Halakhic Positivism, Lecture 3
This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.
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Table of Contents
- Going outside the system of rules in Jewish law
- Gödel, Turing, and the limitations of axiomatic systems
- Jewish law as a “Gödelian” system or an open system, and adding rules from outside
- Artificial intelligence as a halakhic decisor, and authority versus truth
- Mathematical proof through changing the definition: the intersection of convex shapes
- The gap between everyday and formal concepts, and the claim that the difficulty gets swept into the definitions
- Definition as non-arbitrary, conceptualization as observation in the world of ideas, and the difference between mathematics and physics
- Deduction that adds no information, the dust hidden in the assumptions, and models that do not describe life
- A fortiori reasoning, “two hundred includes one hundred,” and the limits of logical necessity in Jewish law and in law
- The Vandervelde law in Belgium: a failure of mathematically formalizing a norm
- “2+3=5” in the laboratory and vector calculus: what is being tested is the physical model
- John Stuart Mill, hidden induction, and Descartes as an attempt at a carpet with no dust under it
Summary
General Overview
The text argues that when making decisions in Jewish law, it is impossible to make do with acting only “within the system” of rules, because systems of rules may create loops, leave questions unresolved, or be limited in principle, similarly to results in logic and computability. The argument relies on analogies to Gödel’s incompleteness theorems and Turing’s halting problem, and develops the idea that the solution sometimes comes from moving to a meta-level where the rules themselves are examined. It then argues that mathematics and logic do not solve “real-life” problems except after one performs conceptualization/modeling that sweeps the difficulty into definitions and basic assumptions, and that the same pattern appears in law and in Jewish law, so positivism is naive.
Going outside the system of rules in Jewish law
The speaker presents both an obligation and an ability to go outside the system of rules when making decisions in Jewish law, because operating within the rules may preserve the deadlock that the rules themselves create. He describes paradoxes that force us not to act within the rule system, because the rule system is what puts us into the loop, and therefore one has to examine the rules themselves, their weight, and whether they can or cannot be applied in certain contexts. He illustrates this through the paradox of “matzah made from the new grain,” where examining the rules opens some possibility of escaping the loop, while acting internally leaves the problem stuck.
Gödel, Turing, and the limitations of axiomatic systems
The speaker compares the deadlock in Jewish law to phenomena in logic and computer science, where working within an axiomatic system can get stuck on claims whose truth or falsity cannot be decided, or that are true but unprovable, within the framework of the strong and weak incompleteness theorems. He notes the analogy to the halting problem for Turing machines, where a machine operating by defined rules does not halt for certain questions and therefore does not return an answer. He describes how he was personally troubled by the question of how Gödel’s theorem was proved if it states that there is a true claim that cannot be proved within the system, and explains that the proof is done outside the system, in the meta-language, where one constructs a sentence and proves constructively that it is true and unprovable within the system. He concludes that systems of rules of the type similar to arithmetic are necessarily limited, because there will always be a true claim in them that cannot be proved within them, and even adding consistent axioms will at most solve that particular claim but will create a new unprovable claim in the new system.
Jewish law as a “Gödelian” system or an open system, and adding rules from outside
The speaker raises the possibility that Jewish law may be either a “Gödelian” system in which there will always remain questions without an internal answer, or an open system in which one keeps adding more and more rules from outside in order to solve problems. He illustrates such an addition with the rule “passive omission is preferable” in situations of conflict between one positive commandment and another, or between one prohibition and another, and argues that this rule has no explicit source in the Torah, is not a law given to Moses at Sinai, and is not a hermeneutic principle, but rather comes from logic. He distinguishes between situations of a loop and situations in which the problem simply remains open without priority, and illustrates this through “Buridan’s donkey” as a model of paralysis of the symmetrical-choice type. He connects this to the question of whether “the Torah of the Lord is perfect” means complete, and suggests that a complete system cannot be “Gödelian”; therefore, if Jewish law is complete, it contains an element that is not axiomatic and cannot be presented as a set of axioms from which answers to all questions can be derived.
Artificial intelligence as a halakhic decisor, and authority versus truth
The speaker raises the question of whether a mechanical system or artificial intelligence could serve as a halakhic decisor, through the example of a proposed doctoral topic on the status of artificial intelligence in Jewish law. He presents the formulation of a question: if artificial intelligence disagrees with Rabbi Ovadia, “whose view does Jewish law follow?”, and responds that the question is based on a mistake, because the issue is not who is more authoritative but what is correct. He tells a story about Amos from Mishmeret STaM, who would sometimes ask Rabbi Nissim Karelitz and Rabbi Wosner and already knew in advance that one would permit and the other would prohibit, and concludes that the choice of whom to ask already determines the answer, so it is better to study the topic and decide. He presents reliance on “this one said this and that one said that” as something that leads to deadlock and loops.
Mathematical proof through changing the definition: the intersection of convex shapes
The speaker presents a mathematical example of the proof that the intersection of two convex shapes is convex, and describes how the attempt to prove this intuitively gets stuck mainly at the “seam” between the boundaries of the shapes. He compares this to the difficulty in the four-color problem and computerized proofs, and emphasizes that the decisive move is not checking all the possibilities but choosing a precise definition. He gives a mathematical definition of convexity: a convex shape is a shape such that for any two points inside it, the straight line connecting them lies entirely inside the shape; and he shows that once this definition is in place, the proof becomes simple. He describes his shock that a “trivial” proof becomes possible only after the definition, and concludes that the main idea is not the proof but the definition.
The gap between everyday and formal concepts, and the claim that the difficulty gets swept into the definitions
The speaker argues that the mathematical proof does not necessarily solve the “real-life” question, because the original question was asked about an intuitive concept of convexity, whereas the proof relates to a formal concept in an abstract mathematical world. He asks who proved that the formal definition exactly overlaps with what is intuitively perceived as a “convex shape,” and maintains that there is no way to prove such an overlap, because the everyday concept is not defined, and the need for a definition arises precisely from this difficulty. He describes the process of definition as “skipping over” the hard stages and sweeping “dust” under the carpet, so that from that point onward deductive work becomes relatively easy. He concludes that mathematics does not solve real problems, but breaks them into a part that cannot be handled mathematically and gets pushed into the definitions and assumptions, and a part that can then be proved afterward.
Definition as non-arbitrary, conceptualization as observation in the world of ideas, and the difference between mathematics and physics
The speaker rejects the claim that mathematical definitions are arbitrary, and argues that a good definition captures an everyday concept and produces a fruitful concept that enables the proof of many theorems. He states that definition is not “freestyle” but the result of the ability to distinguish and conceptualize, and compares this to observation: for physicists, observations give rise to formalization, while for mathematicians the parallel to observation is conceptualization of the world of ideas. He responds to a question about the analogue in Jewish law and argues that the transition to Jewish law resembles mathematical conceptualization more than physical formalization, because in Jewish law there is no observation extracting data but rather the construction of concepts out of a canonical world of sources. He connects all this to a critique of positivism as a naive method that assumes one can solve problems by purely logical means, and stresses that even in mathematics this works only after the “black work” of definition.
Deduction that adds no information, the dust hidden in the assumptions, and models that do not describe life
The speaker returns to the claim that deduction does not generate information beyond what is already found in the premises, and illustrates this through the example “all men are mortal” and “Socrates is a man,” where the conclusion is already contained in the premises. He emphasizes that the essential information lies in the basic principles and definitions, and that logical inference only brings it from potentiality to actuality, as happens in mathematics and also in physics after equations and concepts are formulated. He stresses that in physics too, the question “who said the equation really describes the world” is a basic assumption that is not mathematics, so the distinction between mathematics and physics is in some sense an illusion, because mathematics too has a non-logical component hidden at the beginning of the process. He brings Fermat’s theorem as an example of a conclusion contained in the assumptions, and generalizes that mathematical models of life hide problematic steps that are not solved logically.
A fortiori reasoning, “two hundred includes one hundred,” and the limits of logical necessity in Jewish law and in law
The speaker cites Adolf Schwarz, who argues that a fortiori reasoning is a deductive syllogism, and distinguishes between types of a fortiori arguments. He describes an a fortiori argument of the type “two hundred includes one hundred,” where the relation is one of real inclusion, such as “if one is liable for opening, then all the more so for digging” regarding a pit in the public domain, where digging includes opening plus an additional act. He gives another example of “one who passes some of his children to Molech” versus “all of his children to Molech,” and explains that this is an a fortiori argument of inclusion; yet in practice there are interpretations that exempt one who passes all his children. He notes the discussion of “we do not derive punishments from logical inference” and the Maharsha in the second edition in Bava Kamma, who suggests that in monetary law we do derive punishment from logical inference when there is no refutation, but then goes on to present a legal example showing that even an a fortiori argument of inclusion is not compelling once one moves into real life.
The Vandervelde law in Belgium: a failure of mathematically formalizing a norm
The speaker brings up the “Vandervelde law” in Belgium, which forbids selling two liters of wine in order to prevent workers from wasting their wages in the pub, and describes a case in which a customer asked for ten liters and the argument was that ten should be permitted because the law only forbids two. He quotes Chaim Perelman, who reports that they ruled in favor of the buyer, and explains this by means of purposive interpretation: the prohibition was intended for drinking on the premises, not for purchasing for storage or investment. He concludes that the model “if 2 is forbidden, then any quantity containing 2 is forbidden” is a mathematical model that does not necessarily describe the legal situation correctly, and that in the transition from life to the model, assumptions and purposes were absorbed that do not follow from the formalism. He uses the example to argue that mathematics and logic do not solve the essential problem but expose the fact that the fault lies in the model and in the assumptions.
“2+3=5” in the laboratory and vector calculus: what is being tested is the physical model
The speaker describes how he asked whether “two plus three equals five” is a scientific law that can be refuted empirically, and argues that in practice no failed experiment would lead us to abandon arithmetic, but rather to assume a fault in the experiment or in the model. He argues that the reasonable conclusion would be that the arithmetic model is not suitable for describing the phenomenon being tested, not that the mathematics is wrong. He compares this to mechanics: two forces of 10 newtons in perpendicular directions do not add up to 20 but to a resultant force of about 14 and something, and what is refuted is the assumption that forces add arithmetically, not mathematics itself; therefore vector calculus is invented as the correct model. He concludes that whoever inserts the information and the formalization into the system controls the conclusions, because the conclusions add no information beyond the assumptions.
John Stuart Mill, hidden induction, and Descartes as an attempt at a carpet with no dust under it
The speaker presents John Stuart Mill’s challenge to deduction, according to which the certainty of the conclusion depends on the certainty of the premises, and the premise “all men are mortal” rests on induction, which is not certain. He uses the example “our forefather Jacob did not die” to show that the problem is not in the logical step but in the generalization and in the assumptions. He mentions Descartes’ project and the cogito as an attempt to find claims about the world that do not depend on observation and on an infinite regress of assumptions, and explains that “I think” was perceived by him as a foundation with no dust underneath it, because even doubt or negation are forms of thought. He places this within the tension between rationalism and empiricism, and argues that these attempts were meant to ground certain conclusions in intellectual tools alone, but concludes that the only certain conclusion is that there are no certain conclusions.
Full Transcript
[Speaker B] We were
[Rabbi Michael Abraham] in the topic of positivism, and I was basically talking about the obligation and the ability to go outside the system when we make decisions in Jewish law. I tried to demonstrate this through various paradoxes that force us not to act within the system of rules, because the system of rules is what puts us into the loop. So therefore, basically, you have to go outside the system of rules and try to examine the rules themselves: what weight they have, and whether they can or cannot be applied in a certain place. We saw this through the paradox of matzah from the new grain, where basically if you act within the rules, you stay in a loop the whole time, but if you examine the rules, there is some possibility of getting out of it. I said that something similar happens in logic and in computer science, in computability, where again, when we work within a system of rules—say, Gödel’s theorem in logic says that if we work within some system of rules, an axiomatic system, we can get stuck on a question that cannot be determined to be true or false, or that can be determined but cannot be proved. There is the strong and the weak incompleteness theorem. And the same thing is analogous—mathematicians showed this in the halting problem of Turing machines. It’s basically the same thing. A Turing machine is simply a machine that thinks within a defined system of rules, and then there are questions for which that machine does not halt. Meaning, it does not give us an answer. And I said that what bothered me for a long time was: how do we know, in Gödel’s theorem—the theorem says that within the logical system there is a statement that one can construct which is true and unprovable. And it always bothered me: how did they prove Gödel’s theorem? After all, when Gödel’s theorem was proved, they proved that this statement is correct and they proved that it is unprovable. But if they prove that it is correct—what’s called “to prove”—what does that mean? So at some point I understood that they proved it outside the system. Meaning, there is a system of rules within which we operate, the axiomatic system, so the concept of proof itself has to be formulated by means of those rules. We operate inside the system of rules, there is a very precise definition of what a proof is. So a proof of that kind cannot be found for Gödel’s theorem. But in the meta-language, when we go outside the system and think about it, we can prove Gödel’s theorem and we can prove—it’s a constructive proof—you build a certain sentence and prove that it is true and that it is unprovable within the system, but you prove this outside the system. And that basically means—and philosophers later dealt with this, I think often not very accurately; I hope I’m doing it accurately, I’m not a mathematician either—but it seems to me there are many mistakes in philosophers’ treatment of Gödel’s theorem. The claim is that wherever you work with some system of rules, and it has to be more or less similar to the system of arithmetic, there has to be some analogy between them, then necessarily there is within it a proposition that is true and unprovable. And that means that systems of rules, at least of this type, are limited systems of rules. Meaning, they will never have an answer to all questions—an answer or a proof for all questions. And if we apply this, then regarding Jewish law as well I said one can wonder whether it is a Gödelian system—that is, a system such that if we operate within it, we will never have an answer to certain questions—or whether it is an open system, a system where we constantly add more and more rules from outside, and in that way solve the problem. I illustrated this through that paradox of matzah from the new grain, how we basically bring in some ad hoc rules from home in order to solve problems of that sort. By the way, just as an aside, in Gödel’s theorem that kind of thing doesn’t help. In Gödel’s theorem, say there is a given axiomatic system and I showed that there is a certain proposition that cannot be proved within the system. I can always add some further assumption, as long as it is consistent with the other assumptions or axioms, and then prove that proposition. Except that then, in the new system—including the given system plus the assumptions I added—in the new system there will be a new proposition that cannot be proved. Meaning, it doesn’t help. You can’t get out of it that way. So there is a deeper question here: whether the halakhic system is essentially open. Meaning, this is not something that can even be presented as a set of rules, and then perhaps there can be an answer to everything. I’ll just give an example so that we’re not speaking in huge abstractions; I need to remind myself a bit. Suppose I have a conflict between a positive commandment and a prohibition, then the rule is: a positive commandment overrides a prohibition. What happens when I have a conflict between one positive commandment and another, or between one prohibition and another? In such a situation there is no halakhic rule telling us what to do. So what do the sages say? “Passive omission is preferable.” The conflict is always between doing something or not doing something, so don’t do it. The rule is that passive omission is preferable. Where does that rule come from? Does it have a source in the Torah? No. Is it a law given to Moses at Sinai? Does it come from the hermeneutic principles? Where does this rule come from, that passive omission is preferable? From logic. Meaning, we say: if the system is stuck, then apparently logic says do nothing, because you need a reason in order to do something. If you have no reason, then you do nothing. Here is an example of a rule that we add at a point where we are stuck, where the system cannot give us a solution. Okay? I talked about the fact that there are several types of unsolvable situations. There is a system that puts me into a loop; there is a system that leaves the problem open. Like one positive commandment versus another. One positive commandment versus another is not a loop. There’s no loop here. Basically, you can do this, you can do that, there is no priority of one over the other. Buridan’s donkey dies of hunger when it gets stuck in a problem of that sort. It gets stuck in a problem of the loop type, so it wouldn’t die of hunger—it would die of exhaustion. It would simply keep running between the troughs until it died of exhaustion. And therefore this is an interesting question: is the Torah of the Lord perfect—perfect in the sense of complete? A complete system would have to be a non-Gödelian system, because a Gödelian system is not complete. So if you take this all the way seriously—again, of course I’m not claiming that this comes out of the verse “the Torah of the Lord is perfect.” But if it really is complete, that means there is something in it that is not axiomatic. Meaning, there is something in it that I cannot build as a set of axioms and derive from those axioms the answers to all questions. One of the implications is: can there be a mechanical system, an artificial intelligence, that serves as a halakhic decisor? Someone met with me yesterday—he was looking for a topic for a doctorate in something related to Torah and science—and he told me he had thought maybe to ask a question of that type. What is the status of artificial intelligence? If artificial intelligence disagrees with Rabbi Ovadia, then whose view does Jewish law follow? Meaning, you teach some software, train it to issue halakhic rulings, it gives you some ruling in a certain area, and this goes against what Rabbi Ovadia wrote or what someone else wrote. So the question is whether I see this as a dispute among decisors, what the relation is, how one relates to such a thing. I told him that in my view his question is based on a mistake—I’m adding that in parentheses. The question is based on a mistake because his assumption says that when I see two opinions, what I need to do is decide who is more authoritative: Rabbi Ovadia or the software. But that’s not true. You see two opinions, you need to decide what is correct, and that’s what you should do. What difference does it make who said that opinion—whether it was the automated system or Rabbi Ovadia? I have a friend, Amos, who was once a respondent in Pardes Hanna on behalf of Mishmeret STaM, for questions about Torah scrolls, tefillin, and mezuzot. And he told me that there was a certain question that came up not infrequently, and when he got stuck and didn’t know, he would go ask, say, Rabbi Nissim Karelitz and Rabbi Wosner. He told me: look, there’s a certain question that has come up for me several times already, and I know that whenever I asked Rabbi Nissim, he permitted it, and Rabbi Wosner forbade it. And I already know this, I’m familiar with it—so whom should I go ask now? Meaning, if I decide—what?
[Speaker D] The one asking, what does he want to hear?
[Rabbi Michael Abraham] Can’t he tell him both? The question is whom to go ask, because by doing that I determine the answer. So I told him, what do you mean—don’t go ask anyone. Study the topic, decide what is correct, and answer him what is correct in your opinion. This whole concept of ruling because this one said this and that one said that—in my view that is exactly what leads to this kind of deadlock, of problems.
[Speaker C] Fine, that’s an answer if someone asks you; if he asks someone else he’ll get a different answer, right? And that too is a loop, because he also knows that.
[Rabbi Michael Abraham] Here too he would know what answer he’s going to get. All right, in any case, okay. So at the end of last time I started with some example I gave you, just as an example to think about—I don’t know if you did. At least one person did, I know, he sent me the correct answer. I’ll go over it a bit because through it I want to demonstrate the next point. The claim is basically this—an example from a mathematical theorem. We have shapes that in mathematics are defined as convex shapes and concave shapes. Convex shapes—yes, intuitively, convex shapes are shapes… a belly, a circle—that’s a convex shape, because everywhere its edge or boundary turns outward, like a belly. In contrast, a concave shape is a shape like this. Yes, now here you can see there’s a kind of inward belly. Fine. Now, of course, this shape too in this area is convex. But in this area it is concave. The definition—the mathematical definition—is that a convex shape is convex everywhere. Fine? If the shape is not convex everywhere, even if it is convex in part, it is called a concave shape or a non-convex shape. Fine? Now, for example, a triangle is also a convex shape. A triangle has no bulges. A straight line also—there are no bulges—so that too is a convex shape. Now the question was whether one can prove that the intersection of two convex shapes is also convex. Now this intersection—take two convex shapes, put one on top of the other, they have a common area, the intersection. The question is how one can prove that this common area is also convex. And that is true for any two convex shapes. Meaning, this is a general theorem. Now when you think about it just naively, just in a plain ordinary way, it’s very hard to prove this. I think—I don’t know—for me it was very hard to prove. You should understand that it seems a bit—it seems trivial, because every piece of the boundaries of the intersection is itself part of a convex shape. So this will be convex, this will be convex, and this will be convex—yes, that’s obvious. The problem is always at the seam. Yes, the seam between them—how can I make sure that there too the property of convexity is preserved, meaning it doesn’t break? Okay? Then you have to think about all the possible situations and see whether we can cover them all.
[Speaker E] And show that it always comes out convex—that it never breaks.
[Rabbi Michael Abraham] You know, it’s like the four-color problem, a controversial problem in mathematics—whether every two-dimensional map, a geographical map, two-dimensional, can be colored with four colors in such a way that no two adjacent regions ever have the same color—that is, the same color will never appear on both sides of a border. On every border, the two sides will always have different colors. So there was some hypothesis saying that it’s possible. Meaning, that four colors are enough to color any map of any kind whatsoever. But there isn’t really a proof for it, only a proof of the type—
[Speaker B] the kind they did with a computer.
[Rabbi Michael Abraham] Yes, exactly, that’s a computer proof. And so mathematicians get very upset about it, because there’s no such thing—a computer, that’s not legitimate. But that’s not exactly it. Meaning, the computer there did not actually serve an essential role. That’s an important point. Because there may be places where the computer does essential work. Overall, the important step toward the proof there was that they divided the—I don’t know how many—17,000 types of maps or types of borders that could arise, and now you go through them one by one and show that in all these types, once you’ve reduced it to a finite number of possibilities, then you can prove it straightforwardly. Just go through this possibility, show it works there, go through that one, go through that one, and show that it’s always possible, and that’s it. You can do that. Now, only 17,000 possibilities—maybe there are some mathematicians still sitting in a cave somewhere still working on it, but usually that’s not how it goes, so you run a computer. The computer just does it, goes through all the possibilities. So there was an elegant mathematical step in itself before they got to the computer, namely that they managed to show there is a finite number of possibilities. So here too, in principle, if we went in the simple way, we would have had to think what kinds of intersections can arise between various convex shapes and try to see whether we can get a finite number of possibilities, and then for each one prove that indeed the intersection is convex. But fortunately there were mathematicians who did not agree to sit in a cave; instead they proved it in the following way. Basically what’s missing here—and this is the point that matters to me—the proof demonstrates this. Pay attention to the pattern of thought. That’s the essence of the proof. In order to prove this, first of all I have to define the concept of a convex shape. What I did earlier was just hand-waving, and I would immediately get a zero on an exam if I said something like that. Meaning, what is a convex shape? A shape with a belly facing outward, or something like that—that’s not how one works. You need a precise definition. So the precise definition mathematicians suggest is, say, that a convex shape is a shape such that for any two points in it, if we connect them with a straight line, the whole line is inside the shape. Think about it: for example, here there are two points that if you connect them with a straight line, part of the line will be outside, right? Here there is no such thing, and neither in a triangle. Therefore a circle or a triangle are convex shapes. Fine? It basically makes sense. Meaning, this definition of a convex shape more or less corresponds to what we perceive as a convex shape. Fine? That’s all. From here on it’s absolutely trivial. Really. That’s what’s beautiful here—the definition does all the work. Good. Now let’s prove it. I want to prove that the intersection is a convex shape, right? So I put two points here, connect them with a straight line, and I want to show that this entire line lies in the intersection, so that the intersection is a convex shape. Now these two points lie in the circle; a circle is convex, squarely so, this thing. So the two points—the line connecting them—also lies entirely inside the circle, because the circle is a convex shape, right? Now these two points also lie in the triangle, because whatever lies in the intersection lies in both shapes. But the triangle too is a convex shape, so for those two points, the line connecting them lies in the triangle as well. Now if that line lies both in the triangle and in the circle, then that line belongs to the intersection. So therefore it lies in the intersection. So for any two points inside the intersection, the whole line connecting them also lies in the intersection; therefore the intersection is a convex shape. Now this looks like real hocus-pocus. When I first saw it I was literally in shock. Why? Because I had spent quite a lot of time on all the possibilities, trying to think, and I was really only twenty or twenty-two when I found some used book dealing with problems in topology. So it asked the question like that and I said—it was some kind of learning book for young people—so it said okay, now think about it, and afterward read the proof. I tried to think, and I couldn’t succeed in any way. It seemed impossible to me. I was dying already to know what was on the other side of the page, and then I saw this thing and said to myself: wait, what is going on here? I mean, how could I fail to prove something so simple? And the answer is: I didn’t make the definition. I couldn’t prove it because I didn’t define a convex shape. I thought about it in a homespun way, in that simple intuitive way of a convex shape, and I tried by that route to see whether I could prove it. Now without defining the concept sharply, I can’t do the work—I can’t prove it. The point here is actually not in the proof; the point here is in the definition. Meaning, if you make the definition correctly, very often the proof is simple. Then—yes?
[Speaker B] Not really from the side that deals with my field, but why do you say “if you make the definition”? Basically, the question assumes there is some agreed-upon definition behind it. When you say the student generated the definition, that makes it sound like it’s freestyle. It doesn’t seem to me that mathematics is freestyle.
[Rabbi Michael Abraham] I completely agree. That’s exactly the point I’m about to get to. That’s where I’m heading. It’s not really a question. Okay. The problem that arose for me afterward was whether we really proved the theorem. And the answer is no. Why not? Because when I read the theorem on the first page, I was thinking of it in terms of what I perceive as a convex shape. The natural concept that everyone understands, not in terms of the mathematical definition of a convex shape—that only came afterward. Right? I tell myself, yes, I more or less understand what a convex shape is, and I also understand what the intersection of convex shapes is: I put them on top of one another and then ask myself whether the intersection is also a convex shape. That is a question about life; it’s a question about life, about our world, okay? It’s not a question in mathematics; it’s a question in our world. If you take any two such shapes and put them on top of one another, does a convex shape really come out? The answer the mathematician gave me does not deal with our world; it deals with a mathematical world, an abstract Platonic world. Why? Why do I say that? Because who said that the definition I gave—that for any two points, the line connecting them lies entirely inside the shape—who said that this definition overlaps with what we intuitively perceive as a convex shape? The mathematician did not prove that, right? Who said that the proof he proved is the theorem I am looking for? After all, in order to make that claim I basically need to say that the definition he proposed fits perfectly with what I intuitively understand to be a convex shape. If that is true, then there is some correspondence, so what you do here will also be true there. Okay? There is some one-to-one mapping. So everything you do here, you also do there. But no one proved that. Not only did no one prove it—no one can prove it. How would you prove it? You would get stuck at exactly the same point. You would get stuck there because the everyday concept is undefined; we do not know how to prove things on the basis of it. That is exactly why we created the formal concept, meaning the definition. So the desire to prove that the intuitive concept is equivalent to or overlaps with the formal concept is probably impossible. More than that, I’m saying that my whole problem in proving this theorem was actually this transition. Meaning, if I present it this way, then I would describe the process of proof like this: I encounter a question from life—I have convex shapes, and is their intersection also convex, right? So I want to prove that their intersection is also convex. What do I do? First step: I define what a convex shape is, right? That was the first step I took. So here too—how did it suddenly become easy? Because we hid all the hard stages inside the definition. Meaning, the step from the intuitive concept to the definition was actually a leap over all the hard stages that I didn’t know how to solve. Now from the definition onward there’s no problem; I can solve it, it’s very easy. But I didn’t really solve the problem. I solved a mathematical problem that I believe is equivalent to my real-life problem. But believing is nice—on that you don’t get a hundred on a mathematics exam. Belief is for the study hall, not for the mathematics faculty. So basically what I’m claiming is that I did not solve the problem. What did I do in this mathematical process? I split the difficulty of the problem into two components. One component I do not know how to handle mathematically. That whole part I pushed into the definition—I swept under the rug all the dust I don’t know how to deal with, and put a carpet over it. Now everything is fine, everything is clean, I walk on the carpet and get straight to the destination. In other words, I took all the hard components in this problem, hid them inside the definition, and from that point on it’s straightforward. There’s no problem now; you can prove things easily. Why? Because you ignored all the hard things, you skipped over them. That was the difficulty. The basic difficulty in proving this theorem was not in the mathematical step; the mathematical step is easy. The essential difficulty was how to translate this intuitive concept of a convex shape into some concept I know how to use mathematically, something well-defined. Now this step of definition is actually the essential step in mathematics. Whoever does it intelligently—and here I’m getting close to your question—whoever does it intelligently sweeps all the dust under the carpet. From that point onward the room is clean. Now you can come and inspect, hold roll call, everything is clean because it’s all under the carpet. Everything I don’t know how to deal with is under the carpet; it is inside the definition. So what does that actually mean? It basically means that mathematics cannot solve any real problem. We talked about the hot-air balloon, right? The mathematician who says something perfectly precise and therefore helps us with nothing. Here too it is exactly the same thing. Mathematics cannot really solve substantive problems. What it can do is distinguish the difficulty in the problem into two components: one component that we do not know how to handle, so let’s put it into the definitions; and the second component is that from the definition onward I can prove the theorems for you. And of course even there sometimes it takes no small amount of effort and intelligence and all that, but those are the easier parts. The hard parts—the mathematical art is simply to sweep them under the carpet. Whoever makes the definition correctly hides as much of the hard material as possible under the carpet, and then finds a definition that wraps around all the difficulties, and from there on you can do whatever you want. Now I come to your question. What does this actually mean? Mathematicians often, if you ask them, will say that the definition they propose is arbitrary. I simply defined this concept, and from that point on I can prove various claims about the concept I defined. But clearly that is not true. The mathematicians are just describing only the second part of their work. Creating good concepts—that is the real genius. Because if he did that, then mathematicians will come afterward and prove a hundred thousand theorems about the concepts he defined. Because that is what is called a fruitful concept. It is a concept about which one can prove many things, an interesting concept. What does that mean? That basically you succeeded in taking concepts from our everyday life, fitting them into a mathematical template, a mathematical definition, and from that point on this becomes an interesting field because one can prove things about it. The first question I asked was not interesting for a mathematician. You can’t do anything with it. I don’t know what such a shape is. Once you defined it well, the convex shape, defined it well, now it became an interesting question because it has the potential for proving theorems. You can prove all kinds of things about it. That means you made a good separation between the dust and the later work—between the parts we do not know how to deal with and the parts we do know how to deal with. Therefore, if you ask me whether the intersection of every two convex shapes is a convex shape, my answer is: I don’t know. Why?
[Speaker B] Because I can’t prove it. Maybe there is no dust.
[Rabbi Michael Abraham] But look—you can see there is dust. When I ask you the question, you didn’t know how to answer it. So that means there is dust; something here needs cleaning. Now someone comes with a definition—a vacuum cleaner—and suddenly you can solve the problem easily. What does that mean? That there was some dust here that someone removed. The initial difficulty we encountered is what proves that there is dust.
[Speaker F] So because of the difficulty in defining, that doesn’t mean that…
[Rabbi Michael Abraham] That’s the dust! In this case, that’s the dust. Meaning, the ability to define things is no less important—and maybe much more important—than the ability to prove things on the basis of those definitions. And that basically means, coming back now to your question, that a definition is not something arbitrary, contrary to what mathematicians often say. It’s not freestyle, like you said. So what is it? A definition is supposed to capture an everyday concept. In other words, there’s some everyday concept that we think about intuitively, and then someone comes and proposes a formal definition for it. And the assumption is that the definition really does define the concept I have in mind when I think about a convex shape. And I believe that the definition here does that, but I believe that it does that. There’s no way to prove it. And so I’m going to try to show from here as well, the way we showed through Wittgenstein and the way I showed through all kinds of other examples, that mathematics, or proof, or positivistic moves—yes, deductive logic—will not really succeed in solving problems from life. At most, only after we’ve turned them into a model. Once we’ve turned them into a model, then we can work with mathematics and solve the problem. So yes.
[Speaker C] The same thing with a physical definition—for example, you experience green, and the physicist talks about this or that wavelength.
[Rabbi Michael Abraham] There I don’t think that’s a definition. No, there it’s a fact. Meaning, the fact is that when I experience green, the wave hitting my eye has a certain wavelength. So here this isn’t a question of definition. It’s true that once you know that you’re
[Speaker G] formalizing it,
[Rabbi Michael Abraham] formalizing it, but it’s a formalization that is the result of observations and not of definition. Mathematicians don’t deal with observations; mathematicians deal with ideas. So there you can’t talk about observations; you have to talk about conceptualization. The parallel to observation אצל physicists is conceptualization among mathematicians. Meaning, you take some everyday concept that people understand and think about, and then the mathematician comes and conceptualizes it. That’s a kind of observation of the world of ideas; it’s an observation of the world, but that’s true—it’s a kind of ability like observing the physical world. And the definition is really the result of observation, that’s the point; it’s not a construction. It’s not something a person builds out of nothing from himself, but rather some kind of observation, where I know how to discern what is actually standing before my eyes in the world of ideas.
[Speaker G] Just before you move to the analogy of what you’re saying, the question is whether the analogy is more like physical formalization or mathematical conceptualization when you move to Jewish law.
[Rabbi Michael Abraham] Obviously the mathematical one,
[Speaker B] But we said everyone can ask one question.
[Rabbi Michael Abraham] Someone brought schnitzel for the second question. In any case, the claim is that this is similar to work in mathematics, because in Jewish law too you have no ability to observe the world and extract some datum from it. It’s a conceptualization of ideas in the halakhic world. In that sense it resembles mathematics and not physics. You could say that, for example, law books or whatever, the canonical sources, are perhaps a kind of facts that we observe and from which we try to extract the definitions. So there, through that, maybe you could see it as a kind of observational work, I don’t know. In any case, this too demonstrates that positivism is basically a very naive method. It’s naive because it thinks that with pure logical means one can solve problems. It doesn’t work in law, it doesn’t work in Jewish law, and as we just saw, it doesn’t even work in mathematics. It works in mathematics only after you’ve done the dirty work. From that point on everything is fine. Mathematics is the way, and that’s why mathematics is used in physics and in other fields, but it isn’t physics itself. Physics is everything there that isn’t mathematics. The dirty work is done by the physicist. After you formulate the laws and the concepts properly, after you define them well, from there on it’s already mathematics. The halakhic decisor, in this matter, is at the first stage. From there on, it’s only bringing into actuality things that are already present within the conceptual world, and I remind you that this is exactly the point we saw when we spoke about the relation between analogy, induction, and deduction. And I said that in deduction—the joke about that mathematician who doesn’t help us at all—deduction, when I say that all human beings are mortal and Socrates is a human being, then the conclusion that Socrates is mortal was really already contained within the premises; it didn’t teach us anything new. Because if I know that all human beings are mortal, then obviously Socrates in particular is one of them, so he too is mortal. Meaning, logical deduction doesn’t teach me anything that wasn’t already in the premises, which is exactly the same phenomenon I described earlier, because it basically means that all the information in this system is already in the premises. What logic or mathematics or positivistic thinking does, essentially, is to extract from my system of premises more and more information that is already contained within them. Never to add additional information beyond what was there. And from every direction we keep seeing the same thing: systems from life cannot be handled with deductive tools, with logical tools. Unless we make a good representation, make a good definition. In the halakhic-legal world and in the mathematical world, in this sense it’s the same thing. And this distinction, that mathematics is something different from physics, is in a certain sense an illusion. Because in mathematics too there’s some part that isn’t logical, it isn’t valid. It’s just that that part is always hidden behind the assumptions and initial definitions. From there onward everything is clean, but that’s true in physics too. In physics as well, once you have the Schrödinger equation, all that remains is to solve it in one problem or another. But who said that this equation really describes the world? That’s the assumption of physics. Not that if there is a Schrödinger equation then this is its solution—that’s mathematics. In other words, all the information, all the physics, all the substantive information that there is in this matter, is in the equations, in the concepts, in the fundamental principles; and the use from that point on is only mathematics or logic, meaning those are the “trivial” parts, in quotation marks. Sometimes they’re very hard, but they’re the parts that can be carried out, meaning they’re necessary, they’re certain, like Fermat’s theorem. So to say that this is trivial is a bit exaggerated. Apparently even those who checked the article didn’t really manage to understand the move there. But still, after they checked it, they understood that everything is there. Meaning everything is there. It’s sometimes a very difficult effort, but after you make that effort you see that everything was already inside the premises. Meaning there’s no leap there. That’s what it means that there is a proof here. And what does that mean? It means that this whole result of Fermat’s theorem is basically contained in the premises. It’s not that they proved it; it’s that it is in the premises, and whoever accepts the premises will also accept the result. And so it is with every mathematical theory. Of course it’s in the premises. Right. That’s what they proved. But how did you adopt the premises? You adopted them. And that’s true of every mathematical model for something from life: it always hides behind it a great many problematic steps that we don’t know how to handle, and therefore we made a model. Once we have the model, everything looks terribly simple to us. Let me maybe give you an example before I move to the next stage. I’ll give you an example. There are some statements about an a fortiori argument. There are some statements that an a fortiori argument is really a kind of logical deduction; it’s a necessary argument. Adolf Schwarz has—he was a scholar of hermeneutical principles, long ago at the rabbinical seminary in Vienna. He wrote several books on the hermeneutical principles. Among other things, he makes this claim, that basically an a fortiori argument is a syllogism, a deductive argument. Now here there’s room to distinguish between several kinds of a fortiori arguments. There is an a fortiori argument called “included in two hundred is one hundred.” “Included in two hundred is one hundred” is the mathematical, logical a fortiori argument. What does that mean? For example: if one is liable for opening, then all the more so for digging, right? A pit in the public domain. The Torah says, “If a man opens a pit or if a man digs a pit.” The Talmud asks: if he is liable for opening, then all the more so for digging. Why does it need to say “if he opens” and also “if he digs”? “If he opens” would be enough. Why? Because if by opening an existing pit he is liable, then when he creates the pit itself he will certainly be liable. A fortiori. But this is a special kind of a fortiori argument. It’s not like, “Behold, the children of Israel have not listened to me, so how will Pharaoh listen to me, and I am of uncircumcised lips?” That too is an a fortiori argument; it’s one of the ten a fortiori arguments in the Torah. But it’s a different kind of a fortiori argument. Because the relation between the children of Israel and Pharaoh is not a relation of inclusion. It’s clear to me that the children of Israel would be more obedient to me than Pharaoh. And if even the obedient ones won’t listen to me, then Pharaoh, who surely won’t obey me, surely won’t listen to me.
[Speaker H] That’s an a fortiori argument. The case of the pit… or the a fortiori argument could be that the children of Israel want to listen because it will be good for them, and Pharaoh…
[Rabbi Michael Abraham] It doesn’t matter, but they’re still more obedient, for whatever reason. Digging and opening a pit is a different kind of a fortiori argument. Because there, opening is literally included within digging. It’s not that digging is more severe than opening, but that the act of digging includes within it the act of opening plus something else. That’s what is called “included in two hundred is one hundred.” Meaning, when I dig a pit ten handbreadths deep in the public domain, then in particular I also removed the top layer when I dug the whole pit. Removing the top layer is opening a pit. So when I dig, I’m not doing an act that is more severe than opening; I am doing opening. And something more. That’s what is called “included in two hundred is one hundred.” Meaning, the relation of severity is a relation of inclusion, inclusion with a kaf, yes? I contain. The broader action contains the narrower action. Or I once spoke about this way back at the beginning of our meetings, I spoke about that Kesef Mishneh that talks about one who passes some of his children to Molech, and not all of his children to Molech. One who passes all his children to Molech is exempt. One who passes some of his children to Molech is liable to death. Now this is an a fortiori argument. If someone passes some of his children, then someone who passes all of his children is certainly liable, and that is an a fortiori argument of “included in two hundred is one hundred,” right? And nevertheless, in their exposition, the Sages exempt one who passes all of his children. So these are a fortiori arguments which are—a moment, I’ll come back to it—these are a fortiori arguments of “included in two hundred is one hundred.” Seemingly, the a fortiori argument of Pharaoh and Israel—“Behold, Israel did not listen to me, so how will Pharaoh listen to me”—is certainly not deduction. There’s no necessity here at all. You assume that Pharaoh is less obedient than Israel, and since this is less, why—if a cat encounters I don’t know what, a dog, then it runs away from it. After that it encounters a lion or a lioness; from the lion it would probably want even more to run away. But then it meets a lioness, and now it has met a lioness for the first time, it doesn’t know what to do with this thing, whether it threatens him or doesn’t threaten him. So it makes an a fortiori argument. It says: if that little and not very frightening dog tried to tear me apart and I ran away, then this lioness, which looks even bigger and even more frightening, surely I need to run away from. This is the kind of a fortiori argument that is not necessary. Maybe this lioness is just a big friendly puppy? There are כאלה. I have a cousin who has a dog the size of an elephant, and it jumps on you like some little puppy, and you don’t know where to bury yourself, it practically buries you in the ground when it jumps on you, and it comes to pet you and wants cuddles. So that’s a kind of a fortiori argument, and clearly it’s not logical deduction. It could be right, it could be wrong. So that’s an a fortiori argument that is not deduction. But an a fortiori argument of “included in two hundred is one hundred” is an a fortiori argument that seemingly is actual deduction. Why? Because obviously, if one is liable for opening, then all the more so for digging. For digging he is liable not because it is more severe than opening, but because he did opening. After all, one is liable for opening; when he dug, in particular he also did opening, so he should be liable on account of the opening that is in it. You don’t need to get to the point that digging—maybe digging is not more severe than opening, I don’t know—but let him be liable not for the digging; let him be liable for the opening involved in it, and for opening one is certainly liable. So that’s complete deduction. Right? Or passing some of his children and all of his children. What do you mean? One who passes all his children to Molech should have been liable. Why? You tell me that all his children may not be more severe than some of his children, but in practice he also passed some of his children; he just also sacrificed all the rest. So make him liable for the fact that he passed some of his children. That is an a fortiori argument of “included in two hundred is one hundred.”
[Speaker I] And now it turns out that there too there are verses teaching that we do not punish based on an inference. Leprosy, leprosy over the whole body.
[Rabbi Michael Abraham] Yes, I said I would
[Speaker I] get to that, this matter, leprosy, leprosy over the whole body.
[Rabbi Michael Abraham] The Kesef Mishneh explains there why not, and I’m about to get to that now. Right, that’s the point. It turns out that even this a fortiori argument, which is seemingly complete deduction—even it is not deduction, even it doesn’t work. The fact is that one is not punished for passing all his children to Molech. The whole question is how this works logically, because after all it is logically necessary. And if it’s logically necessary then it also has to appear in Jewish law.
[Speaker B] The purpose of punishment is something entirely different, right.
[Rabbi Michael Abraham] That’s exactly the question. The moment we take our model and apply it to life, new things always arise. Meaning, let’s take for example—I think I once brought this example—the Belgian law about selling wine, the Vandervelde law, I think I mentioned it once. There’s some place like that, I don’t know if you know it, Vandervelde, some place in Belgium, and there’s a law called the Vandervelde law. This law tells us—and I’m pretty sure I’m saying this correctly—that it was forbidden to sell people two liters of wine. Two liters of wine one was not allowed to sell. What was the idea? The idea was that the worker would bring his weekly wages home and would buy wine with it at the neighborhood pub and wouldn’t bring money home, and there would be nothing to eat. So the legislator said: drink a glass of wine, drink two glasses of wine, but two liters of wine you may not sell. Fine. A worker comes to the bar, one horse walks into a bar, as they say, and he asks the bartender, “Please bring me ten liters of wine.” “I’m sorry, there’s a law, I’m not allowed to sell you two liters.” “I didn’t ask for two liters, I asked for ten. Ten is permitted, two is forbidden.” And that is “included in two hundred is one hundred.” What do you mean? If ten is permitted, then when I sell you ten I am also selling you two. If I sell you another eight, I’ve violated the prohibition another four times, so because of that it’s permitted? And they went to court. That’s how it’s described by a Jew named Chaim Perelman, who was, I think, a professor of law in Brussels, and—I don’t know if he still was because it was a long time ago—he dealt with legal rhetoric, he was an expert in legal rhetoric, and he brings this example. And he says they got to court and the judge ruled in favor of the buyer. This is an a fortiori argument of “included in two hundred is one hundred.” You have to understand, there is no refutation of it, and there can’t be a refutation. When you sell ten liters, in particular you sold two. The Maharshal in the second edition on Bava Kamma—the Maharshal says that if one is liable for opening, then all the more so for digging, because it is an a fortiori argument of “included in two hundred is one hundred,” so here monetary liability can indeed be derived from inference. Because the reason we do not punish based on inference is that perhaps there is a refutation of the a fortiori argument, but in this kind of a fortiori argument there can’t be a refutation—it’s an a fortiori argument of “included in two hundred is one hundred.” Although there’s a dispute there between the Mekhilta and the Babylonian Talmud; he too notices this point, but he claims that the one who says that in this case we do derive monetary liability from inference, it doesn’t mean that we always derive monetary liability from inference; in this case we do, because here there’s no refutation of the a fortiori argument. If one is liable for opening, then all the more so for digging—one is liable by virtue of the opening that is in it, so obviously he should be liable. Now the Vandervelde law shows that this is not true. Meaning, there is a refutation of
[Speaker E] that ruling that ruled that way,
[Rabbi Michael Abraham] or maybe not, because at the end of the day I don’t know whether it survived appeal, but he explained there why not. What was going on there? The law, as I said before—the rationale of the law. The question is whether we use purposive interpretation, but he did use purposive interpretation, and the claim was that if you allow people to buy two liters of wine then basically they won’t bring money home, the workers won’t bring money home. Now someone comes and says: friends, I want to invest in the wine industry. What’s forbidden? There’s a Basic Law: Freedom of Occupation. I want to invest in the wine industry, I want to buy ten liters of wine to put in the cellar. How can you forbid me such a thing? Who will sell wine if no one is allowed to have more than two liters of wine? Meaning, I want to keep ten liters of wine in the cellar, and the judge said: that the law does not prohibit. If you buy wine for drinking on the spot, don’t drink two liters; drink two glasses and bring the money home. But if you’re taking savings from home—after all, that’s not a week’s wages. Buying ten liters of wine is half a year’s wages, I don’t know, two months, I don’t know how much. Fine? That’s okay. If you want to take your savings and get into wine, that is permitted; the halakhic decisor—the legislator—did not prohibit such a thing. Okay, what does that mean? That regardless of whether you can argue about whether purposive interpretation should be used or not used—this is a dispute in many legal systems, they always use it, the question is how much—but what is really behind this? What’s behind it is exactly the question of the purpose of punishment. When we take the model, in the mathematical model this is “included in two hundred is one hundred.” There is no refutation of such an a fortiori argument, “included in two hundred is one hundred.” But a mathematical model is not life. When we want to take this law and model it mathematically, we say: it is forbidden to sell two liters of wine; for every x, if x is greater than two, it is forbidden to sell x. Fine? That is basically the model, because if x is greater than two then it contains two as well. Fine? He says no, but that’s already a mathematical model, and this mathematical model is not necessarily a working one. Why? Because you assumed all kinds of assumptions when you did the formalization, when you passed from life to the model—there you swept away all the dust, there you inserted all the assumptions. From there onward everything looks mathematical, everything necessary, everything clear, everything is singing to you, everything is dancing to you. But in the background there are steps that you took and that have no justification, and the problems are always there. The problems are always there. And therefore people live under the illusion that mathematics or logic will solve their problems. Mathematics and logic never solve substantive problems. Mathematics and logic only help you sharpen where the problem is, at most, nothing beyond that. Sometimes that helps solve it, but mathematics won’t solve it; the sharpening will then help solve it. Whenever we do—I think I once mentioned this example of vectors, I spoke about it once, sometime, when I was teaching mechanics at the university. I opened the first exercise session and asked them whether the law two plus three equals five is a scientific law. An arithmetic law, yes? Two plus three equals five. And the definition is that a scientific law is a law that can be subjected to empirical falsification. Fine? Meaning, a law that I can test in a lab is a scientific law. A law that, if it passes the tests, then it’s fine for the time being, and if it doesn’t pass the tests, then it has been refuted—that’s a scientific law. A law that cannot be subjected to an empirical test is not a scientific law, a law that cannot be refuted, yes? That’s Popper, so it’s not a scientific law. Now the question is whether two plus three equals five is a scientific law. So people said yes—we talked about this—take two apples and put them in a basket, take another three apples and put them too in the basket, count how many you have altogether. If you get five, then you have proved the law that two plus three equals five. Suppose it came out minus two, or twelve. Fine? So what conclusion would we draw? Let’s be honest for a second: what conclusion would we draw—that two plus three does not equal five? That we made a mistake in the experiment, right? Or that there were oranges in the basket beforehand, or holes in the basket, I don’t know exactly what. We would never in our lives give up the claim that two plus three equals five; rather, we would assume that something went wrong in our experiment. This means that de facto the rule two plus three equals five cannot be refuted, because every time we find that it doesn’t hold up under an empirical test we’ll find excuses, we will never give it up, we’ll never refute it. Why really? Because at most, suppose I found no excuse at all—what should the conclusion be? Not that two plus three does not equal five, but that when adding oranges into a basket, the mathematical model that describes this is not arithmetic. Meaning, to describe the addition of oranges into a basket, it is not correct to describe it through two plus three equals—go see what it equals. That is not a good model for what we are doing here in the lab when we add oranges into the basket. That’s the furthest-reaching conclusion; I think no one would draw it, but that is the furthest-reaching conclusion one could arrive at. We would never give up the idea that two plus three equals five. What is the meaning of that conclusion? That conclusion basically says that two plus three always equals five, but that doesn’t help us at all. When we want to apply it to something in life, there is always some assumption that this model is a good model for what happens in life. That assumption is an assumption in physics, not in logic and not in mathematics. Logic and mathematics never deal with life; you always have to put the information in, feed the information into the system before you start operating the mathematical machine, the computer. And whoever puts the information in is the one who controls the conclusions. The information contained in the premises is what basically appears in the conclusions. Why did I bring this in an introduction to mechanics? Because I told them: there is a body standing here, and now a force of ten newtons is acting on it northward and a force of ten newtons is acting on it eastward. Now I ask: what is the resultant force acting on the body? A vector? Yes, so that’s fourteen point something, right? It’s the diagonal, fine? Square root of ten squared plus ten squared, so that’s fourteen point something. So have we refuted the law that ten plus ten equals twenty? Ten plus ten and the result is not twenty, the result is fourteen point something. So there you have it—we did an experiment and refuted the mathematical law that ten plus ten equals twenty. The answer is no. What did we refute? We refuted the assumption in physics that says that arithmetic is a good model for describing the action of forces. It isn’t true, and in fact that’s why vector calculus was invented. Vector calculus is the correct mathematical model for describing the action of forces. So this means that what stood the test in this experiment, what stood the test here, was not the mathematics of the matter; it was the assumptions that brought us to the definitions and the fundamental principles. They are the dust that we swept away on the way to the definition. Meaning, if we were to find here, for example, someone would now bring an example of two convex shapes whose intersection is not convex, people would tear their hair out. How can that be? We have a proof, it has to be true. Because if not, where’s the bug? The bug is in the assumption that this definition correctly describes our concept of a convex shape. We hide that here, and so it seems to us that everything is necessary, everything mathematical, everything clear. But that isn’t true—we’re simply hiding the problematic parts in our argument, and in life we often get slapped in the face. Because it’s like the Vandervelde law: we hid everything inside the model. Two is included in ten, so what do you mean? It can’t be. What do you mean it can’t be? Because you decided that the model is two and ten, that the mathematical model correctly describes the legal situation—but it doesn’t, it doesn’t describe it correctly. And your problem is that suddenly you got a contradiction, you got a result that doesn’t fit your model, so that doesn’t mean that what was in your calculation was wrong. The calculation is perfectly exact. Two is inside ten, yes, that’s certain. So what then? The model you built is not a good model for describing the legal situation, or for describing physics, or for describing some factual situation we are dealing with. Okay? Good. So when we spoke about the relation between deduction, induction, and analogy, I said there’s Mill’s challenge to deduction, John Stuart Mill. He says that if all human beings are mortal and Socrates is a human being, the conclusion is that Socrates is mortal. So seemingly our feeling is that the conclusion is certain, but Stuart Mill says that isn’t true: the conclusion is certain only if the premises are certain. Now how do you know the premise that all human beings are mortal? You saw how many
[Speaker J] people die and not
[Rabbi Michael Abraham] you didn’t see all of them die, because otherwise you wouldn’t be seeing anymore either. Meaning, you saw some people die, you generalized, you reached the conclusion that all human beings are mortal, and now you say okay, then Socrates too in particular. So he says that basically every deduction is built on some hidden induction. Right? Your conclusion that Socrates is mortal passed along the way through some principle that is itself the result of induction, but induction is something speculative; it is not certain that it is correct. So therefore you can never really be certain of the conclusion. Deduction is an illusion, says Stuart Mill. Now that’s the same argument as the one I’ve been making here from every direction. What is he basically saying? He’s basically saying that the certain part, the logical part, the mathematical part, is the second half of the way. Once you have arrived at these premises, when you infer the conclusion everything is certain, everything is necessary, everything is absolute. But your premises hide all the dust inside them. That’s where the problems may be. Suddenly you’ll get to Elijah the Prophet or to “our father Jacob did not die”—“Was it for nothing that the embalmers embalmed him and the mourners mourned him?”—“our father Jacob did not die,” so how can you tell me all human beings are mortal? Ah, right? Indeed the premise was wrong. If all human beings are mortal, then of course Socrates too is mortal. But that all human beings are mortal is the result of a generalization. And a generalization is not necessarily correct; either I was right or I wasn’t. That’s where the dust is. The dust is in the definitions and in the basic assumptions. And there is always dust. I once mentioned Descartes’ project. Descartes, in his cogito—yes, I think therefore I am, the principle of the cogito—basically tried to get around this problem. The project—those who understand a little of the nuances of this philosophical process, this historical-philosophical process, understand that Descartes did not want to prove that God exists. Descartes tried to show that it is possible to be nourished by proofs. He tried to find something of which I can be sure that it is true without any dust under the rug. Without observations, without assumptions, without generalizations, without anything. Simply by pure conceptual analysis. And to show that there is such a thing. So the ontological proof for the existence of God is an attempt to do that. Descartes’ proof of my own existence—yes, Descartes’ cogito, I think therefore I am—is an attempt to do such a thing. Basically all the arguments of this kind, all of which apparently fall into the same failure, are arguments that try to bypass the dust. They try to say: I can present here a rug with no dust under it, and from the rug onward show that it is true and there is nothing under the rug. Meaning, a conclusion that requires no assumption in the background. For example, the conclusion that every proposition is either true or not true. You understand that this is what in logic is called a tautology, meaning something that assumes no premises, right? It is true from itself in conceptual analysis. There are tautologies and things that are true from themselves; you don’t need to assume anything in order to conclude that they are true. But there is nothing that is a claim about the world that is like that. And Descartes tried to show that there is, and Anselm also tried to show that there is. The existence of God—that’s Anselm. My own existence—that’s Descartes. But what they were really trying to do was to prove something on the basis of a rug with no dust under it. Meaning, to build a model that does not depend on mapping life onto the model. Meaning, if I show that it is true in the model, then that means it is true in life, and that was exactly the problem in this kind of argument. Maybe you can also see it—I won’t begin today the discussion of a fortiori argument; that will be next time. But maybe you can see it if you look a bit more closely at Descartes’ argument. Descartes says, yes, I think therefore I am. Seemingly this is a trivial argument, yes—if I don’t exist then who thinks? Obviously, right? Meaning, if I think then obviously I exist. But that’s not Descartes’ argument. People make a mistake when they think that’s Descartes’ argument, because if that were so then also I walk therefore I am. There’s nothing special about my thinking. Obviously, if I don’t exist then I also can’t walk, I also can’t sit, I can’t eat either— I eat therefore I am. There are people for whom that is the essential cogito. So the claim is that there is nevertheless something special in “I think.” “I think” is the rug. From that I infer the conclusion that I exist—that is the mathematical step. The question is whether there is dust underneath. Now with “I walk” there is dust underneath. With “I think” there is no dust underneath. Why? Because if I think that I am not thinking, that too is a thought. So in any case I am thinking. Meaning, you can’t do that with walking. So “I walk therefore I am” is a correct argument, but it is based on some assumption that you may accept and may not accept—either I am walking or I am not walking, right? Therefore it is a bad argument, an argument that would not help Descartes, because it is an argument with dust under the rug. Whereas if I assume that I am not thinking, that too is a thought. Therefore I am thinking. So “I am thinking” is a rug without dust. And that is necessarily true; there is no assumption here about the mapping from life to the premise. And if on that basis I can prove that I exist, then that means I made the move from the rug to the conclusion and there is no dust underneath at all.
[Speaker J] What about “I breathe”? What? I breathe, therefore I exist.
[Rabbi Michael Abraham] Same thing. It could be that you’re not breathing.
[Speaker J] So I don’t exist?
[Rabbi Michael Abraham] No, who said so? You’re assuming a premise from biology that if one doesn’t breathe, one doesn’t exist. But before we get to accepting biology, you have a long road to go through. No, it does not follow logically. “I breathe therefore I am; I do not breathe therefore I do not exist” is a claim from biology. He was looking for claims that are not scientific, but philosophical claims, because claims in biology are once again a matter of observation. We observe and we know that human beings do not exist if they don’t breathe. I’m looking for something that doesn’t depend on observation. Observation is the dust under the rug. Do you understand? I’m looking for something that doesn’t depend on observation, that has no dust underneath. Now, all arguments of this sort are basically arguments trying to protect or ground what is called rationalism. Rationalism is our ability to arrive, by means of reason, by logical means, at claims about the world.
[Speaker C] And that stands against
[Rabbi Michael Abraham] empiricism, because empiricism basically doesn’t believe in that. Empiricism thinks that you can’t learn anything about the world unless you’ve made an observation. Fine? That’s basically what underlies modern science. And Descartes, in the sixteenth century—the end of the sixteenth and beginning of the seventeenth century—this is basically the death throes of rationalism. That phrase… was that Descartes’ phrase? Fine, it’s an aphorism, but the question is how we ground this thing. So Descartes’ effort was to fight against that. The attempt to show that it is possible, even by the tools of reason alone, to arrive at factual conclusions about the world, such as that God exists, such as that I exist, or all kinds of things of that type, without observations. Because observations can always be doubted. Okay, but if there is a rug with no dust underneath, the doubt is always: who says you put the dust under the rug, and who says your observation was conceptualized correctly, or entered correctly into the definition? Okay? But if I start from a definition that assumes nothing about reality and from it infer some conclusion, then I have done the rationalist work. I have shown that by means of reason I can reach results—which is basically, once again from another angle, that same positivistic conception that basically says that with logic I can arrive at conclusions, or in other words, that there are certain conclusions. But there aren’t. Except for this, of course, as I once said—there aren’t. That is the only certain conclusion: that there are no certain conclusions. Good.