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26: Infinity Shades of Grey (Paradox)
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A paradox is characterized either by a logical problem that does not have a single dominant expert solution, or by a set of logical steps that seem to lead somehow from sanity to insanity. This happens when a problem is either ill-defined, or challenges the status quo. The thing that all paradoxes, however, have in common is that they increase our understanding of the phenomena which bore them. So what are some examples of paradox? How does one go about resolving it? And what have we learned from paradox?
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Transcript
Introduction to Paradoxes
00:00:00
Speaker
A paradox is characterized either by a logical problem that does not have a single dominant expert solution, or by a set of logical steps that seem to lead somehow from sanity into insanity. This happens when a problem is either ill-defined or challenges the status quo. The thing that all paradoxes, however, have in common is that they increase our understanding of the phenomena which bore them. So what are some examples of paradox? How does one go about resolving it? And what have we learned from paradox? All of this end more on this episode of Breaking Math.
Host Introductions and Call for Questions
00:00:30
Speaker
shades of gray.
00:00:37
Speaker
I'm Jonathan. And I'm Gabriel. You can find us on Facebook at breakingmathpodcasts, of Patreon at breakingmathpodcast, Twitter at breakingmathpod, Gmail at breakingmathpodcast at gmail.com, or at breakingmathpodcast.com. Yeah, and we'd love to hear from you very soon. We're going to implement our series of ask math questions. We want to do that this week, but we want to give you guys an opportunity to think of some good questions.
00:01:00
Speaker
Yes. So we've got today's episode on paradoxes and also any of our old episodes. If you've got questions, send them in and we will give you a well researched answer. And with us on,
Exploration of Achilles and Tortoise Paradox
00:01:09
Speaker
we have Adam and Adam, you've been on the episode before, right? Uh, one I have. I'm asking it. Like, I don't know. What's an example of the paradox. Maybe that you learned in school. They thought was pretty interesting. It's one you're going to cover later in the podcast, the half plus one fourth. Like you keep adding things in infant infinitesimal amount of times. It's just one.
00:01:30
Speaker
Okay. So you're talking about the fact that one half plus one fourth, plus one eighth at an item is equal to one. Yeah. So, and, uh, why is that a, how's that a, and that's a good one, but why is that a paradox to you? It's so counterintuitive. Like you're continuing, like, I know it's a little less each time, but you're, you can do that in infinite amount of times. And it's one, it's just one that seems very counterintuitive to, uh, the way your brain naturally works. Yeah. And that's really what the essence of paradox is. It's our human brains not being perfect.
00:01:59
Speaker
Paradoxes are, gosh, I think it would be underserving them to say they're a lot of fun, which they certainly are. They can also be mind-numbingly frustrating. And as you'll see later in this episode, there are examples of paradoxes. There are also examples of puzzles that were once paradoxes, but now are considered to be thoroughly solved. We will be discussing both of them on this episode.
00:02:25
Speaker
First on this episode, we're going to be covering a couple of Zeno's paradoxes. One, Achilles and the Tortoise, and the other one, the Fletcher's Paradox.
00:02:32
Speaker
which is also known as the arrow paradox. The third paradox that we will be discussing today is a more recent one formed by mathematician and physicist James Clerk Maxwell. This is a paradox that seems to contradict the second law of thermodynamics, which we will also be discussing in today's episode. The first group of paradoxes which we'll be talking about today are ones that have to do with the infinitesimal division of real-world events such as time and space and come to us via the ancient Greek mathematician philosopher Zeno.
00:03:00
Speaker
There are 10 such paradoxes, but we'll focus today on two, the Fletcher's or Arrow Paradox and the Achilles and Tortoise Paradox. We will start today with the paradox of Achilles and the Tortoise. You'll find there are similarities with the two paradoxes that we'll be discussing from Zeno.
00:03:15
Speaker
The story of the Achilles and the Tortoise is pretty simple.
Zeno's Paradoxes Explained
00:03:19
Speaker
So Achilles, one thing you should know about him is that he's an all-star athlete, except for his... I guess you could shoot his ankle and he's bad, but otherwise he's basically like Hercules, kinda. I know that some Greek history buffs are gonna be angry about that.
00:03:35
Speaker
but basically he's yeah he's an all-star athlete so he's raising a turtle Achilles and the tortoise are racing and the tortoise is given an arbitrary head start it doesn't matter how big of a head start so let's just say he's he's within eyesight of Achilles
00:03:50
Speaker
and so anyway the race starts but by the time Achilles reaches where the tortoise reached the tortoise has moved because the tortoise is moving just like Achilles is moving maybe a little bit slower but it's still moving so by the time Achilles reaches the place where the tortoise was at the tortoise is already ahead of him so now Achilles runs to where the tortoise now is but then by the time he reaches that place the tortoise moved a little bit more
00:04:13
Speaker
and that happens over and over ad infinitum. And the conclusion, the paradoxical conclusion, the conclusion that we tell is incorrect, is that Achilles can never surpass the tortoise. When I first read this paradox, something about it really bothered me, and I think that may be the same for many readers and listeners, but putting my finger on it and explaining what about it really bothered me was the challenge. In this episode, we will attempt to explain that challenge.
00:04:42
Speaker
Yeah, and to some people the answer might seem pretty obvious. They'll be like, well, it's just like what Adam said at the beginning of the podcast. It's like one plus one half plus one fourth and you get a finite number. And so the, because the second time Achilles runs for the tortoise, he doesn't, it doesn't take as long as the first time. The first time the tortoise was just with an eyesight and the next time maybe he's a stone's throw away.
00:05:04
Speaker
And again, if you're following the logic of Zeno, he seems to indicate that although Achilles is continually getting closer and closer to the tortoise, as Jonathan said, he never ever passes him. Adam? Can I ask a question? Yes. Did this paradox come about when there was a basic understanding of just the way like velocity and acceleration work?
00:05:26
Speaker
No, that didn't really come about until Galileo. But at the same time, the ancient Greeks, if you look at Euclid, they divided space up infinitesimally all the time. It's just they had a problem with dividing time up infinitesimally. Oh,
Convergent and Divergent Series
00:05:40
Speaker
interesting. That's interesting, the time versus space way of looking at it and analyzing it. Yeah. What an interesting paradox in general.
00:05:48
Speaker
And we've given you the answer pretty much. It's just that the distances get shorter and shorter, so you could divide it up infinitesimally and add them all together. One other way, if I may, one other way of thinking about the answer is when you can understand conceptually the difference between convergent series and divergent series of infinite steps.
00:06:09
Speaker
An example of a divergent series would be adding up all of the real numbers, starting at 1, and then adding 2, and adding 3, etc., etc., all the way to infinity. The sum would be infinitely large.
00:06:20
Speaker
And if you're wondering divergent mean, it basically means either infinitely larger going between two or more options. And as Adam had said earlier, an example of a convergent series would be summing up all of the elements of the series one half plus one fourth plus one eighth plus one over two X as essentially as X goes from one to infinity. That number does converge to the number one.
00:06:48
Speaker
Yeah, or 2 to the x. But yeah, it does converge to 1. And same thing with like 1 plus 1 fifth that converges to, I think, 6 fifths or something like that. Yeah.
00:07:02
Speaker
So even though we can talk about an infinite amount of times, apparently that the tortoise gets some amount of distance ahead of Achilles in any given amount of time, those infinities still converge, and they converge very quickly. So as we all know, Achilles certainly will pass the tortoise, as our common sense does and should tell us.
00:07:25
Speaker
Yeah, also I meant five-fourths, not six-fifths. And a modern twist, just to show you the, and this is related to a paradox found in Godel and Sherbach, just to show you the source of confusion, if you're confused about where they were confused, is the following. Suppose you had a genie, and this genie can answer any question you have, but the way this genie operates is by having its own genie, and its genie is half as big as itself.
00:07:52
Speaker
and he asks the genie the question now that genie asks its genie the question which is half as big and is twice as fast so basically you ask genie one which has genie two which has genie three all the way up to genie infinity now somehow the answer comes out of this
00:08:08
Speaker
And the infinite infinity of genie that's not mathematical at all to say that tells infinity minus first even less mathematical genie the answer and it goes all the way back down the chain to you. And the question might be, where did the message come from? And in this case, it's the exact same paradox style, basically.
00:08:29
Speaker
just makes you think of a, the same thing. Like when you think of the series half plus one fourth, et cetera, what's, what's the last number? Just out of curiosity, where does it end? And it, it doesn't. Yeah, there's no last number. If there was a last number, there'd be one after it. It's real weird to think about. Yeah, it absolutely is. So even though these paradoxes are considered puzzles, that is to say, even though they're considered solved, it still is, um, unsettling. Is that the word to try to think about infinity?
00:08:56
Speaker
Yeah, and a reason why it might be unsettling is because these singularities are difficult to detect, these false singularities. Some arrive in general relativity. There is a
False Singularities and Ancient Greek Paradoxes
00:09:08
Speaker
singularity the size of the Schwarzschild radius in Cartesian coordinates, the Schwarzschild radius being the radius of the black hole with a certain object's mass. And people thought for a while that it was a real singularity that the universe had.
00:09:22
Speaker
until they change the coordinates around. It turns out that only real singularity happens at the point at the center of the black hole. I love that term, false singularity. That sounds like, I wouldn't say a punk band, maybe. I think it sounds like a documentary about white crystals, Helio. False singularity. I like that. I'm going to, I'm going to keep that in mind for, for, for some kind of a media presentation of, uh, what was the false singularity with the black hole?
00:09:48
Speaker
Oh, yeah. Well, basically, there's a mathematical singularity in Cartesian coordinates. So when you're trying to figure out math that happens around an object, a very massive object, there's a point at which it seems like the math breaks down. That's a radius at a black hole with the same mass of that object would be. And it turns out the only real singularity, the only singularity that exists to the universe is at the point at the center of the mass.
00:10:17
Speaker
And the and just to show you why that's the case you do math inside of black holes, even though you can't get information out of that you can get Still stuff going on. We just can't see it. Yeah, I think that that part of the whole reason why this this just is is hard to grasp and
00:10:34
Speaker
and just the term I used earlier was unsettling was it has to do with the idea of accountability and obviously infinity is not countable but the fact that we are dealing with time and we absolutely in our everyday life always talk about time with a beginning and an end and a given duration that is not zero within that time you can have an infinite amount of divisions so it's a little bit of sleight of hand but it's something you just gotta kinda don't look directly at the paradox
00:11:04
Speaker
And Gabriel just used the word countable. And for all the mathematicians out there, he meant that in the colloquial sense, not the countably infinite sense, because for the non-mathematicians out there, there's countable infinities and uncountable infinities, but we're not going to go into that. Whoo-wee, Jonathan, you just helped me dodge a bullet there. I almost had an internet firestorm of corrections. Whoo-wee. So what's the second paradox we're going to be doing? Adam, do you want to read the second paradox?
00:11:34
Speaker
Second paradox is the arrow paradox. Aristotle's recounting. When the arrow is in a place just its own size, it is at rest. But at every moment of flight, the arrow is in a place that is its own size. Therefore, at every moment of flight, the arrow is actually at rest.
00:11:50
Speaker
Now, let me break down the place just as own size thing. What they're saying is, is that at any incident time, the arrow doesn't move forward at any measurable amount during that. It moves zero ahead during that zero amount of time.
00:12:06
Speaker
I'd like to actually say this paradox again in one other way that it was described to me. We all know that when you shoot an arrow, it goes from A to B in a given amount of time. Call it anything. Let's say two seconds. It takes two seconds to leave the bow and reach the target.
00:12:24
Speaker
And then, as was alluded to earlier, you can divide those two seconds into an arbitrarily small amount of time or even the distance that
The Arrow Paradox and Infinitesimal Calculus
00:12:35
Speaker
it travels. You can travel into an arbitrary small of time. If you pick a size that's one-tenth of that, it would be 0.2
00:12:44
Speaker
seconds, but it would also be a much smaller distance. And of course, as you go down and down and down, you can approach zero. But for the purpose of this paradox, we'll say at a how many seconds pass while the arrow passes a length of zero. I know that that's a bit of a mind bender here. But yeah, essentially, what we're talking about here is infinitesimal calculus. Isn't that right?
00:13:08
Speaker
Yeah, and St. Thomas the Coin has had a really good insight into this problem. He said, quote, instance are not made of time, for time is not made of instance any more than a magnitude is made of points, as we have already proved. Hence, it does not follow that a thing is not a motion in a given time, just because it is not a motion at any instant of that time.
00:13:29
Speaker
and he was getting at the whole dividing zero by zero thing because yes if you define something being at rest as something that does not move over a certain amount of time then yeah over zero amount of time it does not move however we're dividing zero by zero and that's where the paradox come actually a lot of paradoxes come from dividing zero by zero or something or infinity by infinity
00:13:51
Speaker
You know, I like that a lot actually. I tried to solve this one on my own and basically the logic that I wrote was if you are allowed to, this is very similar to what you just said, if you're allowed to say, if you're allowed to
00:14:08
Speaker
divide time or length into an infinitesimal, then you're also allowed to multiply it by infinity to reach the duration. So it's sort of like if you had a scale and you had an infinitely heavy weight on one side, it's not that you can't balance it, you just need an infinitely heavy weight on the other side and you have a perfect balance.
00:14:27
Speaker
Yeah, and since we're dividing zero by zero, and since the first zero is space and the second zero is time, dividing space by time, you get velocity if you're dividing one dimension of space. And so if we keep the same units, velocity, and we just assume dividing zero by zero, we get that velocity. And if we assume that velocity is part of the arrow, then yeah, we have a solution to the paradox, but it requires a shift in thought.
00:14:50
Speaker
entirely. It's an old Greek paradox that we weren't going to mention, but I'll mention right now, that reminds me of this one is the pile of rocks paradox. What happens if you have a pile of rocks and take away run rock? You're left with a pile of rocks, right? Yes. So if you do that a bunch of times, when do you get to the point where you don't have a pile of rocks anymore? You would have to have a given in definition of a pile, something like a minimum of three rocks would be a pile.
00:15:14
Speaker
Yeah, but only with that definition. And that's our modern solution to it, but the Greek side is a paradox. Yeah, so there these again, as I said earlier, these are considered solved paradoxes. So they're now puzzles, they're no longer considered paradoxes, but they're still very entertaining to wrap your mind around and try to solve them on your own.
00:15:32
Speaker
You know, I wanted to say as a quick side note while researching for this episode, I found out that as we know, for anyone who's taking physics and calculus, we know that the derivative of position is velocity. We know that the derivative of velocity is acceleration. Some of us know what is the derivative of acceleration. Oh, mega acceleration.
00:15:57
Speaker
Not quite. It's a quantity called jerk. Yes, it's a quality, which is interesting. But then what's also interesting is, well, what's the derivative of jerk? I forget. One of them is called snap. And I think that sometimes mathematicians and physicists are humorous. The derivative of snap is oftentimes called crackle. And the derivative of crackle is called pop as a homage to Rice Krispies. Now you know.
00:16:23
Speaker
Oh, yeah, it's fun. I thought that it was higher up than that, but don't give it to him. If you want to know what Adam is referring to, it's a very funny college humor video called. Oh, it's the one about Reverend Oates. Yeah, just search up Reverend Oates. Yes, yes. Yeah, it's worth watching. It's about cereal for adults and adult mascots who are sad.
00:16:49
Speaker
So the next paradox we're going to cover is called Maxwell's demon paradox, but we have a whole lot to cover before we can tell you about the paradox. That's right. This paradox is essentially something that violates the second law of thermodynamics. And
Thermodynamics Overview
00:17:01
Speaker
in order to prepare ourselves, we'll be talking a lot about specifically what thermodynamics is and how we have come to know and understand it and what laws of physics we've formulated to explain it.
00:17:14
Speaker
Now simply put, the study of thermodynamics is a study of heat, but really it's a lot more than that. The laws of thermodynamics can tell us a lot about the fate of the universe. They could tell us indirectly about the difference between order and disorder. And the modern thermodynamics relies on the kinetic theory of gases. And all that is saying is that gas is made of particles.
00:17:40
Speaker
And the reason why it's called that is because this predates the theory of atoms. Because before the theory of atoms, people just thought, oh yeah, we're just going to pretend that gas is made out of a bunch of bouncing balls. And afterwards, people said, oh wait, it probably is made out of that.
00:17:57
Speaker
That's fascinating. I had no idea about that actually. I had no idea. So in thermodynamics, one of the things that we'll be talking about is Kelvin temperature. Kelvin temperature of course is named after Lord Kelvin who described it. A Kelvin essentially is a temperature scale that is the same size
00:18:16
Speaker
It's a temperature scale that is the same size as the Celsius scale. However, it starts at Kelvin zero. And in this universe, it is impossible to have a temperature colder than zero.
00:18:32
Speaker
And the reason why is because the way that this was found out is, and we're going to talk about balloons later, and we're going to use a lot of stuff about balloons. But if you put a balloon in the freezer, it'll get smaller.
00:18:48
Speaker
and uh if you get in and if you measure the amount that it gets smaller and if you get something colder like liquid nitrogen it'll get even smaller but it can't get smaller than zero um and if it did go to zero it would be a kelvin zero so we know what kelvin zero is by extrapolating and we've gotten really close to kelvin zero like i think we've gotten
00:19:09
Speaker
three nano degrees above Kelvin zero, but not exactly. And so the difference between Kelvin and Celsius is 273.15 degrees, which means that since ice would be at 273.15 degrees in Kelvin, but zero in Celsius, twice that would be hot enough to melt lead.
00:19:31
Speaker
Wow, that is that is immense. The just the sheer scale of these phenomena is just breathtaking. It's something that I don't really feel like I pay a lot of attention to. But when you consider it, it's quite amazing. So Adam, do you have any questions about Kelvin zero? How come we can't get to Kelvin zero?
00:19:50
Speaker
Well, it's because it would take essentially an infinite amount of energy because we'll talk about refrigeration in just a second. Refrigerators work by creating a heat that basically like for, you'll see how it works in just a second. It sucks heat out of the space where you don't want heat or really allows heat from the space you don't want heat to enter into the other place.
00:20:14
Speaker
But to get to Kelvin zero, we'd have to suck out every little bit of heat. It's like, and the reason really, really reason why it wouldn't work is because since you would have these things occupying infinitely small volume, they would be the same particle and that would violate the Fermi exclusion principle amongst other things. So yeah. What's the temperature like in a black hole?
00:20:39
Speaker
This episode is all about paradoxes and how they occur in math and physics, and one of the topics we cover has to do with energy. To that end, our partner Brilliant.org has a course about solar energy, which is how solar energy is converted into useful work. I love how the course starts by taking you through solar thermal, one of the earliest and easily understood forms of solar energy, all the way to advanced theoretical concepts.
00:20:59
Speaker
To support your educational journey in math and physics, go to brilliant.org slash breaking math and sign up for free. The first 200 breaking math listeners get 20% off the annual subscription, which we've been using. And now back to the episode. The temperature in a black hole is hot. That's a phenomenal question. Well, the temperature of a black hole is different than the temperature in a black hole.
00:21:22
Speaker
The
Black Hole Thermodynamics
00:21:23
Speaker
temperature in a black hole is basically how much stuff is entered. It's related to that. But the temperature of a black hole on the surface of it is just enough to let the, let the black hole evaporate. Yeah. Wow. Adam, we should have you on more often. Those are actually phenomenal questions. Cause now suddenly I'm thinking about well, if objects inside black holes are accelerating infinitely, would it be an infinite temperature? You know, I mean, this is just.
00:21:47
Speaker
Oh yeah. I mean, that's something that's interesting. There's a lot of paradoxes surrounding black holes that we, I wish we didn't have time to get to. Yeah. One of these days we'll talk about, uh, some of the current ideas about black holes, including things like the firewall, which was proposed, I think in the last 15 years, I believe.
00:22:03
Speaker
Now one thing we need to make clear is the difference between heat and temperature. If you've ever burned yourself with steam, you'll know that it's different than getting something out of the oven. When you get something out of the oven, you're putting your hands in air that's very, very hot, like 300 degrees, but it doesn't have very much heat. It has a high temperature, but not very much heat. So when you burn yourself with steam, even though it's only a little bit above 100 degrees Celsius, it imparts a lot of energy, a lot of heat.
00:22:31
Speaker
Uh, so in that specific to every material, how much heat it has. And that's the difference between heat and temperature. Now we need to explain a little bit about heat engines. Um, and one example we're going to use is a car. Um, Adam, when you drive a car after a while and let's say you park somewhere and you put your hand on the hood, uh, how does that hood feel? Um, my hand would probably get stuck to the hood from being melted.
00:22:57
Speaker
Yeah, it gets really hot. And that's how a car generates movement is it creates heat. Uh, and then the difference between that heat and the heat outs and the temperature outside creates the movement in the car. So if you put a car on the sun, it would not run. Can you elaborate? Why? Cause I, I thought maybe heat wasn't the direct cause, but it was the expansion within the cylinder.
00:23:21
Speaker
It is the expansion within the cylinder, but it's, so let's see, how does the car work? So you explode this stuff and that creates an increase in volume in the cylinder, but you couldn't, but if you were already hot, you couldn't create a difference in volume by exploding it, you would just stay the same temperature.
00:23:46
Speaker
Is that sort of like why a car engine would not work on, say, Mercury, because it's already hot? Yeah. I mean, maybe not Mercury, because I'm not sure if Mercury is hot enough, but definitely the sun. Or Venus. Something. Yeah. Yeah. Interesting. Yeah. That's a real interesting idea.
00:24:03
Speaker
And so this is the concept of a heat engine, and a heat engine is almost like a particle. You could treat it like a particle when you're doing physics. It's simple. So, Adam, let's say you have a 300 Kelvin reservoir, like in our example in the car, the stuff outside of the car is a reservoir, and the heat that it creates inside is a different reservoir, and the difference creates the movement.
00:24:28
Speaker
But let's say you have a 300 Kelvin reservoir connected to a 100 Kelvin reservoir through an ideal heat engine, right? Okay, I think I'm tracking. Reservoir meaning like a container? Yeah, like a container full of molecules that are at 300K. And one such that you could take heat from it and it would stay the same temperature. Okay. So basically a very, very big reservoir. Now if you connected them with a heat engine, so we're in like
00:24:55
Speaker
When me and Gabriel were preparing for the podcast we found it a little bit difficult to imagine a perfect heat engine So I came up with a visualization that'll help you in this Just imagine a straw with a wheel on it and the wheel turns when you connect the straw between two different temperatures That's a heat engine
00:25:12
Speaker
Say that again. It's a straw that has a wheel on it. And if you connect the straw between two different temperatures. So like, let's say you have a hot thing of water and a cold thing of water and you connect them using the straw. The wheel in the middle will turn because the water is flowing. Not because the water is flowing because the heat is flowing. Okay.
00:25:30
Speaker
So it's, or you could imagine a string if you don't want to imagine a straw. Okay. So work is done by the difference in temperature. If you have a 300 Kelvin reservoir and a 100 Kelvin reservoir, then you can imagine it like dollars. So if you have $300 and you give a hundred dollars to the hundred Kelvin reservoir, how many dollars do you have left? $200. And that's what could go into doing work. Okay. So we say that engine is at two thirds efficiency, 200 over 300. I got to keep thinking about that.
00:25:58
Speaker
So you can make the efficiency go up, for example, by connecting a 300-caliber reservoir to a 50-caliber reservoir. Essentially, it's the difference between the reservoirs. And this illustrates the example of how heat wants to reach an equilibrium with its environment. Yeah, exactly. So that's what a heat engine does. It turns differences in heat, not heat by itself, but a difference in heat, and we'll talk about this in a second, into work. Just like voltage on a battery. A current will only flow if there's a difference in voltage.
00:26:28
Speaker
Yeah. Does this have anything to do with why circuits, why they keep like data centers cold for all the circuitry and the temperature of the electronics?
00:26:37
Speaker
I think that the reason why they do that has to do with, um, you don't want stuff. Uh, you don't want things like jiggling around because it could cause bit flips and stuff, but I'm not totally sure. It's a good question. Yeah. In other words, it's not because it is a heat engine as much as you're trying to stabilize the, uh, computing equipment. Okay. So it's not related to an increase in efficiency or anything.
00:27:01
Speaker
Well, it can increase efficiency just because when you're overclocking a processor, it'll get too hot. So maybe indirectly, there is a heat engine there. That's a very good example, because computers are essentially information machines, and information and heat are directly related. OK.
00:27:21
Speaker
So refrigeration is basically a heat engine in reverse. So let's say you take the straw and you connected the straw that's the heat engine connected between two equal temperature reservoirs and you turn the wheel instead of having the wheel turn you turn the wheel. It'll refrigerate one end and it'll heat up the other end. Got it. So now we're going to talk about the first two laws of thermodynamics.
00:27:45
Speaker
So the first law of thermodynamics is that energy is conserved, and I've heard this law also called the law of conservation of energy.
00:27:53
Speaker
Oh yeah, and that's just saying that any process, you have the same energy that you started out with. And this is different than useful energy. Useful energy is very closely related to entropy. Oh, and actually on that note, and again, this is from the days when I taught eighth grade physical science, we would talk about this where even in the example of a car engine, a heat engine of sorts,
00:28:17
Speaker
in a car engine, it does transfer chemical energy into mechanical energy as well as thermal energy. So if you were to add up all of the energies that you get at the end, you would have the amount of energy that was in the beginning, but it's just now different forms of energy.
00:28:35
Speaker
Yeah, the chemical energy from the arrangement of atoms in the gasoline molecules turns into heat, and then the heat turns into movement. And everything else radiates outward in terms of light, infrared light, which is the light that comes off of things that are hot.
00:28:52
Speaker
Exactly. So yeah, it is certainly not a one-to-one. There's always a loss of energy. You don't get the same amount of energy in propelling the vehicle or driving it forward as you do that exists at the beginning of the reaction.
Second Law of Thermodynamics
00:29:06
Speaker
Yeah, we'll lost in useful energy because the same amount of the energy stays the same, but it just radiates into the into the universe. We've got another law of thermodynamics. The second law of thermodynamics says that heat cannot be converted into work without a difference in temperature. We actually talked about this earlier in the in the episode.
00:29:24
Speaker
Yeah, and the reason why is pretty simple. You remember the straw that would cool things down, right, Adam? Yeah. So let's say that we had this magical engine that could turn heat into work without a difference in temperature. So it's just a wheel that you put in the cold reservoir and then it starts turning by itself.
00:29:43
Speaker
That sounds magic already, doesn't it? Now let's say we connect that wheel to the wheel on the straw and we connect the straw between the cold and the hot reservoir and we turn the wheel the right way. What would happen is that the heat would flow from the cold to the hot reservoir because when the wheel turns, the cold reservoir gets colder and the hot one gets hotter since you're reversing the direction of the wheel.
00:30:07
Speaker
And so another way, so that's all to say that another way of stating the second law is that heat cannot flow from cold to hot by itself. And another, and it's pretty obvious when you put a egg onto a frying pan, the frying pan doesn't get hotter while the egg freezes, right? Last time I checked, that's correct. Yeah, that happened to be weird. Gabriel, what is heat in the kinetic theory of gases or temperature at least? Energy.
00:30:36
Speaker
Yeah, and energy, the average amount of kinetic energy or basically the square of the speed of the atoms involved. So this makes a lot of sense because if you had two rooms, one with a lot of hot, really fast atoms and one with cold atoms, and you put the separator up, all of it would get medium speed atoms. So you would seem like the heat went from the hot to the cold reservoir on average using kinetic theory of gases.
00:31:03
Speaker
So that's three different ways of stating the second law, that entropy must increase or stuff must go from hot to cold, that he cannot be converted into work without a difference in temperature. And finally, what we're going to talk about before we talk about the actual paradox itself is Brownian motion. And we have a quote here by the Roman philosopher Lucretius.
00:31:29
Speaker
Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways. Their dancing is an actual indication of underlying movements of matter that are hidden from our sight. It originates with the atoms which move of themselves, i.e. spontaneously.
00:31:48
Speaker
Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses so that those bodies are in motion that we see in sunbeams moved by blows that remain invisible.
00:32:07
Speaker
And that's the same reasoning basically that Einstein used about why Brownian motion happens. And Brownian motion, just to sum that up, is when you see particles moving in the sunlight and they look like they're moving around randomly, or when stuff diffuses in a liquid, that's Brownian motion. And to understand Brownian motion, you have to understand the gas law.
00:32:33
Speaker
So the pressure times the volume of a specific collection of gas is equal to the gas constant, which is just a constant that we're using for proportionality, times the number of gas particles, or any kind of particles, times the temperature of those particles. Now those don't have to be atoms, they can be bowling balls.
00:32:52
Speaker
But what we have to know about atoms is that they jitter around all the time. It's just that that's not useful jittering. We can't convert that, for reasons we will show in a second, into useful energy. This is a really great illustration. I've seen many examples of this. I think that when I was in middle school, I would oftentimes have the job of cleaning the erasers.
00:33:13
Speaker
Now, I know a lot of classrooms don't use typical chalk erasers anymore, but essentially if you have two erasers that are for a chalkboard and you clap them together, you get all kinds of dust. And of course, in the example above from Lucretius, when you have sunlight pouring through them, pouring through a cloud of chalk dust, you can very, very easily see the jittering. This is just fascinating. And then from that, you get examples like Brownian motion. That's really cool.
00:33:40
Speaker
So I think now you have all the physical background that you need to understand the paradox. Without further ado, let's read the Maxwell's Demon Paradox. Suppose you separate
Maxwell's Demon Thought Experiment
00:33:52
Speaker
two equal temperature containers with a thin wall, but one constructed such that heat cannot flow between them. In the middle of this wall, we install a door and we man it with a demon.
00:34:04
Speaker
The demon waits for a fast particle from the left container to hit the door, and right before the particle hits the door, he opens it up and lets it into the right container. He does the same thing for slow moving particles from the right container to the left container.
00:34:19
Speaker
Over time, the left reservoir gets colder and the right container gets hotter and hotter. But since the door can be operated with a very small amount of energy, there is not enough work to show for it. Essentially, that's saying that the work that's done in opening the door is minuscule. It doesn't really amount to much in this analogy, in this paradox.
00:34:42
Speaker
Yeah, and it's a weird paradox because it is really easy. If I have a door, it takes less energy to open the door than it takes for a barrel to hit the door. But if I close the door, the barrel can't go through the door. The door is strong enough.
00:35:01
Speaker
So the barrel in this case are like the atoms. And one thing that we didn't mention in the beginning is that, yes, hot stuff has fast-moving molecules, but it does also have slow-moving molecules and very, very fast-moving molecules, but on average they're fast. Yeah. You know what I'd love to do with this paradox? I'd actually like to see if I can't write a computer program to simulate it somehow.
00:35:23
Speaker
So I have a dumb question. In my mind, I'm thinking that energy is conserved. And so even though we're separating the hot and the cold, in my mind, they would still average out to the same temperature. So it doesn't seem like a paradox. See, it's only a paradox for the second law of thermodynamics, not for the conservation of energy. Those are super laws. The second law being.
00:35:43
Speaker
The second law being that you can't get useful work without a difference in temperature, which is the same thing as saying that heat cannot flow from cold into hot by itself. Okay. Yeah. And this is a very interesting paradox. So essentially one, one room is getting constantly hotter and one room is getting constantly colder, even though the only thing that's happening is that molecules are changing the room based on their temperature.
00:36:09
Speaker
Yeah, but various student observations, they do have the same total energy. It's just that one side is getting less energy and one side is getting more energy. I see. Is there anything to be said about the energy required to move the door? I mean, I said that... Oh yeah, you can make it almost arbitrarily small. Okay. And the reason why this is a problem is because let's say we connect a heat engine between the cold and the hot reservoir. We get useful heat. Let's say that it takes a very small amount of energy to run the demon.
00:36:36
Speaker
But if the demon creates a very big difference in heat between the two reservoirs Then we could run an engine with that difference in heat until the reservoirs became the same temperature Then we just use the demon again over and over again and we get something for nothing which is impossible I see
00:36:52
Speaker
So essentially there is something wrong with it. I mean, I think that right now we can say that we know the paradox can't exist, but we can't quite explain why. And again, we've already said that whatever energy that the demon is using, we can make it arbitrarily small.
00:37:07
Speaker
Yeah, and the solution right now relies on the second law being correct, which there's more probability in most people's opinions that even quantum field theory is more likely to be incorrect than the second law. The second law is so perfect. It's never violated in nature, ever, because if we were, then we'd be able to do things that we cannot do.
00:37:32
Speaker
I mean, look at the gecko. The gecko exploits Van der Waals forces between atoms to stick to walls. Evolution finds a way of utilizing everything. And if it can't find a way to violate the second law, it's probably not violated. Wow. You know, I'd be curious. We should look at some of the proposed solutions to this paradox.
00:37:50
Speaker
Well, the solutions basically say that the amount of entropy within the demon's brain is equal to the difference that it's creating. So it's basically saying that you can't have an arbitrarily small amount of energy because of the brain of the demon.
00:38:08
Speaker
The way that it measures the particle because the particles have to be measured white right before they're Open the door because if you're not measured then the door is just opening randomly. So all that process must take enough energy to Show to account for the difference. That is a one tough paradox You know, I mean I wonder who proposed that solution and again, although that solution very well may be exactly right. It's
00:38:35
Speaker
for me it seems impossibly far too intuitively think of that you know what i mean like like this paradox is i'll take it on faith i'm gonna pray on that i got a tricky solution relevant to the 21st century if uh 21st century is all about automating human or demon tasks
00:38:52
Speaker
Then what if you create a spring-loaded trap door such that it would be opened in one direction by a fast-moving particle, but the settings would be such that the slow-moving particles wouldn't affect it? So then the demon is out of the way and the door just allows this motion. Will you explain that one more time?
00:39:08
Speaker
So it's a it's a door that can open in one direction. It's like a cat door kind of thing that would only goes one way. Yeah. So the settings are such that if a slow moving particle hit it, it'd be spring loaded and it would be it would be calibrated so that only a fast moving particle would go through the door. Well, you would go with the spring loaded. It's like you would still have to wind up the spring.
00:39:31
Speaker
you would lose also in the fast-moving particle hitting the door, although it does have the energy to open it, it would lose some momentum in pushing the door open. What if you made it such that it did lose some momentum, but not so much so that it achieved the speed of the slower one, so it became a medium speed particle? Well, that could work if you could wind up the thing for free. But since you can't wind it up for free, because with any spring-loaded thing, you have to wind it up again, right?
00:40:01
Speaker
uh you wouldn't have to wind it it would just be a oh yeah yeah yeah like i i i see what you're saying yeah like like an expanded spring that its natural state is expanded you know what i mean like we you you squeeze it and like it restores back to its initial so how does that only let fast-moving particles through
00:40:17
Speaker
Uh, I guess what I'm saying is this would work at least in one direction and it would allow the, uh, fast moving particles into the slow, slow tank. Yeah. All right. Um, and then, and that you could hypothetically configure another one such that if a fast moving particle hit the other door, uh, if it's moving with a certain velocity, then it like closes a second door, but if it's moving slow enough, then it slips past. So what does the spring attached to, uh, the side of the room, probably in the door.
00:40:45
Speaker
All right. So you're saying, help me out here. So explain exactly how the door is set up. So the door is like, so like the front door to the house, uh, when you unlatch the handle, you can push it open and then it'll shut. If it cat paws that door, it's not going anywhere. But if I push the door, the door opens. Oh, and, um, I see, I see what you're saying. Um, so you're saying that if a molecule goes with enough force, then it's gonna go through, but slow down.
00:41:15
Speaker
Yeah, it would be calibrated such that a fast moving particle would be able to open it and would become a medium speed particle. Not quite as slow as the slow moving ones, but a slow one would not affect it. So you could have fast particles going from tank A to tank B, but you wouldn't have slow particles. Yeah. I think in that, I think the solution to that one, I'll have to think about it a little bit more, but I'm pretty sure the solution is the same to the solution for the Brownian ratchet. And I'll explain the Brownian ratchet real quick. This is another Maxwell paradox, isn't that right?
00:41:42
Speaker
Yeah, it's another style of paradox. And Adam told me if this relates a little bit to what you're saying. So Brownian motion, all particles are moving around randomly, blah, blah, blah. So a ratchet is a gear that has a little stopper on it, so the gear can only turn in one direction. So let's say these molecules are hitting the gear. Sometimes the gear is going to turn right and sometimes it's going to turn left, right?
00:42:06
Speaker
So when it turns right, the ratchet lets it turn right, right? But when it tries to turn left, the ratchet stops. So the net movement of the ratchet should be in one direction, which creates work. But the reason why this cannot work is because the ratchet and the little thing that prevents the ratchet from moving the opposite direction are also being bombarded with atoms at the same rate.
00:42:32
Speaker
So the ratchet would fail about 50% of the time, so no useful work could be done. Why would it fail? Because everything's bouncing around so much. Because to create something such that the motion of the ratchet would go forward with something as small as atoms bombarding it, the scale that we're talking about is small enough so that the device would fail.
Brownian Ratchet and Perpetual Motion Paradoxes
00:42:57
Speaker
This was proven by Feynman in the 60s. Fail in what sense?
00:43:01
Speaker
Fail in the sense that it would bounce it would bounce off of the ratchet and the ratchet would be allowed to turn freely in the opposite direction. So the ratchet the it would mechanically fail. Yeah, there's no there's absolutely no way of building a device or a ratchet that wouldn't fail this situation.
00:43:19
Speaker
So the equivalent might be to create a cheat, like at our scale might be to create a large, cheap plastic ratchet and bowling balls are hitting it. I just want to understand in which it's failing. But remember that the ratchet is also being hit by bowling balls and is itself moving by itself because remember everything with heat moves like jitters around. And the rat and the little pin that's holding a ratchet in place is also jittering around all over the place.
00:43:45
Speaker
But what's cool about the Brownian Ratchet is that if there is a difference in temperature between the stuff and the ratchet, then this just works as a heat engine. So maybe with your example, with the door swinging open for fast particles, I'm pretty sure even in that case, the heat would be transferred into the spring and they would have to dissipate in some way.
00:44:13
Speaker
but I'll have an answer for you by next week. Yeah. Okay. Hopefully this shouldn't turn into a whole thing. Yeah, no problem. So, um, again, just, just clarify that answer is pretty well, well accepted. The brownie and ratchet one. Uh, no, for, for Maxwell's demon.
00:44:28
Speaker
Oh yeah, that's pretty much the accepted answer. There's been a lot of research done into it, but there's not like one, as far as I understand, genuine, perfect answer like there is for the Brownian Ratchet. And there's actually some Maxwell demons which are unsolved to this day, and they're called Maxwell zombies.
Maxwell Zombies and Unresolved Paradoxes
00:44:48
Speaker
And they're even trickier than the Maxwell demon because they're a lot like Adam's example in that they have mechanical things that have almost no intelligence, but they still
00:44:57
Speaker
Create differences and a lot of them are solved but some of them are not. Oh, can we share any of those? Yeah, totally one of these paradoxes is called Duncan's paradox and You can look this up on Wikipedia, but there's been some laboratory tests that seem to support the second law being Violated which means that we haven't looked closely enough at why that's happening So I encourage everyone to look up Duncan's paradox
00:45:26
Speaker
Whether it be the laws of thermodynamics, or perception of space and time, or just our general intuition, phenomena which are sufficiently hard to understand, will be get paradoxes, and this dynamic will never change as long as we are still alive. Paradoxes are, in some ways, the front lines of mathematical research, and should never be ignored. I'm Jonathan. And I'm Gabriel. And this has been Breaking Math. With us today, we head on. Adam. And Adam, is there anything you want to say to the audience or plug, or whatever?
00:45:54
Speaker
Uh, no, this is a great episode. I think paradoxes are one of the most, uh, interesting things that you bump into as a very young student in school. So I think
Personal Story: Building a Perpetual Motion Machine
00:46:02
Speaker
it's the most compelling part of getting people into science and math. If, if I may, Adam, you recently shared during our last break, uh, a story about a middle school experiment that you did. I was hoping that you could share that with our listeners.
00:46:13
Speaker
I think anyone in middle school can relate to how freaking weird magnets are, and that's the perfect introduction into STEM education. I tried to create a perpetual motion machine by creating sort of a circular frame, and then within that circular frame was a smaller circle that rested on a skateboard bearing, and I put little magnets in it at 45 degree angles, eight of them around it, and the idea was
00:46:36
Speaker
When you slide that smaller wheel inside the larger one, the attraction of the magnets would cause the wheel to spin in one direction. What I failed to realize at that time is when the magnets move slightly past alignment, when an inside magnet moves slightly past the alignment with the outside magnet, there's an attraction there that pulls it back. I only saw the attraction moving in one direction because of an elementary understanding of the north and south poles and how that would affect the circular magnets. I knew this to say it didn't work, but
00:47:03
Speaker
The lesson learned there I think is just your interface as a human with the paradoxes of mother nature and science are kind of mind boggling in how unintuitive they are with the rest of our experience. So it's a great introduction into the field and study of STEM.
00:47:18
Speaker
and tan scaffold. There ain't no such thing as a free lunch. You can find us on Facebook at breaking math podcast. You could buy us a drink at patreon.com slash breaking math podcast. And these are the drinks that'll help us get you more content. They're magical lixirs, I guess. You can find us on Twitter at breaking math pod. You could check out our applets at breaking math podcast.com. And that's our info for contacting us.
00:47:47
Speaker
Oh, please send in questions. Yes. Any, any questions? And again, these are not just about, about paradoxes, but anything that you'd like to answer about math, including math education, we'd love to talk about math education as well. Yeah. So if you've ever wanted to know why there's like,
Closing Call for Listener Questions
00:48:09
Speaker
why certain numbers show up in nature more than others if you wanna know why a certain proof in a book works the way that it does. Anything really, we'll try to answer it or get you answers.