 Well, we'll just go through this. So basically, the theme is just a general theme of philosophy. And part of philosophy is dealing with the critical examination of worldviews and conceptual schemes. And one of the goals of philosophy is to identify conceptual worldviews or schemes which may mislead us. So I'm going to look at a certain subset of those. The theme I'm going to explore, and it's a loose theme. I'm using this theme to join together a lot of different things, is how pictures can conceive us in the form of bad analogies. So that's going to say you have to actually think about some things. And you need answer. Actually, my own mentor, who I was his TA, embarrassed me in front of all his students with this following question. So this is actually the philosopher Norman Malcolm and his memoirs, reports that my favorite philosopher, Ludwig Wittgenstein, once asked the following slightly modified question by me. So suppose your task was to wind a cable as tightly as possible all the way around the earth. And we can imagine that's at the equator. You discover you barely have enough cable to do so. And you can just bring the two ends together after going all the way around the earth. But then by chance, you happen to see about six inches of cord left on the ground, which you splice into the existing cable. So the question Wittgenstein posed was how much slack does that add to the cable? Or in other words, more specifically, what is the increase in the radius of the circular loop of the cable? So for my students, you already know I'm going to be talking about math. I don't apologize for that because philosophy can extend itself from just the English language to reasoning with symbolic. And clearly, the language of mathematics is going to be something that we can use if available to us. And it's all baby mathematics, trust me. So here is some important information. The radius R of the earth is about 3,959 miles, which is 250,842,240 inches. I did that. That's what the calculator says. The circumference C in the circle is 2 pi R, R radius. And this implies the length of our imaginary cable is about 1,576,88,277 inches. Furthermore, so just out of curiosity, how much slack does adding that 6 inches give us? None. It actually gets tighter when we try to splice it. Well, we're actually going to imagine this splice is perfect. So we've actually added 6 inches to the original circumference. So reality is swept side, so you're perfectly adding 6 inches. So it seems like it's going to be very little, to be quite honest with you. So any other thoughts about how much it would be? Would you just add 6 to that 1,510? Yeah, but it would just give us 6 inches of slack. It would make it perfect if you added 6 inches. So you just shh all the way, barely, barely, barely. And you'd add 6 more inches. So how much are you going to be able to lift off the earth after that? Oh, no clue. I didn't have a clue either. It's a circumference that's got a radius measured with a radius that's 6 inches greater than the original one. Well, interesting. Interesting, I mean, this is what I actually thought about. I said, well, the temptation is to compare the length of the new cable, 6 inches, compared to the total circumference, which is this huge number right here, 1.5 billion. And it turns out this, I did a little Google Slurge, and just by curiosity, it turns out to be slightly less than the radius of a typical atom. And that's exactly what I said in the lecture. It's like, it's not going to add anything to the radius. Well, maybe, I mean, atomic measurements. But it actually turns out to be wrong. That's a wrong picture. The radius increases by just under an inch. So 6 inches was too much, not at all the size of the diameter of an atom. And it's way too little. So this could be shown, and we're not going to do it this way, using the language and rules of algebra. And this is just a start. I'm just going to start, because that's not the way I'm going to do it. I'm going to use a picture. So our first circumference is 2 pi r1. We add 6 inches to the circumference. So that's going to be 2 pi times the radius 2. So 2 pi r1 plus 6, then just replace c1 with 2 pi r1 plus 6, is equal to 2 pi r2. So 6 is equal to 2 pi r2 minus 2 pi r1 factor out the 2 pi and solve for r2 minus r1. But that's not the way we're going to do it. We're going to do it by a picture, because that's a misleading picture. And there actually happens to be a really interesting picture that allows you to reason through this. There's trust me, there's a reason I'm doing all this. So I'm assuming that everyone in here knows what the graph of a line is through the origin. If you don't, y is equal to m, where m is the slope, and x is the x-axis variable. So what we're going to do is just compare two things. All I need to do is just reinterpret variables. y will be our c. Our slope will be our 2 pi, and our r will be our x. So that gives me a line of slope 2 pi r, 2 pi rather, that goes through the origin. I have the picture for you, because this is due to Mr. René Descartes. We'd love to talk about him because I think therefore I am. But this picture of the world is his other contribution, which is a major contribution. Modern mathematicians would be quite lost without it. And this is it right here. So this line right here goes up 2 pi, which is about 6.28. And one unit. So that's the slope. So it opens the rise divided by the run. So the run is just 1, 2 pi divided by run. So we just go up to 6 inches, come over, find the intersection here, come down, and there we see it's just almost an increase of 1 inches. So this is the correct picture to look at. And you can just ignore the abstract rules. So that's my first puzzle, the puzzle of the bad analogy just impairing the total inches of increase with the total circumference length. So here's my next puzzle. It's not really a puzzle. It's just to think about this. But there's going to be interesting picture. Some of you, if you've been in my logic class, or maybe you, Cliff, I've talked about some of these. So as ancient knowledge of the world grew, the question of the sphericity of the earth became a matter of intense interest. So this is before people realized that the earth was actually spherical around, if you want to call it that way. And there actually were arguments, clearly, arguments that did not involve sailing around the earth or anything like that, both for and against a spherical earth. The one I want to talk about is the argument against the antipods. So an antipod is, I always call it the upside down people, but it's people that would be on the exact opposite side of an earth if it were supposed to be spherical. And a 13th century Christian author like Tantius asked or made the following observation. Is there anyone so senseless as to believe that there are men whose footsteps are higher than their heads, that the crops and trees grow downward, that the rain and snow and hail fall upward toward the earth? And I am at a loss to say of those who, when they have once aired, steadily preserve and they're following and defend one thing by another. So this is an ancient argument. It's called the argument against the antipods. It actually gives the prize to this picture. So another thing that antipods hear. And I actually tried to figure out this word in Greek and Antoacy. And even my huge Liddell Scott Lexicon did not help me. But the problem seems to come from an absolute view of down. So here is down. And someone living down here would fall off. And that's just not possible. And the rain would fall up. And we all know that's just absurd, as a matter of fact. So absurd that I am at a loss to say of those when they have once aired, steadily preserved and they're following and defend one thing after another. Well, the part of the point is, is this is when we try to figure out the nature of the world purely with logical thinking. And clearly, if this assumption of absolute down holds, that argument's really, really good. We could say that's an axiom. Once assumed, the whole abial of the sphericity of the earth becomes problematic. And maybe we can just change the picture of what down is and take care of the problem. And that's exactly, if I get that one more time, that's exactly what we can do. So we reinterpret down as being pointing toward the center of the earth. And now that that false picture of going down is gone, now suddenly the spherical earth makes sense. Now there's something even more interesting in this because this picture is due to our concept of gravity. And there's something else going on with gravity these days that we won't get to in this lecture. But it makes all of these questions really interesting, especially with respect to the next problem I'm going to have or show you. This is called the parallel postulate. So the Greek mathematician Euclid, who we have quite a long time ago, starts out his famous system of deductive knowledge in the elements by means of five axioms. Many times they're called postulates, but now we like to call them axioms. And the fifth axiom has been of important significance since antiquity. And this is the version of the axiom given two points and the line drawn through them. And I'll show you a picture of this in just a second. And another point not on that line. Then there exists exactly one line through this point parallel to the first line. So let's look at the picture. So, yeah, you can see it. I actually make these a little bit bigger. So here's my line. And as you know, you should remember, two points determine the union line. So here's the third point that's not on that line. So his question of being this, I made those points bigger fearing that you wouldn't be able to see them, that you can't. So look at all of the different lines that we can draw through this point. There's actually an infinite number of them. His axiom states that all of these, there is exactly one that is parallel to this one. And indeed, it's this one right here that's parallel to it. So the question is for the ancients was, whoops, not whether that fifth axiom was true or not. Because if you look at that picture, to go back, it seems self-evident. If you slightly angle this one line, unique line that's parallel to this one, just a little bit, and your lines extend on forever, it's eventually going to come down and intersect. The only way it couldn't is that it's going the exact same direction. So here are some huge examples. This one almost immediately intersects. And all of the others will eventually too. If you can extend those lines, but I can't do it because I only have a finite number of space. So the controversy surrounding this axiom was not whether it was true, because its truth by that picture seems self-evident. The question was whether it could be deduced from the previous four axioms. And literally the history of geometry is full of erroneous proofs showing or trying to show falsely that the fifth axiom can actually be deduced from the previous four, and I didn't even bother to list the previous four. It doesn't matter. But then in the mid-19th century, several mathematicians almost simultaneously, Gauss, Bolle, and Ljubljewski, independently produced geometries consistent with the previous four axioms. But for the fifth axiom was false. These geometries are now known typically as non-euclidean geometries. And the simplest one to picture, these are really, if you actually do a Google search on non-euclidean geometries, you can get quite a lot of interesting geometries out there. It's not near what you went there, even some fractal geometries. But the easiest for us to see and the one that's important is just Griemann's spherical geometry. So now that was the question. In this geometry, instead of having a flat surface, the surface is a sphere. So we're no longer in flat land, we're in a sphere. And these are curves on that sphere. So don't think about cutting or short cuts. You can't go through the surface or extend outside. This is our surface. So I asked the question, which lines are straight and which are curves? Line, I should have put in scare quotes there because a line has a very special mathematical definition that a curve doesn't have. And you guessed it, so I think I can go to the next line and give you a little bit more information. The questions are answered by extending and abstracting. That's what philosophy loves to do. We have this case here. Can we abstract it to other cases? It also works well in this world of mathematics and geometry. And we can answer the question by the unifying concept of distance. And this is a word you've probably not seen, but I tell you what it means, it's morphogonality. And here, I'm just gonna tell you, it's being at a right angle. But of course, as we go to higher and higher dimensions, angle ceases. Angle is something that is a unique measurement between two lines, right? So how do I measure an angle even in three dimensions? Well, we actually can, but we don't need to go into that. This pours over quite well, morphogonality. And that just means being at a right angle. And I won't go into detail on how that goes. So given two points, the curve with the least distance between them is a straight line. And any other line is not straight. So that makes sense. I mean, if I have two points here back to Euclidean space, which is not this, I can either go straight to them, I'm not a very good drawer, or I can go this way. And there's an infinite number of paths. And the one that is uniquely the least of all distances is a line. Everything else is curved in some way. So the longitudinal lines, those going north and south of these right here, turn out to be the real lines. Those are straight in this geometry. And of course, the equatorial line. Here is the question. Given what I've just said, suppose this is like, say, the UK. And this is some place, maybe in Maine, I have no idea. What's the straightest path between this point here and that point there on the earth? I don't know, you can't do that. Remember, we're in a spherical geometry and we can't go through it. You can imagine this is an impenetrable crystalline. Go to the north pole. Go to the north pole, hang left. Hey, well, at least you're trying, at least you're thinking, I like that. Because, you know, Dustin said, well, we gotta go along a line. And I already told you a line is, by definition, the shortest distance between two points. And if you also, I mean, I don't expect to do, this is a lot of information. These, I say, are not lines. Only the equator is a line. So this possibly cannot be the shortest distance between two points. It's a distance that ancient mariners like to travel because this goes to west, but it's not the shortest distance between here and here. To help you visualize that, and here you can see, I've tried to reproduce it here, the real lines are called GADSs or Great Circles. In any Great Circle, cuts the sphere exactly in half. So clearly, to get from here to here, that's not the geodesic, it's not a straight line. Because if I cut along this, I've not cut the sphere in half. So it's not intuitive, and we were misled by a picture in a certain sense. The whole point of this is it introduces the concept of non-euclidean geometry, which is absolutely essential to understanding the nature of reality, which philosophers love to look at. So we can actually extend the concepts of parallel lines. So let me actually explain why this is called a denial of a parallel postulate. So we're gonna extend the concept of parallel lines to be lines which intersect another line orthogonally, meaning at a right angle. Then the longitude lines, so if I had these lines right here, I'm gonna call this one the equator, okay? The longitudinal lines, right here and here, both intersect the equator at a right angle. So that would mean these two lines are intersecting a third line parallel to that. And these two lines actually intersect in two points. Here and here at the end. So in this particular geometry, all parallel lines intersect at one point. Actually two points is what I just said. So that's the reason we say that's the denial of the fifth postulate. There is actually an infinite number of lines parallel to the other line. So one of the reasons why back to my theme that that was seen to be impossible is this picture was stuck in our heads. But we had to extend and abstract that picture to a different context. And I can see why people seeing this picture was very resistant to actually accept the non-Euclidean geometries of Lovachevsky-Bolyae and Riemann in the 19th century. But of course the math usually takes care of a good argument. So here we're gonna actually watch an eight minute video, you know, like videos. And you'll see why I'm going this direction, hopefully. This is on PBS. I thought I was gonna spend the night watching PBS. And I saw this on camera and watched this. It is eight minutes long. And I think it's for people with ADD. He says it's short and he's gonna go slow, but he doesn't. So you'll see. Link. Today's episode is about space, time, and the nature of reality. My name is Gabe, and this time it really is space time. If you pay attention, this episode is going to blow your mind. So we're gonna take it slow. What is space time? Exactly. Before I get into that, I need you to do something for me. Give up your intuition about how time and space work. And first, you bring life to the system. Hold on to those intuitions for dear life. Don't worry. That's normal. This is challenging for everyone, even Einstein. Ready? Okay. Space time refers to whatever external reality underlies our collective experiences of the space between things and the time between events. Why can't space and time just be reality? Why add space time as an extra concept? Here's why. Suppose two observers are moving relative to each other. And particles count as observers. Fact. Those observers don't agree about how much time passes between events. Fact. They don't fully agree on how much space there is between things at any given moment. Fact. They don't even agree on the chronological order of all events. And yet, each observer measures things properly and is entirely consistent, which means neither of them is wrong. Now that sounds absurd, but it's true. Plenty of other resources, some of which we will do in the description, discuss these discrepancies and the experimental evidence for it. For today, we'll just take them at base value and focus on what they imply about the nature of reality. Because if you think about it, some of the implications are staggering. Take this disagreement about sequence of events, for instance. It is severe. If two observers can't agree on the sequence of events, it means that at present, someone's past is in someone else's future. For nearby events, the effect is microscopic, but so what? Any disagreement means that there is no universal division of events into past, present, and future, which opens major philosophical cans of warrants for things like free will and our belief that we can change the future. So, is everyone's experience of the universe entirely subjective or phrased another way? If time and space as we usually conceive of them aren't part of objective reality, then what is causality? Let me explain. A good starting point for objective reality is universal agreement. And above it, all observers do agree about this space. It's called the space-time interval or space-time separation between two effects. Even though two observers in relative motion will measure different distances and different elapsed time between the same two events, they always agree about the space-time interval between those events. Now, if everyone agrees about space-time intervals, they must signify something. But what? We'll notice that since it involves subtraction, a space-time interval could be positive zero or negative. When it's positive, nothing can get from one event to the other, and there are always observers who disagree about which one happens first. When it's zero or negative, signals or things can get from one event to the other, and everyone agrees on their sequence. So, it appears that the space-time interval between events A and B tells you whether A can influence B. In other words, even though we can't agree about past, present, future, time, or distance, we all appear to agree about causality. Now, that may seem counterintuitive, but normally we think that time is responsible for causality, but actually it's the other way around. To the extent that we agree about temporal anything, it's only because of causality. Causality is what's real. So what does causality have to do with space-time? As it turns out, everything. See, shortly after relativity first came out, a former math professor of Einstein, named Hermann Mankowski, noticed that the space-time interval resembles a weird version of a distance formula in what's called a non-independent space. So, he proposed the following radical idea. Maybe reality is not a three-dimensional space that evolves in time. Instead, it's a four-dimensional, non-Euclidean mathematical space that's just there. No, no time. That four-D mathematical space is space-time. Its points correspond to events, all events everywhere, everywhere. And in this view, only things that correspond to geometric relations in that four-D space are objectively real, like, for instance, causal relations. They correspond to space-time intervals, which are geometric relations, a non-Euclidean version of distances between points. In contrast, our experiences and measurements of time and space don't correspond to anything per se. They're more like the x, y-grade beams in math class, useful for talking about the board, but arbitrary and inherently meaningless. The board, its points and geometric facts are simply there, whether we put axes on that board or not. So, aren't you objectively real? And kind of, if you are the sequence of all events in which you are present, then you are a geometric object in space-time, a line-second, joining the points representing the events of your birth and your death. Do you rule along that line-second? No, no, you are the line-second. There's no motion through space-time. It's not this kind of space. It's tenseless. And your future isn't merely predetermined. It already exists. There's some sand trying to express what space-time is without misleading you, but I think the following gets the flavor right. Imagine we're all being a flip-book big graph paper. We agree on the events of the story, but we don't agree where they happen on the page, on how many pages there are between events, or even on the order of some of those events. And yet, we're all reading the same book, only there's no graph on the paper, there are no pages, and there is no book. All of that is just an imposition our brains make in order to perceive whatever it is. So why do we perceive reality in such a vividly spatial and temporal way? Good question. No one really knows. So have I told you all there is to know about space-time? No, far from it. All of this has just been a loose introduction to what's called flat space-time. Once general relativity enters the mix, we'll find that there are many possible space-time with different geometries, making it hard to ascertain which one this is. So we gotta crawl up before we walk. We will get to that fun stuff eventually though, so subscribe. And as always, the comments are for you real questions. I'll do my best to answer them at the next causally connected point of space-time. Lastly, we asked whether NASA could start a zombie apocalypse. Yeah, we'll just do that. We'll just do that. And he does actually. The space-space zombie outbreak, as soon as a more virulent organism would actually spread better. First of all, that assumption is unnecessary. It's enough for the bug to just be more harmful and harder to fight off with your space-depressed immune system than second, as Nicholas Garrison pointed out. Okay, so that, did you see what's going on? So subscribe to there, you can see these. So that was really quite a bit, right? Now I have to figure out how to... A lot of buzzwords there. Clearly philosophically connected. Talking about reality, the existence of free will. That we do not move through time. There is an external mathematical way to see reality in which things are not temple. Which is really interesting. If you're interested in this, you can do a Google search on BBC. You know, all of the BBC, everyone who is my student knows that. And do what time is it? And if you can't find it, I think I can give you a DPD. It goes into this in quite a bit more detail. Let's go on to the next slide. I'm gonna try to explain some of this. And tie it into what I was saying earlier. So that video introduces the concept of distance in non-Euclidean geometry. So in the Euclidean geometry, distance is always determined by the square root of the sum of the coordinates squared. I'm gonna give you two examples. But we're just down to one dimension. 10 is equal to the square root of 10 squared. Two dimensions. Here we go along four and up three. And the distance is five. That's probably the most well-known factor in interval. And five is equal to the sum of four squared, which is 16, plus three squared, which is nine, that's 25. So that's the familiar cases in the Euclidean geometry. But he actually discusses something that I correct, believe it or not. Yeah. The spacetime interval, he does this, I think to make it simple. And he introduces this right here. The minus sign is the important part. This is a new distance moment. It has to do with non-Euclidean geometry called a Rens geometry. And this is the distance moment. And it's this minus sign that he makes such a interesting part about it. There's quite a bit to be said. But then you say, well, if the delta x squared, the change, and this is just down to one dimension, this is the spatial dimension, this is the temporal dimension. If the delta x squared is less than this quantity here, I have the square root of the negative. And you may have told the square root of the negatives don't exist, but they do. For other reasons, mathematicians, they abstracted the real numbers so that we could have solutions to all polynomial equations so we have the complex numbers. But I'm not gonna do that. I just want to point out this thing for the philosophers in here who are waiting for me to philosophy. The recognized authority, as a matter of fact, for the philosophers, Hillary Popham, you've all known, from Harvard. Actually, I read this book because he recommended it. Sir Archibald Rieler is the well-known U.S. authority on gravitation. He has a book about this thick, and you really need to know quite a bit of math to understand gravity, gravity under the nuance, in the in-view, basically writes a book called Space-Time Physics, which is an excellent introduction to this special relativity, what he was just talking about there in space-time. Because the whole point of that is, if you think about it, it has some major implications. If we have events A, B, and C, so in our frame of reference, I see event A happen first, then event B happen after A, and then event C. So A happens, then follows B, then follows C. It's easy to construct another sequence of someone moving with respect, where they actually see C first, then A and then B. So just the whole concept of sequence of temporal events seems to be relative. His point in that video is there is a mathematical construct or in C in geometry in which everyone agrees is objective, and it's the freaking space-time interval. It's something that has to be calculated, that everyone agrees, it becomes an invariant with respect to frame of reference, is how we say it. But this is why he actually says with respect to the minus sign, the person who was just on the video spoke about it. The invariance of the space-time interval evidences the unity of space and time while also preserving in the formulas minus sign the distinction between the two. So in a certain sense, from this mathematical construct, we actually get ontology. In other words, the very existence between space and time is preserved by a mathematical formula and a minus sign that denotes the distance in non-Euclidean geometry, which I think is fascinating. I'm gonna stop it there at 38 slides. Trust me, I was prepared to go way in depth on those things, what is my next slide? Yeah, so pictures which represent reality. I mean, I can go on or I can have stop and pause for questions. What do you want me to do? Intertain or you want to throw things at me? That's not rotten tomatoes. I'm gonna say for movies, is it just questions to throw at you? That makes you less interested. You don't have any rotten tomatoes there, do you? Let's just go on then, if you don't have any. I want to order you, Cliff. You always do. Well, okay, it's not in this case. Oh, okay. Well, this is actually stint really fast. And then we'll be done. Seriously, I'm gonna get through these 38 slides if you can kill us with it, okay? So I wanna deal with pictures that mirror reality. So I got these sketches, artist sketches of, you know, wanted people. And the view here is, you know, this mirrors something that is real. And we kinda have that view that a picture mirrors reality. And believe it or not, not too bad. I can definitely see a resemblance there. Actually, again, well, take off this cap. That kinda looks the same. And you'll laugh at this last one. It'll arise right on, right? So here we have this concept of pictures that mirror reality. But here's an interesting question. What happens when we have a picture that is supposed to mirror reality, but as things stand, we have no independent access to the reality. And as a result, no direct way to determine the accuracy of the picture. And I pick, I'm gonna turn off the lights for this. I pick this one purposely. Because here is a picture of the last supper. And it's a famous picture that you find in a people in Cusco, Peru. Now here, the problem is, if there is a problem, is, well, this is a picture and it's supposed to mirror reality, but this reality was in the past. So is there any sense in which this picture is accurate? And we don't have access to that reality that's in the past. Now here, we kinda know it definitely does not mirror reality. And it's not because Jesus is glowing or that people did not wear that garb, but there's something else that's really sticking out that makes me love this picture. Is it the table? It's like the angle of the table goes down and it makes everything with TV in sense. Yeah, I mean, it's definitely past there. They're kind of getting into that where they have perspective. This is a little bit of a perspective. That, what they're eating right there. You wanna know what that is? No, I was looking at that, too. What is that? It's wheat, it's guinea pig. So this was painted by Perugia Perugians who actually assumed that what people ate the time of Jesus is what they ate and everyone eats guinea pig. So they're gonna have big guinea pig meat, too. And that's a famous picture in one of the cathedrals in Cusco. But this concept, and I'm going somewhere with this, the pictures mirror reality that we don't have access to that independent reality check reminds me of a recent article I read in Time Magazine talking about remote viewing. And this is just what I picture did for about Wikipedia. It's the practice of seeking impressions about a distant or unseen target using subjective means such as extra sensory perception. It was popularized in the 1990s when the U.S. government spent 20 million starting in 1975 in an attempt to use this remote viewing to somehow help in our spy and our remote intelligence. But again, this is a problem, right? The remote viewer says they see X and they might draw a picture or describe it. But we don't have an independent picture of what they're describing to say that they're remotely accurate or not. So this is an interesting problem. When you take away the ability to correspond with this is what is real, but I can't see it, it might actually lead to this type of bad reasoning. 20 million dollars back in the 70s was more than 20 million dollars, not. And it was probably furthered again and again and again because people were saying this is a picture of what's going on and they didn't have independent access, it seemed likely, et cetera. And I just wanted to throw that in. And then there's this one. Everyone has seen this picture of your student of mine, I believe. And the question here is it gives rise to this view that our picture and our experience of the world somehow accesses an independent reality. But we can't access it because all we can see is what we see, right? We can't get behind that. So this leads to, in this case, what can you do to subtract out what you're seeing? You believe that really these things aren't moving and in my class as we've discussed ways where we can actually show that it's not due to the computer, that it's actually due to the way we see it. But we can't get beyond this reality. So that leads to an interesting, historically significant concept of philosophy that there is the world that we perceive and there's the world that lies behind it. We have no access to that world that lies behind it. The world we perceive is what is real. The phenomena and the phenomena. And finally, one last question here that I'm gonna pose and I'm actually going to get through all 38 slides. This is supposed to be a picture, notice the structure here of silicon atoms. So I'm gonna pose to you here. These could never be seen by the eyes. Are you seeing a picture of silicon atoms or a picture of the laws of physics? So that's the question. Well, that's what they really, really look like. We have no independent means to see what silicon atoms look like. And we know they're not blue. That's a concept that doesn't apply to an atom. And how do we know why do they look this way? Just like our laws of physics say this here. Well, it might very well be that we're programming the laws of physics into the very instrument that detects them. And if it gives us any other picture, we tweak it until we get this way. So this is an interesting question. Is this a picture of reality, which leads to the other question, is our mathematical picture of reality correct? But thankfully in science, we have ways to test that and to see, sorry, the picture's just gone now, and to answer questions as to whether this is a correct picture. But in that case, the picture's correctness is determined by a different type of criteria. Done. Okay, any questions? Do you have one, David? Okay, so next. What the other criteria is, maybe an object's behavior? An object's behavior is predicted by the laws of physics. That we can actually, the time element we can run forward and we can see what's gonna happen. That's what's really cool. And if you're interested, my plan is next semester, if you're curious about the level and this concept of infinity, I'm gonna do an examination on the continuum hypothesis. So it'll make you feel smart when you go to it, right? You can talk about things and use cool symbols to refresh your plans. But that's the plan for next lecture. So out of no questions, thank you for coming. Before you leave, I'd like to make a shameless plug or a discussion that I'm gonna have tomorrow at around the same time downstairs in 116. Information, it's on religion, questions, and facts about world religion. And I will be there. Is anyone more surprised if you fill up the table and fly over the door? Yeah, fly over the door. Highly recommended in this time. Mission C is very modest. It's nothing. We're gonna test the hardest questions you have about religion. We're gonna test the hardest questions you have about religion. So if you're my students, which is everyone in here, because the others do, but make sure you sign this if you want to explain it. Class year in.