Doubt and Probability—in Halakhah, in Jewish Thought, and in General – Lesson 1
This transcription 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
- Opening the series: doubt and probability
- A cultural dichotomy between religious certainty and secular doubt
- Doubt and certainty are not values but cognitive facts
- Truth versus certainty: a categorical distinction
- Truth and falsehood depend on the claiming subject
- Practical certainty versus philosophical certainty, and the confusion with necessity
- Anselm and the ontological proof: certainty versus necessary existence
- There are no tools that bring certainty: empirical versus a priori
- Mathematics and logic: certainty of derivation within a framework
- Certainty as a fiction and the move to tools for deciding under doubt
- The maturation process: childhood, adolescence, crisis, and three exits
- Fundamentalism as refusal of criticism and the danger in it
- Synthetic maturity and rules for conduct under uncertainty
- Closing questions: Godel and the triangle example, and doubt in life-saving cases
Summary
General Overview
The series opens with the subject of doubt and probability, מתוך a desire to sharpen and renew points in relation to an earlier series on statistics, and from the central claim that we have certainty about nothing, so there is no point in turning either certainty or doubt into values. The speaker presents a cultural dichotomy between a religious ideal of certainty and a secular ideal of doubt, and argues that both are mistaken because doubt and certainty belong to the cognitive plane, not the plane of values. He distinguishes between truth and certainty and develops the idea that scientific-empirical and philosophical-logical tools do not provide certainty, and therefore we need to develop rules for rational conduct in states of doubt; probability, in that context, is part of those tools.
Opening the Series: Doubt and Probability
The speaker opens a new series on doubt and probability, and notes that there was previously a related series on statistics, but he is continuing the subject with new emphases and different directions. He argues that there is no study hall without innovation and emphasizes the distinction between repetition and genuinely new lectures. He presents the move as an attempt to build a general framework for the concepts of doubt.
A Cultural Dichotomy Between Religious Certainty and Secular Doubt
The speaker describes a modern religious outlook that sets certainty up as an ideal and sees doubt as destabilizing and negative. He also describes a secular outlook that grants doubt the status of a value and an aspiration, and he identifies a difficulty in the conversation between the two sides because each side struggles to understand the other’s language. He argues that fear of conclusions and dogmatism leads people to cling to doubt as a cultural ideal.
Doubt and Certainty Are Not Values but Cognitive Facts
The speaker argues that there is no value in being in doubt, though there is value in critical thinking aimed at forming a position. He states that there is certainty about nothing, and whoever thinks he has certainty is probably mistaken, without turning that into a moral criticism. He concludes that doubt and certainty do not belong to the realm of values but to the cognitive realm, and distinguishes between questions of what ought to be done and questions of what is true or not true.
Truth Versus Certainty: a Categorical Distinction
The speaker argues that the common identification of truth with certainty is mistaken both factually and categorically. He defines truth as correspondence between a claim and a state of affairs in the world, and defines certainty as a person’s state of consciousness with respect to a claim. He illustrates that there can be truth without certainty and certainty without truth, and explains that certainty belongs to epistemology while truth is tied to ontology, while correcting that truth is a property of claims rather than of facts.
Truth and Falsehood Depend on the Claiming Subject
The speaker explains that without human beings there would be no truth and falsehood, only facts, because truth and falsehood arise from comparing the subject’s claim to the state of affairs in the world. He presents the question of truth as whether an observation correctly describes the world, and the question of certainty as how sure the person is of the truth of that observation. He emphasizes that certainty relates only to the human being and not to the world itself.
Practical Certainty Versus Philosophical Certainty, and the Confusion with Necessity
The speaker distinguishes between practical certainty, which is a very high probability that people treat as certainty, and philosophical-logical certainty, which is unattainable. He describes the possibility of producing a feeling of certainty about false things, and therefore says that the feeling of certainty is no guarantee of truth. He adds a note about the confusion between the concept of certainty and the concept of necessity, and explains that necessity is a claim about reality, whereas certainty is a claim about the person.
Anselm and the Ontological Proof: Certainty Versus Necessary Existence
The speaker brings in Anselm of Canterbury and his book Proslogion to illustrate the difference between proving existence and proving the necessity of existence. He argues that many commentators are mistaken when they think Anselm is merely repeating himself, because in one chapter he is speaking about the believer’s certainty and in another about a claim regarding God as necessary existence. He illustrates this with a phone, whose existence may be practically certain but is not necessary.
There Are No Tools That Bring Certainty: Empirical Versus a Priori
The speaker presents two “toolboxes” for claims about reality: one empirical-observational, and the other philosophical-logical-mathematical and a priori. He argues that observation is not certain even at the level of individual cases because of errors of perception, and therefore generalizations and laws of nature are certainly not certain either. He presents Popper as arguing that science does not provide truth but only non-refutation, and adds the criticism that Popper took that to an extreme, though the point about scientific uncertainty is correct.
Mathematics and Logic: Certainty of Derivation Within a Framework
The speaker argues that mathematical and logical certainty is certainty about a relation of derivation between premises and conclusions, not certainty about the premises themselves or about claims concerning the world. He explains that mathematical propositions are always conditional within an axiomatic framework, and illustrates this with the sum of the angles of a triangle, which depends on Euclidean geometry. He refers to Godel’s theorem to illustrate that “there is no proof” means there is no proof within a given system, whereas in a meta-system one may prove it, and he warns against drawing hasty philosophical conclusions from Godel’s theorem.
Certainty as a Fiction and the Move to Tools for Deciding Under Doubt
The speaker concludes that the concept of certainty is inaccessible to us and exists, at most, as practical language or as an error, and therefore all human situations are situations of doubt in the sense of lack of certainty. He argues that from this follows the importance of tools for dealing with situations of doubt, including statistical-probabilistic tools and halakhic / of Jewish law rules of “rules of conduct and rules of clarification.” He notes that there is a difference between evenly balanced doubt and majority, but even majority is a rule for deciding under doubt, because there is still no certainty which option is correct.
The Maturation Process: Childhood, Adolescence, Crisis, and Three Exits
The speaker describes a stage of childhood in which authority is accepted dogmatically, and a stage of adolescence in which proofs are demanded, followed by a crisis in which it becomes clear that every proof rests on premises and there is no certainty. He says the crisis is created by combining the claim that only what is certain is acceptable with the claim that there is no certainty, and he presents three exits: skepticism, which accepts both and concludes that nothing is acceptable; fundamentalism, which gives up the claim that there is no certainty and adopts a source of knowledge beyond criticism; and synthetic maturity, which gives up the demand that only what is certain is acceptable and is willing to live with truths that are not certain.
Fundamentalism as Refusal of Criticism and the Danger in It
The speaker defines fundamentalism as a position that does not subject itself to critical thought, whether its source is a suit, a rabbi, parents, or supra-rational charisma. He argues that the danger is not only in violent fundamentalism but in the very philosophical structure that allows principles to be adopted without criticism, even if those principles are currently moral. He says the concern is that fundamentalism can easily flip over into other principles in exactly the same way.
Synthetic Maturity and Rules for Conduct Under Uncertainty
The speaker presents synthetic maturity as a position that accepts claims even without proof and without certainty, and therefore the central question becomes how one makes decisions under conditions of uncertainty. He explains that the adolescent may interpret the synthetic adult as dogmatic like a child, but the difference is that the adult has principles and rules for rational conduct in situations of doubt rather than blind acceptance. He connects this to the maturity required in fields that are not mathematics, and concludes that the series will deal with rules for conduct under doubt, with probability as a central but not exclusive component.
Closing Questions: Godel and the Triangle Example, and Doubt in Life-Saving Cases
The speaker answers that the example of Godel is not crucial to what follows but was meant to show that mathematical proof is always within a given framework, and therefore mathematical truth is hypothetical. He confirms that the example is essentially the same as the sum of the angles in a triangle, which depends on Euclidean space. He accepts a halakhic / of Jewish law comment that doubtful danger to life overrides the Sabbath, and notes that this will bring the discussion back to the distinction between practical certainty and philosophical certainty, and that the laws of physics are only practical certainty, “certainty for all practical purposes.”
Full Transcript
[Speaker A] All right, good morning. We’re
[Rabbi Michael Abraham] starting a new series today. I sent out the title and a bit of a description about doubt and probability, and then someone wrote to me afterward that maybe two or three years ago, not all that long ago, there was some series on similar topics, on statistics. Now, truth be told, I looked at it a bit and thought it was different from what I had planned. It’s not completely different, but still I decided not to change the topic. There’s no study hall without innovation. I’ll try to take it in somewhat different emphases and slightly different directions.
[Speaker A] Besides, it’s not the same thing to review a tractate one hundred times and to review it one hundred and one times.
[Rabbi Michael Abraham] Okay, except that for that you don’t need lectures. Reviews you can also do on your own. Anyway, as I said, our topic is doubt and probability, and I want maybe to begin with some general framework for the concepts of doubt. In our world, in recent generations, some sort of dichotomy has arisen between religious conceptions and secular conceptions. In religious conceptions, the ideal is certainty. Meaning, you’re supposed to arrive at certainty, to cling completely to your faith / belief, and doubts are perceived as something destabilizing, something reprehensible, something negative. By contrast, in the world—and I think this is partly a rebellion against religious conceptions, but by now it’s become very, very entrenched, I think, in secular culture—there is some kind of ideal or value in doubt. Meaning, doubt has gone from being an existential condition that you need to cope with to being, in a certain sense, a condition that you even aspire to. In other words, you want to be in doubt. There is some kind of value in not holding firm positions and in understanding that there are various options and things of that sort. And the struggles between these two sides seem a bit like a dead end. One side simply can’t understand the other at all. In a place where doubt is an existential condition, then people who speak in a decisive language just don’t fit in for you at all. They’re not conversation partners. You can’t understand what they want, where they draw their certainty from—and maybe the reverse as well. I think that a balanced view of the world, of philosophy, of the world of ideas, says that both sides are mistaken. Both sides are mistaken because, first of all, I don’t think there is any value in being in doubt. There is value in critical thinking, meaning there is value in checking the options carefully, in listening to different opinions, but all that is done in order to form a position in the end. If it’s done as an end in itself, I can’t really see great value in that. And I think that often people’s fear of conclusions leads them to cling to the state of doubt. They’re unwilling to accept conclusions because the state of doubt has become some sort of ideal. Part of that is that someone who has conclusions is also considered dangerous. He’s decisive, dogmatic, domineering, coercive—so having conclusions is seen as a dangerous condition. And therefore doubt has acquired such an idealized or important status in the culture around us. On the other hand, when I speak about doubt in its constructive sense, beyond the fact that I assume there are several possibilities between which I am uncertain—if there’s only one possibility, then there’s nothing here to be uncertain between—I also assume that even if I reached some conclusion, chose one of the possibilities, that choice probably does not bring me to certainty. In other words, I can’t really hold any position with certainty. And in that sense I really think—well, here I want to argue that one can see in such a view some kind of ideal: to understand that there is no certainty in anything. In my view, that too is not an ideal; it’s simply a fact of life. We cannot have certainty about anything. Whoever thinks he does have certainty about something is probably mistaken—except about the thing itself, as I’ve often said, except about this thing itself, that there is no certainty about anything. But he’s simply mistaken. It’s not that he’s wicked; it’s not that there is some moral or evaluative criticism of him. It’s just a mistake. Therefore I oppose, in general, looking at certainty or doubt as positive or negative values. And that’s common to both sides. Both sides see certainty and doubt as belonging to the evaluative plane. One chooses certainty and condemns doubt; the other chooses doubt and condemns certainty. I neither choose nor condemn either one. I simply say: we have no certainty. We are condemned to be in some state of doubt, but that does not mean we don’t choose a position. We just don’t have certainty about it, and that is the fact. Not because someone who has certainty is blameworthy or praiseworthy, but simply because he is mistaken. There is no such thing. Therefore this whole way of looking at states of doubt and certainty as belonging to the realm of values is, in my opinion, a mistake. The concepts of doubt and certainty belong to the cognitive and intellectual plane, not to the plane of values. That is, the question whether there is truth or there isn’t truth is a question whose answer is an answer of fact. It has nothing to do with evaluative questions. Values deal with what ought and ought not to be done. Facts deal with what is true and what is not true. So these are two different worlds. I don’t think it is right to bring the concepts of doubt and certainty into the world of values. Now, I want to speak, really, about the relation between certainty and truth. Usually—or often—people identify the concept of truth with the concept of certainty. Meaning, when you’ve arrived at the truth, that means you have certainty about something. Now, first of all, as I said before, I don’t think there is certainty about anything. But beyond that, I think this identification is mistaken even in the categorical sense. Here I’m saying that this identification is mistaken in what I’d call the factual sense. There simply is no certainty about anything. Factually. But beyond that, I’m saying there is also a categorical mistake here. Why? Because the concept of truth and the concept of certainty belong to different semantic fields. The concept of truth belongs to the question of what the state of affairs in the world really is. If a certain claim that describes some state of affairs in the world is true, that means there is a correspondence between the content of the claim and the state of affairs it describes. Say, if I say, “There is a table in front of me,” then if there is a table in front of me, that is a true claim; if there is no table in front of me, that is a false claim. So the concepts of truth and falsehood belong to the state of affairs in the world and to the correspondence between my claim and that state of affairs. The concept of certainty is not connected to the world at all. The concept of certainty speaks about me. It is an epistemic concept and not an ontic one. Ontology is the theory of being, of what exists in the world. Epistemology is the theory of knowledge, meaning how I know what is happening in the world. When I say I have certainty about something, that is a claim about me, not about the world. I say, “I am certain that there is a table in front of me.” That is not a claim about the world at all. The claim about the world is that there is a table here. That is the claim about the world. The claim “I am certain that there is a table here” is a claim about me. I am in a state of consciousness called certainty. I am sure of the correctness of some claim. Now, these are really two things that are somewhat related to one another, but not connected necessarily at all. There can be a situation in which I have certainty about a certain thing, but it will not be true. There can be a situation in which the thing is true and I have no certainty about it. Right? There is no necessary connection between the two. For example, a blind person cannot be sure that there is a table here. Maybe he feels around and assumes there is a table here, but he isn’t sure. Yet the truth is that there is a table here. The truth, in terms of the world, is that there is a table here. That means this is a situation in which there is truth even though the person has no certainty about it. Right? That doesn’t mean there is no truth here. It means he has no certainty about that truth. The opposite case is a situation in which I do have certainty. Consciously, I am in a state where I have certainty about something, but that something is not true. I am sure that it is true, but I am mistaken. It is not true. I am simply living in error. So that is a situation in which there is certainty but no truth. Therefore this confusion between the concept of certainty and the concept of truth—beyond the fact that truths need not be certain and in fact also cannot be certain—is a confusion in the categorical sense. Quite simply. These concepts deal with completely different worlds. Certainty deals with me; truth deals with the world.
[Speaker C] What? What has to happen for my certainty to also become truth? What do I have to do? Because I want to be in a state where, when I’m certain of something, I want to know that it’s also the truth.
[Rabbi Michael Abraham] I don’t think there’s anything you can do. Bring evidence? Sure, bring evidence, but evidence… you know, nothing is certain. We’ll talk more about these things later; I’ve spoken about them before as well. But no proof will help you. Even with something you see with your own eyes, you still can’t really have full certainty. Eyes sometimes deceive us too, okay? Logic does not bring you any truth. Someone here asked about logic. Logic does not bring you any truth; it derives conclusions from premises. But the question is always: what is the status of the premises? So, in truth, we have no tools to arrive at truth—to certainty, sorry. And therefore, beyond the fact that a person who thinks that some truth he possesses is certain is probably mistaken, even if he were right, the truth in it and the certainty in it are unrelated to one another. That’s what I want to claim. In other words, I don’t just want to claim that he is mistaken, but that the concepts belong to different worlds, to different conceptual spheres. Yes, certainty belongs to the world of epistemology, the world of my cognition—how much I know that this is true—whereas truth belongs to metaphysics, to the world—not necessarily to metaphysics, okay, to ontology—to the world itself, to what is happening in the world itself. Now, that isn’t entirely precise, because when I talk about truth, truth is not a property of a state of affairs in the world; truth is a property of claims. So when I say that something is true, the concept of truth applies to a claim, not to a fact. There is no such thing as a true fact. If a fact is a fact, then it is a fact. There is no such thing as a true fact. A claim about a fact can be true. In other words, the claim is true if the content of the claim corresponds to the fact it describes. But the fact as such—there is no such thing, it is not “true.” So the concept of truth does indeed relate to the state of affairs in the world, but not entirely. It actually relates to claims about states of affairs in the world. Okay? So in a certain sense, both truth and certainty come closer to the human being. The world as such does not speak in terms of certainties and truths. The world is the world; what is in it is in it, what is not in it is not in it. It’s not about that. I, as an observer of the world, can ask myself whether my observation is true—that’s one question. In other words, does my observation really describe the state of affairs in the world? That is the question of truth. And I can ask myself how sure I am of the truth of that observation, and that is the question of certainty. So you see that the distance, or the distinction, that I made between truth and certainty was too extreme. In fact, both truth and certainty are, in a certain sense, statements about the person and not about the object. They are statements about the human being. The question is whether a person holds a true claim or not, and whether he is sure of the claim or not. Those are two questions about the state of the person, or about something connected to the person. The world itself proceeds as it proceeds. There is no truth there—if there were no human beings in the world, there would be no truth and falsehood in the world. One has to understand that. There would be facts. The facts would be what they would be. The concepts of truth and falsehood always speak about what claims I make—I as a human being, as an observer, as a subject looking at the world. I make some claim, and that claim is supposed to correspond to something, to some state of affairs in the world. If it corresponds, then it is true; if it does not correspond, then it is false. But correspondence between what and what? If there is a world and there are no subjects observing it, then there are not two things to match against one another. There is the world. What is in it is in it, and what is not in it is not in it. All the concepts of truth and falsehood are born as a result of my making some kind of comparison. A comparison between the stance or observation of the subject—the human being who looks at the world—and the state of affairs in the world itself. That comparison is what gives rise to the concepts of truth and falsehood. If it fits, then truth; if it doesn’t fit, then falsehood. The concept of certainty is not connected to the world at all. The concept of certainty relates only to me. The question is how sure I am of a certain claim, a certain truth claim. So it may be true, but whether I’m sure of it—that is a different question.
[Speaker D] I’m not sure what “certain” means anymore. We already said that there’s no certainty about laws of nature. But ask anyone whether there is gravity, and everyone will say, “I’m sure it exists.” From the standpoint of philosophy I can’t prove it. But I really believe it. I don’t think there’s a person who says, “I’m not sure—if I jump off the roof maybe I’ll fall and maybe I won’t. I can’t prove it.”
[Rabbi Michael Abraham] No, so here I think we need to distinguish between two things. One thing is practical certainty versus what we might call philosophical-logical certainty. Practical states of certainty—of course human beings have those. What we call certainty means a very, very high probability. It can always be that I’m mistaken, but the chance that I’m mistaken is negligible. And therefore, as far as I’m concerned, I don’t take it into account at all. That’s what I call certainty in ordinary language. Earlier I was speaking on the philosophical plane. On the philosophical plane, even if a person lives with the feeling that something is certain, that is practical certainty. It is not certainty in the philosophical sense. And at the principled level it may turn out that he is mistaken, even though he feels certainty in that matter. Yes, today by various means you can bring people to a state where they will feel certainty about all sorts of things that never happened and were never created. You can induce in a person the feeling that he is meeting his uncle, even though his uncle isn’t there at all—he already passed away, actually. But you stimulate in his brain the memories or the state as if the image of his uncle is standing in front of his eyes, and from his perspective he is sure that his uncle—he is meeting his uncle right now. Therefore the feeling of certainty does not necessarily testify to real certainty. And even certainty as a feeling is practical certainty, not philosophical certainty. Beyond that, I’ve mentioned in other contexts—and this is just a parenthetical note—that people often also confuse the concept of certainty with the concept of necessity. When I say that something is necessarily true, that basically means it is certain, right? Often in language we mix these two things together, but that’s a confusion. It’s not the same thing. Certainty, as I said before, is a statement about me—how sure I am of something, how certain the thing is for me. Necessity, by contrast, is a claim about the things themselves. For example, I spoke about this in the context of the ontological proof for the existence of God. There a lot of people confuse these two things, and Anselm, in my opinion, devotes a chapter to this in his book Proslogion. Anselm of Canterbury—that’s the scholar who formulated the ontological proof for the existence of God. When I say that the existence of God is necessary, what that means is that there cannot be a world without God. In other words, God is a necessary existent. When I say that I am certain that there is a God, that is a claim about me, not about God. The claim of necessity is a claim about the world itself; it has nothing to do with me. That in the world itself it is necessary that there be God—that is, there cannot be a situation in which God does not exist. That is a claim about objective reality, about God, about the world. It has nothing to do with me at all. Even if I were not here, that necessity would still be a necessity. When I say that I am certain that there is a God, that is a claim about me. If I were not here, that claim would have no meaning. It is my claim when I say that it is certain there is a God. And that is really the difference between the two… Yes, in the Proslogion, when Anselm formulates the ontological proof, in the first chapter he proves that God exists. He proves it logically. Then in the second chapter he moves on to prove that His existence is necessary. And many of his commentators, it seems to me even most of them, think that he is merely repeating himself in different words. But that is a categorical mistake. It’s simply not correct. In the first chapter he showed that, from his perspective, the existence of God was certain. Again, I don’t think he was right, but that’s what he claimed. He wanted to prove that the existence of God was certain from his perspective. In other words, that it cannot be that I won’t believe in God. In the second chapter he is claiming something about God, not about me: that God’s existence is not like the existence of everything else, which is contingent—possible, maybe yes, maybe no—but that God’s existence is necessary. In other words, He is necessary existence. And that is a claim about reality itself; it has nothing to do with me. The question of whether I am certain is different. Therefore, if I proved the existence of something, that does not mean that the existence of that thing is necessary. Suppose I prove the existence of this phone. Obviously the existence of this phone is not necessary. There could have been a world in which this phone was never manufactured. It is not a necessity of reality that there be such a phone, or that there be a phone at all. The reality is that there is such a phone. This truth—you know what, I’m even willing, on the practical level, to say that this truth is certain: there is such a phone. But that does not mean that it is necessary that there be such a phone. “It is necessary that there be such a phone” is a claim about reality—that there could not have been such a reality in which such a phone had not been manufactured. That is an entirely different claim. So there are many things that get mixed together in this field of certainty, truth, and doubt, and I think it is very important to distinguish among them. Now, as I said before, beyond the categorical distinction between certainty and truth, I also made a certain claim, and the claim is that our truths are by their very nature uncertain. We have no ability to reach certainty. Now here again it’s subtle; it’s connected to the distinctions I made earlier. I am not making a psychological claim. What Professor Turkel said before, I agree with: there are people who will tell you they have certainty about something. Clearly there are such people; that’s a fact I can’t deny—a certain fact I can’t deny. But that is not the certainty I’m talking about. That is psychological certainty. I am speaking about philosophical certainty. In other words, a person who arrives at a philosophically certain conclusion. What’s the difference? A person who has psychological certainty—I can undermine it, show him that he is deceiving himself. That he is living under an illusion; it’s not really certain. Maybe it’s true, but it isn’t certain. And the fact that you feel a sense of certainty means you are fooling yourself. When I say that something is certain in the philosophical sense, I am making claims—this is not on the psychological plane. I am claiming that this thing, my knowledge of it, is certain. I can show why my knowledge of it is certain. This is not a report of a psychological mental state, that I feel certainty about it; or not only that report. There is a philosophical claim here saying that this is a claim one can hold with certainty. Okay? So when people come and say, “Yes, I am sure of this, I have certainty about this,” of course there are many people who feel certainty about all sorts of things. I’m not denying that. But that is not what I am calling certainty here. Certainty in the philosophical sense is some claim that has justification. It is not a description of a psychological state, a description of a feeling—“I have a feeling of certainty”—but rather a philosophical claim, really a meta-philosophical claim, that in regard to this claim I have justification for holding it with certainty, I have good proofs, because I have whatever reasons to hold it with certainty.
[Speaker D] So according to that, certainty exists only in logic and mathematics. Beyond that there’s nothing that’s certain.
[Rabbi Michael Abraham] Correct. And even in logic and mathematics—and I’ll get to this in a moment—even in logic and mathematics it’s certainty of derivation. It’s not certainty in the claim itself, but in the path from the premises to the theorem, to the conclusion. But the premises themselves, and the conclusion itself, are not certain. So even logic and mathematics do not really give us certainty, because no claim you make is a certain claim—not even in mathematics. Unless you say that if the premises are such-and-such, then the conclusion is such-and-such. That you can say with certainty.
[Speaker D] They start from the axioms and go from there.
[Rabbi Michael Abraham] Yes, yes. Based on the axioms you can claim all kinds of things with certainty. Those are hypothetical claims, right? If this is true, then I can say with certainty that this too is true. But it is necessarily always an if-then. It is never certainty about the claim in itself. Fine, you could say that the if-then claim is itself a claim, right? The implication that A entails B, that if A then B, is itself a claim. And about that claim I may perhaps say that I have certainty about that claim, but it is not a claim about the world; it is a claim about a relation between claims about the world. So when I speak about certainty, I mean certainty regarding claims about the world—also about the world of values, about the factual physical world—but I am speaking about certainty regarding claims about the world. Now why do I say that there really is no certainty? So, this. Here’s the point: basically we have two principal toolboxes with which I can approach claims about reality. One toolbox is what we might call the philosophical, logical, mathematical toolbox—the a priori toolbox. And the observational scientific toolbox, the empirical toolbox. Those are the two toolboxes. Neither of these toolboxes can bring me to certainty. Why? That’s what I just said. The observational toolbox does not bring me to certainty on two levels. First, when I make observations in the scientific world, I observe particular cases and build from them a generalization and from that create a law of nature. Now even in the particular cases that I observed, I may have been mistaken, maybe I didn’t see correctly, maybe I didn’t interpret correctly, all kinds of things of that sort. So even there certainty is not required. There are even visual illusions in our own seeing, like a mirage or other optical illusions, where we are sure we see something and it isn’t true. “Hearing cannot be greater than seeing”—the Talmud says that sight is the best kind of legal evidence. Hearsay testimony is considered weaker testimony. But that does not mean sight is perfect; it means it is better than hearing. I have said more than once that this saying attributed to Rabbi Kook or to the Rebbe of Vizhnitz—”better to fail in baseless love than in baseless hatred”—always makes me think of the response that it is better not to fail in either one, neither in baseless love nor in baseless hatred. But the claim that A is preferable to B does not mean that A is perfect; it means it is preferable to B. It does not mean A is perfect. “Hearing cannot be greater than seeing” does not mean sight is perfect; it means sight is better than hearing. But sight too is not perfect. Eyewitness testimony—every day in court and in religious courts, eyewitness testimonies turn out to be incorrect. That can certainly happen. Therefore even the basic observations that we make in the scientific context cannot be called certain. All the more so when we speak about laws of nature or the generalizations that we build on those observations. From those observations we build generalizations and general laws, and we cannot possibly have certainty in those general laws if even in the particular observations we have no certainty, and afterward comes the generalization itself, which is another step about which we have no certainty. Therefore science cannot give us any certainty. Popper formulated this in an extreme way in the opposite direction. Popper argued that science cannot even give us truth, not just that it cannot give us certainty. All it can tell us is that something has not yet been refuted. That is to say, a scientific claim is not true in any sense, except in the sense that it has still not been refuted. In other words, all attempts to refute it have failed so far—that is all I can say about a good scientific theory. But that is an extreme approach. Against him, many have already argued that science can also confirm a theory—not prove it, but confirm it—and therefore science also tells us something about truth and not merely that it has not been refuted. But it is clear that what science says—and here it seems to me that Popper put his finger on a correct point and, as usual, took it too far, like positivists generally do—is that scientific truth is not certain. And therefore the scientific observational tools cannot give us truth. That is with regard to the empirical observational toolbox. Then there is the philosophical and logical toolbox, the mathematical one. Logic and mathematics are branches of philosophy, the formal branches of philosophy. In the sense that they make claims about the world not necessarily by virtue of observation but by virtue of reason, judgment, logic—whatever you want to call it—and therefore this is a different toolbox, not the scientific observational toolbox. This toolbox, or at least part of it, logic and mathematics, is often distinguished by the fact that the claims there are certain. And many times people say, “We have tools to reach certainty, and that is logic and mathematics.” And from here, by the way, comes the childish aspiration, I would say, to prove everything, because we are used to the fact that in mathematics and logic we do not accept a claim unless we have a proof for it; you don’t go to the grocery store with conjectures. Well, sometimes you do. Mathematicians know that many times when you publish an intelligent conjecture, that too is a mathematical achievement. That is to say, it isn’t… It is true that mathematics wants, and is willing to recognize, only things that are proven, but an intelligent conjecture is a very important mathematical step, often perhaps even more than the proof. Because if you have an intelligent conjecture, afterward people will look for the ways and somehow succeed in proving it. But to come up with the conjecture itself requires some kind of creativity or elevation of spirit or a deeper grasp of reality—or in any case of the reality or ideas that mathematics deals with. In any event, the problem with logical and mathematical certainty is like what I said before: mathematics and logic deal with deriving conclusions from premises, and therefore one cannot say either about the premises or about the conclusion that they are certain—sorry, not that they are true, but that they are certain. Nothing is certain. The only thing certain in mathematics and logic is the relation between the conclusion and the premises. That relation is certain; the premises are not certain, and the conclusion is not certain. Therefore if you ask a mathematician what the sum of the angles in a triangle is, he says: wearing my mathematician’s hat, I have to tell you I have no answer. Tell me what your assumptions are. If your assumptions are Euclidean, then the sum of the angles is one hundred and eighty degrees. If your assumptions are different, then the sum of the angles is, I don’t know what, two plus three i, something complex—it doesn’t matter, I’m just fooling around. The point is that every answer of a mathematician basically depends on assumptions. You cannot state what is called an apodictic proposition, an unconditional proposition, not hypothetical, an absolute proposition. Every proposition a mathematician states is supposed to be conditional. If your assumptions are such-and-such, then this is the conclusion. But he cannot tell you any conclusion as such as a mathematical claim. The conclusion by itself is not a mathematical claim. A mathematical claim is always given within some framework, a framework determined by the assumptions. Therefore yes, even with Godel’s theorem—I often got Godel’s theorem in mathematics wrong. Kurt Godel was a very famous logician in the twentieth century, and Godel’s theorem showed that in certain systems that satisfy certain conditions there is a proposition that is necessarily true and cannot be proved. Now it took me time—I didn’t understand this thing, it really shook me. How can that be? After all, he proved that claim. More than that, his proof is what mathematicians call a constructive proof. That is to say, you can prove that there is a solution to a problem, prove by various techniques that there is a solution to such-and-such a differential equation, there is a solution and it is unique. An existence theorem and an existence-and-uniqueness theorem, all kinds of theorems in mathematics that prove the existence of a solution. That is one kind of proof. There is another proof called a constructive proof. A constructive proof means I simply build the solution. I show you that if this is the equation, I can simply construct the solution. Once I have constructed the solution for you, that is a proof that a solution exists. By the way, it is not a proof that it is unique, but it is a proof that a solution exists, okay? But Godel, absurdly enough, when he proved his claim that in a system similar to number theory—never mind—there is a proposition that is necessarily true and unprovable, he proved that claim. He proved it by constructing the proposition. He simply constructed that proposition and showed that it is necessarily true and cannot be proved. And I exploded when I first encountered this; I did not understand the matter. He proved this proposition and he constructed it. Isn’t that a proof? So how can he say that this proposition has no proof? Here he showed the proof. He constructed the proposition and proved that it is necessarily true. That’s a proof! So how can he say that it is necessarily true but has no proof? Until someone explained to me—I don’t remember who it was, this was many years ago—that when they say it has no proof, they mean it has no proof within the framework of the theory under discussion. Within the system of number theory there is no proof for it. In the meta-system, when you look at the system from outside, you can prove Godel’s theorem constructively. I can prove to you that the proposition he constructed is one that is necessarily true, but that proof is not considered a proof within the system. It is a proof in the meta-language; it is a proof from my external perspective on the system. If I formulate a proof within the assumptions of the—so this is, I think, a beautiful illustration of the power of mathematics and of its limitations. When mathematics says something, it always says it within the framework of a particular system. And therefore when mathematics says there is no proof, it means there is no proof within that particular system. That is what “there is no proof” means. It may be that in some more general, more external system, I will indeed find a proof of that thing. And therefore, for example, philosophical implications drawn from Godel’s theorem are a bit dangerous. It is very popular to do that, but it is a bit dangerous. Mathematicians often get annoyed when people do these things, because to infer from this that there is a difference, for example, between truth and provability—which is basically the inviting conclusion from Godel’s theorem, that there may be something true even though it has no proof, which is a philosophically very far-reaching idea—that is in fact an incorrect conclusion from the mathematical theorem. Because this proposition does have a proof; it is just not a proof within the system. For example, one of the conditions the system has to satisfy is that it have a countable number of assumptions—a discrete number, like the natural numbers, say—and not something larger; there are larger infinities. Fine? Now if I adopt another system that does not have a countable number of assumptions but a larger number of assumptions, then there I can indeed prove the claims about that system, and therefore it is not correct.
[Speaker D] What the theorem says is that if I add this to my assumptions, there will be something else that I still won’t be able to prove. There will always be something I can’t prove. I can add more and more axioms and I can prove more things, but there will always be something
[Rabbi Michael Abraham] that I
[Speaker D] can’t prove.
[Rabbi Michael Abraham] I am not a mathematician—far be it from me—but from what I learned, that isn’t so. Because when you can add additional assumptions, you create a new system to which Godel’s theorem will also apply. But that new system too has to satisfy the conditions. Godel’s theorem has conditions. And among the conditions is that there be a countable number of assumptions. Therefore, if you take a system and you do not merely add another number of assumptions to it—because then it is still countable—but instead you say, no, this is a system with a continuum of assumptions, the cardinality of the continuum is the number of its assumptions, then Godel’s theorem is simply not true there—that is how I understood it, at least. And then indeed there will be no Godel theorem. It is not that Godel’s theorem applies to every system. The system has to satisfy certain assumptions, certain conditions. So that is why I say the philosophical conclusions that people often draw from Godel’s theorem are hasty. But for our purposes, again, I truly am not a mathematician, but what I want to say is that I am using this to illustrate the limitation of mathematics: mathematics always speaks within a given framework, within a given conceptual structure based on assumptions, derivation rules, and a language, on definitions. And therefore only within that framework can I speak about certainty. But when I speak about certainty within that framework, it is always certainty of derivation: that from within the framework I can infer or derive that conclusion with certainty. But the certainty is always about the hypothetical move, that if this is true then that too is true. But I cannot speak with certainty about any specific factual claim. For example, the claim that the sum of the angles in a triangle is 180 degrees is not a certain claim. Why? In a Euclidean world it is a certain claim, but who says the world is Euclidean? Einstein, for example, says it isn’t—our world. Okay? So the mathematical certainty still exists, only mathematical certainty is not about the proposition “the sum of the angles in a triangle is 180 degrees,” but about the proposition “if space is Euclidean, then the sum of the angles in a triangle is 180 degrees.” About that hypothetical proposition there is certainty. But when I am interested in factual claims—not hypothetical claims but what I called apodictic claims, apodictic claims, maybe, I don’t know exactly how to say it—then there is no certainty. Mathematics and logic too cannot give me certainty. So what happens is that beyond the distinction between truth and certainty that I made earlier, the categorical, conceptual distinction, I want to argue that we do not have certainty. The concept of certainty is a fiction. The concept of certainty is some kind of ideal that is inaccessible to us. That is to say, it does not exist in our world. There is no such thing; talk about certainty is at most talk on the practical plane, or simply a mistake. We cannot have certainty about anything. As I said before, that includes this claim itself, that we cannot have certainty about anything. In other words, the concept of certainty is, in a certain sense, some theoretical concept that one can talk about in terms of how close I am to it, how close I am to it—but I do not get there, right? You won’t see it, and you won’t come near it, or something like that. In other words, it is not something truly accessible to us, and therefore concepts of certainty do not really exist in our world. And that brings us to the field of our discussion. Fine—so if we have no certainty, then what do we do?
[Speaker C] We have no certainty.
[Rabbi Michael Abraham] Usually our inclination is to look for certain truths. What is certain, I accept; what is not certain, I do not accept. But if we have no certainty, we are condemned to live in uncertainty. What are we supposed to do with that? So that is what are called states of doubt. States of doubt are states in which we have no certainty—that is to say, all states in the universe, everything, all human states are states of doubt, according to what I have said up to now. And because of that, it is so important to deal with tools for coping with states of doubt. The tools for coping with states of doubt can be statistical, probabilistic tools; they can be other tools, it doesn’t matter—what in halakhic language are called rules of conduct and rules of clarification—but we will get to that. In any case, these are tools for coping with states of doubt, and since all our states are states of doubt, it is very important to understand these tools: what they mean, what the differences between them are, what they give us. They never give us certainty, but they do give us the ability to maneuver within a state of doubt, within a state of incomplete knowledge.
[Speaker D] Is there a difference between a fifty-fifty doubt and a ninety-nine-to-one doubt?
[Rabbi Michael Abraham] Obviously, obviously there is a difference. In Jewish law that is the difference between majority and doubt. But majority too is a rule for deciding in cases of doubt, or a rule of conduct in cases of doubt. In other words, a state of doubt as I have defined it here is a state in which there is no certainty. That is called doubt. It is not necessarily fifty-fifty. What is called doubt in Jewish law is fifty-fifty or—as we will see later—other situations that are de facto fifty-fifty. That is, sometimes the fifty-fifty is a methodological decision, not the result of a calculation. But in Jewish law, doubt is balanced doubt. When the doubt is not balanced, we enter the realm of majority. But majority too, from my standpoint right now, is a rule of conduct or a rule of decision in cases of doubt, because it is still a state in which I have to decide between different possibilities and I do not have certainty which of the possibilities is correct. Now this very claim of mine—that I need to develop tools for coping with states of doubt—also assumes something. This is not a simple thing in itself. Today I am trying to expose all the assumptions that we hide behind things that we are somehow very used to saying, but we do not notice that behind them there are often all kinds of assumptions that are not so trivial. So here, I think I have already described this more than once, I will do it briefly—the process of maturation, right? Because I want to get to how one deals with doubts. So I often describe the maturation of a person, or the maturation of our civilization, in parallel, as a three-stage process. There are three stages. The first stage—I am speaking about a person right now—the first stage is childhood, where the child basically takes what adults tell him for granted, right? You ask your mother why the sun rises every morning, and she tells you because the earth rotates on its axis and therefore every morning the sun rises. Okay, I understand. In other words, it does not occur to him to ask, and who told you that the earth rotates? He says fine, if Mom said so she probably knows—or the teacher, or the rabbi, or whoever it may be. Children generally accept things dogmatically. What adults—the all-knowing ones—tell them, that is true as far as they are concerned. That is the stage of childhood. Then comes adolescent rebellion, when the teenager begins to ask his mother: wait, but who told you that the earth rotates? So she may bring him some evidence of that, and then he will ask her: and who told you that? In short, in adolescent rebellion the teenager demands proof or justification for claims. He is not prepared to accept them dogmatically. Just because someone said it is not enough for me to accept it. I want proof. Then comes the crisis at the end of adolescence, the transition, the crisis of maturity, and the teenager comes to understand that in fact he has proof of nothing. Because every proof brought to him is always based on certain assumptions, and when he asks, “And why do you accept those assumptions?” they will at most bring him another proof based on other assumptions. But in the end it is turtles all the way down. In other words, you cannot produce truth that does not depend on some assumptions which are themselves ungrounded or themselves unproven. And then the teenager enters a crisis as a result of what I described earlier. We cannot reach certainty. He expects certainty. For him, truth is certainty. If you have proof, I will accept it; if not, then no. And he expects proofs so that he can have certainty and then be able to accept it. He is sure he will be a rational person, not like his stupid parents, but instead he will accept only things that are proven, only things that are certain—not like them, who are such dogmatists like children, accepting things just because someone said so. But at some point he reaches the conclusion that he can have certainty about nothing. About nothing. The two toolboxes, the philosophical and the observational and scientific one, as we saw, offer no way to reach certainty. What does one do in such a situation? And this is exactly—you understand that at this point we are exactly at the point where I was five minutes ago, when I reached the conclusion that there is certainty about nothing, and therefore one needs to formulate tools of conduct or decision for states of doubt. But here I have already assumed something: that when I am in a state of doubt there are tools, or there should be tools developed, that will help me behave or make decisions in states of doubt. It is not at all obvious that this is the necessary way out. That teenager in crisis who says nothing can be certain, and he is prepared to accept only things that are certain—what do you do in that situation? So as I said, the crisis is created by a combination of two assumptions or two claims. One claim is that only a proven thing is admissible, only a certain thing is admissible. And the second claim is that nothing is certain.
[Speaker D] As a result, nothing is admissible.
[Rabbi Michael Abraham] That is what creates the crisis, the combination of those two claims. And one can get out of this in one of three ways—let’s say four, but the fourth is not relevant. You can adopt both assumptions, you can give up assumption A, or you can give up assumption B. Those are three possibilities, combinatorially there are no more. So if I adopt both assumptions, I stay with both of them—what is the conclusion? If only a certain thing is admissible and nothing is certain, then the conclusion is that nothing is admissible. The conclusion is skepticism. Nothing is admissible. And with that conclusion there is no point in formulating tools for deciding or making decisions in states of doubt. There is no such thing as decisions; nothing—there is nothing to do with such a state. That’s it, lost. You are condemned to remain in doubt all your life with no ability to cope with it. You are a skeptic. Nothing is any more correct than its opposite. That is one possibility. That is if you remain with the two assumptions that create the crisis. If you give up the assumption that nothing is certain, the second assumption, you say no, there are certain things. How? But the observational toolbox does not give us certainty, and neither does the logical, philosophical, mathematical toolbox—so what does? How do you reach certainty if not through those two toolboxes? And the answer is fundamentalism. Fundamentalism is the second form of maturation. There is skepticism—that is one form of maturity—and fundamentalism, that is the second. What is fundamentalism? I have some source of authority or source of knowledge that is beyond all doubt. It is not logical, not philosophical, not mathematical, not scientific, not observational. It is something else. Some sort of spiritual charisma above logic, above intellect, I do not know exactly what, but I have no doubt about it; it gives me certainty, and therefore I can accept these things as true because I have certainty. Notice: the fundamentalist retains the teenager’s assumption that only a certain thing is admissible. He just claims, unlike the teenager, that there are certain things. The teenager is wrong that nothing is certain. No—there are things. We have the ability to reach certainty. How? Through something, I don’t know what, supernatural, supra-rational, supra-intellectual. Okay? That is what I call fundamentalism. Why is it fundamentalism? Because fundamentalism is—I’m talking now at the philosophical level—fundamentalism is a position that does not subject itself to the test of critical thought. There is no way to criticize it. Not logic, not science, not philosophy—nothing will shake it. It is simply true because the caliph said so, or because the Rebbe said so, or because I don’t know what, someone—or my parents—said so. It doesn’t matter. From my perspective, it is beyond all doubt, not exposed to intellectual criticism or critical thinking. That is called fundamentalism. Why is fundamentalism dangerous? I don’t need to tell you that. But on the principled level there can also be a non-dangerous fundamentalism. That is to say, there is a kind of fundamentalism in which a person clings to morality without any willingness to subject it to critical thinking. He is a moral person, a fanatic, okay? Like Mother Teresa, if you like. So there is no problem living next to her; on the contrary, it can even be very pleasant—although people who are too good are also a bit of a problem, they can be annoying. But that is not the fundamentalism people mean in our everyday discourse. Still, it is fundamentalism in the philosophical sense. Why? Because it is a position not willing to subject itself to critical thinking. In that sense it is fundamentalist. Of course, the fundamentalism that troubles people more is when the position one is unwilling to subject to critical thinking is a violent position. It is the position that one must cut off the head of anyone who does not belong to our group. And if that is what you are not willing to debate, not willing to submit to critical thought, that really is dangerous. Therefore people fear fundamentalism. But I fear philosophical fundamentalism, not just practical fundamentalism. I am afraid of Mother Teresa too. Not because she will kill me—maybe she will too—but why? Because if she is a fundamentalist, then tomorrow morning she may suddenly adopt, in a fundamentalist way, other principles too. Like who was it who said this, Groucho Marx? Or maybe Bernard Shaw, I don’t remember—”I have many principles, and if you don’t like them, I have others.” In other words, the claim is that fundamentalism as such is the acceptance or adoption of a certain position without being willing to subject it to critical thought, to the criticism of reason. Okay? That is called fundamentalism. Fundamentalism is the second form of maturation. The third form of maturation—simply because it is what remains, the only possibility left—is to give up the first assumption, not the second. To give up the assumption that only a certain thing is admissible. And that is what I call synthetic maturity. It is a maturity that says I am prepared to accept as true also claims that are not certain. Unlike the teenager who asks his parents: who told you? Prove it. Do you have proof? If you don’t have proof, I won’t accept it. Then I have not matured. I accept also things for which I have no proof. And then of course the question arises—and only here does the question I asked earlier arise: okay, so how do you make decisions under conditions of uncertainty? In other words, if indeed you are prepared to accept truths even though you have no certainty about them and no proof for them, then tell me on what basis. What are the rules or principles that help you conduct yourself in a state of doubt, in a state of uncertainty? And perhaps from another angle the importance of this will be clearer. When the child asks his father, his mother: who told you? Prove it. Fine? What is he actually saying? The teenager asks, sorry. He really remembers his childhood. In childhood he accepted things dogmatically, simply because someone said so. And if someone said it and he had a long enough beard, then it is probably true. Okay? Now he has sobered up from that. Now he is already a teenager, already older, already understands, no longer accepts things without proof. He wants things with proof. Okay? He is going to be a rational person, not an idiot like his parents. Therefore he wants proofs. Now his father—he comes to his father and says, who told you the sun will rise tomorrow? Maybe the earth won’t rotate? So his father says: look, I don’t know who said so, but common sense. In other words, never mind, such-and-such principles make me think it will happen tomorrow as well. Not with certainty, because the father is a synthetic father, but not with certainty—still, it is probably true. Okay? Let us call it practical certainty, what I mentioned earlier. Okay? The teenager of course will not accept that. Why won’t he accept it? Because when the teenager looks at his father saying this, how does he interpret it? He understands that his father… his father is basically stuck in childhood. His father adopts things without having proof, because he is simply dogmatic. But I have already woken up from that; I have matured; I already know that I am not willing to accept things just like that—I want proofs. So when his father tells him, look, I don’t have proof, but it is so, the sun will rise tomorrow too, then the child says: come on, nonsense, this father is stuck in childhood—in other words, he is a baby, because he accepts things dogmatically. Where is the teenager’s mistake? The teenager’s mistake is that when he looks at the adult, he looks at him through glasses that recognize only two states: childhood and adolescence. When he sees his father in front of him, he says: this father accepts things without proof, he is probably stuck in childhood. But this father, if he is a synthetic father, comes from a different starting point, from a mature synthetic starting point according to which he is prepared to accept things without proof. But this is not childish dogmatism; I do not accept things merely because someone said so. Rather, I have principles for how I am supposed to conduct myself in states of doubt, and they help me maneuver despite the lack of knowledge, to make decisions under conditions of uncertainty. That is the difference between the father and the child. The child accepts things simply because someone said so—that is dogmatic, and indeed not the right way to act. The father will not accept just anything because someone said so; rather, he activates general systems that instruct him how to make decisions, how to think under conditions of uncertainty. In that sense he is different from the dogmatic child. Now the teenager, who has gone through only childhood and adolescence, looks at the father and of course sees him as a child—he accepts things without proof. He does not understand, because he has not yet passed through this stage, that the father did not regress from adolescence to childhood; the father advanced to the stage of synthetic maturity. And since each of us has to shave on his own beard, right? Nobody learns from his parents; we always learn from our own mistakes; we have to fail, and “a person does not understand words of Torah unless he has first stumbled over them.” So there is no choice; we will have to go through the process of maturation ourselves, and we will not accept the common sense of our father or our teacher or whoever it may be, and we will have to reach these conclusions on our own. But in the end, the important point is that the teenager’s question really is a good question. After all, you—the synthetic adult—accept things that have no proof, that you have no certainty about, so how are you different from the dogmatic child? He too simply accepted things because someone said so, dogmatically, so you too are simply dogmatic. What is the difference between you? There is one and only one difference: the synthetic mature father has rules for how one conducts oneself in states of doubt. He does not accept things merely as a dogmatic matter; rather, he has principles which admittedly do not give him certainty, because he is still in a state of uncertainty, a state of incomplete information, a state of doubt, but he has principles that help him act rationally, sensibly, within such a state. And these are the kinds of things the child or the teenager cannot accept, cannot grasp; they have not yet reached that stage of ripening. Okay? By the way, just as an aside—and here I will finish—I once wondered why all geniuses at age sixteen always do a doctorate in mathematics. Today I am talking a lot about mathematics. Why not in literature, or physics, or I don’t know, psychology? I think mathematics is a certain kind of talent that even teenagers can succeed in. That is to say, you do not necessarily have to be intellectually ripe to be good at mathematics. To be a researcher maybe yes, but to be good at mathematics, if you have a good grasp of the form of this logical way of thinking, you can be a genius in mathematics at sixteen. You can hardly be a genius in literature or physics at those ages. These are fields that, precisely because they are not mathematical, are not certain, and they require a certain kind of intellectual maturity to say sensible things in them. Therefore synthetic maturity quite naturally comes together with some kind of emotional or intellectual maturity. That is not accidental; there is something here—mathematics even a teenager can understand. I prove to him that if x equals y and y equals z, then x equals z—every child understands that. He will not always discover it on his own; he needs the teacher’s help, and after the teacher explains it to him, he will understand—there is no problem understanding it. But to understand an idea in literature, or a psychological idea, or an idea in physics—that will be harder for a child; he needs some kind of maturity. In that sense, synthetic maturity really does focus not only on the synthetic but also on the maturity. That is to say, it requires a certain kind of ripeness to understand that there are states in which I do not operate with certainty. Truth is not certain, and yet there are right ways to act and wrong ways to act. And here our whole topic really enters. All the rules for how we conduct ourselves in states of doubt. Probability is of course an important part of those rules, but not the only part. These rules are basically rules that help us be a synthetic adult. How do I conduct myself within states of doubt? That is essentially our subject. Okay, we’ll stop here. If there are comments or questions, then you can ask.
[Speaker E] Earlier you spoke about mathematical certainty through Godel’s proof, and I didn’t quite understand. Is that something critical for understanding the rest of the series, or is it just an example?
[Rabbi Michael Abraham] It is an example; it is not critical. The point I was trying to show through it, simply put, was that when mathematicians speak about proof, they mean proof within a given framework of definitions, assumptions, derivation rules, and so on. There is no such thing as proof, period. Proof is always context-dependent, and therefore mathematical truth is always hypothetical truth, even though we are used to thinking that mathematics is the queen of certainty, right? The best tool for reaching certainties. They are certainties within a given framework.
[Speaker F] Is that the same idea as the sum of the angles in a triangle?
[Rabbi Michael Abraham] Yes. The sum of the angles in a triangle is one hundred and eighty degrees within a Euclidean framework, with the assumptions and definitions and properties of Euclidean space. We usually don’t say that explicitly; mathematicians put it on the table when they lay out the axioms. But we are usually accustomed to talking about it as though it were an obvious truth, self-evident. Yes, it’s obvious, the sum of the angles in a triangle is one hundred and eighty degrees. No, it is not obvious, and it is not necessarily true either. It depends on the assumptions. In ordinary Euclidean space, that is how it is.
[Speaker D] Just a comment from the side of Jewish law: we say that even a doubt regarding danger to life overrides the Sabbath. And I was always taught—not every doubt. If the doubt is sufficiently small, if the doubt is one in a million, it does not override the Sabbath.
[Rabbi Michael Abraham] We will get to that, because it will bring us back to the distinction between practical certainty and philosophical certainty. There are situations where at the philosophical level there is doubt, but there is practical certainty. That is good enough. The laws of physics, for example—I have practical certainty about them. It is not philosophical certainty; they may be wrong. But I have practical certainty. Practical certainty is certainty for all practical purposes. So yes, we will talk about that more, of course, we’ll get to it. Okay then, Sabbath peace, good tidings
[Speaker D] good ones, goodbye.