Learning from Experience – Lesson 4
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Table of Contents
- Synthetic maturation, intuition, and control
- Francis Bacon and the critique of “from facts to theory”
- Dividing the world of facts into scientific domains
- Borges, idealism, and the arbitrariness of grouping phenomena
- Leibniz, the identity of indiscernibles, and the critique of “object = set of properties”
- Infinitely many theories for the same set of facts and the need for Occam’s razor
- Critique of Wikipedia, Popper, and the methodological interpretation of Occam’s razor
- Actualism, informativism, and a statistical decision in favor of the essential status of the razor
- Intuition, learning, Chomsky, and Wittgenstein
- Questions at the end: evolution and Popper
Summary
General overview
The series presents a consistent claim that science does not advance from “neutral” observations to theory, but from the very beginning requires assumptions, intuitions, and rational rules that guide the collection of facts, their division into domains, and the choice between competing theories. The presentation uses Francis Bacon as a model of naive empiricism: “collect facts and then find a theory,” and rejects that model through three cumulative arguments: you cannot choose relevant facts without some theoretical idea, you cannot define scientific domains without some theoretical idea, and even after you are given a set of facts there are infinitely many fitting theories, so a principle like Occam’s razor is required. From this it is argued that Occam’s razor is an essential principle of correctness, not merely a methodological recommendation, and that there is a statistical way to show that the actualist-Popperian position is mistaken.
Synthetic maturation, intuition, and control
The synthetic adult advances within a framework similar to a child who holds onto an example, but does not accept it as certainty; instead he relies on intuition while understanding that control, cross-checking, and elimination are needed. The approach highlights that progress is not blind adherence to what one has been told, but the use of tools that reduce error even when there is no certainty.
Francis Bacon and the critique of “from facts to theory”
The presentation attributes to Bacon, and to a view still common today, the idea that science begins by collecting facts and only afterward formulates a theory that explains them. The claim is that you cannot even determine which facts to look at without some prior idea of what the theory might be, because there are infinitely many facts and you cannot chase all of them. The examples of Napoleon at Waterloo and Semmelweis and childbed fever illustrate that one begins with an abstract, amorphous idea, uses it to choose facts that may be relevant, improves the theory in light of the facts, and goes back and forth in a movement of “running and returning” between theory and facts.
Dividing the world of facts into scientific domains
The division into physics, chemistry, and biology, and even the internal divisions within physics such as gravitation, electromagnetism, and thermodynamics, cannot be derived from “facts” alone. The example of Newton shows that tides, falling bodies, and planetary orbits would not have been seen as belonging to the same domain before the unifying idea of attraction between bodies. The presentation stresses that from a given number of facts one can build a huge number of subsets—two to the thousandth power for one thousand facts—and therefore, without theoretical direction, there is no way to “sniff out” which facts belong together in one domain. The division itself emerges in a process of correcting the theory and swapping facts in and out of the group.
Borges, idealism, and the arbitrariness of grouping phenomena
Borges’s story “Tlön, Uqbar, Orbis Tertius” is brought as an analogy: in an idealist world where there are no objects but only phenomena, one can invent nouns at will for any arbitrary collection of phenomena, so that every division is possible and there is no natural criterion that groups items into the “right” subsets. The analogy is transferred to scientific facts: without a unifying idea of a law or theory, every subset of facts can be a “domain,” and inquiry gets stuck in endless divisions without progress.
Leibniz, the identity of indiscernibles, and the critique of “object = set of properties”
Leibniz’s principle of the identity of indiscernibles is presented together with his proof: if two objects have the same set of properties, then they are the same object, because each one has the property of “not being the other.” The presentation argues that Leibniz is mistaken because he assumes that an object is nothing but the set of its properties, whereas one may assume an object as the bearer of properties, not reducible to the totality of those properties. The discussion continues with the claim that logically there is no obstacle to two entities with identical properties being two distinct things, and a physical illustration is brought through the distinction between fermions and bosons, including the claim that in bosonic systems and in Bose-Einstein condensation identical particles can appear in the same state.
Infinitely many theories for the same set of facts and the need for Occam’s razor
Even after choosing relevant facts and assuming that they belong to one law, there are still infinitely many theories that fit them. The demonstration is given through a graph of force versus acceleration: five measurement points can fit both a straight line, as in Newton’s second law, and many other curves, so “the line” is the theory and the points are the facts, and there is no unambiguous move from the points to the line. Choosing the straight line is presented as a decision based on simplicity—that is, on a non-observational but a priori principle of reason—and that choice may later be overturned if further measurements refute it.
Critique of Wikipedia, Popper, and the methodological interpretation of Occam’s razor
A quotation is brought from Hebrew Wikipedia on Occam’s razor, including the claim that one should not see it as a rule of truth but as a pragmatic recommendation, supposedly helping us choose between theories that are “correct for the time being” rather than between truth and falsehood, relying on Karl Popper and the idea of falsification. The presentation states that the number of mistakes there is “astonishing,” and that this is a common but mistaken view because it misses the role of intuition and the possibility that Occam’s razor is an essential principle.
Actualism, informativism, and a statistical decision in favor of the essential status of the razor
A distinction is presented, from Ze’ev Bechler, between actualism and informativism: actualism identifies truth with the facts actually measured and sees theory only as an organizing tool, whereas informativism sees theory as carrying true information about the world. The presentation argues that Occam’s razor is a criterion of correctness, not only of efficiency, and that there is a scientific-statistical decision between the positions: if simplicity is unrelated to correctness, then the chance that the simple theory’s prediction will succeed in the next experiment is effectively zero relative to the infinity of possibilities; but in practice science does advance and produces a substantial percentage of successful predictions, so actualism is mistaken. From this it is argued that the intuition that “the simpler is also the correct one” is a tool with real validity in research, even if not always certain, and that without it science could not progress at all.
Intuition, learning, Chomsky, and Wittgenstein
It is argued that intuitions cannot be explained merely as the product of accumulated experience, because experience itself relies on prior intuitions, and if we were a “tabula rasa” we could not even begin to sort facts, divide domains, or choose a theory. Chomsky’s argument about language acquisition is brought in, according to which there must be an embedded linguistic capacity that allows generalization beyond examples; otherwise one could derive infinitely many “languages” from any set of examples. Also brought in is Wittgenstein’s motif of “following a rule,” through the example of a number series—3, 5, 7—that has no single necessary continuation, along with the connection to psychometric tests, where the “correct answer” reflects convention and perceived simplicity rather than one uniquely necessary truth.
Questions at the end: evolution and Popper
The possibility is raised that intuition is a product of evolution and therefore adapted to reality, and this is accepted as an open possibility “for the sake of the discussion.” It is said that the example of a straight line is only an illustration, and that the same principle applies even when the “simple” option is a parabola or a sine wave. The fundamental line of argument is set against Popper: Popper presents a theory as one that has merely “not been refuted,” whereas here it is argued that the simple theory is not on a par with all the others and is not “a shot in the dark,” but has real priority backed by the predictive successes of science.
Full Transcript
[Rabbi Michael Abraham] Okay, we’re in learning from experience, the series on learning from experience. I’ll remind you where we stopped last time. I was really talking about the way one can progress within a framework, or within the synthetic conception—the synthetic maturation on the map I sketched out—and what the difference is between that and the child. The child kind of proceeds dogmatically: whatever he is told, he accepts without proof; that’s exactly what the adolescent rebukes him for. But the synthetic adult, apparently, does the same thing. And then I explained that no, he uses intuition, but he also understands that it’s not certain, and therefore some kind of control is needed—cross-checking, elimination. And then we started discussing Francis Bacon’s scientific logic, and the claim was basically that there is some system of logical rules that forms a framework for scientific thinking. This is not deductive logic, which belongs to the mathematical-philosophical branch, but rather induction, elimination, tools of that kind that serve us in the scientific context. But in Francis Bacon’s picture of the development of science, it really could give the impression—and that’s how he himself thought, and as I said, many still think this way today—that science works by first collecting facts and then trying to find a theory that explains them. In other words, it goes from facts to theory. Last time I tried to show that this cannot be correct, because without having at least some preliminary idea of what the theory might be, we have no way to determine which facts to look at. I can—there are infinitely many facts. We spoke about Napoleon in some battle or another that he lost, say at Waterloo, and the question was how to explain the victory of the British—or the English—and the Prussians. And I said that one could look at countless facts, so you simply can’t proceed that way. How do I test, or how do I decide, which facts to focus on, which facts could be relevant—even if later they turn out not to be, they at least could be relevant—and which facts I can ignore right from the start, without even trying to check whether they are relevant or not? There is no way to do that unless you have some idea of what the theory might be. Once you understand what the theory could be, you understand which facts could be relevant and which facts are clearly not relevant. Sometimes you’ll make a mistake, but in general it’s an important instrument. Because if you can’t sort out some subset from among the infinite facts, then you’ll spend your whole life chasing facts—infinitely many facts. You have to know which facts to focus on in order at least to reduce the group of relevant facts so that you can move forward, and then later make corrections. So through this I showed, both historically and scientifically, with the example of Semmelweis and childbed fever, those two examples—which really point to exactly the same thing, even though neither one knows about the other—that in both of them you see that the movement is not from facts to theory. We begin with an abstract, amorphous idea of what the theory might be, approach the collection of facts when we already know how to isolate which facts might be relevant and which might not, look at those facts, check which of them really has an effect and which does not, or what can explain things, go back to the theory, revise it, go back again to the facts, to the theory, to the facts, to the theory—and that is really the scientific movement, constantly trying to improve the theory more and more through further attempts, through collecting further facts. But the movement is definitely not from facts to theory; rather, it is a back-and-forth between facts and theory, where the beginning is actually in theory and not in facts. So that’s what we did last time. Now I want to add two more angles, so that altogether we’ll have three angles describing the same phenomenon. The first angle, as I said before: we have infinitely many facts—how do we choose which facts to focus on in order to build our theory, in order to find the explanation we’re looking for? That’s what I did last time. Now I want to ask the question: how do I divide the world of facts into different scientific domains? And the next question will be: given that I have a particular set of facts, there are still infinitely many theories that can explain it—how do I sift out the correct theory? Occam’s razor, basically. So these three things are really supposed to show us—I’m just trying to show the context so we don’t lose our way here—these three arguments try to show us that facts are not a neutral thing. They are not something theory-independent, something you begin with and then move on to theory. No. Theory is involved within the discussion of the facts themselves. So we saw that before in the question of how I select the relevant group from among all the facts. Now I’ll ask a similar question, but a bit different. When we approach the attempt to examine the boundaries between different scientific domains, we can ask ourselves how this demarcation arises at all. How do I know that physics is one domain, biology is one domain, chemistry is one domain—or even within physics, how do I know that this particular collection of phenomena belongs to one law, say gravitation, and this collection belongs to another law, electromagnetism, and this collection to thermodynamics, and so on? In other words, even within physics there is a division into domains, and each domain deals with a certain collection of facts. Now how can I know how to divide the collection of facts into those subsets, each of which constitutes a domain? You have to understand that this question is very similar to the previous one. Just think for a moment, for example, about Newton when he formulated the law of gravitation. Newton’s law of gravitation explains, among other things, the phenomenon of tides, the falling of bodies to the earth, and the paths of the stars—the motions of celestial bodies. Now before we know the theory of gravitation, nobody would have thought that these three phenomena belong to the same domain or are explained by the same law. Right? What connection is there between tides and the paths of the stars—those ellipses, or the circles with epicycles—and the falling of bodies to the earth? What connection is there between these phenomena? You would assign one to one domain, the second to another domain, and the third to yet another. Why assume that all three belong together, or are explained by the same law, belong to the same domain—the domain called gravitational theory? There is no way to know that. So if I now want to build the law of gravitation, and I’m trying, I’m searching for some law that will explain a collection of facts, imagine that I have, I don’t know, a thousand facts. Then how many groups of facts can be built out of a thousand facts? Yes, does anyone know?
[Speaker B] One thousand factorial.
[Rabbi Michael Abraham] No. Two to the thousandth power. Because in each group, each fact either appears or doesn’t appear. First fact, second fact, third fact, up to one thousand. So it’s two times two times two—two to the thousandth power groups. Let’s say there are three facts, okay? A, B, and C. There’s one group that is fact A, a second group that is fact B, a third group that is fact C. A fourth group is facts A and B, a fifth group is facts B and C, a sixth group is facts A and C, and a seventh group is all three facts A, B, and C. If you add the empty group, you get eight—two cubed. So when there are three facts it’s two cubed; when there are a thousand facts it’s two to the thousandth power. Okay, two to the thousandth power—a crazy number. Simply unimaginable. A completely crazy number. Right, two to the tenth is about a thousand, so this is a thousand to the hundredth power. In other words, one with three hundred zeros. Just a crazy number of possibilities. Now I want to know, from—and that’s just a thousand facts, which is a joke, there are far more facts than that, right? A thousand facts is nothing. How many facts are there in the world? Impossible to imagine—infinitely many. So really it’s two to the infinite power, the number of domains I can generate from these facts. If I had no idea whatsoever, and I had to sort out from among the facts in the world the facts that belong to the same domain—there’s no end, there are infinite ways to do it. No way to do it. How can I nevertheless somehow sniff out that facts number 1, 13, 54, 110, 257, 402, and so on belong to the same domain, and that I should look for a theory that explains those facts? And of course there are all sorts of other collections of facts, each of which constitutes another scientific domain or another scientific theory. There is no way to know that those facts belong to the same group unless I have some idea of the theory that will explain them. Because otherwise, what connects those facts? So for example with Newton, clearly, in order to gather together tides, the falling of objects to the earth, and the paths of the stars, you have to understand that all of these somehow relate to attraction between bodies. You already have to think in advance that there is probably some attractive force between bodies. Then you say: okay, let’s see, maybe this is between bodies with mass. Then he says: tides—the water is drawn by the moon, and all sorts of things like that. And then he begins to understand how the whole business is organized. Then he says: there is such a domain as the theory of gravitation. The same thing applies to electromagnetism, thermodynamics, quantum theory—it doesn’t matter which domain you want. Quantum theory actually isn’t that, never mind, I mean domains of content in physics, not methodologies. Quantum belongs to methodology. So my claim here is basically that the division of the set of facts in the world into subsets of facts that divide science into its domains is a division that cannot be explained under the Baconian assumption that I first collect facts and then build a theory. No. I could not have collected the facts unless I had some idea of what the theory might be. You see, this is exactly what we saw last time; only last time I was talking about an explanation for a particular phenomenon, and now I’m talking about the division of the world of facts and of science into different scientific domains. But the logic is exactly the same logic. It reminds me, by association, a little of—there are very few writers about whom I can say they are obvious geniuses, clearly, beyond doubt. One of them is Borges. Borges, the Argentinian writer, who I think didn’t win the Nobel Prize—didn’t he? I don’t know how that can be. In any case, the man was a genius, obviously a total genius. Now in one of his stories, called “Tlön, Uqbar, Orbis Tertius”—there is a collection of Borges stories called Fictions in Hebrew, which Yoram Bronowski edited and translated. He edited and translated the collection and the stories in it. And I think that story is the first one in that collection, “Tlön, Uqbar, Orbis Tertius.” The story tells of some planet—this is of course fiction from Borges’s fevered, extravagant imagination—but it is presented as a description of a piece of history. There is a planet called Tlön, governed by Berkeley the idealist, the idealist philosopher, together with various people there, and it is a planet ruled by the idealist conception. A conception that says that in fact objects do not exist, or the world does not exist; all that exists is only my consciousness, but that consciousness has no basis in the world itself. I’ll mute this for a second. Right, these cognitions have no basis in the world itself. This is the view called idealism. We talked about it when I discussed empiricism. Berkeley was the most extreme empiricist, and therefore he was forced to erase the world altogether, because nothing is really given with complete empirical certainty. The fact that I see something does not mean it exists, and once there is no observation, then an empiricist cannot accept its existence. So in short, it’s a totally wild description of that world, how it behaves, and so on. Among other things he says that they found an encyclopedia that describes something of this world of Tlön, and so on. Among other things he says that in one of the languages spoken in that world there was a language without nouns. You couldn’t say, for example, “the moon rose over the river,” because after all there are no objects—we’re idealists. What should you say? “The mooning above the flow of water,” above the flowing. “Water” is already a noun, right? “The mooning above the watery flow.” Okay? Because you speak only about phenomena that you experience in your world of consciousness. The objects that these phenomena are supposedly describing—those objects do not exist; only the phenomena exist. So you are really describing all of reality through predicates without nouns. Right? So that’s a fascinating language. It’s a brilliant description of idealism and its implications. I think that story really is a masterpiece. In any case, that’s one example, and in another language of that world—and this is the example that concerns us—look what a genius the man is. So he says that nouns are of course fictions in that language. In that language there were nouns. In the southern hemisphere and the northern hemisphere, in one there were no nouns, and in the other there were. The one without nouns is what I said before: “the mooning above the watery flow.” In other words, you can say neither “moon” nor “water,” because those are nouns. But in the other hemisphere there were nouns. Yet the nouns are fictions, because objects do not exist. We merely want to collect the phenomena, so we sort of assign them to some fictive objects, and we call that nouns. But then he says: in fact you can define a noun by any collection of phenomena you like. In other words, the cry of a bird in the distance together with, I don’t know, the depth of the pit in my courtyard—let’s call that “yekomforkan.” That is the object yekomforkan, because after all I’m only collecting phenomena, but they are not really phenomena describing a particular object, because there are no objects. So I simply collect phenomena in a fictive way, and the group of phenomena defines a fictive object. So there is no reason to reject just some collection of phenomena that have no connection to one another, and that subset of phenomena, yes, that collection of phenomena, is for me what is called an object, and we will give it a name—let’s call it yekomforkan. Fine? That’s the name of this collection of phenomena. Because after all there is no—in our language, it’s clear to us why “bird” is a sensible noun. It has wings, it’s this color, it has a digestive system, I don’t know, it has certain navigational abilities—a set of properties. But all those properties characterize the bird, that particular bird. So it makes sense to use the term “bird” as something that really unifies this collection of properties. These are the properties of that object. But in a world where there are no objects, what is supposed to unify the collection of properties, so that this whole will give me some noun? Every subset of properties will in fact be a noun. Maybe one will be more useful and one less useful, but basically there is no way to determine a true division of the world of properties and assign each such group to a separate object, because there are no objects. So make any division you like of the world of phenomena into all those two-to-the-thousand, yes, groups of phenomena. Each of those groups is in fact supposed to have a name, and that name, for me, is called an object. It’s a fiction, of course, because there is nothing here that exists in the world. But that is what is called an object—a collection of properties, that is what I call an object. So the language in this idealist world, a world that does not believe in the existence of objects, was composed of a huge collection of nouns in a completely arbitrary way. Every collection of characteristic events is defined as a noun. That reminds me—yes, before saying what it reminds me of: do you understand the connection to what I’m talking about here? Because now translate this into facts, what I said before. The facts I described earlier—I can in fact take any collection of them, or any subset of the whole set of facts, and define that as a scientific domain. Right? I have no way to determine which subset it makes sense to define as belonging to the same domain and which does not. So why is there still some sense to it? Why is it possible to move forward at all? Because I have an intuition about what the nature of the law might be that would explain this collection of facts. So naturally I understand that probably this set of facts belongs to one domain. There will be corrections, but broadly speaking I have a pretty good starting point for beginning the process of scientific inquiry. Then the inquiry advances: I remove some facts, add others, improve the theory, return to the facts, and so on—again this back-and-forth. Okay? But in Borges’s context, of course, there is no way to move forward, because you have no way to test empirically which objects exist and which do not. There are no objects. So there you remain at the first level. Every collection of properties is an object, no matter which one. And in that sense it is a crazy absurdity, but for us it shows just how much, if you do not have some idea of what unifies the items, you will not be able to divide a set of items into subsets. There are infinite ways to divide it; you have no way to move forward. You’ll keep going around in circles forever with endless divisions of this world of items, and you won’t be able to make progress in any way. Why I started saying what it reminds me of—it reminds me of something we once discussed, I don’t remember in what context: Leibniz’s principle of the identity of indiscernibles. Leibniz argued that there cannot be two objects that have exactly the same set of characteristics. Why? Because if it’s the same set of characteristics, then it is simply the same object. They are not two different objects; it’s the same object itself. And he had a proof of this. What is the proof? That if we assume that these are really two different objects—not different in their properties, but two objects and not one—yet each of the two objects has exactly the same set of properties, then object A has the property of not being B, and B has the property of not being A. So they do not have the same set of properties. Which is what was to be proved. Proven by contradiction. Let us assume they really are two separate objects. If they are two separate objects, their set of properties is not identical. But if we assumed that the set of properties is identical, then these two objects must not be two, but one. Where is he wrong? He is wrong because he assumes that the object is nothing but the collection of its properties, that there is nothing whatsoever in the object beyond its collection of properties. Therefore he says: if there are two identical sets of properties, then it is the same object itself, because the object is the collection of properties. But if I say that the object is not the collection of properties—the collection of properties are properties of the object, but there is some object, some thing, that bears these properties, that these properties describe. It is not the totality of the properties; it is the object whose properties these are. If that is so, then there is no obstacle at all to saying that there are two different objects, and each of them has exactly the same set of properties. So what defines them as two? The fact that they are two. Not a difference in properties. You do not need a difference in properties to define them as two; rather, the fact that they are two objects. The object itself is other than that object. Not different from that object in the sense that it has another property, but separate from that object. It is not the same object; it is another object. Leibniz assumes that the object is nothing but a collection of properties—which in my eyes is a silly assumption—really parallel to Borges’s idealism. I’ve just now remembered a nice analogy. Because Borges’s idealism says exactly the same thing: a collection of properties, that is what is called an object; there is nothing there beyond the collection of properties. Then indeed in Borges’s world, the idealists in Borges, it is obvious that a collection of properties defines an object, so if there is a collection of properties that contains exactly the same properties as this collection, then it’s not another collection, it’s the same collection, simply the same object. The set of properties is the object. Okay. So—
[Speaker C] What separates the two objects? If there’s nothing at all distinguishing the two objects, then it’s the same object.
[Rabbi Michael Abraham] No, if there’s no difference in properties, that doesn’t make it the same object.
[Speaker C] So how do you grasp intellectually that these are two objects if there is nothing different between them? They’re the same thing.
[Rabbi Michael Abraham] What do you mean? But there are two.
[Speaker C] And that word—that’s a nice phrase, “there are two,” but there’s nothing behind those words.
[Rabbi Michael Abraham] It’s not just a word—
[Speaker C] “Two” is not an empty word, but it’s an empty set. The Rabbi can’t explain what makes it two. We called it two. Would the Rabbi say it’s four? If he said it’s four, that would stand just as well as saying it’s two.
[Rabbi Michael Abraham] Maybe I should explain it to you by means of a property? What does it mean to explain it to you?
[Speaker C] No, in any other way. That’s the point, that’s the only way. No, but how does the Rabbi grasp this intellectually, not just say a sentence?
[Rabbi Michael Abraham] I grasp it perfectly well intellectually, I have no problem at all. We all understand the difference between two and one.
[Speaker C] The characteristics of the electrons—their location in space—is exactly the same location of every electron and every proton; all the characteristics of the two bodies are exactly the same, and the Rabbi says they are two.
[Rabbi Michael Abraham] Including the location in space, and they would still be two.
[Speaker C] But the Rabbi surely knows that he doesn’t really grasp intellectually that this is possible, and nevertheless he calls it a sentence and then we’ve solved the problem.
[Rabbi Michael Abraham] I grasp it completely intellectually. Listen, I grasp it completely intellectually, and it’s amazingly simple. There isn’t the slightest problem here. Let’s take two electrons—wait, wait, listen. I have two electrons that are in exactly the same place, they have the same charge, the same mass, everything is the same. With electrons that can’t happen because they are fermions, but for the sake of the discussion—that is a physical problem, not a logical one. In physics they distinguish between bosons and fermions. Fermions are particles such that two of them cannot have the same set of properties; bosons in physics are different particles that can have exactly the same set of properties. By the way, there’s your proof—in physics we know such particles.
[Speaker C] When they are at the same moment, the same second, the same time, the same place—and these are two different ones. Right. Is that something one can grasp intellectually?
[Rabbi Michael Abraham] Completely. Now I’ll explain why. Because I see that what is in that same place has a charge that is twice the charge of one unit, so that means there are two units here, that’s all. But they have exactly the same properties—the same charge, the same mass, in the same place, at the same time. Fine, but in that same place at that same time there sits a particle whose mass is two m and whose charge is two e, that’s all. What’s the problem? Why can’t one grasp that? One can grasp it very easily.
[Speaker C] So that’s already something else. Together they created a new body that has two charges.
[Rabbi Michael Abraham] Who told you that it’s a new body? We know them separately, and we know that it’s not a new body. When they are in different places, we see here there is charge e and there there is charge e.
[Speaker C] And if I insist on defining them as one body?
[Rabbi Michael Abraham] You can define whatever you want, but these two bodies, when they are in different places, and then they gather into the same place, still their charge is two e. There is no reason to assume that this created a new body unless you are talking about a collective body. I have no problem with that. But that collective body is composed of those two bodies; there are two bodies here.
[Speaker C] The Rabbi is talking about two bodies located at exactly the same position in the universe, and the Rabbi says he grasps intellectually how that could be?
[Rabbi Michael Abraham] Of course. What is it that you don’t grasp? It’s simple.
[Speaker C] I don’t think intellectually it’s any different from if I were to ask the Rabbi, “Can the Rabbi imagine a state where there is no space at all, no universe at all?” Don’t bring me other examples.
[Rabbi Michael Abraham] It’s the same—
[Speaker C] thing, exactly the same thing. After all, we can’t grasp that intellectually, and we can’t grasp this either.
[Rabbi Michael Abraham] Do you realize that you’re begging the question? You bring me an example that cannot be grasped intellectually, and then you try to demonstrate through that that this too cannot be grasped intellectually, and you tell me, “It’s the same thing.” But that’s exactly what I’m arguing with you about: that it’s not the same thing. That one cannot be grasped intellectually, and this one can. What’s the problem? It’s so simple, I just explained it. I just explained it. You have two parts. You need to understand that the fact that two particles cannot be in the same place—that’s not a problem in logic, that’s a problem in physics. Logically there is absolutely no problem with two—fine.
[Speaker C] What do you mean, two particles? I don’t understand. How can one grasp—where have you ever encountered in the universe, in our reality—
[Rabbi Michael Abraham] Because I understand. It doesn’t matter where I’ve encountered it. I understand it because there is no logical problem in it. If it doesn’t exist in the world—
[Speaker C] But part of the definition of a universe, of space, is that something occupies space. And to say that the same thing—no, then what is space at all?
[Rabbi Michael Abraham] Wait a second. When I explain to you that there are two particles here in the same place, there is no problem at all understanding that logically. It may be that physically this is forbidden, because physics forbids it, and so I will never encounter it. So what? Even if I never encounter it, it is completely understandable. By the way, I do encounter it too—physics does not forbid it. With rigid particles like electrons, physics forbids it, but with particles like photons, physics does not forbid it. Two identical photons can be in the same place, at the same point in time, with exactly the same property, exactly the same properties. That’s all. Physics allows it, recognizes that state, describes it, and there is lots of work in physics about this. There is no problem at all. And that is true of every boson, every bosonic system. What is called Bose-Einstein condensation in physics is a system of bosons all located in the same place, in the same state, with the same properties, at the same time—and we count how many bosons there are.
[Speaker C] So that means I haven’t encountered any boson, but I wouldn’t be surprised if in the end they say—
[Rabbi Michael Abraham] that it’s really a wave and not— You don’t need to encounter it. In order to understand, you don’t need to encounter it. What? Only materialists think—and even they don’t really think this way—that in order to understand you have to encounter. You don’t have to encounter anything. I understand plenty of things without encountering them. If they sit well with reason and make sense, everything is fine, I understand it. Whether I encounter it or not is a question in physics: does physics allow it or not allow it? In this case physics also allows it, but that doesn’t matter. You will encounter it if you use measuring instruments. You don’t see bosons with your eyes, but with measuring instruments you can detect them. What’s the problem? I don’t see any problem here in understanding it. One simply needs to define properly what it means to understand, that’s all. Okay, now back to our topic…
[Speaker C] Rabbi, rabbi, if we take, for example, two stones that have—why bosons? Let’s take two stones. Two stones that have exactly the same characteristics in every respect. Suppose, by some bosonic ability, we managed to unify them into the same space, and they’re simply in the same place, with the same characteristics, both the same weight, the same color, everything exactly the same. Now would the Rabbi still say that these are two different stones, even though they’re in the same place in space and have the same characteristics and nothing has changed from what was there before?
[Rabbi Michael Abraham] Of course they’re two different stones, of course. Of course. The whole problem is only in physics. Physics forbids two stones from being in the same place. So what? Logically there’s no problem at all. If physics didn’t forbid it, then they would be in the same place and there’d be no problem with that. Like photons, that’s all. Two photons are in the same place because they have no mass, so they don’t exert force on one another, and therefore there’s no prohibition against their being in the same place. That’s all. There’s no problem with it. You have to get out of the material picture, right? Not all entities in the world are matter. Because matter has this property in physics that it occupies space and doesn’t allow other matter to be in that same place. Fine, but there are things that aren’t matter in that sense, and they can be in the same state, in the same place, everything the same, no problem at all. Okay, so we’ve really gone off track here, that was just an association. My claim is that when Leibniz talks about an object as a collection of properties, that’s exactly the same thing as Borges’ description of the idealists. It’s a bundle of properties without there being an object that possesses the properties. Okay? Good, so to our matter. Basically my claim—this is the second claim. The first claim was what we saw in the previous class: I have infinitely many facts, and if I want to explain why Napoleon lost at Waterloo, then I need to know which facts to focus on. Because with infinitely many facts I’ll never get anywhere. I’ll check fact after fact after fact, I’ll never finish checking the facts, so when will I get to building the theory? Okay, so you can’t make scientific progress if you don’t have some way to filter and focus on a subset of the facts. Therefore I need the theory, or some initial idea about the theory, in order to extract the relevant facts. That was the argument from last time. The argument this time deals with dividing the facts that exist in the world into different scientific fields, into different subjects, by different scientific domains—and that too can’t be done unless I have some initial idea about the theory that I’m going to discover after I divide the facts into different fields. Now I want to move to a third argument that says the same thing, and this is a situation in which I already have a given set of facts, I’ve already divided the facts. Now I have a given set of facts, right? And now the question is how I get from those facts to the theory. It turns out that for every given set of facts there are infinitely many theories that can explain it.
[Speaker D] Simple. The proof of this is very, very simple. Look at the graph I’m sharing now.
[Rabbi Michael Abraham] Look here. What am I measuring here—what is this doing here? I’m making here a graph of the force, F, against the acceleration. Okay? The X-axis is acceleration, the Y-axis is force. And I’m measuring the relation between the force acting on a body and the acceleration it develops. Okay? Now Newton’s second law says that the relation is direct: F equals MA. That is, mass multiplied by acceleration gives the force. So there’s a straight line for the relation between force and acceleration; the slope of the line is the mass. Okay? The tangent of the angle of the line is the mass. This is Newton’s second law, that force equals mass times acceleration. Now I—I’m Newton, meaning I don’t yet know the second law, I haven’t discovered it yet—but now I want to check the relation between force and acceleration, and I already understand that the relation between force and acceleration probably gives me some domain in physics, in this case the second law, okay? It’s governed by the same law, the relations between applying forces, the values of forces, and the accelerations of different masses. Fine? I’ve already divided the facts, I’ve concluded that this set of facts is supposed to be explained by one law. So I’ve already gotten through all the—meaning I identified the relevant facts, I divided the scientific world, and I already know that this set of facts is supposed to be explained by one law. I still haven’t solved the problem. Why not? Look at the empty circles in order to understand this. You see? This is one, this is two, this is the third, this is the fourth, this is the fifth. Each such circle marks a pair of force and acceleration—say the force has this value here and the acceleration is five, okay? For some reason I didn’t mark the values of the force here. Fine? These aren’t values of acceleration, they’re serial numbers. This is point number one, point number two, point number three, four, and five. And at each such point there is a pair of numbers: acceleration and force. Okay? So I took five measurements—ignore the black dots, I’m talking now about the empty dots. Okay? I took five measurements of force and acceleration, got certain results, and I plot them on this graph. Now I ask myself: what is the line that describes the general law? You understand that searching for the law that explains these facts is basically searching for the line. Right? The line is the law. The facts are the points that I measured, and the law is the line. Now I have these five points, and I ask: what is the line that connects them? How many lines can be drawn that connect these five points? Infinitely many. Right? Two of them appear here. The straight line is the black one, the solid black line, and the second line is this dashed line, you see? That line also passes through the five points, and of course there are infinitely many like that. Infinitely many—not even countable. Okay? There are many lines that thread through these five points. Now we’re already in serious trouble. We’ve already found the relevant facts, we’ve already divided the set of facts into facts that are supposed to be explained on the basis of one law. I’ve already done everything. Impossible steps, of course, but I did them. Okay? Now look—it’s still impossible to move from the facts to the theory. Why? Because even for a given set of facts, after I’ve already divided the facts, there are infinitely many theories that can explain it. So how do I choose? The correct theory? How do I know how to generalize correctly from the set of facts that I measured? You understand that this is a problem on top of a problem on top of a problem. There are infinitely many possibilities. Okay, so what’s the answer here? The answer is—this is what every scientist will tell you, this isn’t a philosophical answer, it’s a practical answer, right? Every scientist will tell you: leave me alone, for heaven’s sake, the line is the straight line. Don’t confuse me. Ask him why. Maybe it’s the dashed line? After all, if all the facts you know are only the five points you measured, they fit every line that threads through all of them. So why do you choose דווקא the solid straight line and not the dashed line? That’s no less good an explanation, or any of the infinitely many other lines. Why do you choose the straight line? So he’ll tell you: because it’s the simplest. Right? A straight line is the simplest line. The minimal assumption. Let’s put it in mathematical language: we’ll say that each such line requires a certain number of parameters in order to define it; a straight line requires the fewest parameters, right? It requires two; if it passes through the origin then only one. Y equals AX plus B is a straight line. Once we have a parabola, then it’s AX squared plus BX plus C—that’s three parameters. If we have something more complicated, four, five, six parameters, and so on. So if we go by the number of parameters, then the simplest line is a straight line. It’s the minimum number of parameters. If it passes through the origin then Y equals AX without B. Okay? One parameter, that’s the simplest. Therefore we choose the straight line. That’s the answer. Now I ask: what is that answer based on? Observation—everything it gave me was these five points. The values of these five points. The move from the facts I observed to the theory involves some assumption here, namely that I choose the simplest option, because after all there are infinitely many theories that will explain this. Out of the infinitely many theories that explain it, I choose the simplest theory. But that itself is an assumption that has no observational basis; it’s an assumption of reason. It’s a theoretical assumption, an a priori assumption; it isn’t the result of observation. By the way, it also doesn’t always work. Sometimes the straight line really isn’t the correct line and we discover that we were mistaken. Suppose we took another measurement, say here, at this acceleration, and we discovered that the force is this. Let’s say, okay? Then that would refute the assumption that the straight line is the correct line. Okay? It’s not certain that it’s correct. But in the absence of refutations, we will always choose the simplest line. In this case, the straight line. What fits a parabola, we’ll choose the parabola and not something more sophisticated, and so on. We will always choose the simplest line. And the question is: where does that principle itself come from—that the simple line is also the correct one? That is an assumption of reason. It is not a result of observation. On the contrary, observation, or the facts, reveal to us that physics is not always so simple. Quantum theory, relativity theory, all sorts of complicated physical theories are really not simple at all. You can’t say that we learn from experience that a scientific theory is a simple theory. So why do we assume that among the theories that explain the facts, we should take the simplest one—or that at least it is the best guess, the most likely to turn out correct? That is an assumption of reason. So once again we see that looking at the facts and going from them to the theory is a very, very superficial and mistaken way of looking at things. First, we need to assume things about the theory already when we look at the facts. After we’ve already chosen which facts to look at and used theory implicitly, returning from them to the theory requires us to use Occam’s razor—and perhaps other things too, but at least Occam’s razor—to choose the simplest theory. And that too is an assumption of reason, not an assumption that comes out of observation. When we speak about science as observational, meaning based on observations, we’re throwing dust in our own eyes. Science begins with observations, but you can’t say that it is based on observations, meaning that observations alone yield the final result. Absolutely not. There are many, many, many more assumptions here—assumptions that precede or come after the observation, but you need them too in order to arrive at the theory in the end. Basically what this says is that the Baconian picture cannot be correct. That pure observational science—the idea that we stick to observation and know what observation gives us—is wrong. There is no scientific theory in the world that is the result of observation alone, without requiring additional assumptions, assumptions that we bring from home, assumptions of our logic, philosophical assumptions, whatever you want to call them, which are not the result of measurement or observation. And as I said in the previous class—someone asked about this—there are those who explain that the intuition which adds these assumptions is itself the result of accumulated experience. That’s the accepted view, that it is the result of accumulated experience, but that is of course utter nonsense. Psychologists can say such a thing because psychologists don’t understand mathematics. But anyone who knows mathematics understands that this cannot be true. It can’t be true because if we were born tabula rasa, then experience could not give us anything, for the reasons I listed here. Because experience itself is always based on intuitions. Experience cannot build the intuitions, because if there weren’t intuitions at its basis, I couldn’t get anywhere with it. I would go around in circles, looking for infinitely many facts; after I found a subset of facts, there would be infinitely many theories; I wouldn’t know how to divide the facts into groups; I couldn’t move a millimeter if I stuck only to the facts. And anyone who thinks that all our intuitions are the result of facts is talking nonsense—of learning from facts, from observation, he’s talking nonsense. By the way, this is Chomsky’s claim regarding the development of language—I mentioned this, I don’t remember when—regarding the development of language, he says that it’s impossible for language to develop only through learning, that I learn the language and that’s how I acquire the ability to use it. There must be within us, embedded in us, some linguistic faculty. Something that is not a result of what I receive—of the facts I receive or the examples I receive. Something has to be inside me, some kind of linguistic intuition that enables me to build the skill of using language out of the examples I encounter. Because otherwise I couldn’t do anything with the examples I encountered. Because unless I simply repeat them themselves—but every other sentence I build, you can’t say that it comes from the examples. From those examples I could generalize in infinitely many ways and generate infinitely many languages. And therefore it is clear that I have some linguistic intuitions that help me build my linguistic ability. The examples given to me when people teach me a language, by themselves, could never teach me a language. Yes, this is the problem that Wittgenstein calls following a rule. Wittgenstein says—and I spoke about this once too—
[Speaker D] Wittgenstein basically says, suppose I give you a sequence of numbers. One—let’s say three, five, seven—and I tell you to complete it. So what is the next number? Three, five, seven, and?
[Speaker F] Anything is possible. Can’t hear? Anything is possible. Why? Because you can do—as the Rabbi already said—you can do a calculation that yields any result.
[Rabbi Michael Abraham] Okay, the first answer you’d get, say on psychometric tests, is nine. It’s nine, of course. Right? Anyone who doesn’t answer nine loses the points on the test. But why not eleven? Nine assumes that I’m going by the odd numbers: three, five, seven, nine, right? But I can also say that I’m going by the prime numbers: three, five, seven, and the next prime is eleven; nine is not prime. Nine is three squared. So three, five, seven are three prime numbers, and the next prime is eleven. But of course even that doesn’t have to be it. I can also tell you minus two and a third, and I’d have no problem showing you the rule according to which in the first place it’s three, in the second place it’s five, in the third place it’s seven, and in the fourth place it’s minus two and a third. You can build such a rule without much trouble; it’s four equations with four unknowns—or never mind the details right now, but I can. It’s possible to build such a rule. And of course any number you want. So there is no correct answer here. So how do I know that on the psychometric test I’m supposed to complete it with nine? Because that is somehow the simplest in some sense. And the simplest is not necessarily the most correct. Eleven is no less correct than nine. And minus two and a third is also no less correct than nine. It may be less simple, less natural, less immediately obvious to us. But that is only a result of how we are built. For us, nine is the simplest, but some creature may come along for whom eleven really is the simplest, or some creature from outer space for whom minus two and a third is the simplest. And on the psychometric test they give on his planet—complete, yes, on his planet—then indeed anyone who doesn’t write minus two and a third but writes nine fails the psychometric test. Why? Because the psychometric test doesn’t check whether you’re right. The psychometric test checks whether you think like I do, whether you’re conventional. Anyone who passes the psychometric test with a good score—that means he is conformist in his way of thinking. It means he thinks exactly what he is expected to think. It doesn’t mean that someone who fails the psychometric test is creative; sometimes he’s just stupid. But a creative person will fail the psychometric test—unless he knows what is expected of him despite being creative, and then he’ll answer in a non-creative way. Okay? In other words, there’s an interesting point here. We need to pay attention to this. Psychometric tests basically check whether you are built like me. And rightly so, by the way. When you go study at a university, then when the lecturer teaches he assumes all kinds of things about how one should explain to students so that they understand. If there is a student who is built differently, he won’t know how to explain it to him. That student won’t understand the explanations. Therefore the university is right when it checks the students to see whether they think according to the standard pattern, so that the lecturer will be able to teach them. That’s what they check there; they don’t check talent. They check…
[Speaker H] Does there even exist a test that checks creativity or not?
[Rabbi Michael Abraham] I don’t know. I guess yes, but I’m not sure, I don’t know. You have to be creative to answer that question. Okay, so I return to our topic. So this basically means that even when I have a given set of facts, the move from them to the theory requires additions that are not empirical. Various insights such as, in this case, Occam’s razor. Now here I want to make one more remark, connected somewhat to things we talked about earlier, but I’ll use this example that I used. Regarding Occam’s razor there are two possibilities… one second.
[Speaker I] One…
[Rabbi Michael Abraham] You see my website now, right? Not the previous file. Right?
[Speaker E] We see a file.
[Rabbi Michael Abraham] What? Not the graph—you see the site.
[Speaker E] No, no—yes, yes.
[Rabbi Michael Abraham] Okay. So look here at the end. This is a column I wrote about Occam’s razor, and I went over the Wikipedia entry on Occam’s razor, and the amount of nonsense written there is just hard to grasp. So much nonsense and so many mistakes—it’s amazing. You see Word.
[Speaker G] What? We see Word, not the site.
[Rabbi Michael Abraham] Ah, then wait, then apparently—this is what I asked earlier—so now I’ll share it. There. Now you see it, right? Right. Here’s the quote from Wikipedia. Okay, Wikipedia sums up, after bringing many refutations of Occam’s razor and so on—none of the refutations is correct, never mind that now—but really this is an entire entry that is one long mistake from beginning to end, and what is most fascinating is that… this entry really summarizes what philosophers write about Occam’s razor. It’s not that the writer invented it. Everyone talks nonsense. And the reason everyone talks nonsense in this context is because they don’t understand the option of intuition, as I’ll explain in a moment. So look how he sums it up. “One should not treat Occam’s principle as a rule or law, but only as a pragmatic recommendation. If we treat it as a rule, we will use it to choose a certain theory on account of its being simpler. According to Karl Popper, if one day an observation is found that does not fit that theory, then that theory immediately becomes refuted and cancelled.” Fine? “But according to that same Popper, this would also immediately refute and cancel Occam’s principle itself, on account of which we earlier chose the refuted theory.” Now look what he writes here. “A common mistake is to claim”—that common mistake is of course the pure truth—“a common mistake is to claim that Occam’s razor is meant to provide a tool for choosing between true and false theories. But that is not so. The razor helps us choose, from among different true theories, the simpler theory in terms of explanation, economy, and content, and to use it for the continuation of the scientific path. But what about the other theories, those that were rejected? After all, they too are correct for the time being. Why reject correct theories merely because they are more complex and difficult to understand? Many argue that the role of a scientist is to rule out and reject false theories, not complicated theories. Galileo mocked Occam’s razor by saying that one should reject all science books and choose only the letters of the alphabet, since with them one can explain anything, and they are simpler than science books. Still, Occam’s principle should not be rejected completely.” He of course accepts all these foolish claims, and then he sums up: “Still, Occam’s principle should not be rejected completely. It is useful in fields such as the didactic field. It is easier to explain and teach theories and concepts in a simpler form than in a complicated way. Choosing a simpler theory can also reduce the costs of implementing the theory as compared with a more complicated one.” Yes, so here I go after him in the next paragraph, never mind that now, but this is what is supposed to summarize the up-to-date philosophical and scientific knowledge about Occam’s razor for all readers. And this is, by the way, a faithful summary of the source. This really is the accepted view in the world of philosophy. It’s all nonsense from Aleph to Tav, and it can be proven mathematically. In other words, this isn’t my philosophical position as opposed to another one; it is simply nonsense, and it can be proven mathematically that it is nonsense, and it is simply not true. And why? Now I’ll do this briefly. What really lies behind the matter? There are two ways to understand Occam’s razor. In the simple view—let’s go back for a moment to the graph, okay?
[Speaker D] Here, this is our graph. So in this graph we saw that through the set of five circles—
[Rabbi Michael Abraham] the empty ones—you can pass infinitely many lines. The straight line is the simplest line. So Occam’s razor tells us: let’s choose the theory of the straight line because it’s the simplest. What does the fellow from Wikipedia, or the fellows from Wikipedia, claim? That of course this theory is no more correct than the dashed line; it is only simpler. Because as far as the information we currently have goes, it is these five facts that we measured. We don’t know anything else. So for the moment both theories, the dashed and the solid, are equally correct. So the theory of the straight line is only simpler than the dashed line; it is not more correct than the dashed line. That is basically his claim. This is the view of Occam’s razor as a methodological principle. It doesn’t tell us what the truth is; it tells us how it is preferable to proceed in the most efficient and simplest way. That’s all. Or in other words: why use a complicated theory if I have a simple theory that explains all the known facts in exactly the same way? Why make things complicated for no reason? Let’s use the simple theory. But not because it is more correct—rather because there’s no reason to complicate things and it’s more convenient to use. Okay? That is the methodological thesis or the methodological interpretation of Occam’s razor. What I propose is the substantive interpretation of Occam’s razor. I claim that we choose the straight line because it is more correct. Occam’s razor is not a methodological rule; it is a rule that discriminates between theories. Between correct and incorrect theories. Exactly what he said was the common mistake—he said on Wikipedia that this is a common mistake—so this common mistake is of course the pure truth, which I am now going to prove. Let’s think for a moment. Usually the conception—what stands behind these two conceptions—is what Ze’ev Bechler talks about. Ze’ev Bechler was a professor of philosophy of science—already emeritus, I think—at Tel Aviv University, and he speaks about actualism and informativism. What does that mean? Actualism is a philosophical approach in philosophy of science that says that only the facts we measured are true. What is actually present before our eyes—the facts we measured—that is the truth. The theory is neither true nor false; it is only our way of organizing the facts in the best way we can, the most convenient and simple way. Actualism parallels the methodological view of Occam’s razor, namely that it is only a methodological rule, not a rule of truth and falsehood. Informativism is the view that says that the theory contains information about the world; it is correct. It is not empty of information; it is correct. Meaning that the theory is not only the simplest explanation of the known facts, but also the most correct explanation of the known facts. And so basically there is a dispute here over how to relate to scientific theory. Actualism, or the methodological approach to Occam’s razor, says this: scientific theory is merely the most efficient way for us to organize the facts, but it claims nothing about the world. When I say that there is a force of gravity, I have not made a claim about the world; it is not that there really is a force of gravity in the world. Rather, assuming that there is a force of gravity will organize the known facts for me in the simplest and most efficient usable way. That’s all. I am not really claiming that there is a force of gravity in the world. By contrast, informativism, or the substantive approach to Occam’s razor, says that if I conclude that I have a theory of gravity, I am claiming that there is a force of gravity in the world. And therefore these facts really occur—they occur by virtue of it. Or in other words, in the context of this graph, I am claiming that the straight line is the correct law. It is more correct than the dashed line. Why? After all, I have only five facts, and those five facts are explained both by the dashed line and by the solid line. So on what basis do I assume that the solid line is more correct? I claim: because it is simpler. The simple is a criterion of correctness. Not only of efficiency and simplicity, methodological efficiency, but a criterion of correctness. Now here, in the conception—if you read Bechler’s book, Three Copernican Revolutions—I read it in the hospital in some intense two-day stretch, I was hospitalized, I went through it from cover to cover, otherwise I don’t know if I would have gotten through it. But throughout the book—and afterward I even corresponded with him a bit about it because I had criticism of the book—he argues in favor of informativism but doesn’t bring even a single proof for it throughout the book. He only vilifies actualism, how ugly it is and how wicked it is and how tendentious it is, but he doesn’t bring any proof in favor of informativism. I told him he did half the job, and in my book Two Wagons I do the second half. Or in At Times That Exist and That Do Not Exist—there this graph appears for the first time—this graph does the second half, the proof for informativism. And now I am going to prove informativism. Because Bechler assumed, and this is the general assumption in the scientific world, that this dispute between actualism and informativism—or between viewing Occam’s razor as a methodological principle and viewing it as a substantive principle—cannot really be decided. It’s just two philosophical views: you can assume this, you can assume that, there’s no way to decide. Why is there no way to decide? Let’s try to think. Suppose I measured and took one more measurement—here you see point seven—I took another measurement here and found this line, okay? This point, sorry. I took a measurement at this acceleration and found that the force is this. Okay, so I found this point. In that measurement I showed that the straight line is the correct line and the dashed line is not correct. What’s the problem? There, I showed that the straight line is the correct line. But of course that does not decide in favor of informativism. Why? Because the actualist will say: what you showed is not that the straight line is correct; you showed that this line is not correct. But now let us draw a new line that threads through the five—the five circular points and this point—and you’ll see that there are still infinitely many curved lines that can be drawn through all these points. So all you did was rule out some of the lines, but I still have infinitely many other lines left. You didn’t show me that the straight line is the correct one. You just caused me to choose a simple theory relative to a different set of possibilities. Or, if you measured this point and measured it here, not on the straight line, then you refuted the straight line. But of course you did not prove the dashed line, because through that point too one can still draw infinitely many lines that thread through it as well. Therefore, the scientific process itself can be explained in exactly the same way by an actualist and by an informativist. The scientific process conforms equally well to these two philosophical views. It will not be able to decide between them. Therefore it is commonly thought in the world that this dispute is a philosophical dispute. There is no way to decide it. But that is of course a mistake. There is a way to decide it, and a very, very simple one. I’ll say it now briefly—maybe next class, since we really have to finish—but just to complete it, to close the circle. Look. Let’s now try a hypothetical thought. Suppose I have these five circular points. Okay? I measured them. We said there are infinitely many lines that thread through these five points. Among them are the straight line, this dashed line, and many other lines. Now I ask—not really ask, I’m not doing an experiment and checking what comes out. I’m asking you now: let’s bet. I do an experiment at point seven, at this acceleration.
[Speaker D] What force will come out? Will the force come out as this or as that? A bet. What are you betting on? So let’s do the calculation.
[Rabbi Michael Abraham] If you are actualists, the bet should be that the probability is equal. Or in other words, the probability that the point will fall on the straight line is zero. There are infinitely many lines. There are infinitely many possibilities. And the straight line is not more correct, from the actualist’s point of view. It is simply more convenient. But there is no connection between the convenient or the simple and the correct. So what is the probability that the straight line is the correct one? Zero. One over the number of possible lines. Therefore the probability that the result will be this is negligible. It is zero. In the actualist picture, right? What is the probability in the informativist picture? If Occam’s razor reveals to me the correct theory, not the simple theory—close to one hundred? I don’t know if close to one hundred, but not zero. In other words, I’m saying: Occam’s razor isn’t certain. I’m only saying it won’t be a shot in the dark. So the probability might be thirty percent, might be seventy percent, whatever—but not zero. Because there are indeed cases in which the straight line that we posit in the experiment fails. That happens from time to time. But it doesn’t fail in all experiments. There are some tens of percent of cases in which it will succeed. Which means that the straight line is, with a high probability, a more correct line than all the other lines. Not certain—but more correct. So here you have a scientific criterion, not a philosophical one, that can decide who is right: the actualist or the informativist. Let’s do many experiments on laws where for the time being the straight line fits, and check what the next experiment gives me. If in the next experiment all the following experiments fail, and this does not fall on the straight line, then apparently the actualist is right. But if in a significant, statistically clear percentage of these experiments the next point also falls on a straight line, then that means the informativist is right—and this is a scientific proof. It isn’t even a philosophical argument; it is a scientific, statistical proof. Now if you check in the history of science how many scientific experiments succeeded—that is, in how many of them the prediction was indeed confirmed in the lab—of course not one hundred percent. There were quite a few mistakes, but also not zero. If this happened in zero percent of cases, then today our science would be the science of Adam. We would not have moved a millimeter. If science advances, that means we succeed in building theories that give us not-bad predictions about what will happen. Not every prediction of every theory fails. But according to the actualist, what should really happen is that every prediction of every theory should fail. Because the probability that it succeeds is zero. There are infinitely many theories. I chose the simplest one out of infinitely many possible ones. So no experiment can succeed according to the actualist. But there are experiments in the history of science that succeeded—quite a few experiments. Some would say most of them, but never mind, a high percentage. Significantly above zero. That is enough for me. It means that the actualist is mistaken, which is what had to be proven. This dispute is simply a pseudo-dispute; it is a collection of mistakes by many, many philosophers, statisticians, mathematicians. Everyone who wrote about Occam’s razor, as summarized in Wikipedia, all think this way—and all are wrong. A simple mathematical and scientific, statistical proof that they are wrong. What does this mean for us, basically, to close the circle? It means that when I choose the straight line, it is simply because my intuition tells me that the simple is also the correct. Or in other words, what does “simple” mean for me? Why is the straight line simple? Because it fits my intuition better. Which means that intuition is a good tool for making claims about reality. That is basically what the substantive interpretation of Occam’s razor says. Actualism thinks that intuition is a subjective tool; it says nothing about reality, it merely serves us in organizing the facts. And therefore it says that the rule of Occam’s razor is a methodological rule, not a substantive rule. And what it misses—the common mistake that it attacks—this mistake actually assumes that this is a substantive rule. Why? Because it is a rule that comes out of intuition. And intuition, true, has no observational basis, but despite having no observational basis, it has validity; it works. Not always, but it works. It is not a shot in the dark. And the entire history of science proves it. And again, this basically means that our intuition, or the philosophical principles that we use beyond the observations, are an inseparable part of science, and without them science also would not have succeeded. We wouldn’t have advanced. Okay, I’ll stop here. Maybe I’ll elaborate on this a bit next time, because I feel I did it a bit quickly. I’m not sure how clear it was, because not everyone has the same mathematical or scientific background, so this has to be done a bit carefully. I just wanted at least to manage to close the circle. Comments or questions?
[Speaker F] Can I ask whether—
[Speaker J] Is it possible that this intuition is the fruit of evolution, which built us in such a way that our intuitions would fit reality?
[Rabbi Michael Abraham] That can be discussed. If intuition is a product of evolution, maybe. For the purposes of the discussion right now, maybe yes.
[Speaker F] The straight line is probably the correct line if we measured enough times and statistically got—
[Rabbi Michael Abraham] A straight line between two points is speculation. That’s a weak scientific generalization. But five, ten points—that’s already much stronger.
[Speaker F] So because we measured and we have enough observations for a number of cases that lie on the straight line, and therefore—
[Rabbi Michael Abraham] Now again, even if it isn’t a straight line but a parabola, the argument is the same argument. I demonstrated it with a straight line, but not all the laws of science are straight lines. But still, the scientific law is the simplest way to stitch together all the results. Sometimes it’s a parabola, sometimes it’s a sine. But those results too can be stitched together in infinitely many ways. So if I measure the predictions of the sine or the parabola and discover that they also fit the next experiment in a significant percentage of cases, that means the same thing I said before; I just demonstrated it with a straight line because it’s simpler.
[Speaker F] So even if there are infinitely many possibilities, that still doesn’t rule out the fact that this is something statistically weighty?
[Rabbi Michael Abraham] Yes. Not statistically weighty, but having statistical weight. It doesn’t have weight because of the statistics; it has weight because of the intuition, but its weight means that statistically it will be correct in a significant percentage.
[Speaker F] Here the element of intuition enters in.
[Rabbi Michael Abraham] I understand.
[Speaker J] Exactly. In the Rabbi’s words this actually somewhat contradicts—or kind of comes against—Popper too? Who says that you don’t prove the theory, only—
[Rabbi Michael Abraham] Of course. Popper is the greatest of the actualists. Because for Popper, precisely for that reason, a scientific theory is not correct; it simply has not yet been refuted. All you can do is refute a theory. Why? Because from his point of view every theory is a generalization, and there are infinitely many generalizations. Therefore there is no way to know which theory is correct; there is only a way to know which theory is not correct. And of course he is wrong. There is no way to know with certainty that a theory is correct—that, I agree. But it is not true that this is a shot in the dark, meaning that it is equivalent to all the other theories that explain the same set of facts. Fine, from the three arguments I brought here, from each one separately you can see how much empiricism is an illusion. This idea that we can build only on observation and from that build all our science—that is simply nonsense and a lack of understanding. Thought is always involved here. Rationalism has not left us; it is with us all the time. Okay, we’ll stop here. Goodbye, have a peaceful Sabbath.
[Speaker B] Thank you very much, goodbye, thank you very much.