Zeno's Paradox - Numberphile 

I love the way he says tortoise

Zeno: motion is an illusion! See how I use maths to prove it! Diogenes the Cynic: *gets up and walks away*

This video is not about maths, it is about someone having found his true passion in life. The sparkle in his eyes while explaining is priceless.

Physicists say that the least possible lenght is "Planck's length " of the order 10^-35 & nothing can be smaller than that(or the rules of space-time break down) So, both physics and mathematics explain the phenomenon but in a different way.

You never stop clapping, there are only longer intervals between your claps.

So I’ve watched pi nearly became 3.2, then squaring the circle, and now....... is it just me?

Also, for irrational measures, planck's length kind of solves the perfect circle or the square root of two on the triangle issue. You only have to go somewhere around 60 decimals deep if the circle or triangle was the size of the observable universe and the precision you had to reach was the planck's length. This would result in a perfect precision for physical reality with a finite number of decimals despite mathematically the number could never be written fully.

I love the random cuts to the clap 😂

I love how they always insert the clap scene 😂

1:44 , 2:10 , 2:18 , 2:24 , 3:56 , 5:26 , 6:06 , 7:48 , 7:54 , 7:58 , you are welcome

There are many answers to Xenos paradox actually. My favorite is from Thomas Aquinas. He says that “Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."

2:10 Yes there is a force field stopping your hands meeting, on the atomic level your hands don't meet !

Who would win, xeno or one clappy boi

2:24 3:56 5:27 6:06 7:48 7:54 7:58 Just 'cause I can.

That is the weirdest pronunciation of a tortoise I have ever heard 😂

Came here to try and understand better Gojo's Infinity power.

For a physicist, It is impossible divide space an infinite number of times as in Zeno's paradox. Eventually you'd hit Plank length (1.616199(97)×10−35 metres) and this is the absolute, rock bottom basement for reality. This is smaller than all elementary particles (quarks and electrons) and deep into the boiling soup of zero point. Think of a boiling pan of water, but in this case the water is raw energy. To give you a rough idea how really minuscule we are talking here, imagine a dot about 0.1mm in size (which is at or near the smallest the unaided human eye can see, apparently) if this dot was then magnified in size to be as large as the observable universe, then inside that universe-sized "dot", the Planck length would be roughly the size of an actual 0.1mm dot. This is Worlds end on a nano scale and anything "smaller" has no meaning. Beyond this "here be dragons"

*sees james* *clicks on video*

Shout out to Gojo-sensei and Dr. Grime for helping me me learn something today! It's important to study, you know

I'm reminded of the Frank & Ernest cartoon in which one ancient Greek is telling another, "Zeno isn't coming into work today, and you should hear his excuse!"

never heard someone pronounce tortoise like it's spelled

specially in physics your hands never clap, because there will be always a distance that separate the atoms of your hands. and yes planck length..

1:48 Gojo watching this like: 👀📝....

I just wanted to hear about a guy who “solved” the squaring a circle problem

I love his pronunciation of tortoise.

You say "tortoise" the way Benedict Cumberbatch says "penguin".

Been watching this channel for quite a few years now. I've probably seen this episode 4 or so times. I have never really appreciated it's simplicity until i studied SR and sums. Glad to have this channel as always

You still clap because, even though you're halving the distance you move each step, you're also doubling the speed at which each step is taken. At the very "last" infinitesimal step you're effectively moving at an infinite speed.

Always knew I was a paradox.

the Zeno's paradox is just a representation that mathematics is not perfect. maybe in math you can assume that infinity exists, but in the real world it doesn't. This paradox can only be interpreted if you are thinking only mathematics, and not physics.

*Two problems* I find with these paradoxes: *1) They try to infinitely divide the finite.* *2) They try to finite the infinite.* This may be summed up as follows: *These paradoxes try to equate finite with infinite.*

The problem with all paradoxes of Zeno's form is that time decreases with each additional step, infinite steps taken in infinitesimal time will equal terminate after a finite interval.

Well, if you think about it, you can hear a sound of hands clapping but the hands never meet. the sound is from the movement of the air between the surfaces. You can't put your hands on anything at the atomic level which means that maths is right and it isn't really a paradox, it's just the way it is.

Dr James grime is my favourite on this channel

Once you hit the plank length you’re there. But additionally this paradox just shows that our understanding of a 3D physical space is actually just how humans model the world in our minds.

Hi Dr. Grime. I know you made this almost seven years ago, but I still had to comment. I wanted to comment from my own viewpoint on your Zeno's paradox question. How would a physicist solve this? Well, I don't consider myself a physicist, but my bachelor's degree is in physics. So I could give it a go... On an exam once, we were given a similar problem that went as follows: Two trains are on the same tracks 100 miles apart, heading towards each other. The first train is travelling at a constant speed of 20 miles per hour, and the second train is travelling at a constant 30 miles per hour. On the very front of the first train sits a hummingbird. The hummingbird can fly at an average speed of 60 miles per hour. The hummingbird flies at it's average speed from the first train to the second train. Then the moment it reaches the second train, it immediately turns around and flies back to the first train, all the while maintaining it's average speed of 60 mph. The bird continues flying back and forth between the trains until the two trains meet (let's not discuss the potential fiery crash). Calculate the distance that the hummingbird flies by the time the trains meet. ****************Spoilers - in case you want to solve this yourself - Do not read below************** Obviously, this problem is very similar to the problems stated in this video. The bird seems to fly in ever decreasing distances in an infinite summed series. However, I learned that in physics, the answer is often found, not by forcing your way through infinite series (not if you don't have to), but by looking at the problem from a different perspective. If you ignore the problem of the bird completely for a moment, you can focus on the trains. With the two trains at a distance of 100 miles and a constant complimentary converging speed of 50 miles per hour, how long will it take the two trains to meet? The answer is easy - it will take 2 hours. Now look at the bird. Regardless of the crazy path it flies or how many times it goes back and forth between the trains, you know that the average speed of the bird is 60 mph. So if this bird flies at 60 miles an hour for 2 hours, how far will it fly? 120 miles. We don't need to worry about the infinite series. We cut straight to the answer. That's how (I believe) physicists think. The same applies for Achilles and the tortoise. Instead of going through the infinite series, let's just create an equation. A = the speed of Achilles running T = the speed of the tortoise t = the time period since Achilles started running So, A * t = The distance that Achilles has run (speed times time) T * t + 100 = The distance that the tortoise has gone, given it's 100 m head start. So if we can assume that Achilles will pass the tortoise at some point, then we can set their two distances equal: A * t = T * t + 100 We can solve for the time. Both t values will become tP, the amount of time it takes Achilles to pass the tortoise (assuming we can guess the speeds of both the man and tortoise). Then we simply solve for tP and plug in the values. A*tP - T*tP = 100 (A - T) * tP = 100 tP = 100/(A - T) Assume Achilles can run at approximately 4 meters per second. Assume that the tortoise can run at approximately 0.25 meters per second tP = 100 / ( 4 - 0.25) = 100 / 3.75 = 26.67 seconds At 26.67 seconds, Achilles will pass the tortoise. It's all in the way you look at the problem. From a physicist's standpoint, we view infinite series as another tool in our mathematical toolbox. If it helps us to solve our problem, we use it, but if it only makes the problem harder, we try something else.

No. You can't divide by an infinitely smaller number in physics. The Planck Length is the smallest measurement. You cannot travel a portion of a planck length, you must travel the entire planck length.

I love how the majority of the people that watch these videos and comment on them are actually pretty smart, except for me. I just love watching things that make me really confused and question everything. I have no idea how any of this stuff works.

Damn, that example of the triangle with the irrational hypotenuse is a perfect description of how an infinite task can be completed. :)

I really appreciate these videos about paradoxes. They validate all the arguments I have with people who suppose themselves too smart for paradox

Real questions: why does he call it a TOITOIS?

Zeno's Paradox but every time James mentions clapping he claps

It's like clapping on a microscopic level when you're having the distance. Material disappears when your down to the stomach level of space. Then start having that space and so on.

the sequel to the sequel, amazing...

i like how they keep cutting to james clapping his hands

I understand people from different English speaking countries pronounce certain things differently... but TOITOIS?

This was a great explanation and I appreciate that Dr. Grime concedes that he does not have a physical solution to this paradox.

In the Achilles paradox by dividing scale by 100, you maintain the exact original problem but scaling it down. So you’re calculating a value where the distance is getting infinitely close to 0 but never reaching it, 0 would be the point at which Achilles would pass the tortoise. It follows a reciprocal function.

Zeno obviously just wanted to start a world war.

Since the atoms in your hands never touch, the second paradox is completely solvable.

1:00 in my head a bit after that time "just have him sprint 20 meters instead of 10, in the time it takes him to sprint 20 meters, the tortoise will have only moved 2 meters, so Achilles will be ahead by 18 meters"

The most incredible thing about math, to me, is how I could possibly get through Calculus, and yet not really understand even 1/10 of the lower math...if that math is actually lower. A brilliant friend of mine who is now a doctor doing research for a cure for cancer, had trouble with probability. I guess that's why he went into medicine maybe, instead of physics, which was his major. Thanks for posting. Always a treat.

while you guys are arguing about Planck lengths, the answer is no, they will never touch because there is a magnetic field created by the protons, since protons are only positive they repel each other, therefore, they will never touch anyway

A toytoise?

With regard to Achilles and the tortoise ... It is true that Achilles will be running to where the tortoise has already been, but only for a limited time, because the distance between where the tortoise has already been and where Achilles is, changes by a factor of (V(Achilles) - V(tortoise))* time, so that it eventually reduces to zero and then increases again with Achilles in the lead. So there is no paradox. In order for it to be the case that Achilles is ALWAYS heading toward where the tortoise has already been, Achilles' velocity would have to be slowing (at an exponential rate), which is not the case in the scenario.

Moving your hands until they clap: If you move them at a constant speed, then when they get to half the distance, they do so in half the time. When they get half of the remaining distance, they do so in half of the time of the previous segment, since your speed is unchanged. So, by factoring in the speed as the "third leg of a stool", halving the distance and/or time allows the infinite series to complete. And it makes sense to the real world when they hear the clapping sound from your hands.

Doesn't the planck length "solve" both of these paradoxes?

all I care about is how he say tortoise and it is pretty funny

Nice! Super glad you guys FINALLY got to a episode, one done about Zeno. I’m a huge fan of Zeno actualy so naturaly thanks. Hes that warrior prinecess guy , Yea thanxz !!

Charles Dodgson (aka Lewis Carroll) wrote an entertaining short paper on this topic called "What the Tortoise Said to Achilles" where an erudite tortoise demonstrate his ability to beat the famous warrior Achilles by running philosophical rings around him.

there's the minimal distance unit "Planck length" equal to 1.616199(97)×10^−35 metres.

Yes the turtle moves 1 meter but Archilles moved at the same time as the turtle in real life, so yes he would catch up because they aren't taking turns running they are running at the same time... Am I missing something here or is this not really a paradox?

That recurring clap though 👏🏼🤣

Thus, we find that the distortion of the measured particle by the gravitational field of the particle used for measurement prevents the resolution of arbitrarily small structures. Resolution is bounded by the Planck length, which is about 1/(10 to the 33) cm . The Planck length thus plays the role of a minimal length. (See: Fermilab Today, Friday , Nov 1, 2013)

Drink a shot everytime he claps his hands

scientifically speaking no, the shortest possible distance is Planck length so every time you move your movement is defined as a number of planck lengths you are moving, and it cannot be defined as less. also their is Plank time which is the time it takes to travel 1 Plank length if you were going at the speed of light.

I know I’m 6 years late but I want to say well done on simply explaining a calc 2 subject that I struggled with when i took the class

In physics its not a paradox. I like this explanation: The takeaway is this: motion from one place to another is possible, and it’s because of the explicit physical relationship between distance, velocity and time that we can learn exactly how motion occurs in a quantitative sense. Yes, in order to cover the full distance from one location to another, you have to first cover half that distance, then half the remaining distance, then half of what’s left, etc. But the time it takes to do so also halves, and so motion over a finite distance always takes only a finite amount of time for any object in motion. Although this is still an interesting exercise for mathematicians and philosophers, not only is the solution reliant on physics, but physicists have even extended it to quantum phenomena, where a new quantum Zeno effect - not a paradox, but a suppression of purely quantum effects - emerges. As in all scientific fields, the Universe itself is the final arbiter of how reality behaves. Thanks to physics, we at last understand how.

For this question, you need to ask us physicists. The paradox arises because of a wrong assumption, namely that space can be infinitely divided. However, there is a minimal distance: the Planck length of roughly 10^(-35) meters. Going below it has no meaning and violates quantum mechanics. At some point, when the hands (or Achilles) are 1 Planck length apart, they do not go to 1/2 Planck length, but all the way. It is perhaps easier to picture if you think of the universe as a three dimensional screen made from Planck sized cubic pixels (voxels), so everything that moves around the universe essentially jumps from voxel to voxel. In superstring theory, these voxels are the Calabi-Yau manifolds, which themselves are higher dimensional but are arranged in a 3D grid. As you move your hand around, each particle in it jumps from manifold to manifold in this grid, giving the impression of a 3 dimensional space.

That's a fast fucking tortoise.

In the first period of time Aquiles reach the position where the tortoise was, but he had already moved a distance which is smaller than the distance Aquiles moved, so if that period of time repeats Aquiles would reach the tortoise

OMG... It turned out so easy to resolve Zenon' paradox... Just to take into account a time to pass each next shorter piece of the path... I was so blinded))... Thanks!!!

acording to modern physics it seems you can't get a unit of length smaller the the plank length and a unit of time smaller than a plank second. so the number operation for running or clapping and the number of halves would be finite for a given length.

SOLUTION: Because when you half the distance an infinite number of times you will reach the diameter of an atom and then after that the planck length and since there is nothing shorter than the planck length it means that when the distance between your hands reaches planck length the distances can't be halfed anymore and thus the hands clap in the next step and since this means that the clap paradox isn't a paradox Zeno's paradox can't be a paradox cause it was made with the premise that the clap paradox actually was a paradox.

I tend to reply to Zeno's paradox with "you're not keeping your unit of time constant. Once you do that, the entire paradox falls apart."

I see that in a lot on jrpgs, you see, it usually is an special attack that half the oponent's life, this happens until the half is no longer an integer number.

i liked how you constantly added the clap clip, it was funny

I have to be honest here. I am a bigger nerd than I thought.

It's not a paradox. Infinity just isn't a real thing; it's purely a concept.

7:22 that sound gives me goosebumps

You can also say "how is a complete circle possible?" Because it's an infinite number of angles, or how is a complete line possible because it's an infinite number of dots. Just some thoughts I had during the video.

Plank length would be the limit of spacial division, correct?

This man defeated the RU-vid algorithm.... I started off with Pi = 3.2 and now I’m here. I’m watching math videos... like HoW?!?!

I don't know why people are complaining about your pronunciation of tortoise - you're one of the few people I've come across online who says it right!

In this situation, you can't half the distance of the atom. When you approach the atom limit, then you simply clap the hand.

technically your hands never touch, nothing ever touches.

I notice this every time I drive. When you pull up to a stopped car at the red light, you will keep slowing down until stopped, otherwise you would crash into the stationary car in front of you. Your velocity relative to the stationary car is getting slower and slower, but at some point it changes from "having velocity" to "being stopped". I alway try to make the change from moving to not-moving as smooth as possible, trying to get my speed down to imperceptible ranges before stopping altogether. It's harder thank you'd think.

Thank you for this, 10 years ago i never appreciated the effort and work that went into these videos

There was a Greek philosopher (can't remember which one) that said that the world and motion must be made of steps, because when you move your arm, your arm must travel through an infinite number of positions, but that's impossible, so there must be steps. Or as we can think of it today as images in a movie or tv film. Obviously he must be wrong somewhere, but where....

His enthusiasm is contagious and so does his non-rhotic tortoise

His expression shows how passionate he is about the subject. I wish i would have that traits in me. Sadly my throughout academics teachers kills it.

Answer to the last question about physics: There is a smallest possible distance and a smallest possible time called the planck time and the planck lenght. This makes it a non-infinite process that’s why it works in the real world. Perfect pi and perfect root 2 is impossible to create in the real world. It’s really a mathematical concept. Nothing irl behaves perfectly, as i should in maths!

The tortoys is fast

Tour toys.

*Mathematician:* Can't work out a sum *Mathematician's mom:* Look Sum, if you won't stop behaving i'm calling your parents to pick you up

5:00 So very well explained!👌🤩 Definitely understand a little better now, and will be passing the time clap example on.

Toytoise.

I went from hating math to loving it... Power of youtube.

2:06 that "force field" between his hands is the perfect boundary between theoretical mathematics and practical mathematics

I adore James' accent, and nowhere more than in his beautiful pronunciation of "Tor-toyss" ❤🐢