Platonism – Lecture 38

Table of Contents

• Scientific Platonism, abduction, and an infinite number of generalizations
• Actualism versus informativism and a statistical decision based on the history of science
• A skeptical reservation about using statistics and an internal reply to the actualist
• Theoretical entities and scientific Platonism
• What exists according to the actualist: law versus force, facts versus causality
• Intuition as intellectual perception of theoretical entities
• Practical implication: there is no single decisive experiment, but there is cumulative decision
• The relation to positivism and a Platonic thesis with no direct practical implication
• Qualification: scientific Platonism as “weak Platonism,” not the World of Ideas
• Moving on to mathematical Platonism and the non-empirical status of mathematics
• Physics versus mathematics: empirical refutation is always physical
• Mathematical Platonism: discovery versus invention, and criticism of conceptual confusions
• The tree falling in the forest, sound versus a pressure wave, and a mistaken analogy to mathematics
• Properties and events are not entities: speed, color, and the mistake of identifying them with Platonic existence
• Open ending and a request for a Kabbalistic expansion

Summary

General Overview

The text presents an argument that from the history of science one can empirically decide between actualism, which sees scientific theory as merely a convenient organization of observations, and informativism, which sees theory as making a claim about the world and about the existence of theoretical entities. The argument rests on the fact that from the same observations one can derive infinitely many generalizations; therefore, according to actualism, the chance that a theory will be confirmed by an additional experiment is negligible, whereas in practice a significant percentage of experiments confirm theories over time. From this, the text argues for a statistical advantage in favor of informativism and a certain kind of scientific Platonism, while qualifying that this proof concerns theoretical entities in the physical world and not Ideas in the World of Ideas. Later, the distinction between physics and mathematics is presented; it is argued that mathematical claims cannot be empirically refuted; and criticism is offered of discussions of mathematical Platonism that, according to the speaker, mix together different questions, such as the existence of mathematical entities and the question whether mathematics is discovered or invented.

Scientific Platonism, abduction, and an infinite number of generalizations

The text describes a graph with five observations marked as hollow circles, and presents two possible theories that fit those five points equally well: a continuous straight line and a dashed line. The text states that from those same observations one can propose infinitely many lines and theories connecting the points, and therefore generalization from observations is not determined by the data themselves. The text points out that at other places, such as observation six and seven, a difference between the theories would appear, and so the next experiment becomes a test point between different generalizations.

Actualism versus informativism and a statistical decision based on the history of science

The text defines an actualist as someone who claims that scientific statements are statements about us and not about the world, and an informativist as someone who claims that theoretical statements are statements about the world. The text argues that both sides will usually choose the simpler theory, such as the straight line, but the actualist sees simplicity merely as a matter of convenience, while the informativist sees simplicity as an indication of truth, within the framework of Occam’s razor or judging favorably. The text argues that according to actualism, the chance that a chosen theory will be confirmed in an additional experiment is effectively zero, because it is one out of an infinite number of possible generalizations, whereas according to informativism the chance is not zero. The text claims that the history of science shows in practice that many experiments confirm hypotheses rather than refuting them almost all the time, and therefore the very existence of a non-zero success rate is statistical proof in favor of informativism. The text concludes that if actualism were correct, then every experiment would refute the current theory, scientific knowledge would never accumulate, and we would be left with “the science of Adam.”

A skeptical reservation about using statistics and an internal reply to the actualist

The text suggests that an actualist could insist and argue that statistics too is a theory “about us” and not about the world, and therefore one should not use it to prove a claim about the world. The text argues that such insistence leads to extreme skepticism that nothing can really be done with. The text adds that even in the actualist’s world, science is conducted using the same techniques, so he too uses statistics as a thinking tool. From that it follows that the statistical argument obligates him at least to adopt informativism within his own mode of thought, even if he adds that this adoption is a statement about himself and not about the world.

Theoretical entities and scientific Platonism

The text presents the debate between actualism and informativism as a debate about the existence of theoretical entities such as field, force, particle, electron, and wave function, which are not directly observed but only through interpretation of measurement results. The text states that the actualist sees theoretical entities as fictive concepts that merely organize the facts in consciousness, while the informativist sees them as entities whose existence is disclosed by scientific generalization. The text presents informativism as a position with a Platonic tint because it attributes real existence to abstract things in theory, and concludes that if the statistical decision in favor of informativism is correct, then this is an empirical proof in favor of Platonism in the scientific context.

What exists according to the actualist: law versus force, facts versus causality

The text states that according to the actualist, what exists is the actual things before our eyes and directly observed phenomena, such as a table, a phone, a moving car, and a burning fire. The text explains that the actualist accepts descriptive laws like “the law of gravitation” as a simple and convenient descriptive tool, but does not accept “gravitational force” as an entity in the world, only as a convenient manner of speaking. The text shows that the question “why do bodies fall” leads, for the actualist, to a rejection of causality in the style of Hume, so that all that remains is observed facts without commitment to a causal mechanism as an entity.

Intuition as intellectual perception of theoretical entities

The text offers an explanation of how an informativist “sees” theoretical entities that are not visible to the eyes, through intuition, which is a cognitive faculty and not a faculty of thought. The text says that the informativist sees with the mind’s eye gravitational force, an electron, and an electromagnetic field, just as the actualist does not doubt what he sees with his eyes. It defines this as a retrospective justification and not as a decisive argument, and places the decisive argument in the empirical-statistical success of the predictions of theories throughout history.

Practical implication: there is no single decisive experiment, but there is cumulative decision

The text states that there is no specific experiment that will decide between actualism and informativism, because both sides agree that a theory can be refuted and that sometimes a theory can be confirmed “by chance” in a single experiment. The text argues that the practical implication is statistical at the level of the collection of experiments: according to actualism, almost no experiment should succeed, whereas according to informativism one should see “tens of percent” of successes. The text presents an example of “spurious correlations” using the sticker “Rabin has no mandate to return the Golan,” and shows that an individual person can always claim consistency, but the division of the population into two groups instead of four provides statistical evidence that something non-incidental is going on. It parallels this to deciding between the theses by looking at the overall distribution rather than at a single case.

The relation to positivism and a Platonic thesis with no direct practical implication

The text distinguishes between a positivist approach, which identifies meaning in a difference only if there is an experiment that separates between theories, and a Platonic approach, which allows two theories to be different even without a direct empirical practical implication. The text argues that if theory is only a way of describing facts, then without a fact that distinguishes between them there is no difference; but if theory describes an abstract structure, then there may be two different structures between which no physical experiment can decide, and yet they are still two distinct things.

Qualification: scientific Platonism as “weak Platonism,” not the World of Ideas

The text qualifies that the decision in favor of the existence of gravitational force, the electron, and the electromagnetic field is a claim about physical entities in our world, entities that have interactions and move measuring devices, and not about Ideas in the World of Ideas. The text states that original Platonism deals with the question whether the Idea of the electron or the Idea of horseness exists, not whether electrons or horses exist. The text argues that one can be an informativist and Aristotelian, that is, accept the existence of theoretical entities in the world without accepting the existence of Ideas as entities in the World of Ideas. The text adds a principle of “philosophical uncertainty,” according to which the more certainty one seeks, the weaker the content of the conclusion becomes, and therefore scientific evidence will lead to a weaker kind of Platonism.

Moving on to mathematical Platonism and the non-empirical status of mathematics

The text presents mathematical Platonism as closer to philosophical Platonism because it deals with numbers and triangles as Ideas and not with elusive physical entities. The text argues that there will be no empirical proof here, because mathematics, and even metamathematics, are not established empirically. It illustrates that “two plus three equals five” does not depend on experiment, because even if an experiment with nuts yielded a different result, the conclusion would be that there was an error in the experiment or in the physical situation, not an error in mathematics.

Physics versus mathematics: empirical refutation is always physical

The text demonstrates that in mechanics “five plus five does not equal ten” when one combines perpendicular forces and gets a resultant of five root two, and concludes that arithmetic has not been refuted; rather, what has been refuted is the physical claim that forces combine by arithmetic addition. The text argues that when Einstein shows that physical space is not Euclidean, he is not refuting Euclidean geometry as a mathematical theory, but rather refuting the physical assumption that our space is described by it. The text states that the mathematician develops many mathematical theories, and the physicist chooses which of them fits the world, and therefore mathematics itself does not deal with the world but with “abstract truths.”

Mathematical Platonism: discovery versus invention, and criticism of conceptual confusions

The text presents the question whether mathematics is invention or discovery as a central question, and offers a non-Platonic position according to which mathematics investigates the form of our thinking, as opposed to a Platonic position according to which mathematical truths exist and we discover them. The text argues that many discussions of mathematical Platonism are “grinding words” because they mix together several different questions, and that some of the arguments brought in favor of Platonism do not bear on the question of the Platonic existence of mathematical entities. The text attributes a certain opening of the discussion to a video by Jeff Dekofsky and to “column 434” on the speaker’s website, and presents the question: “Are the claims of mathematics true even if they had not yet been discovered?”

The tree falling in the forest, sound versus a pressure wave, and a mistaken analogy to mathematics

The text states that to the question whether a tree falling in the forest makes a sound, the answer is no, because sound is a cognitive phenomenon, whereas in physics there is an air pressure wave even without an ear and eardrum. The text argues that the parallel question in mathematics is not about “sound” but about “the pressure wave,” meaning the objective truth of propositions like the Pythagorean theorem even before they were proved. The text states that clearly the Pythagorean theorem was true even before Pythagoras proved it, and that its objective truth in itself does not decide the question of the existence of triangles in the World of Ideas.

Properties and events are not entities: speed, color, and the mistake of identifying them with Platonic existence

The text presents the example of a car traveling at 80 kilometers per hour and distinguishes between an entity like a car and a property or event like speed, the color green, tallness, and kindness. The text argues that properties and events are not entities, but they can still be objectively true even without an observer, and therefore the “discovery” of a property does not require the existence of that property as an entity. The text applies this distinction to mathematics and argues that one can say that the Pythagorean theorem is objectively true without committing oneself to the existence of a “triangle” or an “Idea of the triangle” as an existing entity in the World of Ideas, because the theorem describes a relation that always holds in a right triangle in Euclidean space.

Open ending and a request for a Kabbalistic expansion

The text stops after laying “the tools on the table” and declares that the continuation of the discussion will be postponed until next time. The text notes that mathematics is the final topic in the series, with the possibility of a concluding remark. The text includes a request to later incorporate reference to “Kabbalistic senses,” such as kindness and judgment and the ten sefirot, and the speaker notes to himself to think about it.

In the previous lecture we talked about Platonism. We started talking about scientific Platonism, and I tried to show that scientific abduction, or a scientific theory that we create from observations, is actually not a statement about us but about the world. Maybe I’ll briefly review the main points so we can continue. I’m taking us back to this graph that has been accompanying us. Okay, so you see this graph in the drawing. Let’s say I’m presenting five results there. Those are the hollow circles: one, two, three, four, five. Five such results that I measured. And now I ask: what will the general theory be? The general theory is the line that connects those points. Usually in such a situation we’ll place the solid straight line. That line, and we’ll treat it as the theory. But now someone could come along and say: no, maybe this line is also the theory, because it too stitches together all the points we measured. Both lines pass through these five hollow points. They’ll differ in other places. For example, observation six. If the straight line is correct, then the force will be this one. If the dashed line is correct, then the force will be that one. And likewise here, observation seven. If the dashed line is correct, then this is the force. If the straight line is correct, then this is the force. So in fact there can be infinitely many proposals that fit any set of observations we’ve made. Meaning, once I have some finite quantity of observations and I now ask what the general theory is, what generalization we make on the basis of the observations, the answer can be infinitely many generalizations. The number of lines that fit these five points is infinite.

Now, I said that basically the dispute between the actualist and the informativist—an actualist is someone who says scientific claims are claims about us and not about the world. And the informativist says that theoretical claims are claims about the world. Usually people understand this as a philosophical dispute that can’t be decided, certainly not empirically. Rather, these are two ways of looking at the scientific process. In both of these ways the process proceeds in the same manner, and so in practice there’s no real difference. There’s no way to decide which of these two pictures is correct, actualism or informativism. And I argued, in light of the analysis here, that there is a way to do this, and even an empirical way.

And what I said was the following: if I ask an actualist—here are the five measurements, okay? What is the likelihood that the straight line—even the actualist will say that the straight line is his theory, and so will the informativist. The actualist will say that the straight line is his theory because it’s the simplest, so why choose something complicated if I have something simple? But not because this simple thing is also true. By contrast, the informativist will say that I choose the straight line because the straight line is more likely to be the correct one. Not necessarily, but it’s not a shot in the dark; it doesn’t carry the same weight as every other line. Now I’m saying: if that’s the dispute, then they’re really disagreeing over the question—both choose the straight line, but they disagree over the question of what the chances are that this choice really reflects the truth. The actualist will say: zero. The probability is one divided by the number of possible lines. By contrast, the informativist will say that the probability is not zero—I don’t know, forty percent, seventy percent, whatever it is, but not zero. In other words, simplicity has an advantage. Simplicity—we talked about Ockham’s razor, or giving the benefit of the doubt. Simplicity is a criterion for truth. That’s the dispute.

Now I claim that if that’s the dispute, then it can be decided by looking at the history of science. Because I ask myself: how many scientific experiments actually confirmed the hypothesis they were testing? Say I reached the conclusion that it’s a straight line. Now I did experiment number six. According to the actualist, the chance of getting this result is zero. You could get this result, or here, or here, here, here, here—all possible results. And this one isn’t special. It’s just simpler, but not special. So the chance of getting it is zero. Same here, right—the chance of getting this result, if I choose result seven, the chance of getting this result is zero. But in practice, when we carry out experiments and we have a theory that we test in that experiment, it’s not true that in zero percent of cases it works out. Because then it would have to be that no theory in the world survives more than one experiment. Every experiment would refute the theory I currently hold, and I’d have to replace the theory, because after all there’s no chance that this theory is correct according to the actualist; it’s simply the most convenient one among those available to me. Fine—but if “most convenient” isn’t an indication that it’s “most true,” then in the next experiment there’s no chance at all that it will be verified, that it will be confirmed. And therefore I’d expect that if I surveyed all the scientific experiments ever done in the history of science, I would discover that basically one hundred percent of them—or ninety-nine point nine nine nine nine percent of them—refuted the hypothesis they tested. That’s not the case. I don’t know what percentage of experiments succeeded, but it certainly isn’t zero percent. I don’t know—fifty percent, eighty percent, I don’t know how much, but it isn’t zero.

That itself is very clear statistical proof in favor of informativism, because it basically says that when I conclude that the straight line is the correct line, it isn’t a shot in the dark. The straight line really has a better chance of being correct than any other line—which the actualist does not agree with. Because if the probability were the same probability, then in terms of results, in every experiment I carried out there would be no chance that the result would actually land on the line where I expect it to land, because it’s just some line that is convenient for me; it isn’t a correct line in any sense. And therefore I claim that one can really decide empirically that informativism is correct and not actualism. That’s basically the claim.

Now I want to get to the genuinely Platonic significance of this, but before that maybe one more comment. The actualist might insist and say: what do you mean, what proof have you given me here? You’ve given me a proof based on statistics, right? You say: what’s the probability that the straight line is the correct one? I say zero, you say, I don’t know, sixty percent, and when we examine the experiments carried out throughout history, it’s not zero—dozens of percent succeeded. So it’s not zero. He says: but statistics itself is also a theory of ours and not about the world. You’re using statistics in this proof, but maybe statistics too is a thesis about us and not about the world, and therefore you can’t use it to prove a claim about the world. So someone who insists at that level—of course we simply reach completely skeptical territory. There’s nothing to do with such a skeptical claim.

But I do want to make a comment. I mentioned that both the actualist and the informativist use the same techniques. Science proceeds in exactly the same way even in the actualist’s world. Therefore he too uses statistics; he just claims that statistics is simply a way I think, not something that is truly correct as a description of the world. It’s my way of thinking. Fine. So in terms of your way of thinking, I’ve managed to show you that informativism is correct. Do you understand what I’m saying? Maybe I haven’t managed to prove to you that objectively informativism is correct, but from your perspective there is no such thing as “objectively.” From within your form of thinking I want to claim that not only is this how science should be conducted as we in fact conduct it—because on that both the actualist and the informativist agree—I want to claim that from within your own way of thinking, informativism is correct. So maybe it isn’t correct in the world, but you yourself should adopt not only the scientific mode of conduct that is accepted by both sides; you should also adopt the informativist philosophy. You’ll just add afterward, as a note, that this adoption is only a statement about me and not about the world. Because after all the actualist too uses statistics—he doesn’t dispute that. He only claims that statistics is our way of thinking; it isn’t something true in the world itself.

We’ll get to mathematical Platonism in a moment. So I’m saying fine—at least in terms of your way of thinking I can show you that you must adopt an informativist position, and that’s enough for me, because beyond that, to the world itself, one really can’t get with these tools. I can’t really prove it about the world itself, but I can show him that at least he should adopt an informative conception even on the philosophical level, not only on the scientific level; he’ll just adopt it as a statement about himself, not about the world. That’s a somewhat subtle distinction, but it seems to me it’s a correct claim. In any case, that’s on the principled level.

Now I want to look a bit at this move and try to understand what it says. After all, our series deals with Platonism. What does this move really say? The actualist in effect claims that scientific theory is not a claim about the world. The informativist says: scientific theory is a claim about the world. Let’s talk about theoretical entities. The theoretical entities—for example field, force, particle, electron. All these things are things no one has ever directly observed. These are theoretical entities. The theory revealed to us the existence of these entities. They are not entities that I can inspect under a microscope or with my eye or with some measuring device, because the measuring device gives me a result which I interpret in terms of the theoretical concepts or the theoretical entities. And therefore the dispute between the actualist and the informativist is over the question of the existence of theoretical entities.

The actualist says: theoretical entities—particle, force, field, and so on—do not really exist in the world. This is simply my way of organizing in my own mind the facts I observed. Those facts are facts about the world. The theory is a theory that exists only within me. In other words, I merely organize the facts for myself by means of these theoretical concepts, but I am not claiming that there really is an electron in the world, or that there is a gravitational force in the world, or an electromagnetic field, or a wave function, or whatever—the theoretical entities we know in science. The informativist claims that the facts and the generalization based on the facts are my way of discovering the existence of theoretical entities. In other words, I discover that there really are electrons in the world, and there really is an electromagnetic field in the world, and there really is a gravitational force in the world, and all the theoretical entities that we formulate within the framework of scientific theory. The claim is that these are entities; they exist in the world. You can’t see them with the eye because that’s their nature, but they exist in the world and I can posit that as a factual claim about the world.

Now this is why this thing is called a dispute about scientific Platonism. Here we return and connect to the move of Platonism. The informativist basically claims that theoretical entities have existence. These are not Ideas, not categories in the Aristotelian sense, but actual entities. Maybe they exist in the world of Ideas—or in this case, by the way, the electron does not exist in the world of Ideas; the concept of the electron exists in the world of Ideas. The electron exists in the world itself, only it exists in a way my senses do not grasp. I cannot grasp it with the senses. It is too small to see, whatever the reason, I cannot grasp it with my senses. Or gravitational force—I cannot grasp it with my senses. I do not see gravitational force; I see its phenomenal expressions, that bodies move. So I assume a force acts on them, but I do not see the force itself. So the informativist claims that my conclusion that these theoretical entities exist is a factual claim about the world. Or in other words, he is expressing here a position one could call a Platonic position, because it is a position saying that these abstract things I created in my theory are entities that really exist in the world. And the actualist says these are not entities that exist in the world; this is simply a set of fictitious concepts I built because it is convenient for me to think by means of them. That’s all, but they do not exist in the world.

Do you see why this connects to the dispute about Platonism? And if I previously proved that the informativist is right, then basically there is here a proof of Platonism in the scientific context. In other words, I prove that theoretical entities really are existing entities; there is here a proof in favor of Platonism. So in that sense this really may bring the philosophical discussion we have had until now to some kind of climax. Because the philosophical discussion I’ve had up to now dealt with philosophical arguments. I can bring arguments in favor of Platonism or arguments against Platonism. I tend toward Platonism for various philosophical reasons that I’ve laid out throughout this series. Here I claim that I have empirical proof for Platonism. I’m showing you, from the way science operates, from the history of science, that this itself statistically proves that Platonism is correct—that is, that theoretical entities are existing entities and not merely concepts I use to handle the world of appearances.

What does the actualist say does exist in the world? You hear? According to the actualist, if these are only our concepts, what does exist in the world in these areas? Actualist, not activist. Actualist means he claims—the actual. What is actually before my eyes. What is actually before our eyes is what exists. In other words, there is a table in the world, there is a phone in the world because I see it, and there are phenomena I have observed—a car driving, fire burning, and so on. Those things he does not deny. He only denies the existence of theoretical entities that are not actually present before my eyes, but that I arrive at through scientific generalization or scientific abduction. That, he says, is a claim about me. Like Hume—all generalizations are an act you perform for yourself; the world owes you nothing. It was here before.

So then does gravity exist or not exist? Does gravity exist or not exist? Do electrons exist? The theory of gravity is no problem; the actualist accepts that. I understand, but in reality he says there is no such thing? I don’t understand. Gravitational force, not the law of gravity. The law of gravity he accepts. Gravitational force speaks about some entity in the world. He does not accept that as an entity in the world. It’s only a convenient way to talk about the law of gravity. It’s convenient for me to think of it as though there is a force moving bodies, but it’s not that there really is a force there. It’s just convenient for me to think that way because that’s how I’m built.

And why then do bodies fall to the ground? Causality is, again, a problem that Hume denied. I simply see the facts. The question of why is also a question that has been thrown out of the world I’m used to thinking in—as though every single thing needs a cause. He won’t accept that either. Why does he accept the law of gravity? Why does the actualist accept the law of gravity? For now, because it’s the simplest one there is. But who said it’s true? Nobody said it’s true. He does not say it’s true. He says it’s the simplest law that describes the facts, and therefore I use it. Why use a complicated law if there is a simple law? But you stake your life on it—you get on an airplane. That’s the claim I made on the philosophical level, that I don’t think there is a true actualist. Fine, but I’m talking now about the actualist position. The actualist position says it isn’t true; it’s only the most convenient thing to use. I agree—I said those arguments against actualism.

But something else Rabbi said today: it still hasn’t been proved that there is an Idea of the electron. There is an electron, okay, but— Right, right. That’s the distinction I mentioned a moment ago, and I’ll return to it in a second. You’re right, in just a moment.

So basically what I want to say is that at this stage, when I arrived at Platonism in the scientific context, apparently we’ve advanced one more step. And that step basically says that I have empirical, statistical evidence for Platonism. It’s no longer a philosophical question; it’s now a question that has been settled by scientific tools. I am able to show you by scientific tools that Plato was right and Aristotle was not.

Maybe I’ll say one more sentence before I qualify what I just said. Basically when I ask myself, okay, but how do I really know that these things exist? After all, you don’t see them with your eyes. And the fact that I make generalizations there—like the actualist said—there are infinitely many possible generalizations. The fact that one of them looks simpler to me, why should I assume that because of that it is true? I say: first of all, factually I see that it is true. Because in fact a significant percentage of experiments succeed; they don’t refute the hypothesis. So first of all I have factual evidence. Maybe I have no a priori reason to say that what seems simple to me is true, but I have an empirical reason—I simply see that it works. It is in fact true.

Now I say: since that’s the case, I ask myself, okay, but why really? Or how am I really able to see things that are not visible? How do I manage to see that they are correct? Here the claim is indeed—and I won’t go into this again because we’ve discussed it in other contexts, maybe in this series too I think we discussed it—the talk about intuition as a faculty for knowing the world. Intuition is not a faculty of thought—I’ve elaborated on this in many places—it is not a faculty of thought; it is a faculty of cognition. And the claim is that I can look at the world and see the theoretical entities—see them not with my eyes but with the mind’s eye. But I simply see them. So just as I do not doubt what I see—the actualist does not doubt what he sees—the informativist does not doubt what he sees with the mind’s eye. And with my mind’s eye I see that there is gravitational force, and I see that there is an electron and an electromagnetic field, even though these are not visible to the eyes. They are visible to the mind’s eye. But from his perspective this is a kind of seeing. And since it is a kind of seeing, the result is a claim about the world. And therefore it is justification. But that’s not an argument; it’s only retrospective justification. The argument is what I showed you empirically—that it works. Empirically I can prove that informativism is correct. You only ask me: how can that be, after all the eyes don’t give me this. So then what does? In other words, how can it be that this works? The de facto answer says: apparently we have some intellectual faculty, called intuition, for grasping theoretical structures, for grasping generalizations, for grasping existence—the existence of theoretical entities—and that’s how we really manage to reach the correct description. But if I had brought this without the proof, then the actualist would have said: that’s how you think; I think we have no such faculty. So what, that’s not an argument.

So is there a practical difference between the two theories? In other words, can you bring a case where according to this theory it will be one way and according to that one—? No, and that is exactly the problem. That’s why everyone thinks this is a philosophical dispute, because there is no practical difference such that if you conduct an experiment you could decide. There isn’t. Science proceeds in the same way. You can ask the kind of questions that were asked here earlier, and I asked them too: why do you get on an airplane? If the chance is ninety-nine point nine nine nine that it will crash, I wouldn’t get on it. The fact that you get on it means you probably believe the laws on the basis of which they built the airplane and fly it. Fine, but then you can say I’m inconsistent. That’s not a claim by virtue of a fact in the world; I’m only showing you that you’re inconsistent, because you say you’re an actualist, but the truth is you’re not an actualist. Fine, then he says I’m not an actualist, but I want to argue in favor of the thesis called actualism. I’m inconsistent. So that itself is not an argument.

The only claim I manage to raise against this is the claim I made earlier, the claim that in the history of science the percentage of experiments that succeeded is not zero. And that itself, I think, is empirical evidence against actualism. In other words, if you ask whether there is a practical difference—there is a statistical difference. The question is what percentage of experiments will succeed; that is the practical difference. If you do many experiments in many scientific fields, according to the actualist I would expect that no experiment succeeds. And according to the informativist I expect that dozens of percent of the experiments will succeed. And that, it seems to me, the history of science has decided. Because if actualism were correct, then today we would still be holding the science of Adam the first man. Because every theory I raised would be refuted in the next experiment; I would raise another theory, and it too would be refuted in the next experiment, and no theory would survive, and I would not build scientific knowledge—having a correct theory and building more and more on top of it, and thus scientific knowledge accumulates. You can’t build on anything. Every theory you build would be refuted by the next experiment performed. Obviously. Therefore the history of science is, I think, the practical difference people are looking for. Yes, that’s the catch here.

In other words, there is no specific experiment that I’ll be able to do, but the history of science—I’ll give you an example, it reminds me of an example I’ve spoken about more than once. I’m talking, say, about spurious correlations. Right? “Rabin has no mandate to return the Golan.” I once saw that sticker on a car: “Rabin has no mandate to return the Golan.” And I asked myself—yes, during the Rabin government—I asked myself whether the owner of the car was in favor of returning the Golan or against returning the Golan. And the answer is that I can’t know. Why can’t I know? Because when he says “Rabin has no mandate to return the Golan,” this can be interpreted in two ways.

It can be interpreted as: Rabin has no moral mandate to do this, because before the election he promised he would not do it, and if he wants to change his position he should return to the public and receive a mandate for his new position. It is not moral to change what you promised before the election. That is a question of political morality. The second question is a question about the matter itself: are you in favor of an agreement with the Syrians or against an agreement with the Syrians? Today there are no Syrians anymore, but then there were. So the question whether you are in favor of an agreement with the Syrians or against one is a different question; it is independent. So I would have expected there to be four groups in the population. How many groups were there? Two, of course, right? All those who were in favor of an agreement also said there was no problem, it was moral for Rabin to change his position; and those who were against an agreement also said it was not moral for Rabin to change his position, even though there is no real connection between the questions. The group that says, “I’m in favor of an agreement, but you’re right, Rabin has no mandate because he promised otherwise before the election; he has to return to the public”—such a group did not exist. And conversely, a group that says, “It is completely fine morally, although I oppose an agreement with the Syrians”—such a group also did not exist. Only two out of the four groups existed.

And therefore this is proof that we act in a crooked or irrational way—intellectually crooked, if you like. Because we basically create a connection between questions that in truth have no connection, because we have an interest in advancing the agenda we believe in. In other words, it starts from the political question: are you in favor of an agreement or against an agreement? And you drag the moral question after the political question. In other words, if someone is against an agreement, then he will also say Rabin is immoral if he does it because he promised otherwise. So he drags in the moral question in order to support the agenda he wants. And vice versa: if you are in favor of an agreement, then you’ll say, what do you mean, things seen from there are not seen from here, and a prime minister has the right to change what he promised before the election. And again, you harness the moral question—or the specific answer to the moral question—for the sake of the agenda you desire. Therefore this is clear proof that our argument is conducted in a non-substantive and unfair way.

Now I ask: someone will come and say— Rabbi, one second—just a second—can I ask? Sorry. Continue. For the picture to be complete, someone will come and say: okay, I think Rabin does have a moral mandate to change his position, and I’m also in favor of an agreement with the Syrians. Can I say to him that he is dishonest? Absolutely not. That is a legitimate group. In other words, it could be that someone answers yes to the moral question and yes to the political question. That is not an inconsistent group. There’s no problem at all. It is one of the four groups that could appear. And every person who says that can always say: I belong to that group where, by chance, the two questions really do reach compatible conclusions—or the opposite, it doesn’t matter.

So where is my proof that the discourse is unfair? From the fact that statistically, if you take all the people, there are only two groups and not four. I cannot claim of any specific individual that he is dishonest. Because every specific individual whose answers to the two questions are correlated can say: true, I considered them independently, and these are the two answers I reached. Right? So I can’t claim that he is inconsistent. That is a position that definitely is one of the four positions that can appear here. So he says: I hold that position—what’s the problem? But when I do statistics on all human beings, and I see that if I ask a million people, say, I see that half a million hold this correlation and half a million hold the other correlation, but no one holds either of the two anti-correlative positions—then I say: statistically I see that something here is not honest. But I cannot accuse each individual person of dishonesty. It may be that he really does hold those two positions because that is in fact what he thinks independently about each one.

So why do I accuse someone of bribery? If someone took a bribe and judged someone favorably— No, no. Leave it. Shmuel, that takes us to other regions. I think this argument is clear; everyone understands it. Leave bribery aside. We could answer that too, but it would take us elsewhere.

In the end, the same claim applies here. I said to Ezra earlier—he asked whether there is a practical difference, whether there is an experiment that could decide between informativism and actualism. My answer is: there is no such experiment. No specific case will decide it, because either the theory was refuted or it was not refuted. The informativist also agrees that a theory can be refuted. And the actualist also agrees that a theory can, by chance, despite the low probability, be confirmed in a particular experiment. Therefore no specific experiment can decide between actualism and informativism. But when you look at the totality of all scientific experiments performed to date and ask what the statistics are, what is the distribution of successful versus unsuccessful experiments—once you see that the distribution is eighty-twenty, then you say: okay, statistically I have evidence. In other words, no single case by itself will constitute evidence, because with respect to every single case both the actualist and the informativist will agree. But the overall statistics show that the informativist is correct. Exactly like what I said about spurious correlations.

That’s what misleads all the people who deal with this issue, because they are looking for an experiment that will show it. There is no experiment that will show it. Therefore they think it’s a philosophical question; it is not a scientific question because it cannot be decided by an experiment. But that’s not true. It can be decided if you look at the collection of all scientific experiments that have been performed—not a particular experiment, but if you look at the totality of all the experiments and do statistics, that will decide the question. And therefore, in my view, this is a question that can be decided empirically, scientifically. It is not a philosophical question, even though there is no specific experiment I can propose that will decide it. That’s the trick here.

Okay. But according to your approach in Talmudic topics, your approach was that when I don’t find a serious practical difference, that’s a sign that the two theories are probably the same thing. No, that was not my approach. That is the positivist approach. I claimed not that—on the contrary. I claimed there is a difference between them even if I don’t find any practical difference. A practical difference for a woman’s betrothal. Yes, that I remember. So the positivist approach—the approach of Rabbi Chaim or those who are always looking for practical differences—I don’t know if Rabbi Chaim actually belongs there; it’s not clear that he does. But those who claim that without a practical difference it is basically the same thesis—that is the positivist approach, the approach that says only if there is an experiment is there truly a difference between theories. I argued, from a Platonic perspective: no. Two theories between which there is no practical difference can still be different. Why? If you understand theory as just a way to describe the facts, then if there is no fact that separates the two theories, then it is the same theory. But if you understand theory in a Platonic way—that is, as describing some abstract theoretical structure—then there can be two abstract theoretical structures such that no physical experiment in the world can decide between them, and still they are two different things. That is exactly the practical difference I brought between Platonism and Aristotelianism.

Okay. Now Shmuel, you wanted to ask earlier. Yes. I wanted to ask: if you defined the actualist as someone who believes what he sees, can one make the distinction—instead of doing the distinction statistically or by looking for a practical difference—can one just say that he believes in the experiments he sees. He puts the plate on the table, he sees it doesn’t fall. He gets on an airplane, he sees this and that. Bring him an experiment with the Higgs particle and he says: listen, just as I don’t believe in the particle, I don’t believe in the experiment—I don’t see it with my eye. So if in any case what for him is, let’s call it, almost certainly— Fine. If the distinction is in what he sees, then also— But he does believe that the needle of the ammeter moved, right? He sees that. So if the theory tells you that if you do such-and-such an experiment the needle of the ammeter will move, and he sees that the needle of the ammeter moved, then from his perspective the abstract theoretical prediction was realized in the laboratory in a way I can see. So he sees it too.

Who said that’s the explanation for the ammeter needle moving? Because that’s the fact I gave; otherwise, if that were not the explanation, then it should have failed. That is exactly the statistical argument. No, maybe there is another explanation—back to your graphs. But if maybe there is another explanation, then I say okay, don’t check this experiment; check the statistics of all the experiments performed so far. And you can’t say about all of them “maybe there’s another explanation,” because then the statistics are against you. Because if there were always maybe another explanation, I would expect it to fail almost all the time, but the fact is that in most cases it succeeds. Exactly the statistical argument that says: true, about each experiment by itself I will not manage to convince the actualist. In a specific experiment he will always tell me, “it’s a coincidence.” Therefore I say: let’s look not at one specific experiment but at the collection of all the experiments ever done in the history of science. And here I have statistical evidence.

No, regarding the Higgs particle? Yes, also regarding theoretical entities. After all, many experiments have been done—at least since there have been theoretical entities, which is in modern science—that test things that are abstract, that are theoretical, and you see that this succeeds not in zero percent of cases. I don’t know in how many, but it isn’t zero. So that means there is something here that is not a shot in the dark. In other words, yes, these theoretical predictions have statistical success. That’s exactly the claim. Even though no single experiment by itself can decide it. That’s exactly the trick I’m using here. In other words, I’m saying: this appears to be a philosophical question because no experiment can decide it, but on the other hand I can decide it empirically, and therefore it is not a philosophical question. How? Not through a specific experiment—no experiment will decide it—but by looking at the totality of experiments that have been done.

Fine. Now one final comment before I move to mathematical Platonism. This is really what I also remarked earlier and others remarked earlier. What I showed here is Platonism with limited liability. Because what I showed here is that gravitational force, the electron, and an electromagnetic field are existing entities in the world. But these entities are not Ideas existing in the world of Ideas; they are physical entities. They interact with other physical entities and they move measuring devices. In other words, I’m not talking here about observing Ideas in the Platonic world of Ideas. This is observation of the abstract structures underlying our world. The claim is that in our world there exist abstract, elusive entities of this kind that cannot be seen by the senses, and yet they exist. So one can call this Platonism; there is a Platonic flavor to this dispute, but it does not really decide the original question of Platonism.

Because if original Platonism deals with the question not of whether electrons exist, but whether the Idea of the electron is an existing being in the world of Ideas—understand the difference between the questions. Informativism and actualism dispute the question whether the electron that just passed here and was measured by the device is an entity that really exists here in the world, or whether it’s only some—I don’t know—something moved the measuring device here, I call it an electron because it’s convenient for me, but the claim is not really that an electron actually passed here. That’s the dispute between the actualist and the informativist. But the Platonist says something else. The informativist says: an electron passed here, fine, so we settled the question of informativism versus actualism. But the question of Platonism goes one step further in abstraction, and it says: okay, so I reached the conclusion that there are electrons in the world. One says there is an electron in the world, yes, as the poet said. But I ask the question whether the Idea of the electron is an existing being. Electrons exist just as horses exist, but the Idea of horseness is in the world of Ideas. The dispute between Plato and Aristotle is about that.

It is not about entities. Say an angel, for example. Suppose angels exist. That is not a Platonic position. Aristotle too could admit that angels exist, if there is some indication; whatever, he would agree that an angel exists. The Idea of angelhood, or angelness as such—that is something existing in the world of Ideas. The concrete angel is an entity, admittedly abstract, but an entity that exists in our world. By contrast, the Idea of angelhood is what the dispute between Plato and Aristotle is about, and no experiment can show that. An experiment can perhaps show that angels exist. It cannot show that angelhood is an existing thing and not merely some fiction we created for our convenience.

Therefore it is true that our move now into the question of scientific Platonism advanced us in the sense that we now have scientific proof in favor of a quasi-Platonic claim. But the claim I proved is not really the Platonic claim itself. Rather, it is a quasi-Platonic claim that theoretical entities exist in this world. Whether there are Ideas existing in the world of Ideas—that I have no empirical way to decide. That is a philosophical question between Plato and Aristotle and all that we discussed throughout the series. Therefore I somewhat qualify the significance of the things we’ve arrived at. We’ve arrived at the conclusion that scientific Platonism is probably correct, but scientific Platonism is not really Platonism in the philosophical sense. It is half-Platonism, something with a Platonic coloring. Someone who is not a Platonist will probably tend to reject it, but that is not necessary. It could be that someone will be an informativist and Aristotelian. In other words, he agrees that there are electrons and fields and forces and so on; he does not agree that the Idea of force or the Idea of the particle or something like that exists, that these are beings existing in the world of Ideas. He claims this is only a conceptual world that we created, because by means of it we organize reality. All the particles passing through here—let’s give them a common name, all of them are electrons. But it is not that the Idea of the electron is something that exists somewhere.

Okay. So there is some achievement here in scientific Platonism—I managed to prove something—but what I proved is weaker than the pure Platonic claim. And that is of course how the world works. If you succeed in making progress, you pay for it somewhere else. There is always a price. In other words, you manage to attain scientific proof, but the claim you prove will probably be weaker. In other words, the more the tool gives you higher certainty, the less information you get from it, because as I’ve said more than once, there is a kind of philosophical uncertainty principle. In other words, if you want full certainty, you have no information. If you want all the information, you have no certainty. You always have to pay for information in the currency of certainty. If you lower the threshold of certainty and accept claims of sixty percent, you’ll have more information. If you want only claims of ninety percent and up, you’ll have less information. If you want claims of one hundred percent, you have no information at all—there is no claim of one hundred percent. It’s a kind of game between these two things.

So here the same thing happens. You want scientific evidence for Platonism? I’ll get you scientific evidence, but the Platonism I’ll prove to you will be a weaker Platonism. In other words, if you want to advance and be more confident or more certain in the conclusion, you’ll pay for it in content, in the information included in that conclusion. Okay. So that’s it regarding scientific Platonism.

I want to move to mathematical Platonism. What’s the idea, what does the discussion advance for us? First of all, just to see another aspect of the question of Platonism. But because of the last comment I made here, notice the significance of the discussion. Say I manage to show that a triangle is an Idea existing as a Platonic Idea that really exists. If I manage to show that, then I’ve really shown Platonism. This is not Platonism in the same sense as scientific Platonism. Because scientific Platonism says that electrons exist in the world. These are entities that exist in our world. They are not Ideas that exist in the world of Ideas. But if I manage—or somehow reach the conclusion—that the number five exists, or that triangles, the Idea of triangularity, exists, then I really am talking about Ideas in the world of Ideas. These are not abstract entities in our world; these are Ideas. And in that sense mathematical Platonism is much closer to the question of philosophical Platonism than what we did before with scientific Platonism. Scientific Platonism does not bring me all the way to Platonism at the philosophical level. Mathematical Platonism is exactly like philosophical Platonism. It’s the same thing, and I don’t think that’s accidental. So that’s good; perhaps we’ll talk a little about that too.

So I want to deal a bit with mathematical Platonism. Naturally, you can already understand that I won’t have here a proof, an empirical proof, because—well, it’s not even mathematics, it’s meta-mathematics. Mathematics has no empirical proof, so meta-mathematics certainly won’t have empirical proof. But that’s exactly the point. Since I’m talking about a philosophical question, about Platonism in the philosophical sense and not the scientific one, then obviously the tools I’ll be able to bring in favor of this matter will not have scientific strength; they’ll be weaker, because the claim is more far-reaching. The same uncertainty principle I spoke about earlier. So here we do indeed return to the regular Platonic move, and don’t expect scientific evidence in favor of this.

Why does mathematics have no empirical proof? How would I prove, say, that two plus three equals five? How can I prove that empirically? I take two nuts, put them on a plate, take another three nuts, put them on the plate, and count how many I have in total. Suppose it came out seven—what would your conclusion be? That two plus three equals seven? That you’ve refuted the claim that two plus three equals five? I promise you not. Your conclusion would be that there was probably some mistake in the experiment—there happened to be two nuts on the plate already and I didn’t notice. Why? Because it is obvious to you that two plus three equals five is not an empirical factual result. It is a result of analyzing ideas; it is an a priori analytic result.

But isn’t this similar to what Rabbi said earlier about statistics? In what sense? Basically a statistical proof and not an empirical one, that usually that’s okay. That’s exactly what I wanted to ask. An actualist can claim that statistics as a mathematical theory, or probability as a mathematical theory, is indeed a claim about us and not about the world. And I said that I would get to mathematical Platonism in a moment, and that subject really belongs there, not here. So here—I’ve arrived. Someone remarked that earlier? No, I wanted to ask exactly: you said that factually one cannot prove mathematics as— Refute. Let’s talk about refuting. In science we refute theories; we do not prove them. No, but two plus three—I do an experiment, I do infinitely many experiments, and it always comes out five, so why is that not a proof? Like before. No, that may indeed be proof on the scientific level. It is proof that two plus three equals five in physics. But the claim in mathematics that two plus three equals five is not built on experiment. It is confirmed in experiment, but that’s not where we get it from. We get it from understanding what two is, what three is, what plus is, and what five is. Where we get it from is fine, but we do have empirical proof. No—we have mathematical proof, not scientific proof. The claim is a claim in mathematics, not in physics.

I’ll maybe bring the example—yes, this question with nuts or oranges or whatever, two plus three equals five—I asked it when I started teaching mechanics at Bar-Ilan. So I asked the students there whether, in their opinion, two plus three equals five is a scientific claim. That is, whether one can do an experiment that would refute that claim. It seems to me someone there raised this—and if not, I raised it—let’s take nuts into a bowl and see: add two and then three and count. If it comes out five, good; if it comes out six, then we’ve refuted it. And then I said: but that’s not true, because if it came out six, and even if it came out six infinitely many times, you would not give up on two plus three equaling five. You would remain with two plus three equals five and say: okay, then apparently there is something about nuts such that whenever there are five nuts, a sixth is born.

Okay, okay, I understand. Now, why did I bring this at the beginning of a mechanics course? Because in mechanics I told them: let me show you that five plus five does not equal ten. Apply a force of five newtons northward and another force acting on the body of five newtons eastward. What is the total force acting on the body, the resultant force? Not ten, but five root two, right? The resultant of the forces is according to vector addition. The length of the diagonal. Yes, so it’s five root two, that’s seven point something. Okay, so here, I’ve refuted the claim that five plus five equals ten. No, I haven’t refuted that claim. I’m still convinced that five plus five equals ten. No, because this isn’t plus. Ah—what I’ve refuted is not a claim in mathematics but a claim in physics: that adding forces is described by arithmetic addition. That’s false. The theory of arithmetic does not fit the relations between forces. For the relations between forces I need vector addition, not arithmetic addition. But notice: that is a claim in physics. In other words, arithmetic is a mathematical theory, and vectors, vector calculus—that too is a mathematical theory. But the claim as to which calculation is appropriate for adding forces—that is a claim in physics. And that claim can be refuted by experiment. But no experiment will refute the mathematical claim. At most it will show that that mathematical theory does not fit the scenario I examined in the experiment. Another mathematical theory is needed for the physics I measured in the experiment. Exactly.

Therefore whenever something is refutable, it is basically something in physics, not mathematics. And that’s what people don’t understand. For example, when Einstein showed that our world is non-Euclidean, that our space is non-Euclidean, that it is not Euclid’s geometry that they teach in high school—what does that mean? Did he refute Euclidean geometry? Of course not. We still study Euclidean geometry to this day. Why? Because it is approximately correct at low masses and low velocities? No, because it is absolutely correct. It is absolutely correct not about our world, but as a mathematical theory. Now the question in physics is which geometry, which mathematical theory, is appropriate for describing our physical space. But that is a question a physicist asks, not a mathematician. The mathematician prepares all the geometric theories; he has an arsenal of mathematical theories. Now the physicist must come and say which of those theories he chooses as fitting what happens in the world. But that choice is made by the physicist, not the mathematician.

And therefore when I perform an experiment and discover that once people thought the world was Euclidean and now it turns out not to be, I have not refuted any mathematical claim. I have refuted the physical assumption that our world is described by Euclidean geometry, and I have not refuted Euclidean geometry itself. So in what sense do I say that Euclidean geometry itself is true? It is true in some mathematical sense; it is not a claim about the world. Okay? Therefore what I want to say is that whenever I am dealing with claims about the world, I am actually in the domain of physics, not mathematics. Even though I use formulas and theorems and mathematical claims, I am in fact operating here wearing my physicist’s hat. I am testing this mathematical theory in an experiment. And if the experiment shows me that it is not correct, I’ll throw it out. I’ll throw it out not from the mathematics books, but from the physics books. I’ll say that in physics this is not the right theory. The mathematician will go on studying it as usual—on the contrary, they especially like theories that have no application.

I hear again and again from math students—and I have a few such people in my family—how mathematicians get terribly excited when they study some esoteric subject that has no application in any field, and people explain to them, “No, in physics this has lots of applications, it’s really a super practical thing.” Meanwhile no physicist has ever heard of it and it has no application. But mathematicians really enjoy seeing their esoteric things as something that is also realized in the world. Fine, but obviously even if they had no applications in physics at all, they would continue to study it in mathematics, and that is perfectly fine. That’s how it should be. Because mathematics does not deal with the world. Mathematics deals with certain truths that are unrelated to our world—abstract truths. And that now brings us to the question of the nature of mathematics, and that is the question of mathematical Platonism.

Because the dispute regarding mathematics is actually very similar to all the disputes we’ve dealt with so far. There too there are questions like: is mathematics an invention or a discovery? Do I discover the laws of mathematics or do I invent the laws of mathematics? In other words, if there were human beings here, or creatures that are not human, who think in different patterns, then there would be no mathematics, because mathematics describes how I think, not things that happen in the world. If I’m talking about things that happen in the world, that’s physics. Notice: clearly mathematics fits what happens in the world. But that is a claim in physics. That is not a claim in mathematics. Mathematics as such does not deal with the world. Therefore the question is: so what does it deal with? Some will say: well, if it doesn’t deal with the world, then apparently it deals with us. We are simply investigating the structure of our own thought. That is mathematical inquiry. Okay? We are investigating the human intellect, our own. By contrast, there is a Platonic claim that says no: mathematical truths are truths that exist in the world and we discover them. We do not create them; we discover them. Just as we discover physics, so too we discover mathematics. Fine, mathematics does not exist in our world but perhaps in the world of Ideas, yet we discover it and do not create it. So mathematics is not an invention but a discovery. That is basically the question of mathematical Platonism.

Okay. Now I want to do this a bit more precisely, because my feeling is that discussions of mathematical Platonism are usually just grinding words for nothing. The people who engage in them do not distinguish between several different questions one can ask here, and I claim that one question is unsolvable and cannot be answered with ordinary tools—the philosophical question of Platonism—and most of the questions they deal with are not that question at all. They are other questions which for some reason seem to them to be the same question, but they are not. And there the answer is simple and obvious, so there’s no point fussing with it. In short, discussions of mathematical Platonism always smell to me of conceptual confusion. You bring arguments in favor of Platonism, but those arguments actually deal with a different question, not with the question of Platonism, which you happen to identify with the question of Platonism unjustifiably. And then when we return to the actual question of Platonism, after I’ve shown you these are two different questions, you’ll be left without arguments. Either you’re a Platonist or you’re not. But all those arguments are irrelevant.

So let me start showing you this. It all begins with Shmuel—he’s really to blame for what will now happen. He once sent me a video of a man named Dkovsky—actually Jeff Dkovsky—who devotes that video to the question of mathematical Platonism. So he begins—there’s an article on my website, article 434, I’ll send a link later, and there I describe the video critically, yes? I follow what happens in the video with criticism. So Dkovsky begins with the question: are the claims of mathematics true even without having been discovered? Right? If a tree falls in a forest, does it make a sound? So he asks the same question about mathematics. Are mathematical claims true even without having been discovered?

You know what, maybe since the tree falling in the forest has come up here, let’s take a look at it for a moment. I’ve talked about it already, but I want to make the comparison between the questions. The famous question: if a tree falls in the forest and nobody is there, does it make a sound? The answer is of course no. Clearly it does not make a sound. What it does is move air; there is a pressure wave in the air as a result of the fall. Now if there is an eardrum there, an ear with an eardrum, and the pressure wave hits the eardrum, then in that person’s consciousness a sound will be formed. Sound is a cognitive phenomenon; the pressure wave is a physical phenomenon. This is in the subject and that is in the object. Okay? Therefore as long as there is no eardrum there, there is a physical pressure wave there, but there is no sound. Sound is a phenomenon.

Now if you are a Platonist—right? What’s the connection here to Platonism? So I say there is no connection to Platonism. As for the question of sound, the answer is: it does not exist. It exists only within my consciousness. It is not a Platonic Idea; it is a mental phenomenon, a cognitive phenomenon—the phenomenon of sound. Okay? But I am not thereby arriving at any skepticism, because I say that when a tree falls in the forest, clearly a pressure wave is created there in the air. That is an objective claim; it is not a claim about me. Translating that into sound is a statement about me. But the statement that a pressure wave is created there in the air—that is a claim about reality itself, and it is a true claim about physical reality.

When I ask the same question in mathematics, I am basically asking the counterpart of the question about the pressure wave in the air. Was the pressure wave in the air there even if I’m not there? And there the answer is obviously yes. Clearly a pressure wave is created in the air. I won’t hear it if I don’t put my ear there, but a pressure wave is created there. Now when I ask whether mathematics is an invention or a discovery, that is the counterpart to the question whether there is a pressure wave there, not whether there is a sound there.

Now let’s see. Say I’m talking about the Pythagorean theorem. Okay? So think for a moment about this question. What does it mean to ask whether mathematics is an invention or a discovery? Does it mean that before the Pythagorean theorem, the Pythagorean theorem was also true? That’s the question. Before Pythagoras came and proved it. Was it still true? Obviously yes. Does anyone say no? What kind of nonsense is that? What does that have to do with the Platonic question? The Platonic question is whether triangles or Pythagorean relations are Ideas that exist in the world of Ideas. That is not connected to the question whether the Pythagorean theorem was true before Pythagoras proved it. Even if it does not exist as an Idea in the world of Ideas, it is obvious to everyone that it is certainly true, and it was true even before Pythagoras proved it. This relation between side lengths in a right triangle is objectively true. Again—not in the world, not in physics, but in triangles, Ideas of triangles. Okay? That relation is objectively true.

I’ll try to illustrate this more, just because we’ll soon have to finish and I don’t want to leave this hanging in the air. Look: when I speak about a car, I can ask whether a car really exists or whether it is only an image in my consciousness. Fine, the accepted views are that the car certainly also exists. It’s not that there is only an image in my consciousness. Now I ask: what about the speed of the car? The car is traveling at 80 kilometers per hour. Is the speed of the car a speed that exists even if I’m not observing the car, or only if I observe it does it have a speed of 80 kilometers per hour? I assume we would agree that it has a speed of 80 kilometers per hour even if I’m not observing it. I may not know it if I’m not observing it, but it has a speed of 80 kilometers per hour.

Now notice: the fact that the car has a speed of 80 kilometers per hour does not mean that 80 kilometers per hour is an existing Idea, that it has objective existence. 80 kilometers per hour is a description or characteristic of the car. The car is traveling at 80 kilometers per hour—that is not an entity, not even an abstract entity. Okay? In other words, attributes of things are, on the one hand, not entities. The green color of the table. The green color is not an entity. It is a characteristic of the table. The goodness of a person. The goodness of a person is not an entity. It is a property of the person; the person is an entity. A kind-hearted person has the property that he is kind-hearted. A tall person has the property that he is tall. But his tallness is not an entity. It is a description or characteristic of a being.

Does that mean it is subjective? Does it mean that if I don’t look at it then it doesn’t exist? Obviously not. Descriptions or characteristics too have objective standing. They do not exist in the same sense in which the table exists or the car exists, because there I am talking about entities and here I am talking about properties, events. The event of the car’s traveling. The event is not an entity, but it is not true that I invented the car’s traveling, that it exists only in my consciousness. It exists in the world itself—exists in the sense that the event occurs in the world itself, not in the sense that it exists in the same way objects exist. But the question whether this thing is an invention or a discovery is not connected to the question whether it exists or does not exist. Do you understand the distinction I’m making? In other words, I can say that the car’s speed is 80 kilometers per hour even if I don’t measure it. When I measured the car I discovered something; I didn’t invent something when I measured the car’s speed. Does that mean that the speed of the car is an entity, that it exists like entities exist? No. The speed of the car is an event. It is a property of the car. This event that it is traveling, this property that it is traveling at 80 kilometers per hour. Events and properties are not entities. They are not entities, but on the other hand they are not something subjective. They are something that is objectively true. Not objectively existing, but objectively true. The occurrence happens in the world, not in my consciousness. But that does not mean that an occurrence is an entity.

Is it clear what I’m saying here? Because in all these discussions—look at the article afterward, I’ll send the links to the articles—in all these discussions there is a complete confusion between these two questions. The Platonic question is whether mathematics, the beings of mathematics and the relations between them, exist, and whether mathematics is an invention or a discovery. I can say that mathematical beings do not exist—I am not a Platonist. There is no such thing as a triangle or the Idea of triangle. There is a concrete triangle; if I drew it then there is a triangle. But the Idea of triangle has no existence in the Platonic sense. That is a conceptualization by human beings. And still the Pythagorean theorem is a theorem that is objectively true. It is objectively true because every triangle you draw in Euclidean space—a right triangle—has the property that the lengths of its sides satisfy the Pythagorean theorem. Do you see? The question whether this is a discovery or an invention is a different question from the Platonic question of whether it exists, whether the beings of mathematics exist in the Platonic world of Ideas or not, whether these are properties of my thought or whether these are beings that exist in the world. That is an entirely different question.

Now it is an entirely different question in one direction. In other words, if triangles are existing beings—if I am a Platonist—then clearly mathematical truths are objective truths, right? Because they describe those triangles that exist in the world of Ideas independently of me; they have the property of the Pythagorean theorem. But the reverse is not true. In other words, if I say that the Pythagorean theorem is an objective truth and not merely my invention or my way of thinking, that does not necessarily mean that there are triangles in the world of Ideas, that there is a triangle in the world of Ideas. It describes relations between mathematical properties or mathematical ideas, and those relations are objectively true, not because I am built in a certain way.

Okay, I’ll stop here, because I’ve only put the tools on the board. We’ll have to discuss this next time. Any comments or questions?

Rabbi, is this the last topic in the series? What? Mathematics? You said this is the last topic. Right, this is the last topic. It could be that in the end I’ll still have some concluding remark, one more comment of some kind—we’ll see—but yes, this is already almost the end.

I wanted to ask whether, if possible, you could bring in—if you have something to say about it—Kabbalistic meanings. Once you spoke about how kindness and judgment are— Okay, I hear. I’ll try to think about that. I mean, I intended that you’d maybe start blending things here—if you could, like when you once talked about kindness and judgment and identified them as really two properties of reality that even in actual concrete reality do exist, and in many teachings people speak about them too, and maybe also, I don’t know, the ten sefirot, things of that kind. I’m not knowledgeable enough, but if you have something to say about that I wanted to ask. Okay, I’m making a note to think about it. It could be we’ll touch on that indeed, okay? It sounds like something that definitely belongs to the flow here. Thank you.

Okay, that’s it? Good, then have a peaceful Sabbath. Goodbye.