outside the shape. So that means it is concave. Okay? That is the definition.

Now, once we’ve defined that, we’re done. Why? Because look now at this thing.

[Speaker E] What is this software?

[Rabbi Michael Abraham] It’s Zoom’s whiteboard. Zoom lets you share a whiteboard. This is the intersection of two convex shapes, and I want to prove that it’s convex, right? In order to prove it I have to put two points here, right? And connect them with a line, and prove that if the two points are in this shape, then the line connecting them is also in this shape. But that’s trivial. Because if the two points belong to the intersection, then the two points are in this shape and in this shape too, right? That’s what intersection means. Now if the two points belong to this shape and it is convex, then the line connecting them belongs to this shape as well. And for the same reason, the line connecting them belongs to this shape as well, because this shape too is convex. But if the line belongs to both this one and that one, then the line is in the intersection.

[Speaker E] So it’s not mathematical, it’s logical.

[Rabbi Michael Abraham] It’s mathematics. Logical thinking is the kind of thinking used in mathematics. It’s mathematics. Never mind; it’s not important. It’s mathematics, that’s not the issue here. So in the end what I want to say is that after I defined the concept of a convex shape, from that point on proving this claim is a one-line theorem. But before I defined it, I understood what a convex shape was, right? Something with the belly outward and all those hand-waving motions, but I couldn’t manage to prove that the intersection of two convex shapes is also convex. Why not? Because there is no mechanical way to do it. Once you create a definition for a convex shape, suddenly you can already understand in what systematic way you’re supposed to proceed. What do you need to prove? You need to prove that if two points are in the intersection, the line connecting them is also in the intersection, right? That’s straightforward. It has to be—that’s completely mechanical. That is the definition of a convex shape. If you want to prove that the intersection is a convex shape, that’s what you need to prove. Completely mechanical; there is nothing creative here.

Now, how do I prove that? To prove that it lies in the intersection, I need to prove that it lies in shape A and in shape B, right? To prove that it lies in shape A, I need to say that these two points lie in shape A; by its convexity the line connecting them lies in shape A. And of course for shape B the same thing. So if it lies in A and in B, it lies in the intersection. Which is what had to be proven. Completely mechanical. After the creative idea of proposing such a definition for the concept of a convex shape, from then on everything can be completely mechanical; it’s already easy. But at the beginning there is always some step that is a creative step. Therefore if you gave a computer this theorem to prove, without defining for it what a convex shape is, it would not be able to prove it. How would it even begin to deal with such a thing? The problem isn’t even defined from the computer’s point of view until you define the concept of a convex shape. There is no way to explain to an ordinary computer—again, not artificial intelligence, don’t jump ahead on me now, I’m talking about an ordinary computer—there is no way to explain to an ordinary computer what a convex shape is. All you can do is give it a definition. So a computer will not be able to solve the problem. By contrast, I can think of a definition, and after I’ve thought of a definition I have a simple way to prove it.

That is exactly the meaning of thinking about the meaning. What is a convex shape, really? I try to think about the matter: what does that mean? It probably means that it has no inward belly, meaning that every line connecting two points stays inside the shape. But that is creative thought, thought about meaning—what does “convex” mean? You cannot do that by mechanical manipulations.

As an aside, I’ll tell you: did we really prove that the intersection of any two convex shapes is convex? I claimed that we did not. Why? Because we did not prove that the mathematical definition of a convex shape is equivalent to the ordinary everyday definition of a convex shape. That everything an ordinary person on the street would call a convex shape falls under the mathematical definition and vice versa—that the two are equivalent. Intuitively, this mathematical definition seems like a good definition to us, but we have no proof that everything an ordinary person would call a convex shape would also be called convex by the mathematical definition. So you understand that what we really proved is just a mathematical theorem. In terms of life, if you ask me in everyday life whether the intersection of two convex shapes will be convex—I don’t know. It depends on whether the mathematical definition properly captures the everyday concept of a convex shape.

[Speaker C] But the everyday concept is also informal.

[Rabbi Michael Abraham] Right, it’s also informal.

[Speaker C] No, it’s the same thing, so there’s no distinction between an informal concept and an everyday concept; they’re both entirely informal.

[Rabbi Michael Abraham] I didn’t make a distinction between them.

[Speaker C] No, you’re saying the definition might not match.

[Rabbi Michael Abraham] Who said—

[Speaker C] That the definition fits the ordinary concept?

[Rabbi Michael Abraham] That the mathematical definition matches the ordinary, everyday definition. As long as we haven’t proved that, we haven’t really proved the ordinary theorem; we’ve proved the mathematical theorem. So in general one should know that exercises of this kind never solve the problem; they merely sweep the difficulties under the rug. My difficulty in proving the ordinary problem—how does the mathematician solve it? He says: look, sweep all your difficulties under the rug. For the moment give me a definition above the rug of what you call convex, and from there on I’ll continue mathematically—no problem. But how did you arrive at that definition? Who said that definition really describes the problem?

[Speaker C] What do you mean? The definition is the ordinary consciousness of what convex and concave are—it’s the same thing; I didn’t invent something else.

[Rabbi Michael Abraham] The ordinary definition is: it has a belly outward. That’s the ordinary definition.

[Speaker C] Which is an empty statement. What is a belly outward? What is a belly? And what is outward? Where is the outside and where is the inside?

[Rabbi Michael Abraham] It’s all the same intuition. The inside of the shape is within the boundary, and “outward” means when you look from the outside. That’s exactly the point. If the question is a question about everyday life, you haven’t solved it. You solved a mathematical problem that in your view is equivalent, but you didn’t prove that it is equivalent. You think it is equivalent; it seems reasonable to you that it is equivalent. You didn’t prove it. Therefore you cannot prove the everyday problem; you can only prove the mathematical problem. But really, that’s in parentheses—it’s not connected to our main line of thought.

[Speaker E] It’s like we—like what we do in mapping. What? Like what we do in mapping. In mapping every child sees that here there’s a hill, here there’s an ascent, here there’s a descent, and so on. Right? If we define it mathematically—that two points that are on the hill, in the intersection, right?—that will give you some other cut of the hill. But in principle that’s only mathematical; right, informally you can’t prove it. But—

[Rabbi Michael Abraham] I didn’t understand what needs to be proved here and what—

[Speaker E] Because where are these shapes used? In topology. Topology—what is topology? It’s mapping the terrain.

[Rabbi Michael Abraham] That’s topography.

[Speaker D] Yes, that’s topography.

[Rabbi Michael Abraham] Ah, you’re talking—

[Speaker E] About topography, no.

[Rabbi Michael Abraham] No, I’m talking about topology.

[Speaker E] I wanted to make a comparison between the two.

[Rabbi Michael Abraham] That point doesn’t matter to me here; it’s irrelevant. Okay, so what? Let me summarize for a moment what we saw today. What I wanted to show is the relation between mechanical thinking and creative thinking, or between the thinking of a machine that only performs a calculation and a being that relates to semantics and thinks about it—not calculation, but thought. In the end it will also perform a calculation, but the calculation is induced by thought. Okay? There is a difference between the two. In a mechanical operation, it may come out the same and there may be no advantage of one over the other and vice versa. But in an operation that requires creativity, there will often be a great advantage to thinking in semantics over thinking in syntax. Right? When a person wants to write a poem, I assume that a speaker of Chinese will do it better than the person sitting in the Chinese Room, even though in plain prose, let’s say, they’ll speak the same way. Because ordinary conversation is mechanical: you ask me how I am—I say, fine. You ask me what time it is—two ten. Fine, that’s a mechanical operation. A mechanical operation can be done with syntax alone, so it doesn’t matter whether you understand or don’t understand; perform the equivalent mechanical operations and you’ll reach the same answer. But when you tell me to write a poem, that’s not a mechanical operation, it’s a creative one. And the man in the Chinese Room won’t be able to write a poem. He’ll write something that isn’t a poem, or some dry thing and maybe call it a poem. Fine? To do something creative, you need to understand what you are doing in order to create. Creativity cannot work with syntax alone.

Okay, that’s it for now. Any comments or questions? More power to you.

[Speaker D] A small question, may I ask, Rabbi? Yes, yes. I wanted to know—this point really bothered me—the claim that I must have judgment, or cannot be deterministic, in order to count as intelligent. And I wanted to know: understanding—when I understand something, is that intelligence? I really feel like I’m missing the piece about understanding a chain—

[Rabbi Michael Abraham] If you invest judgment in that understanding, then it is intelligence. And if it just comes to you in a—

[Speaker D] Passive way—

[Rabbi Michael Abraham] Completely passive way, someone explained it to you and you understood, then that is not intelligence.

[Speaker D] Wow, I really can’t understand why it has to be that way.

[Rabbi Michael Abraham] If you—I gave the extreme example—do you think water has intelligence?

[Speaker D] I don’t think that’s a good example, the water one. I mean—why? Why? Because with water, all we’re really doing is describing the activity, the results, the implications.

[Rabbi Michael Abraham] But what is water? Again, we’re back to what is real and what isn’t real—like what you asked about the U’s. There’s no truth or falsity here; with water too, all we’re doing is describing its motion. We describe it by the fact that it moves physically. That’s just a mode of description, like the mathematical mode of description. They’re completely equivalent.

[Speaker D] I don’t know, I can’t understand why. Why?

[Rabbi Michael Abraham] Later on—when I talk about analog computers—I think it will be clearer. We’ll talk about a capacitor and an inductor as a differentiator and integrator. I mean, I come—

[Speaker D] My feeling is that when I talk to someone—just, the Rabbi gave examples of conversation in the context of the Turing test, if I’m not mistaken—when I identify whether it’s an intelligent entity or not, a computer… my feeling is that when anyone is told this story, they understand that… when he’s speaking to a computer, it won’t be an intelligent being because he isn’t talking to someone. I mean, there’s nobody there.

[Rabbi Michael Abraham] What does “there’s nobody there” mean? Because there is no awareness behind—

[Speaker D] The movement of the lips.

[Rabbi Michael Abraham] There is movement of the lips. Think of a computer that looks exactly like a human being—it moves its lips and talks to you like a human being, but behind the speech there is no cognition. There is no entity.

[Speaker D] Exactly.

[Speaker C] What is an entity? What is cognition? That’s not an answer—you can’t run away again to words. What is awareness and what is cognition? It’s not simple at all.

[Rabbi Michael Abraham] Earlier you asked what awareness is; I said it’s reflection.

[Speaker C] And what is reflection? What is reflection?

[Rabbi Michael Abraham] Do you want to drag me into an infinite chain?

[Speaker C] No—maybe yes, so what? Does the fact that there’s an infinite chain solve it?

[Rabbi Michael Abraham] I’m not going on that journey.

[Speaker C] No, but that’s the question of questions. Obviously that’s the question of questions.

[Rabbi Michael Abraham] No, it’s not the question of questions at all. It’s a very simple question—what is awareness? What? The simplest question in the world. What is it?

[Speaker C] Does the Rabbi know how to define, fully and clearly, what awareness is?

[Rabbi Michael Abraham] Of course—reflection.

[Speaker C] What is the color red? No—when a person is aware, he feels something, right? What does it mean that he feels something?

[Rabbi Michael Abraham] Awareness is reflection. I explained it, that’s it—that’s the definition.

[Speaker C] Wait. Reflection means thinking about myself, if I understand correctly.

[Rabbi Michael Abraham] Correct.

[Speaker C] Correct. And if I don’t have that—and an animal that doesn’t engage in reflection, does it not have feelings? Does it not feel?

[Rabbi Michael Abraham] I didn’t say it doesn’t have feelings. I’m talking about awareness. Why is awareness—

[Speaker C] Specifically reflection?

[Rabbi Michael Abraham] There can be an animal that feels pain. There can be an animal—

[Speaker D] That is aware—

[Speaker C] Of the fact that it hurts—it’s not the same thing.

[Rabbi Michael Abraham] Why must—there may be no awareness.

[Speaker C] Why is awareness—why is self-awareness so relevant to us? I don’t know—you asked what awareness is, I answered what awareness is. No—regarding the computer, regarding some mathematical or logical calculation. Why do I care about reflection? There’s no need for any reflection in this story.

[Rabbi Michael Abraham] Of course there isn’t. That’s exactly the point.

[Speaker C] You can—

[Rabbi Michael Abraham] Do it in a completely mechanical way. You don’t need awareness. It’s just that if you do it mechanically, you’re not intelligent.

[Speaker C] And if I have—say I manage to teach a cat to do some mathematical calculation, and it does it without reflection because it has no reflection, it doesn’t know it exists—then it’s not intelligent, but you are the intelligent one—

[Rabbi Michael Abraham] Because you taught it.

[Speaker C] But then is it already identical to a computer? Is it already identical to AI?

[Rabbi Michael Abraham] Yes. As for AI, we’ll talk about it. For now, to a computer. Even worse.

[Speaker D] Does the Rabbi distinguish between consciousness and awareness? Hm? Does the Rabbi distinguish between consciousness and awareness? I don’t know. So I’m asking why I have to think—all that it feels, mental events that happen in something, in a cat, no matter in whom—is that not enough for it to be intelligent? I mean, I don’t need it to think about itself; it’s enough that it has some experience that is a mental event.

[Rabbi Michael Abraham] That’s not enough. It has to choose a tool in order to solve the problem. It has to choose a tool in order to solve the problem, that’s all. What does “choose” mean?

[Speaker C] Does it have free choice? Do we have free choice? Yes. What—also in mathematics we have free choice?

[Rabbi Michael Abraham] Of course. Here—the proof in topology—do you think someone arrived at that without judgment? Completely mechanically?

[Speaker C] But how is that connected to free choice?

[Rabbi Michael Abraham] It may be completely deterministic. Judgment—what is free choice? I said: in the intellectual domain it’s called judgment; in the value domain it’s called choice.

[Speaker C] What does judgment mean?

[Rabbi Michael Abraham] Judgment means that you choose from your toolbox and your set of possibilities the most suitable tool in order to solve the problem.

[Speaker C] But how did that happen? It’s a decision. And the decision is based on what?

[Rabbi Michael Abraham] On nothing. On the fact that it seems preferable to me.

[Speaker C] Just like that? You rolled a die? No—you activated some thinking about—

[Rabbi Michael Abraham] Reality, and somehow there were—

[Speaker C] Stochastic processes that brought us to a decision.

[Rabbi Michael Abraham] You’re taking me back to an old question that I’ve dealt with many times, and I assume we’ve talked about it too. There are three mechanisms, not two. Not everything that is non-deterministic is random. There is random, there is deterministic, and there is choice or judgment. That is a third thing. And that third thing has no cause, and it is not random.

[Speaker C] What, it has no purpose? Doesn’t it have that reason the Rabbi talks about?

[Rabbi Michael Abraham] It has a purpose; it has no cause. And what purpose does it have? The purpose is to solve the equation or solve the problem—that’s the purpose.

[Speaker C] But it lacks the need for the solution, so it has a cause to seek it.

[Rabbi Michael Abraham] Why? It wants the solution. It chooses a tool that will lead it where it wants—that is a teleological explanation.

[Speaker H] But why one tool rather than another?

[Rabbi Michael Abraham] Doesn’t matter. It chooses because that tool really leads it there. A person who is not intelligent will choose a tool and fail to reach the solution because he isn’t intelligent; he didn’t choose the right tool.

[Speaker H] No, but he decided to choose one tool and not another—

[Rabbi Michael Abraham] For a reason—or because he was smart enough to understand that such a tool would probably bring him to the solution.

[Speaker C] No, I don’t see it. You see that they deliberate? That’s completely mechanical.

[Rabbi Michael Abraham] I don’t see. Maybe yes, maybe no. I don’t see. It looks as though they deliberate—that’s what I meant.

[Speaker C] “Deliberate” is a nice word. You see they have many possibilities. They can go north, south—

[Rabbi Michael Abraham] They see the wind, they see… A machine too can deliberate in that sense. It can say: check this, check that, and decide what to do. But it does not decide freely.

[Speaker C] What does “freely” mean? How can there be freedom here? How do you suddenly escape from the causality that envelops all of reality into some purpose that itself is, in the end, also a cause?

[Rabbi Michael Abraham] So all right—

[Speaker C] About value-based free choice—

[Rabbi Michael Abraham] But in mathematics too?

[Speaker C] The same thing.

[Rabbi Michael Abraham] In this area—look at my column 35. There I spell it out. That’s what I know how to say about it. There’s also 175, I think, but mainly 35.

[Speaker D] Your column 35, yes. Rabbi, I have to go back to what was earlier. Experience. Something mental—like perception—

[Rabbi Michael Abraham] Experience depends how you define it. There can be a creature that feels pain—

[Speaker D] And still—

[Rabbi Michael Abraham] Yet is not aware.

[Speaker D] Excellent. But isn’t it also true that what the Rabbi said about materialism—that it cannot create consciousness—it also cannot create experience before it creates awareness?

[Rabbi Michael Abraham] Okay, right. So what follows from that?

[Speaker D] No, because it seems to me that when someone who has experience tries to speak, or wants to speak, or wants conversation because he needs something—even without free choice, because he’s compelled to it—I feel that most people, when they look at… when I try to speak to someone who has experience, whether or not consciousness—I want to distinguish between consciousness—why does it matter that he has experience?

[Rabbi Michael Abraham] I didn’t understand why it matters that he has experience.

[Speaker D] Because in the sense that there’s somebody there. There’s somebody there, meaning there is a figure that experiences. Words. The fact that he is mental is just—

[Rabbi Michael Abraham] All that happens in his brain also happens in the computer—exactly the same thing. Except that what happens in his brain is accompanied by what is called an epiphenomenon, a secondary accompanying phenomenon—namely what you call sensations. Okay, so what? I want the sensations to operate the brain, not merely to be operated by the brain.

[Speaker D] Yes, but it’s not only sensations; there’s somebody there. There’s somebody here.

[Rabbi Michael Abraham] I don’t know what “somebody” means. Fine—call it somebody, what do I care? It doesn’t matter.

[Speaker D] And when I talk to somebody, I feel that I’m talking to somebody.

[Rabbi Michael Abraham] Fine, you’re talking to somebody, good health to you. What does that have to do with us? I’m claiming that this “somebody” is not free, has no choice, and therefore also has no moral standing like a human being does.

[Speaker D] Call him somebody, don’t call him somebody. Yes, but I don’t feel that in a Turing test I would say I’m talking to somebody, because there’s no consciousness here.

[Rabbi Michael Abraham] Again, you’re going back to “somebody,” so define “somebody” however you want. If you define it the way you do, then it’s somebody. I have no problem with that.

[Speaker D] And there isn’t that in a Turing machine? And there isn’t that in a Turing machine.

[Rabbi Michael Abraham] Right. In a Turing machine there are no feelings either. But I’m claiming that feelings are not the important thing here.

[Speaker D] Yes, not only feelings exactly. Somebody—something that experiences, inside of which there is this whole thing of feelings.

[Rabbi Michael Abraham] No, not even that. You need awareness and you need freedom.

[Speaker D] Okay, more power to you, thank you so much Rabbi.

[Speaker C] Rabbi, Rabbi, this point that the Rabbi just distinguished between the feeling of pain and awareness of pain—that happens to me a bit at work when I use laughing gas. Sometimes the patient is sedated and does feel pain but is not aware; he won’t remember it when he wakes up. But I still feel uncomfortable with that, and it hurts me when it hurts him, because there is a creature here that feels pain. True, in terms of awareness he won’t remember it. A cat that feels pain, even if it has no awareness—wait, Rabbi, Rabbi, but I still feel distress that it feels pain even though it’s not aware of it, because I know there is subconscious, sub-aware psychic life that is… I’m completely with you, Shmuel. So there is a creature here—even if it can’t be that it will affect him consciously in some complex that appears ten years from now. Why do I care that he isn’t consciously aware of it?

[Rabbi Michael Abraham] Consciousness is only the tip of the iceberg of the psychic life.

[Speaker C] “Tip of the iceberg” is just words.

[Rabbi Michael Abraham] When there is a cat that has no awareness but it hurts, I won’t hurt it because I don’t want to hurt creatures that feel pain. But the fact that it lacks awareness means that it is not on the moral playing field. That is, it has no right that I not hurt it. I have an obligation not to hurt it.

[Speaker C] On what assumption is the Rabbi basing this? Do I have a moral right to hurt a creature just because it has no awareness of it? On what moral basis does that exist?

[Rabbi Michael Abraham] First of all, on what logical basis are you putting those words in my mouth? I didn’t say that. What I said was not that I have a right to hurt it, but that it has no right that I not hurt it. That’s not the same thing.

[Speaker C] Do you have an obligation not to hurt it? Yes. Fine, so I’m not talking about its right. I deny the existence of rights altogether—not for humans and not for animals.

[Rabbi Michael Abraham] There are no rights at all. Another basis for argument. Another one among many. Fine.

[Speaker C] Okay, have a peaceful Sabbath. Thank you very much. Peaceful Sabbath.