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Here are 17 paradoxes in physics and math. If you'd like a longer explanation of any of them, I have full videos of all of them, which you can find in the description. (texts chime)
The pole in the barn paradox. Imagine you are running toward a barn that's 10 meters in length, holding a pole that's 20 meters in length. Will there be any point that the entire pole is inside the barn? Well, of course not. But now you try again, and this time you're running at 90% the speed of light. Special relativity tells us that when objects travel at near light speeds, they contract in length, as seen by an observer. When you do the math, your pole is now 8.73 meters long. So there will be some time when the entire pole is inside the barn. But relativity also tells us that all motion is relative. If we look at things from your reference frame, we could just as easily say that you are standing still and the barn is moving toward you at 90% the speed of light. Therefore, it's the barn that's length contracted. From this frame of reference, there's no point when your whole pole is inside the barn. The fact that the pole fits in the barn in one frame of reference and not in the other is the pole in the barn paradox. So which one is right? Well, counterintuitively both are right. Another result of special relativity is that there is no absolute time. Two events can be simultaneous in one frame of reference, like both ends of the pole being inside the barn, and happen at different times in another frame of reference. This is called the relativity of simultaneity. (texts chime)
Aristotle's wheel Paradox. Aristotle's wheel paradox is the observation that when you roll a smaller circle embedded inside a bigger circle both circles travel the same distance. Do both circles have the same circumference? Well, that doesn't make sense. We all know that a bigger circle has a bigger circumference, so how do they travel the same distance? The trick lies in the fact that the smaller circle is being dragged. It's not just doing rolling motion like the bigger circle, it's moving horizontally as well. It's just really hard to see. This insight helped inform the fact that all lines have the same number of points. Even the points of a smaller portion of a line can be mapped to all the points on the original line. (texts chime)
Braess's paradox. The next paradox shows us the weird and counterintuitive behavior of equilibrium. Here we have a water bottle filled with water, which is acting as a weight. It's hanging from a bar connected to two springs and two strings. Now the way that the springs and strings are connected is a bit hard to see and it's very important. So listen carefully. Tied to the bar is a regular piece of string, which doesn't stretch or anything fancy like that. It's then tied to the lower spring, which is connected to the weight. Another spring is also tied to the bar and then tied to the lower string, which is connected to the weight too. However, and this part is very important, the two springs are linked together. You'll see that both the strings are actually slack at the moment. There's no tension pulling on them. This is because they're both longer than the springs right now. And, as the springs are linked together, the strings are kind of just hanging there. At the moment, our spring string setup has a total length of about 795 millimeters from the bar to the pencil. Now, if I unlink the springs, what do you think will happen? Will the weight, A, fall; B, rise; or C, stay the same? Pause the video now and try to figure it out for yourself. (drumroll beating) (tumbler clunks) Congratulations to those of you who chose B, the weight rose. If you didn't get it, don't worry, you're not alone. My intuition tells me that the weight will fall. The strings were slack, so if we unlink the springs, the string should stretch to full length, making the setup longer, and so lowering the weight. But that's not what happened. See, when the springs were linked, they alone were bearing the entire weight of the water bottle. There was more tension pulling on them, making them stretch more. When they are unlinked, the weight divides equally between each pair of string and spring link so now each one only bears half the weight of the bottle. So even though there was a lengthening in the system here, the reduced weight on the springs caused them to compress and therefore become shorter, making the total effect and overall shortening of the system and the weight rising. (texts chime)
Moravec's paradox. Moravec's paradox is the observation that tasks that humans find easy, like sitting in a chair, understanding jokes, or grasping objects, are very difficult for robots and computers. Whereas tasks we find hard, like complex algebra and chess, computers find very easy. Things have definitely changed in the past few years with AI and robotics advancing like crazy. But it had been this way for a long time. For example, in 1997, IBM's Deep Blue defeated World Chess Champion Garry Kasparov, yet no robot could reliably walk across a messy room or pick up a variety of objects. This paradox arises because human skills we consider simple are the ones that come naturally to us because they're deeply rooted in unconscious processing. But these processes took years of evolution. Whereas tasks we find more difficult and mentally draining, like complex mathematics, usually involve very explicit rules, well-defined goals and logical reasoning, which are straightforward to program into a computer. (texts chime)
Gabriel's horn. Gabriel's horn is a famously strange shape that has an infinite surface area but finite volume. It's created by taking the graph of the function Y equals one over X and rotating it around the X-axis. This creates a horn shape that's infinitely long. Because it has infinite surface area, you could never finish painting the inside of it. But because it has a finite volume, you could fill it with a finite amount of paint. (texts chime)
The Raven Paradox. The Raven Paradox says that the observation of a white shoe supports the hypothesis that all ravens are black. Imagine your hypothesis is all ravens are black. To test this out, you'd go out and look at ravens. Every black raven you see supports your hypothesis. But the statement "all ravens are black" is logically equivalent to the statement "all non-black things are not ravens." So according to logic, every time you observe something that isn't black and also isn't a raven, like a white shoe, it should count as evidence that all ravens are black. The paradox is that observing a white shoe seems to have absolutely nothing to do with black ravens yet, according to logic, it does. This paradox isn't really resolved, but the most common view among philosophers of science is that while observing a white shoe technically does support the hypothesis that all ravens are black. It does so in an incredibly weak way that we should pretty much just ignore it. (shoe thuds) (texts chime)
Olbers's paradox. Olbers's paradox asks, "If the universe is filled with stars, why is it dark at night?" In 1823, the German astronomer Heinrich Wilhelm Matthias Olbers argued that if stars are spread infinitely in space, in every direction, a line of sight should eventually encounter a star, making the night sky as bright as the sun. Sure, stars further away appear dimmer. But since the number of stars increases with distance, their collective brightness should compensate. Yet the night sky is dark. Olbers's paradox remained unsolved for over a century. And the resolution came when Edwin Hubble discovered the universe is expanding. Galaxies are all racing away from each other, stretching starlight into wavelengths our eyes can't see, a phenomenon known as Redshift. Furthermore, because the speed of light is finite, light from very, very distant galaxies hasn't had time to reach us yet. And, in fact, some light never will. (texts chime)
The Muon paradox. Keeping on the theme of historical paradoxes, this paradox helped prove Einstein's special theory of relativity. Muons are tiny subatomic particles created when cosmic rays collide with Earth's atmosphere, about kilometers above Earth's surface. They only live for around 2.2 microseconds before decaying. They travel toward Earth at speeds of up to 99% the speed of light. Even at these speeds, because of their incredibly short lifetime, they should only travel about 660 meters before decaying. So how many would you expect to detect on Earth's surface? Well, almost none, right? But here's a muon detector on Earth's surface. Every time it flashes, that's a muon being detected. And if you hold out the palm of your hand, about one muon passes through it every second. How does so many muons last the whole journey down to Earth? This is the muon paradox. And the resolution comes from two-core ideas from special relativity. The first is time dilation, the phenomenon that when an object moves close to the speed of light, time runs slower for that object relative to a stationary observer. This is summed up by the phrase "moving clocks run slow." So from our frame of reference on Earth, the muon's clocks are running much slower than ours. Therefore, their lifetime is significantly extended, giving them plenty of time to reach Earth's surface. The second core idea from special relativity is length contraction, which we already met in the pole in the barn paradox. From the muon's viewpoint, the Earth is traveling toward it at 99% the speed of light and the distance of Earth's atmosphere contracts. So to sum up, from our frame of reference on Earth, the muons have plenty of time to reach Earth because of time dilation. And from the muon's frame of reference, the Earth's atmosphere experiences length contraction. So the muon only needs its short lifetime of 2.2 microseconds to cover the distance. (texts chime)
Zeno's dichotomy. It wouldn't be a YouTube paradox video without some of Zeno's paradoxes. But more than just being fun thought experiments, thinking about them actually helped to develop calculus. Take a look at this one. If you want to travel one state, the distance covered in ancient Greek foot races, do you agree that you first must travel halfway there? Yes. Do you agree that you must travel to the halfway point between your current position and the one-state mark before you reach the end? Yes, you need to go a further quarter of a state on your way to finishing your journey. But once you're there, there's another halfway point for you to reach. Now an eighth of a state in front of you. Where are you going with this, Zeno? But then halfway again is a 16th of a state, and so on. Once you reach that halfway point, there'll be another one in front of you no matter how far you go. Before you can reach your final endpoint, you must first reach the infinitely many half points that come before it. How do you manage that? It seems that what the logic is telling us is that the journey takes infinitely many steps to complete, therefore it, and in fact all motion, is simply impossible. But you're moving right now. It's an illusion. This paradox challenged philosophers for centuries and led mathematicians like Newton and Leibniz to the concept of a limit: to provide a rigorous way for understanding how an infinite number of sums can sum up to a finite number. Today, the limit is the core idea at the heart of calculus. (texts chime)
Cavalieri's widthless lines. Staying with the calculus theme, this next paradox was a stepping stone to the modern idea of integration. In the early 17th century, Italian mathematician Bonaventura Cavalieri wondered how to find the area of complicated shapes, shapes that didn't have a neat formula. He wondered if we could use the formulas for shapes we already knew to calculate the area of shapes we didn't know. For example, this semicircle has a known formula of half pi r squared. He imagined the semicircle as being made up of lots and lots of widthless lines and then created a new shape by only moving the lines. Because the same lines were used to make the new shape, it should have the same area as the original shape. This idea was immediately paradoxical to mathematicians at the time. How could stacking infinitely many widthless things result in a finite measurable area? No matter how many zeros you add together, you always get zero, right? It also didn't always work. According to Cavalieri's theory, we can fit any line from this triangle into this one, which would suggest that the two triangles are the same size, which we can see isn't true. Even though Cavalieri's theory wasn't quite right, thinking about it led to modern integral calculus. To calculate the area of a complicated shape rather than the lines having a width of zero, we use the concept of the limit and ask what happens when the width of the lines approaches zero. (texts chime)
Zeno's Nerf gun. And to finish up our calculus-themed paradoxes, here's one that helped with the modern idea of the derivative. Imagine a Nerf bullet in flight. At any single instant, the Nerf bullet occupies exactly one position in space. But occupying a position in space at a single instant means that it's not moving at that instant. It's motionless. Since this logic can apply to every moment of its flight, the Nerf bullet is motionless at every single instant. But how can a series of motionless instance create motion? Thinking about paradoxes like this led Newton and Leibniz to the idea of the derivative, which says that to find the instantaneous velocity of an object, you take the limit of its average velocity over an increasingly small time interval. (texts chime)
Russell's paradox. Russell's paradox is probably the most famous mathematical paradox, because it uprooted the foundations of math. It's commonly stated in terms of the barber paradox. Imagine there's a town with a barber. His job is to shave all and only those who do not shave themselves. Does the barber shave himself? If he doesn't shave himself, then he does. And if he does shave himself, then he doesn't. At the time of Russell's paradox, the theory of sets was being proposed as the foundations of mathematics. Sets are a collection of things, like the set of all natural numbers or the set of all penguins. You can also have sets of sets. For example, the set of all sets of bird species. There exists the set of all penguins, the set of all seagulls, the set of all pigeons, and the sets of all other bird species. If you collect them all together and put them into a set, this is the set of all sets of bird species. Russell asked, "Consider the set of all sets that are not members of themselves. Is that set a member of itself? If it is a member of itself, then it isn't. And if it isn't a member of itself, then it is." Both answers lead to a contradiction, which, as we all know, is disastrous in mathematics. (texts chime)
Berry's paradox. What is the smallest positive integer that cannot be described in fewer than 15 English words? At first glance, this seems like a straightforward question. There are infinitely many numbers and only finitely many ways to describe them with a limited number of words. So there must be such a number. But wait, isn't this a description of the smallest positive integer that cannot be described in fewer than 15 English words? And it's only 14 words. This is Berry's paradox. And it isn't just wordplay, it has huge implications for data compression and the way we represent information. Complexity is the number of bits needed to describe an object. Sometimes we can compress the number of bits while keeping all the information, a process known as data compression. Data compression is very useful when it comes to data storage and information representation. But it's not always obvious when a string of data can be compressed. Computer scientists wondered if we could write a program that could take in any string of information and compress it into the shortest possible string of ones and zeros. But it turns out that such a program is impossible, because it would lead to contradictions like Berry's paradox. (texts chime)
Galileo's paradox. Galileo's paradox is an observation about the counterintuitive properties of infinite sets of numbers. Take the set of all the natural numbers, one, two, three, four, five, et cetera, and the set of all square numbers, 1, 4, 9, 16, 25, et cetera. Galileo noticed that there seem to be fewer square numbers than natural numbers since square numbers appear less frequently. However, every natural number can be mapped to one square number. One with one, two with four, three with nine, 4 with 16, 5 with 25, and so on. For any natural number, there will always be a square number it can be mapped to. This pairing suggests the two sets have the same size, or cardinality. Galileo found this paradoxical because it challenged the notion that a part of a set should always be smaller than the whole. This paradox was resolved when Georg Cantor developed his theory of infinite sets and showed that a subset of an infinite set can have the same size as the original set. (texts chime)
The Ross-Littlewood paradox. Imagine you have an infinitely large vase and an infinite number of ping pong balls all labeled by a number, one, two, three, four, five, and so on. The time is exactly one minute before noon. You add balls 1 to 10 to the vase and remove ball 1. At exactly half a minute until noon, you add balls 11 to 20 and remove ball 2. At a quarter of a minute to noon, you add balls 21 to 30 and remove ball 3. You keep going like this, adding more balls and removing one at each time division until the clock reaches noon. How many balls are in the vase at noon? The most obvious answer is that there are infinitely many balls in the vase. We're effectively adding nine balls to the vase at each time step. And there are infinitely many times steps if we assume that time is infinitely divisible. Infinity times nine is infinity. But hang on, we're not just adding nine balls each time step, we're adding 10 balls and removing 1. Infinity times one is also infinity. We remove a ball from the vase infinitely many times, so there should be zero balls left in the vase at noon. In fact, every ball that's put in the vase can be mapped to a ball that was taken out. This is the Ross-Littlewood paradox. And, once again, it highlights the counterintuitive nature of infinite sets. (texts chime)
The dome paradox. The dome paradox describes a special scenario of a ball sitting perfectly on the apex of a dome. Newton's laws tell us that assuming a perfect frictionless world free of outside disturbances the ball should just balance there for all eternity, unless acted upon by a force. But according to the equations that describe this scenario, we can derive a solution where the ball moves off the dome completely spontaneously without a cause. This strange scenario is known as Norton's dome, and it challenges determinism, the idea that if we know the exact conditions of something in the present, we can predict its exact conditions in the future. This idea is at the heart of classical physics, so it's interesting that there's such a simple example that seems to defy it. (texts chime)
The time reversibility paradox. (balls clunking) The time reversibility paradox says that we should see time flowing backwards just as often as we see time flowing forwards. Newton's laws are what's called time reversible, meaning they don't change whether you play them forward or backwards. So in principle, we shouldn't be able to differentiate between the forward and backward directions of time. But obviously this isn't what we observe at all. There's a clear direction of time. This discrepancy between what Newton's laws say we should see and what we actually see puzzled physicists for a long time, until Ludwig Boltzmann came along and resolved it. He said that Newton's laws hold at the microscopic level, but the arrow of time is a macroscopic phenomenon. Because entropy is always increasing, there are just so many more ways for a system to be in a higher entropy state than a low one. Therefore, it's just overwhelmingly probable that systems will tend to states of higher entropy. This increase in entropy is what we perceive as the arrow of time. Today, the fact that entropy will always increase in a closed system is known as the second law of thermodynamics. But then Boltzmann's colleague Josef Loschmidt was like, "Wait, that doesn't make sense. You say it's more probable for a system to evolve from a state of low probability to a state of high probability. However, for every state that evolves toward higher entropy, there is an opposite state that evolves toward lower entropy. All one has to do is flip the directions of the velocities of the molecules and they will all go back the way they came. This means that there are equally many states evolving toward lower entropy as there are evolving toward higher entropy. So how can you say it's more probable for a system to be increasing in entropy when there are literally an equal number of states decreasing in entropy as there are states increasing in entropy?" This is the time reversibility paradox, also known as Loschmidt's paradox. Logically, time should flow backwards just as often as it flows forwards. So why doesn't it? While Loschmidt's argument makes sense if we assume we're in equilibrium, but our universe just happened to start in an extremely low entropy state at the Big Bang, why our universe started in such a low entropy state is one of the biggest unsolved questions in cosmology. Do you know any other math or physics paradoxes that weren't in this video or haven't been covered on YouTube? Let me know some in the comments below. I hope you enjoyed the video, and I'll see you next time. Bye.