Platonism – Lesson 39

Table of Contents

• The distinction between scientific Platonism and mathematical Platonism
• Discovery versus invention, and the critique of how the question is framed
• Mathematical language versus mathematical entities
• Object and description: the example of the “center of gravity” and its implications
• A critique of Gadi Taub and the possibility of different languages for the same physics
• Quantum theory as languages of description and transitions between representations
• Zeno’s arrow paradox and velocity at a moment in time
• Operational definition, the Doppler effect, and measuring velocity
• Objectivity without an “entity”: velocity as a property and not a thing
• Mathematics as philosophy and the mathematician’s experience as Platonic
• “The world is written in the language of mathematics” versus Pythagoreanism
• The fit between mathematical theories and physics years after their development
• Three levels of claim about mathematics: truth, independence from application, and Platonic existence
• Uses, non-uses, and negative and complex numbers
• Contradictory theories and objective truth conditional on assumptions

Summary

General overview

The text distinguishes between scientific Platonism, which deals with the question of whether theoretical entities such as an electron or an electric field exist in our physical world, and mathematical Platonism, which returns to the ordinary Platonic meaning and asks whether general entities such as a triangle or a number exist in the world of ideas. The speaker argues that the debate over “invention or discovery” with respect to mathematical truths is confused, because propositions such as the Pythagorean theorem were true even before human beings existed, whereas invention may refer to the language or representations through which mathematics is described. He presents a central distinction between mathematical entities and relations on the one hand, and the languages and conceptual tools used to describe them on the other, and applies this distinction to physics as well and to different ways of formulating the same truths. He adds a distinction between the objective truth of a mathematical theory and the question of the Platonic existence of mathematical entities, and states that he himself is a Platonist, while arguing that the fit between mathematics and physics proves mainly objectivity, not necessarily Platonic existence.

The distinction between scientific Platonism and mathematical Platonism

The text states that the Platonism of physics asks whether theoretical entities such as an electron, gravitational force, or the wave function exist in our world, or whether they are merely tools for organizing phenomena. The text argues that even if one accepts this kind of “Platonism,” these are still material-physical entities in the world, and therefore the use of the term Platonism here is borrowed. The text states that mathematical Platonism asks about the existence of general entities such as triangularity or horseness in the world of ideas, and that this is the original dispute between Plato and Aristotle.

Discovery versus invention and the critique of how the question is framed

The text presents the question of whether mathematical claims were true before they were discovered, framing it as discovery versus invention, but argues that this discussion is “nonsense” and that the answer is obviously in favor of discovery. The text argues that the Pythagorean theorem was true before Pythagoras and even before human beings existed, just as a table can exist even if there is no one to see it. The text states that Fermat’s conjecture was true even before Andrew Wiles proved it, and that Wiles did not “create” the theorem but rather proved and discovered it.

Mathematical language versus mathematical entities

The text presents a distinction between the language used to describe mathematical objects and the objects themselves, illustrating this by means of representing a point on a plane in Cartesian coordinates as opposed to polar coordinates. The text argues that systems of representation such as that of Descartes are, in a certain sense, the invention of a language, whereas basic concepts such as angle, distance, projection, and point are not human creations but conceptualizations of something real. The text uses a comparison to Chomsky and argues that languages such as Hebrew or English are human creations, whereas linguistic ability is an innate capacity and not a human invention.

Object and description: the example of the “center of gravity” and its implications

The text brings an example from analytic philosophy of “the center of gravity of the solar system on such-and-such a date at such-and-such an hour” and argues that this whole complicated description ultimately points to a particular point in space. The text states that understanding the description requires a complex conceptual system, but the object itself may be simple, and thus the difference between the object and the language that describes it becomes evident.

A critique of Gadi Taub and the possibility of different languages for the same physics

The text mentions Gadi Taub and his book “The Stooping Revolt,” and his claim that he would not want to fly on a plane built on “feminine physics,” and agrees that there is correct physics and incorrect physics, but disagrees with him on the point about language. The text argues that the same physics can be described in different languages, similar to different bases for describing numbers, and therefore a possible gender difference could be a difference in the style of language and description rather than in the physics itself. The text adds that it is even possible in principle to handle physics using conceptual tools that are not mathematics, among creatures with a different perceptual structure, without that changing the physics itself.

Quantum theory as languages of description and transitions between representations

The text describes how in quantum theory there are different languages for describing the same reality, such as description in position space as opposed to description in velocity/momentum space, and that there is a mathematical transition between the representations. The text states that the same physical state can be described in both languages, but that a sharp description in position translates into a complex description in velocity space, and connects this to the uncertainty principle. The text suggests that we are “creatures who think in position,” but there could be creatures who “think in velocity,” seeing the same physics through different conceptual lenses.

Zeno’s arrow paradox and velocity at a moment in time

The text presents Zeno’s arrow paradox and rejects the formulation “at every moment it stands in a different place,” replacing it with “at every moment it is in a different place,” while distinguishing between “stands” as having zero velocity and “is” as position alone. The text argues that there is velocity even at a single instant, and that the need for a time interval in order to calculate velocity is a computational need, not an ontological claim about the existence of velocity. The text argues that this solution removes the paradox, because the arrow “at every moment is also moving” in the sense that it has velocity, even if at the point itself it does not “change place.”

Operational definition, the Doppler effect, and measuring velocity

The text defines an “operational definition” as a way to calculate or measure a quantity, and emphasizes that this is different from the essence of the quantity. The text brings the Doppler effect as an example showing that velocity can be measured in a way that conceptually depends on an encounter at a single moment in time, thereby illustrating that a body has velocity at a moment in time even if ordinary measurement requires an interval. The text compares this to the fact that the existence of things does not depend on our ability to perceive them, just as a dog can hear frequencies that human beings cannot hear.

Objectivity without an “entity”: velocity as a property and not a thing

The text argues that the question of whether something is “objective” is not equivalent to the question of whether it is an existing “entity,” and illustrates this by the velocity of a car as a property or event in the world, not as an independent entity. The text states that velocity is not a hallucination and is not subjective, and that measuring it is discovery even if there is no separate entity here. It uses this to argue that the fact that mathematics is objective and discovered does not by itself settle the question of the Platonic existence of mathematical entities.

Mathematics as philosophy and the mathematician’s experience as Platonic

The text argues that mathematics is a branch of philosophy in a metaphysical sense, because it deals with entities not accessible to physical observation and is attained through thought and intuition. The text describes how, in the Platonic conception, understanding the Pythagorean theorem is “seeing with the eyes of the intellect” the idea of a right triangle, while using a concrete drawing only as a means. The text recounts that the speaker conducted a poll in a mathematics seminar at Bar-Ilan and all the faculty identified themselves as Platonists, and argues that the experience of everyday mathematical work naturally leads to a view of discovering objective entities.

“The world is written in the language of mathematics” versus Pythagoreanism

The text distinguishes between Pythagoras and the Pythagoreans, who claimed that everything is mathematics and that one can arrive at the laws of nature without observations, and the modern position according to which physics is well described in mathematical language after observation. The text illustrates this by comparing a mathematical theorem such as the sum of the angles in a triangle with Newton’s second law \(F=ma\), and argues that the former can be proved without observation whereas the latter requires measurements even though it is formulated mathematically. The text states that the physical equation is physics written in mathematical language, not mathematics in itself.

The fit between mathematical theories and physics years after their development

The text presents a double wonder: the suitability of mathematical language for describing physical relations, and the fact that mathematical fields created without any known application later become central tools in physics. The text gives examples such as tensor calculus in general relativity, group theory and Hilbert spaces in quantum theory, and argues that this “calls for explanation” if one assumes that everything is a matter of accidental invention. The text concludes that this fit supports the idea that mathematical theories are objectively true, but warns that this still does not require the Platonic existence of mathematical entities.

Three levels of claim about mathematics: truth, independence from application, and Platonic existence

The text presents two claims and one question: the first claim is that every mathematical theory is an objective truth, in the sense that if somewhere in the world there is a context that satisfies its assumptions, then its theorems will hold there. The second claim is that a mathematical theory is true even if there is no context anywhere in the world that actually realizes it, and the text illustrates this with the image “every ball is round even if there are no balls in the world.” The third question is the question of Platonism: do mathematical entities and relations exist in the world of ideas? The text states that the speaker is a Platonist, but grounds the justification for this in the general arguments of philosophical Platonism and not only in the fit between mathematics and physics.

Uses, non-uses, and negative and complex numbers

The text notes that mathematicians tend to exaggerate claims about “lots of applications,” even though most mathematics is not applied in physics, and argues that the absence of application does not detract from the objectivity of mathematical truth. The text answers a question about negative numbers and argues that they have many applications, giving the example of “holes” in physics and in semiconductors as a way of replacing a description in terms of minus electrons with a description in terms of holes, and attributes this to “Dirac’s trick.” The text argues that complex numbers have applications as a technique, and suggests that perhaps the quantum wave function is an example of a physical entity whose values are complex.

Contradictory theories and objective truth conditional on assumptions

The text states that contradictory mathematical theories can exist because they proceed from contradictory assumptions, and that both are objectively true in a conditional sense: if the assumptions hold, the theorems hold. The text illustrates this with Euclidean versus non-Euclidean geometry, and argues that in our world space cannot fit both at once, but each is correct relative to models that satisfy its assumptions. The text concludes by saying that the arguments about predictive success and fit with physics point to the correctness and objectivity of the method, but are not identical with the third claim about the existence of entities in the world of ideas.

Okay, last time I began the discussion of the question of the Platonism of mathematics, and the background to it was really the previous discussion, where I dealt with the Platonism of physics or of science, and I said that there is a difference between the two discussions. The discussion about the Platonism of physics is really a discussion about our world, not about the Platonic world of ideas. We are basically dealing with the question whether an electron is an entity that really exists in the world, or whether it is just something we invented in order to organize the phenomena for ourselves. And the same goes for an electric field, or whatever, or a gravitational force, or a wave function, or all kinds of theoretical entities. But even if I am a Platonist in the scientific sense, meaning if I think that the theoretical entities, the abstract entities, exist, we are talking about entities that exist in our world. An electron is a particle running around here in the world. The question whether it exists or does not exist is only because I do not see it, but in principle, assuming that I claim it exists, then that is a claim about a material object that exists in our world. Therefore the concept of Platonism in the scientific context is a somewhat borrowed concept, I would say. It is called Platonism because there too we speak about the existence of entities that we concluded exist by means of some theoretical analysis. But in mathematics, when we deal with mathematical Platonism, there we return to Platonism in its ordinary sense. Meaning, when I ask the question whether a triangle is an existing entity—not a particular triangle that I happen to see in front of my eyes right now, but triangle as such, in general—is it an existing entity, that is a question about the Platonic world of ideas, not about our world. Triangularity, right, or horse-ness, these are not entities that exist in our world but in the world of ideas, and this is really the original dispute between Plato and Aristotle. So when we now come to discuss mathematical Platonism, we are basically returning to that discussion that has accompanied us all along about Platonism in general. And here I want to say a bit more—I made the initial distinction last time, but I’ll repeat it in order to get back into the matter. Basically, the question is whether the claims of mathematics were true even before they were discovered. Right? Is it discovery or invention? Some ask whether numbers or certain polygons are entities. Meaning, not of course a particular triangle. A particular triangle that I drew on a sheet of paper in front of me, yes, that is an entity. But I am talking about the concept triangle, or various other questions. I use some video that Shmuel once sent me: is mathematics an artificial structure or a universal truth? Is it a human product or a divine creation? All kinds of things of that sort, all of which—you can find all these formulations in texts, videos, or articles that deal with the question of the Platonism of mathematics. But as I began to show last time, these formulations are really not equivalent to one another. They are several completely different questions. The question whether mathematics is invention or discovery, to my mind—I don’t know, people keep grinding water over this for so long—and it is simply nonsense. Obviously it is discovery and not invention. Meaning, if I want to say: if I draw a right triangle, is the relation between the sides—Pythagoras’ theorem, right, that the sum of the squares of the legs equals the square of the hypotenuse—was that not true before it was discovered? Obviously it was. I mean, obviously Pythagoras did not invent that theorem but discovered it. I really do not understand how one can even discuss this ridiculous question of whether this is an invention or a discovery. Of course—even if I don’t speak only about Pythagoras, but go back to before there were human beings in the world at all, when there was no human intellect, no human thought—was Pythagoras’ theorem true then? To me it is obvious that yes. Of course yes. So what if no one was there to think it? This table here too would not be seen by anyone if there were no human beings in the world, or if there were no creatures with eyes in the world. So because of that, would this table not exist? I cannot even understand this discussion; in my opinion it is simply confused people. The difference—there is one point that may be easy to miss, actually two points. The first point is the point about language. Mathematical language can certainly be seen as a human creation. One can argue about it, but this is certainly a reasonable position. What do I mean? Let me take an example. Suppose I am looking at a point in the plane, and I can describe it in Cartesian form, give its x and y components, the projections onto the two axes. Okay? You see? Sorry for the barbaric drawing. Okay, this is the x and y axis, and here is some point. Okay? Now one representation of this point is through its projections on the axes. This is its x and this is its y. So this is, I don’t know, say two and this is one and a half. Okay, suppose. Fine? So these are the representations of this point in Cartesian language, which is a language that Descartes basically created, and here I can perhaps say invented—that is why I bring this example. In Cartesian language, then, the representation of this point looks like this: two comma, the x component is two and the y component is one and a half. There is another language in which this point can be described, and that is polar language. Here there is the angle theta, and here there is the distance from the origin, which is r. The distance from the origin is this, r, and the angle is theta. So I can certainly describe this point in that way as well, if I give its r and its theta. Now you understand that these are simply two different ways of describing the same point, right? I can describe it in Descartes’ language, Cartesian language—Descartes is Cartesius in Latin, and therefore it is called the Cartesian coordinate system. So I can describe it in Cartesian language this way and I can describe it in polar language that way. Of course there are more languages in which I can describe it; there are infinitely many languages in which I can describe it. As for the language in which I describe it, there is definitely room for the claim that this language is a human creation. It could have been described in all sorts of ways, and as long as someone had not created that language, had not conceptualized that language, then perhaps one could claim that the language did not exist until it was invented. But the concept angle that we use here, the concept angle or the concept distance or the concept projection or something like that—these are the basic concepts with which I build those languages—these are mathematical concepts that, quite simply, human beings did not create. Human beings merely conceptualized them, discovered them, but these are concepts that already existed. You can explain the different languages to people while using concepts such as projection, axis, and so on, distance, angle, and so forth. These concepts are concepts shared by all of us. The use of those concepts to create a language—Cartesian, polar, or whatever, any other language—that may be the creation of a language like Hebrew, English, or whatever, and here one can indeed say that this is a human creation. By the way, Chomsky, the well-known Jewish-American linguist—somewhat Israeli in the past through his parents, really—wants to claim that people are born with certain linguistic capacities. Meaning, there is some linguistic ability that is built into us. That we do not learn; it is in us. We use that ability to learn a particular language: English, Hebrew, Portuguese. So here I think it parallels the distinction I made here. Basically I am saying that the language—Portuguese, English, Hebrew—would not have existed if people had not created them. So language can certainly be called an invention rather than a discovery. But linguistic capacity, linguistic ability, is not something we invented. Linguistic ability is something that existed before we were here. In this case one can perhaps say that it is built into us, but it is built in—we did not create it. Therefore linguistic ability is not something that, it seems to me, we can say we created. Okay, so that is a first distinction that is important to make: the distinction between the language through which we present mathematical entities and the mathematical entities themselves. Say, a point. The concept point, which we just represented in those two different languages, is a mathematical concept, and the discussion about the Platonism of mathematics concerns that, not the question whether I present it cartesianly or present it in polar form. The example they often give in philosophy—analytic philosophy—is the center of gravity of the solar system on such-and-such date at such-and-such hour. Suppose we describe something like that. What does this whole complicated sentence actually describe? A certain point in space. You can point to it with a finger: that one. That point is the center of gravity of the solar system on such-and-such date and such-and-such hour. But all this grand description is just a way of pointing to a certain point in space. In order to understand the description, I need to understand many concepts that are not at all simple: gravity, center of gravity, the solar system, dates, hours—many, many things. Meaning, a small child would not understand this description. But if you point with your finger to this point in space, then he understands: there is a point in space. In order to see that point in space, he does not need to know the whole conceptual system that I used in the description of the point. And again, this is the difference between the object and the language in which I describe it. The object in itself can be a simple object, objective, and so on. The language I choose to describe it can be a language that is my invention or the invention of some human group that uses it. I do not think I mentioned this, maybe I’ll mention it here. Gadi Taub has a book—from the period before he started his move rightward, when he still strongly claimed to be a leftist—and he wrote a book called The Bent Rebellion. And there he brings various quotations; I think he was actually the one who made accessible to us this American phenomenon, this utterly insane American craziness of everyone with his own truth, and the undermining of all the most basic concepts—today it is already genders and so on—and in short, what is called the new critique. Among other things he brings there various feminists who claim that physics is a male field, and therefore it is no wonder that women do not succeed in it. And Gadi Taub writes there that he would not want to fly on an airplane built on the basis of female physics. Meaning, in short, there is correct physics and incorrect physics; there is no such thing as male physics and female physics. If you invent a different physics, it will simply be incorrect physics. And if you build airplanes on the basis of a different physics, they will crash, or you will not be able to build them at all. So he tried to ridicule this claim that it is somehow a masculine field. It always reminds me of something I once heard: an interview on the radio with the principal of a Tel Aviv school, who was very, very proud of the fact that he did not allow computer science to be taught in the school, because in his experience girls succeed less than boys. Therefore he would not allow computer science studies into the school. A wonderful solution. And I think one can stop teaching altogether, and then everyone will succeed perfectly. So yes, it very much reminds me of Gadi Taub’s argument—but here I actually want to disagree with Taub. I want to say that he too did not make the distinction I spoke about earlier. There is—one can claim—that the correct physics we know today could have been described in another language. It would not be a different physics. It would be the same physics described in a different language. We know, for example, that worlds of numbers can be described in many languages. There is decimal, binary, hexadecimal, any base you like. That is another language for describing the world of numbers. And it is entirely possible that there are even languages that are not by base at all, but I don’t know, something completely different. That does not mean you cannot describe all the mathematical or physical truths in this new language. And they might look completely different. I’ll tell you more than that. It may be that all our physics today, where we are so accustomed to using mathematical language when dealing with problems in physics, maybe those problems in physics could also be solved without mathematics. Maybe if there were creatures with entirely different kinds of minds, they would somehow manage to deal with the same physics. It would be exactly the same physics. But they would deal with it in languages, or with cognitive tools, a completely different conceptual system. You might say that their physics would not be built on the concepts—the basic concepts with which we build our physics—of space, speed, acceleration, force, mass, time, and so on. Those are the basic concepts on which we build our physics. There could be another world in which they build physics on the basis of completely different concepts. And they could build an airplane on the basis of their physics, and it could indeed—maybe it would be a completely different airplane—but it could fly. And maybe it would be the same airplane as ours, only the calculation and design papers would not look like ours; they would look completely different. Just as there is translation between Hebrew and English—you can design an airplane in Hebrew, in English, in Japanese, or in any other language. One might be able to design an airplane in mathematical language and also in completely different languages. Therefore there is room for the claim that the language in which we today describe physics is a language developed by men. And if women had been more involved in it from the outset, maybe it would have developed into a different language, and then perhaps women would have had more success in physics. That is entirely possible. And it would be the same physics, and it would build the same airplanes, and the airplanes would fly. The same airplanes—or different ones—but they would fly. Therefore I think Gadi Taub too is confusing the question whether it is different physics or a different language for describing the same physics. And in this sense—and I think a great many of the discussions around the Platonism of mathematics fall at exactly this point—because one can say that mathematical language is our invention. Yes, mathematical language is something we invented. But mathematical truths—say, perhaps one could describe Pythagoras’ theorem in a completely different language, without speaking at all about sides, angles, and triangles, but someone might see it in some other way. Yet there would be some relation between the concepts in his language which, when translated into our language, would look like Pythagoras’ theorem. And that is really what I want to claim. It may be that in his world there are no angles, no sides, no triangles, no nothing. He does not think in that way. But he would have other concepts whose interrelations would be some expression of Pythagoras’ theorem. Therefore I think it is very important to distinguish between the claims themselves and the language in which I describe those claims. In the physical world, in quantum theory, we are accustomed to speaking about different languages in which reality can be described, and there it is built into the theory itself. Meaning, one can describe a particle through the position in which it is found, and that is a language of position, and then all the properties in that language are functions of place, of x y z, of space. And one can speak about a language of velocities or momentum, in physical language—let us speak about velocities so it will be clearer—and describe everything as a function of velocities, not as a function of positions. Now in quantum theory it is well known that these are two languages that do not speak to each other. You have to choose: either you use this language or you use that one. But everything you can describe in this language, you can describe in that language. If you describe a particle that is in a very specific place, then in the language of position you say it is at this place, x equals two. In the language of velocity you would need a sum of infinitely many functions, each describing a different velocity, and you would describe the same thing. You can describe the same thing in the language of velocities, only there the description will be more complicated. So one can move from language to language and describe the same things in completely different languages. And in quantum theory specifically there is the uncertainty principle, which says that once you have a certain position, in the language of velocity you will not be able to describe it with one function—a function describing one specific velocity—but you will need many functions with different velocities. That is already a theorem in physics. But on the conceptual level, in terms of forms of description, there is no obstacle to there being different languages that describe exactly the same thing. And if there were a person who thinks in terms of positions—and by the way, we are creatures who think in terms of positions—then we would see the whole world in the language of place. But there might be completely different creatures who think in the language of velocities, let us call it that, or momenta, and they would see the world in a completely different way. Therefore—I once illustrated this in an article about Zeno’s arrow. Zeno the Greek from Elea presented various paradoxes concerning the concept of motion, and all those paradoxes are basically connected in one way or another—many of them, not all, but many—to the distinctions I am making here. For example, he says: look at a flying arrow. At every moment you look at it, it is in a different place. Right? It stands here, and then stands here, and then stands here, so at every moment it stands in a different place. So the question is: when does it move? When does it pass from here to there? That is the paradox, Zeno’s flying arrow paradox. And I wrote an article in which I proposed a solution to this paradox, and my claim was basically that it is not correct to say that at every moment it stands in another place. What is correct to say is that at every moment it is in another place. There is a difference between stands and is found. To say that a body stands in some place is to say that its velocity is zero. To say that a body is found in some place means that its position is there, but it can be there while in motion. And what confuses people so much—why this paradox confuses people so much—by the way, in my opinion it was not solved until my article; all the solutions offered for it, connected with calculus, in my opinion do not solve the paradox. And what I think confuses people is that they think one cannot speak of the velocity of a body at a point, at a spatial point or a point in time. A body—if you want to speak of the velocity of a body, you need to look at an interval, a stretch of time as short as you like, but still some interval. At a single point in time, a body cannot have velocity. And I claimed that this is not correct. Therefore the point is that if the body is at some place, then it must also be standing, because at some place, at a point in time or at a point in space, it has no velocity; it stands. That is why people get entangled with this paradox of the flying arrow. But that is a mistake. The point is that when we calculate velocity, we calculate it by change in place divided by change in time it took to traverse that distance. Okay, so in order to calculate the velocity I need to look at the body over an interval of time or over an interval of space, but that is only for the sake of the calculation. Once I have finished the calculation, the body has a velocity at a point in time. Every point. Every point in time you give me, I can tell you what the velocity of the body is at that point in time. If you ask physicists, they will usually tell you that this is a fiction. It is the velocity of the body over a very small interval around the point I am talking about. Not true. The velocity of the body is a velocity at that specific point in time. To calculate it I need to look at the body over a stretch of time. That is a problem of how to calculate. But the result of the calculation is velocity at a point; it is not velocity over an interval. It is velocity at a point. Therefore I think there is such a thing as the velocity of a body at a point in time. And if so, the paradox of the flying arrow disappears. Why? Because you ask me: the body is in another place at every moment—when does it pass from place to place? When does it move? The answer is: at every moment you look at it, it is also moving. It does not change place at that point in time—that is clear, that cannot happen—but it has velocity nonetheless. It will be expressed at the next moment in time. And at the next moment in time it will be found in another place. So it does not change place at a point in time, obviously that cannot happen. But it has velocity even at a point in time. And what lies behind this—and from there I moved on to quantum theory and relativity—is what I think is the conceptual or philosophical basis of the uncertainty principle in quantum theory. Or complementarity, really—uncertainty is already something more specific—but this principle that says you cannot speak about velocity and position simultaneously. Meaning, you cannot really know velocity and position simultaneously. Because in order to know velocity you need an interval of space or an interval of time. And in order to know position—when you look at a flying arrow—then whoever looks at the film, it depends how he looks. If you look at it through the eyes of position, then at every moment you see it in another place, and you cannot know its velocity. In order to know its velocity you need to see how much time passed and how much distance it traversed and divide one by the other. But if you look at it at one point in time, you will not be able to know its velocity. But someone who looks through glasses of velocity—he will be able to see, at one point in time, the velocity of the body and will not be able to know its place. And it is hard for us to imagine this because that would be a creature not built like us. We are built—we wear glasses of position. But in principle there can be creatures—and quantum theory gives us a certain description of how they would perceive the world—who wear glasses of velocity or momentum and not of place. And these are completely different glasses, but they see the same physics we see. Exactly the same thing. But they would describe it in an entirely different conceptual language, in a language of velocities and not in a language of positions. And of course if there are completely different creatures, then from their point of view there might even be languages I cannot imagine at all. Completely different languages. Therefore, with the question of Platonism in mathematics, when we discuss it, it is very important to distinguish between whether I am speaking about the mathematical entities and the relations between them, and whether I am speaking about the language in which I describe the mathematical truths. The language may be a human invention. But the mathematical truths that the language describes are certainly not an invention but a discovery. That is obvious. Rabbi, what is the definition of the velocity that the Rabbi said exists at that point? For the particle at that point, what is the definition of its velocity? The definition is: if you wait a tiny bit of time, how much distance it will cover. That is what is called an operational definition, a definition of how to calculate a magnitude. Fine, but the magnitude itself exists even at a point in time. To calculate it or measure it you need to look at it over a certain time interval, not at a single point in time. But that is only a problem of how to calculate. Once you finish the calculation, the result is velocity at a point in time. There is an example that I like in this context, though one needs to discuss philosophically whether it is correct. You know that there is what is called the Doppler effect. I think once police radar was based on this; I think today they changed the technology because there were problems with it, if I remember correctly. In any case, the Doppler effect means that if I send a beam of light at a moving body, like a car, and the car is moving at some velocity, then the beam of light will return at a different frequency from the frequency at which I sent it. There is a difference in frequency. The difference between the frequencies of the incoming beam and the returning beam is proportional to the velocity of the body. That is how they measured the speeds of cars with radar. One could compare the wavelength of the beam I sent to the wavelength of the returning beam, and the gap in frequencies between the two beams could give me the speed of the car. Now when one looks at it this way, at least in the simple description, the beam hits the car at a single point in time and returns. That means that we are measuring the speed of the car in a way that requires an encounter at one point in time with the car. Usually when we measure speed, we measure where the car is here, and after a second where it is then, divide the distance it traveled by the time of a second, and then we know its speed. Right? For that you need to observe the car over some interval of time. But in the Doppler effect—again, I am speaking conceptually, not really at the practical level—but on the conceptual level, the beam I send meets the car at one point in time. And still I can know what the car’s speed was at that point. So that illustrates very nicely the fact that in principle the car has speed even at a point in time. Only in the standard methods of calculation or measurement, I have to follow the car over a segment of time. But those are just computational constraints. Here, in the Doppler effect, I can do it at a point in time, measure it at a point in time, not follow it over some interval, but just one point. I meet the car at one point and I know its speed. So all these things are demonstrations that there can be different mathematical languages, and the languages may indeed be inventions or conventions, but the effects those languages describe—whether Pythagoras’ theorem, theorems of geometry, and so on—are certainly discoveries and not inventions. It may be that someone who did not hit upon that language would not discover the mathematical truths either, because that language is convenient for describing mathematical or physical truths. So he would not discover it, but it would still be there. And if I taught it to him, he too would understand it. I think I gave this example—I no longer remember—of tribes that have primitive counting systems, the one-two-many system. Right? Tribes that count one, two, and many. Those are the numbers they have: one, two, and many. There are articles describing encounters with people who live in such a culture. They ask them various questions—for example, they place before them three batteries and five batteries and ask them where there are more, and they do not know how to answer. Or not two—three and five, because two they do have, one, two, and many. But three and five are both in the category of many. So when you compare three to five, they do not know where there is more. But between three and twenty they do know, even though both three and twenty are within the range of many, and yet they know that. Meaning that the concepts of comparison and magnitude exist for them. They do not have the language of numbers by means of which they can conceptualize it and handle it more precisely. But of course one can teach them, and they are no less intelligent, and they will grasp it and know everything I know. Meaning the lack here is a lack of language, but the mathematical truths—that this is greater than that—even if they did not have the language to describe why twenty is more than three, they understood that twenty is more than three. Therefore the claim that twenty is more than three is not a claim they invented when they learned the language; it is a claim they discovered. The language—calling this three and this twenty and so on, all sorts of numbers—that language may perhaps be an invention. I don’t know, maybe. But that is only—the language is not important for our purposes. It may have practical importance, but it is not important for the principled discussion. For the principled discussion, what matters are the mathematical claims themselves. Rabbi, Rabbi, yes. If God decided to stop motion in the world, to stop time, then would that same arrow of Zeno still, at that point in time, have velocity according to the Rabbi? There would no longer be any way to calculate it. Correct, I think yes. It would be frozen at a certain velocity and not move. Think about a body moving at a certain velocity; I throw it and it hits a wall. Now when it hit the wall it had a certain velocity, but the wall does not let it translate that velocity into accumulated distance. It cannot continue to accumulate distance because the wall stops it. So that means it has a velocity that does not manage to pass from potential to actuality through change of place. So it will come out as heat, it will bounce back, depending on whether the collision is elastic, plastic, or whatever. Meaning, this potential called velocity does not always pass into actuality by my accumulating more distance in the future. And you are describing another kind of barrier, a barrier of time, not a barrier of distance. Fine, motion needs both distance to move into and time within which to move. If one of those axes—either time or distance—stops, then the velocity I have right now will not manage to be translated into continued motion. Okay, but that still does not mean I do not have velocity right now. So maybe our definition of velocity is not sufficiently precise. No, I think our definition is an operational definition, a definition of how one calculates velocity. And that is what is so confusing, because it is not the definition of what velocity is; it is a definition of how one calculates velocity. Therefore the claim is that to calculate velocity you need an interval, but velocity in itself exists even at a point. Many times there are certain things you cannot hear if you do not have the proper tools of perception, but that does not mean they are not there. Okay, a dog hears frequencies we do not hear. So what, because we do not hear them, does that mean they are not there? They are there; we just do not hear them. Meaning, there is something we need to do in order to perceive the thing, but that does not mean the thing does not exist if we have not done what is needed to perceive it. Okay, so I want to move on. The second point that is important to understand here is that when we speak about the question whether mathematics exists in some objective sense, let us call it that, or subjectively—right?—whether it is our invention, or intersubjectively, of all human beings but only among human beings and not in the world itself—I think that question too is an imaginary question. I mean, obviously it is in the world itself, but that still does not bring us to the Platonic question. Because when we speak about things that are objective, that does not always mean speaking about whether they exist. I spoke about this at the end of the previous class. Suppose we speak about the speed of a car. The speed of the car is some speed. Now, is the speed an entity? Not the car—the speed of the car. Is it an entity? No, it is not an entity, right? It is a property of the car, or a state of the car, or an event that happens to the car, but it is not an entity. Does that mean the speed is subjective because it is not an entity, because it does not exist in the world? No, it exists in the world. It exists in the sense that it is an occurrence in the world. It is not an entity in the world, but it is in the world. It is not my hallucination. When I measure the speed of a car, I do not invent the speed of the car; I discover the speed of the car, although the speed of the car is not an existing entity. Here too Plato knows that the speed of the car is not an entity; the speed of the car is a property of the car. But as a property of the car, it is not my invention. Therefore, when I measure that speed, I discover something and do not invent something. You see that the question whether mathematics is invention or discovery and the question whether mathematics exists in the Platonic sense are different questions. The first question—the answer is obviously yes, obviously yes. Pythagoras’ theorem is certainly invention and not discovery… certainly discovery and not invention, and all the theorems of geometry and all the theorems of mathematics are certainly discoveries and not inventions, there is no doubt about that. They were true before they were proved or found. Fermat’s conjecture—even before Andrew Wiles proved it—it too was true. We just did not yet know how to prove it. Does anyone imagine that Andrew Wiles created Fermat’s theorem? He did not create Fermat’s theorem; he discovered or proved Fermat’s theorem. So the answer to that is simple. The Platonic question is whether Pythagoras’ theorem represents some kind of idea—say, a right triangle is an idea that exists, like horse-ness, right?—that exists in some sense in a Platonic world, the Platonic world of ideas. That is the question of Platonism. And that question is not similar to scientific Platonism, which asks whether the electron exists. Because the question whether the electron exists is the question whether this particular electron of ours exists here in the world. That is a question about a certain specific material object that exists in the world, a physical object. I just cannot observe it directly, so it is presented as a Platonic question. But the right triangle—not the right triangle here on the paper, but right-triangle-ness, right, the idea of the right triangle—that is like the idea of horse-ness. The question is whether it exists in the Platonic world of ideas, not in our world. That is the Platonic question. And this brings me back again to something I have said several times before, namely that mathematics is a branch of philosophy. When philosophy speaks about metaphysics, it speaks about the existence of things—the existence of God, of angels, of abstract worlds of one sort or another—that are not accessible to the tools of physics, and one deals with this by tools that are not tools of observation but tools of thought or intellectual cognition. I spoke about intuition as a kind of cognition, and in mathematics it is like that too. In the Platonic conception, when I understand Pythagoras’ theorem, I am basically observing the concept of the right triangle with the eyes of my intellect. I am looking at the Platonic idea of the right triangle. Now of course I usually do this through a concrete right triangle that I draw on paper. But obviously I am not looking at that particular triangle, because I am investigating the property of right triangles as such, not of this one triangle. I want to know what the property of a right triangle is by virtue of its being such. A right triangle without knowing anything else about it. Here too it has a certain size and a certain color and it is in a certain place and on a certain paper. That does not interest me. What interests me is what characterizes right-triangle-ness, like horse-ness in relation to horses. Okay? You see that this is exactly the question of philosophical Platonism. Scientific Platonism is simply a particular case of the question of philosophical Platonism. What is stronger in mathematical Platonism is that perhaps it is easier to illustrate the Platonic position, which usually seems like some kind of mysticism and people are put off by it. In the mathematical world, I think if you ask—I once did a poll in some seminar I gave in mathematics at Bar-Ilan, and I asked the mathematics faculty there who among them is a Platonist. Everyone. Meaning, they all perceive mathematics as consisting of existing things, and of course that we discover them and do not invent them. That is the straightforward experience of the mathematician. I assume there are some who, after you corner them and they think to themselves and conceptualize and so on, will say no, no, no, no—they retreat from it because it looks mystical. But the experience that accompanies them in their ongoing work seems to me to be a Platonic experience. And when you can say things so precise that everyone agrees with them and they can be proved, this points much more strongly to the conception that you are actually investigating existing, objective things on which we all agree. And from here it is very easy to arrive at the conclusion that these are in fact existing entities, the Platonic conclusion. Therefore I think that although the question of Platonism in mathematics is a particular case of philosophical Platonism, here it is easier to feel or become convinced that we really are speaking about some kind of shadow world, a world of ideas, which we investigate. It is like an observational science in which we observe those ideas and try to determine their properties. That is the meaning of Platonism in mathematics. Now I want to add a few more points on this matter. Many people—I sent the two columns on WhatsApp following the previous class, so I will not send them again, you can read them there; there I spell this out more. But many mathematicians and philosophers of mathematics rely on the fact that the world speaks the language of mathematics. Physics is written in the language of mathematics. Many physicists and mathematicians are very impressed by this phenomenon and see in it some expression of the fact that mathematics exists in some sense, that it is part of physics, that it is Platonic. Now here one first needs to distinguish between two things. The first person who really spoke in this language was Pythagoras and the Pythagoreans, a somewhat mystical Greek sect who held that numbers are existing things and that ratios between whole numbers basically build all the relations in the universe—a kind of mathematical aesthetics. The whole universe seemed to them like some sort of mathematical aesthetic. In a certain sense they really were thousands of years ahead of their time, but they went far beyond what we are talking about today. From Pythagoras’ point of view there was really no such field as physics. Everything was mathematics. Meaning, if we knew the mathematical laws all the way to the end, we would know all the physics. There would be no need for observation. He understands—he understood—that the laws of physics are merely mathematical relations between things. And one can arrive at them through proofs and mathematical tools, and then one would not need observation at all. Not observation at all. And that is of course taking things too far. And that is basically saying there is no such thing as science—that everything is mathematics, or there is no such thing as mathematics because everything is physics, it does not matter—but you do not need observation. That is extreme rationalism against empiricism. It means that it is enough to think in order to know everything that happens in the world. You do not need to open your eyes. You do not need to observe. I want to distinguish between that claim—which I completely reject. Obviously one needs observations, and obviously there is such a thing as physics as distinct from mathematics. But what is confusing in this context is that the physics we discover through observation is described in mathematical language. Meaning, all the relations we know in physics are relations that we write in the most natural way in the language of mathematics. Therefore a great many people confuse the claim that the world is written in the language of mathematics with Pythagoreanism. It is not the same thing. Because when I say that the world is written in the language of mathematics, I am basically saying that after I have made the observations and discovered the relations—say, between acceleration and force, Newton’s second law, that force equals m a, force equals mass times acceleration—look how I write it: with an equation, F equals m times a. So here I have variables: force, mass, and acceleration. I have an equation that describes relations among the variables. Is that equation mathematics? The answer is no. That equation is physics. That equation is physics. The language in which I describe this physical truth or this physical claim is mathematical language. The relations between force, mass, and acceleration are not relations I can discover through mathematical analysis. I need observation in order to see that this really is the relation between force and acceleration. But I describe the results of observation in mathematical language. That is not the same thing as saying that everything comes without observation, simply from analysis. Think, for example, about how I discover that the sum of the angles in a triangle is one hundred and eighty degrees. I do not need observation for that. I can discover that with my eyes closed. I can prove it to you. So there is a difference between the law that the sum of the angles in a triangle is one hundred and eighty degrees and the law that F equals m a, force equals mass times acceleration, even though both are described in the language of arithmetic. Something times something equals something, or something plus something plus something equals something, one hundred and eighty. Alpha plus beta plus gamma equals one hundred and eighty. Or F equals m times a. What is the difference? Both are mathematical formulas, one with addition and one with multiplication—what difference does it make? But there is a fundamental difference. To alpha plus beta plus gamma equals one hundred and eighty—the sum of the angles in a triangle is one hundred and eighty degrees—I arrive without observation. Once I understand what an angle is and what addition of angles is, I know that the sum of those angles is one hundred and eighty. I do not need observation for that. Therefore the second statement is a theorem in mathematics, that the sum of the angles in a triangle is one hundred and eighty degrees. But the first statement, F equals m a, is also written in the language of mathematics, but in order to know it I need observations. Therefore it is physics. And Pythagoras thought that if I understood well enough what force is, what mass is, and what acceleration is, I would not need to make measurements at all. From analysis of the concepts F, m, and a—force, mass, and acceleration—I would be able to know that force equals mass times acceleration. Meaning that in the world there is no domain at all that requires observation. Everything is abstract thought. And that is of course very optimistic, and I think no physicist today would agree with that. So what does this fascinating phenomenon actually mean, that the world is written in the language of mathematics? In this context, I now want to speak not about Pythagoras’ assumption but about the claim made today: that the world is written in the language of mathematics. The intention is that all the relations among physical parameters or variables can be described in mathematical language—described naturally, compellingly, and precisely in mathematical language. And that is not self-evident. There is definitely room to wonder why this happens. But in this context they bring another interesting fact—and again, see the sources and the people who made the different arguments in the two columns I sent; at the moment I am speaking only in broad strokes because I want to finish this today—they speak about how many times a certain mathematical theory is created and has no application at all, and then years later physicists arrive at some issue or context where the most natural thing to use is that already existing mathematical theory. And there are quite a few examples of this: tensor calculus for general relativity, group theory for quantum theory, Hilbert spaces for quantum theory, all kinds of things. There are several excellent examples of this. Geometry is perhaps a less good example. Geometry clearly began in the world and only afterward became a mathematical field, but its motivation and its basic definitions were brought from the world, that is obvious. Only afterward did they abstract it and turn it into a mathematical field. But group theory did not come from the world, nor did set theory, nor Hilbert spaces, vector spaces, and things of that sort. And all these have uses in physics. And their uses in physics were discovered or arose many years after the mathematical field had already existed. The arsenal of mathematical tools was already standing there ready for use by physicists. And that really does call for explanation, because it means that we did not draw those mathematical fields out of physics—as opposed to geometry, for example. We did not draw them out of physics; on the contrary, physics uses those mathematical fields. So where did we get them from? Some say we invented them. But if we invented them, what are the chances that this invention would fit exactly a need that would emerge in physics a hundred years later? One can invent countless things. Why would these inventions turn out to be so useful many years later when they were not invented for that purpose at all? That is already a coincidence that calls for explanation. It is not self-evident. So in this context we really have two puzzles. The first puzzle is: how is it that the language of mathematics fits so beautifully to describe physics in the first place? Mainly physics, though it is true also of chemistry and a bit less of biology, but in other fields too, though in physics it is most prominent. That is the first puzzle. The second puzzle is that not only does it fit, but the mathematical tools were not only formulated and conceptualized for the sake of physics—which itself is interesting, that one can do this and it succeeds—but on the contrary, we did it beforehand, and only after a hundred years it turned out that the system of tools prepared in advance was suitable for treating current physical problems. And that doubles the wonder. So how does one explain this? Does this mean—and from here various philosophers or mathematicians infer a Platonist conclusion—that mathematics exists, that it does not come out of physics, because, after all, they found this mathematics earlier. Okay? But note, here too I sharpen the point and say: it does not mean that the entities of mathematics exist in the Platonic sense, not necessarily. It means that the relations among them are objective relations. But remember the speed of the car. Relations among entities can be objective even if the entities do not exist, or even if the existence of the entities—perhaps the entities are even material, it does not matter. The objective existence of the relations does not mean the relations themselves exist as entities. Therefore what I want to say, I will summarize it this way: one can ask three questions here in the context of mathematical Platonism. The first question—or rather, the first claim I want to make; it is not a question. Two claims and a question. The first claim I want to make is this: every mathematical theory is an objective truth. First claim. Here I see no room for discussion at all. There are discussions, but in my opinion that is just confusion. It is obvious that every mathematical claim is an objective truth. We did not invent it; we discovered it. What does that mean in the most basic sense? It means that if there is found in the world some relation—whether in physics or some other domain—that satisfies the assumptions of that mathematical theory, then all the theorems of that mathematical theory will be true about it. For example, if our space in the world is Euclidean space, then I can assume that the theorems of Euclidean geometry are true of it. And if I draw triangles in the world, I tell you that the sum of their angles will be one hundred and eighty. How do I know? Because I have a proof in mathematics. So what? What does that have to do with the world? Mathematics is something abstract. The claim is that if the assumptions of Euclidean geometry hold in the world, then the theorems of Euclidean geometry are also true of that context in the world. In that sense, the mathematical theory is true; it is discovery and not invention. But note, now I move to the next step. My next step says that this mathematical theory is true even if I have not found any context in physics that serves as a model for it. Meaning, that satisfies its assumptions and therefore also its theorems. Even if I have not—not only have I not found one, there isn’t one. Still my claim is that the mathematical relations of that mathematical theory are true. It is discovery and not invention. The fact that if there were something in the world it would satisfy the assumptions and therefore the theorems—that is only an indication. But it is not required. Meaning, even if there is no such thing in the world, it is still a hypothetical matter: if there were, then it would satisfy the theorems. I am not assuming there has to be. Meaning, there may not be. And still I say that this mathematical theory is discovery and not invention; it is an objective truth. That is the second claim. Meaning, the first claim is that if there is something in the world, then it will satisfy the theorems as well. The second claim is: yes, but even if there never will be such a thing in the world, and there is no context at all in which it is applicable, the theory is still discovery and not invention. The third question is the question of Platonism. The third question is whether these mathematical relations and mathematical entities are entities that exist in some sense in the Platonic world of ideas. And that brings us back to all the discussions we had about Platonism in general, with all the arguments I raised for and against it. All those arguments can be made here too. You can see in those two columns that I apply them here. Just an anecdote I hear from the people in my family—I have several mathematicians there—is that mathematics lecturers really, really enjoy it whenever they teach some theorem in mathematics or some mathematical phenomenon and then say, yes, and this has many applications in physics, physicists use this in countless ways. Usually that is nonsense; it has no use at all. They heard that once some application of the thing could perhaps represent some mathematical theory, but most things in mathematics remain in mathematics and do not have the slightest use in physics. It is true that there are quite a few things that do have use in physics, but they are still a tiny fraction of all mathematical discoveries. Therefore it is entirely reasonable that there are mathematical discoveries that do not have, and will not have, and probably never will have any application. Does that mean they are not objective truth? I think they are objective truth in exactly the same sense. There does not need to be an application; what is needed is that if there is an application, the mathematical theorems will hold of it. That is the condition. Therefore there is no need that there actually be such an application. Say, every ball is round even if there are no balls in the world. Because if there were a ball in the world, then it would be round. So even if there are no balls in the world, I can still say that every ball is round. Okay? Something of that kind. Well, that is more or less what I wanted to say about mathematical Platonism. If you want expansions, they are in the two columns I sent on WhatsApp following the previous class. I will have one more meeting and with that I will conclude the Platonism series, the whole series on Platonism. That is it. If there are comments or questions? Thank you very much, Happy Hanukkah. Happy Hanukkah. Are negative numbers an example of a mathematical domain that has no application in the world? It has many applications. What do you mean, no application? I mean, when dealing with minuses and things like that. It has many applications. Even complex numbers have application in the world. It is just that with complex numbers, the application is an application of techniques. It is not that there is some i in the world running around—i, the square root of minus one, right?—running around. For engineers, by the way, it is j; for mathematicians and physicists it is i. So the technique of complex numbers is a super-useful technique, with countless applications. Maybe the quantum wave function. That might be the only example of a mathematical entity that itself is a complex number—not that complex numbers are a technique by means of which it is easy to handle problems in physics. Maybe the wave function is a good example of that. But negative numbers—for example, in physics there are holes, there are particles and holes. Holes are the absence of particles. Instead of saying I have minus five electrons, I say I have five holes, but that is the same thing; it is basically a negative number. And in semiconductors, many times it is convenient to work that way: if there are many particles and only a few holes, then it is preferable to look at the motion of the holes rather than the motion of the particles—that is Dirac’s trick. Better to look at the movement of the holes than at the movement of the particles. Then I am basically turning the negative number into the number I use. Instead of saying here are minus five electrons, I say I have five holes. I wanted one more question, thank you. If you can sharpen a bit: last week we spoke about the proof, about the claim that the very fact that one can predict results proves that people hold that there is some kind of truth, that they believe in the method. They believe in this formula and are not surprised by the result. That is not a theological argument, it is a philosophical argument. And I do not want to prove that people believe in the method; I want to prove that the method is correct. Because if the method were not correct, there could not be so many good predictions. And based on that too, okay, that is where I wanted to get—but what is the relation between that and what you said today, that theories were developed that were not useful, but somehow in the end it turned out that after some years the theory was used? It is the same argument. The same argument, only I want to say that this argument only shows that the mathematical theory is an objective truth. But it does not mean that the mathematical entities or these relations exist in some Platonic sense. That is the second question, not the third. Are both of them the same claim, or is there some gap here? No, they are two different claims. That is what I am saying. I said there are three claims. The first claim is that mathematics—meaning, one can say mathematics is an invention, but that is nonsense. Mathematics is discovery. But still, even if it is discovery, one can speak about it on three levels. One level is that it establishes some necessary relation among things, so that if in the world there are things satisfying the assumptions, then all the theorems of that mathematical theory will also be true there. The second claim is that this mathematical theory is true even if there is no domain in the world in which it is instantiated. There is no domain in the world that is a model of this mathematical theory. And the third claim is that this mathematical theory is true in the Platonic sense, meaning that it exists in the Platonic world of ideas. The mathematical entities—triangle, group, set, things like that—exist in some Platonic world, and the relations between them are the laws of nature of the Platonic world of ideas. Okay? That is the third claim. In support of that claim, I think it is true by the way, I am a Platonist, but in support of it I can bring only the arguments for Platonism that I discussed throughout the whole series, philosophical Platonism. Okay? It does not follow from this argument about the suitability of mathematical theories for physical problems later on. That only means that mathematics is correct. It is objective; it is discovery and not invention. Does it exist in some sense? That does not have to be. If something is instantiated, that does not mean it exists as an entity; it means it is true. Therefore I said that in scientific Platonism, which was the previous subject in this series, in the previous classes, scientific Platonism does not speak about the Platonic world of ideas. It speaks about whether I formulate correct generalizations of the laws of nature in our world. Or the electron that I say exists—that is a particle or object that exists in our world, not in the Platonic world of ideas. Because there I am really speaking only about the first two questions I asked regarding mathematics. The third question is something else. In the third question, several contradictory theories can in principle coexist, right? Yes, obviously. Because those theories are derived from contradictory assumptions, so they arrive at contradictory conclusions. There is no problem. There can be non-Euclidean geometry and Euclidean geometry, and both can be objectively true. Why, in what sense are they true? That if there is something that satisfies the assumptions of non-Euclidean geometry, then the theorems of non-Euclidean geometry will hold there as well. And if there is something that satisfies the assumptions of Euclidean geometry, then the theorems of Euclidean geometry will hold as well. Say, if our space is Euclidean space, then the geometric figures in it will satisfy Euclidean geometry. But after Einstein we know that our space is non-Euclidean geometry, so the properties of non-Euclidean geometry hold in it, and there is no contradiction. Both are objectively true. Objectively true in the Platonic world of ideas. In our world, either it is Euclidean or it is not. Either it fits this theory or it fits that theory. It cannot fit both. I understand, thank you very much. Okay then, Shabbat shalom, Happy Hanukkah.