Zeno's Paradox - Numberphile
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- čas přidán 4. 06. 2024
- Dr James Grime is back and talking about tortoises.
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In many ways this video follows on from • How Pi was nearly chan... and then • Squaring the Circle - ...
James Grime's website is: singingbanana.com
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I love the way he says tortoise
😂😂😂 me2
+Booker DeWitt I've never heard a British person say that word before. It's... it's really absurd. I can't get over it.
+Ryan N Not all British people pronounce tortoise like he does
+Ryan N I've never heard _any_ Brit pronounce tortoise like he does.
+Xultrain Kaos you've not heard very many Brits then. I've never heard a Briton say it any other way than this.
Zeno: motion is an illusion! See how I use maths to prove it!
Diogenes the Cynic: *gets up and walks away*
George Berkeley: matter is an illusion, nothing can be asserted to exist.
Samuel Johnson: [kicks stone] I refute it thus!
He actually walked around in a circle.
@@triambakeshwar8766 Calculus: "Im about to destroy this man's whole career"
@@ruhaanb6 lol
@@ruhaanb6 how does calculus connect to the circle? Or were u talking abt just in general to the main comment?
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.
It's about maths.
Hi sir
Its certainly about maths
It's certainly not about Chinese checkers.
It’s… it’s about maths
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.
I was thinking of that too. This essentially breaks the paradox because in real life it's a finite number of steps, so we are able to complete the task.
What we are talking about here is basically the resolution of universe, the smallest unit we can work on.
@@Ally5141 Yes
@@darklord9813 Planck's Length is the smallest unit of measurement but Zeno's Paradox kinda proves there has to be the smallest distance overall. Either that or we still don't understand how our world works because there can't be paradoxes like that in real life.
If that is so it would prove infinitely small doesn't exist and it makes you think if infinitely large doesn't exist too.
@@Ally5141 Infinity is a real thing(not number for sure) in mathematics
But in physics it's just not that easy to define infinity due to our finite universe.
And if you go further down like smaller than the Planck's length, we don't actually know what happens there but it's quantum mechanics for you
You never stop clapping, there are only longer intervals between your claps.
Exactly like how a ruler has smaller measurements centemeters and millimeters but it's still 6 inches
You only stop clapping when you die.
I agree, all we are doing is infinite analysis of finite terms.
@@IIxIxIv Or do we?
Suppose a person only claps once?
So I’ve watched pi nearly became 3.2, then squaring the circle, and now....... is it just me?
Mark Pointer me
Nice one, CraftQueenJr ! Lol!
I did the same 0_0
It's because these three videos are explicitly related to each other, and could easily be in one single video and make as much sense as they do separately. Additionally, by nature, we want to see the end to any beginning (provided the subject of said beginning interests us), and so we continue a series we've started.
Basically, you have human nature combined with your personal interests to blame for watching.
(import a comment here)
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.
And that's why quantum physics is the sh*t
Thank you sir
I was thinking something like that, and your hands could only get so close before the atoms repel eachother
I a thinking along the same lines. Planck length is like the pixel size of the universe. It just jumps from one Planck length to the next. Therefore Achilles can overtake the turtle, as soon as the difference reaches the Planck length
@@adamfanning9412 Maybe his hands never actually do touch, we just see the ripples/shadow of the illusion of touch in other mediums of air, sight, memory etc.
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
OMG I can't stop laughing at this :'D
just press 2 several times.. keep pushing!!
It is like a forced meme.
Hahahaa
Hahahaa
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."
Do you have the reference? I would love to check it out.
Interesting
and yet the Universe, that created time, seems to have been created out of 'instants', and once created, seems to be able to cross the threshold again into Instants/Singularities - sort of like defining Limits to be true (i.e. 3.99999.. =4) even though it clearly isn't, yet it is.
@@wallstreetoneil Time wasn't created as you can't say there is something "before" time itself.
Thomas Aquinas is based
2:10
Yes there is a force field stopping your hands meeting, on the atomic level your hands don't meet !
but if you clap at the speed of light they will touch even on an atomic level.
@@qwertyslapil6957 i just looked it up.
the two hands were technically never apart from each other.
even if you cut of one hand and bring it to mars they would still being effecting each other.
They have a video that explains why this statement is false.
@@alspezial2747 nothing can travel at speed of light
@@ashutoshchouhan8380
photons can travel at the speed of light. i think electrons as well but am not sure.
what you mean is that nothing with mass can go that fast.
Who would win, xeno or one clappy boi
Clman4 *zeno
Clappy zeno
one defaulty boi
Me
Up high... anyone?
2:24 3:56 5:27 6:06 7:48 7:54 7:58
Just 'cause I can.
I see what you did there...... ;)
+Finn Underwood XD we have the same minds here...*clap*
+Finn Underwood Looks like I was bored enough to find out what you meant :P
+Kevin Huynh
i don't get it
+Lastrevio all the times he clapped
That is the weirdest pronunciation of a tortoise I have ever heard 😂
Tortoyce
He pronounced it exactly as it's written
@@TheArtheanos Well at least he didn't pronounce Achilles like it's written.
Toytoyce
I agree
Came here to try and understand better Gojo's Infinity power.
Oh wow I was preparing my presentation about infinity and found jujutsu kaisen about 3 months ago ... It's quite ironic how you found this because of gojou and how I found gojou because of infinity .. and now we all have crushes on gojo
@@mathematicsguru97 i know right when he said force field I instantly thought of Gojo ,Gege is a genius in character building,incorporating maths in anime really made it fabulous
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*
+Gabriel Espinoza Don't we all?
Accurate
+Gabriel Espinoza i came here for the "tortoise"
+Gabriel Espinoza you forgot "unzips pants"
so tru
Shout out to Gojo-sensei and Dr. Grime for helping me me learn something today! It's important to study, you know
literally the same. I just wanted to understand Gojo's power but I end up learning Set Theory, Zeno's paradox, Continuum, absolute convergence, Riemann series theorem, and also religious beliefs in the Buddhism like Enlightment and The profound understanding of thyself.
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
I had to rewatch the first 2 minutes because I was so distracted by that
Something about his pronunciation of that makes me happy
Thor toys
Claytonian JP Lol
Claytonian JP What about "Tur-diss"?
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 could explain it to you, but I'd need an infinite amount of time and a pen with an infinitesimally small nib.
I love his pronunciation of tortoise.
You say "tortoise" the way Benedict Cumberbatch says "penguin".
true
He says "toytoyz" that's so cute :)
안녕하세요!
He says it the way it's spelled
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
I just seen this and I ran. I thought wait a minute? Didn't I just see a nice vid on this. But somehow all this sum+scary math...
I have dabbled in it due to chem and bio.
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.
How are you even moving each step? Let's say we have a distance of 10. I'm at 0 and I want to start moving, what is my first step? Do I go to 1? I can't because I have to cover 1/2, 1/4, 1/8.... first. Unless there's a distance that can't be divided I won't be able to move at all.
Always knew I was a paradox.
Zeno is paradox, true
Very funny
Zeno Silva 😂😂😂😂😂😂😂
Killua
Zeno Silva we all knew it on some level. Congrats
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.*
Thanks for the random bold buddy
The paradox is solved anyway. Assuming it's true, we can still find the answer- through the infinite series mathematics that they do *in this video.*
Pay attention to what they're saying, not what you think they're saying. Achilles isn't real and he never raced a tortoise, but these aren't "problems with the paradox." The paradox asks how we can create a reference frame of infinite subdivision and have the maths still work. If your answer is "we can't, actually," you're missing the point of the video.
To me, the solution will be found in understanding infinity. How can something not end?
Thank you sir
@@blacktimhoward4322everyone knows writing in bold makes you smarter!
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.
It's not time that is of relevance (time doesn't exist anyway) it is acceleration (or rather deceleration).
The paradox relates to the concept of instants of time. The idea is that it takes instants of time for objects to move from one place to the next, but if those instants get smaller and smaller, it would take an infinite number of instants to reach the final destination. The answer to the paradox, as the video alluded to at the end, has to do with whether or not space and time are quantized, which is still unresolved.
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
Lyle Sargent mine too
Mine too
You ever seen Cliff Stoll?
Matt Shap They are equally awesome.
Matt is my favorite to listen to.
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.
Thank you for the problem! I'm an adult relearning math and tried to have a go at the question, and even though it is easy I'm glad I figured it out on my own using the algebra skills I've been working on(I find solving easy enough, but setting up the equations/relating them to stuff in the wild is something I need to improve on). I started by just disregarding the bird at first. If I'm not mistaken the time to the trains impact is a system of equations? So m(h)=30h and m(h)=100-20h (m of h is mile marker at impact in h ours). Once I got 2 hours till collision, I started overthinking the bird part of the equation until I realized that it doesn't matter, the bird has 2 hours to fly at a rate of 60mph so 120 miles. Thanks again for the insight!
Hello Ken, much respect to you for having completed a bachelors degree in a difficult subject like physics. I don't think Zeno's intention of elucidating this paradox was to subject people to tricky math problems, though. More or less, it's bringing light to the fact that you are indisputably traversing an infinite number of points through space and how peculiar it is. Logically, it shouldn't be possible. Yet, it is. When you add time to the paradox, it's even more puzzling.
At the end of the day you're just finding an approximate answer then. The true answer is dfferent.
Always loved physics intuition, but got into engineering for the business at uni 😔
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.
Connor Skudlarek This is a theoretical scenario. You could change it from a distance question to a simple number question if you like.
RustyGold No, no. In the video he said something like, "I don't know if that's how physics works or not." I'm addressing that specifically.
Connor Skudlarek I thought that the planck length was just the smallest measurable length?
***** Without going into detail (which I'm really not qualified for anyway), the Planck length is the smallest measurable length in the standard model. It shouldn't be possible to travel half of a Planck length in the standard model.
That's why since physics follows the standard model, we can't say that it can travel infinitely shorter distances. If we ignore the standard model of physics, then practically anything is possible. But under the current understanding of physics, you cannot travel less than a Planck length and have it make any difference in your position within space.
In reality, the vibration of atoms moves many Planck lengths. So technically it really shouldn't be possible to move just one Planck length anyhow (not for atoms, anyway).
All of this is up to snuff, unless I'm mistaken in which case I'd appreciate if someone showed me the error of my ways. :)
Ah, thanks for clearing things up
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
some people just dont care, get over it
Real questions: why does he call it a TOITOIS?
That's how he was taught.
Zeno's Paradox but every time James mentions clapping he claps
zcolucci numberphile but every time James repeats himself he claps
Zcolucci this meme was a little advanced for that time
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.
There is a physical solution. Planck's constant. Time and space are not infinitely divisible
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.
Yeah, but he did not want anyone to arrive to the battlefield :-D
Better yet he just wanted to die.Since he was considered a 0 he was willing to prove he wasn't and take those that numbered themselves with him.
Perhaps the numbered should of considered Zeno's true value as he grew to consume them!!!! Lol 🌀
@@shannonchuprevich3021 maybe he tried to convince himself (wierdly) that he can not die however close he comes to death.
legendary nacho what? No
A war where the bullet never reaches its target?
Since the atoms in your hands never touch, the second paradox is completely solvable.
Those two things don't have anything to do with one another.
I agree totally, but if we remove the constraints of physics ( in the sense of divisibility of matter and time) then is their a logical answer?
Once it is small enough, they will be what we consider touching, however, the hands will never physically meet.
They may well not touch, but the electromagnetic field of those atoms most definitely DOES touch, it's kind of a semantic argument, but one could apply the exact same argument to the electromagnetic fields of those atoms and end up with the same problem.
Not at all, that completely misses the point. If the motion ceases just before your hands touch due to electrical resistance exactly the same paradox applies to the shorter distance.
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
"Tawrtoise"
Berak Clan greatest pronunciation of any word ever
Berak Clan It's British English and it's right
Adittya C British pronunciations are rarely right. They don't even pronounce the R at the end of words that end with -er.
SilentBudgie English pronunciations and spellings came first - plus different things can be pronounced differently depending on language, so like c in Russian is pronounced always like a s.
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 but that's in the physical world. That's why so many different fields of study are intrigued by the paradox
No, it is more accurate to imagine the universe as pixelated down to 1 Planck length. Less than that there is no "travel" because the very physicality of length breaks down.
At distances smaller than 1 Planck length, one location is indistinguishable from another because the standard model of physics dissolves into quantum mechanics
he specifically asked for physicists at the end of the video because he was curious how this works in physics
It's the same case with the time, two events that occur within less than 1 Planck time are considered to happen at the same time.
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 👏🏼🤣
Ikr!
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
@Mr. H *pours two shots*
take a shot every time he halfs the distance
Yuu
I actually did this. 😥🥃👏🏻
@Mr. H or double the amount...
😂😣🥃🤤
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.
Actually, there is no proof/evidence yet that the planck length has any physical meaning. It's only hypothesized to be the shortest possible distance. Right now, it's just a relationship between 3 constants (gravitational constant, Planck's constant, and the speed of light) that results in a value with distance units. It has no real meaning yet.
TheAzaka7 well actually their is tons of evidence suggesting that their is a plank length the part which people disagree with is what is the size of a plank length. their is a fair amount of evidence pointing to 10^-32m, and most scientists agree, but some do disagree.
martinshoosterman The proof of the planck length is the exact reason why the zeno paradox is false. You can move. There is motion. If it took an infinite amount of time to travel an infinite amount of space, there would be no motion, as you would never stop attempting to move from your start (point A) to the infinitely divided point adjacent (point B) (This is where rationality breaks down, when infinity is involved). Thus, there must be a minimum length for something to move, (from point A to point B) so that there is a limited time that it'd take to move that distance (and thus, any other calculable distance from that) and thus motion would be finite and would exist as it does today. This length, which is proven to exist by the realisation that reality is how it is, is called the Planck length.
martinshoosterman I believe it's 1.6x10^(-35)m
Bliss Woven well its more than that though. plank length is not just their to solve a paradox of movement if a wave leangth goes smaller then planck leangth the thing emitting the light will turn into a black whole. smaller than 1 plank leangth and no laws of physics work. non at all.
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.
I said this 3 weeks ago....in much fewer words: Space is quantized. I've thought this for years, no calculus or infinities needed, yet maybe I don't communicate well...now lets see if you can prove to me time exist. Because personally, It's superfluous and I think it's only needed to store imaginary values but not necessarily a real thing (vector). Ie we can't have all the information in the universe ergo time makes math work easier ... I've been trying to formalize this but haven't
I simply scrolled down into the comment section thinking of this, and unsuprisingly it's there.
Oddly enough, I came up with this theory in middle school in a very basic conceptual sense, without all the complicated numbers and Planck. I've done a little playing around with the idea, and it helps me to know this is where the idea of (something jumping instantly between 2 points) comes from.
Looks like I'm gonna have to get into quantum stuff if I wanna go any further, lol...
Andy Payne Read up on the philosophy of time. I mean this constructively. Most physicists (including me) struggle with the nature of time and, unlike the notion of quantised space, we're not even close to some united opinion. The two most common interpretations of time (typically termed theory A and B) are both invalidated by fundamental properties of either general relativity or quantum mechanics.
I'm fairly sure there wasn't a notion of quantized space in Zeno's time.
That's a fast fucking tortoise.
Alan Jay Asking the real questions
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.
isn’t it that we can get a number less than planck’s constant but we just don’t use them in experiments?
@@Ryan-ee5lp from my understanding that's the theoretical limit of the fabric of the universe itself. It's the idea that the universe of space-time has a theoretical smallest peace that can exist. It's quantized not continuous
The clapping example is easy, because while it could be considered infinite, each step increases the velocity twice, so the more you divide, the faster it goes up until something catches up.
@@TheChrisey i don't follow. please clarify.
the planck units are just scales where effects our theory cant describe show up, so we dont go beyond them. but they are not limits
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 think about it the same way
Why does it fall 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.
When it comes down to it, it doesn’t really matter. A Line is just a connection between two points
@@gravinboginagis6568 But that connection is a set of infinite points
xolotltolox and what is a point
xolotltolox how is that any different? A connection of infinite points is an infinite amount of connections between two points.
A true circle is impossible, only representations of circles are posible in the real world
Plank length would be the limit of spacial division, correct?
Yus!
tonyrosam what I was thinking...
In classical physics yes. You could say that you could have smaller lengths if you try to apply it to quantum physics but those lengths would be useless unless you could more accurately ascertain what happens at that level seeing as our understanding of space and time break down at that level.
Planck lenght is the distance a photon at lightspeed travels during planck time. Any time shorter than planck time doesn't exist because it becomes indistinguishable from zero. And since every physical interaction needs a force to be in place and any boson of a force travels not faster than lightspeed, planck lenght is the absolute limit of physical interaction. Nothing can happen below those limits.
@dreamyrhodes not at all... Plank space (and time) simply are the space and time scale at which we expect to notice both quantum AND gravitational effects. It has nothing special to it however, it's just a scale where gravity becomes important. Yes, we don't know a theory for quantum gravity yet, but our understanding of quantum gravity doesn't imply that there's nothing smaller than plank leght/time, it's just that we don't know how to describe something that small.
This man defeated the CZcams 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....
Why he must be wrong?
His enthusiasm is contagious and so does his non-rhotic tortoise
toitoise
Hai sir
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
gtg fast
toy toys
Paul da playful panda I didn't understand this comment until I understood this comment.
Tour toys.
안녕하세요!
Danny Burke stop
*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" ❤🐢