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Hello, my friends, and welcome to Origins. My name's Don Chapman. It's a privilege to be your host. During this program, we showcase interesting guests who present evidence from science along with other important facts validating the truth of creation and the accuracy of the Bible. Today's guest is Dr. Jason Lyle. He has a Ph.D. in astrophysics from the University of Colorado, and before that, he had an undergraduate degree in physics and astronomy from Ohio-Westland University, where he also minored in mathematics. And on today's program, we're actually talking about one of his favorite areas of math. Dr. Lyle, welcome to Origins. Thanks. Thanks for having me on. So you love to do math. I do. Yeah, and I hope after today you'll love it too. Okay, well, we have an interesting title, the Secret Code of Creation. I love secrets. Are you going to reveal one? I am, yes. There is a secret code that God has built into math. We don't think about God creating math a lot, but we think of Him creating animals and planets and things like that. But God's responsible for math as well, and I want to show you one of the secret codes that God has built into math that was only discovered really in the 1980s. I have to tell you, this is really fascinating. I'm excited. All right. Well, we need to start with some of the kind of mundane details, but then we'll get to the good stuff, so stay with me. We need to start with sets. A set is a collection of elements with a common property to find. And so, for example, in mathematics you can have certain sets of numbers, and some numbers will be included in the set, and other numbers will be excluded from the set. So, for example, you might have the set of even numbers, and in that set you're going to have those numbers that you see there, but you're going to have the other ones that are excluded. That makes sense. And, of course, you could have the set of negative numbers, in which case a different set of numbers would be included and others would be excluded. Now, those ones are easy because you can tell just by looking at the number whether it belongs to the set or not. We're going to deal with a set that you can't tell just by looking at the number. You have to do a little bit of thinking, and it's called the Mandelbrot set. Now, the Mandelbrot set is the set of all numbers that obey this particular formula that we have on the screen there, whereby the number in question is C. C is the number that we're checking to see if it's a part of the Mandelbrot set. And then Z is another number. Actually, Z is a sequence of numbers, so you have Z1, Z2, Z3. We just call that ZN. And when that other number, if it gets larger and larger and larger, then that means C is not part of the Mandelbrot set. But if the sequence of Zs stays small, then C is part of the Mandelbrot set. It sounds very complicated, so I'm going to give you two quick examples here just so you can see how this works. All right. So we're going to ask, is the number one part of the Mandelbrot set? And so in this case, C is equal to one. And we stick it into that little formula, Z squared plus one. Now, the first Z always starts as zero. And so we're going to initially have zero squared plus one, which is one. And so that becomes the new value of Z then. And so we stick that back in the formula. And now we have one squared is one plus one is two. That's pretty easy. And then we stick that back in. Two squared, four plus one is five, and so on. And we stick that back in. Five squared. OK. 25 plus 126. And then we stick that back in. 26 squared, big number plus one, even bigger number. And so you can see what's happening with the value of Z there. It's getting bigger and bigger and bigger and bigger, right? And so that means the number C is not, the number one is not part of the Mandelbrot set because the sequence... The number gets too big. Exactly. The number, the sequence of Z gets large. Let's try one other example here because I know it's a little tedious. Let's try negative one. Is negative one part of the Mandelbrot set? And so we'll put that in the formula. So now it's going to be Z squared minus one, right? And that's going to equal negative one initially. And so we stick that negative one back in. Now, negative one squared would be positive one minus one is... Zero. Zero, yeah. And so we stick that back in and we have zero squared minus one is negative one. Stick that back in. And you see what's happening there. You go zero, negative one, zero. It stays small. It stays small. So is the number negative one part of the Mandelbrot set? Probably. Yes, it is. Yes, it is because it stays small. You got it. And of course we can do that with any other number. You can check and see whether any number belongs to the Mandelbrot set. There's one other complication and then it starts to get really cool. The Mandelbrot set also includes what are the so-called imaginary numbers. I hate that term because they do exist. They're real in that sense. But an imaginary number is the square root of a negative number. And that's a little bit hard for us to understand, but the so-called... We abbreviate it by a lowercase i. That's the imaginary number that when you square it, you get negative one. And they do exist. It's just that they... That's just a name, imaginary. And people say, well, how could that be? Because how could you have a number squared equaling a negative? Because we know that a positive number squared gives you another positive number. We know that a negative number squared gives you a positive number. Imaginary numbers can't be zero because zero squared is zero. So how do you have a number that's not negative, not positive, and not zero? And that's a little tricky for us. But if you think about it, when you were a kid, you probably thought negative numbers didn't make any sense. How can you have less than zero, right? But you get a little older, you get a bank account, and suddenly negative numbers make a lot of sense, don't they? And so it's the same way with these imaginary numbers. And again, they do exist. It's just most adults don't have experience with them, and so they're a little counterintuitive. The way you can think about this is if you have the number line here, positive numbers would be to the right of zero. Negative numbers would be to the left of zero. You can think of an imaginary number as being sort of above zero. It's not to the right. It's not positive. It's not to the left. It's not negative. And it's not zero. OK? And so we'll put the imaginary numbers along this other axis, and you can multiply them by different numbers to get a whole sequence of imaginary numbers. And you can also have what are called complex numbers, where they're off axis. They have a real component, which would be sort of their x position, and an imaginary component, which would be their y position. And so any number you can represent on a plane with its real component, and it's an imaginary component. And the Mandelbrot set also includes these complex numbers. Now we already checked some numbers. We found that one is not part of the Mandelbrot set. If we'd have checked zero, we'd have found that it is part of the Mandelbrot set. We checked negative one and found that it is, because the sequence of z stayed small. And if we checked these other values, we'd find that negative two is, negative three is not, and so on and so forth. And we could check the imaginary numbers as well, and we would find that some of them are, some of them are not. I'm looking, is there a pattern here? Is there some kind of pattern to the Mandelbrot set? And one way to discover this is to plot them on this number plane. And so, for example, we'll plot all the numbers that do belong to the Mandelbrot set. We'll plot them in black. And so we found that these ones do belong to the Mandelbrot set. And the ones that are not part of the Mandelbrot set, we'll make them a bright color like red, for example. And we'll see that as we plot more and more of these, a shape develops. Now, of course, it'd be tedious to go through all of these, but computers don't mind. And so we'll run the computer, and we'll have it check all these different points, and we'll find, lo and behold, as we plot more and more points, a shape develops. And the shape is not probably what you would expect. This is the shape that develops. And so let me go up to the board, and we'll take a look at the shape. And so what we have here is actually a map of those points that belong to the Mandelbrot set. All the points that belong are in black, and the points that do not belong to the Mandelbrot set are in red. And points that are very close to being on the Mandelbrot set but are not quite on it are a color that's kind of a bright yellow color. So it gives it really, you can actually see the shape quite a bit better there. And so it's a really interesting shape. You have what's called a cardioid feature here, that kind of heart shape. And you have these little circles that grow off of it, perfect circles, one of them centered right on negative one, just centered right there. The shape has really interesting mathematical properties, which we'll take a look at here. For example, you can see this circle branches off into sort of three. There's one stem and then two branches, right? So it's three. And then this next one down, you see one, two, three, four, five. Oh, and the next one is seven. And then it goes on, nine, 11, 13. All the odd numbers, all the way down to infinity, odd infinity, apparently, isn't that wild? Each one has two more spikes than the previous. And on this side, you have all the numbers, the evens and the odds. Obviously, it branches off into four and so on. So it knows how to count somehow, which is rather remarkable. And even more than that, it knows how to add. Because if we take the five here and the three there, five plus three is eight. Lo and behold, that's exactly how many you have in that middle one. Every one of them is that way. Every one of them has the number that's the addition of those two in between it. So it's really rather remarkable. But one of the things I find really interesting about this shape is if we zoom in on this little spike out here. You can see there's a little bump on that spike. And if we zoom in on that, lo and behold, it's another one. It's another version of the original. You can see it's got that same cardioid shape here, and it's got the circle and so on. And it's a little different. It's got some extra spikes growing off of it. We zoomed in on a spike, and we got a baby that's got extra spikes growing off of it. And you'll notice it has a little bump on it too. Well, let's see what that is. And well, lo and behold, it's another one. Yeah. And so how about that? It's amazing. It seems to repeat indefinitely. And of course, if you look there, what do you see? Well, there's another one. And apparently, this continues infinitely. And so a shape like this that repeats itself on smaller and smaller scales is called a fractal. And it sort of means broken. It's kind of like a broken shape really. But it's really remarkable, very beautiful. Who would have guessed that a map of that very simple formula, z squared plus c, gives you this incredibly complex and beautiful shape. Amazing. And I thought what we'd do is explore some of the aspects of this. We're going to go down to this valley here and take a look at some of the interesting aspects of this map of the Mandelbrot set. For example, when we go off here, you can see over here we've got what they call the valley of the seahorses. And you can see why they do look a little bit like seahorses, kind of upside down seahorses there. And of course, the colors are arbitrary. I can make the colors anything I want. But the shape I did not make and you did not make. The shape is discovered as we find which points belong to the set. We find if we zoom in, for example, on this valley of the seahorses, incredibly complex shapes. Now what you're seeing here, because this is kind of a bright color over here, that means it's very close to the Mandelbrot set. So what's happening is the black points are coming out and forming a very fine thread that's wiggly. And it's sort of infinitely wiggly and it wiggles around like that. And so every place that's bright, it's close to a point that's black. It's just some of them are smaller than a pixel even. So all we're doing here is zooming in on this map of which points belong to the Mandelbrot set. And it's going to be such a beautiful shape that the Lord has built into numbers. Remarkable. And as we zoom in on this shape, you see this wonderful spiral pattern. Now I found you can zoom in on the center of that until your heart's content and it just keeps going and keeps going infinitely. And it gives us a little window into the mind of God. Because human beings did not create this shape. This is something that we discovered as we had computers determine which points belong to the Mandelbrot set. So let's go off axis and zoom in on these threads of this kind of spider web structure. And we'll find that these webs are made up of more spider webs, amazingly. And again, it kind of repeats itself. And then you have this bright spot there and a bright spot there. And in the middle, you have kind of a structure. Look at that. Look what's in the middle there. It turns out it's another baby Mandelbrot set, amazingly. And of course, now it's got this extra stuff growing around it because we zoomed in on a spider web. And you can see there's all kinds of extra spider webs growing around the Mandelbrot set there. Rather remarkable. Of course, it's got a little baby, too, that you can see there on this spike. So isn't it phenomenal that that shape would... It is, unbelievably so. Yeah. And this was... Of course, this was an amazing discovery back in the 1980s is when these shapes were discovered. So let's go back into the valley of the seahorses. We went to the right to the seahorses last time. Let's go to the left, which they call the valley of the double spirals. And you can see here these wonderful galaxy-type spiral shapes. And it's a double spiral. And by that, I mean it's actually two threads around each other. If you follow this thread around, all the way around like that, you can see, ah, there's one in between them. See? This thread and this thread are the same. This thread and this thread are the same. So it's two. It's a double spiral. And we go into this thing, and we see all kinds of interesting structure. We see the spider web-type spirals. We see more galaxy-type spirals. And we see... I call them bow ties. They look like a little bow tie there in the middle. And if we zoom in on that bow tie, we find it goes from two to four. And if we go in even more, it'll go to, you know, 16 and so on. And right in the middle, another baby Mandelbrot. How about that? And it's rotated a little bit because we zoomed in on a spiral, and so it's spiraled around many times. But basically, this is connected to the original shape by a very thin and infinitely wiggly black line that connects to the original. So it's remarkable, the intricacy of this shape. And of course, you could spend a lifetime exploring this map, and it's infinitely complex. So you'd never run out of things to discover. And this is something that no human being sat down and painted. Now, I think I'm going to make a bump here. And no, no, this shape is built into numbers by the creator of numbers. It's built into math by God, remarkably. So let's go over here, what we call the Valley of the Elephants. This is a fun one. And you can see why they call it that. Once we get a little bit closer, it looks a bit like a series of elephants marching along one after the other. You see that? Yeah. They have even a trunk, and their trunk kind of curls up like that. And so I thought, well, that's kind of interesting. Let's zoom in on this curly trunk and see what the elephants look like up close. And this is a single spiral, because you can see this strand is the same as all of them. There's not another strand in between. So it's a single strand, a single spiral. And we zoom into that, and we find spider web-type structures, and we find galaxy spirals, and we find bow ties. But this time, the bow ties are made up of single spirals instead of double spirals. Again, the baby seems to inherit the properties of the parent that it grows off of, interestingly. And if we go down into the middle, you find, lo and behold, another baby mandelbrot set. Isn't that remarkable? It is. And who would have guessed that that would be there? Unbelievable. Yeah. And so, again, anything that repeats itself like that is a fractal. There's other kinds of fractals, but this is the one that I first learned about. And so it's the one that I like to share with people. So again, it's really just remarkable what we can learn from this shape. So let's give a little bit of thought then. What does all this mean? That's what we need to think about. You know, we need to take a break, Jason. This is fascinating. I'm anxious to see the application of this and what it does all mean. We'll be right back. Don't you go away. We are back with Dr. Jason Lyle. Jason, you've introduced something that kind of has us mesmerized here on the set. We're talking about fractals. We're talking about numerical formulas. And somehow when you graph them, it becomes an amazing thing that God has done, a thing of beauty. Absolutely. And just maps of these very simply defined sets. When you map them and see which points belong, you get a shape that is infinitely complex and has a tendency to repeat itself on smaller and smaller scales and actually gets more complex as you go in. Man-made objects, you know, things that we build, they get simpler as you go in. Are we peeking into the mind of God? I think that's exactly what we're doing because numbers, I think, are a reflection of the way God thinks about quantity. And so I'd like to think a little bit about this. What does all this mean? What causes the beauty in fractals? What causes the complexity in fractals? And of course, I'm going to suggest that God is responsible for that. Not everybody agrees with me on that. Of course, there are those people who reject the existence of God, but I want you to think through how would this make sense in an atheistic world view. I don't think it would make sense at all. What causes the beauty in fractals? Is it the man-made color scheme? Now, I admit I chose the colors in terms of, and of course, we changed the colors a few times just to keep things interesting. But that's not really what makes it beautiful. I think that can enhance the beauty of it a bit. But I've plotted it in grayscale, and it's still beautiful. The shape is intrinsically beautiful. And human beings didn't make the shape. Did the computer create the beauty? Well, again, all the computer did was run that formula. It just did quickly something that you and I were doing for the first couple of points. You could do that by hand. It would take forever, but you could do it by hand. The computer just does it quickly. It didn't create the beauty. It merely discovers it. It didn't create beauty any more than a microscope creates bacteria. The microscope allows us to see the bacteria efficiently, and the computer allows us to see the shape efficiently. Did people make this shape? No. No human being sat down and said, well, I think I want to draw kind of a heart shape here and then a circle here, and I'll make it repeat infinitely. Take an infinite amount of time to create something that's infinitely complex. And in fact, as a matter of history, people were surprised when the shape was discovered. Now, you wouldn't be surprised by something that you yourself made. No. That wouldn't make any sense. The beauty appears to be built into math. Built into math. Now, what causes the complexity in fractals? Not only are they beautiful, they're very complex. They repeat infinitely. They have these mathematical properties. They can apparently add and things like that. Remarkable. Did the computer create it? Well, again, the computer just plotted it because we could do it all by hand, theoretically. Did human beings create it? Again, we were surprised by this. It's not something that we created. Did the formula create it? Well, I found by experience you can change that formula and you still get these other types of fractals. And so the formula didn't really create it. The formula simply reveals it. The formula reveals this complexity that appears to be built into math. So again, the beauty and the complexity somehow are built into numbers by the creator of numbers. And that's remarkable. So what is math? Math is the study of the relationship between numbers. Okay. Well, what are numbers? You know, it's funny. Some of these things that we think, well, what are numbers? It's hard to define. What is a number? It's hard to define that. I looked through a number of different dictionaries and I think probably the best definition that I found is that numbers are a concept of quantity. A concept of quantity. A concept, of course, is something that exists in the mind. And numbers are conceptual. They exist, they're a way of thinking about quantity. They're abstract in nature. Numbers are not something that's physical like this desk where I can touch it and feel it, knock on it, and so on. Numbers are abstract. They're not made up of atoms. They exist in the mind. Now, people say, well, I can write down a number. Here's the number three. Well, that's not really the number three. That's a representation of the number three. Because if you write down the number three and then I destroy that, the number three doesn't cease to exist, does it? No. No, no. People don't suddenly count one, two, four. Just the expression of it. Yeah, exactly. It's an instance of a number. It's a representation of it. So written numerals are not numbers. They're representations of numbers. Laws of math are conceptual. The relationship between numbers is not something that, again, is physical that you can touch and everything. It's something that exists in a mind. So where do these laws of mathematics come from that govern the relationship between numbers? Two plus two equals four, and it does that in Europe, and it does that in the United States. It's the same everywhere. Why is it that we have the same laws? You have different civil laws in Europe than you do in the United States, different speed limits and things like that. Laws of math are not like that. Did laws of math evolve? And the answer is, of course not. I mean, it doesn't make any sense to think, well, two plus two equals four today. But millions of years ago, two plus two equals three, and it evolved into four. It's way up. No, that doesn't work. Numbers don't evolve. No, no. Number seven has always been the number seven. It didn't evolve from three or four or something like that. They don't even say they're absolutes. They're absolutes, absolutely. Absolutely absolutes. Were they created by people? Well, again, numbers weren't created by people. And people say, well, we created the notation. Sure, we created the notation. No doubt about that. But in terms of the actual number. The concept behind it, we did not create. It's there. Two plus two equals four. We're discovering what's there. Yeah. I mean, two plus two equals four before human beings discovered it. That's right. And so, of course, it's not something that were created by people. Do they come from the universe? Well, the universe is physical. The universe is made of atoms. And numbers, we decided, are conceptual. They exist in a mind. Now, the universe is in a mind. And so they can't have come from the universe. I'm going to suggest they stem from the mind of God. Amen. And in fact, the properties of the laws of mathematics make sense in light of the fact that they come from the mind of God. Because laws of mathematics are conceptual. They exist in a mind, the mind of God. They're universal. They apply everywhere. Why? Because God is sovereign over the entire universe. And so His laws govern the entire universe. They're invariant, meaning they don't change with time. It's not like two plus two equals four today. Sure, but on Fridays, all bets are off. No. They're the same throughout time because God does not change with time. He's beyond time. And they're exceptionless because, you see, God is sovereign over everything. There's nothing that escapes His will and His mind. And so for that reason, laws of mathematics have these properties. And we all believe that. We all know that. But it only makes sense if they exist in the mind of God, if they're an expression of the way God thinks about quantities. So God's thoughts are conceptual, all thoughts are. God is omnipresent. God does not change with time. God is sovereign. The characteristics of God and His thinking make sense of what we find in numbers. And so you see, somebody who rejects the existence of God has a pretty bad dilemma on his hands. The naturalist, the one who says, no, nature's all that there is, there is no God. Because he has to explain two things. First of all, he has to explain the fact that laws of mathematics are conceptual. They exist in a mind. But laws of mathematics existed before people. And therefore it can't be a human mind. And so the naturalist has a problem. He's going to say, well, laws of mathematics are something that people created. No, no. The planets obeyed laws of math long before, you know, human beings. In the secular view, millions of years before human beings came around. So laws of math existed before people, which means they can't be from a human mind. And yet they are conceptual, meaning they come from a mind. And so you see, the naturalist has a horrible dilemma on his hands. He knows on the one hand they're conceptual, but in his view, the only minds that exist are the minds of human beings and other physical creatures. Yeah, exactly. And so you see, it just doesn't make sense in an atheistic worldview. It only makes sense in the Christian worldview that we should have these laws that exist in the mind of God that are expressed in the world that we observe. And when we do mathematics, what we're really doing is we're thinking God's thoughts after Him. Yes. We're actually repeating, in a sense, what God has thought about the one who's... I've heard that's always been good science. Yeah, it really is. Yeah. It really is. This is amazing to me. I've done over 200 origin shows, and you've taken me to a place we've never gone. Suppose that I'm sitting here as a naturalist. How would I respond to what you've said? You know, most of them haven't really thought about that. Most naturalists have not thought about why it is...what is mathematics? You know, most people will think about that. You'll have something you learn in school, and it's useful, and it's practical. But what is it? There are laws of math, and they're conceptual. They're not material, and they've existed before man existed. And they're universal, and they don't change with time. Now, how does the naturalist account for that? I want to suggest to you the naturalist cannot account for that. There is no secular explanation for this. When you think about it, in the world of biology, in the world of geology, you've got creationists and evolutionists, right? You've got creation biologists, evolution biologists. You've got creation geologists, you've got evolution geologists. When it comes to mathematics, there's only creationists. Do you realize that? There is no such thing as evolutionary math. And so, although the Darwinists can invoke evolution, natural selection, and whatever over millions of years to account, perhaps, for the variety of organisms we see on earth, that won't work for something like the Mandelbrot set. Because that existed, and it doesn't change with time. It's always been as it is today, because God's thoughts have always been as they are today. I have to tell you, this is one of the most amazing and fascinating proofs for the existence of God that I've ever heard. And it seems to be almost the one that the other side hasn't thought enough about to even have a way to refute it. Yeah, I don't think there's any response to this. I think it's bulletproof, really. God is always accurate. The depth of his intelligence is beyond what we can plummet, but the beauty is there at the same time. That's one of the amazing things to me about God. And you've given us an amazing picture of that. Yeah, I hope that we've given people just a little glimpse, a little glimpse, into the mind of God.