Is infinity real or just mathematical?

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Speaker 1: Hey, Katie, how many pets do you have living with you at home? That depends on your definition. Does it include all living creatures in my house? Does it include the mites in my eyelashes? Maybe not, like all the microobs and the tardy grades and all the bacteria in your gut? What about my husband? That's your call. Depends on your relationship, personal choice. Okay, then I've got a dog named Cookie, I've got to fish. I'm going to name them right now, called Bill and Head. So three or four, depending on the definition of my husband. All right, well that's more than I've got. And I got a question for you about pets. At what point do you say you have too many pets? I think that's the wrong way to look at it. Once you have, you know, two or three, like one more, there's no difference, right, It's the cat principle, like once you have two cats, there's no difference between two cats for cats tin cats. I think that sounds dangerous. You're gonna mathematically induce yourself to having an infinite number of cats. Exactly if you already have infinite pets, you can add more and more and still have infinite pets. Hi, I'm Daniel, I'm a particle physicist, and I have an infinite number of questions about the nature of our universe. And I am Katie Golden. I like pets, but I am not a particle physicist, and I still have infinite questions about the universe and about how many cats I can have and get away with. How many cats can there be in the universe? Could we have an infinite universe field with infinite cats. I think that if you replace the sort of building blocks of the universe with cats, it would be a lot cuter. But yeah, I come to the show with a little bit of background in a psychology and evolutionary biology, and that's why I like to talk about cats all the time. And welcome then to the podcast Daniel and Jorge Explain the Universe with its Infinite variety of Cats, a podcast in which we talk about all the crazy, amazing things that are out there in the universe. Not just the things in the universe. We talk about the very nature of the universe. We talk about the universe that's in our minds, that mental model of what's going on outside, and we talk about the real, actual universe, if it actually exists, and what it means what's in it, how it all works, and we try to explain all of it to you. We tackled the biggest questions, the even bigger questions, and the absolute biggest of big questions on the podcast. I think we can do all of that in an hour. I think we can. You know, I was thinking as we were talking about this question of an infinite universe is filled with infinite cats. And you know, probably there was some like ancient Greek philosopher who first raised this question. You know how the Greek philosophers had like the craziest ideas about how the universe might work, and basically every idea that is out there, some old dude in a robe thought of his first thousands of years ago. Yeah, just Plato surrounded by his cats thinking about infinity as they're all me owing. The thing that bugs me about the Greeks is that they had so many ideas that even though only like one or two of them turned out to be correct. Like Democratus, he's the first one to name the adam and think about the universe is made of tiny bits. People give him credit for that because he had a right idea, but he had like a million wrong totally idea. If you just like spew ideas, eventually one of them is going to be right. Statistically speaking, exactly, It's like Nostradamus, like he predicted everything because he made a bunch of false predictions as well, exactly. And so if I wrote down literally every possible thing that could happen, if I made an infinite number of predictions, then one of them would be correct. I could predict the whole future of the university if I had an infinite number predictions, and one of those would be the actual theories of events of the future, right exactly. If you think about, like if you make a random noise in your head, like, the chances are someone else is going to think that same random noise in your head at some point in an infinite universe. And that's what we're gonna talk about today on the podcast. These big concepts, the concepts that are bigger than big, the ones that are so hard that we can only grapple with them in our minds. Today's episode will be focused on this concept of infinity. Katie, what do you think of when I say the word infinity? Just real big, you know, really really big, actually, you know, in a serious answer to that. When I was a kid first learning about infinity, it was almost like the concepts of infinity and nothing were very similar to me because both kind of made my brain feel fuzzy. So thinking about an infinite amount of stuff and thinking about the absence of stuff, it's like a similar feeling to me, where it's like my brain just gets this prickly static feeling when I try to think too hard about it. It's amazing that you can hold an idea in your mind that also sort of confuses your mind. I also remember learning about infinity as a kid, and you know, my dad was really into math, and so this is a very simple proof of like, all right, give me a number. I can give you a bigger number, and then I can give you a bigger number than that shows you pretty quickly that there's like no biggest number. And to me that was a crazy concept lying in bed thinking about super big numbers and imagining, well, you know, there's always one bigger than that. So it's super fun to think about, like how your brain gets stretched by the ideas that you stick inside of it. Yeah, it's like, how do you fit a concept like infinity inside of a finite brain with a finite number of neurons, so you can't It's not something that you can really conceive of without using some kind of shortcut, because our brains aren't infinite, so you can't fit infinity in there. Well, I haven't met you in person, so I don't know if your brain is infinite or not, but I definitely know that mine is finite in its capacity. But this really goes to the heart of what we're trying to do with the whole project of physics or of science, which is to understand that external universe out there and represent it in our minds. And so we do that often using mathematics. Right, we have seen so many beautiful moments when crazy things happening out there in the universe can be represented with mathematical representations in our minds. Newton's laws of motion, for example, tell us about the crazy motions of all the planets and sort of lock them into place to some mathematical rigor. So we know that mathematics is very helpful in helping us understand what's going on out there, that we can represent the craziness, the insanity, the beauty, the complexity of the chaos of the universe using simpler mathematical representations in our minds. But if we know that everything that's physical has to be mathematical, do we know that everything that's mathematical has to be physical? Like? Is everything that you can imagine physically possible? Right? Right? Because you can have something in math that's theoretical, but it's not necessarily a tangible thing that you can like observe or hold in your hands, Like can you pick up a chunk of infinity and have that in your hand? Yeah, And it seems like math is definitely sort of larger than physics. A very simple example is I can come up with a theory of particles that's wrong. It would be mathematically correct, it could work, you know, it doesn't generate nonsense or whatever. It just doesn't describe our universe. So there are more theories. There are more mathematical theories of the universe than actual physical laws that govern the universe. Right, And so far we don't even think we found the one. We have lots of really nice theories of the universe that are beautiful but are probably not correct. So it seems like not everything that's mathematical has to be physical, right. I think for me. One of the simplest examples is, just as you were saying before, things that are perfect, like a circle, like a perfect circle. You can hold that idea in your mind somehow there's a representation in your brain to this concept of a perfect idealized circle. But I've never seen a perfect circle in reality of you. Ah, well, when I eat enough food, maybe, but no, I guess that any kind of drawing or depiction of a circle, even if it looks perfect, if you look at it with like an electron microscope, you're gonna find imperfections. Yeah, and everything is made out of particles, out of atoms at a discrete chunks. Right, So you can't have a perfectly continuous circle in reality. But of course you can hold one in your mind, and you can write down in just a few letters the equation for one right x where it puss wise square, it equals four. That's the equation that describes a perfect circle. But where is that circle? Right? Have the equation? I have the idea in my mind is this set of neurons in my brain that's now representing that concept, But there is no circle actually out there in the universe. It's sort of hard to grapple with and even in our brains, like that circle that you can kind of visualize is just made up of the firing of your neurons, and so is that circle in your brain even perfect? Who knows? Who knows? And we'll dig into that in another episode. But today I want to focus on this biggest concept in mathematics, infinity, because it's something that pops up all the time in physics. It's something that really tickles our brain and makes us ask these really big questions. So today on the podcast, we'll be asking is infinity real? Is it physical? Or is it just mathematical? And people have some answers to this question, which we're going to listen to right now. That's right. Thank you very much to those of you who answered my tough physics questions with no preparation for the podcast. If you like to participate for a future virtual Person on the Street interviews, please write to me two questions at Daniel and Jorge dot com. I promise it's fun. Here's what people have to say. I think that infinity is definitely real. I I believe that the universe is infinite in its size, so that there is the real infinite in the universe that we live in. And also if we believe in the Einstein's theory about black holes, that also inside a black hole we have a singularity that is a point of infinite density. So there we go. We have another very real infinite The only situation in which I can think of um from a physics point of view, where infinity might be real is if the universe has been internal, if it's been around forever. Maybe if there is a singularity in black holes and that might also be considered a point of infinite density. I guess, well, that was very interesting. I do like the idea of the universe being eternal. I think that's kind of one of my concepts of infinity, is that existence of something some you know, universe, even if it's not our universe, but just the existence of stuff has to be infinite, right, Like, there can't be a start and an endpoint to stuff being around. There can't be such an appeal to naturalism. Right, you're just saying, like it's obvious that it must be this way. You're basically a Greek philosopher. That's how they argued. Also, right, I don't know whether to be insulted or not. But yeah, I mean that's how I feel. This is not based on science. It's just how I feel of like, well, we can't just have stuff come out of nothing, go do its thing for a while and then end and then there'll be nothing again. But there's not any scientific reason for that. That's coming from my human feelings of existence. And maybe even that's what I want to believe, because it's very comforting to think there's an infinite universe that doesn't die, and I think it tells you something about yourself and how your mind works and your understanding of the universe to answer these questions that it feels natural to you that the universe is infinite, or that it's not that there are infinities or that there aren't. But remember, of course, that the process of science is to question all that intuition, is to go the opposite direction and to start from the data and say, well, universe, tell us what's out there, and then from that will build our model of the universe, instead of saying like, well, this just has to be this way. But it's hard to sort of understand where those limits are. You can't like question every single thing all the time. You can do experiments every day about every tiny detail. So it's hard to know sometimes when a question is this hard to grapple with exactly how to figure it out. And that's what I think we hear our listeners grappling with whether it can be, whether it must be, whether it can't be. Tricky stuff. Are you saying that I, Katie Golden, could be wrong about one of the most difficult concepts in the universe. I'm saying, basically, everybody is probably wrong about almost everything you know. But those are the best parts of science, right When your intuition is confronted with data, that's when you learn something. That's when the universe is telling you are n't I get why you think this is true, but it turns out it's not. Works a totally different way, And that's why we do science, right, That's why we have this methodological approach to revealing the truth about the universe, rather than just you know, eating olives and lying around in robes and saying maybe it's made of tiny cats, as fun as that is. So say I'm Plato and I'm trying to conceive of infinity. Am I just thinking of like a really really long line of olives? What's going on with that. No, you're right, the Greeks actually do have records of thinking about infinity. And you know, probably people before that thought about infinity. Who knows, But the earliest records we have about infinity go back to about six hundred BC. And you know, there are some examples there are people thinking about infinity, not necessarily in terms of an infinite number of olives. But for example Zeno's paradox. You know, when you say I'm gonna walk first halfway home, and then how much distance do I have to get to the rest of my house, Well half the distance, and before I can get home, I have to then walk half of that distance. Then I have to walk half of that distance and half of that distance, And there's no limit to the number of times you can divide the rest of your distance between yourself and your home. So in principle, there's an infinite number of steps you need to take between yourself and your house if every step is half as long as the previous one. This is Zeno's famous paradox. Of course, right, if there's an infinite number of steps between you and your house, how do you ever get there? You can't, But of course you do. This is how I approach eating a cookie, because I think, well, I'm not going to eat this whole cookie, so I'll just eat half of this cookie. And I'm like, well, I want a little more of this cookie, so we'll eat half of the half of the cookie, and then half of that half and so on and so that way I get to say, like, well, I didn't eat the whole cookie, even though there's only an infintestimally small chrome left of the cookie. Yeah, that's a mathematical proof of converging series. Right. You could have an infinite number of terms and still have it converged to a finite number. Unfortunately, my cookie eating strategy is a divergent series. I started with great intentions. I'm like, I'm just gonna have one, and then I'm like, well I had one, so mighty will just have another one. And then I'm like, well I've had two, so i can just grab a couple more. And then I'm like, might it will just finished the box? Because we all know where this is going, so let's just get there right now. But the Greeks really struggled with this idea. They didn't know if infinity was real, and they had two different concepts. They talked about actual infinity and potential infinity, and they separated these two concepts. They're like, well, this potential infinity, we can think about that for like proofs and geometry and stuff like that. But you know, actual infinity, that's something they actually shied away from. They didn't even want to talk about it. They try to avoid it ever coming up in their proofs and in their logic and in their thought. That like saying sort of a naughty word, like what about actual infinity? And then you get kicked out of class, out of Greek class. Yeah, and they called it the horror of the infinite. They did all these proofs where you could have, you know, used infinity as a mathematical step, but they took all these extra steps to make these proofs happen without needing to resort infinity, without infinity appearing anywhere in their math, because they thought, well, that must be nonsense. That's so interesting. So they had a concept of the potential for infinity, but actually discussing infinity itself was taboo because it makes your brain feel bad. Because I think they thought it wasn't real, you know. I think they thought as a mathematical curiosity, and if you invoked it, if you had to invoke it, then it's somehow made your proofs invalid. And now, of course, you know, we have the same sort of relationship with infinity, Like it's a thing. We can think about it, we can understand it, but it is difficult for us to wrap our minds around, at least to all sorts of like counterintuitive concepts, and infinity has many, many interesting layers. You know. The basic notion of infinity is just the one we talked about before, Like the number of numbers. Right, if you start from zero and you keep going, you'll never get to a point where you can't add another number. Our number system just has that flexibility is no limit, so that you can always make a bigger number. And that's like a really powerful concept. It also makes me wonder sometimes, like what's the biggest number any human has ever thought of? Right? There isn't the biggest number in principle that there might be a biggest number anybody has ever like written down or held in their head. Yeah, I mean, but then once you find that out, you can just add one to it and then you've got the record. Right, you're the human who's thought the biggest number ever. That would be super cool. Yeah, like a googleplex, a google plex plus one, that's it. You're the chain, So that's super fun. And there's all these really interesting exercises to get your brain used to the counterintuitive nature of infinity, because the mathematics of infinity are quite slippery, right, you can think about infinity plus one, but like infinity plus one really is just infinity. One of my favorite examples comes from David Hilbert. The guy invented Hilbert Space like hundred years ago, and this is called Hilbert's Hotel. And in Hilbert's Hotel, he's got a hotel with an infinite number of rooms, right, but he's also got an infinite number of guests. So every room in the hotel has somebody in it. It's all occupied. And you might think, well, if I had a thousand rooms and a thousand guests, then I'm filled up, and if somebody shows up and asked for a room, I would have to say no. But Hilbert's Hotel is different because infinity is really different from other numbers. If he's in the office, he's got infinite rooms and infinite guests, and somebody shows up, he can make room. What he has to do is this special trick and sell everybody to pack up, leave your room, and move over to the next room. So the person in room number one goes to number two. Person room number five hundred forty seven goes through room five hundred, and in this way, room number one now becomes vacant. What happens at the very very end, like in room infinity, will room infinity goes through Room infinity plus one, right, and I think, well that room didn't exist before. Well it did because there were an infinite number of rooms. And so in this weird way you can still make room even if you had an infinite number of guests. But you know, it sounds like this is going to only get a two point five stars on Yell because Albert's making you pack up and move your rooms all the time. Terrible stay. Yeah, and he can even accommodate an infinite number of guests, Like if an infinite number of people show up and want rooms, he can just say, all right, everybody, leave your room and go to twice your room number right to then person in room four goes to room eight for example, leaving three rooms open, and so there's a lot of flexibility there left in infinity. It's really fascinating, but of course it feels unfa of all, this doesn't feel like something that could actually exist because hotels have a certain number of rooms. Even if there are very very large numbers, there is a number, and so that shows you that, like infinity really not a number in the same way that other big numbers are. Right, when we try to apply something like infinity to something that in our human minds is a finite thing, like physical space of a hotel rooms, you know, those little mints they put under your pillow, it's very difficult to actually try to translate that concept into infinity, Like I can't think of infinity mints. That's just not you know, infinite meus. Like, although that would be a good brand for those little hotel ments in the infinity, we are copyrighting that right here. And you can play other fun games to really blow your mind, because it turns out that there are different sizes of infinity. Like you know, I think a whole lot of infinity is everything is the biggest thing. There can't be anything bigger than that. But you can't actually take an infinite list and then make something bigger than that infinite list. There's another really clever example in mathematics, like say, for example, you write down all of the numbers between one and two. How many numbers are there between one and two? Well, is an infinite number because you can always add another digit to subdivice, like Zeno's paradox. If you have one point seven and one point eight, you can always go one point seven five, and then one point seven two five and one point seven one seven five or whatever. So there's an infinite number of numbers there because you can always take half of a half exactly. So we have an infinite number of numbers, and each of those numbers can have an infinite number of digits. So between one and two there's an infinite number of numbers, each of which have an infinite number of digits, right, just like pie. And so that's amazing just to think about. Now, if you take that list, I can always find a number that's not on that list. There's always one more number between one and two that's not on that list. How do you do that? Well, it's sort of just like taking the biggest number and adding one which you need. Is a recipe for generating a new number that's not on that list, and that's not actually that hard. You just take the first digit from the first number and you add one to it, and you take the second digit from the second number and you add one to it, and the third digit from the third number, and you add one to it, and then you're making a number which disagrees with every number on your list in one of the digits. So it wasn't on your list, which means that you had an infinite number of numbers on your list, but you can make it bigger. And so that's an example of a kind of infinity we call uncountable infinity, where you can't like associate all the things on your list with the natural numbers. With one, six, seven, it's bigger than the infinity of all the numbers. So, for those of us who are less numerically inclined, you're saying, basically, you could have a pilot cats, but it's an infinite pilot cat, like, well, this is infinity cats. But then you find just one more cat and it looks so cute that you have to add it to the pile of infinity cats that you're hoarding, And now you have an even bigger pile of cats, but they're both piles of infinite cats, that's right. And you can have a pile of cats that's so infinite that you can't even assign them all numbers. You can't give them all unique numbers. There's more cats on that list than there are numbers in the infinite number line. At a certain point, you just gotta start calling them all minute, mitten seventy two, Come get your pill. That's a lot of being a cat mom. So there's these crazy concept we have countable infinity, which is like the natural numbers, and then we have uncountable infinities, which are even bigger and hard to grapple with. Well, I think one thing we can grapple with, though, is a very finite break before we talk more about the infinity of infinity after the break, which won't be infinity. So don't worry, and we are back, And I hope that break did not feel like an infinity being separated from our lovely voices telling you about infinity. We are having a lot of fun thinking and talking about infinities, and so far we've been talking about infinity sort of mathematically right in terms of like what the number means and how big it is and how hard that is to get your mind around. But now let's talk about infinity in physics, because physics and theoretical physics specifically has infinities all the time, right like their infinities all over the place in theoretical physics. And so first let's talk about infinities and theoretical physics, and then we'll figure out whether any of those are actually real out there in the universe. So when I think of physics, I think of you know, you drop a ball off of a build, and hey, you can figure out how fast that ball is going, or even particle physics, you are looking at the interaction of particles with each other. It's hard for me to conceptualize what infinity looks like. How do you have infinity tall building with the infinity balls dropping off of it, or how do you have infinity particles interacting with each other? Yeah, that's really as hard to think about, but it entered into physics very very early with Newton. Like, as you're saying, Newton is thinking of like an apple falling off of the tree or the motion of something. In order to solve those physics problems, he had to invent a new kind of mathematics, calculus. Calculus is adding up what we call infinitesimals, lots of tiny little contributions. You know, for example, when the apple is falling from the tree, it's not moving a constant velocity, is not falling with the same speed all the time. It's falling faster and then faster, and then faster. And so to do those calculations to figure out like how long does to take the apple to fall from the tree and hit the ground means adding up lots of changing velocities. So we have to invent a new kind of mathematics called calculus, which allows you to add up an infinite number of little slices of time to get the right answer. And that's the way calculus works. It's like the limit of taking a big sum when that limit becomes infinite, when you have an infinite number of things you're adding up. And so calculus is all about infinitesimals. It's everywhere around us. So here's a potentially stupid question. If you had an infinitely tall tree, and you had an apple fall from an infinitely tall branch onto Newton, who was infinitely below that apple, with that apple without when resistance become infinitely fast and just pulverized Newton into infinitely nothing. Well, if we're living in this universe, then there's a speed limit and so it can't go infinitely fast. Right, that's a really fascinating question about infinite because in theoretical physics, you can't have infinite velocity. Right, there is a speed limit to the universe. You can't go infinite velocity. So if he's infinitely far away from the apple, then they would never hit him. So is there some kind of like space cops that are like space highway patrol enforcing this space speed limit, because that seems like they're really harsh and my mellow, No, it's just a fact of the universe. But fascinatingly, there is no limit on the energy of a particle, and so these particles can't go infinitely fast, but they can have infinite energy. Like if you build a particle accelerator, you can just keep pumping these particles to have higher and higher energy even though they don't go faster and faster. The relationship between energy and speed changes as you approach the speed of light, and you can like double the amount of energy a particle has and it just goes a tiny little bit faster. So there is no limit to the energy a particle can have, but there is a limit to the speed it can have, which is interesting sort of counterplay there between finite and infinite. So what is the universal speed limit? Is it like a hundred miles per hour or like KOs per hour? I don't want to be too exclusive to our European listeners, it's a three times ten the eight meters per second, So three hundred million meters per second is the speed limit for the speed of light in a vacuum, of course, and that's the fastest that anything can move relative to any other object in the universe. But again, no limit on the energy, which is amazing. Wow. Well that's good to know though, so we don't get a speeding ticket. And theoretical physics also deals with infinity in terms of the particles. As you were saying, you know what happens when two particles bump against each other, Well, they're not like physically actually touching each other. In our model, we have them passing particles back and forth like two electrons repel each other. How while they do it by emitting a photon and passing that photon from one to the other, which gives it a little push. It's a little momentum, it's firm. But it's not just one photon. The electron is constantly emitting photons. Most of the photons aren't like big, heavy duty photons they're gonna do anything. Most of them are very very very low energy, infinitesimally small energy. But then there's an infinite number of them. So every photon is constantly emitting an infinite number of tiny, tiny, tiny photons. So that's like if you have a kitten who's got a very weak little swipe of the paw, but if you have infinity little kittens, it's gonna have an impact on you exactly. And these infinite little photons or kittens just add up to a normal number, and they don't like give you a divergence. They don't give you an infinite amount of force between the electrons. That's the key that sometimes in these calculations you have an infinite number of steps in the calculation, but you still get a finite answer, right. The electron pushes against the other electron with a certain finite number, and that's sort of like Zeno's paradox, it tells you that the infinity isn't real. It's just an artifact of how you're doing the calculation. You can take any finite number and slice it up into infinite pieces doesn't mean there's a real infinity there. It just means that you've cut it into an infinite number of pieces, because, like if you cut a single cookie into infinitely small pieces, it's still just a single cookie unfortunately exactly and now probably doesn't even taste as good, because you've separated all the ingredients, right, And it's that magical mixture of ingredients that makes a cookie what it is. And often in theoretical physics, when you do get a real infinity, when all the particles add up to give you, like an infinite force between the two electrons, that's when you know you've done something wrong that we call a divergence, right, because you can have an infinite series that does diverge. Say, for example, you just add up all the natural numbers between one and infinity, that some if you add them all up, is going to be infinity. If you add up you know, one and then plus a half and a fourth and an eighth, that's not going to be an infinite number that's going to current verge. And so in physics, when we do a calculation and it gives us infinity, we usually think of that as a failure. It's sort of like the Greeks, you know, like actual infinities, like the horror of the infinite, because we don't think that particles push against each other with infinite force. So when buzz light year says to infinity and beyond, that's like horrific. Physicists we're like, no, don't go there, fix your calculation. You must have missed a minus sign or something. And so it does pop up all the time, and usually it's a sign that something is wrong. You've done your calculation wrong, and so your series is like diverged instead of converged. You know, you missed a minus sign maybe. And this is how we know, for example, that a theory isn't working, because we don't see these infinities in nature. You don't see particles with infinite electric charge or pushing against other particles with infinite forces. And so that's why like the lack of seeing infinity in the physical universe tells us that we can't have a cropping up in our theoretical concepts as well. That also happens in biology. You have like a Fibonacci sequence perhaps where you get these fractals in nature, like you'll see them in plants often where or like even if you see like a cauliflower, or if you have ever had a romanesco it's this very beautiful green plant that has tastes like cauliflower. It's quite quite delicious when you fried up, and it has a natural fractal that occurs with it. And with mathematical fractals, they are theoretically infinite, but of course there's not infinite amount of romanesco. So even though that fractal is repeating and is you know, following the mathematical model of like of an infinite pattern, at a certain point it tapers off. There is a smallest, like little particle of romanesco that we're not even getting to the particle physics. You're getting to biology, which has a much larger cutoff for how small an individual unit, because it can only be as small as like a cell. Yeah, exactly, And so I love that you just said the phrase particle of romanesquo. You're right, we don't see those infinities and so we have to sort of like try to cut them out of our theory. But you know, in some cases we don't know, And we have infinities in theories, we don't know if they are real or they're just mathematical. I think one of the most famous ones are infinities in general relativity. Right, general relativity tells us that space is curved, and that it gets curved when you're in the presence of mass or energy, and if you have a huge amount of mass, incredibly compact mass, then the curvature of space, which determines how strong gravity feels, can get infinite. And that's what we mean by, for example, a singularity. Singularity is infinite curvature, essentially infinite gravitational force at that point. So does that mean that a black hole is evidence that there is actual tangible infinity. We just don't know, because we don't know if there's a real singularity inside there. There's a singularity in the theory, and that singularity predicts black holes, and we see black holes. But also we don't think those singularities are probably there because they're in conflict with quantum mechanics, which says you can't have an infinitely dense point at a certain location. So we're really pretty confused about what's going on inside black holes, and we'd love to see one because if an answer not just the question of you know, our infinity is real, but also tell us you know how general relativity works. But if you talk to people who do general relativity, they don't say this is a physical prediction. They don't say general relativity predicts the existence of physical infinities and singularities. They say this is a failure of the theory. This is they say, we can't predict what happens there because infinite curvature is like nonsense. It's like the theory breaks down. It's like you have a hole in your fabric of space where you don't know what to do. Instead of like actually predicting an infinite singularity, I mean that seems kind of like a bummer. And I guess until we can get a big enough black hole where it's safe to kind of float around on like a space inflatable life jacket and then just kind of cruise right in, Like, how would you even see, though, whether there's an infinite singularity even if you could cruise around in a black hole like how would you identify that. Yeah, we actually need a fun podcast episode a few months ago about seeing singularities. They're called naked singularities. And you could try to take a black hole that's spinning and make it spin faster and faster, so the event horizon sort of shrinks and it might reveal the singularity, but nobody knows, and that would take a crazy amount of energy. Until then, though, we are trying to come up with a better theory, one that can describe what's going on in the inside of a black hole and be consistent with quantum mechanics, and the category of theories are called quantum gravity. And some of these theories are pretty good, but most of them fail because they have infinities. Like you try to calculate what happens when you have two big masses they get close together, and they predict infinite horses and you're like, well, we know that's not right, and so these theories break down, like it doesn't agree with what we see, and so their infinities are also a failure. So we have infinities in these theories. But again it's the horror of the infinite is the failure of the theory. We're trying to remove those infinities so we can make predictions for what happens in the actual physical world instead. Okay, so we don't have a theory yet that doesn't have that mistake, that oopsie of infinity. But we are going to take a quick break about a minute, so everyone listening out there. Daniel, I'll get out a calculator. Let's see if we can get this before we get back from our quick break. All right, we're back. Has anyone come up with the theory without infinity as an answer? We should totally crowd source the theory of quantum gravity. All our listeners, all that brain power together approaches infinite genius exactly. And so we've been wrestling with the concept of infinity. We know it's a mathematical idea. We can do equations with it. We use it all the time in our mathematics. We even have it inside our physical theories as we're like doing calculus and summing over an infinite number of particles. But really the question is whether it's just something in our minds or whether it actually exists out there in the universe as well. So, Katie, you were talking earlier about the infinity of time, like whether the universe could go on forever, right, which I like the idea of because it seems to indicate to me there's an infinite amount of possibilities, such as a universe in which I Katie forty one b is surrounded by an infinite number of cats. But I do like the idea of a universe going on forever because it feels, you know, the idea of the death of the universe is maybe even scarier to me than my own mortality. In some ways, the idea that things could just stop existing is quite terrifying, right, And it is terrifying, and it's hard to imagine not existing anymore. But we all know that it happens, right There are people who were alive and are now not alive anymore. And so you can apply that same sort of concept to the universe and wonder like, even though the universe exists now, doesn't mean it always will. And the biggest argument against the universe being infinite in time to the future is that we don't know if the universe was infinite in time in the past, right, Like, we look back at the history of the universe and we see what looks like a beginning fourteen billion years ago the universe began. We don't know if there was anything before that, or if the universe actually literally started there. A lot of theories of general relativity have like the clock beginning at that moment, and before that there was no clock, there was no universe, there was no space, there was nothing. Doesn't mean we know how that happened. But in that concept, the universe has a finite age so far, why would it make sense for it to then go on for infinity. It's interesting because this is sort of what I was saying earlier, where trying to think about infinity gives me that same brain tingle sensation as trying to think about nothingness. So trying to think about nothing existing before the universe started and then nothing existing after, I would almost say that's a harder concept for me to think about than infinity, like thinking of like, well, how could just there suddenly be an explosion of things out of nothing and then how could all that stuff suddenly become nothing again? And what does that even mean? And it was hard for people to accept this idea before we understood that the universe was expanding, and that suggested that it was once younger and denser and hotter, and therefore might have had a beginning. People thought the universe was infinitely old, and that was just a much more natural idea. But of course can't just accept physics because it sounds natural. It has to be right. And so now we've accepted the concept that maybe the universe has a finite age so far, so we just don't know, like will the universe go on forever or will it come to an end? And there's some really weird possibilities, like it might be that the universe doesn't come to a dramatic end. It might be the universe doesn't like reverse the Big Bang and end up as a big crunch. One possibility is that the universe just sort of like spreads out into an evenly tempered muck so that everything is the same temperature. You don't have like hot spots and cold spots anymore. This is called the heat death in the universe. So in the universe is maximally disordered. You have maximum entropy, and there's some folks out there that think that entropy and time are so closely connected that it's entropy that's driving time forward. And when you get to this heat death, this equally spaced muck of the universe, that time itself comes to an end, that nothing can happen anymore. And so it might be that the universe has a dramatic beginning and then very undramatic sort of end like a lion, and then France is out like muck, like entropy muck. So that's a possibility, right. We just can't say whether the universe will exist forever forward in time. We don't know if it existed forever backwards in time, but current thinking is probably not. But you know, there are other dimensions, of course, of infinity. One of my favorites is thinking about the universe is infinite in space, right, Like if you look out into the universe, can you point your finger which makes a line, and how that line continue out through space and never ever stop right, never like loop back on itself or hit a wall or something. Does that line actually traverse space literally forever? That's what I thought you could do. If you've got a strong enough flashlight and pointed up at the sky, it's like, take that space. All this light is going to go on forever, but probably not. Well, if the universe is infinite in space and in time, then that photon from your flashlight will literally go on forever. I guess it could get absorbed by something, probably a black hole between me and infinity, a black hole made of somebody else's infinite number of cats that collapse on themselves. But this is like a true infinity, right, Regardless of what else is going on in the universe, it makes some sense for the universe to be infinite. It might make more sense for it to be infinite than for it to have an edge or for it to be like looped or closed in on itself. In our theories of the universe, we typically assume that the universe is infinite because it just currently feels like the most natural thing, right, because I've seen like images of maybe the universe is like a doughnut or a cone. And then the hard part for me to wrap around a non infinite universe is like, well, what's outside of that doughnut or cone? I can't my brain just again, it starts to break down when I try to think, well, what would be outside of a universe? Like, what would be that? Nothing? Like? I think that's another I don't remember which Greek philosopher came up with this, but I thought that was another thing of like, well, once you get to the edge of the universe and you pull apart the veil, what's beyond that? Right? And we just don't know. But the fascinating thing is that there's only one answer we could possibly know. Like, we could discover an edge to the universe. We could discover that the university's finite, that it loops around itself or has some weird deformity at the edge, or something like that, but we could never prove that it's infinite. Right, How could you prove that the universe is infinite? It's like proving that it doesn't have an edge. You can't prove a negative. It could just be that the edges beyond where you thought it was, right, because every time you get a government grant to send out like a spaceship to the edge of space, then it's like okay, but there's more, and then you have to write another grant, And so you would need an infinite number of grants to get to the infinite edge of the universe, and nobody wants to go through that. I do feel like I've really a number of grants of my economic career and I've not received an infinite number of dollars. So it's a really fun concept. And you know, some people suggest that you might be able to prove that the universe is infinite if you can come up with a theory of cosmology that describes the universe and is accurate and requires that the universe is infinite, if your theory only works for an infinite universe and you can exclude all other possible theories. But I'm not sure philosophically that's even possible. So this is another like potential infinity. But it's tantalizing because we can never actually know whether the universe is infinite. That Galileo said this hundreds of years ago in some famous letter, that mankind would never be able to know whether the universe is infinite, And so this is like a centuries old puzzle about the infinity of space in the universe. I like to think that because you look through this telescope. You know, one of the first people to use a telescope, he just sees all these stars out there. It's like, yeah, there's no way we're sorting this now. Now, maybe that was in his grand proposal. He's like, folks, I need even a number of dollars, got a lot of work to do up here, and then he probably got sent to jail. But you don't need to look out into space to see potential infinities. Like we were talking about how there's an infinite number of numbers between the numbers one and two. You can just hold your fingers apart and say, well, how many pieces of space are there between my fingers? How many unique locations? How many different possible spots could a particle be at between my fingers? And if space is continuous, then there's an infinite number of places for a particle to be between your fingers. I'm staring at my hand right now and like pulling my fingers apart. I'm just really glad nobody's here right now see me doing this, because like, hey, how's it going? What you doing there? I'm just doing physics, and like, okay, that's where you look physic shut up a science. And so, if, as general relativity suggests, the universe is continuous, then you can divide that space an infinite number of times, and there are real locations for every one of those numbers. Every subdivision is real. But you know, quantum mechanics says not so fast. It says that everything in the universe so far seems to be quantized, and a lot of theories out there suggests that space itself might be quantized. That space might be made out of tiny little space pixels that are so small that you can't see them. It's like a retina screen. They might be like ten to the negative thirty five meters wide. It doesn't really matter how small they are as long as there's a minimum distance. That means that the space between your fingers is not an infinite number of locations, but a large and finite number of possible locations, like little slots that particles can fit into. Thinking about space pixels just makes me feel like we're some aliens computer game, and maybe we are. Maybe our universe is finite, but it's just a simulation inside an infinite universe. That's what I'm thinking. But even if our universe is finite, we still wouldn't know if there are infinite finite universe is right, that's right. Another dimension of infinity is whether our universe itself is all that there is. Right. We think of universe like uni means like all of it, the one thing. But these days we have the concept of the multiverse where it's possible that there are other universes just like ours. In fact, there might be an infinite number of them. This concept comes from like a struggle to understand why our universe is so weird, like why is the speed of light the way that it is. Why are the various numbers the parameters that control physics seem to have kind of weird values. And they answered by saying, well, maybe there's an infinite number of universes and so that all the values are explored and we just happen to be in this one. And you know, that's a fun idea, and it's another concept where it's easy to like create infinity in your mind, but it's not something we can ever really study because as these other universes in the multiverse aren't places we can interact with, there are things we can see, we can't ever get evidence of their existence, and so it might be real there might be other universes out there, but I'm not sure that's something we could ever prove or really know. It seems like the only quote unquote evidence we would have of another universe is just if we can't figure something out about our own universe. So we think that maybe there's another universe somehow interacting with our universe, but we would have no way of measuring that at all, because we're stuck inside our universe like a bunch of fish and a fish bowl, and then we're bonking into another fish bowl, but we can never interact with the fish and that other fish bowl. We only know Maybe I think maybe they could be there, if it's the only way we could explain how our fish bowl is getting jostled around. Yeah, there are some fun ideas about how other bubble universes might have been made near hours and when they were very young, and they could have bumped into each other and left a little imprint like these circles on the cosmic microwave background radiation is left over glow from just after the Big Bang, But nobody's ever observed any evidence of that, so it's just a fun idea so far. So that's another possible infinity that could really be real, but we might never actually know. One that's closest to my heart is this question of infinite number of particles, like how many particles are there in my finger? Well, you know, something like Avagajo's number of atoms and then those are made of electrons and protons and neutrons which are made of quarks. But then you can ask like, well, what are those electrons and quarks made out of? One of the electron is made out of particles, which are made out of particles, which are made out of particles which are made out of particles, And it goes infinitely far down to infinite testimal particles. Then you could say, hey, I made out of an infinite number of particles. Yeah. I mean, it's the cookie problem again, where if you start to cut apart a particle like a cookie, it's not that the particle itself is infinitely large, but it's made up of infinitely nesting particles like a Russian nesting doll. But they're sub atomic particles. Yeah, and so that's certainly a possibility, Like, we can't rule that out. We call the electron and the corks the fundamental particles, meaning that they're not divisible, only because we haven't seen anything smaller inside them, not because we have any evidence that they're actually fundamental. We just haven't built a big enough particle collider to smash them open and see what's inside. In fact, we strongly suspect there are smaller particles inside them that would explain lots of the weird patterns that we see in the universe and in those particles. But we also suspect that there might be the smallest particle that if you keep cracking them open far enough down you get down to the smallest bit. Quantum mechanics again says that everything in the universe is discreet, not continuous, that it comes in grains or in chunks. So we suspect that maybe it's a vibrating string or something else, but at very very short distances there is a smallest thing, which would mean you're made out of a finite number of those little vibrating strings or tiny cosmic kittens or whatever. It's so hard to conceptualize a smallest particle because in the physical world that we interact with, even when we find like a smallest building block like you mentioned, like a pixel, we know that pixel actually you can break apart, like it does have little elements, Like when you have a pixel on a screen, it's actually made up of Well, now I don't know how computers work, so I'm just gonna assume it's it's made out of you know, light coming through a little square on your screen. That's exactly how they do it. They build thingies. They put a bunch of little thingies on your screen and little tiny lights that come through the thingies. I'm sure someone who knows how a computer screen work could explain that better. But you know, it's the same thing like with the human brain when you break it up, like the smallest unit of a thought is the action that happens in a synapse, like from one end of a neuron to another. But then you can break that apart because you can look at the molecular structure of a neurotransmitter. I mean you could say though that like if you look at the space between two neurons and the synaps, that could potentially be broken down into infinitely smaller halves of a space. So maybe there's an infinity whenever we have a single thought, because you know, you have an infinite haves that your neurotransmitters have to traverse over your synaps. But again it's kind of like that would according to the mathematics converge, but it's also sort of the goal, Like we feel that if we found the smallest unit, we'd be seeing something like the source code of the universe would reveal something deep and true. Everything above that is just like how those pieces tend to fit together and turn into stuff, right like ice cream is not fundamentalists and teach you something about the universe, regardless of how delicious it is. We want to figure out what is the smallest thing, so we can like look at the deepest nature of the universe. But we don't know if there is the smallest thing or if it just goes on forever. Alright, So we've talked about the potential infinities. I think the most promising ones are the spatial infinities, like maybe the universe really is infinite and goes on forever, or the temporal ones, like maybe the universe can last forever. I think it's pretty unlikely that space can be infinitely divided into bits, or that particles can be infinitely broken into smaller particles. But I think there really is a chance that the universe is infinitely big. But what's amazing to me is that for all these hundreds and thousands of years that we've been thinking about infinity, that we've had it in our minds, we've still never seen actual infinity. No human has ever witnessed laid eyes on anything that is truly infinite. It might be out there and there are universe hiding from us, waiting inside black holes or out past the edge, but we have never seen it. I mean, if you ever stare into a dog's eyes though, and see the love of there, I'd say that's infinite. All right. Well, there you having, folks. If you are a seeker of infinity, then get an infinite number of pets and feel an infinite amount of love from all of them. Until then, keep infiniting in your mind and let us know if you see it out there in the universe. Thanks very much Katie for joining us again on the podcast. Thank you so much for having me. This was infinitely fun. And don't forget to check out her other awesome podcast, Creature Feature, which she talks about the infinite, amazing, nous and complexity of biology. And until then, thank you very much for sharing us your curiosity and tune in next time. Thanks for listening, and remember that Daniel and Jorge explained. The Universe is a production of I Heart Radio. 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