Probability and Statistics – Lecture 4

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[Rabbi Michael Abraham] What you just described, what you just described, is exactly the deterministic position that we have no choice. Because what they’re basically saying, the determinists, is that although we live with the illusion of choice, as if it’s conscious, in fact things happen in the unconscious and consciousness is just an illusion layered on top of that. And therefore—but those things that are generated unconsciously are generated by the brain; that is exactly the deterministic process they’re talking about. When people say we have “unconscious choice,” in quotation marks, what they really mean is that we have no choice. That’s the claim, basically. They use the term unconscious choice because that’s what we call those acts, but in fact the concept of unconscious choice is a concept from everyday language; what it really means is that we have no choice.

Now, what happens is that afterward—I discuss this in my book on neuroscience—I talk there about Libet’s experiments, and the main claim I raise there, one of the claims—not something I invented, but one of the claims I describe there—is that there is a difference between picking and choosing. The experiment, Libet’s button-press experiment, is an experiment of picking. What does that mean? After all, it makes no practical difference at what moment you press the button—press whenever you feel like it. So in that situation there is no real deliberation or choice and so on; you simply decide to press the button according to whatever signal pops up in your brain. By contrast, if you were to do the same experiment on an act of choosing—that is, where there are two options that have value-significance, and you need to decide whether you choose A or choose B—there my claim in the book was that the signal, what’s called the RP, the readiness potential, would not precede your decision. They would not be able to predict it. That’s what I argued in the book. And the experiment had not yet been done at that stage. Someone from the Hebrew University who was in touch with me told me that they were working on such an experiment there.

Now, I argued that if such an experiment were done, then the readiness potential would not precede the decision. In other words: everything is in the hands of Heaven except the fear of Heaven, so to speak. Now what happened is that, surprisingly—and I only found this out years later—but a year after the book came out, or maybe a year or two after it came out, a very, very broad experiment was published with dozens of scientists from around the world, headed, incidentally, by an Israeli scientist in the United States. Liad Mudrik from here also took part; now she’s at Tel Aviv University, and she has a lecture about this matter. And they came up with a wonderful idea for how to do such an experiment, because it’s very complicated to do an experiment on choosing, since basically you need to allow a person to make immoral decisions in the experiment and then check whether you can predict what he’ll decide to do or not—but that goes against the rules of ethics. You can’t let a person do immoral actions; the Helsinki committee wouldn’t approve an experiment like that, even if in principle it could be done.

So they came up with an idea—the details don’t matter right now—they came up with a beautiful idea for how to do an experiment on moral choice. They talk about a situation where you have to decide which fund your money will go to, to donate to different organizations, all of which are legal, but some of which you think are immoral, let’s say. So here this is a question of choosing and not of picking—not a selection where the details don’t matter right now, I’d have to go into more detail there. And not to my amazement but to my delight, from my point of view, they discovered that when you test this on problems of choosing, there is no prior readiness potential; you can’t predict the result. And that shattered this myth of Libet’s experiments, as if Libet’s experiments prove determinism. They didn’t prove it even then, but this experiment showed very clearly—again, there are surely discussions after it, I’m not up to date on everything happening there—it shows very clearly that Libet’s whole concept is mistaken. And the root of the problem is the failure to distinguish between picking and choosing.

Choosing is precisely those value-laden questions about which you consciously deliberate, whereas picking is something else—it’s a trivial decision, like when to press a button. And in a trivial decision, if some signal pops up in your brain, then of course you’ll press the button, because you have an urge to press and no reason not to press, since no value issue is involved. So whenever the signal arises, you’ll press. Therefore obviously there the signal will precede the action. But in an action that has moral value-significance, then in such a situation, even if a signal arises in you, who says you’ll act on it? The signal will tell you to do something immoral, and you’ll veto it, in Libet’s language. That is, you won’t act according to what the signal is trying to persuade you to do. And therefore precisely there, in those places where you have judgment, deliberation, and value-based decision, there the signal will not predict the result. That’s basically the finding. That was my claim in the book, but in the end it turned out to be an empirical finding. And this very clearly strengthens the distinction I mentioned to you earlier.

I’m bringing in this whole story because of the distinction I mentioned to you earlier between picking and choosing. Actions about which I supposedly make decisions—like when to press a button—that’s an illusion. I’m not making any decision; I’m driven by completely deterministic processes of my brain, and that’s what is called picking. Choosing is a situation in which I weigh things in my mind, I deliberate, and I decide one way or the other. That happens with moral value questions. It could be, for example, that someone will come and say maybe it’s also true of economic questions—any place where there is deliberation, not necessarily moral deliberation, but what to invest in on the stock market. I have considerations this way, considerations that way. One could also say there that we are dealing with choosing and not picking. I’m not sure about that, because in the end, with economic judgment, it’s a calculation. It doesn’t matter that the calculation doesn’t produce a single unequivocal result and there can be disputes about it, but it’s not the result of a choice; rather, the calculation according to your assumptions leads to this result, and according to his assumptions leads to a different result. But in the end, in the end, you’re making a calculation.

Personally I tend to think—again, this is my own personal view—I personally tend to think that choosing exists only when the dilemma is value-based. Meaning, if there is some other dilemma, like what to invest in on the stock market, that isn’t choosing; in my opinion that could be predicted by a signal. But that—I don’t know, that’s just a hypothesis. Okay, let’s return to our matter.

So the claim I basically want to make is that there are situations—wait—there are situations in reality that people see as ontic doubts, as ambiguities, but in fact they are doubts and not… And that is what I called a pseudo-ontic doubt. Now what I want to show—I was in the middle of showing—is that in other contexts of reality, of life, of thought, we encounter the same type of definition, or confusion if you like, where people see something as though there were ambiguity in reality itself, when the truth is that it is only an epistemic doubt. So one example I mentioned is rolling dice, or dragging benches across the ground. Another thing I mentioned was evolution.

What happens with evolution? In evolution—why am I talking here not about evolutionary science, but about evolutionary anti-theology, or evolutionary anti-theology? What is the whole dispute between neo-Darwinians and believers about? The claim basically rests on the physico-theological argument. The physico-theological argument says that a complex thing does not arise by itself. There has to be something that created it, someone who created it. And therefore, if the world is complex, then there is God, because God created it. And the assumption is that a complex thing does not arise by itself without a guiding hand. The neo-Darwinians say: evolution shows you that yes, it does—it arose without a guiding hand. Here, there is an evolutionary process with no guiding hand at all, and increasingly sophisticated or increasingly complex creatures emerge. In other words, ordered things, complex things, can emerge in a natural process without a guiding hand. That is basically the weak point of the dispute between believers and neo-Darwinians.

Now what the neo-Darwinians assume in the subtext, though, is that the evolutionary process is a random process, a statistical process. In other words, if a tiger comes by here, then the ape that appeared here will not survive; the tiger will prey on it. If by chance the tiger chose to go in a different direction, then this ape will survive. In other words, the survival of the mutations that arise depends on accidental circumstances, chance circumstances. The same applies to the mutations that happen in the DNA itself. In other words, how do all sorts of mutations arise in the evolutionary process? That too is generally perceived as a matter of chance, random, random. And therefore the claim is basically that there is no guiding hand here; probability or statistics alone—what Dawkins keeps repeating all the time—probability and statistics do the job better than the Holy One, blessed be He. You don’t need the Holy One, blessed be He, in order to explain how life arises and keeps improving; statistics does it. That is basically the claim.

Now what I argued in the book Does God Play Dice is that this is a mistake of biologists—as opposed to physicists. Meaning, biologists usually—and rightly so from their perspective—when they deal with evolutionary processes, or even with mutations in DNA molecules, treat them as random processes. But there is nothing random there. These are not scales on which ambiguity can appear. These are not quantum scales at the level of a single electron, or microns at most. These are scales of living creatures: a lion, a tiger, an ape, even an ant. An ant is a huge scale; there is no quantum phenomenon at the scale of an ant, okay? And therefore even at the scale of a cell, by the way—a single living cell, yes—even there it’s a huge scale. A neuron, yes? A neuron is a nerve cell. Quantum phenomena in a neuron—that’s a matter of dispute and there are discussions about whether yes or no. In any event, the claim is that this is really something pseudo-ontic, not something ontic. What does that mean?

From the biologists’ perspective they are right, because for them it is a random process: either the tiger chose to go here or the tiger chose to go there; if it was hungry it will prey on the ape, if it wasn’t hungry it won’t prey on the ape. In other words, these are chance matters. So the whole idea is that chance can produce what we attribute to the Holy One, blessed be He. It can generate life by itself too, even without an involved hand, in a completely random process. Except that this is not true—that it is a random process. It is like rolling a die: a completely non-random process. The physiological processes that caused the tiger to go right or left, or to be hungry or not hungry, are all dictated deterministically. From the moment of the creation of the world until today, it would in principle have been possible to know whether this tiger would go right or left already at the moment of creation of the world—I’m smoothing things over a bit here, but in principle.

But when, at the physical scale, I would want to calculate this thing, it’s such a complicated calculation—much more than calculating on which face a die will land, and even there we don’t do the calculation. So here too we’re dealing with a deterministic calculation; there is no ambiguity in reality. It’s a deterministic calculation. But it is so complicated and depends on so, so many things across the whole time scale, the entire timeline, that there is no chance of making such a calculation. So what do we do? We say: let’s assume that this thing is random, like a die, and use probability. I have a one-sixth probability for each face; I have a probability that the tiger will go here, or there. I use probability even though there is in truth nothing random here. These are not random processes. Exactly as I use probability with respect to a die or a coin or whatever, or roulette. Okay, there is nothing random there. So in fact what we have here is not ambiguity in reality itself, but an epistemic doubt. I don’t know what’s happening there in reality, but in reality itself the Holy One, blessed be He, knows exactly what is happening. Reality itself is completely dictated in a deterministic way.

Now understand that if that is so, then the whole sting of the neo-Darwinian argument against believers collapses. It doesn’t prove that there is God, but it knocks out the argument against. Why? Because what they want to claim is that you don’t need a guiding hand; probability does the work. But if the probability here is not essential, if the probability here exists only because we don’t know, while in fact we are dealing with a completely deterministic process, then what comes out is that the laws of nature did all the work in an entirely deterministic way. So if that’s the case, whoever created the laws of nature actually did this whole thing. The laws of nature do not arise by evolution. And therefore, in my opinion, the mistake in the neo-Darwinian argument stems from the fact that the people dealing with Darwinism and evolution are biologists and not physicists. And biologists are accustomed to treating these processes—again, not because physicists are necessarily smarter, though maybe that’s true too, I don’t know, but I’m not talking about that—but because we’re dealing with a different scale. At the biologists’ scale they are right to look at it as a random process.

Just as someone who wants to discuss the properties of a die will need to use probability. If he takes a physicist and asks him what will happen with the die, he’s crazy—he won’t get any result; he has no chance. It’s an extremely complicated and impossible calculation; you don’t have the data, you can’t reach conclusions. So if you want to treat the outcomes of a die roll scientifically, you need a statistician, not a physicist. In exactly the same way, if you want to treat evolution scientifically, you need a biologist, not a physicist. Because the physicist cannot calculate what happens inside a tiger. Do you know how many particles there are in one tiger? With three particles I can’t do calculations in physics, so with I don’t know, ten to the I-don’t-know-how-many hundred particles—I don’t know how many particles there are in a tiger—who can make such a calculation? The fastest quantum computers on earth won’t be able to do such a thing. For one tiger, not to mention all the animals and the weather and everything else you’d need to know in order to describe an evolutionary process deterministically.

So the biologists rightly say: forget it, let’s ignore the microscopic determinism for the moment and look at the large-scale phenomenon, like rolling a die. And basically this is a random phenomenon, and therefore we’ll use probabilistic statistical tools. And they’re right; that’s how biology should be done. I’m not coming to teach biologists how to do biology correctly. But when we draw philosophical conclusions from this, then we need to remember that the use of statistics here is due to epistemic doubt, not due to ontic ambiguity. This is exactly this state of confusion—and this time it’s confusion among experts, not among people in the street. Scientists get confused about this at every step. A lot of people I spoke to explained to me: what do you mean, it’s all random, it’s all chance, what nonsense are you talking, you simply don’t know what you’re talking about—there simply is no such thing as chance. And therefore the claim is that even experts can fall into this—fall into it in the sense not within biology, because in biology it doesn’t interfere; they conduct biology correctly. But when they draw philosophical conclusions and forget that what you’re calling random here is only random for methodological reasons—you treat it as a random process, as a chance process—it isn’t really a chance process. And then suddenly you draw conclusions because it’s a chance process, and from that you infer all kinds of conclusions. Just like the example I brought—if you remember—with that group, what’s the name of that group that invented chaos, I forgot.

[Speaker D] Excuse me, Rabbi, excuse me—why, why do you refuse to accept the possibility that whether the tiger turns right or left is actually random? It’s not necessarily—

[Rabbi Michael Abraham] At that scale there is no randomness. This is not quantum theory.

[Speaker D] But it’s not only quantum theory that—

[Rabbi Michael Abraham] What besides quantum theory is random in the world, truly random?

[Speaker D] Maybe, maybe whether the tiger turns right or left.

[Rabbi Michael Abraham] Why? Why on earth? The tiger is a collection of particles, and I know exactly what they do.

[Speaker D] But maybe another tiger in the same state would have turned right with the same history?

[Rabbi Michael Abraham] No, absolutely not. And by the way, nobody believes that either. Nobody believes that; nobody makes that claim. If another tiger turns right and not left, it’s simply because his brain is different from the other tiger’s brain. If they had the same brain structure, everything being exactly the same, they would do the same thing. That’s agreed upon by everybody, it’s obvious.

[Speaker C] Rabbi, but maybe Buridan’s donkey.

[Rabbi Michael Abraham] Buridan’s donkey, which dies between the two troughs, is exactly because of that fact. Yes, sorry.

[Speaker C] Rabbi, but perhaps the constraint of those who raise the evolution claim as a challenge to the physico-theological proof is that they say: you don’t need any complexity, you need some six constants and a few laws of nature, maybe eternal ones, and then it all rolls on by itself and eventually a very complex creature like a human being, like all creatures, comes into existence. It’s not that the complexity itself gives us the proof that there was someone thinking, but if we see that it can—Rabbi himself could have pressed a button to trigger the Big Bang, and within the framework of the eternal laws of nature, after fourteen billion years, Rabbi appeared. So what here is complex?

[Rabbi Michael Abraham] Wait, I don’t want to get into the theological discussion; that’s not our subject. I brought this only as an example; I answered it in the book as well, but I brought it here only as an example in order to show the difference between epistemic and ontic doubt, and pseudo-ontic doubt. So this is an example in which even professionals, even scientists, can fall and treat an epistemic doubt as though it were ontic. Those are the situations I call pseudo-ontic doubts. Okay? This is just an illustration. I don’t want to get into the physico-theological argument now; it’s not our topic.

I just want to bring an interesting source from Rabbi Shlomo Zalman in Minchat Shlomo. He speaks there about the prohibition against eating worms. Yes, so he says: “This is also what was written in Shevut Tzion, section 28, and it was also brought in Imrei Binah, the laws of meat and milk, in the name of a certain gaon, and in Darkhei Teshuvah 84, and it was brought from the author of Beit Ephraim, that regarding a worm to which one pays no attention, it is considered merely an unintentional act. And even though there is no exemption of unintentional act in the case of forbidden fats because one derives pleasure, here it is different, because the pleasure is only from the fruit and not from the worm.” In short, there is here a discussion of acting unintentionally and without intent, and for some reason he mixes the concepts. But at the basic level, there are arguments that want to say that if I eat a fruit and there is a worm inside it, this is permitted, and one does not need to inspect for worms. Why? Because I’m eating the fruit, and eating the worm happens without intent. Okay?

Now of course the question arises: but this is inevitable. If there are worms in the fruit, then true, it’s unintentional, but it’s inevitable. So he says: “Even though this is like a doubtful inevitable consequence regarding the past, which is not considered unintentional, as explained by Rabbi Akiva Eiger in Yoreh De’ah.” Rabbi Akiva Eiger, whom we saw, yes? What is he saying? He’s saying: the worms inside the fruit are really a doubtful inevitable consequence, which according to Rabbi Akiva Eiger is forbidden. A Torah-level doubt is ruled stringently. Why? Because either there are worms in the fruit or there aren’t. If there are worms in the fruit, then if I eat the fruit I have also eaten the worms. I just don’t know whether there are worms in the fruit or not. That is an epistemic doubt, not an ontic doubt. And if it is an epistemic doubt, then it should be forbidden, so this is a doubtful inevitable consequence. Because the fact that you don’t know whether there are worms in the fruit or not—the claim was that if there is a doubt whether there are worms in the fruit, then it’s not inevitable that if you eat the fruit you’ll eat the worm too, because maybe there is no worm. But that’s not true. According to Rabbi Akiva Eiger, it is a doubtful inevitable consequence. A doubtful inevitable consequence—that is, an epistemic doubt—really should be forbidden.

So how can it suddenly be permitted to eat a fruit when you have a doubt whether it contains worms? So he says as follows: “Nevertheless it appears that if the verification can only be made with very great effort, that is considered like an ex post facto situation, and in our case it is indeed considered as something done afterward, at the time of eating, by an act, and as an unintentional act and without intent, which is permitted. For even dragging a bed and the like, one could also know beforehand by means of a great expert, and nevertheless it is permitted. And since it is permitted to the restaurant owner, the same is known to apply to others.”

What is he saying? He’s saying exactly as we noted earlier: when you drag a bench across the ground, an expert can tell you whether a groove will be made there or not. But that’s something for experts, and even the experts are not certain they can do the whole calculation and know it with certainty. In any case, it’s a matter difficult to determine; not every ordinary person can just look and know. Rabbi Shlomo Zalman’s claim is that in a place where checking whether there are worms in the fruit is difficult—only for experts, or with a microscope, or whatever, all kinds of things like that—then you really don’t have to worry about it. Why not? Why should it matter that the inspection is difficult? He wants to claim that if inspection is difficult, then from my perspective this is like an ontic doubt and not an epistemic doubt. Why? After all, in the end, either there are worms in the fruit or there aren’t. What difference does it make whether it’s hard to check or not hard to check? This is a doubtful inevitable consequence and therefore it should be forbidden to do it. What difference does it make whether it’s hard or easy to check? Don’t enter doubtful situations.

When checking is possible, this comes up in the context of a rabbinic-level doubt. With a rabbinic-level doubt we rule leniently. So regarding that, the halakhic decisors write that if you can check, don’t rely leniently on the principle that a rabbinic-level doubt is treated leniently; rather, check. If checking is difficult and involves great effort, then they say: all right, you may rely on the rule that a rabbinic-level doubt is treated leniently. That is all with rabbinic-level doubts. But here, with eating worms, this is a Torah-level doubt. So why should I care whether checking is difficult or not? Bottom line, there is a doubt whether there is a worm in the fruit or not. This is a doubtful inevitable consequence; it should be forbidden. What difference does it make that checking involves great effort?

It seems to me that what he means to say is not precisely the point of the effort, but rather that an inspection carried out by an expert is, for the layman, what is called an ontic doubt. Like the ground. In principle, as we saw, the ground too presents only an epistemic doubt. But only an expert can know whether a groove will be formed or not. Therefore the person in the street, when you ask him, says: look, as for the ground, it’s fifty-fifty—either a groove will be formed or it won’t, it’s not clear. He doesn’t relate to it as something he doesn’t know; he relates to it as something in the ground itself. Ask the expert, of course, and he’ll say: what do you mean? It’s only that he doesn’t know; I can tell you whether a groove will be formed or not. But that’s when you ask an expert. When you ask the layman, then in a case where it can be checked but the checking requires an expert, the assumption is that the layman sees it as an ontic doubt.

[Speaker D] And that’s probably, yes, and that’s probably what he means when he writes “as something done afterward.” It’s not a doubtful inevitable consequence regarding the past, but regarding what comes later.

[Rabbi Michael Abraham] Exactly. And it’s not a doubtful inevitable consequence—what is called an epistemic doubt regarding the past—but rather an ontic doubt, like a doubt about the future, the question whether this will happen or won’t happen. All right? Not that I don’t know, but that it’s not clear whether it will happen or not happen. And what do you mean, it’s not clear? It’s not clear to you; to the expert it is clear. Fine—but if it’s unclear to me and clear only to the expert, then I as an ordinary person see this reality as a reality that is itself doubtful, ambiguous, not merely a doubt. And therefore such a thing is not inevitable.

This is exactly what I wanted to argue with the whole notion of doubt, because you can see that his proof is the very same proof. After all, the proof he brings is precisely from dragging a bed across the ground, and he says that there too an expert can know whether a groove will be formed or not. Exactly the question from which we started. And therefore I defined the pseudo-ontic doubt. That is exactly what is written here, in my opinion. True, he speaks in terms of effort, and I’m not entirely… What would happen in a case where, say, there are flies—there is doubt whether there are flies in the box, but only an expert could know whether they’re there or not. They’re hard to see, or I don’t know exactly for what reason. Still, if I ask the average person in the street, he would certainly tell me this is an epistemic doubt. Right? Either there are flies or there aren’t; that is obvious to him. So the fact that an expert is needed does not always mean that the average person sees it as an ontic doubt. Very often that’s true, but not always. With the ground it’s true, but with flies—even if we imagine that some kind of expert is needed to check whether there are flies there or not—that doesn’t mean maybe there are and maybe there aren’t in an ontic sense. It’s obvious that either there are or there aren’t; I just don’t know. Every layman will tell you that, even if perhaps an expert is needed to check. Therefore the criterion of expertise is not exactly identical with the criterion I discussed earlier, but it seems to me that that is what he means. It seems to me he means something similar to that. At least that’s how I understand what he’s writing here.

[Speaker C] Well now, can’t we understand his intention simply as saying that the Holy One, blessed be He, did not command us to do things that cannot be done in a… discovering bacteria, single-celled organisms—we can’t.

[Rabbi Michael Abraham] No, again. I’ll repeat what I said at the beginning, at the beginning of the previous lesson or maybe the one before that, I don’t remember. The exemption for an unintentional act is not an exemption based on lack of blame. It is the absence of a transgression. Meaning, if I did not intend it, then it is not that because I did not intend it I’m not guilty and therefore they exempt me. No. If I committed the act unintentionally, then that is not a transgression. It’s not that I have a claim for exemption. Okay? In legal terms they distinguish between a justification and an excuse—I don’t remember the exact terms already. Ezra, do you remember? I don’t know. It’s really the parallel distinction. A justification of—I don’t remember exactly what it’s called. Compulsion and justification, I think, or something like that. Meaning, compulsion is a claim of exemption; justification means I did not commit a transgression, it was justified. Yes, that’s basically the claim here.

Now, if that is so, then when I say that it is very hard to check, then what exactly are you suggesting? That because it is very hard to check, they don’t want to require it of me, so I’m exempt? That is not the kind of exemption involved in an unintentional act. The exemption of an unintentional act is not because they have pity on me and don’t demand that I do complicated things. Rather, an unintentional act is not a prohibited act at all. If I did it without intent, then it is not a prohibited act. Therefore I think that in this context the effort involved in checking cannot be interpreted on the level of blame. And therefore the fact that they don’t require me to do hard work, or expert work, or to bring an expert—whatever it may be—in order to check, that sounds like a claim of exemption. I’m not guilty; they don’t want to hassle me too much. But the exemption of an unintentional act does not operate on that plane at all. An unintentional act means there is no transgression. No transgression was committed here. Okay?

Now I just want to add one more point, and with this I’ll finish this part. I want to go back for a moment to prayer about the past. There was someone on the site who asked about this—I think even following the lesson, I don’t remember anymore who it was. I—

[Speaker E] What? It was me.

[Rabbi Michael Abraham] Eliyahu. The question is whether this can also explain prayer about the past. After all, what did I ask there about prayer regarding the past? I asked: what’s the difference between a fetus before forty days and after forty days? Before forty days, it’s permitted to pray that the fetus will be male, and after forty days it’s already forbidden, because then it’s as if it’s already fixed; it’s a prayer for a miracle, or a prayer about the past. And that is forbidden. So Eliyahu suggested: maybe the explanation here is like the explanation I gave in the topic / passage of an inevitable-but-unintended result under uncertainty. Maybe before forty days it’s also fixed; it’s clear that it’s fixed, and the uncertainty is only epistemic. But it’s pseudo-ontic. Because in people’s minds, or at least in the time of the Sages, people thought that until forty days it really was an open question. And therefore, even though in truth it isn’t an open question, since it’s pseudo-ontic, that’s also enough for me to treat it as ontic. And therefore it would be possible to pray about it. Because in reality itself there can still be two possibilities, and if the Holy One, blessed be He, brings about one of them, that’s not called deviating from reality. So it would be permitted to pray for that.

So I answered you that this doesn’t sound plausible to me—not, not plausible to me. It’s an interesting suggestion, but it doesn’t sound plausible to me. Because with prayer about the past, the discussion is a theoretical one. It’s not a question of what the person thinks. After all, in the end the person doesn’t know whether the Holy One, blessed be He, answered the prayer or not. In the end a boy was born. Does that mean that the Holy One, blessed be He, acted against nature? It could be that from the outset there was already a boy there. Right? Even I don’t know whether the Holy One, blessed be He, intervened or not. So why should it matter whether it’s pseudo-ontic or just ordinary epistemic uncertainty? In any case, if in reality itself the Holy One, blessed be He, can do it without intervening, then I’m allowed to ask for it. But if it requires intervention in the laws of nature, then I’m not allowed to ask for it. My claim is that prayer about the past is determined by actual reality, not by human opinion. Because human opinion here is irrelevant, since even if people think that the Holy One, blessed be He, did not intervene before forty days, I still don’t know whether He did it. After all, once a boy is born in the end, I don’t know whether that’s because the Holy One, blessed be He, intervened or because from the outset there was already a boy there. So in any case, from the standpoint of awareness, between epistemic uncertainties that are pseudo-ontic—where you say they’re like ontic—and ordinary epistemic uncertainties, which are epistemic uncertainty and about which prayer is forbidden, I think that anything that is truly epistemic—I don’t care how people view it—if the truth is that it is epistemic, then you cannot pray about it. That’s the claim. Meaning: regarding prayer about the past, I don’t think this third category, the pseudo-ontic one, is relevant. There it’s either ontic or epistemic, and in my view the third category is not relevant.

Okay, I’ve finished that section. Now I want to start dealing—up to now I’ve been talking about concepts of doubt. Now I want to start touching on questions of probability and statistics. A few points—again, obviously we’re not going to do a full course in this field here, but still a few important points, without mathematical formalism, just the modes of thought that underlie probabilistic and statistical analysis, which are important for a number of things.

First of all, I want to distinguish between plausibility and probability. That’s a distinction that a great many people, including experts, don’t make, and in my view it leads to all kinds of mistakes. Look, for example, people talk about the question of what the probability is that God exists. So the atheist will say the probability is very low. In order to make a probabilistic calculation, you need to have a sample space, a mechanism of calculation, a distribution, and then you can perform a calculation and tell me the result, what the probability is—a number between zero and one, or between zero and one hundred if you want percentages, it doesn’t matter. There is no way to do that calculation here.

How do we make probabilistic calculations for a die? We know that a die has six possible outcomes, from one to six. We know that the die—let’s say for the sake of discussion—we know that the die is fair, so the chance of each of the possibilities is equal. Given those two pieces of data, I can do a probabilistic calculation and tell you what the probability is of getting three sixes in a row. Okay? I can give you the number: one over 216. Okay? Why? Because I have the number of possible outcomes and I have the likelihood of each outcome, the distribution. And because of that, I can do probabilistic calculations and then I tell you: look, getting three sixes in a row is very implausible. One in 216—that’s very implausible. So that’s a probabilistic calculation.

But when the atheist tells me, “It’s very implausible that God exists,” that is not the result of a probabilistic calculation; it’s an assumption. His assumption is that it isn’t plausible. Legitimate, by the way—assumptions are completely legitimate. But that’s plausibility, not probability. And people very often mix up plausibility and probability. You need to know that plausibility consists of assumptions, and assumptions ought to be examined, especially compared to the plausibility of the opposite assumption. With probability, by contrast, you have a calculation. You can know that if the probability is such-and-such, then it’s implausible that it would happen.

So the first distinction I want to present here is the distinction between plausibility and probability, where in the background probability is always the result of a mathematical calculation. That means: you need to know how many possibilities there are, what the chance is for each possibility—that’s called a distribution. What is the chance of getting each of the numbers? That’s called a distribution. And then you can do calculations. If you have no basis for a calculation and only intuition—something seems plausible, something doesn’t—then all you can talk about is plausibility, not probability.

Now look, for example, at another illustration of this. Somebody claims that the probability of a certain event is zero. It seems to me—if I think I’m right, I’ve thought about this a few times and it seems to me that I am right, though maybe it would be worth asking experts greater than I am—if one reaches the conclusion that the probability of a certain event is zero, absolute zero, not very small but zero, then it is never probability but plausibility. Never. Why? Because zero probability is always—after all, probability is always the result of products, sums, and the like. There are no negative probabilities. Right? So in that sense, you only get to zero through multiplication. Okay? Now when you multiply several probabilities in order to… then as a result you can say that the final thing also has zero chance. Okay. Now when you multiply several probabilities, in order for the result to be zero, one of the factors has to be zero. Meaning, one stage in the process has to have zero chance. And then you can say that the result too has zero chance. Okay, but how do you know that some particular thing has zero chance? Exactly for the reason I just gave: there is never a way to calculate it. Unless you can present this event as a product in which one of the components is zero—and then I’ll ask about that component. In the end, in the end, you will always get to something of the form: such a thing cannot happen. That is an intuition, a basic assumption, I’m not exactly sure what—but it is not the result of a probabilistic calculation.

Therefore, when one arrives at the statement that a given event has probability zero, you should know that we are dealing here with plausibility, not probability. In other words, we are dealing here with an intuitive judgment, not the result of a calculation. It can of course be the result of a calculation in the sense that one of the factors is zero, but then go back and ask about that factor—why is it zero? In the end, in the end, you will come to: I assume that such a thing cannot happen. And if such a thing cannot happen, then I say its chance is zero. But it’s not correct to say that its chance is zero, because zero is not a chance. Zero is not a chance in the probabilistic sense. You cannot get to zero by a probabilistic calculation. Zero is simply the assumption that such a thing is impossible.

And that’s an interesting point, because many times a person says, “For me, the chance that God exists is zero.” That’s what the atheist says. So even though we have a complex world, and a complex world does not create itself, the alternative that God exists and created the world is, for me, worse than the alternative that this world came about by chance. Because the chance that this world came about by chance is very, very small, but it exists. And the possibility that there is a God who created the world is zero. That’s what many atheists have told me. And then I ask them: the result of what calculation is this zero of yours? Where did this zero come from? From nowhere. Meaning, you assume that it cannot be that God exists. You have every right to assume that, everything is fine. But you cannot compare possibilities here. You are simply assuming the conclusion. You assume there is no God, and therefore the conclusion is that there is no God. Fine, okay, that’s perfectly all right; you may assume that. But don’t present it as an argument. Exactly—that’s not a calculation.

When you make the comparison between those two things, it is not a probabilistic comparison. Not calculations. Very often they present it that way, but that is a mistake. They present it that way because it is very convenient to present it that way, because probability sounds terribly persuasive. To say, “It just seems to me,” or “Intuitively this is what I think”—okay, so you think that and I think differently. That sounds much weaker. But in fact what stands behind it is: that’s just how I think. And it’s not the result of a calculation.

A probabilistic calculation never gives zero. There is no such thing; it cannot give zero. If there are a million possibilities and I ask what the chance is of one of them, then the chance is one in a million—very, very small, but still a chance that is the result of a calculation. The calculation is one divided by the number of possibilities. That can come out of a calculation. But a zero chance cannot come out of a calculation.

By the way, even—say—think back to what we said earlier about evolution. Think, for example, about the constants, right? The constants of nature. The speed of light, the dielectric constant, all kinds of—there is the gravitational constant. There are all sorts of fixed constants like that in physics. And the values of those constants are essentially what create the conditions that are responsible for the existence of an evolutionary process. Now the question is: what is the chance that the constant G, right, the gravitational constant, would be exactly this number and no other? And suppose it’s an exact number down to the last digit, infinitely many digits, completely exact. What is the chance?

So the chance of getting one specific number, say in the interval between zero and one—the chance of getting one particular number between zero and one is zero. Right? Because I have infinitely many numbers. Uncountably infinite, even; I’m talking about the continuum between zero and one. So the chance of each one of the numbers is zero. But in probability theory they do not say that the chance is zero. There is no such thing as zero chance. Because in such a situation it turns out that every result that could be obtained has zero chance—but if it was obtained, how can its chance be zero? A result with zero chance cannot be obtained. A result with a very, very small chance can be obtained. But a result with zero chance cannot be obtained. And if I conduct a random selection among those numbers, some number will come out. So that means it is not probabilistically defined. The sample space is not defined, the distribution is not defined—the “distribution” that gives zero probability to each of the possibilities when I have infinitely many possibilities is not a mathematically well-defined distribution function. Therefore a mathematician will tell you that this is not a probabilistic claim. To randomly choose one number out of a continuum and say that the chance of obtaining it is zero—they will tell you that such a randomization cannot be defined. That is not what is defined as a random selection in probability theory. Okay? So you see from another angle why zero chance is never the result of a probabilistic calculation.

And still I can say: in my eyes it is very implausible—I don’t think one can say zero—but very implausible that this thing would happen, because my intuition tells me it is implausible. But that does not mean the probability is zero; rather, the plausibility is very low. That I can say. Okay? But to say “zero probability” is an oxymoron. There is no such thing as zero probability. If it is zero, it is not probability.

Okay, so that’s regarding plausibility versus probability. Another thing: when you have, say, a die—we talked about a fair die and an unfair die. What is the difference between a fair die and an unfair one? Both belong to the world of probability. You can do calculations with both of them; everything is fine. Probability deals with both fair dice and unfair dice. The difference is that with a fair die the distribution is uniform, and each face has the same chance—in this case one-sixth. With an unfair die, there are different chances. Say one face has a chance three times that of all the others. Fine? Or say three times all the others—then basically each of the others has probability one-eighth, except for one face, whose probability is three-eighths. Okay? That’s what you get. That’s an unfair die.

But if I know the fact that the die is unfair and I know the different chances of getting each result, there is no problem at all; I can still do probabilistic calculations. If you ask me what my chances are of getting three ones in a row—no problem. It’s 0.375 cubed, which is three-eighths cubed. That’s all. Meaning, the fact that the die is unfair does not take the discussion out of the realm of probability. So what is the difference? The distribution is different. And that is a very important point. When we do a probabilistic calculation, we always need to know the distribution function. And many times people are not aware of this, and they do probabilistic calculations as if they are implicitly assuming some distribution function, without even noticing it, but they have no way of knowing that this is really the distribution function.

For example, if we have many possibilities and we know nothing at all about those possibilities, then we assume a uniform distribution, that all possibilities have the same weight. But who told you that? Maybe in fact we are dealing with an unfair die, or with a non-uniform distribution. You cannot know. So say: in the absence of other information, I assume a uniform distribution. Fine, you are allowed to assume that. You just need to understand that this is essentially some kind of assumption. You assumed it, and under that assumption you can do the calculations. That means that in order to do a probabilistic calculation, you need to know all the possibilities and what the chance is of each of those possibilities. Even if the chances are not uniform, I don’t care, but you need to tell me what the chance is for each possibility. That is what is called the distribution function. Okay? And therefore only then can I speak in terms of probability or probabilistic calculations, or statistics, and so on.

One more distinction: the distinction between rarity and anomaly. If I toss a coin, say—or throw a die, let’s say—a hundred times. Fine. Now each result can be written as some vector, right? Two, five, one, one, three, five, six, four, and so on—a thousand-place vector, where in each place there is a number between one and six. That is the result vector, okay? Now the vector six, six, six, six—if a fair die comes up six a thousand times. Okay? The chance of that happening is very, very small, right? It won’t happen. But what is the chance of any other sequence of results? The same chance. Also very small. Also one-sixth to the thousandth power. Every sequence of results you get has a probability of one-sixth to the thousandth, which is an astronomically small number, right, one over something astronomical.

So why do we treat the result six, six, six, six as implausible or as special? Meaning, if someone throws a die and gets a thousand sixes, I’m going to say: something smells here. This person has some kind of control over the die; this can’t be. But any other result that happened with the die has the same probability. So why don’t I say there too that he has some kind of control over the die? The difference is that the random series—one, two, and so on, the alternating series—is random. It is indeed very rare, but it is not unique or anomalous. By contrast, six, six, six, six is a series that is both rare and anomalous. And that is a very important point, because you should understand that this is actually something probability theory does not really deal with. Because the probability of every sequence of a thousand such results is exactly the same. The result six, six, six, six and any other thousand-entry vector have exactly the same probability.

So why, if someone won the lottery a hundred times in a row, you understand that after the third time the police would already be knocking on his door, right? It’s obvious he’s cheating, right? But that is exactly the same probability as his winning the first drawing, I win the second, and you win the third. Assuming they’re independent, the probability of that is one over the number of people in the world cubed, or the number of participants in the lottery cubed—the same probability. So why in the first case would the police knock on his door, while in the second case the police would ignore it? Aside from the fact that they don’t know statistics, which is also true, but even if they did know statistics, they would still ignore it. Why? Because the first case is not only rare, it is also anomalous. Rare means low probability, but low probability still doesn’t mean miracle. When something with low probability happens, that is not necessarily something requiring an explanation, or something so implausible that you have to look for the catch. No, not every time something of low probability happens is there a catch.

When I throw a die a thousand times, every result I get will be a result of low probability. And therefore the fact that I got a low-probability result should not surprise me in any way; I knew in advance that I would get such a result. But if I get a thousand sixes, then the result is anomalous, not just rare, not just low-probability—and the anomaly arouses suspicion. Now how to translate this into probabilistic calculations—you can translate it in some way, but that’s not important at the moment. I only want us to put our finger on the difference between rare and anomalous.

To clarify this matter, I’ll bring an example I’ve given before. There’s a book called Choose a Number, a sort of thriller called Choose a Number. And the thriller opens with a man who receives a letter. He opens the envelope, from an anonymous sender he doesn’t know. He opens the envelope. Inside it there is a small envelope and a letter. He reads the letter. The letter says this: choose any number you like between one and a thousand, then open the small envelope; inside it I wrote down the number you chose, because I have the ability to know what people will choose, okay? If you see that there is a match between what you chose and what is written in the envelope inside, then you understand that I am a man with supernatural powers, the ability to know what people are thinking. Send me a thousand dollars and I’ll make you a millionaire; I have supernatural powers, okay?

The man guessed—I don’t know—seven hundred and twelve. He opens the small envelope: seven hundred and twelve. He can’t believe his ears—seven hundred and twelve. Well, immediately he sends the man a thousand dollars, obviously—it’s worth the investment. With powers like that, giving me the winning lottery numbers in future drawings is easy for him.

What turns out in the end—sorry to anyone who’s about to read the book, here, what’s that called, I’m blanking on the term—to reveal the ending, spoiler,

[Speaker D] Spoiler, I’m giving you a spoiler.

[Rabbi Michael Abraham] What turns out in the end? The man sent ten thousand such envelopes. And in each envelope he inserted a number between one and a thousand, okay? Now if ten thousand people did this experiment and each randomly chose a number between one and a thousand, then ten of them, on average, would randomly choose the same number that was written in the small envelope they received, right? Each of those ten would send him a thousand dollars—good, so he made ten thousand dollars. Not bad.

What’s the catch here? The catch is that indeed the chance of this happening is very small. The chance that there will be a match between the number I choose and what is written in the envelope is very, very small. But if I send ten thousand envelopes, then obviously even something whose chance is one in—yes, sorry, can you hear me now? I don’t know why my internet keeps cutting out. Okay, now yes. So if you send ten thousand envelopes, then the very implausible event becomes very plausible. It is not anomalous. It is plausible, but it is not anomalous. It is not anomalous because out of ten thousand people there will be ten such cases; that is simply expected. Many times, what turns the rare into the non-anomalous is repetition of the experiment. And when I do this experiment many, many times, then true, the result is a rare result, but if it was obtained, it is not anomalous because I repeated it so many times—so then this thing is plausible, it’s plausible that it would happen. Okay? That is basically the claim.

That’s regarding rarity and anomaly. There are, for example—I spoke about this in my book The First Existent—regarding the fulfillment of prophecies. One of the claims is that there are prophecies in the Torah that come true, and that this is evidence of its truth, evidence of its divine origin, because the Holy One, blessed be He, basically knows in advance what will happen. Now when you examine the fulfillment of prophecies, it’s a very complicated business. Why? Because if the prophecy is not a rare prophecy, then the fact that it came true is no big deal.

Yes, if the Torah prophesies that at some point the Jewish people will number ten thousand people, or I don’t know, a million people—fine. Fine. Obviously, wait a generation, two, three, five, eventually we’ll get to a million people. That’s not something unexpected, or some prophecy that indicates supernatural powers. One of the things that illustrates this is all kinds of prophecies of the Oracle of Delphi, or the oracle of Bar-Ilan—David Passig, yes, which is about the same thing. I think the Oracle of Delphi predicts better than he does. Maybe he got a lower salary. In any case, these “prophets” always predict very, very general things, so that whatever happens in the future will fit what they predicted. Okay? And when you predict something that isn’t rare, the prophecy is no great feat. Meaning, okay, that’s something that…

For example, one of the predictions Passig brought to the country—someone sent it to me excitedly and asked what I thought. The editor of Tzofeh asked me what I thought. He predicted there about the success of the State of Israel in some decade or other, that it would be very economically successful and so on. I said to him: what does “very successful” mean? In what part of the relevant decade? What exactly are the criteria? Successful in what sense? He didn’t say anything that can really be pinned down or refuted. And when you say vague things, you can always explain afterward that you were right.

Therefore, with the prophecies in the Torah, for example, when you examine them, you need to understand very well how sharp they are, or what would have to happen in order to show you that they were not fulfilled. Because if nothing can show you that they were not fulfilled, then the fact that they were fulfilled proves nothing either. Because fine, they’re always fulfilled. It’s not… Therefore, to predict something non-rare is no big feat. But there are times when I predict something rare, and that still does not necessarily call for explanation if it is not anomalous. Fine? For example, if I predict something very rare, but over the course of, say, three thousand years of history, then at some point it will happen. So that too may not raise any question, may not require any explanation.

Usually, though, the rare that is not anomalous—the rare that is not anomalous is, for example, throwing a die. I throw a die a thousand times and get some vector of a thousand places: one, two, one, five, one, four, three, six, and so on. A thousand results between one and six. Okay? That should not surprise me; no result I get should surprise me. Why? Because it’s rare but not anomalous. What happens if someone comes and tells me in advance that that is the series of results I’m going to get? Then of course that is something entirely different. That certainly calls for explanation, and you have to look for one—how did that happen? Why? Because once the person predicted in advance that this thing would happen, it becomes anomalous and not merely rare.

Six, six, six, six is anomalous simply by virtue of the fact that it is the same number. Therefore, although its probability is the same as any other vector, it is rare but not—sorry, it is both rare and anomalous. Other vectors are rare but not anomalous. But because it is marked by some special structure. However, if someone predicts in advance a thousand random die results, then he has essentially marked that vector as a special vector. Now, if I threw the die a thousand times and got exactly that set of results, that means those results are also anomalous, not just rare, because someone marked them in advance. Yes, and therefore prophecy in the Torah really does have a completely different significance even regarding something that is rare but not anomalous. If the Torah predicts it in advance, that definitely has a different meaning.

[Speaker E] But mathematically it’s exactly the same thing, no? No. Because if I predict, the chance that it will land ten times on six, and the chance that it will land on one, two, one, four, five, one—

[Rabbi Michael Abraham] Yes, something else—ten times, it’s the same thing. Again: if I predict in advance that it will land a hundred times on six, or I predict in advance that it will land on one, two, one, four, five, one, yes, something else—it’s the same thing. But if I don’t predict the six, six, six—if it just came out six, six, six, without a prediction—then that is both anomalous and rare. But if a different result came out without a prediction, it is rare but not anomalous. However, for that other result, if there was a prediction about it, then the prediction makes it anomalous too, not just rare. The police will come. Because if there was a prediction, that means I marked that vector as a special vector. So in essence it becomes like six, six, six, six, which is a special vector, except that it is a special vector without needing to mark it. And this one becomes a special vector because I marked it.

And then what happens—what happens with anomaly? After all, they are all equally rare. “Anomalous” means that all the other rare results are similar to one another, or not similar, they have no special structure. But six, six, six, six has a special structure in addition—besides the rarity, it also has a special structure. It is special, and therefore the question is whether it happened, as opposed to any other thing among all the other possibilities. It stands against all the possibilities—not against each one separately, but against all the rest collectively. So here it is clear that six, six, six, six is implausible. But if I mark one of the other possibilities, then now I have turned the experiment into an experiment of whether that particular result will happen or all the others. Meaning, here too it now becomes one result against all the others, and therefore once again this is something anomalous and not merely rare.

[Speaker E] Rabbi, I didn’t understand—even apart from prediction now. When the throw lands three times on six, that has the same chance as landing on one, two, and six. Correct.

[Rabbi Michael Abraham] So—

[Speaker E] Why are we astonished that it lands three times on six and not astonished that—

[Rabbi Michael Abraham] —it lands on one,

[Speaker E] two, and six? That’s what I’m asking you. Yes, so what’s the answer?

[Rabbi Michael Abraham] My answer is that we are astonished by it because here it is not only rare but also anomalous.

[Speaker E] So it’s not a mathematical astonishment, so to speak?

[Rabbi Michael Abraham] It’s exactly the same thing. No, I’m saying, you can translate it into mathematics, and the translation is roughly this—I’m saying it very crudely. The translation is that if you measure six, six, six, you measure it against all the other possibilities. Not against one of them. Because it is special. So what is the chance of getting six, six, six? Nobody asks himself what the chance is of getting one, one, three, six, four, five, three, six. There’s nothing special there. So when you assess six, six, six, you assess it against all the others. Then mathematically its chance is small. Every other result is measured against one particular other result like it, not against all the others, because it is not special relative to all the others. But if I marked it in advance, if there was a prophet who predicted it in advance, then he is saying: no, no, here there is one result that stands against all the others. Now you need to check whether I am right or wrong. Wrong means any of the other possibilities. Right means only this one. So the prior prediction turned it into something like six, six, six, six.

[Speaker C] Rabbi, suppose—what the Rabbi once told us about the miracle that happened to him on the trip to Yeruham, from the accident in Jerusalem, if I remember correctly—so if, suppose, at the moment the accident happened the Rabbi began to pray that there would be some solution to that mess, and before he even finished the sentence the prayer was already answered, wouldn’t that marking turn it into a miracle?

[Rabbi Michael Abraham] If in all the other cases… they were not answered, and mine was answered, then that has no significance. Right. Okay?

[Speaker D] It’s just that you’re more righteous.

[Rabbi Michael Abraham] Maybe. Maybe yes—that’s the question. After all, people want to argue that this is exactly the faith-based claim, so to speak, of providence—that the reason it happened is because I am more righteous. But against that I argue that maybe you’re right, but even if I weren’t more righteous, it would still happen to one person, so you have no proof that it happened because I am more righteous. Because the other alternative is equally correct. I’m not claiming that it did not happen because I’m more righteous; rather, I’m saying there is no proof that it happened because I’m more righteous. Because the alternative probability is the same probability, so you have no proof from that.

[Speaker E] So I didn’t understand: are we amazed by three sixes because it’s intuitive or because it’s mathematical?

[Rabbi Michael Abraham] So I said: in the end, in the end, it’s mathematical. Because when you—but the mathematics comes from the fact that when you ask yourself whether six, six, six came out or something else came out, then you say: you compare six, six, six with all the other possibilities. The chance of it is very small. If it happened, then it’s a miracle.

[Speaker E] One in 216? What? Is that one in 216?

[Rabbi Michael Abraham] If it’s three times.

[Speaker E] Yes. So that’s like one, two, and four? Correct.

[Rabbi Michael Abraham] But six—that is the chance. But let me talk now about anomaly, not rarity. Anomaly means—now I’m translating it into numbers—anomaly means that six, six, six stands against all the other results, not against one, two, four. It is special. And now I ask: what is the chance that a special result will occur? One over 216. But if I got one, two, and four, and now I say wow, unbelievable, I got one, two, four—it’s a one in 216 chance. You understand that that is nonsense. Because one, two, four is no different from three, one, five.

[Speaker E] I understand intuitively, but mathematically I still don’t see the difference.

[Rabbi Michael Abraham] The difference is that this one is special and that one isn’t. The specialness means that when you gather the probabilities, all the other possibilities form one pool that stands opposite the six, six, six, and therefore the six is tiny. But one, two, four does not stand opposite all the rest; it stands opposite each of the others individually. Against each of the others it is exactly the same as they are. There is nothing special there.

[Speaker E] So in the end the chance of six, six, six is not one in 216?

[Rabbi Michael Abraham] Yes, it is. But when you are astonished, you are not asking what the chance is. You are asking what the chance is relative to the alternative, right? But the alternative too—every result that could happen is one in 216, so what is there to be astonished about when it happens?

[Speaker E] Yes exactly, so I don’t…

[Rabbi Michael Abraham] If you ask what the chance of this is relative to the alternative. Now let’s formulate the alternative. If you ask what the chance is of six, six, six versus everything else—it’s one in 216. But if you got one, two, four, wow, unbelievable, one in 216. What do you mean one in—every result that came out would have come out with probability one in 216. If it were special relative to all the others, then you’d say: ah, it’s one in 216, and every other result is part of all the other possibilities, which is 215 out of 216. Okay? So our amazement depends on the specialness of the result. And when you translate that into mathematics, the specialness simply tells me: collect all the other results together; that’s basically one basket. It’s not each result individually. When you compare that basket to the special result, then it really does call for explanation. Fine.

[Speaker E] And what do you say about the question I asked this morning about medicine and Bayes’ theorem?

[Rabbi Michael Abraham] Well, there? I said…

[Speaker E] No, but in the end I suggested that test reliability isn’t like you think, that test reliability of ninety-nine—

[Rabbi Michael Abraham] No, no, I answered, I answered there. Maybe you didn’t see it; I answered in the evening because I wasn’t near the computer. I said: that’s not correct. The reliability of a test is always the question of how many of the sick it will detect. First of all because that’s how the term is used. Otherwise none of these problems of Bayes’ reversal would exist at all. All the problems arise from the fact that the reliability of a test is always defined as how many of the sick it will detect, and not: if the test came back positive, what is the chance that he is sick. By the way, I’ll still talk about this in this series too.

Moreover, why do they in fact use that datum? Because the other datum cannot be measured. When you test a diagnostic test, when you want to check the reliability of a test, you take sick people whom you know are sick by other means. Okay? You put all of them through the test, and you see how many of them come out positive. So you have the datum of how many of the sick the test detects. That can be checked. That is how test reliability is checked. But how will you check, given that the test says someone is sick, what the chance is that he is sick? There is no way to check that. After all, that very test is what tells you whether he is sick or not. Therefore it is not accidental that whenever people talk about test reliability, they speak about it in sense B, not in sense A. Okay? Obviously, if someone has a datum of type A, somehow they managed to measure it, then no problem—then there’s no question at all and everything is fine. But that’s beside the point. I’m talking about the usual situation, where they give you test reliability in this sense: how many of the sick it will detect. Fine?

[Speaker E] Okay, because I heard from an expert here in the hospital—I heard from an expert that test reliability means that given a hundred people, the test will reveal the prevalence of the epidemic, the prevalence of the disease. For example, if the disease prevalence is one in a hundred, then when you test a hundred people you’ll get one sick and ninety-nine healthy.

[Rabbi Michael Abraham] So what? What test is that? One that is correct ninety-nine percent of the time? Yes. It seems to me you didn’t understand him correctly. What are you talking about? I don’t…

[Speaker E] I’ll check with other experts, but… and also, also—

[Rabbi Michael Abraham] And if it identifies one sick person and it’s not the right one, then what?

[Speaker E] No, it will be the right one. With a probability of ninety-nine.

[Rabbi Michael Abraham] No, again. If you say to me: I gave a hundred people the test, one of them is sick, and the test showed me that one is sick. Which one? I don’t know—one of them.

[Speaker E] One came out. No, no, you don’t know that there is one sick person among the hundred. The disease prevalence is one in a hundred.

[Rabbi Michael Abraham] The disease prevalence is one in a hundred.

[Speaker E] Yes, exactly. So now you test a hundred people. Well? With probability ninety-nine, the test will reveal that there is one sick person among the hundred.

[Rabbi Michael Abraham] I don’t understand—so you’re not defining the ninety-nine, you’re assuming the ninety-nine. That cannot be the definition of ninety-nine percent reliability. You’re building it into the definition. I’m asking: what is the definition of the statement “the test is ninety-nine percent reliable”?

[Speaker E] So the definition is that when you test a hundred people, it will reveal the prevalence of the epidemic. If the epidemic prevalence is one in fifty, then it will reveal two sick people.

[Rabbi Michael Abraham] If… what? What are you talking about? If it reveals the true prevalence of the epidemic, and assuming it identifies the actually sick people, then why isn’t that a hundred percent? And why isn’t it ten percent? I don’t know, it has nothing to do with it. It’s simply a definition—or either you didn’t understand him or he is mistaken, I don’t… it can’t be. Where, where did ninety-nine come from here?

[Speaker E] No, there is a ninety-nine percent chance that in terms of the prevalence of the epidemic, the test will reveal the prevalence of the epidemic.

[Rabbi Michael Abraham] Again, so you are building the ninety-nine into the definition again. Don’t build it into the definition. I want you to give me the definition, and from that derive the number ninety-nine percent. You won’t succeed. You have a test, you know that the epidemic prevalence is one in a hundred. You run a hundred people through the test and one comes out sick. Can you show me from that what it means that the test reliability is ninety-nine percent?

[Speaker E] No, no, never, absolutely not. But you already know in advance what the epidemic reliability—what the test reliability—is.

[Rabbi Michael Abraham] Then that’s not the definition of test reliability. You are using the concept of the reliability of the test. I am asking what the definition of the concept “test reliability” is. You are using a datum that already assumes some given reliability. I’m asking what it is. What does that reliability mean? Fine.

[Speaker C] Rabbi, one more small question. Just on the principled level: if, say, we accept the assumption that there is infinity—that time is infinite and the universe is infinite and there are infinitely many universes—then wouldn’t expanding the framework on that level basically uproot any possibility of real anomaly, I mean miracle in the sense of statistical anomaly? Because okay, what happened to you is anomalous, but over infinite time in infinite universes…

[Rabbi Michael Abraham] That’s the anthropic principle, so what? Okay. That’s the claim of the anthropic principle. So then there are no miracles—

[Speaker C] In that sense, in the sense of anomaly?

[Rabbi Michael Abraham] Anyone who accepts the anthropic argument, in my view, it’s nonsense. That’s the claim of the anthropic principle. They say: why are you so amazed that there’s a special world here? There are billions and billions and billions of other universes that aren’t special, so one turned out special. Right. My claim is: who creates those universes? Second point: I haven’t seen them, so why assume they exist? These are just ad hoc assumptions that are very implausible. If you rolled a six on a die a thousand times, you’d say, something’s fishy here. Why is something fishy? It could be that they rolled billions of other dice around the world a thousand times, and only with yours did it come up six a thousand times. So what are you so surprised about?

[Speaker C] That’s exactly what I asked.

[Rabbi Michael Abraham] And that, of course, is not true. Okay, all right, friends, Shabbat shalom, and goodbye.

[Speaker C] Shabbat shalom, goodbye, thank you, Shabbat shalom, thank you.