Halachic Positivism, Lecture 2

Table of Contents

• Paradoxes in Jewish law and the difficulty of seeing Jewish law as a set of rules
• The loop of matzah from the new crop, spending a fifth of one’s assets, and a positive commandment overriding a prohibition
• The completeness of Jewish law, laws of doubt, and passive omission
• Freedom of choice, active doubt and passive doubt, and the proposal of a lottery
• Interpretation as stepping outside the system and the expansion of the rules over the generations
• The Oral Torah, computers, and the Church-Turing thesis
• Dispute in Jewish law and the claim that the halakhic decisor does not work like a computer
• Turing, the halting problem, and Gödel’s theorem as a parallel to meta-language
• Positivism: completeness versus a set of rules, and the critique of the axiomatic project
• Returning a lost item in a bank: identifying marks, number, the way it was placed, the law of the land, and interpretation of concepts
• Wittgenstein: the illusion of following rules and the example of the series 1-2-3-4-5
• The psychometric exam, scientific language, and “male” physics versus alternative formulations
• Convex and concave, and a theorem about the intersection of convex shapes

Summary

General Overview

The text argues that Jewish law cannot be understood as a closed set of rules that allows for a mechanical calculation of halakhic rulings, because it contains paradoxical situations in which the rules do not provide a decision, and reaching a decision requires stepping out into interpretation that adds a layer not derived from the rules themselves. It presents the tension between the completeness of Jewish law and halakhic positivism, and compares this to Gödel’s theorem, Turing’s halting problem, and Wittgenstein’s critique of the very idea of “following a rule.” It also demonstrates Jewish law’s dependence on interpretation through a practical example involving the return of a lost item in a bank, and concludes with a mathematical question about the intersection of convex shapes as preparation for what follows.

Paradoxes in Jewish law and the difficulty of seeing Jewish law as a set of rules

The speaker presents a possible model of Jewish law as a system of rules in which every ruling is the product of an internal calculation within the rules. The speaker argues that paradoxes create a problem for this model, especially if one assumes that “the Torah of God is perfect” in the logical sense of systemic completeness that gives an answer to every relevant question. As an example, the speaker brings Tosafot in Bava Metzia regarding one’s father’s lost item, one’s rabbi’s lost item, one’s own lost item, and one’s father’s honor, and argues that the rules there do not provide a solution from within the system. The speaker distinguishes between a contradiction among rules and a failure of application, and argues that in these cases there is no contradiction but rather non-transitivity that creates a situation with no practical decision.

The loop of matzah from the new crop, spending a fifth of one’s assets, and a positive commandment overriding a prohibition

The speaker describes a case on Passover eve in which a person has no grain from the old crop at a reasonable price and is required to buy flour for matzah, while grain from the new crop is forbidden before the day of the waving offering. The speaker presents a loop in which, on the one hand, a positive commandment overriding a prohibition might justify eating matzah from the new crop, while on the other hand one can avoid the prohibition by buying from the old crop even at a very high price, since to avoid violating a prohibition a person must spend all his assets. The speaker argues that if one buys from the old crop, it turns out that one is spending all one’s money in order to fulfill a positive commandment, which is not required, because for a positive commandment a person need not spend more than a fifth of his assets. That then creates room to say one should not eat matzah at all, which brings back the side that would allow eating from the new crop, and so a loop is created.

The completeness of Jewish law, laws of doubt, and passive omission

The speaker notes that one could argue that Jewish law is not required to answer every question, but clarifies that many assume Jewish law is supposed to be complete in the sense that even if there is no substantive answer, there is still a “halakhic solution” for how to act. The speaker lists general halakhic tools such as Torah-level doubt being treated stringently, rabbinic-level doubt leniently, a positive commandment overriding a prohibition, and passive omission being preferable in balanced conflicts. The speaker argues that in the case of the lost items, “passive omission is preferable” is not a solution, because it is equivalent to “Buridan’s donkey” and to leaving all the lost items to sink, and he sees this as an illogical result showing that there is no methodological solution from within the rules.

Freedom of choice, active doubt and passive doubt, and the proposal of a lottery

A participant argues that there are situations in which Jewish law gives no answer because the authority is given to the person to choose, and the speaker responds that this makes sense in a case of two lost items belonging to one’s father, where saving one does not carry the “negative price” of violating another rule. The speaker distinguishes between passive doubt, in which there is no violation in any of the available options, and active doubt, in which every choice has a negative cost in relation to another rule. The speaker suggests that a lottery, or the idea that “permission is given,” requires justification, because it is not clear why one is allowed to choose a side that has a negative price when every choice involves violating some other rule.

Interpretation as stepping outside the system and the expansion of the rules over the generations

The speaker argues that dilemmas such as the lost items or matzah from the new crop can be solved only if one is willing to step outside the system of rules and interpret, weigh, or decide from a meta-systemic point of view. He says the price of completeness is that Jewish law is not a closed set of rules, because such decisions are not the product of calculation but of interpretation that creates a new rule “from home” rather than “from Sinai.” The speaker explains that once one interprets, the interpretation can then be translated into a written rule, and in that way the halakhic system keeps branching out over the generations, creating the impression that it is becoming more like an axiomatic system. He presents this as an illusion produced by the fact that earlier interpretations have become binding rules, but with every new malfunction one returns to the need for further interpretation.

The Oral Torah, computers, and the Church-Turing thesis

The speaker agrees that one can connect this idea of completion to the Oral Torah, but argues that even the Oral Torah cannot be a closed set of rules, because interpretation is a condition for decision there as well. He defines a system of rules as something you could hand to a computer and it would produce an output, and argues that if the human being is a “computer” according to the Church-Turing thesis, then in paradoxical cases no answer emerges from the computation. He says that after the halakhic decisor solves the problem and adds a rule, that can then be programmed, but in the next case there will once again be a need for a move that is not computational.

Dispute in Jewish law and the claim that the halakhic decisor does not work like a computer

The speaker says that one implication of this is that different halakhic decisors can give different solutions, and the phenomenon of dispute in Jewish law fits with that. He adds that dispute in itself is not proof, because one could claim that one side erred and the other was right, just as in mathematics, but “loop” situations make it possible to show that the problem is not merely a computational mistake but a limitation of the model of a closed rule-system. In this context he mentions Maimonides on disputes and comments that the loop is a case from which one can “prove” the limitation.

Turing, the halting problem, and Gödel’s theorem as a parallel to meta-language

The speaker presents Turing as an attempt to define the concept of computation systematically, together with the claim that anything a computer can do, a Turing machine can do, given unlimited time. He describes the halting problem as a question about a machine that receives a description of another machine and is required to determine whether it halts, and argues that there is no machine that gives an answer for every machine as to whether it halts or not. He compares this to Gödel’s theorem about certain kinds of axiomatic systems equivalent to number theory, and argues that such systems are not complete and contain true statements that cannot be proven within the system. He explains that he knows the statement is “true” only by means of a proof from outside the system, and that this reflects the need to step outside Jewish law as a rule-system in order to achieve full decision.

Positivism: completeness versus a set of rules, and the critique of the axiomatic project

The speaker defines positivism as an approach willing to discuss claims whose concepts are well-defined and can be proven according to a set of rules within a given axiomatic system, and says this approach attracted people and was extended to legal theory and other fields. He mentions Tarski and Carnap as examples of attempts to build axiomatic systems for biology, physics, and every field, and presents the positivist thesis as the claim that every field is an axiomatic system that covers it completely. He presents Gödel’s theorem as a blow to positivism and brings up Russell and Whitehead’s Principia Mathematica as an attempt to ground all of mathematics completely, arguing that Gödel showed the project was doomed to fail in principle. He formulates a dilemma for the halakhic positivist: if Jewish law is a closed set of rules, then there will be questions with no answer, and if one wants a system that decides every case, then one must assume that Jewish law is not a closed set of rules.

Returning a lost item in a bank: identifying marks, number, the way it was placed, the law of the land, and interpretation of concepts

The speaker gives a practical case in which someone found 2,000 shekels in the entrance area of a closed bank, left a note, and after four days a man came and claimed it was his and knew the amount. The speaker says that according to the strict law, “number is not an identifying mark,” and adds that the place is problematic because the loser claimed he had placed it near one machine while the finder found it near another. He notes the possibility of deciding according to civil law by force of “the law of the land is the law” or established custom, and adds that the loser was a Hasid and therefore asked for a “Torah judgment,” while a rabbinical judge argued that beyond the letter of the law it should be returned, and even that “one compels beyond the letter of the law.” The speaker responds that going beyond the letter of the law belongs to a case where one knows it is his but there is despair of recovery, not to a case where there is no identifying mark, because then the claim also arises: “Maybe it belongs to someone else.” He describes how the judge showed him a Shakh who holds that an amount is in fact an identifying mark, and the speaker comments that this goes against the Talmudic law and that the Beit Yosef objects. He adds that even if an amount is an identifying mark when there is a location, here there is a mistake about the location, so the amount is not an identifying mark. He analyzes the concept of “the way it was placed” and argues that it is not simple, noting the law that if something appears to have been deliberately placed one may not take it, but if one already took it, “it is yours,” and concludes that even in a very detailed rule-system, decisions depend on interpretation and on defining concepts.

Wittgenstein: the illusion of following rules and the example of the series 1-2-3-4-5

The speaker presents Wittgenstein as claiming that the concept of acting according to a rule is an illusion: not only is a system of rules not always enough, but there is no such thing as purely following a rule. He gives the example of the sequence one-two-three-four-five and asks what the next number is, presenting six as one possibility but also nineteen as another by means of a polynomial that fits the five points and generates a different continuation. He argues that the question of who is “right” shifts into the question of what is “simpler,” and that simplicity depends on the structure of the mind, so one can imagine an alien for whom nineteen is the simpler continuation. He argues that learning rules is always done through examples, and that there is no top-down transfer of a rule without a base of examples, which are themselves open to many possible rules, so there is no rule that is understood “from itself.”

The psychometric exam, scientific language, and “male” physics versus alternative formulations

The speaker says that the psychometric exam does not test wisdom but rather suitability for the mode of thinking the exam writers want, and he presents this as legitimate because it is the mode of thinking assumed in the university. He mentions Gadi Taub in the book The Crouching Revolt and the feminist critique that physics was built in a masculine way and therefore women succeed less in it, and quotes the statement about “an airplane built with feminine physics,” while qualifying it and presenting the principled possibility that the language for describing physics could change without changing the facts. He gives the example of defining velocity through differences in place and time, or through the Doppler effect, and mentions the principle of complementarity in quantum theory and the possibility of describing through position or through momentum, as well as the distinction between Cartesian and polar coordinates. He concludes that rules are a kind of language, and transmitting a rule depends on the person opposite you “understanding exactly the language”; otherwise the rule will not be received in the way you intended.

Convex and concave, and a theorem about the intersection of convex shapes

The speaker defines a convex shape as one that has its “belly facing outward” in every direction, and a concave shape as one that has a point or part where the “belly” caves inward. He presents a mathematical theorem according to which the intersection of two convex shapes is convex, and expands that this is also true for the intersection of many convex shapes. He ends with the question, “Who says that’s true?” and suggests discussing the proof next time in order to illustrate his point.

Last time we talked about paradoxes, and I tried to use that discussion to explain why, fundamentally, you can’t view Jewish law as a set of rules. Meaning, if I see Jewish law as a set of rules, and all the work is basically done inside the rule-system—that is, it’s a calculation that uses rules in order to reach the halakhic conclusion—then that’s supposedly a possible model for Jewish law. There may even be people who actually understand it that way: that there’s some set of rules that we operate within. But if, for example, we run into a paradox, then that creates a problem. Now, the problem doesn’t necessarily have to bother us, because it could be that Jewish law really doesn’t have an answer to every question. I mentioned this last time: the assumption that “the Torah of God is perfect,” meaning that “perfect” means complete. Now, the concept of completeness is really a concept from logic. The completeness of an axiomatic system means that it knows how to give an answer to every relevant question, say, within its framework or something like that. In Jewish law, if we assume that it is complete, then a problem really does arise. Because as I showed, your father’s lost object, your rabbi’s lost object, and your own lost object, and honoring your father—Tosafot there in tractate Bava Metzia—that’s a problem with no solution inside the rules.

I also mentioned another problem, of using the new crop, if you remember—that was at the end of the class. Right: a person is on the eve of Passover and doesn’t have flour, and he needs to buy flour to bake matzah. On the eve of Passover it is, of course, still before the day of the Omer offering, so it is forbidden to eat from the new crop, from newly harvested grain. But the old grain costs a great deal of money, and for a positive commandment a person is not required to spend more than one-fifth of his wealth. Therefore there would seem to be room to say that he should still buy forbidden new grain. Why? Because a positive commandment overrides a prohibition. In other words, the positive commandment of eating matzah overrides the prohibition of the new crop. But then of course the question arose: but I do have a solution—I can buy old grain, and then I won’t need flour from the new crop, thank you very much, and then I won’t need flour from the new crop. You’ll say that I’m spending a huge amount of money on it, say half my wealth. Fine—but in order not to violate a prohibition, a person has to spend all his wealth too. So in that case, really, he’s supposed to buy old grain.

But if I buy old grain, then in effect I’ve spent all my wealth in order to fulfill a positive commandment, to eat matzah. That I’m not obligated to do. So supposedly I just won’t eat matzah at all. If I won’t eat matzah at all, then why? Let’s just eat from the new crop, and a positive commandment overrides a prohibition, and there’s justification for eating from the new crop, and so on. In other words, there’s a loop.

As I said before, there are two kinds of contradictions you can talk about in the framework of a rule-system. You can talk about contradiction between the rules. In this case there is no contradiction between the rules. There are the rules—say even your father’s lost object, your own lost object, and your rabbi’s lost object—it’s not that those rules contradict one another; they’re just not transitive. Meaning, they don’t preserve a relation such that if A is preferable to B and B is preferable to C, then A is preferable to C. There can be a situation where A is preferable to B, B is preferable to C, and C is preferable to A. That isn’t a contradiction. There are principles that are not transitive; things do not have to be transitive. Okay? So that is not a contradiction between the rules.

The problem arises at the level of application. Meaning, when we have, in effect, three lost objects in the river, or two lost objects and honoring your father, and now he has to decide which of the three things to deal with. In that case, the rule-system gives no answer. So what if it gives no answer? Fine, then there’s a question in Jewish law that it doesn’t answer. So that’s the question I talked about before: does Jewish law have to be complete? Is the Torah of God perfect? Is Jewish law supposed to give an answer to every problem?

You have to understand that even if I say Jewish law is complete, that doesn’t mean that this set of three rules is supposed to answer the question. It could be that the answer comes from some other rule—what you do in a case of doubt, that passive omission is preferable, or rules like that. That too is part of Jewish law. In other words, the completeness of Jewish law is not something so far-fetched; on the contrary, I think it’s the natural assumption. If you ask people, they’ll tell you Jewish law is supposed to be complete. Complete in the sense that even if there’s no answer, there is a halakhic solution—what to do. Fine, that too is a halakhic directive. In the laws of doubt: in a Torah-level doubt, rule stringently; in a rabbinic-level doubt, rule leniently. A positive commandment overrides a prohibition. In conflicts, passive omission is preferable if there is an even conflict between two possibilities. Those too are halakhic solutions.

Now the question is whether Jewish law is complete in that sense. And again I say: in the case of the lost objects, I think there’s also no solution outside the rules. Meaning, there’s no “passive omission is preferable” here. I talked about passive omission being preferable—that’s Buridan’s donkey: to leave all the lost objects there to sink and not save any of them. That is obviously not logical. Meaning, at least one of them must be saved. There’s a dilemma as to which one, but it makes no sense to leave both lost objects to sink. Therefore “passive omission is preferable” is not a solution here, and in fact there doesn’t seem to be a solution from the laws of doubt or some methodological solution beyond the solution that is the calculation within the rules themselves. And therefore this challenges the assumption that Jewish law is really a complete system. Because here we are—we’ve found a question to which Jewish law gives no answer.

Yes.

I think I missed something you already said earlier, but there are a great many situations where Jewish law doesn’t give an answer because discretion is left in human hands. In this case, couldn’t the halakhic answer be: it’s indifferent—you must save what you can save, but you’re allowed to choose? Like when I walk into a bakery and choose between a croissant and—

You mean to draw lots?

Not to draw lots. The discretion is left to the person acting. I have to save, but…

Where does that come from?

I’ll give you an example: two lost objects belonging to my father. This one fell here and that one fell there. I can’t save them both. I choose. Jewish law doesn’t tell me which one.

Of course, because the difference is huge. With two lost objects of your father, those are two possibilities such that even if you save one of them, you haven’t violated any prohibition, you haven’t violated any other rule. Whichever one you save, this is a passive doubt. No, this is a passive doubt—you can’t save them both. It has nothing to do with being obligated to save both. It’s a passive doubt. There is active doubt and passive doubt. There is doubt—like in probability. Sometimes I decide there’s a fifty-percent probability of something because I know there are two possibilities here. Say, like a fixed prohibition: I have two pieces, one is forbidden fat and one is permitted fat, so if I took one piece, what’s the chance that it’s forbidden fat? Fifty percent, because I know one here is forbidden and one is permitted.

There is a situation of one piece, and I have no idea what it is. Again, the probability is fifty percent because I don’t know what it is. That is passive doubt; the previous doubt is active doubt. I have a positive reason in this direction and a positive reason in that direction. Now when it comes to two lost objects of my father, that is passive doubt. I am not violating any rule if I save this one or if I save that one. Whatever I manage to save, I save. But here, every side I choose will involve violating a rule. Meaning, if I choose to save my own lost object, I’ve violated the rule of honoring my father. If I choose—yes—or regarding spending one’s money, if I spent all my wealth—not to spend all my wealth on a positive commandment. In other words, here there’s a problem because every side I take carries a negative cost, unlike the case of two lost objects of one’s father.

Therefore a possible way out could be, for example, to draw lots. I don’t know what…

And the discretion is left to the rescuer.

Fine, but I’m saying: “discretion” is still something that requires explanation. Where does that come from? Because the cost is the same in all directions. The cost is the same, but who said I’m allowed to choose a side that has a negative cost? Fine, I’m saying one could have said that, but it doesn’t emerge from the examples of your father’s two lost objects.

Okay, let’s continue here for another second. So there are detailed rules down to nano-details. That assumption doesn’t need to hold in order to view the system as broadly complete. Meaning, it doesn’t have to answer every tiny detail.

Why? Why not?

If you assume Jewish law is supposed to guide me in every halakhic question—which is what he just said—then there is a test. Now Jewish law doesn’t give rules…

Yes, now you can—there is choice and you need to…

I’m saying again: if the doubt is passive doubt, or if there are two permitted options. I want to eat breakfast. The question is whether to eat a roll or bread. Jewish law allows me to eat whatever I want, of course—there’s no problem there. In Tosafot’s case, theoretically Jewish law says it doesn’t provide an answer to these tiny details, and in such situations one has to choose.

Okay, so I’m saying: then you’re saying Jewish law is not complete.

Jewish law is complete.

No, no, because down to those tiny details, that’s already…

Fine, that’s called not complete. Meaning, there are questions about which Jewish law does not answer. So it doesn’t matter anymore whether you call it complete or incomplete; in practice there are questions to which Jewish law does not answer.

Jewish law does answer—it says that now you can choose.

Fine, that’s the previous discussion. I said: maybe, I don’t know. It doesn’t emerge from the case of two options.

What I heard is that Gödel’s theorem says the system isn’t complete. So over there that doesn’t scare anybody.

Right. Even mathematics is not a complete system—or at least systems like number theory. And the theorem says that in every system of axioms there will always be statements that you can’t prove in number theory, yes.

Not every system, right.

Fine, but I’ll comment on that in a second. What I said last time is that nevertheless you can find a solution to these dilemmas, like with the lost objects or with matzah from the new crop, if you go outside the system. Meaning, if I’m willing to look at the rule-system from the outside—not to work inside the rule-system and simply apply it and see what it does, but if I have some freedom to step outside, to look at the rule-system, and say for example to weigh the rules, to decide which of them is more important or something like that—which by definition is some viewpoint from outside the rule-system, to give some interpretation to the rules—then in that case you can say that Jewish law is indeed complete, but the price is that Jewish law is not a rule-system. Meaning, you have to create a new rule.

Yes, that’s what he suggested before.

Right, so I’m adding it. That’s not much of a trick. Fine—then it’s not a given set of rules. I can always add the rule. By the way, paradoxes will also arise after you add that rule. It’s like Gödel’s theorem.

But reasoning is part of the system.

What?

Reasoning?

Yes, but lines of reasoning are not rules. Here the question is where these things enter. I’ll comment on that in a moment.

The point I showed with matzah from the new crop is that in the case of matzah from the new crop, my claim was that in fact the solution is to eat matzah from the new crop. There were three possibilities: either not to eat matzah at all, or to eat matzah from old grain and pay a lot of money, or to eat matzah from the new crop at a reasonable cost. My claim is that the solution is to eat matzah from the new crop. Why? Because why not eat matzah from the new crop? Because I have the option of spending all my wealth to buy old grain and avoid the prohibition of the new crop. But this prohibition of the new crop is overridden by a positive commandment. Once it is overridden by a positive commandment, who says one must spend all his wealth to avoid a prohibition when the positive commandment overrides it? If I have a way of avoiding a prohibition—an ordinary prohibition that I am violating—by spending all my wealth, say I can only eat either pork or something terribly expensive—fine? But there, eating pork is just a plain prohibition; there is no positive commandment overriding it. So I say I have to spend all my wealth not to eat pork.

But here, when I eat matzah from the new crop, I’m not violating any prohibition at all, because a positive commandment overrides a prohibition. Consequently, there is no obligation to spend all my wealth in order to avoid that prohibition, because that prohibition is not a prohibition in this context.

Now this is a kind of consideration—I have no source for it. There is “passive omission is preferable.”

What? Like with the shofar.

Yes, but in this case the solution of passive omission won’t help, because passive omission here means leaving all the lost objects, or in the matzah case not eating anything. Okay, but I think that in this case it’s not correct to say passive omission is preferable, because there is a decision within the rules. So what I want to say is: why is the consideration you just mentioned outside the system? A consideration that uses rules?

It doesn’t use rules; it interprets the rules. After you interpret the rules, if you now use them with your interpretation, you’ll have a solution. But think of it this way: if you programmed a computer with these three rules, okay? You’d say to the computer: tell me what to do, I’ve got this matzah problem now, what should I do? The computer wouldn’t stop; it would remain in an infinite loop. Right? Why? Because a computer doesn’t know how to leave the rules. It works inside the rule-system it was programmed with.

What I’m saying here is one of two things: either I look at the system of Jewish law as a system of closed rules—but then it cannot be complete, as in these cases and many others—or, if I do want to assume it’s complete, or I do assume it’s complete, then I have to assume that it probably isn’t a closed set of rules. Now of course after I make this interpretation, I can now also write down this rule: when a prohibition is overridden by a positive commandment, one need not spend all his wealth because of it. Now there’s an additional rule for future generations, and they’ll be able to solve the problem within the rules. But that doesn’t matter; they’ll run into the new problem, and with the new problem they’ll have to interpret again. Meaning, ultimately I brought this rule from home; I didn’t receive it at Sinai.

Okay, so that’s why I do not regard such a thing as a rule. For me this is interpretation, not a rule, even though this interpretation, once you translate it—you can write it in the form of a rule. Okay? It’s always like this: when I say things that seem strange to people, they say, “Tell me, where is that written?” So if you want, I’ll write it for you, and then it’ll be written. So what. If someone wrote it, then it stops being strange? Meaning, if he wrote it before me, then I’m writing it now. So what.

Depends.

Depends on what?

If it’s written in the Talmud.

No, fine—I’m talking about written in responsa or in…

Yes.

The claim is: clearly, in the end I can translate everything into rules, and therefore the halakhic rule-system really does branch out over the years, or over the generations. There are more and more and more rules. Once there were very few; today there are millions. Why? Exactly because of this. Because every interpretation and every intervention by a halakhic decisor, or a medieval authority, or whatever sacred book it may be, ultimately itself becomes a rule, joins Jewish law, and now there is a broader rule-system. Next time we get stuck with the rules, what do we do? We interpret. That interpretation, again, we’ll translate it and turn it into a rule, we’ll write it down, and now it will also be written, and now we have a broader set of rules. And so it continues. Therefore the rule-system really does keep branching out over the generations.

Okay, and therefore Jewish law too gives us the impression that over time it becomes more and more similar to an axiomatic system, to a system of rules. Why? Because everything that was interpretation over the generations eventually entered into Jewish law and became some kind of binding rule. Even though originally there is no source for those rules; they are a product of interpretation. Therefore this feeling that Jewish law is a system of rules is an illusion. It is a system of rules after you have done all the work of interpretation—and even now, from here on, when you get stuck on the next problem, you will again have to do the same work. Meaning, it is not a closed system.

The question you started with was whether the Torah is perfect or not. We spoke once, when we were talking about forced interpretation and so on—maybe that is exactly the meaning of the Oral Torah: that it has to complete the Torah.

I completely agree, completely agree. But I’m saying: it has to complete it. Meaning, it isn’t complete.

So it is complete when you include not only the Written Torah but also the Oral Torah.

No, and not even rules. Because even the Oral Torah cannot be a set of rules. Because if it were, then again it would be a set of rules after the interpretation was done. Translate it, write a rule from it, and it will be a set of rules—but that’s no trick. And the point—why do I say it’s no trick? Because the answer I arrived at was not the result of a calculation. For me, a system of rules means: give it to a computer and the computer will produce the result. Now if I were a computer—say the Church-Turing thesis, that a human being is basically some kind of axiomatic system, a kind of computer—if I were a computer, I would have had no answer to this question. After I did the work, I can program the computer with the rule I just made, and of course now it will be able, by calculation, to produce the answer. But with the next question where it gets stuck, it won’t be able to. So that is what matters to me. True, after I’ve done it, it will be a rule. But I did not get there by calculation. After I got there, I can already present it as a calculation. Okay?

Doesn’t that open a box of branching possibilities, where anyone who accepts some rule—and basically Jewish law can then open up in all kinds of directions?

What do you mean “can open up”? You’re talking as if this were hypothetical. This is daily reality. It happens all the time. It happened and it happens, obviously. That’s exactly the point. Of course, not only in these loop situations. Of course not—if only it were only that. These loop situations are merely the case from which I can prove it. About everything else you can say maybe someone made a mistake. Two people disagree; one made a mistake and the other was right, but there is still a closed rule-system here. With a loop, you can show that this is not so.

That’s what Maimonides says—that the disputes are because, after the fact, the disputes and so on…

Yes, right—that from Sinai there was no dispute, he says, and so on. “The Torah of God is perfect”—how can this be?

Okay, maybe this is not a priori. Meaning, “the Torah of God” plus the rabbinic additions and so on, and what was given to us to develop, and through that some notion of perfection is achieved.

Fine, I have no problem with the semantics. Meaning, in the end the point is that a halakhic decisor does not work like a computer. I’m trying to translate this into simple language. A halakhic decisor does not do the work of a computer.

And every halakhic decisor can give a different solution.

What?

Every halakhic decisor can give a different solution.

That is one of the implications—exactly one of the implications is that, since this is not the operation of a rule-system, situations can certainly arise in which different halakhic decisors give different answers, and we know the phenomenon of dispute in Jewish law. So therefore I say that the phenomenon of dispute by itself does not prove this, because the phenomenon of dispute could have been that one made a mistake and the other was right. One simply got confused; he didn’t do the calculation correctly. Just as someone might give two different answers to a question in mathematics. I wouldn’t say, “Ah, that means the system isn’t closed.” I would say one was right and one was wrong.

There is a dispute about the rules? Or about what the rules are?

Yes, right, and so on. So what I want to say—and this really brings me to comments on Gödel’s theorem and on Turing’s halting problem, which are the two contexts where essentially the same issue appears—Turing, yes, tried to define in a systematic way the act of computation. He basically built computability, what this field is called today. This field is called computability, which is basically to define and construct systematically the concept of computation. And at a certain point he reached the conclusion that the machine he built—and he has a theorem that says that anything a computer can do, a Turing machine can do, even though it’s a terribly simple machine, and to do it requires many, many steps and lots of complexity—but with it one can do everything we call computation.

Provided there is unlimited time.

Yes. And he showed that there are questions a Turing machine will not be able to answer. For example: whether a given Turing machine halts or does not halt. That is why it is called the halting problem. So a machine—a certain machine—receives as input a machine, and we ask the receiving machine whether the machine it is looking at will halt or not halt, meaning whether it will reach an answer in finite time or not. And the receiving system will not know how to answer that. By definition. Or not always—sometimes it will know—but there is no machine that can give an answer, for every machine, whether it halts or not. So that is Turing’s halting problem.

It’s not relevant; it’s a meta-question.

But it is a relevant question within the system. Whether the machine halts or not halts is a legitimate question in this world of Turing machines. Obviously—the whole idea is to generate a meta-question inside the system. A parallel question to that, completely equivalent, is proofs in mathematics—this is what is called Gödel’s theorem. Gödel’s theorem talks about axiomatic systems of a certain kind. Not all axiomatic systems, but axiomatic systems of a certain kind that are equivalent in some way to number theory. There has to be some not-too-large number of assumptions, and then an infinity—but not too large an infinity—of assumptions, and various other conditions have to hold. Under those assumptions, in systems of that kind, one can show—as with number theory, which is one of these systems—that it is not complete. Meaning, there are certain questions that the system will not know how to answer.

Now what is interesting about this is not only that it won’t know how to answer, but that one can construct—one can construct a statement that I can prove is true, and the system won’t know how to give it to me. That is how Gödel’s theorem is actually proved. Meaning, this statement is true but not provable. Meaning, it cannot be proved within the system. For a long time I wondered about this: if it is not provable, how do I know it is true? The answer is that if it were not true, that would lead to a contradiction. So why is that not a proof? It is a proof. The answer is: it is a proof outside the system. It is not a proof within the system.

Meaning, if you do the calculation according to the rules within the rule-system, you will not be able to reach an answer with respect to this statement. But if I am outside the system—if I am in the meta-language and thinking about the system—I can show that if I assume this statement, I arrive at a contradiction. Meaning, it is a statement that must be true even though it has no proof within the system. The proof is outside the system.

And that exactly reflects this point. Because if I want to find an answer—after all, that is what I said about Jewish law—if I want to find an answer to every halakhic question, that means I have to step outside the system. The calculation has to be made not within the rules, as in Gödel, but I have to go outside and see what happens, or think about these things outside the rule-system. Okay?

Therefore I now have to choose: either I have a system that is entirely an axiomatic system, but then it will not be complete. That is the price. You will have to say—and this is the price the positivist pays. The positivist really claims—the halakhic positivist; there is legal positivism, there is philosophical positivism—the halakhic positivist basically claims that Jewish law is a set of rules. Meaning, in principle, if we knew them all—I’m not sure he thinks we know them all—but in principle, if we knew them all, we could program a computer and the computer could give me answers to all questions in a completely mechanical way. And the price the positivist will have to pay is that there will be questions that have no answer. They just won’t have an answer.

Alternatively, what is the second option? The second option is: if you do want a complete system, or a system that gives you some answer in every situation—and not answers of the kind suggested here earlier, draw lots, choose as you wish, whatever, but some answer—then you have to assume that the system is not a set of rules. You have to be a non-positivist. So it’s either completeness or a rule-system. In other words, no—you can’t adopt both. It’s a short blanket—you can’t have both of those things.

Now it’s pretty clear that…

That Jewish law is not a system of rules. Anyone who knows Jewish law knows that.

I’m saying that here there is even a proof of that point—that one can prove it is so.

Now let’s try to get a bit more into this matter of positivism. That is really the topic. Positivism is basically an approach that developed at the end of the nineteenth century, beginning of the twentieth. In its philosophical sense, it basically says that we are prepared to talk about, or discuss—or that there is meaning to—claims whose concepts are well-defined, fully defined, and that can be proved according to some set of rules, meaning within a given axiomatic system; claims that have a proof. Okay, let’s put it that way.

Now this positivism was of course very attractive to people. It later became legal science—we talked about this, I think. They applied it to law, they applied it to everything else. I even have logic books at home by Tarski and Carnap from the school of the positivists, where they create an axiomatic system for biology, an axiomatic system for physics, an axiomatic system for every field. Meaning, the positivist thesis is that basically every field we deal with is an axiomatic system. Meaning, everything can be formulated as a set of rules, with a set of assumptions and derivation rules, and that covers the whole field. And the blow that this received from Gödel’s theorem comes precisely from this problem. Gödel’s theorem struck positivism, because basically—I think I once talked about this—there is the book Principia Mathematica, yes, by Russell and Whitehead, the book no one has read except maybe Russell.

The one who wrote it.

The one who conceived it—I don’t know if he read it. And that book purported to present a complete system that grounds all of mathematics in set theory. And Gödel’s theorem showed, without even examining the book, that it cannot possibly be correct. This is a project doomed to failure from the outset.

Is that because of infinite sizes—aleph-half?

No, no, no—that’s Cantor, something else. So Gödel’s theorem basically says that this ambition—even in mathematics, which is a field where we would expect this much more than in Jewish law or law or something like that, a precise field, a defined field, a field where we would expect it surely to behave in a positivist way—and it turns out that even in mathematics this doesn’t work. Meaning, this is really a very significant conceptual revolution.

So basically what happens is that if I want to relate to Jewish law as a set of rules, then I have two problems, or two limitations. First, that set of rules will not always give an answer; I’ll need some element of interpretation. Second, clearly rules rely on some assumptions, and the question is: where do you get your assumptions from? Okay? More than that: the rules use a system of concepts, and the question is how do you know the meaning of the concepts?

I’ll give you an example of a question that came up for me a few days ago. Some fellow called me and said: listen, I walked into a bank branch that was closed, into the lobby, some sort of lobby with all the automatic machines, I walked in there and found two thousand shekels lying there. Fine—I took it, left a note there saying that whoever lost it could get it back by giving identifying marks, and I took it. And then he tells me that four days later some guy came to him and said: I left it there, and he even told me the amount—he says: two thousand shekels. Okay? So he asked me whether he had to return it to him or not.

So I told him: look, it’s not so simple. Strictly speaking, the Talmud says that number is not an identifying mark. Meaning, one who finds coins—the number of coins is not an identifying mark. If someone says how many coins were there, that is not an identifying mark. Certainly when there is no defined location. Now the location was a bit problematic, because the fellow said he left it next to one machine, while the finder found it next to another machine. So it’s not entirely clear. Maybe someone moved it there, I don’t know exactly, or he forgot—but to say there is an identifying mark here is not so simple.

No one moved it.

Why not? The one who found it took it.

Not rightly.

Not rightly. Someone—an honest person—used that machine, saw it there. He says he put it there and used the machine, he leaves it there—understand, people leave it there in the bank, people go in there, the person will come back in a moment and take it. People are honest. Fine, anyway, this saga didn’t end there. I told him: look, but if the law—I don’t know what the law says—ask some lawyer. If the law says it has to be returned, then it is also explicitly stated in the Shulchan Arukh that the law of the kingdom, or accepted custom, also determines the law in returning lost property, and then that too is binding as a matter of law.

Anyway, the fellow who lost it was some Hasid, so the civil law didn’t bother him all that much. You lose it…

So he says to him: I’m going by Torah law; I think according to Torah law you have to return it to me. The loser says this to the finder. Okay? So the finder says: listen, I spoke to someone, and so on. He says to him: no, according to Torah law one is not obligated. So he went to some rabbinic judge; the loser went to some rabbinic judge, and the judge said to him: what do you mean? Beyond the letter of the law one certainly must return it—even with a lost object, yes—even one can compel beyond the letter of the law.

What? But maybe it belongs to someone else.

Yes. Exactly. So I told him: what, then? So he called me to clarify what’s going on. The other one said it should be returned beyond the letter of the law. It says in the Shulchan Arukh that it should be returned. There is no such thing as beyond the letter of the law here. That only applies to a lost object where the owner despaired of recovering it but there is an identifying mark. Then you know it is his, but you are not obligated to return it because he despaired. So beyond the letter of the law, return it, because you know it is his. But if the lost object has no identifying mark, and assuming that the number or the amount is not an identifying mark, then there is no such thing as beyond the letter of the law—that doesn’t work there.

Anyway, in the end that judge called me yesterday evening and we had a conference call. It was actually interesting. He showed me some passage in the Siftei Kohen, where the Siftei Kohen says that the number—the amount—is an identifying mark. The amount is an identifying mark. That already goes against the law of the Talmud. But it’s written, yes—so this has nothing to do with beyond the letter of the law, obviously; then it comes out that this is the law. But that goes against the Talmudic law. The Beit Yosef disagrees with the Siftei Kohen; I didn’t read this passage in the Siftei Kohen. The Siftei Kohen really says the number is an identifying mark. But I told him: look, a number when there is a location—the statement there is in a context where there is a location—but if he is wrong about the location, then the number is not an identifying mark. Meaning, if he says the number in a certain place, then it is an identifying mark. But he was wrong about the place, so it is not an identifying mark.

So he says to me: yes, but it was placed deliberately. It was placed deliberately because it’s not something that fell. You saw this machine or whatever, he placed it there—it’s not deliberate placement. Deliberate placement is when someone puts something down for a few minutes and is coming right back to take it, or for an hour, or a bag, or whatever it may be. No one leaves two thousand shekels somewhere and four days later comes back to get it. That is not deliberate placement. Even though from the standpoint of the formal definition of deliberate placement, it is: it was placed there, it didn’t fall scattered or something; this is what the Talmud defines as deliberate placement.

You see? I’m just trying to show you plain questions from last night. I dealt with this after I got back from last night’s class. And you see how much these matters are open to interpretation. Because in the end, on the other hand, there is an element of deliberate placement here, because I know the person did place it there—it didn’t fall from him. Now if he placed it there, there is a reasonable chance that he will come back to take it. Fine? True, he did not intend to place it there and return, but after he notices it is gone, he will remember that he probably put it there. So there is an element here of deliberate placement. So it’s not so simple. Okay?

True, but in such a deliberate placement, if you took it, then it’s yours. You are not allowed to take it, but it’s yours. We had long arguments about these matters. The Talmud says, and this is also ruled in the Shulchan Arukh, that with deliberate placement, if there is no identifying mark and only deliberate placement, then you may not take it. Leave it there so that the person who placed it there can come back. But if you already took it, it is yours. That is written in the Shulchan Arukh. Certainly in a doubtful case of deliberate placement, all the more so.

Anyway, I’m trying to show that there are very detailed rules in returning lost property: identifying marks, deliberate placement, despair of recovery—everything is arranged, everything is defined down to the end. You won’t be able to get anything out of it. Meaning, there is hardly any situation, except for really simple situations, where there is a simple answer according to this rule-system. There is no such thing. There is always interpretation, definition of concepts—what is deliberate placement? Definition of concepts. How do you derive your conclusions from… What happens if there is a given number of coins but not in a certain place? And all of this came from the fact that he said he put it somewhere else. If he had said, “I put it here,” then there wouldn’t have been a problem. There would have been a problem, because according to the Siftei Kohen it would be one thing, but the Beit Yosef says that even then it is not an identifying mark, and that is the plain sense of the Talmud—it is not an identifying mark even then. And once there is doubt, then the burden of proof is on the claimant.

But the Beit Yosef—it depends whether he is Sephardi or Ashkenazi, or on the Siftei Kohen.

Fine, that doesn’t interest me. It depends on who is right. I told him: look, as far as I understand, they said they contacted the police and the police said he has to return it. I told him: look, you can check with the police. If the police say you have to return it, then Jewish law also says you have to return it, because civil law is recognized in Jewish law. In monetary law, the law determines the matter, even if according to the law of the Talmud it would not be so.

Okay, so I really want to show this by means of an example that maybe I already mentioned once—I no longer remember what I talked about and what not—perhaps I spoke about this once: Wittgenstein’s example. Wittgenstein talks about following rules. Wittgenstein says that basically the very concept of acting according to a rule is an illusion. Not only is it insufficient, not only does it not always work—it never works. Meaning, there is no such thing as acting according to a rule. There isn’t. We never act according to a rule. That is a much more extreme claim.

He says this: let’s take, for example, the sequence: one, two, three, four, five. What goes here? Psychometric exam. What’s the next number?

Six. Nineteen. Six. Nineteen. Nineteen.

Okay. Someone suggests six. Both of you are right. Right? Because if, say, you define that this is indeed a function of n, with n equals one giving one, n equals two giving two—this is the series, right? If that is the function, the next number is six, right? But if the function is f of n equals something like a plus bn plus cn squared plus dn cubed plus en to the fourth, okay? And now I’ll arrange it so that at five—there are one, two, three, four, there are five coefficients here, okay? Now I’ll arrange it so that when n equals one, yes, f of one—all sets of points can be fitted by a function. The set of one, then, is a plus b plus c plus d plus e, and that has to be one. Right? I want the function to give one. Now f of two has to be two, right? a plus 2b plus 4c plus 8d plus 16e has to be two. Any set of points can…

Yes, now do five equations with five unknowns, and I can always find a set of five such coefficients that will give me whatever sequence you want. One, two, three, four, five, nineteen. One, two, three, four, five, minus two-thirds. One, two, three, four, five, 2i minus 1 as a complex number. Whatever you want—it makes no difference. I can make whatever sequence you want. Now who is right? No one is right. The question is what is simpler.

What is “simpler”? There may be someone whose mind is built differently from yours, and for him the simplest thing is nineteen. He already sees this function: wow, nineteen is next. So who is right?

You’re making the opposite claim. You’re claiming that when you have the rule—that is, when you have the function—you don’t know how to say what the next number is.

No, you’re claiming that you have the function…

No, once you give me that function, I know how to say… if I have the function…

No, wait, I haven’t finished the argument. I haven’t finished the argument. One second. What I’m saying is this: in the end, clearly, the assumption here—as in mathematics—people know that if I have the function, I know what to do. If that is the given function, one, two, three, four, then six, as far as you like. But given these five numbers, I do not know what the function is. There are many possibilities for what the function is, and every such function will give me a different number as the next one.

Now Wittgenstein says: that’s the first datum. Now, says Wittgenstein, okay—let’s now see how rules work in mathematics. Let’s talk about the field that is most supposed to be like this. Fine? How do we work in mathematics? I teach a child, say, to count in class. I say to him: count one, two, three, four, five, six, ten, eleven, twelve, twenty, one hundred thousand, ten thousand, and so on. Let’s say we get to ten thousand, fine? After that he’ll have to continue on his own. Now the question is how will he continue?

Well, give him the function, yes? f of n equals n.

But he doesn’t know how to read what this thing is. He doesn’t know this language. How do you explain to him what is written here? So let’s take an example: one, two, three, four, five, get to ten thousand or something like that; the rest he’ll have to continue by himself. So what did you gain by writing it? Meaning, in the end, even after you have written it, in order to explain the rule you will always teach it through examples, because otherwise the rule has no meaning at all. A rule has meaning only if you explain it through examples. But once the examples do not dictate one unique rule—every set of examples can fit many rules—then the feeling that we are going by rules is an illusion.

But why are you assuming there is no meaning to a rule without an example? I know many rules that do not need examples.

There is not one such rule. Except maybe the rule I just said, that there is no such rule. You’re doing induction. You’re assuming…

No, I think you don’t have to explain every time. No, because you’re not talking about a child now. You don’t need to explain because the first symbols were already explained with examples. After that you build more complex symbols on the basis of the first symbols. But you always begin with an initial set that is only through examples. There is no other way to convey it. We have no way to teach top-down. We always teach bottom-up.

Okay, so you taught him. Now he knows. Now your computer knows.

No, he doesn’t know, because if he is different, then he doesn’t know. My computer knows because it’s Tannenbaum. It’s a clever Tannenbaum.

No, no, it’s not a clever Tannenbaum—it’s a total idiot. So if there were someone else built differently, and I taught him this, the result he would give would not be like the result my computer gives. Induction works because we are all built in the same way. That’s all—just some totally accidental fact. Yes, our minds are simply built in a similar way, our brains are built in a similar way, and so we make similar generalizations. Or as was said here earlier: if this equals n, then it equals n plus one—that works. Yes, but if these symbols “if n then n plus one”—how will you explain them? But these are rules—you seem to want to pull the ground out from under what a rule is.

I don’t want to—it pulls the ground out. It’s not that I want to; that’s just how it is. There is no rule that you understand from itself. You always understand a rule when someone explains it to you. You need to distinguish between what we understand or how we understand, and whether there are rules apart from what we think—like you once talked about Popper or something like that. There are rules outside human beings; the rule is there, even if we could not grasp it. The rule exists.

Not in Jewish law?

No, I’m not talking about—I mean, this function exists. It exists. You are given five such numbers.

No, I’m not talking about the string. You were given the five numbers.

Yes. And what do they mean? Now, there is such a thing, there is.

What is? What exists? I don’t know what that is. You’re not talking about whether you know it. There is some function—by nature there is the function.

No, what is “nature”? There is no nature here. The function is a definition. Explain to me what was defined here. It has nothing to do with nature. It’s a definition. It’s something intellectual, not something in nature. Is a forty-five-degree angle something that exists in nature?

What? A forty-five-degree angle?

There are things cut at a forty-five-degree angle.

And that is a forty-five-degree angle.

No, but that’s exactly the point. Rules are not entities. Rules are things we define. Now the question is what we defined. You cannot tell me “the rule exists, I just don’t know how to define it.” The rule is the product of a definition. If I don’t know how to define it, then there is no rule. A rule is the way in which I convey to you…

But even if you don’t know, there are things you don’t know how to explain that still exist.

No, they don’t exist—except that a person understands. The question is how we grasp it.

No, so I claim not. Meaning, in the end the rule is the result of our definition. If we don’t know how to define it, then there is no rule. It’s not that we don’t know how to define the rule—the rule is only a definition.

Suppose he’s right.

Clearly I’m right, not “suppose I’m right.” Basically Wittgenstein’s claim is that not only does the use of rules limit us and not always give a solution, but the very idea that we think we are working with rules is a self-deception. We are never really working with rules, but with some sort of generalizations on the basis of examples. And with generalizations on the basis of examples, if someone were to come before you—as an example one can give here—someone alien, I don’t know, from another planet, and he is really built in such a way that when you show him one, two, three, four, five, the next one for him is nineteen. His mind is simply built that way. That is the simplest thing for him. Not this—for us this is the simplest—but for him, that is the simplest. That’s how he is built. Everyone can have a different mind. Okay?

Then you will never succeed in teaching him this thing. For him, this thing is a fifth-degree polynomial. Meaning, it is complicated relative to the simple case by five degrees. He won’t be able to grasp it at all. So what does that mean? He is no less right than we are.

That’s the excuse for everyone who fails the psychometric exam. They’re not stupid, their mind is just built differently.

And the psychometric exam—the psychometric exam doesn’t really test whether we’re smart. And that’s true, by the way, I agree with that. Even though it sounds like a postmodern kind of argument. But the psychometric exam doesn’t test whether we’re smart; it tests whether we think the way the test writers want us to think. And that’s perfectly fine, by the way—it’s legitimate, I have no criticism of it. Because when you come to the university, people assume that this is how you think, because this is how they teach there. So now I want to see whether you are skilled in this form of thinking.

There could be someone whom you would define as…

Yes, exactly. So that’s perfectly fine, we have no other way to do it. But it is always possible that he is no less intelligent than I am if you explain it to him in that strange language.

What we once said about Gadi Taub, who wrote in his book The Stooping Revolt—he brings there feminist critiques of physics, that physics was constructed in a masculine way and therefore women are less successful in that field. So he said that he wouldn’t want to get on a plane built on feminine physics.

Still with me—this doesn’t shake me. In any case, by the way, on that matter I don’t agree with him. I don’t agree with him—at least in principle that critique could be right. Why? One second. Because we write physics in a very particular language. But we’re not talking about the facts. The facts of physics are facts—that’s obvious. But the question of what language we use to describe physics—there definitely can be different languages, just as there are Hebrew and English. One can use one language or another.

You can define velocity—we talked about velocity with Zeno’s arrow in one of the recent classes. So I said there that velocity can be defined by difference in place divided by difference in time, and you can also do it with the Doppler effect, which has nothing at all to do with differences. Okay? What does that mean? It doesn’t—we’re talking about the same quantity. Meaning, velocity is something that exists in the world, but the language we use to describe what happens in the world definitely depends on us. Now if there were a language that was more convenient for women to use than another language, then it is entirely possible they would be more successful in physics if physics were formulated in that language. That is not absurd.

The claim is not that it would change the findings of physics, the laws of physics, but that the language would be a different language. If I return once again to this issue of velocity, yes, of Zeno’s arrow that I talked about—I mentioned there the principle of complementarity in quantum theory: that one can look at everything in terms of place, and one can look at everything in terms of velocity, yes, momentum. Okay? And those are two modes of view, and both are equivalent. Now you can definitely see that there are people for whom it is more convenient to calculate things in one way, and people for whom it is more convenient to calculate things in another way. Or if you look at coordinate systems: some people are more comfortable using a Cartesian system, and some people are more comfortable using a polar system. And this is definitely a matter of how our minds work. There is no right and wrong here, smart and not smart. It is only a question of fitting the language to the person.

And therefore in this sense, what we are talking about here is language. And this language—the rules are a kind of language. But when you want to describe the rule that exists, the real rule you want to describe, you will not succeed in putting it into language in an objective way unless the person facing you understands exactly the language you are speaking.

You say it exists.

No—the rule I want to convey, not that it exists. There is a rule that I want to convey to you, a certain definition. I use a certain language for that definition, but you won’t be able to grasp that language unless your mind is built like mine; otherwise you won’t grasp it. Okay?

So therefore Wittgenstein says that not only are rules a limited system, not only is there Gödel’s theorem, and not only will we not always get an answer—even when we do get an answer, it is not the result of mechanical calculation. You’ll say to me: but the computer, yes, the computer does mechanical calculation. The computer doesn’t leave the system; it gives us answers about various things. Right—because there was a programmer. The programmer did all the work, and after the programmer did the work, the computer only does the mechanical parts.

I’ve got three more minutes, I’ll just start something. I’ll try to do it here on the board, right? Good. There is a theorem in mathematics that says there is—before the theorem, there is a distinction between a convex shape and a concave shape. A convex shape is a shape whose belly goes outward. In simple terms, that is a convex shape. And a concave shape is a shape like this, where at some point the belly goes inward—not a belly sticking out, but sunken in. Okay? Now the assumption is that convex means convex in every direction. If it is concave at one point or in part of the points, that is called concave, not convex. That’s how it is defined. Okay?

Now there is a theorem in mathematics that says that the intersection of two convex shapes is convex. Say a triangle is a convex shape and a circle is a convex shape.

A straight intersection?

A triangle is a convex shape and a circle is a convex shape. Their intersection is also a convex shape. Always. Any two convex shapes whatsoever—put them one on top of the other however you want—the intersection will always come out convex. And likewise if you put many convex shapes. The intersection of all the convex shapes, no matter how many, in the end comes out itself as a convex shape.

Now I just want to ask the question and we’ll talk about it next time—if someone can prove it. Who says this is true? Fine. So that’s a good point to end on, and we’ll return—we’ll return next time. Stringently. Now I just want to ask the question and we’ll talk about it next time, if someone can prove it. Who says this is true? Fine, so that’s a good point to end on, and we’ll return—that’s it—we’ll return next time, because through that I’ll want to illustrate what I said.