Why Does 2 + 2 = 4? What Math Teaches Us About Deep Reality (Transcript)

But when you say to me, 2 plus 2 equals 4, I have to say, of course, you’re right about that. Is that the point?

STEPHEN MEYER: I can speak to this from the standpoint of someone who’s worked in the natural sciences and as a philosopher of science. The natural sciences provide empirical observational evidence in support of conclusions and scientists will evaluate particular theories or hypotheses by comparing their explanatory power, 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: Hang on 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 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 characterized as abductive, that kind of detective reasoning that we enjoy when we watch detective shows.

PETER ROBINSON: Yeah, Columbo or someone’s trying to figure out who done it.

STEPHEN MEYER: 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 can give you.

PETER ROBINSON: Good scientists will never say any more than, “The theory is X, Y, Z, and on the best evidence it holds up for now.” Whereas the mathematician feels perfectly confident in saying, “I’ve proven it.”

SERGIU KLAINERMAN: We’ve proven it. We have a proof. We don’t say proof in science. The mathematicians, however, are better than we. When you say you’ve proved something, you mean it.

DAVID BERLINSKI: Yes.

PETER ROBINSON: All right. Okay. So just go ahead.

SERGIU KLAINERMAN: It could be wrong, but somebody would show it to me.

The Unreasonable Effectiveness of Mathematics

PETER ROBINSON: So this brings us to Sergiu’s article in Inference, a magazine that you edit, David. The article is 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, 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, Sergiu 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, or 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 physics, but then later were crucial. And maybe he should speak to that.

DAVID BERLINSKI: 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: 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?

PETER ROBINSON: Two.

DAVID BERLINSKI: 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 of the situation reveals the fact that one and one are 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 category 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 across, but somehow or other, it’s not the answer to the question. And we were all in the position of watching someone cross a large body of water and explaining his success by saying, “Look at my feet, small steps.” That’s where we are.

Discovery or Invention?

PETER ROBINSON: Okay. So Sergiu, well, 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 an invention, a creation of the human mind.”

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

SERGIU KLAINERMAN: Yeah, no, absolutely. I mean, one image that you could make is that of an alpinist that is trying to go on the top of the mountain. Everybody’s doing Alpine metaphors today, but 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 deduction. There is sort of a vision of where you want to go, which is very important.

PETER ROBINSON: Inspiration as well.

SERGIU KLAINERMAN: Right. It has something to do with inspiration. But then, you know, you have something very objective in front of you, right? The stone is what the alpinist, that you have to take into account that you’re not going to fall. You have to touch the stone to know exactly where you are going. And if you don’t, you get into trouble.

Some mathematics is very similar. People have the feeling that everything is deductive. It’s not. I mean, it is very similar in respect to physical sciences. Physical sciences also, you have some idea of, you have, let’s say, an expectation. You make a hypothesis, right? The choice of hypothesis is not a deductive thing. It’s just an insight.

And then you try to show that it fits everything else, all the other experiences. And that’s the process of proving something, right? So you have the process of discovery, but then the process of justification.

PETER ROBINSON: The process of justification.

SERGIU KLAINERMAN: 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 have the chain, the computer can go very fast to it and maybe even check. So there are yet no computers that can check large—I mean, they can check small proofs, but not large ones.

In any case, it’s sort of a very good example. I think that illustrates very well relations between math and physics. Take geometry. So the geometry was the first really theory of the physical world, right? I mean, it’s what it describes.

PETER ROBINSON: We go back to Euclid. Is that what you mean?

SERGIU KLAINERMAN: It goes back. Right. So Euclid tried, obviously, he tried to make sort of mathematical statements out of—but it’s a physical theory, without a doubt, including geometry is a physical theory. Then it developed.

The Freedom and Beauty of Mathematical Discovery

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 physicist would do. You know, they’re interested in problems because they are beautiful or because they feel that it will lead to certain understanding of something else.

And this freedom, really, in the case of geometry, went for, I don’t know, 2,000 years without essentially no connection back to the physical world. I mean, Euclidean geometry was there to start with. But then by the 19th century, you have Gauss, you have Lobachevsky, you have Riemann, you have Minkowski at the beginning of the 20th century.

And then all of a sudden, all that stuff becomes an essential ingredient. I mean, it’s not just the technical, it’s an essential ingredient of spatial and general relativity. It becomes directly applicable to fundamental physical theory. And I think this has happened many, many times, and maybe it’s not very well acknowledged.

The Aesthetics of Mathematical Work

PETER ROBINSON: So can I ask, when you, Sergiu, needless to say, I cannot evaluate your work on my own, because you have done a 2,000-page proof, 2,000 pages of close mathematical reasoning. I could live to 2,000, and I could read a page a day, or a page a year, and it still would escape me, I’m sure. And this was on the stability of some aspect of 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 this? I’m trying to understand what it felt like to you as a working mathematician engaged on a deep, very difficult problem.

SERGIU KLAINERMAN: And the answer is both. I mean, I would take the problem in the first place. I’m a mathematician, I’m not a physicist. I’m interested in, I believe that the best mathematics is connected somehow with physics in complicated ways. So, but there are many problems in physics, and I would pick the one that satisfies my aesthetical feeling as a mathematician.

PETER ROBINSON: Your aesthetical feeling, so explain that. You want something that’s beautiful?

SERGIU KLAINERMAN: Something that I feel is very beautiful, it’s very profound, it gives lots of very interesting questions. Okay, so that’s one aspect. But then it has to be the second, for me at least, there has to be a second aspect, which is that it should say something about the physical world, and in this case, it does, right?

The Kerr Solution and Mathematical Reality

I mean, the Kerr solution, so here’s how it goes, right? You have general relativity, which was well formulated by Einstein at the end of 1915, excuse me. And this was at the time a new theory of gravity. Massive bodies curve what’s called space-time. And then certain solutions were found.

Schwarzschild was the first to found in 1916, immediately a year after, found the so-called Schwarzschild solution, which is a stationary solution with a lot of symmetries that you can actually extract from the theory of relativity, from the Einstein equations, exact formula. And that had led to lots of issues, because it has a singularity. This is connected later on with the Penrose singularity theorem, for which he actually got the Nobel Prize.

PETER ROBINSON: He’s the only mathematician to have gotten a Nobel Prize in physics, despite there’s nobody else.

SERGIU KLAINERMAN: And I guess the math applied so beautifully to the physics and cosmology.

PETER ROBINSON: Right, or to a question that was very important.

SERGIU KLAINERMAN: And then there was a second major development by Kerr, this was 1963, where the Kerr solutions were. Okay, so now you have a Kerr family, which includes Schwarzschild. It’s a large family, depending on two parameters. So these are exact solutions of the Einstein equations, right?

I mean, you know, from a mathematical point of view, they are real, because for me, reality, mathematical reality has to do with the objective fact that these are solutions of an equation, which you can write down conceptually.

Einstein’s Enduring Theory

PETER ROBINSON: And may I interrupt for just a moment? 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. Einstein comes up with general relativity in 1915. And here we are in 2025. And there’s still 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, 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 what it’s real to that extent.

SERGIU KLAINERMAN: So I like to call it a test of reality. 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 Einstein equation, which starts with specific initial conditions, right?

So the issue of stability is now we make small perturbations of the initial conditions. And all of a sudden, we get something entirely different, which has nothing to do with the solution, the Kerr solution, that would be called instability, right? So if the Kerr solution would be unstable, 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 test, it’s a marker for reality. You can say, so it’s, I call it a mathematical test of reality.

The Rationality of Mathematics and Physics

PETER ROBINSON: Let’s write us what’s going on here, Peter.

STEPHEN MEYER: Yes, please. So interesting. Explicate for us.

Well, I’m just as, from a philosophical point of view here, is that there’s a deep assumption 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 type of advanced mathematics.

And so 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 we move into territory. I’m already in over my head, but I continue swimming.

STEPHEN MEYER: It gets deeper. May I offer a simple thing that might help just with, because we got into the field equations of general relativity and the solutions and, but Sergiu started initially talking about geometry. And just the idea that mathematical objects have stable properties. This is why mathematicians regard them as real.

You know, circle has certain basic properties. We know it’s got a circumference and area, and we can calculate these things. And those properties are true for all people who think about circles. There are stable properties that that geometric object has that we can describe mathematically that’s independent of our minds.

And yet the stability of those properties is a token, as Sergiu has explained in our recent conference, it’s a token of reality, of a mind objective reality or a mind independent objective reality. And that’s why mathematicians don’t think that they’re inventing new mathematical formulas. They think almost universally feel that they’re discovering something, not inventing.

Discovery Versus Invention

PETER ROBINSON: Well, there are some who don’t, but what’s interesting is that physicists always refer to mathematics as being an invention of the human mind. That’s what Wigner says.

SERGIU KLAINERMAN: Sorry, Einstein says an invention of the human mind, but Wigner says something very similar. So I’m always surprised to see that.

PETER ROBINSON: Einstein felt he invented the general?

SERGIU KLAINERMAN: No, no, no. Einstein feels that mathematicians invent things, right? He actually called it “a free creation of the human mind.”

DAVID BERLINSKI: By 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 four. And I presume you didn’t either. Kind of at odds with the notion of spontaneous, spontaneously reaching an invention like addition. It doesn’t seem to be an invention at all.

SERGIU KLAINERMAN: But I have a reason why physicists do that because, and I doubt that Newton would have said that. It’s a modern, it’s a modern physicists who are materialists. They do believe that there is just matter and everything, the mind including, has to be determined somehow. But they can’t be right about it.

STEPHEN MEYER: Okay. So here, notice what’s in the dialectic here, the mathematicians who are doing the math that are developing the math typically believe that they are discovering something that is real and independent of their minds, not inventing something like an internal combustion engine or…

SERGIU KLAINERMAN: Exactly. I mean, any invention has to have a starting point, right? So it means that before that starting point, that mathematical fact did not exist, right? Pythagoras theorem was not true before Pythagoras discovered it. It’s kind of ridiculous.

The Challenge to Materialism

PETER ROBINSON: But how can anybody who, how can, there must be complexities here. I’m sure there are complexities. I’m not grasping, but two plus two equals four 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, it’s 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, 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 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 a realm of ideal forms someplace out there. Aquinas comes along 1,500 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.

PETER ROBINSON: David was never more David than in that very moment.

The Fundamental Dilemma

DAVID BERLINSKI: 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, if 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 mean, 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 imponderables. 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’re left in the position that if mathematics is as useful as Sergiu and Steve says, and they’re absolutely right about that, it’s useful in daily life, there is something fundamentally wrong with our idea of the world as a physical system.

Something cannot be right. Something has to give. Either we develop physical theories with no mathematics. Feynman conjectures something of this sort. Or we agree that materialism simply cannot be right. One of the two. But something has to go.

SERGIU KLAINERMAN: But physicists, until now at least, have not given a point of view of physicality.

DAVID BERLINSKI: No. So that’s the issue.

STEPHEN MEYER: But they’re not reckoning with the dilemma that David just described.

SERGIU KLAINERMAN: Not me. I mean, that dilemma has been in the literature for at least 50 years or 60, even 100 years.

Why the Academy Ignores the Dilemma

PETER ROBINSON: I 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, saying, wait a moment, wait a moment, wait a moment. You have a basic decision to make here about the nature of reality. Then how can the academy, how can you academics just ignore it?

SERGIU KLAINERMAN: Okay. So I don’t know why. I don’t want to talk about issues of existence, ideas, platonic ideas. Maybe it’s a step too far. And that may be what we agree.

But I have an operational definition of reality. So reality is consistency of representations of a particular object. I mean, that’s true in physics and that’s true in mathematics. Mathematical objects are real. A mathematician who works on a mathematical problem calculates in this way, calculates in that other way, always get the same result.

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. I mean, there is a way to defend any position in philosophy.

DAVID BERLINSKI: I mean, we could argue that they’re 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.”

The Limits of Physical Reality

But there is something that inhibits a return to Barclayian idealism in that it sounds vaguely preposterous to say the only thing that exists is the mind. It doesn’t comport with the magnificence of the physical theories we’ve talked about. It just doesn’t. That’s not an argument, it’s an observation.

But having said that, we are really in danger of being reduced to an ever narrowing ice flow. The ice flow, as Sergiu 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’re things that make certain kinds of assertions that can be true or false. And if they’re consistent, that is about as good as we can get in terms of the credibility and commendability of a theory. So we’re reduced now and 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 do they 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, it exists in my mind.

That’s irrelevant. 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? Existence is an ontological question. I prefer reality as something that I can work with. Well, existence, I don’t know.

When it comes to the issue of existence, I feel lost. You feel lost? I mean, I don’t know what exists and what does not exist. What intrigues me in all this is the idea that those mathematical objects, whether it’s a quadratic equation or a circle or more advanced forms of mathematics, have stable properties.

PETER ROBINSON: Is this the consistency you’re talking about?

SERGIU KLAINERMAN: I’m talking about reality. Yeah, I prefer to call reality, objectivity, consistency. Yeah, what Sergiu is saying is that these mathematical structures or objects have a reality that’s independent of whether or not I affirm those properties myself.

Mind-Independent Mathematical Reality

STEPHEN MEYER: They’re mind independent and yet they’re conceptual, essentially. They’re not material, they’re conceptual. So it does raise the question, where do they reside? If there are concepts, and this is where we’re moving in this direction by bringing Plato in.

Plato had the idea that there are conceptual realities that exist in some sort of heavenly realm, but Aquinas’ critique of that was, that’s not consistent with our experience. Ideas exist in minds. And so if there are these objective properties 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 originate or reside fundamentally in some transcendent mind. That’s the theistic take on the mathematical realism.

PETER ROBINSON: Steve says math exists in the mind of God, and Sergiu says, no, don’t bother me with that. I’m a working mathematician. All I need is a piece of chalk and a blackboard. Operational definition of reality, right?

SERGIU KLAINERMAN: The notion that somehow everything that’s real, it’s physical, doesn’t make sense to me because mathematical objects are also real. And I think we’re all saying, whichever way you slice this, whether you’re…

DAVID BERLINSKI: Yeah, I think that’s absolutely right. I mean, even if we adopt Barclay’s position that “to be is to be perceived,” we wind up at the same position that a thoroughly consistent and coherent view of the universe simply can’t be physical.

It simply can’t be free.

PETER ROBINSON: Okay. 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. It’s a real aspect of reality itself.

The Mystery of Complex Numbers

Can I revert back to something earlier in the conversation? Sure. Just something from David’s book. He gave a talk in the US on this one time, and it just was so intriguing to me.

I think it was from year book one, two, three, when you were talking about it on the road. And he showed how, I think it was the complex variables, the complex numbers. Remember the square root of negative one from math. There’s this whole mathematical apparatus that’s been developed around complex numbers.

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 it, 140 years? Developed?

Discovered? 140. 140 years before. 200, 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. What is fascinating about the history of the invention of the number I is that, well, it came up in Italy in the 14th century, 14th and 15th century. 16th century. No, earlier.

I think 1450. Yeah. Okay. 1450, a bit earlier maybe.

But in any case, it was based on their desire to solve equations. So they wanted to solve first the quadratic equation. There was a formula already. The Arabs apparently also knew it already.

In any case, they went to the third order equation and they found that it pays to introduce this symbol, square root of minus one, which makes no sense. You cannot take the square root of minus one, 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. They have 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 minus one.

And at some point in the 19th century, it was even made more formal. It was given a geometric definition of square root of minus one. And you have complex numbers and complex functions. And then the enormous amount of applications came out of it.

So square root of minus one is obviously real. It existed before it was discovered. But there’s a very interesting… I made a basic mistake in ninth grade.

When they introduced negative numbers, I stopped paying attention. Negative numbers have nothing to do with reality. But I was wrong then. Peter, it’s not negative.

These are complex numbers. The square root of minus one. Negative numbers are minus one, minus two. Think of debts.

We’re not talking about debts now. We’re talking about a complex number. But here’s the extraordinarily interesting point. You got this weirdo Italian mathematician in the 15th century who figures out that if I introduce the symbol I equals the square root of minus one, 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 minus one in favor of two real numbers and a set of rules for manipulating them.

All of a sudden the square root of minus one is gone. You’re left with what you began with, the real numbers, three and seven, in a certain order and obeying certain rules. A multiplication rule which is essential. So that’s the complex number.

The complex number. Exactly. It’s a new multiplication rule which did not exist in the realm of real, what is called real natural numbers. But that introduces a very interesting analytical point that ontology can be reduced in favor of a system of rules and regulations.

Reducing Ontological Burden

You can reduce the ontological burden of mathematics. You can say, 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 in 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, a biologist… Ontology 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 biologists. 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’s no place in which language originates. And there’s no place in which life originates. They may well be fundamental features of the universe itself.

But nevertheless, mathematics has a history. But it’s infinite. In the past. Right.

I mean, it’s a human. So there is something human about mathematics. There’s a history of our discovery, but not a history to the realities. Exactly.

That we are discovering. The way we discover. Okay. 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 could it ever be anything other than one finger? No. The number is obligatory.”

“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 non-essential properties.

Essential Properties and Accidents

And accidents. Essential properties and accidents. Explain this. Well, I feel as though I’m tiptoeing around…

You’re doing pretty well, Robinson. All right. All right. Numbers represent an essential aspect of reality.

That’s a big deal. Well, it’s a very general statement. I much prefer… God forbid me…

forgive me for introducing Heidegger. I much prefer his formulation. Heidegger… there are very interesting passages in his work, I admit it.

And he says, look, when we look at objects, we cannot separate the oneness of this glass from the object itself. But we can change the color, we can change the shape of it, it’ll still be the same object. But we can’t 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 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.

No, just look at the glass. The glass requires the introduction of a natural number. Two glasses require the introduction of two natural numbers. That is every bit as mysterious as the invocation of complex numbers in quantum field theory.

Every bit as mysterious. And we don’t know quite why. 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 and over again, proves that reality is not purely, not limited to what we can access by our five senses. It’s a hint.

It doesn’t prove. It’s a hint. Oh, now I’ve lost deep ground even from that. Oh, 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 God. I think what David’s getting at is… David won’t do that no matter how much…

I could pin you down and you wouldn’t… I’ll give you sort of an example of something like this. So, you know, immediately after Newton, there was Newtonian mechanics. The idea was that Newtonian mechanics has to explain everything.

Then came Maxwell, the Maxwell equations, and in order to adjust the Maxwell equations to Newtonian mechanics, they needed this notion of ether, right?

Beyond Materialism

So, you know, ether here is that there, at some point, I mean, Einstein’s great insight was we don’t need it, right? So it’s the same thing, I believe. We don’t need this materialistic representation of the world. Just forget about it.

Reality means something rather than that. It’s inconsistent with the most obvious presentation of mathematics itself. It’s obvious that mathematical properties are not material. You can invent a metaphysical system that explains that away.

That’s why it’s not a proof, but… So nobody proved the ether does not exist, right? We just get rid of it. And I think that’s what we should do about it.

STEPHEN MEYER: 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.

DAVID BERLINSKI: 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 ineliminable. 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 I’m prepared to vouchsafe.

PETER ROBINSON: I was hoping you’d wrap up the conversation with the answer.

DAVID BERLINSKI: I find it a great mystery. The sheer existence of mathematics is deeply puzzling. You will agree with every word of that.

The Mind of God and Mathematical Reality

STEPHEN MEYER: Yeah, I totally do. And you’ll agree, but can you take it farther? Well, I just am intrigued with this kind of argument that I recapitulated earlier in the conversation, that mathematical objects have stable properties, therefore they have an objectivity that is independent of our minds, and yet they are conceptual, which suggests by our experience that they must 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 is the deep reason for the mysterious applicability of mathematics to the physical world.

DAVID BERLINSKI: Bear in mind, Steve, that you’re reaching a position very close to Berkeley’s position.

STEPHEN MEYER: To whose?

DAVID BERLINSKI: Berkeley, Bishop Berkeley.

STEPHEN MEYER: I mean, if you say that to be is to be perceived—

DAVID BERLINSKI: Bishop Berkeley, 17th century British—

STEPHEN MEYER: 18th century.

DAVID BERLINSKI: 18th century English churchman and philosopher, who appears possibly most famously in Boswell’s Life of Johnson, when Boswell says—”I refute Berkeley, thus.” Thus, Johnson kicking a rock. But the point is—

STEPHEN MEYER: Is Berkeley to face 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 arguing.

DAVID BERLINSKI: Although he’s certainly not—

STEPHEN MEYER: I’m not a Berkeleyan. Yeah, I don’t think the physical world has a—I think it has a mind, an independence of our mind and of the mind of God who created it. But I think the ultimate source of mathematical reality may well be the mind of God.

But it’s interesting that you are prepared to go further than I think many contemporary analytic philosophers are 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.

DAVID BERLINSKI: At least for math.

STEPHEN MEYER: Yeah. At least for math, because it is so mysterious.

The Beauty Principle in Mathematics

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. Why should beauty enter into this?

STEPHEN MEYER: 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 section of the film that follows, there may be perhaps even more trenchant comments from a couple of the physicists who are 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 to be beautiful to have them consistent with the data, because we can be mistaken about how we’re perceiving the data. But there is an assumption that there’s something mathematically beautiful about reality itself.

SERGIU KLAINERMAN: Intelligibility. Dirac was mesmerized by it.

PETER ROBINSON: Yeah. I’m sorry, who was?

SERGIU KLAINERMAN: Dirac, Paul Dirac.

PETER ROBINSON: Did I say? I said Dirac, yeah. Dirac was mesmerized by it. And you said a moment ago when you were choosing projects on which to work, I think you used the word, is it elegant? Is it beautiful?

SERGIU KLAINERMAN: Aesthetic. Of course. The Kerr solution is an incredible, incredible object. I mean, very, really beautiful. I mean, yeah.

PETER ROBINSON: Are we on another mystery here? An aesthetic mystery?

SERGIU KLAINERMAN: 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 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 way the Maxwell equations were discovered, it was first Faraday who had the three laws of electromagnetism already discovered experimentally.

STEPHEN MEYER: That’s right.

SERGIU KLAINERMAN: But he was not a mathematician at all. So he just left it there and just stated the laws. And it was Maxwell who realized that if you put those statements within mathematics, there is a lack of symmetry. It sort of guided him towards the fourth one, which led to electromagnetism.

STEPHEN MEYER: And all of the technology of the modern world.

PETER ROBINSON: Beauty?

STEPHEN MEYER: Maxwell.

PETER ROBINSON: With Maxwell. Your comment on beauty?

DAVID BERLINSKI: Yeah, I’d like to hear from these guys because, yeah. I kind of reserve that for my tailors. You know, I can tell you that as 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 just ugly, right? I mean, you know, they choose the problems or directions based on…

SERGIU KLAINERMAN: It’s just a party line, Sergiu. There are all sorts of mathematical grabs that we keep hidden. Nobody’s going to tell me turbulence is a beautiful subject.

DAVID BERLINSKI: Oh, it’s a fantastic subject.

SERGIU KLAINERMAN: Fantastic, but not beautiful, clumsy, lumbering.

DAVID BERLINSKI: But the expectation is that we’ll find the beauty in turbulence by…

SERGIU KLAINERMAN: Expectations are easily purchased. Beauty comes at famine prices.

PETER ROBINSON: All right. Kick him because he’s being perverse now. He’s being mischievous.

Newton and the Divine Mind

PETER ROBINSON: Last question. Isaac Newton, the man who gave us mathematics, on astronomy, fluid dynamics, 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 quote.

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 ending now? Are you willing to go that far, Steve?

STEPHEN MEYER: No, of course. I think that’s exactly right. And what Newton illustrates so beautifully is that this principle of intelligibility that we’ve been talking about, that the mathematical rationality that can be developed by mathematicians without necessarily observing nature 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 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 insight and who advanced what was called natural philosophy or science so much in one generation.

Even in one year, his famed Annus Mirabilis where he went home during the plague to remediate his own deficiencies in mathematics and came back having invented the calculus. A heavenly master.

PETER ROBINSON: David?

DAVID BERLINSKI: All right. What can I say? That insight has not been vouchsafed to me. Fair enough.

PETER ROBINSON: Sergiu?

SERGIU KLAINERMAN: Well, okay. So, first of all, I think materialism is just the only explanation of the world should be put in the ashbin of history. In that we both agree. Bringing God in, sure, why not? I mean, that’s another way of looking at the world.

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, meh, and why not? Right.

PETER ROBINSON: Sergiu Klainerman, David Berlinski, and Steve Meyer. Gentlemen, thank you.

STEPHEN MEYER: It’s been our privilege, Peter.

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

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