Doubt and Statistics – Lecture 24
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
- The law of small numbers and the deceptive magic of small samples
- The guessing envelopes, “Oren Zarif,” and the survival of extreme cases
- Futurology, David Passig, the Oracle of Delphi, and vague prophecies
- Three biases join in: emotion, media salience, and an a priori outlook
- Breaking the Silence, difficult experiences, and a non-representative sample
- Nassim Taleb, success in the stock market, and Warren Buffett as a statistical extreme
- Fear, self-protection, the Gulf War, and expected casualties
- A partial defense of Breaking the Silence
- “The writing was on the wall,” hindsight judgment, and the Barcelona example
- Operation Entebbe, Rabbi Shach, and the distinction between a normative decision and estimating probabilities
- Karl Hempel’s raven paradox, Popper, and weak confirmation
- “Signs” around the World Cup, selective choice, and coincidences
- Conclusion and continuation of the course
Summary
General Overview
The text presents the fallacy of drawing conclusions from small samples, what Daniel Kahneman calls “the law of small numbers” and “what you see is all there is,” and argues that small samples are not only a bad basis for inference but are also tempting and misleading because they appear more coherent and more extreme. It adds three biases that worsen this fallacy—emotion, media salience, and an a priori outlook—and demonstrates how they shape public fears and political and moral positions even when the statistics do not support them. It then warns against saying “the writing was on the wall” and against hindsight judgment, sharpens the distinction between conditional probabilities, and shows how a single case can provide weak confirmation but not proof. It concludes with an amusing example of “signs” surrounding Argentina’s World Cup win in order to illustrate the selective picking of coincidences מתוך an infinite set of data, and concludes that one should beware of persuasive conclusions resting on a few examples, salience, and a good story.
The law of small numbers and the deceptive magic of small samples
The text states that people tend to draw conclusions from a small number of examples, and that precisely a small sample is more convincing than a large one because it allows extreme and coherent results to appear “by chance.” It illustrates this through examining a link between eating oranges and cancer, where a random group of three can look like a pattern, whereas in a huge sample the picture gets dispersed. It argues that small samples are “not only mistaken but misleading,” because you can attach a story to them that gives meaning to the results and creates an illusion of systematicity. It mentions examples of illness in small villages in the American Midwest and of success and failure in small schools to show that statistical extremes tend to appear precisely in small groups.
The guessing envelopes, “Oren Zarif,” and the survival of extreme cases
The text describes a scenario in which someone sends ten thousand envelopes, and a few people receive an envelope with the exact number they guessed, and then become convinced that this is “Dumbledore” or a “wizard.” It explains that the few who happened to get the “hit” speak up, while everyone else stays silent and ignores it, and that this is how public figures of healers and miracle-workers in the style of “Oren Zarif” are created. It argues that this is the same statistical phenomenon of a small group that “finds” coherence and on that basis constructs false authority. It presents this as part of the central lesson: one should be especially careful precisely where the examples are most convincing.
Futurology, David Passig, the Oracle of Delphi, and vague prophecies
The text tells of a futurology article by David Passig that was sent to him for review, and describes it as writing that produces wondrous but vague forecasts that cannot be falsified. It compares this vagueness to the Oracle of Delphi, which gives general answers so that any result in hindsight will appear as though it “confirmed” the prophecy. It argues that when you provide “enough predictions,” some small portion will always come true, and that part will turn the predictor into a “certified prophet” in the eyes of those exposed only to the successes. It adds that academics can “wrap everything in numbers and graphs and still remain vague,” so that the scientific appearance does not produce testable content.
Three biases join in: emotion, media salience, and an a priori outlook
The text argues that strong emotion and public-discourse presence cause people to estimate risks irrationally, such as fearing plane crashes or terrorist attacks more than asthma or car accidents despite the differences in casualty numbers. It explains that reports about asthma are not “news,” whereas a plane crash will occupy the media for days and increase the bias. It applies this also to fear of rockets near the Gaza border and to reactions to salient events, and argues that fear is not a function of statistics alone but of salience and experience. It adds that an a priori outlook, such as a prior position against the reform, can turn every detail into a “crazy demon” regardless of what will actually happen.
Breaking the Silence, difficult experiences, and a non-representative sample
The text returns to the opening point about Breaking the Silence and argues that experiences of a few problematic cases can generate an overall and convincing picture because of the law of small numbers, especially when the story “the occupation corrupts” gives it coherence. It emphasizes that the emotional impact of seeing improper behavior makes it hard to say, “this is one case out of a thousand,” and therefore strengthens the generalization. It adds a description in which out of many soldiers there will be a few who encountered many such cases, and they will be the small group that speaks and creates public salience, while most of those who saw little or nothing “sit at home and everything is fine.” It admits that this criticism could apply to him as well, because he too may rely on personal experience that is not a representative sample.
Nassim Taleb, success in the stock market, and Warren Buffett as a statistical extreme
The text brings in Nassim Taleb (The Black Swan) to explain that even if everyone invests “blindly,” statistically there will be a few who succeed for many consecutive years, and they will look like geniuses even though the success may be a product of randomness. It argues that from this one cannot infer that “his successes in themselves” prove that Warren Buffett is a genius, and proposes that the test must be more systematic, by comparing the distribution of success with the statistical expectation. It agrees that there is also a “distribution of talents,” and therefore such tests are not simple either, and concludes that it is very easy to be seduced by a single extreme case and build a baseless generalization on it. It weaves in questions from the audience about morality, metrics, and the possibility of distinguishing between a case and an essence, and clarifies that his claim is mainly about “how it is wrong to infer the conclusion,” not about deciding what is correct.
Fear, self-protection, the Gulf War, and expected casualties
The text recounts that during the Gulf War he “more or less laughed” after it became clear there were no chemical weapons, because the personal chance of being harmed seemed to him “effectively zero,” and he warns against irrational fear of rare events. At the same time, however, he emphasizes that one must protect oneself not because the individual risk justifies it, but because on a collective scale even a one-in-a-million chance across ten million residents creates an expected value of ten casualties, and that has moral and strategic significance. He distinguishes between “there’s no need to be afraid” and “yes, one should defend oneself,” and warns not to draw the wrong behavioral conclusion from his words. In the course of the discussion he raises a debate about military response according to outcome versus according to action, and argues that “you have to respond to what they do, not to what happened,” and even parallels this to his view of legal punishment that focuses on the act rather than the result.
A partial defense of Breaking the Silence
The text argues that the explanation by means of biases and the law of small numbers can, to some extent, “neutralize” the criticism that sees organizations like Breaking the Silence as wickedness and conspiracy, because one can understand them as normative people who genuinely believe in the picture that has formed for them. It suggests that the problem is at most an error in statistical inference and not necessarily malicious intent, while adding the reservation that this very impression too may itself be subject to the law of small numbers.
“The writing was on the wall,” hindsight judgment, and the Barcelona example
The text brings a case of an imam in Barcelona who had been in prison, appealed his deportation, and later headed a terror cell and carried out an attack, after which people claimed that “the writing was on the wall” and that the court was “guilty.” It argues that one must be careful with such an argument, because a court acts on the basis of evidence and rules at the time of the decision, and the fact that an event later occurred does not prove that the original decision was wrong. It emphasizes that one must ask how many similar cases did not lead to an attack, and whether it is right to be stringent with everyone because of one case, and concludes that this is an argument that is emotionally powerful and statistically weak.
Operation Entebbe, Rabbi Shach, and the distinction between a normative decision and estimating probabilities
The text quotes a letter by Rabbi Shach according to which, before Operation Entebbe, he opposed going ahead with the operation because the chances of success were small and the soldiers were at risk, and after the operation succeeded he argued that the success did not prove he had been wrong, because even a 5% chance includes the possibility that the 5% will occur. It distinguishes between a situation in which the dispute was whether to carry out an operation despite a small chance of success and a situation in which the dispute concerned the very estimate of the probability, and argues that in the second case a single success gives some weight, but weak weight, against the estimate of a low chance. It explains that inferring “from success in hindsight” mixes together “assuming they go ahead with the operation, what is the probability of success” and “assuming it succeeded, what is the probability that the decision was justified,” and links this to the difference between the probability of B given A and the probability of A given B. It adds that one can also factor in a priori considerations and intuition alongside statistics, while being careful of the story that dresses small numbers in coherence, as Kahneman warns.
Karl Hempel’s raven paradox, Popper, and weak confirmation
The text presents Karl Popper’s position that a scientific theory must be falsifiable, alongside the counterclaim that although a theory cannot be proven, it can nevertheless be confirmed by matching cases. It brings Karl Hempel’s raven paradox, according to which “all ravens are black” is logically equivalent to “everything that is not black is not a raven,” and therefore even finding a white cloud that is not a raven should confirm the claim about ravens. It argues that this confirmation is real but “very, very weak” because of the countless objects that are not black, and emphasizes that the paradox dissolves once one understands the order of testing and the tiny degree of confirmation involved. It uses this to explain that even one success like Entebbe can give slight confirmation to an estimate of higher chances of success, without turning it into proof.
“Signs” around the World Cup, selective choice, and coincidences
The text presents a collection of supposed “signs” for Argentina’s victory in the last World Cup, including a photo of Maradona from ’86 in which his hand supposedly touches the Qatar flag, the date December 18 appearing in an interview, parallels between the movie Top Gun released in ’86 and its sequel released in 2022, numbers connected to shirts and game times, similar birthdays of the final’s referees, and a calculator calculation that yields 1.0 on the date 18/12. It argues that the total picture seems impressive because one searches for matches after the fact within “millions of possible details,” so that a few striking correspondences will always be found, and that this is the same phenomenon that appears in Torah skips, “codes,” miracle stories around Gaza-border communities, and other statistical miracles. It explains that the fallacy here is not only a small sample but the deliberate creation of a “small numbers” group by checking innumerable parameters and choosing only the successful results, then presenting them as a compelling story. It concludes that it is easy to write a whole book of “a thousand amazing cases” out of billions of tests, and that this is a central reason to be wary of statistical conclusions based on selected coincidences.
Conclusion and continuation of the course
The text notes that it seems the lecture stops here with matters of fallacies, and mentions questions about multiple universes, about updates to scientific results in the newspaper, and about the categorical imperative, and ends with “Shabbat shalom” and the announcement that this was the last lecture until Elul and that an update will be sent then, along with a clarification that the course is continuing.
Full Transcript
[Rabbi Michael Abraham] Last time I dealt a bit with the failures we have when drawing conclusions from cases we’ve encountered, what Daniel Kahneman calls the law of small numbers, or “what you see is all there is.” And we saw there that people tend to draw conclusions from a small number of examples, and in some senses, when you have a small number of examples—even though statistically, of course, it’s not correct to do that—still, when we have a small number of examples, the persuasion is actually stronger than with a large number of examples. Because with a small number of examples there’s a possibility that the results will come out more coherent, since in a small number it can happen that by chance all the results come out in the same direction. Say, I don’t know, I take samples of people and I ask whether eating oranges causes cancer, okay? So if I check groups of three people, it could be that there will be three people who really like eating oranges and also got cancer. If I check this over ten groups of ten thousand people, I may find three who ate oranges and got cancer, but I won’t find that the entire group ate oranges and got cancer. So with small groups, it’s not only mistaken to draw conclusions from small groups, it’s not only mistaken but also misleading. Meaning, a small group—even though it’s not right to draw a conclusion—has a kind of magic that actually causes us to draw conclusions from it, because the results are more extreme and more systematic, especially if I can find a story that gives meaning to those results. And we saw the examples of illness rates in small villages in the American Midwest and so on, or small schools, success and failure in small schools and the like—we saw various examples. I mentioned the example of yes, choose a number, you get an envelope, and they tell you to choose a number between one and a thousand. You open a small envelope inside and it says exactly the number you guessed, and you’re sure that the person who sent you the letter is probably some supreme magician, Dumbledore. But in fact we’re talking about someone who sent ten thousand envelopes, and there’s an expectation of ten people to whom he’ll hit the right guess. Those ten, of course, are immediately convinced that yes, he’s some terrible wizard, and everyone else just keeps quiet because they understand they got another silly letter and ignore it. And that’s how Oren Zarifs are created, and all sorts of divine kabbalists who heal the sick and perform miracles and do—I don’t know what they do—miracle-working in the vineyards. The law of small numbers accompanies us everywhere, at every step, in every direction. I spoke about alternative medicine; in the end we also talked about Breaking the Silence—not in the end, actually that was the beginning of last time’s discussion—and the claim was basically that there’s a problematic tendency to draw conclusions from small numbers. It’s both misleading and incorrect, and it also causes us to arrive at a conclusion even though it’s not correct to infer it. So in that sense, that’s really the lesson I wanted to get across at this stage. I just want to add three more biases that join this matter of the law of small numbers—or maybe before the three biases, maybe one more example. Some—I don’t know how many—maybe fifteen, twenty years ago, something like that, the editor of Tzohar sent me an article by a man named David Passig, or Pasig, or Passig, something like that, who had then come to Israel after a postdoc, and he deals in futurology, forecasting. Yes, he did a doctorate on this issue and brought it to Israel; today he’s a professor at Bar-Ilan. And he wrote some article there, all kinds of wondrous forecasts about the future of the State of Israel, and the editor was very, very enthusiastic about the article and sent it to me just to ask what I thought, because he knew I had, let’s say, a mathematical or scientific background, and he asked what I thought about this article. I read it and my eyes darkened, and afterward I read a few more things and my eyes darkened even more from that person. Basically, we’re talking about a wizard receiving a professor’s salary. You know in ancient Greece there was the Oracle of Delphi—Delphoi, Delphi, depends how you pronounce it—and people would ask it questions, yes, that was their Urim and Tumim. They would ask all kinds of questions and it would answer with responses vague enough that whatever happened would always confirm what had been foretold—yes, what the oracle had said. So if you maintain sufficient vagueness, you can never be refuted. Yes, “you will be surprised over the next twenty years,” some statement of that kind. So you understand that futurology of this sort is not falsifiable. The question is always how surprised, when, at what point, whether there will be situations in which you won’t be surprised. What distinguishes the Oracle of Delphi is that it doesn’t enter into details and more measurable claims, so it doesn’t put itself to the test of falsification. And there have been all kinds of prophets like that, yes—I’m sure, I haven’t checked, but I’m sure that if you check Nostradamus you’ll discover the same thing. Meaning, yes, some collection of a million prophecies, some didn’t come true at all, most are vague enough that one can assume they did or didn’t come true. And in short, there are people to this very day amusing themselves with Nostradamus’s mystical powers. Again, I haven’t checked—I’m betting that’s what I’d find if I checked—but maybe I’m wrong; every case is certainly worth checking in order to determine that. In any case, what I want to claim is that if you provide enough predictions, you are bound to come out looking like certified prophets. Meaning, give a thousand predictions; I assume ten of them will come true. Those ten that come true will make people stare at you as if you have superhuman abilities, and just like that you’ve become a certified prophet. It’s exactly the same kind of phenomenon I talked about last time. What happens in the mystical writings of scientific futurology is the same thing. Meaning, it’s some collection of predictions vague enough—and I’m sure some of them, even with all their vagueness, did not come true and never dreamed of coming true. And even the ones that supposedly did come true—you need to check what exactly counts as having come true. Give me some criterion by which I can check whether you succeeded or failed strongly enough. And in that sense, usually—yes, maybe Rabbi Chaim Kanievsky is also like that—in that sense you won’t find such criteria. You won’t find such criteria. The art of academics is to wrap everything in numbers and graphs and still remain vague. That’s the whole idea. The Oracle of Delphi was less sophisticated; he probably needed to do a doctorate with Passig, because Passig would have taught him how to wrap his vague statements in graphs and numbers. Then you really look like a scientist of stature. But in the end, in the end, you’re not really saying anything. It’s completely vague and amorphous. Some of what you already said came true, some didn’t come true—it says nothing. Meaning, I assume I predict no worse than he does. I and you too—not because of any special ability of mine. And so this law of small numbers accompanies us in so many contexts and in so many places, it confuses so many people, it’s even managed to create academic chairs that pay serious money to the people who hold them, and this whole story is basically the law of small numbers. So one has to be very careful with this phenomenon. Kahneman divides the biases into various types, but I think the sample I’ve brought here is good enough, at least to make us alert when we draw such conclusions. Now I want to add three more points that join this matter of the law of small numbers. These biases, basically—the first is the bias of emotion, the second is the degree of presence in public discourse, the press. And the third is the a priori conception with which we come into the discussion. Each of these biases contributes to the strength of the error, or increases the chance of error in our inference, yes? As Kahneman says, many people will tell you that plane crashes are far more dangerous than asthma, even though of course the latter kills twenty times more people than the former. Okay? I don’t know if he counted this in percentages, because asthma can affect everyone, while plane crashes only affect those who fly, but still I hope the principle is clear. Yes, people’s fear of terrorist attacks is much greater than their fear of driving a car, even though the number of people injured in accidents is many times larger than the number injured in attacks, even in the most dangerous places. So our sense of what is more pronounced and more unequivocal does not necessarily reflect valid statistical judgment. There are many biases that affect the matter. In the case of attacks and plane crashes, there’s the strong emotion attached to it, as opposed to asthma—yes, people die of asthma, but nobody treats those things as, listen, can anyone imagine a news headline in the evening, “today someone died of asthma”? Or “today ten people died of asthma”? Nobody would bother broadcasting that, and if they did, people would yawn during the report. But if you see a plane crash there, they won’t stop talking about it for three days, even though the number of casualties is of course much greater on the asthma side. And media presence is one bias, and emotional intensity—yes, the emotion or experience that accompanies the thing, say a terror attack versus a traffic accident, for example—tilts the scale enormously. People are hysterical about these things even though they endanger us much less. Yes, I never understand people who are afraid to go out because of attacks and things like that. I don’t know—you get into a car to drive, that’s much more dangerous than any terrorist attack you can imagine, but people are completely irrational. Or you know what—even the rockets flying toward the south. Not that I want to put myself in the shoes of the residents near Gaza, but in the end the chance that you’ll be hit by a rocket is zero. It’s zero, even if you’re not in a protected space. So the fear created by these things—and I assume that if I were there I’d be afraid too, it’s not that I think those people are different—it’s a human phenomenon. Fear is a function not of statistics, or not only of statistics, but no less of presence, salience, the talk, the fearmongering in the media, the emotion attached to things, and so on. I don’t even want to get into the current arguments about what supposedly awaits us because of the narrowing of the reasonableness standard, all these apocalyptic visions—I hear descriptions in the press and in various places as though tomorrow morning we’ll have a cruel dictatorship here. Shock. And as you know—I assume you know—I’m sharply opposed to the reform and to the government, but still, there has to be some logic, and it just isn’t there. These scare tactics, this fixation on the small numbers and the terrible frightening cases, do the job, and then people definitely form their views based on these things.
[Speaker B] I think people take it as a whole; they don’t look at the reasonableness standard individually. The reasonableness standard as such is less of a danger, but I think people are always talking about the whole picture.
[Rabbi Michael Abraham] So first of all, even with regard to the whole picture, I have many doubts about its significance, even though I agree—and I also wrote this—but when you hear people today talking about the reasonableness standard, it could be that psychologically the whole picture is sitting in the background, and yet you see people speaking with deep conviction about how this change in the reasonableness standard is going to destroy the State of Israel. There’s something very, very problematic in the way people relate to extreme cases. It’s always the worst-case scenario, of course. And I think part of this really is connected to the law of small numbers and the biases I mentioned here: salience in the discourse, emotion, and of course the a priori positions with which we come to the discussion. If we come to the discussion with an a priori position against the reform, then of course that will increase and intensify my fears about every single detail in it, and then everything looks like some kind of crazed demon. And if I come with a different position, then I understand that maybe there’s concern, but I’m not so alarmed by it. And both of those things don’t have much to do with the question of what will actually happen. So there are many biases that amplify or reduce our expectations after we’ve done the statistical calculation—if we’ve done it at all. I began this whole discussion with Breaking the Silence, and my claim basically was that there too I think what stands in the background—at least in my opinion, and I said that this criticism can be directed at me as well, it is directed at me as well—is that people who encountered a few cases of problematic behavior by soldiers, those few cases create for them a very broad picture, and that picture is affected first of all by the law of small numbers. And the moment you have a small number of cases, then it’s coherent, there’s some story—“the occupation corrupts,” yes, we talked about that—and so it becomes very, very convincing. And add to that the three biases I’ve discussed here. To go through such an experience of improper behavior by a soldier—that’s a difficult experience. A difficult experience. I know that from myself. To see such a thing. And that greatly intensifies the significance of the matter, and afterward it’s very hard to say, okay, this was one soldier one time out of a thousand cases, and we need to be careful before drawing general conclusions, because the emotional and experiential impact is very, very strong. The media salience, of course. And as I said earlier, the a priori position with which I arrive. Yes, if I know that the occupation corrupts, I have some story like that, then obviously—the occupation corrupts, here’s the example, I saw examples of this with my own eyes—and it fits together nicely and creates a very convincing story. Take it even further—if we talk about the Oren Zarif-type examples. Think about soldiers in the IDF who served, say, several years, each one doing what he did. Most of them didn’t encounter improper behavior by soldiers, it seems to me, at least not anything extreme. Some encountered, say, one example. A much smaller group encountered two examples. And maybe three of them encountered seven, eight, ten examples, okay? Now those ten, or however many it is, the ones who encountered ten examples, they basically say: what do you mean? I saw this at every step, it’s a whole story, that’s how things are. Now the other people, who encountered one or two examples, generally don’t go to the media with it. They don’t say, look, we have a wonderful army and report in the media what amazing cases they discovered. They just sit at home and everything is fine. And then a kind of salience is created because there’s a small group of soldiers—again, the law of small numbers—that saw extreme examples. This is exactly the phenomenon we saw regarding cancer, illness rates, and the success of schools in the United States. If there are a lot of soldiers, the chance of finding a small number of soldiers who saw many problematic cases of behavior is high. There will always be three like that. I think I mentioned this—the Lebanese writer Nassim Taleb, The Black Swan, yes, he has several books—he talks about this phenomenon. He says, think about people entering the investment market, yes, the stock exchange. By nature, some profit and some lose. Some profit over a full week. Some profit over two weeks. Some over a month, a year, ten years, fifty years. Of course, the number of people keeps dropping as the duration over which they profit continuously gets longer. Right? There will be many people who profit for a week—say, I don’t know, sixty percent of people will profit over one straight week, forty percent over a straight month, thirty percent over a year, ten percent over ten years, and one percent over fifty years. There’ll be one person who profits continuously for eighty years in the stock market. Now notice, I’m speaking about a situation where, say, everyone is investing blindly, with no skill, no understanding, and no talent. Assuming that’s so, there will still be such a distribution, right? That among them there will be a few who succeed systematically over many years in investing—not because they’re more talented, but just because by chance that’s how it fell to them. After all, there have to be one or two like that out of a thousand. So those are the one or two. Even if everybody is just doing blind lotteries, in the end one of them will succeed over time and all the others will fall away by the roadside. And then what happens—think about what’s his name, the legendary American investor, his name slipped my mind—
[Speaker B] Warren Buffett.
[Rabbi Michael Abraham] Warren Buffett, yes, exactly. Think about Warren Buffett. Nassim Taleb even writes this. Who says he’s talented at all in this matter? It could be that he’s the one who succeeds over the years and his investments succeed over many years. There has to be one such person, even if none of us has any ability or any understanding. So supposedly, the fact that someone succeeds systematically over—listen, someone comes and says, look, I saw Warren Buffett, he’s a genius, everything he invests in succeeds. It’s unbelievable. Yes—but it’s unbelievable when there are millions of people entering the stock market all the time. Out of all of them, won’t one succeed for fifty straight years? One will, statistically, with no ability and nothing at all. And I’m not saying anything about Warren Buffett, by the way. He may well be an economic genius. I’m only saying that there’s no way to infer that he’s an economic genius from his successes. His successes in themselves do not mean he’s an economic genius.
[Speaker C] If that assumption were correct, then there should be some kind of Gaussian distribution or long tail or whatever of successful people. There’s one Warren Buffett, and there should be ten who only have ten billion, and another hundred who only have one billion, and so on.
[Rabbi Michael Abraham] Right, I agree, that’s my next sentence. There are ways to test this more systematically. I’m not claiming there’s no way to know. What I am claiming is that success in itself is not enough to determine that Warren Buffett is a genius. So how do you check it? As Yossi said, there’s definitely room to see whether the number of successful people declines more or less according to the distribution I’d expect—whether Gaussian or not Gaussian, doesn’t matter—but according to the distribution I’d expect, and then I’d say—
[Speaker D] Even if it’s according to talent I’d expect a Gaussian. Can’t hear? Even if it’s not random but according to talent, I’d expect a Gaussian according to the degree of talent.
[Rabbi Michael Abraham] Right, you’re jumping ahead to the sentence after next. In a second, that’s the sentence after next. So the claim is that basically if we check and find that the distribution of people matches the statistical forecasts, then really that’s correct. But if there’s one Warren Buffett and there’s no Gaussian at all, then apparently it’s not just chance. Again, never certain, but apparently. The thing is—good point—there’s also a distribution of talents. And therefore, even if I do find some Gaussian around Warren Buffett—or not around him, rather Warren Buffett is at one end of it, not in the middle of it—but I find some Gaussian of which Warren Buffett is at one end, it could still be a distribution of talents, not a distribution of accidental statistical success. But here, if we want to be more stubborn, then we can check whether the distribution of talents looks like the distribution of statistical success in the stock market. And we can try to see what kind of distribution I’d expect here and what kind of distribution I’d expect there, and see which of the distributions we actually find in reality. But you can already see that this is not a simple test at all, not a simple test at all. Meaning, to say that success itself proves the man is a genius is nonsense. But even more sophisticated, more complex, and more systematic tests—it’s also very hard to draw a simple conclusion from them, a clear conclusion. And so I’m showing you just how much things that are super-convincing—someone who sees a person succeeding in the stock market for fifty years will say he’s not a genius? Obviously he will. Meaning, someone who sees that the sender of the letter guessed exactly the number I guessed—so he’s not a genius or doesn’t have supernatural powers? Obviously yes. You see, these small numbers that we encounter have enormous power, and it’s very easy to get from here to some general conclusion, to make a generalization that has no basis. And therefore one has to be very, very careful about drawing that conclusion. Rabbi, Rabbi? Yes.
[Speaker E] Rabbi, still I’d want once again to qualify this when it comes to moral matters. Meaning, when we see, say, some lone IDF soldier who did something terrible, to say okay, that doesn’t indicate—it doesn’t necessarily indicate anything, because it’s a very extreme statistical outlier—but that’s not necessarily right, because it could be that it didn’t appear against no background at all, that it didn’t come out of nowhere. Therefore when one sinned, as I said, they didn’t say okay, this is a very extreme outlier—
[Rabbi Michael Abraham] With Warren Buffett too, couldn’t it also be the result of talent? Of course it could. Anything could be. The whole question is whether the facts can serve as the indicator, the litmus test, that tells me whether this is incidental or essential. That’s the question. And I’m claiming that the facts usually aren’t enough. Now what is actually true? It could be either way, I don’t know. I didn’t say anything about the question of what is in fact correct. I said how it is wrong to infer the conclusion. What is the correct conclusion? One needs to do more systematic checks and try to arrive at the right conclusion. Maybe, as I said in the previous lecture, maybe I’m wrong and I’m talking about the law of small numbers while I think the IDF behaves reasonably, and actually the people in Breaking the Silence are right. Because I too, at the end of the day, didn’t do statistics—I’m going by my own experience and by what I hear from people. It’s entirely possible that I too am being fed by some unrepresentative sample. That’s why I said: I’m bringing the example of Breaking the Silence from my point of view not because I’m right, but in order to point out the phenomenon. Once I understand the fallacy of such an inference, it’s certainly right for me to apply it to myself as well and be very cautious in my conclusions or in the overall picture I paint based on the examples I’ve encountered. I completely agree. Maybe one more completion, since I mentioned this example. Yes, in the Gulf War, with the missiles that fell here and all those things, I more or less laughed at the whole story after it became clear there were no chemical weapons. If it had been chemical, that would have been very frightening. But once it became clear these were ballistic missiles, it was simply a joke, a statistical joke. And the chance that I would be hit was literally zero. So I didn’t have a drop of fear in that war. Everyone there—or not everyone, I don’t know—many people were shaking and running to shelters, and people in their panic sometimes even lost their lives. I knew someone who lost his life because of panic. He put on the mask and couldn’t open it or I don’t know what, and in the end he died. So on the one hand, I think that’s absolutely right: this fear of frightening events should always, it’s always worth balancing it with cold statistical reasoning. But there’s another side to the coin—this I’m saying not as part of the series, but just to complete the picture so we don’t come away with the wrong conclusions. I do think we should protect ourselves. We shouldn’t be afraid. The fear is baseless; there’s nothing to be afraid of. But yes, one should protect oneself. And why? Not because even a small danger must be guarded against. This danger is so small that there’s really no point in guarding against it. If someone now said there’s a missile attack—not chemical missiles, again I say, not nuclear, but ballistic missiles, regular missiles—from Iran, from Iraq, I’d whistle and go on my merry way. It wouldn’t move me in the slightest in terms of fear. And even the probability that would justify taking steps, say halakhic steps, from the standpoint of “and you shall greatly guard your lives,” the probability is negligible—it doesn’t require any step. The problem is that someone out of the millions of citizens of the State of Israel will be harmed, even if the chance of being harmed is one in a million. If the chance of being harmed is one in a million and we have ten million residents, then the expected number of casualties will be ten. Now ten casualties is a problematic thing, especially when we’re talking about war, because it’s also a blow to morale, and it also has strategic significance, strategic consequences, for people being hurt in such a situation. And so I think that even around Gaza, say, where rockets are fired from Gaza, there’s really no chance of being hit—in other words, the chance of being hit is negligible. There’s no real reason to defend yourself because of that—again, I’m not blaming anyone for the fear. I didn’t experience it, so I can’t say; it could definitely be that I too would be afraid—but the real chance of being hit is negligible. There’s no reason to be afraid of it. Only what? Even if the chance is negligible, there are quite a few thousands, tens of thousands of residents there, and even if the chance is very, very small, one or a few of them will be hit, and that has strategic significance. And therefore I think one should protect oneself, but one should protect oneself while listening to music at the same time—meaning without fear. The fear here—the chance that I will be hit—is negligible. The chance that someone will be hit is almost certain, if we don’t protect ourselves, I mean. Almost certain. Meaning, if the chance is one in a million and there are ten million residents, then ten people will be harmed. I’m just making a toy model here, but only to illustrate the point. So obviously one should take shelter, and that’s why I say: don’t infer from my words that there’s no need to protect oneself and no need to—
[Speaker D] But it’s a little hard to use the categorical imperative here.
[Rabbi Michael Abraham] For self-defense it’s a bit strange, because there isn’t really—unless there’s some value-based consideration toward others. Because basically I’m coming to defend myself. Why should I care—why is it anybody’s business if I don’t care about the risks to myself? I mentioned the categorical imperative. The categorical imperative says that you should do what you would want to become a general law. Again, I’m saying, I can understand that in a value context. I can’t hear. When it concerns my fellow, I can understand
[Speaker D] using the categorical imperative, when it concerns my own self-defense, where I’m basically the only potential victim here.
[Rabbi Michael Abraham] Not you,
[Speaker D] it’s
[Rabbi Michael Abraham] not you, it’s your fellow. Why? Because if you die, the harm is to me, not to you. You’re already dead. The harm is to me. Because in the end, in the state, such-and-such number of people were killed, and that’s a strategic, morale blow, and that will of course require a response—which is also stupid, by the way, that you always have to respond according to what happened. That’s a stupid policy. But it’s also true in the legal world; it’s true everywhere. In my view, you need to respond to what they do, not to what happened. If they shoot, then you need to respond to that regardless of what happened. It’s so funny that everyone talked about some woman in Be’er Sheva who saved her three children or something, and their house was hit and destroyed, and she ran with all three of them to the shelter, didn’t dismiss it or anything, and a lot of commentators there said, ‘She saved us from a war.’ Because if she hadn’t run away and had been hurt, then we’d have four dead—or I don’t know, casualties—so we’d have to go to war, because we have four dead, we have to respond. The stupidest thing I ever heard. What does that have to do with the question of how many of ours were hurt? You need to respond to what they do, not to what happens. If they failed, then they’re less guilty? If they failed, does that mean they won’t try next time? Why is the outcome that happened relevant at all? I don’t know. By the way, that’s also true in the courts. I’ve written this and said it more than once. In the courts too, in my opinion, there is no significance whatsoever to the question of what happened. The whole question is what the criminal did. If a criminal tried to kill and nothing came of it, he should get life imprisonment. Absolutely. A person—completely irrespective of what happened, whether in the end it succeeded or in the end it didn’t succeed. The fact that by luck it didn’t happen doesn’t make him less evil.
[Speaker F] Why? There’s a difference in reality.
[Rabbi Michael Abraham] I know there’s a difference in reality, but why is that relevant to punishment?
[Speaker F] Because in reality he didn’t do it.
[Rabbi Michael Abraham] What do you mean, he didn’t do it? He did do it—he shot.
[Speaker F] Nothing happened, nothing happened. Nothing happened?
[Rabbi Michael Abraham] What does ‘nothing happened’ mean? He fired a gun. You’re not allowed to fire a gun at people.
[Speaker F] So it’s a different punishment, a different level.
[Rabbi Michael Abraham] No, it’s not a different level. It’s murder.
[Speaker F] What murder was here? Was there a murder here?
[Rabbi Michael Abraham] It’s premeditated murder.
[Speaker F] I don’t understand—there was no murder, there was nothing.
[Rabbi Michael Abraham] When you murder, you’re guilty for what you did, not for the outcome. If the Holy One, blessed be He, saved me, that’s not relevant to you. You tried to murder me. You should get life imprisonment, that’s all. The fact that your weapon happened to have a broken firing pin just means that besides being evil, you’re also unlucky—or rather lucky. So what?
[Speaker F] So by luck he’s less evil than someone who’s also evil and also succeeded? But I’m not punishing him for the fact that he
[Rabbi Michael Abraham] was evil. I’m punishing him for the fact that if he killed someone. Evil is one thing—that’s a punishment in itself. He killed someone—that’s another punishment. That’s the mistake. You punish him for what he did, not for what happened. What happened is irrelevant. Why is what happened relevant?
[Speaker F] This is the kind of argument where the rabbi says one thing and I say another, but there’s no proof for either side.
[Rabbi Michael Abraham] It’s not a matter of proof; it’s common sense. Why do I care what happened? Your degree of dangerousness and the degree of need to protect ourselves from you is not changed in the slightest by whether you succeeded or didn’t succeed. Not at all.
[Speaker F] So from what theory of
[Rabbi Michael Abraham] punishment—what logic of punishment could justify your distinction? No theory of punishment in the world.
[Speaker F] On the point that he needs to be punished, I’m not arguing with the rabbi. Of course he needs to be punished. Punished the same way?
[Rabbi Michael Abraham] What does that mean—punished the same way?
[Speaker F] No, I’m saying he should be punished, but he shouldn’t be punished as though
[Rabbi Michael Abraham] he killed someone, because in practice nobody died. Those are two separate things. According to what theory of punishment can you justify that distinction? Look, there are different theories of punishment. There’s punishment whose purpose is to deter you from next time—you understand that according to that, it makes no difference, right? If the purpose of punishment is to deter you or others from doing the same thing, then it doesn’t matter whether you pulled the trigger; we need to deter you from pulling it again. What difference does it make what happened? If punishment is a sanction for the fact that you are evil and deserve punishment for being evil—again. Other than some kind of mysticism, that it’s some sort of atonement, a rectification of the wrong that happened—or I don’t know exactly, some kind of thing like that. That’s the only option that can explain the difference between these two cases, and what does that kind of mysticism have to do with legal theory?
[Speaker G] Rabbi, I think I have an explanation for the difference, if I may. I think first of all that the rabbi basically assumed that the whole purpose of punishment is deterrence of the person who does the act. I’m not convinced.
[Rabbi Michael Abraham] No, no, no, I didn’t assume that.
[Speaker G] Either from the standpoint of justice or from the standpoint of some kind of order, so that I won’t commit some offense because I know I’ll be punished. But not necessarily—maybe punishment is also so that I won’t harm you because you killed my sister. So if you killed my sister, I have a reason to put you in prison for life, because otherwise I’ll go crazy and I’ll start hurting everyone too. If you’re saying that if you didn’t hurt my sister, then what reason do I have to go crazy? For example.
[Rabbi Michael Abraham] You’re telling me that the one who actually killed should be punished more severely—that is, give him three life sentences. I’m suggesting giving life imprisonment also to someone who tried, so what are you telling me?
[Speaker G] Right now you’re talking to me about gradations.
[Rabbi Michael Abraham] Rabbi, but rabbi—
[Speaker G] If the rabbi agrees that there needs to be a different proportion
[Rabbi Michael Abraham] between
[Speaker G] an act and a case of actual result, why not?
[Rabbi Michael Abraham] I’m not talking about proportionality at all. There shouldn’t be any proportionality. Proportionality is a completely irrelevant consideration. In my opinion, for example, a person who stole one shekel, and there’s a chance he’ll go on stealing shekels, should sit in prison for life.
[Speaker F] Punishment isn’t only about punishment.
[Speaker G] But I’m explaining, Rabbi: there is a need for the result, there is a need to look at the result, because sometimes the result of what happened causes further ripples in society. Among other things, the rabbi talked about the small doubt—once you hear that a woman was murdered, then obviously I’ll hear about it too, and I want the deterrence to be stronger for all the people listening. That’s another reason why the result is relevant to punishment. Punishment does not necessarily have to be symmetrical.
[Rabbi Michael Abraham] The moment you sentence someone for attempted murder to life imprisonment, then attempted murder too will make it into the newspaper exactly like murder—it’ll be the same thing, it’ll get the same treatment. I don’t see the difference. Fine, but that’s not our topic, it doesn’t matter right now. For our purposes, what I want to say is that the consideration we need to make is the correction I made earlier, so that we don’t remain with a wrong conclusion—a statistically correct conclusion but a behaviorally incorrect conclusion—which is that yes, you do need to protect yourself even though the probability is very small. And the need to protect yourself is not because—you don’t need to be afraid—and it’s also not because of the duty of caution and ‘you shall greatly guard your lives,’ because these probabilities are so tiny that it’s really neither this nor that. The point is that we need to ensure that collectively the result in which someone gets hurt does not happen. And in order to do that, each one of us needs to protect himself, because we don’t know who will be the one who gets hurt—the one in a million who gets hurt. Therefore all of us have to protect ourselves. So in that sense I just want there not to be a wrong conclusion from the statistical consideration I mentioned earlier. Okay, so that’s regarding that, regarding the law of small numbers and the biases and the various biases. By the way, I maybe want to add one more comment: in this way of looking at things, it actually somewhat neutralizes the common criticisms of organizations like Breaking the Silence, who are often perceived as evil people with schemes—I don’t know—to destroy the IDF or the state or its foreign relations or whatever. But if I’m really right in this interpretation—and again, of course this is a group and surely there are all kinds—but in general I tend to think that most of them are in fact normative people and don’t want to destroy anything. They sincerely believe that this is the correct picture and feel that it is not receiving proper attention and proper treatment, and therefore they go public with it. And in that sense, they at least don’t come out evil. They may come out a little foolish, but all of us in a certain sense are foolish in these ways. And therefore in that sense there is also a kind of charitable judgment here toward these groups, even though I presented criticism of their criticism. But that criticism means that this is not wickedness; at most it’s a mistake in drawing statistical conclusions—or not necessarily wickedness. I don’t know, maybe there are some who are, but I think that generally it doesn’t seem to me that that is usually the case. Again, according to the law of small numbers, from those whom I met. So it could be that that too is subject to the law of small numbers. Now I want to move to another aspect of the issue. There was—yes, I saw a few years ago some report that there was some imam, leader of a terror cell, who carried out an attack in Barcelona. He had been in prison for four years for smuggling hashish from Morocco to Spain, and then he was a candidate for deportation from Spain. He appealed in court, and his appeal was accepted, and for some reason the reasoning was that deportation would limit his freedom of movement. Then shortly afterward he carried out an attack. He headed a terror cell and carried out an attack. He had been sitting there for hashish smuggling; it could be that the hashish was to finance his terrorism, I don’t know exactly, yes? In any case, he sat four years, then they wanted to deport him, and the appeal was accepted and they didn’t deport him. The point that came up after that was that, of course, the writing was on the wall. They said, look, you see, the writing was on the wall—they wanted to deport him from Spain, they didn’t deport him, and we got hit with a terror attack. Yes, I think that’s an example—I wrote about it in my column no. 87, so I use it—but we encounter examples like this almost every day. Yes? I don’t know, some person whom the court treated in a certain lenient way, and afterward he really committed a crime, murdered his wife, I don’t know, did something, and there you go, the writing was on the wall, the court is guilty, it has blood on its hands. Okay. Now here one has to be very careful—again, with the same story. Because in the end, when the court is deliberating, it has before it the correct data for that moment. On the basis of the correct data at that moment, it has to make decisions. The fact that in the end something happened does not necessarily indicate that the decision from the outset was wrong. Because it has the legal rules according to which it has to work, it has the evidence according to which it has to work. We need, for example, to check how many people in such a situation did not go on to carry out terror attacks. And there was one who did. So the question is: because of one person, should we have deported all 1,000 who were in that same situation from Spain because there was one who later carried out an attack? Maybe yes, I don’t know—but you see that it’s already not so simple. And that feeling of absolute justice, that the writing was on the wall, I always warned and I said they were wrong, and all sorts of things of that type—that’s a terribly strong argument on the emotional level and a very weak one on the statistical level. And you have to be very careful with these things that supposedly were always written on the wall. A nice example of this, a nice and somewhat tricky example, can be found in a letter I once read by Rabbi Shach. There he recounts that before Operation Entebbe there was a public debate whether they should really launch the operation. This was an operation with small chances of success, and it was a very complex operation, very far from the country, in Uganda, etc. And the question was a major one: should they launch the operation or not? In the end they launched it, yes, and we know how it ended. But Rabbi Shach then expressed the position that in his opinion it was not right to launch the operation. The chances of success were very small, the chance that soldiers would be hurt was large, and therefore in his opinion it was not right to launch the operation. After they carried out the operation and it succeeded—Yoni Netanyahu was killed, and there was also someone else, I think Dora Bloch, I don’t remember her name anymore, yes, another woman I think died there as a result of the matter, not directly—but all in all it was considered a success by all accounts. So they came to Rabbi Shach and said to him, well, Rabbi, you see, it worked. So Rabbi Shach writes there—he says: what does it prove that it worked? Did I say it wouldn’t work? I said that the chance it would work was very small. If the chance it would work was very small, meaning let’s say, I don’t know, let’s say there was a 5% chance of success. So I say: there’s a 5% chance it will work. The fact that in the end it did work—what does that indicate? Maybe these were the 5% that indeed succeeded. Again, this is the same ‘the writing was on the wall,’ the same effect, yes? In other words, you’re trying to prove to me from what happened afterward—retroactively—that from the start I was wrong, because in fact what happened did not match my expectations. Now beyond my expectations, one has to distinguish here between two situations. First situation: everyone thought from the outset that the chance of success was small; there was no argument about that. The question was whether to carry it out despite the small chance. Okay? Everyone—there was no argument that the chance was small. The question was whether to carry it out despite the small chance. In that case, if that really was the dispute, then Rabbi Shach is completely right. Because the fact that in the end it succeeded—what does that prove? You too thought there was a 5% chance it would succeed, and I too thought there was a 5% chance it would succeed. So those 5% materialized. Okay. But still there was a 95% chance it would not succeed, and I held that given that distribution it was not right to launch the operation. What does this success prove? It proves nothing. That’s the first situation. The second situation: the argument was over that very point. Suppose Rabbi Shach said—I don’t know, because there were different opinions—but suppose Rabbi Shach said that the chance of success was 5%. And someone else said, no way, there’s a 50% chance of success or a 70% chance of success. The argument was over the probabilities themselves. Here it’s already a trickier question, because now a person can come and say: look, you see, it succeeded, so that supports my position that from the outset the chance of success was not so small. Now of course that’s not a proof, because Rabbi Shach too agreed there was a 5% chance it would succeed. But it is an argument that is not completely absurd. Meaning, if this happened ten times, it seems to me we would already agree that it’s a stronger argument, right? If ten times they did it and it succeeded in eight of them, then I think in that case whoever says the chances of success are 70% and not 5% has a case, has an argument. Now true, we had only one experiment. We tried once and succeeded. Does that—does that prove anything? It proves a little, very little, against Rabbi Shach. Very little. Because one case can succeed, can fail, and with one case—again, the law of small numbers. And this is the case that succeeded. You could have done ten more operations afterward and they all could have failed. Therefore that one example is not strong proof against Rabbi Shach. But here it already has some weight, unlike the previous situation, where the dispute was whether it was worth doing, not what the chances of success were. Because you have to understand that this is actually connected to the phenomenon I spoke about, conditional probability. When we ask, before launching an operation, what is the chance that it will succeed, what we are really asking is: assuming we decide to launch the operation, what is the chance it will succeed? The argument against Rabbi Shach is built on the opposite conditional probability: assuming it succeeded, what is the chance that the original decision was justified? And I already gave several examples of the very big difference between the probability of B given A and the probability of A given B. And they are certainly not necessarily proportional to one another. There are all sorts of multiplying factors here that need to be taken into account; we spoke about this at length in several of the previous classes. So in that sense, on the one hand one has to be careful; on the other hand, it’s not an absurd argument. If I have only one case, that’s the only datum I have, and I had to bet whether Rabbi Shach is right that it’s 5% or the one who says it’s 70% is right, I would bet on the 70%. True, I wouldn’t put much money on it, but if I were forced to choose an option, I would choose the 70% option. And in that sense it does prove something—but again, with very little weight. Meaning, I don’t think I would throw out all my statistical calculations or all my statistical assessments because of that consideration.
[Speaker B] I think that, as the rabbi has explained many times, the professionals present the statistics, the analyses, the numbers. The rabbis make the normative decisions. Okay.
[Rabbi Michael Abraham] And that’s what I’m saying—that’s the first dispute.
[Speaker B] Because it sounds more like that’s really what it was, not that Rabbi Shach decided it was ten percent.
[Rabbi Michael Abraham] The second dispute that I described—I’m not getting into the question of what really was; this is a principled discussion. I’m saying, the second situation I described is one in which there was a statistical dispute. That was the dispute. And then the question—regardless of rabbis and non-rabbis, I’m not relating to Rabbi Shach as a rabbi but as someone holding a position. That was his position. So in that sense, I’m saying, if the dispute was the second dispute, then the success has at least some weight. Maybe I’ll give you an example that will illustrate it better. There is what’s called Carl Hempel’s raven paradox—he was a philosopher of science. And he says the following thing. You know that Karl Popper, also a philosopher of science, said that a scientific theory is a theory that can be put to a falsification test. That’s his criterion for the scientific status of a theory. Okay? So for example, the theory that all ravens are black can be put to a falsification test. Check a raven and see: if it’s black, fine; if it’s pink, you’ve falsified the theory. Okay? Therefore it’s a scientific theory, the theory that all ravens are black, because it can be put to a falsification test. Now according to Popper, if I checked a raven and it turned out pink, the theory has been falsified. But if I checked a raven and it turned out black, nothing happened, because that doesn’t prove the theory. Maybe the next raven will be pink. This raven turned out black. So Popper argues that all you can do is falsify a theory, but you can’t prove a scientific theory. Now others, against him—and again, even in his own writings there are various formulations—but others against him argue that true, you can’t prove a scientific theory, but you can confirm it. What does that mean? If I checked a raven and it turned out black, that confirmed the theory that all ravens are black. It didn’t prove it, but it strengthened it a little. If I find another raven and it’s black, and another raven and it’s black, and so on, that will strengthen it more and more. The more experiments I do that support the theory, the more they confirm the theory. Or in other words, contrary to the way Popper presented it, that with a theory there is only the possibility of falsifying it, these critics say: no, there is the possibility of falsifying it, but if it wasn’t falsified, it’s not that nothing happened—it was confirmed. It wasn’t proven, true, Popper is right, it’s not a proof, but it was confirmed. And the more black ravens I find, the more the theory is confirmed. That’s the principled dispute. Now Carl Hempel says there’s a paradox. What’s the paradox? The theory ‘all ravens are black’ is equivalent to the theory ‘everything that is not black is not a raven.’ Agreed? It’s the same thing. If A implies B, then not-B implies not-A. If I say all ravens are black, I’m basically saying that whatever is not black cannot be a raven, because if it were a raven it would be black, since all ravens are black. Right? So the two statements are equivalent; it’s just a positive formulation and a negative formulation of the same theory. ‘All ravens are black’ is the positive formulation; ‘everything that is not black is not a raven’ is the negative formulation. If these two claims are equivalent, then to confirm—assuming you can confirm the first theory—you can also confirm the second theory. Right? There’s no difference; they’re equivalent. So if I now find one cloud that is white, then I have confirmed the theory that everything that is not black is not a raven. Look, here is something white, and it turns out it’s a cloud, not a raven. So that is an example that confirmed the claim that everything that is not black is not a raven. But if so, then it confirmed the theory that all ravens are black, because it is a logically equivalent theory. To confirm the claim that everything that is not black is not a raven is also to confirm the claim that all ravens are black, because these are logically equivalent statements. You confirmed this, you also confirmed that. Then something terribly absurd follows. It follows that every time I checked an object—say, a white cloud, a green tree, a brown chair—I confirmed the theory that all ravens are black. Every such example contributed further confirmation to the theory that all ravens are black. On the face of it, this seems absurd. What, seeing a white cloud confirms the theory that all ravens are black? So Hempel brings this as proof for Popper’s claim that there is no such thing as confirmation. Confirmation is a fiction, just psychology. There is either falsification or proof, and proof is impossible, so there is only falsification. And confirmation—if there were such a thing as confirmation—then finding a white cloud would confirm the theory that all ravens are black. Where is Hempel mistaken? Because actually finding a white cloud is indeed a confirmation of the theory that all ravens are black. True, it is a very, very weak confirmation. Why? Think about the theory ‘everything that is not black is not a raven.’ How many non-black objects are there in the world? Masses of them, right? So if I found one example—and also they are very diverse, of all kinds, the non-black objects. There are living creatures, inanimate objects, all kinds. Right? Therefore, when I find one example, the confirmation of the theory that everything that is not black is not a raven is a minuscule confirmation. It exists, but it is minuscule—really tiny. And at that level it really does also confirm the theory that all ravens are black. But it’s a very, very, very small confirmation. In order truly to confirm it, I would have to go through billions and billions and billions of objects and see in all of them that if they are not black, they are not ravens. If I went through many such cases, that really would confirm the theory that all ravens are black. There is another important point, which is basically the order of the procedure. I don’t take a cloud and check what its color is—that’s worthless. I take a white object and check whether it is a raven. I have to begin with the color and then check whether it’s a raven or not, not begin with an object that is not a raven and then check whether it is not black. Because I claim that everything that is not black is not a raven, not that everything that is not a raven is not black. There are also things that are not ravens and are black. Therefore the independent variable is the color. The dependent variable is the object—whether it is a raven or not a raven. Therefore one has to check objects that are not black when I still don’t know what they are, and then check whether it is a raven or not a raven. If it turns out not to be a raven, that really confirmed the theory. Therefore Hempel’s paradox is not a paradox. Why am I bringing this up? Because here too—why is it so easy to miss the confirmation that is present? Because the confirmation is minuscule. This is one case out of billions and billions and billions, and therefore the confirmation it gives to the general theory is very, very, very small. So that’s why the initial tendency is to say no, that’s nonsense, it doesn’t mean anything. But that’s not correct. On the principled level there is here some kind of small confirmation. The same thing with Operation Entebbe. I’m saying, it’s true that there is a very small chance this will happen, but there is a chance that it will happen, and maybe it happened by chance. So if I did one experiment, that slightly confirmed the claim that there are decent chances of success for such an operation. It confirmed it a little, but it did confirm it. You can’t say it has no meaning whatsoever. If I repeated this a few more times, it would already be a more significant confirmation—more than in the raven case, of course. Therefore, many times we need to be careful when we say there is no confirmation at all; that’s not precise. Sometimes there is a small confirmation, and if we repeat it the confirmation will grow. But it is not correct to say there is no confirmation here at all.
[Speaker D] I just want to say one thing: it’s very hard to prove historical events. Say, take the Munich Agreement, okay? Chamberlain thought it was a good agreement, and Churchill cried out in horror. Maybe Chamberlain was right? Look, even with the Munich Agreement, maybe in five percent of cases it could go wrong, maybe ten percent, maybe this is that case. The Oslo Accords, same thing, all those things. Right, right. So basically history can’t prove anything to us.
[Rabbi Michael Abraham] Not history. The facts in themselves, the bare facts, prove nothing. So what does? What you can do is a more systematic analysis.
[Speaker D] Yes, but we have nothing to compare it to. Out of a thousand Munich Agreements, how many succeeded?
[Rabbi Michael Abraham] There isn’t anything to do here—there isn’t—
[Speaker D] we don’t have research work on such a thing. We can’t do that; these are always different events.
[Rabbi Michael Abraham] There aren’t a thousand Munichs, but there were different agreements that have the same characteristics, and you can try to isolate variables and draw conclusions, while knowing that history isn’t physics. Right. It’s not an exact science.
[Speaker D] But that’s an observation that is
[Rabbi Michael Abraham] a bit more clear-cut than ‘the writing was on the wall.’ ‘The writing was on the wall’ is a very weak claim.
[Speaker D] If I’m not mistaken, Popper really did also claw at history, I think maybe for that reason. Exactly.
[Rabbi Michael Abraham] Nobody escaped his heavy hand. Yes. In any case, regarding psychology he later partially retracted, I think, but I agree with him in any event. But the point—what I want to add here—is another important point. There is room for a priori considerations even without statistics. If I think this is a bad agreement—the Munich Agreement—or that going out on Operation Entebbe has no chance, and I have good reasons for that, that is a consideration that is certainly relevant. All I’m claiming is that what happened afterward, by itself, cannot decide the matter. But if I have good arguments showing why there is a very small chance of succeeding in such an operation, then even if in the end it succeeded, I won’t give up those arguments—certainly not if it happened once. To be too much of an empiricist—to go only on the basis of facts and statistics—no. Intuition has weight, initial considerations have weight, logic has weight, and still one has to be careful in both directions. Because those initial considerations are exactly the story Kahneman warns against. That story that latches onto small numbers and turns them into things that sound terribly sensible. In short, life is complicated. One has to be careful when drawing conclusions. I didn’t come here to give algorithms, but to point out flaws in other algorithms. I want to finish with an especially amusing example of using signs. Signs—do they have substance or not, yes, the Talmud discusses that. So come see a wonderfully amusing example, I’ll share my screen with you. Coincidences on the way to Argentina’s World Cup win. We’re talking about the last World Cup. Okay? So the fellow writes—I assume he’s Argentine, that would fit him—Yaron Rebelsachi, or whatever his name is. Over the past month, since the World Cup victory, the internet has been flooded with screenshots and videos of various signs from the past, mainly from the 1986 World Cup, when Argentina also won, supposedly hinting at Argentina’s future win of the World Cup. So come see a few signs. Signs are signs, as they say. You see Maradona here? Yes? So let’s start with the first example. In the picture above, and in the heart of every Argentine, yes? A few weeks before the World Cup in Qatar began, an image started circulating online of Diego Maradona after winning that World Cup in ’86. The same hand that touched the ball on the way to the goal—the famous Hand of God, yes, because he scored a goal with his hand—appears in the picture touching, believe it or not, the flag of Qatar. Notice, here, do you see it? Apparently it’s—I don’t know what he meant, I don’t know, apparently that’s the flag of Qatar. Meaning this hand is touching—oh no, sorry, apparently it’s this marking here. This hand is touching the flag of Qatar. That’s the hand with which he scored the goal. There, the arrow points to it. Okay? So it can’t be a coincidence, obviously—exactly the flag of Qatar. Next. There’s an interview with Maradona in 2004. Notice what’s written behind him: 18—December in Argentine format. December 18 is of course—written below, what’s written here in Argentine format? It says here: our end-of-year gift. And 18/12 is exactly the date on which they won the World Cup 18 years later. That was in 2004. Eighteen years later, 2022 or 2023, I don’t know, nineteen years later, they won near the end of the year, yes? December 18 is almost, almost the end of the year—‘our end-of-year gift’ with the date December 18 and Maradona, the previous winner. Next.
[Speaker D] He probably wasn’t interviewed on any other date anywhere. Can’t hear? He probably wasn’t interviewed on any other date anywhere.
[Rabbi Michael Abraham] Yes, apparently. Love in the skies. There is an iconic Tom Cruise movie, yes? It was released in 1986 and immediately became a box-office success. The sequel to the film, whose original name is Top Gun, was released, believe it or not, this year—exactly in the years in which Argentina won the World Cup. The nice story is that the 2022 version was supposed to come out already two years earlier, but the coronavirus pandemic led to the film’s delay—and to Argentina’s victory. The coronavirus caused both the film and Argentina’s victory exactly—the sequel to the film that came out in ’86, the previous win. Wait, we’re not done yet. The big star of the final was the goalkeeper of the last final of 2023—2022—Emiliano Dibu Martínez. Yes? Even before the World Cup it was published all through the final: historically, only three goalkeepers did not play with the number one on their back—Fillol in ’78, which was Argentina’s first cup; Pumpido in ’86, which was the second cup; and Dibu Martínez in 2022, the third cup. Yes, clearly a meaningful sign. What about our friend Messi? Another Argentine legend. Messi is giving a motivational talk to the guys in the locker room, and what you see here—the only jersey visible with a number on it is jersey number 18. Exactly the date, December 18, of the World Cup they won. And even that is still not the end. Notice: at what time did Argentina play in the World Cup final in 1986? Twelve noon Argentina time. And at what time was the final on Sunday in the last World Cup? Of course, twelve noon, needless to say. Next, the next example. The referee who officiated the final between Argentina and France, the Polish referee Szymon Marciniak, was born on January 7, 1981. Why is that detail so exciting? Because it turns out there were Argentines who dug through the internet and discovered that the referee in the 1986 final, the Brazilian Romualdo Arppi Filho, was born on January 7, 1939. January 7 and January 7. The two referees. And of course the last case—the calculator. 2022 is the year they won, and 1986 is the previous year they won with Maradona. Divide 2022 by 1986, and what you get is 1.018—like the date 18/12. One minor detail: the victory was not 1-0. But still, it’s beautiful. It was a victory on 18/12. Okay? Now why is this amusing? Because lots of people laugh at this string of coincidences, which on the face of it still seems fairly impressive—though there are better examples and worse examples—but the overall picture sounds impressive, right? But they do exactly the same thing in other contexts. From the miracles of bullets that hit a pocket, to the miracles of rockets flying over the Gaza envelope and not hitting anyone, to Bible codes and finding codes there, and Sadat and his assassin one next to the other. All these things are exactly the same phenomenon. What’s going on here behind all this? Look: when I compare two such games, I have millions of details I could check. The birth date of every one of the players, the birth date of the goalkeeper, the average birth date of the players, the birth date of each player’s mother, of each player’s father, and of each player’s oldest brother. The date of each one’s bar mitzvah, the date of who-knows-what—that’s only dates. Besides that, the players’ heights, the goalkeepers’ heights, the median age of the players and the goalkeepers. Endless data, endless. Do you understand that once I check endless data like this on two games, I will always find ten amazing examples of the kind we just saw, even more amazing—they simply didn’t search enough. I’m sure I can find even more amazing examples. The problem is that there are so many parameters to check that there is no reason in the world why ten of them shouldn’t come out special. And this, by the way, is one of the claims—and I mentioned it when I spoke about lotteries and Bible codes—that basically, how many things can we check in the Torah? Billions of events that happened throughout history, and personalities, etc. Suppose I find among them a thousand with some interesting correlation. Out of billions I will always find a thousand. Now I write a book with a thousand amazing cases I found in the Torah. Who wouldn’t repent immediately afterward? Why? Because I could have checked billions upon billions of people and events and dates and whatever you want. Out of them I found a thousand, which is a reasonable probability—there’s no problem finding a thousand—and those thousand, when I present you with them, are amazing. I find Sadat exactly next to his assassin, and I don’t know what, the names of U.S. presidents in alphabetical order next to each other, and whatever you want. Do you know how many things can be checked? How many facts were there in the history of the world? So what’s the problem in finding a thousand very impressive examples in which Sadat’s assassin falls exactly on Sadat in the text? Especially if I play a bit with the algorithm that searches for it, and that’s what I talked about last time. Now I’m talking about a different aspect—not playing with the algorithm, but selecting significant results. If I select the results from the total set of results, then there’s no problem always finding a thousand significant results. I’ll do a billion trials and choose the thousand significant ones. Okay? Here someone brings proof for Jesus from the Torah. Yes, surely you can find it, no problem. Just as you can find Moses in the Bhagavad Gita, what’s it called, the Indian one, I know, in the Indian writings. So these statistical miracles stem from a very simple fact. Once I have endless possibilities and I can choose which cases to focus on, then I’ll simply check all the cases, take the thousand for which it worked, and write a book with a thousand amazing cases that happened in the Gaza envelope, or in Genesis, or in I don’t know where, in the Yom Kippur War. Okay? And there’s no problem writing such a book of miracles, to thank and praise and glorify His great name for all the miracles He performed, while forgetting the ninety-nine percent of the miracles He didn’t perform. You can always choose significant results if you have a sufficiently large collection of results. And this is a fallacy too—a cousin of the law of small numbers. Meaning, this is a very, very common fallacy, especially in religious thinking. But as you see, the Argentinians’ thinking about soccer is also religious thinking, as is well known. And therefore it’s very easy to fall into this kind of statistical conclusion when I select the results from the whole set of results. This is no longer really the law of small numbers, but rather deliberately manufacturing the small-numbers group. Okay? That’s really just cheating, supposedly. Well, I think I’ll stop here with the topic of fallacies. If there are comments or questions.
[Speaker D] I remember once someone asked me: maybe the universe isn’t so wondrous. Maybe there were infinitely many universes that were created and disappeared and so on, and here we are, one of the ones that happened to succeed.
[Rabbi Michael Abraham] Same argument, right.
[Speaker D] I assume maybe there’ll be discussion of this later, because it’s an argument.
[Rabbi Michael Abraham] Right, a good argument. I wrote about it in God Plays Dice and explained why nevertheless I think it’s not correct. But the statistical argument is a valid argument. The only question is what conclusion follows from it.
[Speaker B] Apropos the universe, now it’s twenty-eight billion years.
[Rabbi Michael Abraham] I can’t hear?
[Speaker B] Apropos the universe, now it’s twenty-eight billion.
[Rabbi Michael Abraham] I still wouldn’t put too much weight on that yet. You know, the fact that a newspaper publishes some result about someone still doesn’t mean very much. Fine, there are lots of results. You have to look into it. Okay, it could be true; it also doesn’t matter all that much, but I’d wait a bit longer past the newspaper headline. Anyone else?
[Speaker G] Rabbi, actually I have a question. The Rabbi mentioned this really just in passing, not in connection with the question, just as an aside. Someone asked him a question about the categorical imperative, and the Rabbi was trying to say that the categorical imperative instructs us to go into shelters, and the Rabbi said: the categorical imperative tells you to go into a shelter because if you die, you’ll harm me. But from the Rabbi’s words it sounded like there’s no issue at all—that is, is there some kind of categorical imperative that also forbids suicide in a case where I’m not harming anyone? Until now I thought there was. Why? Because I don’t want there to be a law that everyone commits suicide. Like, if things are bad for me and…
[Rabbi Michael Abraham] That’s going a bit far, because you can reject that in a few ways. First of all, if everyone commits suicide, so what happened?
[Speaker G] No, assuming that’s a law I don’t want to exist.
[Rabbi Michael Abraham] If so, fine, but assuming I also don’t want everyone to be five foot eleven, should we do leg-shortening surgery so as not to be five foot eleven?
[Speaker G] No, I mean it like this: in a case where I want a society that will be healthy and good and pleasant and so on, then is there some categorical prevention, even—
[Rabbi Michael Abraham] No one will suffer from there not being a society, or from it not being healthy, because no one will be here to suffer from it.
[Speaker G] No, no, but that’s exactly what I’m saying. If suppose I want people…
[Rabbi Michael Abraham] If that assumption holds, then apparently you’re right, and even that can be rejected the way arguments from the categorical imperative are often rejected, because you can say that the general law is that only whoever wants to should commit suicide. I wouldn’t want everyone to commit suicide, but I am willing to allow whoever wants to commit suicide, because I know very few people actually want that. And therefore society would still remain. Yes, I accept that. The question is how to define the general law.
[Speaker G] It’s just different from what the Rabbi suggested regarding running to shelters. Meaning, I have an obligation to go, to run to a shelter, not because I want ten people who would cause—not because of the ten people who would harm the strategic capabilities of the state, but simply because I don’t want ten people in the country to die. If I personally feel like taking a risk with myself, then…
[Rabbi Michael Abraham] No, no. That’s not the categorical imperative. You’re saying: I shouldn’t avoid running to the shelter because I don’t want ten people to die, and that’s in the interest of the state. That has nothing to do with the categorical imperative. Now I’m saying, fine, but if I personally don’t run to the shelter while everyone else does what they do, then it doesn’t depend on what I do, so why should I run to the shelter even after that consideration? That’s where the categorical imperative comes in. Do you understand what I’m saying? I think the argument of the categorical—
[Speaker G] imperative doesn’t really depend on the result, it depends on the act. You’re saying the categorical imperative doesn’t depend on the result but on the act—isn’t that what you were trying to say?
[Rabbi Michael Abraham] It reflects that fact. My claim is that the argument that I need to run to a shelter, or that all of us need to run to shelters so that there won’t be ten casualties, that doesn’t belong to the categorical imperative. That’s a moral command toward my people. The categorical imperative comes in at the next stage, because at the next stage I can ask myself: okay, I’ve accepted that, but now, if I personally don’t run to the shelter, all the others will run because they’re suckers, but I alone won’t run to the shelter, and I also won’t tell anyone so that it won’t influence anyone—so what’s the problem? After all, there’s no chance that I’ll cause harm, so what exactly is the problem with that? Here is where the categorical imperative arrives. The categorical imperative says: fine, but everyone can make the same calculation you’re making, and if everyone made that calculation, then there would be ten casualties and that would be bad. Right. The categorical imperative is not the harm to the ten as public harm; that’s a basic moral consideration. The categorical imperative—or a basic strategic consideration, not a moral one—and the categorical imperative is the next stage.
[Speaker G] The next stage, yes. Yes, all right.
[Speaker B] It’s like taxes.
[Rabbi Michael Abraham] Yes. Anyone else?
[Speaker G] I also wanted to ask about the black swan. This is also a question not so closely related—about the black swan the Rabbi spoke about with Warren Buffett. Actually there is a solution, I think a relatively simple solution: I can decide from now on to examine Warren Buffett specifically five years from now. Right. Yes.
[Rabbi Michael Abraham] That—
[Speaker G] won’t—
[Speaker D] help me.
[Rabbi Michael Abraham] If Warren Buffett, say, stands out on his own after he was supposedly already supposed to have exhausted his run of successes, that’s roughly the same as what I suggested earlier; it’s a similar kind of test. If I continue to track Warren Buffett, that’s basically like saying he does not conform to the expected distribution.
[Speaker G] Yes, I have to say, after I heard this discussion about Warren Buffett, I said: from now on I’m following him specifically; let’s see what happens in another five years.
[Rabbi Michael Abraham] Keep us posted.
[Speaker D] Or the problem is that even if he does reveal something, it still won’t be real—it proves nothing. What? Even if he fails now, and assuming he really was a great genius and not just a random outcome, there’s still no proof.
[Rabbi Michael Abraham] No, he’s saying that if he fails, there’s no proof, but if he succeeds, then yes—not proof, confirmation.
[Speaker D] Confirmation, yes. Because even then, by chance, it could still continue to… here we’ve only removed one more layer.
[Rabbi Michael Abraham] No, come on, don’t throw statistics in the trash. We’re talking about a situation where we do work with statistics. It’s true that statistics don’t give certainty, but statistics are the best tool we have under conditions of uncertainty.
[Speaker D] So up to age eighty, statistics weren’t enough, and then another two years forward—what is the issue, exactly, from this point on? Like, what are another two years going to add for us here from the distribution?
[Rabbi Michael Abraham] First of all, now Warren Buffett is marked out for us—what do you mean? Up until now he was one among many; now he is marked. And now I want to see: this marked individual, I’m tracking him—that’s something entirely different. Okay. Until now I didn’t mark Warren Buffett in advance; rather, Warren Buffett was the person who survived until now, but it could have been anyone else. The moment I mark a concrete person, the test is completely different, or has a completely different significance. It’s like the difference between finding something that isn’t black and checking that it isn’t a raven, versus taking something that isn’t a raven and seeing whether it isn’t black. The first is confirmation; the second isn’t.
[Speaker G] It’s like the Rabbi also talked about this in his physico-theological argument, where he says that the universe is currently special, that there is something distinctive about the universe. So basically I define Warren Buffett as special, and from now on I look specifically at him.
[Rabbi Michael Abraham] He is special in a way that is defined a priori. Yes, that’s the point—not that I’m looking at him, but that his specialness is defined a priori. The claim of many atheists is that the specialness of the universe is not—there’s nothing special about it—like a sequence of die rolls. Every sequence of die rolls is very special. The probability is exactly the same as six six six six, exactly the same. But as I’ve explained more than once, there is a difference between rare and exceptional or special. Every sequence is rare, but not every sequence is exceptional. And our universe too—the fact that it is rare really doesn’t say much. My claim is that it is exceptional, and exceptional is already something a priori.
[Speaker G] Yes. All right, thank you very much, Rabbi.
[Rabbi Michael Abraham] You’re welcome. Okay, that’s it—have a peaceful Sabbath, goodbye. This was the last class until Elul; in Elul an announcement will be sent.
[Speaker B] Peaceful Sabbath.
[Rabbi Michael Abraham] Wait, no, actually not today—wait, yes, yes, today yes. Next week is already Tisha B’Av.
[Speaker H] But does this course continue, or is it over?
[Rabbi Michael Abraham] Yes, yes, we’ll continue. Goodbye.