Faith – Lesson 16

Table of Contents

• Mapping out routes to faith and the transition to the cosmological proof
• The simplistic formulation, the atheist challenge, and the precise formulation of contingency
• Leveraging the failure of infinite regress and the Conan Doyle analogy
• Polemical shifts in position and the need to justify the failure of regress in its own right
• Turtles all the way down: why infinity is not an explanation
• Infinity, time, and space: the problem is conceptual and logical
• The slap example: a physiological chain versus a first cause and determinism
• Stopping at the Big Bang instead of God, and a promise to return to that claim
• Kant: a correct description of the inference, but opposition to extending it beyond experience
• Against Kant: the failure in infinite regress is not experiential but logical
• Necessary proof versus plausibility: criticism of Kant’s target
• A dispute over plausibility, polemical bias, and dependence on assumptions
• The concept of infinity: “a lot” versus a different category altogether
• Potential infinity versus concrete infinity: induction and limit processes
• Chovot HaLevavot, minus infinity, and the asymmetry of “walking” on the number line
• Hilbert’s Hotel: the illusions of concrete infinity and concrete impossibility
• Why infinite regress is not an alternative explanation
• “I have no explanation,” the law of the excluded middle, and the preference for explanation over no explanation
• Kant and the principle of causality: the response that the challenge is actually the argument itself
• Reservations about “self-cause” and preferring “does not need a cause external to it”
• Concluding questions: logic beyond experience, Maharal, infinite series and Zeno, and plausibility versus probability

Summary

General Overview

The text lays out six or seven routes to arriving at faith in God and focuses the lecture on the cosmological proof in its precise formulation, according to which only a contingent thing that is not its own cause requires a cause; therefore the chain must stop at a necessary being. The text argues that an infinite regress is not an explanation but an evasion of explanation, and grounds this through a distinction between concrete infinity and potential infinity, using mathematical and philosophical examples. The text takes issue with Kant, arguing that the problem in infinite regress is logical rather than dependent on experience, and adds that demanding a “necessary” proof is misguided, because a higher degree of plausibility is enough to justify adopting a position. The text expresses reservations about the term “self-cause” and prefers the wording “a being that does not need a cause external to it,” and concludes with students’ questions about experience, logic, convergent series, and the distinction between plausibility and probability.

Mapping out routes to faith and the transition to the cosmological proof

The text divides the routes to the conclusion that God exists into six or seven types, and defines the first type as the ontological proof, based solely on conceptual analysis, with mention of the previous discussion about Anselm. The text presents the second type as the cosmological proof, which begins from a minimal factual assumption—that something exists—rather than from conceptual analysis alone. The text notes that another formulation later on will begin from the assumption that the world is designed, complex, and coordinated, but for now the assumption is simply that something exists.

The simplistic formulation, the atheist challenge, and the precise formulation of contingency

The text presents a simplistic formulation according to which everything that exists needs something that created it, and therefore if a world exists, then something exists that created it. The text describes an atheist challenge according to which if everything has a creator, then the creator of the world also has a creator, leading to an infinite regress that is defined as a failure and collapses the basic assumption of the simplistic formulation. The text argues that the initial formulation is imprecise and even careless, and can therefore be attacked in this way, and it proposes a more precise formulation: only a thing that is not its own cause—that is, a contingent thing whose existence is not necessary—requires a cause that created it. The text states that the world and material objects are contingent, and therefore a cause is required for their existence. It distinguishes that if the cause belongs to the category of contingent things, then it too needs a cause; but if it is necessary, then the chain stops, because there is no need for something that created it, and it “always was” in the sense that its existence is necessary.

Leveraging the failure of infinite regress and the Conan Doyle analogy

The text presents two final possibilities: either an infinite regress of contingent beings, or a non-contingent being at which the chain stops. The text adopts the claim that infinite regress is a failure and uses it to force the alternative of a necessary being, so that the atheist challenge to the simple formulation becomes the basis of the updated argument. The text quotes Conan Doyle in Sherlock Holmes, in The Sign of Four, that once the impossible has been eliminated, whatever remains, however improbable, must be the truth, and applies this in order to argue that the alternative of an infinite chain is impossible, and therefore one must accept a necessary being even if it appears implausible.

Polemical shifts in position and the need to justify the failure of regress in its own right

The text describes a situation in which the atheist first claims that infinite regress is a failure in order to attack the simplistic formulation, and then “changes his skin” and offers infinite regress as an alternative after the formulation is updated. The text attributes this to “the wonders of scholasticism,” in which opposite assumptions are adopted in order to win an argument. The text asks to leave the polemics aside and to clarify for oneself whether one can in fact be convinced that infinite regress is a failure, because without that, the cosmological argument does not hold.

Turtles all the way down: why infinity is not an explanation

The text brings William James’s story about the Greek physicist who explains that the world stands on a turtle, and on another turtle, and another turtle, and ends with “it’s turtles all the way down.” The text argues that the answer is funny because it is not really an answer but an evasion of explanation, since there is no defined point of “down,” and if there is a “down,” one can always ask what the bottom turtle is standing on. The text concludes that an infinite chain does not provide an explanation, but merely keeps claiming that the explanation is “over there,” and therefore infinite regress is problematic as an explanation.

Infinity, time, and space: the problem is conceptual and logical

The text distinguishes between questions about infinity in time or in space and the claim that the difficulty lies in the concept of infinity itself, not in the physical dimension. The text uses “intuitionist” language to distinguish between a concrete use of infinity, which is undefined, and a potential use, which is mathematically defined. The text argues that an infinite chain of causes in the concrete sense does not really say anything, and therefore does not provide an explanation.

The slap example: a physiological chain versus a first cause and determinism

The text presents an example in which Reuven slaps Shimon and gives a physiological chain of hand, muscle, electric current, and field in the brain, and argues that this is not an answer to the question “why,” because it does not identify the initial factor that triggered the chain. The text argues that on Shimon’s assumption there is a first cause, such as Reuven’s free decision, and without that the question is not relevant within a deterministic framework in which there is no “decision,” only a mechanism. The text parallels this to the cosmological argument and argues that referring the explanation to the Big Bang or to a long deterministic chain is not sufficient, because an explanation requires presenting the full chain beginning with an opening stage that does not itself rest on a previous link in the same way.

Stopping at the Big Bang instead of God, and a promise to return to that claim

The text accepts as a possible comment that the atheist may propose stopping the regress at the Big Bang rather than at God, and clarifies that this is a different argument that will be discussed later. The text emphasizes that the analogies at this stage are meant to illustrate why infinite regress is a problem, not yet to undermine the cosmological argument from the opposite direction.

Kant: a correct description of the inference, but opposition to extending it beyond experience

The text reads a passage from Kant’s Critique of Pure Reason that describes an inference from the impossibility of an infinite series of causes in the world of the senses to a first cause. The text notes that Kant correctly describes the updated argument, but claims that the transition is not justified even within experience, and certainly not as an extension beyond experience, where the causal chain cannot be prolonged. The text objects and argues that the question is being asked about the world of experience, and the need for a cause for the world of experience leads to a cause outside experience, where a being outside experience may be one that does not need a cause.

Against Kant: the failure in infinite regress is not experiential but logical

The text argues that the problem in infinite regress does not depend on the fact that we do not encounter it in experience, because one cannot “learn from experience” that an infinite chain is not an explanation, and one cannot in practice trace an infinity of steps. The text concludes that the failure is logical, and therefore applies beyond experience as well, and compares this to the validity of proof by contradiction as a logical mechanism that is not limited to the world of the senses. The text adds that even if, theoretically, an infinite chain outside experience were possible, still, by Occam’s razor, one necessary being seems simpler than an infinite chain of “gods.”

Necessary proof versus plausibility: criticism of Kant’s target

The text argues that Kant is looking for a necessary proof of God’s existence, and therefore points out that the proofs are not necessary; but the speaker himself is satisfied if the claim that God exists is more plausible than its opposite. The text emphasizes that there is certainty about nothing, and therefore even if additional metaphysical alternatives are possible, the question of plausibility still favors a finite chain over an infinite one. The text acknowledges the difficulty of defining “plausible” precisely, and distinguishes between probabilities that have a sample space and plausibilities that do not admit such a calculation.

A dispute over plausibility, polemical bias, and dependence on assumptions

The text argues that most human beings would agree that an infinite chain of gods is less plausible than one God, and sees resistance to this at times as the product of bias against the theological conclusion. The text states that every argument ultimately rests on assumptions, and the choice of assumptions itself rests on plausibility, giving as an example the principle of causality, for which there is no absolute proof according to David Hume, and yet it still seems more plausible than the claim that things happen without a cause. The text suggests a thought experiment in which an explanation in terms of gravitational force is perceived as simpler than an infinite chain of angels, and argues that this judgment reflects ordinary human intuition outside the context of a debate about God.

The concept of infinity: “a lot” versus a different category altogether

The text brings a passage from Hendrik Willem van Loon about an enormous rock and a bird that sharpens its beak on it once every thousand years until the rock wears away, at which point “one day of eternity” will have ended, in order to give a sense of time for “eternity.” The text warns against understanding infinity as simply “a lot,” and argues that infinity is a different conceptual category that cannot be built up by increasing the finite, just as one does not build three dimensions by increasing two dimensions. The text clarifies that one can speak about infinity only carefully and in a defined way.

Potential infinity versus concrete infinity: induction and limit processes

The text uses proof by induction to argue that the method does not prove something “about infinitely many numbers” in the concrete sense, but rather “about any number you choose,” without committing to a positive claim about the amount of numbers. The text presents convergent series such as one-half plus one-quarter plus one-eighth, and argues that the precise formulation is that the sum is “as close as you like to one,” and that what is being discussed is a limit rather than a concrete realization of infinitely many terms. The text concludes that properly defined discourse concerns potential infinity as a limit or a process of approach, whereas concrete infinity creates problems and is undefined.

Chovot HaLevavot, minus infinity, and the asymmetry of “walking” on the number line

The text mentions Chovot HaLevavot, which argues that the world cannot have existed for an infinite amount of time because it would never have arrived at our own time, and argues that this claim uses undefined concepts. The text illustrates that a process beginning at minus infinity and moving rightward is undefined, because one cannot answer when one arrives at a point such as minus 347.5 or plus pi, whereas beginning at zero and progressing toward minus infinity is a defined process in which every finite point can be assigned an arrival time. The text argues that the symmetry between the two directions is an illusion, and that one can say the world has no starting point when looking backward without needing to posit an undefined process of “setting out” from minus infinity.

Hilbert’s Hotel: the illusions of concrete infinity and concrete impossibility

The text brings Hilbert’s Hotel with infinitely many numbered rooms and argues that in the concrete description one can “vacate” a room by moving each guest from room n to room n+1, and can even accommodate infinitely many passengers from infinitely many buses by assignments such as moving everyone to room 2n and allocating rooms according to powers of prime numbers. The text adds a practical variation and argues that one cannot carry out a concrete announcement that will move every guest, because at every given moment all the rooms are occupied and there is no empty room from which to begin in practice. The text concludes that there is no such thing as a hotel with infinitely many rooms in the concrete sense, whereas “as many rooms as you like” is perfectly well-defined language.

Why infinite regress is not an alternative explanation

The text argues that philosophers have difficulty explaining why infinite regress is invalid, and suggests that the root of the issue is the undefined assumption of concrete infinity required in order to “begin” an explanation from minus infinity and arrive at the world. The text distinguishes between looking backward from the world at one more cause and one more cause as a possible description, and giving an explanation, which requires moving forward from a defined starting point, like geometry, which begins from axioms rather than from an infinite chain of “prior axioms.” The text states that infinite regress is not merely wrong, but undefined, and therefore does not provide any explanation at all; hence these are not two symmetrical alternatives.

“I have no explanation,” the law of the excluded middle, and the preference for explanation over no explanation

The text argues that if the alternative to a finite chain is “I have no explanation,” then even so the option with an explanation is preferable, much like an investigative commission into a plane crash would prefer an explanatory hypothesis over the claim that the plane crashed “without explanation.” The text adds that trying to place “I have no explanation” as a logical position does not fit with the law of the excluded middle, because either there is a first cause or there is not; and if there is not, then one gets an infinite regress, which is undefined. The text concludes that one possible weakness that remains is the very assumption that everything needs a cause, but still argues that it is more plausible to prefer a cause over the absence of a cause.

Kant and the principle of causality: the response that the challenge is actually the argument itself

The text attributes to Kant also the claim that the principle of causality applies only to the world of experience, and asks: who says there is causality outside experience? The text replies that the principle of causality is not learned from experience, and Kant himself writes this in continuation of David Hume’s line, and therefore restricting it to experience seems implausible. The text argues that this very distinction is exactly what is required in order to avoid infinite regress: the world within experience has causes, and at some stage one arrives at a being outside experience that has no cause external to it, and that is God. Therefore, Kant’s “challenge” simply repeats the move of the updated argument.

Reservations about “self-cause” and preferring “does not need a cause external to it”

The text states that the concept of a “self-cause” is a medieval concept and really a meaningless phrase, because if a thing creates itself, then it existed before it existed. The text proposes instead speaking of a being whose existence is an inherent necessity and that does not need a cause external to it.

Concluding questions: logic beyond experience, Maharal, infinite series and Zeno, and plausibility versus probability

The text answers a question about realities outside experience by arguing that logic is always valid, and that “either there is a cause or there is not” applies outside experience too, and gives an example from Maharal’s introduction to Gevurot Hashem about the superiority of the sage over the prophet, because a logical principle is true everywhere. The text explains the series one-half plus one-quarter plus one-eighth through Zeno’s Achilles and the tortoise paradox, and argues that the mistake is the assumption that the sum of infinitely many terms must itself be infinite, whereas there are infinite series that converge to a finite sum. The text distinguishes between probability and plausibility, and agrees that what is less plausible may still be true, but argues that in situations with no sample space, considerations of plausibility still guide preference without yielding certainty, and it concludes with the blessing: Sabbath peace.

All right, first of all, where are we holding? So the routes to the conclusion that there is God, yes, to arriving at faith / belief in God, I divided into six or seven types. The first type was the ontological proof, which is based only on conceptual analysis. That was our previous discussion about Anselm and his arguments. The second type is the cosmological proof, which is where we are now, and this proof does not make do with conceptual analysis alone. Rather, it starts from some factual assumptions, where in this case the factual assumption is that something exists. That’s it, and that’s a very minimal assumption. The next formulation will talk about the fact that there is a world that has very specific characteristics—it is designed, it is complex, coordinated—that’s already the next formulation. But for now we are assuming some very minimal factual assumption: that something exists. And then the argument, in the first formulation I presented, basically says: everything that exists must have something that created it; a world exists; therefore something exists that created it. But against that simple, simplistic formulation, an objection arises: okay, if everything that exists needs something that created it, then whatever created the world is also something that exists, and it too needs something that created it. And that basically brings us to an infinite regress. What does that mean? Infinite regress is a fallacy, meaning we are not prepared to accept an infinite chain of explanations, and therefore this is basically a proof by negation, this objection—the atheistic objection, let’s call it that. Can’t hear. Sorry? It is a proof by negation that it is not true that everything needs to have something that created it, because if that were true it would lead us to an infinite regress. Meaning, it knocks down the basic assumption on which the cosmological argument is built. And then I said that this initial simplistic formulation is an imprecise or even sloppy formulation, and therefore indeed it can be attacked in this way. The more precise formulation of the cosmological argument basically says that every thing that is not the cause of itself, every thing that is contingent, yes, whose existence is not necessary, must have a cause that created it. The world is like that, it is contingent. Material objects can exist, can fail to exist; there is no necessity that they exist. And because of that, there must be something that created it—the world, or the objects in it. Now the atheist will come and ask: okay, and what created the one who created the world? My answer: it depends on the nature of that thing that created the world. If it belongs to the same family of things whose existence is not necessary, then correct, it needs something that created it. If it does not belong to that family—meaning if it is something whose existence is necessary—then the chain stops there. There does not need to be anything that created it, because no one created it; it always was, because its existence is necessary. Even if it does belong to the group of contingent beings, meaning that it needs something that created it, I will ask the same question about the one who created the one who created the world, and again there are two possibilities: either it belongs to contingent beings or to necessary beings. In the end, I have two possibilities: either an infinite regress of contingent beings—which is not an option—or at some stage in this process, oops, or at some stage in this process I arrive at a non-contingent being, and there it stops. Because that non-contingent being does not need something else to create it. Now since the atheist taught us that infinite regress is a fallacy, I leverage that very principle and build the proof on it in the updated formulation. Precisely because infinite regress is a fallacy, precisely because of that I say: fine, then there is no choice but to adopt the second alternative. The second alternative is that this regress, this retreat—yes, regress means retreat—that this retreat stops at a certain being that does not need a cause outside itself. And there it stops; that is the only way to stop the chain. Otherwise it is an infinite regress. Therefore, the objection the atheist raised against the simple formulation is itself the assumption that leads me to the conclusion in the updated formulation, the more sophisticated one, the more precise one. Okay? So therefore I think that once I adopt the idea that infinite regress is a fallacy, there is no escape. As I probably mentioned, it always comes up in this context: Conan Doyle writes in Sherlock Holmes, in The Sign of Four, that after you have eliminated the impossible, whatever remains, however improbable, is the truth. So it may be that there is a very, very improbable option, but if its alternative is impossible, then apparently the improbable option is the correct one. So here too, the same thing. You tell me: maybe it’s improbable that there is some object we do not see whose existence is necessary and that created the world? I say fine, but what is your alternative? As Peirce said—what is their alternative? Your alternative is to produce an infinite chain of contingent beings, and that is impossible. Infinite regress is a fallacy, just as you the atheist yourself said against the original formulation—and that is true not only because you said it; infinite regress is a fallacy. And since that is so, there is no choice but to remain with this thesis, even though it seems improbable to you, because the alternative is impossible. And that is basically the more precise formulation of the cosmological proof, the cosmological argument. After that I started getting into the question: why really is infinite regress a fallacy? Now here I’ll note maybe—I don’t remember if I noted this or not—a methodological point. I said that here I sometimes feel like a lecturer in Arachim, giving seminars for people being brought back to repentance, but that’s not what this is; this is really about sharpening the logical move. In the course of the discussion as I presented it, in the original formulation that I gave, the atheist raised against me: “Wait, but infinite regress—you get to infinite regress, and infinite regress is a fallacy.” I updated the formulation, and now suddenly the atheist changes his skin. The alternative he offers is infinite regress. Wait a second—but a moment ago you said infinite regress is a fallacy. These are the wonders of scholasticism, yes? Meaning, you argue and you always have to win, so you adopt the assumptions that allow you to win, where each time you can adopt the opposite assumption, as long as in the end you come out on top, you come out alive from this argument. But I say, forget the atheist; I’m asking regarding myself. Not in the question of how to defeat the atheist, but in the question of whether, for myself, I can really be convinced that infinite regress is a fallacy. Because without that, the argument does not hold water. Now the question is whether that is correct. Meaning, is infinite regress really a fallacy, and why? And I began explaining that at the end of the previous session. I brought that story from William James about the Greek physicist who gives a lecture on what the world stands on—it stands on a giant turtle, Atlas. Then they ask: what does the turtle stand on? On another turtle. And the second turtle stands on a third turtle. And then when they point and ask again, he says: “What don’t you understand? It’s turtles all the way down.” Meaning, there is no point in asking this all the time. Why is an answer of that kind unacceptable? And we all laugh when we hear it—which does not stop anyone afterward from raising the option of infinite regress. But when it’s raised in practice, everyone laughs. So yes, I told you—the wonders of polemics. But why does it really bother us? Why is it funny? It’s funny because “turtles all the way down” is not an answer. “Turtles all the way down” is an evasion of giving an answer. You are basically saying: “there are turtles all the way down”—that means you have no explanation. You can’t really explain to me what the world stands on. Why? Because there is no such thing as “down.” What does “down” mean? If there were such a point called “down,” then I would ask you what the turtle standing there at “down” is standing on. So what do you mean? That there are turtles all the time, there is no such point called “down”; all the way to minus infinity there are turtles there all the time. So basically you are telling me there is no such thing as “down”—or in other words, you did not give me an explanation. You tell me “turtles all the way down”; you yourself are basically saying that in order for this thing to be an explanation, there has to be somewhere down there some kind of down that starts the chain, because otherwise it is not an explanation. But when I look at it again, I see that there cannot be such a concept as “down.” If there were such a concept as “down,” you still wouldn’t have given me an explanation, because I would ask: what is the bottom-most turtle standing on? And you haven’t solved the problem; you’ll have to continue further. In the end you are left with an infinite chain. And what does that mean, the infinite chain? Why is it problematic? It is problematic because you are not really offering me an explanation. Rather, you keep claiming that there is an explanation. I ask you what the explanation is—you say it’s over there. I get there, ask you what the explanation is—you say no, no, it’s over there. In other words, you are avoiding giving an explanation. Is there an explanation—is there a difference between space and time? I understand. If there is something infinite in time, not in space? One before this one, one before this one? I don’t think that’s… There are people here claiming there was some Big Bang, before that there was another one, before that another one. Yes, but that’s a different claim; I’ll get to it later. But the problem I’m raising—and today I’ll sharpen it more—is a problem with the concept of infinity. It’s not connected to time and space. When you talk about infinite time or infinite space, my problem is not with time or space. My problem is with the concept of infinity. When you use the concept of infinity in this particular way—and I’ll get to later how one can use it and how one cannot, you probably know that better than I do—that’s where I get stuck. Because basically when you use—I’ll say it already in the intuitionist language, as it’s called in philosophical thought—when you use the concept of infinity in a concrete sense, you haven’t said anything. When you use the concept of infinity in a potential sense, that’s fine. And it’s defined, at least mathematically. Defined. Whether it’s true or not true, but it’s a defined thing. You said something. When you use the concept of infinity in a concrete meaning, not a potential one, it is not defined. And basically that means you haven’t said anything, or you’ve avoided saying something. And now I want to sharpen this point a little more, because this is really the core of the cosmological argument. Right? Think about other explanations. Reuven slaps Shimon, okay? Shimon says to him, “Why are you hitting me?” So Reuven says, “What do you want? Because my hand flew onto your cheek.” So Shimon asks him, “And why did the hand fly?” So Reuven says, “Because the muscle activated it.” And Shimon doesn’t give up, he says, “And who activated the muscle?” So Reuven says, “An electric current in the nervous system.” So Shimon says, “And what created the current?” So Reuven says, “A voltage or a field in the brain.” Okay? Shimon says, “And who created the field in the brain?” So Reuven gets annoyed with him: “Tell me, do you want me to teach you all of human physiology on one foot? Go to the university, study there.” Good answer? Reuven slapped Shimon. Now, is that a good answer? Seemingly he described the chain exactly. Wonderful answer, no? Everything is fine—he explained exactly why he slapped him. But that is not an answer. Why is that not an answer? Because in the end there has to be, at least on Shimon’s assumption, some initial cause that is responsible for that whole chain. In this case, Reuven’s decision to slap Shimon. After he decided to slap Shimon, he activated a physiological chain—field in the brain, electric current, muscles, nerves, the flight of the hand, and a slap to Shimon. But why should I care about the whole chain in the middle? Tell me what the cause was, what the initial factor was that activated the whole chain. If you tell me that actually this was a deterministic matter all the way back to the Big Bang—yes, if you’re a determinist—then indeed there is no cause. Then Shimon’s question is irrelevant. Because Reuven did not decide to give him a slap; Reuven was operated by deterministic mechanisms and a slap came out. So then indeed there is no point in asking, “Why did you slap me?” But the conversation between Reuven and Shimon is a conversation based on the assumption that there is a cause. In other words, that the world is not deterministic. Or in other words, that somewhere in the middle of the deterministic chain, something was inserted, a stage that has no prior cause. It is an opening stage. And that is Reuven’s free choice, that he decided to slap Shimon. And that is exactly parallel to the move of the cosmological argument. Because to make the claim—to ignore Reuven’s decision and hang it on the Big Bang, which sent electrons flying and from here and here and here created Reuven, and Reuven was created in such a way that if something like this happens then his brain works this way, and his brain activates a current and the current activates the muscles and a slap is produced—that does not explain anything to anyone. It does not satisfy anyone because it is an infinite regress. And if you tell me that at the Big Bang it stops there—the Big Bang has no cause—then I will ask: what in the Big Bang caused this slap? You haven’t solved the problem. And in the end I am asking you: when you want to give me an explanation, you need to present the full explanatory chain. If you tell me, “No, no, continue onward to n plus one, n plus two, n plus three,” then you are not giving me an explanation. You are evading giving an explanation. That is not called explaining. Therefore infinite regress is a fallacy. Now I’ll come back to this in a moment. But deterministic. He’ll say that just as the Big Bang is really its own cause, that’s God. Meaning, just as you say God is His own cause, they too will say the Big Bang is the same thing. No, I’ll talk about that later too—that’s another issue. I haven’t yet made the analogy to the cosmological argument in order to challenge it, but to demonstrate why infinite regress is a problem. You’re only saying fine, the atheist can also stop the regress so that it won’t be infinite—just instead of putting God there, put the Big Bang there. Correct, that’s a different argument, and I’ll get to it in a moment. Okay. Very good—but in a minute. Okay. So before that I just want to read you some passage from Kant. I said that Kant was basically the one who divided the kinds of arguments or proofs, so in the chapter he devotes in the Critique of Pure Reason, the chapter he devotes to the cosmological argument, he says the following. Maybe I’ll share from my book here. Excellent. First. Here, this is a passage from Kant. There is here an inference that proceeds from the impossibility of an infinite series of causes given one after another in the world of the senses, and passes to a first cause. You see? That is the updated formulation I mentioned, because if the two alternatives are either that there is a first cause that itself needs no cause, or that the regress is infinite, then since that is so, once you eliminate the impossible you are left with the alternative. So Kant correctly describes the argument—the updated argument. But now he goes against it; he raises an objection. He says: this is a transition that the principles of the use of reason do not justify even within the world of experience, let alone justify extending it beyond experience, to a domain into which the causal chain cannot at all be extended. Now here, Kant’s claim—and he will return to this later regarding the principle of causality, we’ll get to that too—his claim is basically that this inference is an inference that perhaps, and even that is doubtful for him, but perhaps is valid for the world around us, the world of experience, but you cannot infer from it to things outside the world of experience. Now here I disagree for a reason. First reason: I am applying this inference to the world of experience, here within it. I am asking who created our world. My answer requires something outside the world of experience—God. But the question is a question that was asked about the world of experience, and the world of experience is a world that requires causes. Now you can say yes, but the thing outside our experience perhaps does not need a cause. Exactly—wonderful, that is what I’m claiming; that is why I say that is God. But what will you tell me? No—outside our experience you can speak about an infinite chain of causes, and something that is the cause—otherwise you’re back to what I’m saying. But you want to say that things outside our experience—I have no problem with infinite regress there. Infinite regress attacks us, or is problematic, only in our world of experience, the world accessible to the senses, to experience, to observation. Here I’ll answer what I answered Professor Tomkal before. That’s not correct. My problem has nothing at all to do with experience; it is with the concept of infinity. The concept of infinity, my claim is that this is really a problem in logic, not in physics or science. It’s a problem in logic. When you tell me there is an infinite chain of causes, you are basically evading giving me a cause or an explanation, not giving me an explanation. Therefore it is irrelevant whether it is applied to things in experience or things not in experience. I am basically claiming: you didn’t give me an explanation. Not because from my experience I learned that an infinite chain is not an explanation. It would be utterly absurd to say that. How could one learn from experience that an infinite chain is not an explanation? Did anyone ever try to give an infinite chain of explanation in experience? How did he finish presenting all the infinitely many stages of the chain? That takes a little time—I’m saying this of course sarcastically. In other words, you can never in life trace an infinite chain of explanation. So on the contrary—from the point of view of experience there is no obstacle to such a chain. And the fact that you haven’t encountered it until now is because you cannot encounter an infinite chain. I also never encountered Napoleon, because I didn’t live then. So what does that mean—is that a refutation of Napoleon’s existence? In other words, the fact that I didn’t encounter it in experience is obvious. I cannot encounter an infinite chain in experience. My problem—and this is an argument against Kant, not for Kant—this is an argument against him because it shows that if I treat an infinite chain as a fallacy, the problem is not because it does not fit my experience. The problem is in logic. Logic says that an infinite chain is a fallacy. And if that is so, then what difference does it make whether I apply this to things in experience or to things outside experience? Logic is supposed to be correct always. By the same token you could tell me: you are proving something by negation, but if it pertains to God or to something not in our experience, perhaps even though X is not true, not-X is also not true. How can you prove things by negation? After all, it is not in our experience. Nonsense. Proof by negation is a logical mechanism. A logical mechanism is valid for anything, not only for things in our experience. Therefore to limit it to our experience is simply a misunderstanding, in my opinion, of this principle. Now I’ll say more than that: even if an infinite chain were really possible outside our experience—let’s say theoretically, okay?—still, when I ask what is the simpler explanation: an infinite chain of non-physical causes, which are not from our experience—in other words an infinite chain of gods, each one creating the next, only there is no first one—versus saying there is a finite chain at the beginning of which stands some God who does not need a prior link in the chain. If you ask yourself which of the two is simpler, by Ockham’s razor it is obviously the second. So even if I were to accept the claim that outside experience one can speak about an infinite chain of causes and call all those causes gods, yes? Then an infinite chain of gods—is that the simpler alternative Kant is trying to offer to my thesis that there is only one God? That does not sound simpler, right? It sounds a little strange to offer such a thing as the simpler alternative. But the miss—and this I think is Kant’s fundamental miss in the Critique of Pure Reason—is that Kant tries to look for a necessary proof for the existence of God. And all the arguments he raises are arguments that show the proof is not necessary. But I want to claim that the claim of God’s existence—and I’ve spoken about this more than once—is, in my eyes, plausible, not necessary. It is enough for me to show that it is plausible, or more plausible than its opposite, in order to adopt it. I do not presume certainty. There is certainty in nothing. Therefore if I ask myself now at the level of plausibility—you may be right perhaps; even here I think you’re not right, but suppose you are right that it is not necessary, because there could be an infinite chain of causes outside the world of our experience, metaphysical causes, okay? Suppose that is possible, that it is something one can talk about. Still, when I ask: one metaphysical cause versus an infinite chain of metaphysical causes—which thesis is more plausible? Then the more plausible thesis is that there is one metaphysical link, or five, and not infinitely many. So in any case I do not see what is gained by limiting regress to experience. Okay. There’s a problem with this—the definition of plausible, I learned, depends on the person. There is— I don’t remember his name—the chance that there is a world is tiny because all kinds of constants have to line up exactly and so on. Anthropic, yes. Yes. And therefore they say there’s a multiverse, there are all kinds of excuses. Or that there is God is not plausible. Everyone defines what plausible is—for one person it’s not plausible and for another it is. Yes, I don’t know how to define the concept of plausible. That is basically the subject of the last column I wrote, about Ockham’s razor—that you choose the simplest theory. What is the simplest? Simple is a matter of opinion, meaning… So say in the mathematical context, a linear line is the simplest because it has the minimum number of parameters. An exponential also has a minimum number of parameters, it has one parameter. Never mind—even in mathematics you can argue about what is the simplest. But in this context I claim that this plausibility is a plausibility that I think most human beings, if they are not sucked into a theological debate, will agree with. Obviously an infinite chain of gods is less plausible than one God. Now you can always say no, no, in my eyes an infinite chain of gods is more plausible. If you are a stubborn atheist then fine, then you said it and stayed alive. But when you really ask yourself—forget arguments now and who wins the argument—when you think to yourself between these two options, I think it is quite clear that the option of a finite chain is simpler or shorter, more simple, than an infinite chain. That seems ABC to me. If someone argues with that, let him argue. So what can I do? Why exactly? Why exactly? Why is it more plausible? I don’t know—why? Because it is simpler. Why is a straight line simpler than a parabola? No, if the level is probabilities, say. No, no—I’m not defining probabilities. No, I’m not defining probabilities, I’m defining… Epistemologically, and then you can say why this is more probable than that, but just to say it’s more probable… I’m not talking about probabilities, I’m talking about plausibilities. Probabilities are when you have a sample space and you can count possibilities. Then the thing that has more possibilities is more probable. But here I have no sample space. I cannot define a probability over these possibilities, so I speak about plausibilities and not probabilities. And I thought that any straightforward person—forget it, I don’t know how to define it, I don’t know how to prove it—any straightforward person to whom you present an infinite explanation versus a finite explanation, and ask him which is simpler, will tell you the finite explanation. When will he not say that? When that would prove to him the existence of God. Once it proves to him the existence of God, suddenly he is open to the possibility that maybe the infinite explanation is no less simple or no less plausible. That, in my eyes, is dishonesty. But again, I have no way to argue with stubborn people. But this… Rabbi, I don’t understand. Beyond the fact that the rabbi says it is plausible, the rabbi doesn’t bring some proof, argument, or I don’t know, experiment… If I bring a proof, then about that proof too you’ll ask who says it’s right. No, just to say plausible… No, if the rabbi brings an experiment proving that probabilistically it is like this, then for me that’s a proof. No, that proves nothing. I’m not talking about probabilities. There are no probabilities now… But to say plausible is without proof; it’s just, like… No, for heaven’s sake, everything you say is without proof. When I bring you a proof, you’ll ask about the proof itself who says so. After all, in the end you start from assumptions, right? Now I ask: what determines which assumptions are correct? Plausibility. This seems plausible to me and that doesn’t seem plausible to me. How do you know that everything has a cause? Do you have proof for that? No, you don’t. No one does. And David Hume showed that there is none. And still I think we all agree that everything has a cause. Things don’t happen without a cause. Are you sure of that? No. But it is much more plausible than the claim that things happen without a cause. Right? On that we agree. We have no proof for it. Why? Because of intuition, common experience, I don’t know what to call it. Yes, no, rabbi, that I can prove to a person. That he holds that there is a cause—I can prove that to him. How will you prove it to him? What does prove mean? He will tell you no, things have no cause. Things happen without a cause. No, maybe, but I can prove to him that in his behavior, in his conduct, he relies on the… That’s exactly what I’m telling you: that in his behavior, in his conduct, he also thinks that a simple chain is more plausible than an infinite chain of metaphysical beings. In every other context of life he’ll say that. That the rabbi should prove it to him, but that the rabbi should prove to him—listen, in this behavior and that behavior you hold that the cause is of this kind and it doesn’t continue… We’re repeating ourselves. I’m not here to prove it to him; I don’t know him. Any plausible human being standing around you, if you ask him independent of faith / belief in God—just do an experiment, a psychology experiment, okay? Ask a person a question. Say that something happened to you—you saw some massive object falling to earth. You have one possible explanation, that there is a force of gravity, some abstract entity that pulls it downward, one option. A second option: an infinite chain of angels, one pulling the other, and that’s how the object came down. In your opinion, which explanation is simpler? I promise you that every atheist will say that the first explanation is simpler. Why? Because here God is not in the picture. You are not biased by the conclusion you arrive at. When you look at it objectively, every person will agree to this. Now if someone really disputes this, I have nothing to say to him. Fine, then he thinks demons or angels are something more plausible. But that is not really true; human beings do not think that way. So one can always insist: prove it to me, you didn’t define it, you didn’t prove it—correct, I didn’t prove it. In every other context that I would ask you, you would say it. It’s just stubbornness. Therefore I’m not coming to defeat the atheist; I’m speaking to myself. I ask myself: what seems more plausible to me, this option or that option? My answer to myself is that the option of a finite chain is more plausible. That’s all. If someone else says otherwise, then for him the argument is no good. Fine. I’m not interested; I’m speaking in relation to myself. Okay, so now I really get to the heart of the matter. What is the problem with infinite regress? To understand this a little better, I’ll try to speak a bit about the concept of infinity. I’m obviously not going to give mathematical definitions here, but I’ll try to give you some sense of the problematic nature of this concept and of the solution that mathematicians usually adopt, and I think philosophers adopt it too. So my goal is not mathematics but philosophy. There was a very beautiful passage that I still remember from my childhood. In my childhood I was a history enthusiast, with a rich historical library, and there was a book by Hendrik Willem van Loon. A very famous historian, I think Dutch. Yes, Dutch-American, yes. The Story of Mankind, that’s what it was called. At the beginning of the book, the motto of the book is the passage I borrowed here: “Far to the north in the land called Svithjod stands a rock one hundred leagues high and one hundred leagues wide. Once every thousand years a little bird comes to sharpen its beak on that rock. And when the rock has been worn away in this fashion,” yes, when it is finished completely, “one day of eternity will have passed.” Yes, a historian gives you some feeling for what it means to talk about infinite time. It’s an unimaginable amount of time until one bird, once every thousand years, sharpens its beak on a rock a thousand leagues by a thousand leagues. The rock will be completely finished, worn away entirely—the bird of course will already have been replaced by other birds—but it will wear away to the end, and one day of eternity will have passed. Meaning, the concept of eternity—that gives some sense of what it means to speak of eternity or infinity. Now here is an important point—pay attention. We are usually accustomed to think that infinity is a lot, that infinity is very, very much, the most there is. But that is an incorrect, imprecise intuition. Infinity is something else. It is not “a lot.” It is something different. It is like when I try to explain something three-dimensional by enlarging the concept of two-dimensionality. Yes, take a two-dimensional sheet of paper and make it very, very, very—put something on it, I don’t know, something terribly thick—will you get three-dimensionality? No. You won’t. You cannot build three dimensions out of two. It is a different conceptual world. It is not just a collection of lots and lots of two-dimensionality in order to get three-dimensionality. Okay? Meaning, infinity cannot be built out of many very finite things. Because “many very finite things” would have to be infinitely many finite things, not just many very finite things. But then once again we built infinity by means of infinity. You won’t get out of this. Meaning, there is no way to build infinity by means of things that are not infinite. Okay? Is that what van Loon did? What? Van Loon? I’ll explain in a moment why that’s not exactly what he did. Though fine, he was a historian, not a mathematician. But as I read it at least, that’s not exactly what he did. Because I’ll explain in a moment in what sense one can talk about infinity. Yes, we can talk about infinity, but carefully. So there is basically a claim here that infinity does not mean very, very much. Okay, so what does it mean? Let’s look, for example, at proof by induction. I mentioned earlier intuitionism in mathematics. In the philosophy of mathematics—or even among mathematicians—there are intuitionists, finitists, intuitionists, there are all kinds of schools in mathematical thought. One of the differences between them is in how you treat proof by induction. Proof by induction, of course, is not scientific induction; it is mathematical induction. Mathematical induction is not really a generalization. Scientific induction is a generalization—from particular cases to a rule—which may be right and may not. Mathematical induction is necessarily true; it is not a generalization in the scientific sense. What does it mean? I proved the property I want to prove for n equals 1, for n equals 2, and I also proved that if it is true for some n, then it is also true for the n after it in general. For most mathematicians that suffices as a proof. This is called proof by induction. Why? Because you proved it for n equals 1, 2, 3—it doesn’t matter, 1 is enough—and you proved that if it is true for some n, it is true for the next n. So basically, if it is true for 1, it is true for 2. If it is true for 2, then it is also true for 3. True for 3, then true for 4. You can continue all the time to infinity. Okay? Therefore basically you proved it for every number whatsoever. But the intuitionists basically say that such a proof—some of the intuitionists, the finitists—basically say that such a proof is problematic because it did not prove it for all the numbers, and this is a subtle point. Rather, what did it do? It did not prove it for infinitely many numbers; it proved it for every number you like. Do you understand the difference in formulation? Meaning, any number you give me—say a number with, I don’t know, fifteen digits, write it down on paper—I can go step by step by step in induction and reach it, right? So for it I can prove it—for every number you like, of any size whatsoever—but that is a different statement from saying I proved it for all the numbers or for infinitely many numbers. I proved it for every individual number you give me. In that I did not say how many numbers there are. Right? I did not say something positive about the quantity of numbers. I did not really refer to the concept of infinity—notice. I am not referring to the concept of infinity. I am not talking about how many numbers there are. I am just making the following claim: for every number you give me, this statement is true of it. That’s all. How many numbers there are—that’s your problem. I don’t want to talk about it. Meaning, I do not need to speak about the concept of infinity. So this is the mathematical technique for dealing with the inability to speak about the concept of infinity. I speak about every number you like. And of course I’m saying this very simplistically, but I’m just trying to illustrate the points. So I speak about every finite number you like. But every number I speak about is finite. I never uttered a word here about infinity. Every finite number you give me, I can prove this property about it. Okay? That’s all. So I spoke only about finite numbers. Now you can ask yourself how many such numbers there are. That is another question; I do not want to talk about it, because I don’t know how to talk about it. Infinity is a word I do not understand. Okay? This is a way of avoiding speaking in terms of the concept of infinity. In other processes people speak about limit processes, yes? Or something like that. Suppose, for example, you tell me the series one-half plus one-quarter plus one-eighth plus one-sixteenth and so on—what is its sum? Usually people are used to saying 1. But mathematicians do not formulate it that way. The sum is not 1; rather, the sum is as close as you like to 1. I can show you that no number less than 1 is the sum. How? I simply add enough terms until I pass that number. But I will never say that the sum is 1. In mathematics, of course, one formulates it by saying that the limit of that series, the limit of the sum, is 1. But that is only an expression. I am not talking about the 1 itself; I am talking about the way toward the 1. This difference is the difference between a potential treatment of infinity and a concrete treatment of infinity. There is a difference between speaking about infinity itself and seeing it as some potential limit that I only approach, or that is somehow there in the background, but I never actually say anything about it. The second way is mathematically well-defined; the first way generates problems. When you speak about infinity in a concrete sense, you run into many problems. Therefore the accepted way in mathematics to get around this—and mathematics is simply the domain that deals with this, but the same in philosophy and every field—the correct way to speak about it, in a way that is well-defined, is to speak about infinity in a potential sense, as some limit that I can only keep approaching, not speak about what happens there. I mentioned, for example, Duties of the Heart, which says that it is impossible that the world has existed for infinite time, as Aristotle said—that the world is eternal. Why? Because if you start from that first point there and progress, you will never reach our time. Yes, this is a very common argument among ancient philosophers, but of course it uses undefined concepts. You cannot start from minus infinity and begin moving to the right. That is an undefined process. One indication that it is undefined is that I will ask you: tell me, after how much time will you be at x = -10? You started at x = -infinity and you advance at a rate of one unit per second. You move one unit on the x-axis per second. Fine? Now I ask you: when will you be at x = -347.5? When will you be at x = +pi? You cannot give me any time. It is an undefined process. You will never be at any point. You are always at minus infinity, if one can even speak that way, and that’s it. You are always there and at no point will you ever be. Or in other words, there is no such thing as minus infinity. You are there and will never emerge onto the number line. You will never be on the number line. So you are just talking. It is not really a defined process of walking on the number line. This is unlike the situation where I start from x = 0 and advance left or right—it doesn’t matter. I start from a defined point, not infinity or minus infinity, and I proceed toward infinity or toward minus infinity. That is a well-defined process. If I advance at a rate of one unit per second and I started at x = 0 and turned right and advance one unit per second, every point on the x-axis that you give me, I can tell you at what time I will get there. No problem at all—a perfectly well-defined process. I traverse the whole number line in a fully defined process. There is one thing I cannot tell you: when I will get to x = infinity. Why? Because there is no such thing as x = infinity. I have no problem with that not being defined. There is no such thing—there is nowhere to arrive; it is a fiction. Every x, however large you want, I can tell you when I’ll be there. The only thing I cannot tell you is when… By contrast, in the reverse process of Duties of the Heart, you can tell me when you were at x = -infinity, but about no other x will you be able to tell me when you were there. Or in other words, you cannot speak about that process as a process of moving along the x-axis. You can’t; it is not really a description of a path on the x-axis. It is just a fiction. It is meaningless, undefined words. There is a big difference between going from minus infinity and walking rightward, and going from 0 and walking leftward toward minus infinity. Because when you go from 0 and walk leftward, for every x you give me I will tell you exactly when I will be there. The process is well-defined throughout. It just never reaches its end. Fine, so it doesn’t reach it—so what? I also do not commit myself as to whether it has an end and where that end lies. What I tell you is that for every x you give me, I will tell you at what time I’ll be there. The path is completely defined. But the reverse process is not defined at all. Even though apparently you are talking about the same space between x = 0 and x = -infinity, the only question is whether you are going right or left. So what’s the difference? A very big difference. You cannot go rightward from minus infinity. It is simply an undefined process. That symmetry is an illusion. An illusion. We think one can speak about both this and that, and maybe one is just less clear. No—it is simply undefined speech. You can speak only in the first way, not in the second. And this is a very important point that people often miss. Look—all the medieval philosophers who dealt with this, like Duties of the Heart, miss it. They bring a proof: we would never have arrived here if the world were eternal. I don’t know what. When you speak about an eternal world, you are speaking about it looking backward. You are saying: the world has no beginning point. That can certainly be said. And the argument of Duties of the Heart does not attack that statement. I am claiming that the world has no beginning point. That can be said. It is a logically coherent statement. Whether true or not can be discussed, but there is no problem; it is defined. It goes from now backward. And Duties of the Heart wants to reverse the picture, to go from there forward. You cannot reverse the picture in such a case. Because the x-axis is not really defined on both sides. The x-axis is defined only from one side, and as far left as you want. That’s all. As far as you want. They don’t tell you where you’ll arrive, because you don’t arrive anywhere. Therefore infinity is not very, very, very much—more than every “a lot” you ever thought of. No. Infinity is something else, not defined. The nicest example, of course, for describing these tangles is Hilbert’s hotel. If you want, you can see it on Wikipedia; it has an entry. Hilbert proposed this example to show the illusions of the concept of infinity. Think of a hotel that has infinitely many rooms, okay? They are numbered by the natural numbers: 1, 2, 3, 4, to infinity. A hotel built here, with infinitely many numbered rooms. Okay? You can already see that this is a concrete treatment, not a potential one. Now I say all the rooms are occupied. Now one guest arrives. Sorry, no room. The rooms are occupied. So he says what? Why? What’s the problem? Take the person in room n and move him to n+1. Every person who was in a particular room—I can tell you in what room he is; it is a well-defined process. Room number 1 will remain vacant, and I’ll enter it. No problem at all. An infinite hotel in which all the rooms are occupied—the rooms are not occupied. You can bring in one more person. Everything is fine. Now I say: a bus arrives with infinitely, infinitely many passengers. They want to enter my occupied hotel. The whole hotel is occupied. What do we do in such a case? Hilbert says there is no problem at all. Every person in room n moves to room 2n. Moves to room 2n. The rooms in between, the odd-numbered ones, I assign to the guests from the infinite bus. You say there are infinitely many buses, infinitely numbered—yes, countably infinite. Infinitely numbered buses, each bus containing a countably infinite number of people. No problem at all. Move every person in room n to 2n. All the odd-numbered rooms remain empty. Put the first bus into the powers of the first prime, powers of 3. Put the second bus into the powers of 5, the next prime. Put the third bus into the powers of 7. There is a theorem that there are infinitely many prime numbers. All the powers of a prime number are odd numbers, and none of those numbers overlaps with a power of another prime. Okay? Therefore I choose the primes. And if that is so, I can fit infinitely many buses, each with infinitely many people, into the hotel that is completely full. Into the hotel that is completely full. Now. Now I do a variation. Okay, I return to my full hotel. A person arrives looking for a room. Now I want to do this process concretely, not speak mathematics in general. How do you do it? I want to announce over the loudspeaker: guest number this, move to room that; guest… yes? That’s what I want to do. Impossible. It is impossible. Move the guest from room number 3—to what room? After all, all the rooms are occupied. To what room can you move him? There is no room you can free up. You can say this theoretically, but you cannot actually carry it out concretely. This is another illustration of the point that you may perhaps talk about the concept of infinity when you speak about it in a general way, that it is as large as you wish, but once you try to do something concrete with it, you cannot. Therefore talk about the concept of infinity is very… Think about it, yes, like Duties of the Heart—one can formulate it differently. Suppose there is an infinite chain of rooms, okay? From minus infinity to minus 1, from 1 to infinity, okay? All the rooms are occupied. Now I say: room infinity, move to room infinity plus one. What am I going to tell him? Room infinity plus one is occupied too. All the rooms are occupied. How can I free a particular room for the new guest? There is no practical way to do this, because you cannot treat it as though before you there is a concrete chain of infinitely many occupied rooms. It is a mistake to view it as a concrete chain. Not that there really are infinitely many rooms that I just don’t know how to count but they’re here somehow and each one has a person living in it and now I can start moving them around. That is exactly the proof that this is the wrong way to look at it. There is no such thing as a hotel with infinitely many rooms. It is not that there is such a thing but I just cannot count it. No, there is no such thing. A hotel with infinitely many rooms is undefined speech. A hotel with as many rooms as you like—that is defined speech. As many as you like. What does that mean? Every number you give me, the number of rooms in this hotel is larger. That is well-defined. But to say that it has infinitely many rooms in a positive sense, in a concrete sense, not a potential one—that is impossible. That is basically the meaning of infinity. And therefore the lesson I really want to learn from here is this: when we use the concept of infinity, we have to use it carefully. Mathematicians eat this lesson for breakfast, but people who are not used to this kind of thinking are not aware of it. You really have to use it very carefully. I’m not even talking now about all of Hilbert’s tricks, yes? Which are more numerous, the natural numbers or the even numbers? The same number. Even though the even numbers are a subset of the naturals. Naturals, yes, are 1, 2, 3, all the integers to infinity. And the evens are 2, 4, 6, 8. A subset. Same number. You can make a one-to-one onto correspondence between the two sets. Same number. This reflects the same problems. You cannot speak about infinite concepts in a concrete sense, only in a potential sense. Now, why is this important? I don’t want to get into mathematics, but there are several kinds of infinity—the continuum, the real numbers and the integers. Hilbert’s classification of infinities, the infinite cardinals, is one of the refutations people raise against the intuitionists. Because they show that in Hilbert’s theory one can speak about concrete infinity and say things about it, not only in the sense of a potential limit. I think that is incorrect. In the philosophical sense—I’m not a mathematician—in the philosophical sense that is not a valid claim. In my opinion, Hilbert too is speaking about potential infinities, not concrete ones. He is only saying that the infinite limit of the naturals and of the reals is not the same size. But these are still concepts of limits; they are not concrete concepts. He is not looking at a box in which all the natural numbers are sitting. You cannot look at it that way even in Hilbert’s theory. Fine, but that is really an aside, not for here. In any case, for our purposes, what I really want to say is this. I return to the chain, to infinite regress. Philosophers—when you ask them why infinite regress is a fallacy—you won’t get an answer. I tried, by the way; you won’t get an answer. They don’t really know how to say what it is. It is just obvious, as it were, that infinite regress is not legitimate in philosophical discussion. It seems to me that what lies behind it is this issue. When you want to offer an explanation in the form of an infinite chain, that basically means to me that there is a concrete infinity here, not a potential one. Because in order to give me the explanation, you have to start at minus infinity and come up until you get to the world, yes? Start from the turtle at the bottom, the turtle above it and above it and above it, and the world stands on the top turtle. That is to start from minus infinity and reach the world. An undefined process. You are basically assuming there is concretely a chain of infinitely many turtles. You cannot say such a thing; it is undefined. Meaning, the logic here does not work. Infinite regress is not a possible explanation. It is like saying that one plus one is five. It is simply not true; it is undefined; you haven’t said anything. By contrast, if you tell me I am looking backward from the world. The world sits on a turtle, and that turtle sits on another turtle. That is a well-defined process. Maybe it is right, maybe not, but it is well-defined. It is like going backward from zero toward minus infinity. That can be defined. But it is not an explanation. When you explain a claim X, you have to tell me the whole thing: start from assumptions, derive conclusions from them, from the conclusions derive more conclusions, yes, like geometry, and in the end arrive at the final conclusion, at the statement you want to prove. That is an explanation. To say that this statement is based on a prior statement, and when I ask you what about that one, you say ah, it too is based on a prior statement, and what about that one? Also prior. That is not an explanation. So you did not present an explanation to me; you evaded giving one. It would be like someone thinking of building a geometry that starts from the fact that the sum of the angles in a triangle is 180 degrees. You ask yourself what the explanation for that is. The explanation is the axioms. What is the explanation for the axioms? Prior axioms. And what is the explanation for them? More prior ones. In the end you did not answer me with an explanation. To give me an explanation, you need to begin from the axioms and go forward. Not begin from the claim and go backward. But in order to begin from the axioms and go forward, you need to begin from certain axioms. Not from axioms at the end of the chain at minus infinity. That is nonsense; there are no such axioms there. There is no bottom in the chain of turtles all the way down—there is no “down.” Therefore this statement that I am offering the alternative of an infinite explanatory chain is problematic because it is not defined, not because it is false. It is undefined. There is no explanation; you have not presented an explanation. It is not an alternative at all. There are not two alternatives here. Again, you can say I have no explanation. But you cannot say I do have an explanation—maybe an infinite chain. Now, if you say I have no explanation, then here I will tell you one of two things. If you have one option that gives you an explanation for something, and the second alternative is maybe there is no explanation, then obviously the first option is preferable. I appointed a commission of inquiry to investigate why a plane crashed, and they found that there was a crack in the wing. Now someone proposes a hypothesis, yes? They ruled out all the other possibilities about the wing; the wing burned up. You can’t see whether there was a crack in it or not. The conclusion of the committee was: apparently there was a crack in the wing. Then someone says: no way. Maybe there is no explanation; the plane crashed without explanation. Maybe that is also an option. Would you accept such an option? No. If I have an option that gives an explanation and an option that gives no explanation, I obviously prefer the option with the explanation. Same thing here. If the meaning of the infinite chain is “I have no explanation,” then certainly—even if I accepted such a thing—it is certainly not a preferable alternative to the alternative that says I do have an explanation. And beyond that I want to claim: once I say I have a finite chain and that is the explanation, the fact that the second alternative does not hold water means I have a proof of the first explanation. You can’t say what do you mean, I have no explanation? Either there was a cause or there wasn’t. This is the law of the excluded middle; it’s logic. I have no explanation is a philosophical claim, but I ask you in logic: was there a first cause? You say no. Ah, there was no first cause? Then infinite regress—which is undefined. So where is the claim “I have no explanation” located? Nowhere. It cannot be located. Either there is a first cause in the chain or there isn’t. That is the law of the excluded middle. Either yes or no. There is no third possibility. So to say “I have no explanation” is a meaningless statement. I have proved to you by negation that there is a first cause. Now of course there is still room here to argue, yes, but who says everything needs a cause? Maybe it is not true that everything needs a cause. That is a claim—at least one can make it. But I ask the person honestly: tell me fairly, do you really think that if I offer you a cause for the existence of something and the alternative is that there was no cause, that this alternative is on a par with the first option? If yes, I have nothing to say to you. This argument is not convincing for you. I don’t really believe you think that. Because we usually assume that things happen for reasons. Things do not happen without a cause. But here indeed there is something: there is an alternative, and it is defined, that what happened happened without a cause. Only I think every sensible person who is not infected and does not skew the results just in order not to surrender to the claim that there is God will say: of course I prefer the option that gives a cause over the claim that there is no cause. Notice, I’m no longer speaking about infinite regress; I’m speaking about cause. Now here I’ll conclude perhaps with another claim of Kant that says the same as what he said earlier, also about the concept of cause. Fine, I won’t read it now; I’ll say it by heart. Kant wants to argue as well, not only against infinite regress—what he argued in the previous passage—but also against the assumption that everything must have a cause. He also says that applies only to things in our experience. Who says things outside our experience also need a cause? And here I think this objection is strange. Why? First of all because the principle of causality is not a principle learned from experience, as Kant himself proved—he himself wrote this. He received it from David Hume and accepted it. Therefore it is clear that the principle that assumes everything must have a cause is not a principle that is the result of experience. So why assume it exists only for things in our experience? In principle you can say that; I’m only saying it is not plausible to say it. But I say beyond that: this is exactly what I am claiming. It is not a refutation—it is my argument. The things in our experience need to have a cause. The cause itself is either among the things in our experience, and then it too needs a cause, or it is not among the things in our experience, and then indeed it has no cause because it is necessary—they call it self-caused. So it has no cause, and that is God. Kant’s objection is not an objection; it is the argument. That is how I opened today’s class—when I said that what the atheist objects to in the primitive formulation of the proof, his objection that infinite regress is possible or that not everything has a cause, only things in our experience—that is itself the basis for the updated formulation of the argument. On the contrary, that is exactly what I am saying: if you say everything needs a cause, you get trapped in an infinite regress. Perforce you must exempt things that are not in our experience—certain things not in our experience—but not everything not in our experience. There are things not in our experience that have no cause. That is what I call God. So in what sense is Kant’s claim an objection? It is not an objection; it is my claim. Every thing from our world that is in our experience needs to have a cause. The cause itself perhaps also needs a cause, but at some point I will get to a place that is no longer in our experience, and then indeed we are talking about a being that has no cause. And there I will stop—that is God. Therefore many times objections to this argument stem from misunderstanding. On the contrary, the objection itself is what builds the correct argument. Maybe one more sentence about “self-caused,” when people speak about God as self-caused. The concept of self-caused is a medieval concept. Rabbi, about the previous point? One second. It is a medieval concept, and of course these are words without meaning. A thing cannot be the cause of itself. A thing that is the cause of itself means it created itself, but if it created itself then it existed before it existed. Who created it? You cannot speak about self-caused. Therefore I don’t like the term self-caused. I speak about a being that does not need a cause outside itself. Whose existence is an inherent necessity. It does not need a cause outside itself. The concept of self-caused is not a successful concept of medieval philosophers. Okay yes, I’m done for today. Can I ask, comment? Yes. Regarding what the rabbi said—why realities are not in our experience? What the rabbi spoke about, realities that are not in our experience—but in my opinion, what is there to say there about whether there is a cause or there isn’t? We don’t know what happens there. So all possibilities are equally open, equally incorrect. One thing we do know: either there is a cause or there isn’t. There is no third possibility. Not in places where we have no experience… Even in places where we have no experience. Either there is a cause or there isn’t. There is no third possibility. That is logic. Not connected to experience. Yes, but wait a second. We said either there is a cause, and then I ask what the cause of the cause is, and if there is no cause, I’ve arrived at God. That’s all. Wait a second. The rabbi says that even in places where we have no experience, we have knowledge? No, we have no knowledge; we have logic. Either it is true or it is not true. Logic always exists. You know, in the introduction of the Maharal to Gevurot Hashem he speaks about this. Why is a sage preferable to a prophet? And really, translated into my language, let’s say this: a prophet is a kind of observation, yes, he sees things in spiritual worlds that we do not see. But his range is limited. Every prophet can see only up to a certain spiritual height. The Ari even ranked the heights of the prophets, each one according to which world he reached. But the sage, when he says a logical principle, a rational principle, it is always true all the way up. One plus one equals two everywhere, not only in the world of our experience. If a thing is either true or not true, that is everywhere, not only in the world of our experience. Wisdom reaches higher than observation, than prophecy. Because wisdom is not connected to our experience; it is a superiority that is true in every context and in every place. A small question about the series of one-half plus one-quarter plus one-eighth and so on—I didn’t understand where it is supposed to get. It gets as close to one as you like. No, you said the definition is that it must be more than one. No, less. No matter how many terms you add, you never touch one itself. One itself is the limit after you add infinitely many terms, supposedly—but that is just words. The precise mathematical implementation is that you can get as close to one as you like and never pass it. And how does that work logically? If we keep adding forever, it won’t arrive. That’s the paradox of Achilles and the tortoise. The paradox of Achilles and the tortoise says that Achilles runs ten times faster than the tortoise, say, okay? Now he gives the tortoise a head start of ten meters and races him. Now when Achilles, after one second, makes up the ten-meter lead, the tortoise has already moved another meter forward, right? Another tenth of a second, Achilles covers that meter, but the tortoise is already ten centimeters ahead. When Achilles gets to those ten centimeters, the tortoise has already moved another centimeter ahead. Achilles will never catch the tortoise. Where is the mistake here? The mistake is that the whole race I just described is one and one-ninth seconds of the race. One second plus one-tenth of a second plus one-hundredth of a second plus one-thousandth of a second—that is 1.1111111 seconds of the race. And indeed Achilles will catch the tortoise at one and one-ninth seconds, but from then on he overtakes it. There are infinite series that converge to a finite sum. The fact that you are summing infinitely many terms does not mean the sum must necessarily be infinite. That was Zeno’s mistake; that’s why he thought Achilles and the tortoise was a paradox, because he thought that if you sum infinitely many terms, the sum must be infinite. Not true. You can sum infinitely many terms and get a finite sum. Good. Okay. I also wanted, regarding probability and plausibility—we won’t get into it now—it is true that there is a difference between them, but logically I think there is no difference. Logically no difference. What am I? I’m doing sudoku, I need to decide what is more plausible, that this number goes here or here. Since statistically I have more numbers here, I say here there’s a sixty-percent chance and forty percent there, so I go with the sixty. But I know in everyday life that the true result is sometimes in the forty percent. It’s not according to plausibility but according to the… Up until now you beautifully described the concept of probability. Now I ask: what do you want to compare it to? So I say that plausibility also understands that even the less plausible option will also sometimes occur. Fine, that’s my claim. But my claim is that plausibility and probability are two different things. There are times in the world when we use considerations of plausibility and not considerations of probability, and still, just as you said, I still think the less plausible option is less correct. I don’t have a probabilistic calculation showing that, but still it is less correct. Right, but still there is certainly the possibility that the less plausible one is the correct one. Right, both in probability and in plausibility. Obviously. That’s why I speak about plausibility and not certainty. Okay. Good. Sabbath peace, goodbye. Sabbath peace, thank you very much.