Peter Robinson: 2 plus 2 equals 4 in all places and for all time, 2 plus 2 equals 4. But why? What does math tell us about the nature of reality? David Berlinski, Sergio Cleerman and Stephen Meyer on Uncommon Knowledge now. Welcome to Uncommon Knowledge, recording today in Salzburg, Austria. I'm Peter Robinson. David Berlinski has taught math, philosophy and English at universities, including Stanford, Rutgers, the City University of New York, and the Université Du Paris. I hope you like the pronunciation. Perfect. He is the author of books, including 1, 2, 3 Absolutely Elementary Mathematics and his forthcoming volume, The Perpetual Rose. A native of Romania, Sergio Cleerman is a professor of mathematics at Princeton. In his own words, his current interests include the mathematical theory of black holes, more precisely the rigidity and stability—I'm reading these words without having any idea what they mean—and the dynamic formation of trapped surfaces and singularities, close quote. I'll ask you to explain a little bit of that. Maybe Sergio, the director of the Discovery Institute Center for Science and Culture, Stephen Meyer, started his professional life as a geophysicist. He returned to school, earning a doctorate from Cambridge in the history and Philosophy of Science. He has established himself as one of America's leading thinkers in intelligent design. His most recent book, The Return of the God Hypothesis. David, Sergio, Steve, welcome.

Peter Robinson: In The Return of the God Hypothesis, Steve's latest book, he argues that three relatively recent developments suggest that science needs to return to some notion of the transcendent. And these three developments are the Big Bang, the fine tuning of the universe and the discovery of DNA. After reading Steve's book, a certain very accomplished, well-known mathematician, took Steve aside and said, "You only named three developments that suggest a transcendent mind. There's a fourth." Sergio, what does he mean by that?

Sergio Cleerman: Well, of course, I should say Steve talked about developments and mathematics. Mathematics has been around for thousands of years. So it's not quite fair to compare, but mathematics has, by definition, deals with its own sense of its own reality, which is, I claim, is objective as physical reality. And so, for example, black holes are like that, right? A black hole by definition. We have a mathematical theory or general relativity that predicts black holes. Yeah. But by definition, a black hole cannot be seen. So, nevertheless, we, we can assert its existence. Why? Because general relativity is a consistent theory.

Peter Robinson: So to take this black holes scare the daylights out of me. We'll come back to black holes, I'm sure. But my mind already hurts, is just when hearing about your work on the rigidity. All right. In layman's terms, which is to say, for me, 2 plus 2 equals 4 is real. That's not a figment. It's not an artifact of our mind of mental processes of the accidental processes that might be going on in our neurons. Whether I think it's two plus 2 equals 3 or 5, I'm wrong. Two plus 2 does equal for. And that is objectively real. Therefore, there is a conceptual objective reality that exists in outside us. It's not material. That's not material. And this is actually a big deal.

David Berlinski: Yeah, of course, it's a big deal. I mean, 2 plus 2 equals 4 is an interesting example, but you can derive that biological inference from still more fundamental ideas, which is an exciting and interesting fact, all its own. You don't have to begin by affirming 2 plus 2 equals 4. There I stand. I can do no other. You can say, I have derived that from still more primitive conceptual items. But when you go back and back and back and back and you ask about the initial assumptions, the axioms of the system, about arithmetic, there is no additional defence that you can offer beyond the consistency of the whole, which is a very interesting position to find oneself.

Peter Robinson: So I'm going to quote to you from your book. Not 1, 2, 3. I think this is, I'm hoping this is the same point, because that will indicate that I have actually understood you. Quote, "Neither the numbers nor the operations they make possible, permit an analysis in which they disappear in favor of something more fundamental. It is the numbers that are fundamental. They may be better understood, they may be better described, but they cannot be bettered."

David Berlinski: I I still think that's true. Bear in mind, when you say two plus two equals 4, that's an assertion. Yes. What I'm arguing for in that particular passage? Is that when you go back to the foundations of arithmetic in the expectation or the hope that you can get rid of the numbers, you're going to be very disappointed because they reappear.

Peter Robinson: All right. I'm going to quote David once again, but I put this to the two of you for judgment. I'm assuming he will agree with himself, although, in David's case, this is always a question again from his book 1, 2, 3. "Across the vast range of arguments offered, assessed, embraced, deferred, delayed or defeated, it is only within mathematics that arguments achieve the power to compel allegiance. No philosophical theory has ever shown why this should be so. It is a part of the mystery of mathematics." So, you argue from some philosophical point that derives from Aristotle and has seemed straight, but I can still say, you know, I'm not persuaded. But when you say to me, 2 plus 2 equals 4, I have to.

Stephen Meyer: Of course, you're right about that. I can speak to this from the standpoint of someone whose work in the natural sciences and as a philosopher of science. The natural sciences provide empirical or observational evidence in support of conclusions. And scientists will evaluate particular theories or hypotheses by comparing your explanatory power or their predictive power. But the logical form of those arguments does not render a deductively certain conclusion. You, in the best of cases, will make an inference to a hypothesis, which provides the best explanation.

Peter Robinson: One second. Just distinguish deduction from inference for us.

Stephen Meyer: So, a deductive argument will start with a major premise: "All men are mortal," a minor premise: "Some fact about the world. Socrates is a man," and then a conclusion: "Therefore, Socrates is mortal." And if the premises are true and the reasoning is valid, then the conclusion can be affirmed with some kind with certainty. But in the natural sciences, you start with facts that you've observed about the world, and you want to infer from those facts to either some kind of generalization that would be an inductive argument or to some sort of causal process that might explain what you're seeing around you. Those arguments are typically characterize as abductive, the kind of detective reasoning that we we enjoy when we watch detective shows or yeah, Colombo or someone's trying to figure out who done it. And so those abductive and inductive inferences, when you examine the logical forms, turn out not to give you certainty. They may give you plausibility. They may give you comparative plausibility, where one theory is very much better than another, but they don't give you the kind of certainty that mathematics alone and mathematical logic. A good scientist will never say any more than, "The theory is X, Y, Z. And on the best evidence, it holds up for now." Where a mathematician is feels perfectly confident in saying, "I've proven. We've proven it. We have a proof." A proof in science, the mathematicians. However, are better than we when you say you prove something, you mean it.

Sergio Cleerman: Yes.

Peter Robinson: All right.

Sergio Cleerman: Okay. So just go ahead, it could get wrong. But somebody will show you to me. It's.

Peter Robinson: So. If this brings us to Sergio's article, an Inference magazine that you edit, David. Articles entitled Reflections on an essay by Wigner. Now, Eugene Wigner, I have to set this up, was a 20th century mathematician and physicist. In 1960, he wrote a famous essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Wigner noted his surprise that mathematics, which, after all, goes on in our minds, should prove so useful in describing and even predicting aspects of the physical world. Okay, can you give me a couple of examples of this? I mean, what when I think to myself, wait a minute, so I have a dream in the middle of the night. I wake up, and it turns out it was untrue. But if I do a mathematical equation, and I wake up, it's still true.

Stephen Meyer: Well, Sergio wrote a brilliant essay. And what he showed was that you can start with very simple mathematics and build up to more and more and more complex forms of math essentially deductively. And then those complex forms of math, take the calculus, take differential equations, they map beautifully onto the physical world to describe actual processes that are taking place in nature so that they provide very precise descriptions of things that are going on in nature. And what Wigner is alluding to is the mystery that this, the puzzle, this induces for a lot of physicists. Why should the math that we have developed through a series of deductive steps effectively from our own reasoning map so beautifully to processes that we sometimes haven't even observed yet? David has a number of great examples in 1, 2, 3 of mathematical structures that were developed well before they had any application to to, to physics. But then later, were were crucial. And maybe he should speak to that.

David Berlinski: Yes, well, I mean, it's not. I mean, Eugene Wigner raised a very interesting point. And people have been discussing it. If you look at theoretical physics, the great structures, Newtonian mechanics, general relativity, quantum mechanics. You can't do it without a lot of mathematics. You just need a whole lot of mathematics. And Wigner raised the question is, you know, we need mathematics to do quantum mechanics, but we don't need entomology. How come the bugs don't figure in quantum mechanics, but the numbers and the complex numbers do? And that is a rewarding and a provocative question, but we don't have to turn to quantum mechanics. There's a glass here. There's a glass here. One glass, one glass. How many glasses are in front of you? Two. From where do you derive that assurance, that there are two glasses in front of you? It's not a physical observation, because nothing in the physics that the situation reveals the fact that one in one or two. That is something so one would think that's additional. Now, we can break that all down into smaller steps. And that's what logicians have done in the 20th century. They've shown us that the method of proof can be decomposed into very small steps. In fact, so small, a computer can execute them. And the initial assumptions can be made so general, in fact, so general that they encompass all of mathematics, as in set theory or categories theory for that matter. But we are in all this in a rather an awkward position. I happen to be looking out at a beautiful Alpine lake now. And just imagine, we see somebody on the other shore who begins walking across the water without any assistance whatsoever. He's just crossing, walking one step in front of the other, and he's crossing the lake toward us. And he comes completely dry. He appears in front of the television camera, and we say, "How did you do that?" And he says, "Well, I took very small steps. I took very small steps." Now, our natural reaction would be, that's commendable. And you got to cross. But somehow or other, it's not the answer to the question. And we are all in the position of watching someone cross a large bottle body of water and explaining his success by saying, "Look at my feet. Small steps." That's where we are.

Peter Robinson: Okay. So, so Sergio, let me go back to your essay. I'm quoting you once again. "The mystery Wigner points out, arises, in part from the perennial question of whether mathematics is a science. Advanced by exploration and discovery, like the main physical theories, or whether it is in an invention, a creation of the human mind."

Sergio Cleerman: I argue that mathematics developed through exploration and discovery. That is to say, it is like a nugget in the ground. You find it.

Peter Robinson: Correct. It has its own.

Sergio Cleerman: I argue that mathematics developed through exploration and discovery. That is to say, it is like a nugget in the ground. You find it. Yeah, no, absolutely. I mean, one image that you could make is that of an alchemist that this is going trying to go on the top of the mountain. He has ans doing alpine metaphors today. He has an idea where he wants to go, right? So that's very important. That's part of doing mathematics. It's not just, it's not just deduction. Sort of a vision of where you want to go, which is very important. In which right, has something to do his inspiration. But then then you know, you have something very objective in front of you, right? The stone is of the alchemist that you have to take into account that you're not going to fall. You, you have to touch the stone, know exactly where you are going. And if you don't, you get into trouble. I is very similar. People have the feeling that everything is deductive. It's not. I mean, it is very similar in that respect to physical sciences. Physical sciences also, you have some idea of. You, you have, let's say, an expectation. You make a hypothesis, right, right. The choice of a hypothesis is not a deductive thing. It's just, it's an insight. And then you try to show that that it fits everything else, all the other experiences. So, and that's the process of proving, right? So you have the once you have the process of discovery. But then the process of justification, the process of justification. So mathematics, at the end of the day, it looks like a chain of logical sequences that the computer can also, I mean, once you, you have the chain, the computer can. Go very fast to it and maybe even check. So, there are no yet no computers that can check large. I mean, they can check small, small proofs, but not blood. In any case, it's sort of a very good example. I think that illustrates very world relations between math. Take geometry. So, the geometry was the first really theory of the physical world. Right, I mean, it's what it describes going to Euclid. Is that what it goes right? So you try obviously try to make sort of mathematicalmatic statements out of, but it's a physical theory without doubt. Geometry is a physical theory. Then it, it develop. So here comes up, something specific to mathematics, different from physics. Mathematicians can take a theory. And then develop. Based on very different criteria than a physics is to do. So, you know, they are interesting in problems because they are beautiful or because because they. They feel that it will lead to certain understanding of something. Else. And this freedom, really, in the case of geometry world for, I don't know, 2000 years without essentially no connection back to the physical world. I mean, the Ukrainian geometry was there to start with. But then by the 19th century, you have you have Gauss. Have Lobachevsky. You have Gauss. You have Riemann. Okay. You have Minkowski at the beginning of, of the 20th century. And then all of a sudden, all that stuff becomes essential ingredient. I mean, it's not just, you know, it's not just a technical. It's an essential ingredient of or special and general activity. Right, so comes directly applicable to fundamental physical theory. And I think this has happened many, many times in. And maybe it's not very well. It's not very well acknowledged.

Peter Robinson: So can I ask when you Sergio. Needless to say, I cannot evaluate your work on my own because you have done a 2000 page proof, 2000 pages of close mathematical reasoning. I could have. I could live to 2000 and not. I could read a page a day or page a year. It still would escape me, I'm sure. And this was on the stability of some aspect of Einstein that Einstein. So may I ask. Did you think you were doing a work of art, creating something beautiful? Or did you think you were interrogating reality? Do you see that I'm trying to understand what you what it felt like to you as a working mathematician, regionally.

Sergio Cleerman: It's a very deep, deep, very difficult problem. And the answer is both. I mean, I would take the problem in the first place. I'm a mathematician are not the physicist. I'm interested in I, believe that the best mathematics is connected, somehow physics in complicated ways. So, but there are many problems in physics. And I would pick the one that satisfies my aesthetic feeling as a mathematician.

Peter Robinson: So that's aesthetic feeling. So explain that.

Sergio Cleerman: You want something something I something that I feel is very beautiful. It's very profound. It gives lots of any interesting questions. Okay. So that's one aspect. But then then has to be the for me, at least to be a second aspect, which is that it should say something about the physical world. And in this case, it does. Right, I mean, the Kerr solution. So here how it goes, right? You have general relativity, which was well formulated bench line in at the end of 2015. So 1, 915 X. And this was at the time, a new theory of gravity. Yeah, massive bodies curve. Spit what's called space time. And then. Then certain solutions Schwarzschild was the first to found in 2016 a year, immediately a year after found the so called Schwarzschild solution, which is a stationary solution, was out of symmetries that you can actually extract from from the serial of relativity for the ancient equation is exact formula. And that had LED to lots of issues because it has a singularity. This is connected later on was Penrose singularities, or for which he actually got a Nobel Prize. He's the only mathematician to have gotten a Nobel Prize in physics.

Peter Robinson: Which is what. There's nobody yet in as the math applied so beautifully to the right, or to a question to a question that was very emerging.

Sergio Cleerman: Was very. And then there was a second, second major development by Kerr. This was 1963. Where the Kerr solutions was. Okay. So now you have a Kerr family, which includes Schwarzschild. It. It's a large family, depending on two parameters. So these are exact solutions of the underlying equation, right? I mean, you know, from mathematical point of view, they are real because for me, reality. Mathematical reality has to do with objective fact that these are solutions of an equation. Which you can write down. It's. It's conceptually.

Peter Robinson: And man, just. So if I understand one of the remarkable things about Einstein. By the way, of course, correct me. Jump in. I'm doing baby talk here, because that is the top of my form when it comes to this material. Yeah. Einstein comes up with general relativity in 1915. And here we are in 2025. And there's still experiments. There have been experiments that have been done over the course of the succeeding century as new satellite, new technology makes new experiments. And every single time, its the theory of general relativity is proven out. That is to say, this theory that Einstein came up with on a chalkboard. For a century of experiments now, it turns out to correspond with and predict reality and your work, if we could somehow devise experiments. On black holes, your work would prove out. It's real to that extent. So I like to call it a test of reality.

Sergio Cleerman: So. The fact that the Kerr solution is stable. Right, it's a mathematical statement, but with a lot of physical content, because let's say if it was not stable. So, it's a solution, a correct solution of the national equation, which starts with specific initial conditions. Right. So the issue of stability is now you make more perturbations of the initial conditions, and all of a sudden, you get something entirely different, which is nothing to do with a solution that the Kerr solution. That I would be called instability, right? So if the Kerr solution would be uns, it means it doesn't have any physical meaning, right? Because you know, it doesn't correspond to anything that you can recognize in nature is corresponding to that right? So the stability. The issue of stability is a fundamental issue in. It's a market reality. You can say. So I call it a mathematical test of reality.

Stephen Meyer: Let us what's going on here. So interesting explic, for. Well I'm. Just from a philosophical point of view. Here. Okay. Is that there's a deep assumption that, that, which is mathematically consistent, coherent, stable, is going to give us a guide to physical reality. As if there's a rationality built into the physics that somehow matches the rationality that's at work when we're doing this. This type of advanced mathematics. And so's. That's the Wigner mystery. Why does the reason within match. The rationality of nature, external to us, the reason without.

Peter Robinson: All right. So now now we move into territory. I'm already in over my head, but I continue swimming. It gets deeper.

Stephen Meyer: May offer a simple thing that might help just with because we got into the field equations of general relativity and the solutions and But Sergio started initially talking about geometry. And just the idea that mathematical objects have stable properties. This is why mathematicians regard them as real. A circle has certain basic properties. It's got a circumference area, and we can calculate these things. And those properties are true for all people who think about circles. Their stable properties that that geometric object has that we can describe mathematically. That's independent of our minds. And yet. And yet, the stability of those properties is a token, as Sergio has explained in our recent conference. It's a token of reality of a mind, objective reality, a mind, independent objective reality. And that's why mathematicians don't think that they're inventing. Yeah. New mathematical formulas, I think, almost almost universally feel that they're they're discovering something, not inventing.

David Berlinski: Well, there are some who don't. But, but what's interesting is that physicists always referred to mathematics as being an invention of the human mind. That's what Wigner says. Sorry, Einstein says an invention of human. But Wigner says something very similar. So I'm always surprised to see that. Einstein felt he invented the general. No, no, no, Einstein hears that mathematicians invent things. Right. So he actually called it a free creation of the human mind thread. So far, which he meant what. It's not really clear, because if it's a free creation of the human mind, why are mathematical propositions, so dreadfully necessary? I didn't have much choice about two plus two equals 4. And I presume you didn't either. But there kind of at odds with the notion of spontaneous spontaneously reaching an invention like addition. Doesn't seem to be an invention at all.

Stephen Meyer: But I have a reason why physes do that, because and I doubt that Newton would have said that. No, it's modern. It's modern phys who materialists. They do believe that there is just matter And everything the mind, including has to be determined. They can't be right.

Peter Robinson: About. Here noticeable's in in the dialectic here. The mathematicians, who are doing the the math that are developing the math. Yeah. Typically believe that they are discovering something that is real and independent of their mice, not inventing something like an internal combustion engine, or.

Sergio Cleerman: Exactly. I mean, any invention has to have a starting point, right? So it means that before that starting point, that mathematical fuck did not exist, right? Pythagorean theorem was not true before Pythagoras discovered it. It's kind of ridiculous.

Peter Robinson: But how can anybody who who how. Okay. There must be complexities here. I'm there complexities. I'm not grasping, but 2 plus 2 equals 4 is true for me. It's true for you. It's true for David. No matter. How perverse David may be feeling at any given moment is still true. It's true. And it has been true for all time. Therefore, there is something isn't that, doesn't that just put a stake in the heart of materialism right there? Something exists outside us. Something we can call it reason. We could call it Plato. Okay. So let's go to Plato. If I understand this much in the Republic, Plato draws a distinction between the intelligible world. The sensible world is what we can. See and touch. And the intelligible world is that which we can. Is intelligible to us, but access the we can access through the intellect. Okay. And so that's where he places his ideal forms. There is a circle out there. There is a triangle out there. Plato says it does have an independent existence, but he seems to suggest that there's. There's a realm of ideal forms someplace out there. Aquinas comes along, 1500 years later. And says ideas exist in minds. And something that is true for all of us and for all time, that is intelligible, but immaterial exists in the mind of God, David.

David Berlinski: Yeah, maybe. David was never more David than in that very moment. I can't make any sense of the discussion so far. That's my problem. There is a problem to which Wigner was calling attention, which is a real philosophical problem. That is that mathematics is essential for every physical theory. It cannot be the case. Unless it's a trivial explanation that there is a physical theory that physically explains mathematics. I, if a man proposes to catch a carp by baiting, his hook with a carp, he's engaged in a trivial pursuit. He has the carp. If we need a physical theory that includes mathematics to explain mathematics, we no longer have a physical theory. We have a physical and a mathematical theory. And that's the dilemma. I think the deep dilemma to which Wigner is calling attention. There are certain principles, we'd like to hold on to. It's part of the cultural imperative. We'd like to hold on to this fundamental idea. The world is physical. We live in a physical world. I'm not saying this is a commendable idea. I'm saying it's a cultural imperative because it seems so reassuring. Look we're faced with impo derables. The basic fact, the thing is a material or a physical object. It is very inconvenient, culturally and intellectually to come to the conclusion that in order to understand that physical object, we need a whole lot of non physical facts about mathematics. And in order to explain a whole lot of non physical facts about mathematics, there is no conception of a physical theory without mathematics that can do the explaining. So, we are left in the position that if mathematics is as useful as Sergio and Steve says, and they're absolutely right about that, it's useful in daily life. Yeah. There is something fundamental wrong, fundamentally wrong with our idea of the world as a physical system. Something cannot be right. Something has to get. Either we develop physics physical theories with no mathematics. Feynman conjectured something of this sort. Or we agree that materialism simply cannot be right. One of the two, but something has to go. But Physists, who, until now, at least have not given up on the idea of Ph deal.

Stephen Meyer: No. So that's. That's. That's a issue, but they're not reckoning with the dilemma that David just described. I think. And not me, I mean, that has been in the literature for at least 50 years or 6 years.

Peter Robinson: Even don't understand all of you, bright people. If something's been in the literature for half a century. Which essentially is a stop sign. Yeah. Saying, wait a moment. Wait a moment. Wait a moment. You have a basic decision to make here about the nature of reality. Either. It is then how can the academy, how? Can you academics just ignore it?

Sergio Cleerman: Well, so okay, so I don't know why. And and like David I hold you responsible. I don't want to talk. About. He. Exist issues of existence ideas, platonic ideas. Maybe its it's a step too far. And that maybe we agree. But I have a operational definition of reality. So reality is consistency of representations of a particular object. I mean, that's true in physics. And the extreme mathematics. Mathematic objects are real. A mathematician, who works on, on a mathematical problem. Yeah. Calcs in this way, calculating that other way, always get the same reason. I mean, there's something so obviously objective about what we do, right, that somehow to claim that these are just inventions of the human mind is ridiculous to me.

David Berlinski: I mean. Yeah. There is a way to defend any position and philosophy. I mean, we could argue that there are inventions of the human mind, because the human mind is the only thing that really exists. After all, there's a very noble tradition going right back to Berkeley, which makes exactly that case to be is to be perceived. But? There is something that inhibits a return to Berkeleyan idealism. In that, it sounds vaguely preposterous to say the only thing that exists is a mind. It doesn't comport with the magnificence of the physical theories. We've do about. It just doesn't. That's not an argument. It's an observation.

David Berlinski: Yeah. But having said that, we, we are really in danger of being reduced to an ever narrowing ice flow. The ice flow, as Sergio just mentioned, is the consistency of representations. Well, representations is kind of an obscure term. Why don't we get down to basics? It's the consistency of our theories. Well, what is the theory? Well, we can provide an answer to that. A theory is kind of a large group of sentences. And what are sentences? They are things that make certain kinds of assertions that can be true or false. And if they're consist. That is about as good as we can get in terms of the credibility and commendability of a theory. So, we're reduced now on our ice float is saying, well, we have a representation or a theory about the physical world, and it's consistent. But when we examine it, we find out it is not a physical object. It invokes non physical substances, like mathematical objects. We don't have to say, we need not make a decision. Do they exist in the mind or the exist in the external world? They exist. That's all we need. The number two exists. I don't have to tell you, well, it exists in your mind exists. My, that's in a realm. It exists. That's the determinative statement. And as long as it exists. And we acknowledge, it's not physical. Then, we're left with a position of saying, how come our best view of the physical world incorporates things that are not physical? Why is that?

Sergio Cleerman: Existence is an ontological question, I prefer reality. Is something that I can work with, well existence, I don't know. When, when it comes to issue existence. If he lost. Right. I don't New field exists. F. I mean, I don't know what What doesn't exist.

Stephen Meyer: What intrigues me in all this is the idea that those mathematical objects, whether it's the quadratic equation or a circle or more advanced forms of mathematics have stable properties. They, when when the of consistency, your talking I talk about reality. Yeah, prefer reality objectivity. Yeah, and consistency. Yeah. What Sergio was saying is that these, these mathematical structures or objects have an a reality that's independent of whether or not I affirm those properties myself. Their mind-independent. And, yet they're conceptual. Yeah. Essentially, they're not material or conceptual. So. It does raise the question. Where do they reside in what if there are concepts. And? This is where Wigner we're moving in this direction by by bringing Plato and Plato had the idea that there are conceptual realities that exist in some sort of of heavenly realm. But Aquinas's critique of that was that doesn't make it's not consistent with our experience, ideas exist in minds. And so if there are these objective properties of material of mathematical structures that are independent of our minds. And. If these mathematical structures are themselves conceptual. It implies not that they're floating around someplace. But they, they originate or reside fundamentally in some transcendent mind. That's the theist dictate tape.

Peter Robinson: So Steve on the mathematical realism. Steve says math exists in the mind of God. And Sergio says, "Don't bother me with that. And we're working mathematician. I need is a piece of chalk and and a blackboard an operational definition of reality."

Sergio Cleerman: Right, the notion that somehow everything that's real, it's physical doesn't make sense to me because mathematical objects are also here.

David Berlinski: And I think we're all saying, whichever, you know, when you slice this, whether you're. I think's absolutely right. I mean, even if we adopt Berkeley's position that to be is to be perceived. We right, we wind up at the same position that a thoroughly, consistent and coherent, coherent view of the universe simply can't be physical. Yeah, simply can agree.

Peter Robinson: Okay. That actually. That actually strikes even my little mind as a really quite profound. I'm not going to call it an insight. It's a discovery. Something. It's a real aspect of reality itself. Can I revert back to something earlier in the conversation It's just something that from David's book, he gave gave a talkca in the US on this one time. And it just, it just was so intriguing to me. It was, I think it was from your book 1, 2, 3. When you were talking about it on the road. And it was, he showed how, I think it was the complex variables, a complex numbers. Remember. The square root of negative one from math. There's. This whole mathematical apparatus that's been developed around complex numbers. Yeah. And it seems like it has absolutely nothing to do with anything. But there's a whole body of mathematics on this. I took a course in grad school on complex variables. And David showed that this was invented something like what was 140 or developed discovered 140 years before 00, maybe 200. And then lo and behold, it's absolutely crucial for doing quantum mechanics, which is our most fundamental physical theory or one of them.

Sergio Cleerman: No. But if I can, least correct, what is fascinating about the histo for the invention of the number I I it. But it came up in Italy in the. Yeah. You know 14th century, 14 1th century, no, little earlier, I 1450. Yeah, okay, 1450 a bit earlier, maybe. But in any case, it was based on their desire to solve equation. So. They wanted to solve the quadratic equation. There was a formula already. They Arab apparently also new it order. In any case. They they went the third order equation. And. They found that it it pay to introduce this symbol square root of - 1, which makes no sense. You cannot take the square root of - 1, right? But. They just put it there. And. They made this incredible observation that you can use this number. And at the end. You are getting a solution of a cubic equations, which are all real. Nothing to do with those complex numbers. But they enter into the formula So. This was quite an amazing thing, right? So, then little by little. They got used to this using square root of - 1. And at some point in the 19th century. It was even made more formal, was given a dramatic dramatic definition of scores of - 1, and have complex numbers and complex functions. And. Then the enormous amount of publications came out of it. So. It's it square or - 1 is is obviously real. It existed before it was discovered by.

David Berlinski: There's a very interesting. I I mean the basic in 9 grade. When they introduced negative numbers, I start paying attention. Negative numbers? No. Nothing to do with reality. But I was wrong. Then. Peter is not negative. These are complex numbers. The square root of - 1. Negative numbers of - 1 - 2 think of deaths. We're not talking about deaths now. We're talking about a complex number, but here's the extraordinarily interesting point. You got this. Weird old Italian mathematician of the 15th century, who figures out that if I introduce the symbol I equals the square root of - 1. I can solidify a chain of inference and come out with the right answer. Just absolutely amazing. However, when mathematicians start to think about it, it's entirely possible to get rid of the square root of - 1. Yeah. In favor of two real numbers and a set of rules for manipulating them. All of a sudden. The square root of - 1 is gone. You're left with what you began with the real numbers, 3 and 7 in a certain order and obeying certain rule, a multiplication of rule, which is essential. So. Yes, the complex number. The complex number. It's a new multiplication rule, which was not did not exist in the real of of real what is called real natural numbers. But that that introduce a very interesting analytical point that ontology can be reduced in favor of a system of rules and regulations. You can reduce the ontological burden of mathematics. You can say. Yeah. I'm going to get rid of the numbers in favor of the sets. I'm going to get rid of the complex numbers in favor of ordered pairs of real numbers. I'm going to get rid of the real numbers and favor of convergent sequences. I'm going to get rid of so much. But as you reduce the burden of ontology, you increase the burden of your regulations. So. You actually never get to the point where mathematics appears from nothing. You never get to that point. Like. Biologist just being the affirmation of what's really true. Something substantially real. I mean, biologists love to say, life comes only from life, 19th century It's, obviously, true. Life comes only from life and how it might not come from life is an utter mystery, but it's also true that language comes only from language. And. It's additionally true that mathematics only comes from mathematics. These seem to be processes with which we have a good deal of experience, which have no point of origin. There is no place in which mathematics originates. There is no place in which language originates. And there is no place in which life originates. They may well be fundamental features of the universe itself.

Sergio Cleerman: But. Nevertheless, mathematics has a history. So in this article.

David Berlinski: But, it's infinite. In the past, right, I mean, was human. So. There is something human about ma There's, a history of our discovery, but not a history to the realities.

Sergio Cleerman: Exactly that we are discovering the way we discover right, okay.

Peter Robinson: David, this is you in a recent. This is a conversation you had. I think with Steve in Cambridge. I'm going to quote a passage. "Hold up a finger. Could this finger be a different color? Yes. Could it be slightly longer? Yes. Could it be crooked? Yes. But it could it ever be anything other than one finger? No. The number is obligatory. The number, the number is something the finger essentially has." All right. So. Now we're in the realm of Aristotle and the difference between essence. Essential properties and nones and accidents, essential properties and accidents explain this.

David Berlinski: Well, I feel it'on tiptoeing really well. Rob. All right, right, so numbers represent an essential aspect of reality. That's a big deal.

Sergio Cleerman: Well, it's a very general statement. I much prefer God forbid me, forgive me for introducing Heidegger. I much prefer his formulation. Heidegger, and they're very interesting passages in his work. I admit it, and he says, "Look, when we look at objects. Yeah. We cannot separate the oneness of this glass from the object itself, but we can change the color. We can change the shape of the story, the same object. But. We can say this one object could have been two. That just doesn't go." So. When we talk about physical objects. We're only talking now about physically realizable objects. Their mathematical art aspects are essential to them. What holds for numbers also holds for shape. We don't have to do what Eugene Wigner did and say, "Look at quantum mechanics and the remarkable fact that Hilbert spaces require introduction of complex numbers." Now, just look at the glass. The glass requires the introduction of a natural number. Okay. Two glasses require the introduction of two natural numbers. That is, is every bit as mysterious as the invocation of complex numbers in quantum field theory. Every bit is mysterious, and we don't know quite why.

Peter Robinson: Okay. So all three of you are willing to agree? All three of you are willing to insist that mathematics. The existence of mathematics, the weird way in which mathematics seems to correspond with and help us to investigate reality in a way that is real. I'm using reality over over again, proves. That reality is not purely not limited to what we can access by our five senses.

Stephen Meyer: It's a hint. It doesn't prove it's a hint.

Peter Robinson: Now, I've lost deep ground even from that. So. All we have is a hint. So. Why aren't you, I think Steve is willing to go. Now, here I am putting words into Steve's mouth. Steve is willing to say. We're dealing here with the mind of. Think what David's. And David is David won't do that. No matter how much I you down and you would.

Sergio Cleerman: I'll give you sort of an example of something like this. So, you know, immediately after Newton, there was Newtonian. The idea was the Newtonian mechanic has to explain everything. Then came Maxwell, the Maxwell equations. And. In order to. Adjust. The Maxwell equations to his Newtonian mechanics. You need. They need this notion of ether. Right, so. You know, ether here. Is that there are. At some point, I mean, Einstein's great insight was we don't need it. So. It's the same thing. I believe we don't need this materialistic representation. Just. Forget about it. Reality means something broader than that. It's inconsistent with the most obvious presentation of mathematics itself. It's. It's obvious that mathematical properties are not material. You can. Yeah. Invent a metaphysical system that explains that away. That's. Why it's not a proof. But. So nobody proved the eat is not. Does not exist. Right, we just get rid of it. And I think that's what we should do about it.

David Berlinski: I think I could agree with that. But going back to your last remark, Peter, I think it's just much simpler to say that the mystery is just the existence of mathematics. It's just that, because it's fundamental. We could well say, and, of course, philosophers have well said, we can get rid of the physical world. Metaphysically, that's not a problem. Berkeley showed how everything is a perception or an idea. External world just disappears, but we can't get rid of the mathematical world. That's inel imitable. And its existence is a profound mystery. What. Is it doing there. Why? Do we see things in mathematical terms? Now, I'm not asking this question because I have a secret answer.

Peter Robinson: I'm prepared to aboutge say. I was hoping you wrap up the conversation with the, "I find it a great mystery." The sheer existence of mathematics is deeply puzzling. You will agree with every word of that.

David Berlinski: Yeah, totally and.

Peter Robinson: And you will agree. But can you take it farther.

Stephen Meyer: Yeah. Well, I just am intrigued with this kind of argument that I recapitulated earlier in the conversation that that mathematical objects have stable properties. Therefore, they have an objectivity that is independent of our minds. And yet, they are conceptual, which suggests suggests by our experience that they must. Yeah. Not be floating around somewhere in the Platonic heavens. But rather, it makes more sense to me to think that they ultimately issue from the mind of God. And that, that is the deep reason for the mysterious applicability of mathematics to the physical world.

David Berlinski: There in mind, Steve. Re reaching a position very close to Berkeley's position Berkeley, Bishop I mean, if you say that to be is to be Bishop Berkeley's 17th century British 1818th century English churchman and philosopher, who appears possibly most famously in Boswell's life of Johnson when Boswell Berkeley Johnson kicking a rock. But. The point that Berkeley faced the obvious question, "If. You're not looking at the moon, Einstein discusses this, too. Does. The moon continue to exist." And Berkeley's response was, "Yes. It exists as a thought in the mind of God," which is very close to what Steve was just A gy, although he's certainly not.

Stephen Meyer: I'm not Berkeley. Yeah I. Don't think the physical world is has a, I think it has. Md and independence of our mind of the mind of God who created it. But I think the ultimate source of mathematical reality may well be the mind of God.

David Berlinski: But. It's interesting that you are. Prepared to go further than, I think many contemporary analytic philosophers prepared to go in the direction of a Berkeleyan kind of analysis, which I think is. In this case, the only analysis that makes sense, at least for math. At least for math, because it is so mysterious.

Peter Robinson: Boys, I'm going to attempt for last question here. I'm going to attempt to introduce one new concept. And, that is the concept of beauty. I have no idea how this will go. But. Here's an excerpt from the forthcoming documentary, The Story of Everything. There's something in science called the beauty principle that says, "True theories, often convey a mathematical beauty or structural harmony." Upon looking at their model of the DNA molecule, Francis Crick was quoted as saying, "It's so beautiful. It's got to be right." Explain that.

Stephen Meyer: What should enter into this. We don't really know, but it tends to be what's called a heuristic guide in science, a guide to discovery. And in the, in the section of the film that follows. There were maybe perhaps even more trenchant comments from a couple of the physicists who were saying how often their perception of mathematical beauty had been a guide to discovery. I think it was Paul Dirac, who first said that it's more important for the theories. Beautiful than than to have them consistent with the data, because eventually.

David Berlinski: Yeah.

Stephen Meyer: Well, because we can be mistaken about how we're perceing the data. But there. There is assumption that there's something mathematically beautiful about, about reality itself.

Peter Robinson: Intelligibility. Was mesmerized by who Dirac did I. Was mesmerized by, and you said a moment ago when you were choosing. Yeah. Projects on which to work. You, I think you use the word. I. It elegant. Is it beautiful?

Sergio Cleerman: Of course. So. Why is that the Kerr solution is an incredible, incredible object? I mean, very, really beautiful. I mean, yeah. Are we on another mystery here? An aesthetic mystery?

Sergio Cleerman: Beauty plays a fundamental role, of course, in the way we choose problems, but also in the way we are guiding ourselves towards the truth, I mean, towards the solution of a problem. Somehow, we reject, reject arguments, which are contrived, which are not beautiful. We, don't call them beautiful. Yeah, I mean, it's mysteriously true. And physics, of course, is full of such examples. Maxwell, you know, the the Maxwell. The way the Maxwell equations were discovered. It was first Faraday who had the three laws of like mine that he's already discovered experimentally. That's right. And then, but he was not a mathematician at all. So. He, he just. Okay. He just left it there. Let just stated the laws. And. It was Maxwell who realized that if you put those statements within Mathematics, there is a lack of symmetry. Sort of guided him towards a fourth one. Which led to electromagnets. So in all of us, technology of of the modern world, beauty, Maxwell with Maxwell.

Peter Robinson: You're comment upon beauty. Yeah, I'd like to hear from these guys because, I kind of reserve that for my tailors.

Sergio Cleerman: And, you know, I can tell you that is a working mathematician. It plays absolutely a fundamental role in everything people do. There are very few mathematicians, who would say, "I work on this because it's. It's. It's just ugly," right, I mean, you know, they, they choose the problems or directions. Based on genetic party line survey.

David Berlinski: There are all sorts of mathematical drabs that we keep hidden. Nobody's gonna tell me turbulence is a beautiful subject. Oh. It's. It's a fantastic, fantastic. But not beautiful, clumsy lumbering.

Sergio Cleerman: Well, but, but the expectation is that we'll find.

David Berlinski: Yeah. We find the beauty in 3 as by expectation purchased beauty comes at famine prices.

Peter Robinson: Kickick him because he's being perverse. Now. He's he's being mischievous.

Peter Robinson: Last question, Isaac Newton. The man who gave us mathematics that on astronomy, fluid dynamics and calculus. And Newton explains a lot. Here's a quotation from Newton. "A heavenly master governs all the world as sovereign of the universe." Close. A recognition of the divine or a mind that transcends our own is taken for granted for thousands of years. As recently as Newton. And, then it gets kicked out. Of intellectual life and kicked out of the academy. Does the mystery of mathematics suggest. That. The materialist error was an aberration that ought to, and may be. May be ending now. Are you willing to go that far, Steve?

Stephen Meyer: Of course, I think that's exactly right. And. What Newton illustrates so beautifully is that this, this principle of intelligibility that we've been talking about, that the mathematical rationality that can be developed by mathematicians without necessarily observing nature. Yeah. Does then apply to understanding the rationality that's built into nature. And. This was so crucial to the period of the scientific revolution. When these the systematic methods for interrogating nature for studying nature were developed and culminating in figures that maybe is no figure like Newton, who was so profound in his his insight and who advanced. What was called natural philosophy that or science so much in one generation, even even in one year, his fame and mib ilous, where he. Went home during the plague to remediate his own deficiencies in mathematics and came back having invented the calculus. It. So a heaven Lou masters ter.

David Berlinski: David. All right, what can I say, that insight has not been vouchaf to me.

Peter Robinson: Fair enough. Sergio.

Sergio Cleerman: Well, okay, so, 1s of all, I think material should, I mean, materialism is just the only explanation of of the world should be put in the Ash binov his. So that that that we both agree. Bringing God in. Sure, why not? I mean, that's another way of looking at the world. Whether if there is something else. Well, let's find out. Maybe. There is another explanation. But at this point, I don't see any reason why you should not look at that possibility. So. God exists. Me, and why not?

Peter Robinson: Sergio Cleerman, David Berlinski and Steve Meyer, gentlemen, thank you.

David Berlinski: Then our privilege, Peter.

Peter Robinson: For Uncommon Knowledge, the Hoover Institution and Fox Nation, I'm Peter Robinson.