Zeno's Paradox - Numberphile

[@SavageGreywolf]

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

[@patavinity1262]

George Berkeley: matter is an illusion, nothing can be asserted to exist. Samuel Johnson: [kicks stone] I refute it thus!

[@triambakeshwar8766]

He actually walked around in a circle.

[@ruhaanb6]

@@triambakeshwar8766 Calculus: "Im about to destroy this man's whole career"

[@mihailmilev9909]

@@ruhaanb6 lol

[@mihailmilev9909]

@@ruhaanb6 how does calculus connect to the circle? Or were u talking abt just in general to the main comment?

What the hell's going on, i just casually started to watch one of your videos, and now it seems i can't stop. help.

[@warrenarnold]

ikr you are approaching the end of the movie but you will never reach it XDXD

[@ricardotrujillo2472]

i hear ya bro

[@vitaplex1]

me too. it makes math fun

[@petulapear4377]

I can't stop either :0

[@buxadonoff]

It's the knowledge virus. Lucifer warned us all not to eat it ..

[@thomasgoetz8178]

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.

[@PadraigTomas]

It's about maths.

[@mdsharfuddinmd5710]

Hi sir

[@theblinkingbrownie4654]

Its certainly about maths

[@jd9119]

It's certainly not about Chinese checkers.

[@Dark_Souls_3]

It’s… it’s about maths

[@darklord9813]

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.

[@cetyl2626]

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.

[@Ally5141]

What we are talking about here is basically the resolution of universe, the smallest unit we can work on.

[@darklord9813]

@@Ally5141 Yes

[@Ally5141]

@@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.

[@darklord9813]

@@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

[@vojtech4598]

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

[@Niko-fj4mo]

OMG I can't stop laughing at this :'D

[@fede77]

just press 2 several times.. keep pushing!!

[@MasterYoshidino]

It is like a forced meme.

[@rajajinnah3478]

Hahahaa

[@rajajinnah3478]

Hahahaa

[@gucciseatbelt9500]

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

[@adamsavage2646]

Exactly like how a ruler has smaller measurements centemeters and millimeters but it's still 6 inches

[@IIxIxIv]

You only stop clapping when you die.

[@sai789987]

I agree, all we are doing is infinite analysis of finite terms.

[@spawnterror]

@@IIxIxIv Or do we?

[@anthonynorman7545]

Suppose a person only claps once?

[@howtoandroid7523]

I love how they always insert the clap scene 😂

[@XwoooahX]

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.

[@combatwombatstl5598]

And that's why quantum physics is the sh*t

[@mdsharfuddinmd5710]

Thank you sir

[@adamfanning9412]

I was thinking something like that, and your hands could only get so close before the atoms repel eachother

[@WBSband]

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

[@erawanpencil]

@@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.

[@infinite_dynamics]

I love the way he says tortoise

[@AnteP-dx4my]

😂😂😂 me2

[@2Cerealbox]

+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.

[@SaifKhan-om6tz]

+Ryan N Not all British people pronounce tortoise like he does

[@xultrainkaos9094]

+Ryan N I've never heard _any_ Brit pronounce tortoise like he does.

[@maeglin9252]

+Xultrain Kaos you've not heard very many Brits then. I've never heard a Briton say it any other way than this.

[@clman4]

Who would win, xeno or one clappy boi

[@Nat-br7zo]

Clman4 *zeno

[@blue9139]

Clappy zeno

[@ratiolinho6315]

one defaulty boi

[@x3no750]

Me

[@richmrmac]

Up high... anyone?

[@markpointer2967]

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

[@CraftQueenJr]

Mark Pointer me

[@markpointer2967]

Nice one, CraftQueenJr ! Lol!

[@blue9139]

I did the same 0_0

[@AbstractionDistraction]

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.

[@blue9139]

(import a comment here)

[@hunterbelch2524]

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

[@jameschristopher2540]

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.

[@finn_underwood]

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

[@oooohapenny4707]

I see what you did there...... ;)

[@WheatleyOS]

+Finn Underwood XD we have the same minds here...*clap*

[@kevinhuynh4533]

+Finn Underwood Looks like I was bored enough to find out what you meant :P

[@Lastrevio]

+Kevin Huynh i don't get it

[@oooohapenny4707]

+Lastrevio all the times he clapped

[@johnjackson4511]

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"

[@Deantrey]

You are confusing our models of reality for reality itself. Very common mistake.

[@gabrielespinoza5269]

*sees james* *clicks on video*

[@davidjoffe-hunter7016]

+Gabriel Espinoza Don't we all?

[@elating96]

Accurate

[@Sweeny_de]

+Gabriel Espinoza i came here for the "tortoise"

[@StrangeDuck]

+Gabriel Espinoza you forgot "unzips pants"

[@dioncu3721]

so tru

[@parkerstern1800]

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."

[@loicchabut9156]

Do you have the reference? I would love to check it out.

[@rujotheone]

Interesting

[@wallstreetoneil]

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.

[@Vld45]

@@wallstreetoneil Time wasn't created as you can't say there is something "before" time itself.

[@mbsb1376]

Thomas Aquinas is based

[@kage131]

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.

[@Ryan-ee5lp]

isn’t it that we can get a number less than planck’s constant but we just don’t use them in experiments?

[@kage131]

@@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

[@TheChrisey]

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.

[@kage131]

@@TheChrisey i don't follow. please clarify.

[@gabrielguimaraes9548]

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

[@lylesargent4863]

Dr James grime is my favourite on this channel

[@AdittyaC7]

Lyle Sargent mine too

[@Sooboor]

Mine too

[@mattshap9731]

You ever seen Cliff Stoll?

[@moritzfeiler6978]

Matt Shap They are equally awesome.

[@natanbagley8949]

Matt is my favorite to listen to.

[@Makethings987]

Always knew I was a paradox.

[@99bits46]

Zeno is paradox, true

[@man-with-a-plan]

Very funny

[@fundekaustubh]

Zeno Silva 😂😂😂😂😂😂😂

[@jmccullough975]

Killua

[@bangkokdangerous2507]

Zeno Silva we all knew it on some level. Congrats

[@Muzly]

I love the random cuts to the clap 😂

[@mrpeanutguy4719]

I love his pronunciation of tortoise.

[@rupert93r]

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

[@Normandy-e8i]

true

[@Liliou]

He says "toytoyz" that's so cute :)

[@James-yw9ht]

안녕하세요!

[@lukapopovic5802]

He says it the way it's spelled

[@elisa-wz5mq]

Zeno obviously just wanted to start a world war.

[@olleollesson11]

Yeah, but he did not want anyone to arrive to the battlefield :-D

[@shannonchuprevich3021]

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 🌀

[@spacemaniii1460]

@@shannonchuprevich3021 maybe he tried to convince himself (wierdly) that he can not die however close he comes to death.

[@alexcerullo3143]

legendary nacho what? No

[@rareword]

A war where the bullet never reaches its target?

[@markstevens7699]

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

[@verchiel_8295]

some people just dont care, get over it

[@rolirolster]

Max Planck's Length, beats Zeno's Paradox. Mathematically you can't resolve it, but in real life, there is a point whereby you cannot half the distance between two objects, hence movement is not an infinite process.

[@jw-pc7920]

Absolutely true. But I wonder... the existence of a minimal length leads to the conclusion that we cannot move 'smoothly'. Everything moves in 'steps' - like a bad frame rate - 'teleporting' from the beginning of a Planck-distance to the end of a Planck-distance.

[@rolirolster]

Yes but each frame is something like 0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000054^256 seconds long, so that's pretty smooth :-)

[@marc.ristau]

That’s what my first guess has been too. All your infinite decimals get cut off at one point due to planck length and that makes everything resolvable in the real world. Same with halving the distance. There is a point where you can’t half it anymore and they touch. No paradox at all imo.

[@rolirolster]

So then, describe a distance smaller than the Planck length... and then tell me when your Nobel Prize ceremony is so that I can watch.

[@quangpham4372]

Total noob, genuine question here. Take the right triangle for example. We draw two 1-inch edges, then connect the two loose ends for the sqrt(2)-inch edge. So the line goes on forever because sqrt(2) is an irrational number. Okay let's say there's some point when I cannot lengthen the edge anymore because the drawn part at that step will be smaller than Planck length. Now I have problems. 1. Have I reached the other loose end? Presumably not, because I haven't "finished" the sqrt(2) length. But the line will of course connected together. Secondly when the dots connect (and they will, obviously), does it mean there has to be an ending digit for the sqrt(2)? Or is it so that I cannot exactly connect the two dots, but rather I can only go very very close, and then take another step and have my line overlap the second end? (Sorry for bad English, I'm not used to mathematics terms)

[@matthewmarek8758]

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

[@The_Unexplainer]

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..

[@KingGrio]

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

[@alspezial2747]

but if you clap at the speed of light they will touch even on an atomic level.

[@alspezial2747]

@@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.

[@blacktimhoward4322]

They have a video that explains why this statement is false.

[@ashutoshchouhan8380]

@@alspezial2747 nothing can travel at speed of light

[@alspezial2747]

@@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.

[@blendersmithee2826]

This channel breaks my brain and I love it.

[@zcolucci]

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

[@shealupkes]

zcolucci numberphile but every time James repeats himself he claps

[@peloncano8286]

Zcolucci this meme was a little advanced for that time

[@AJ-Channel]

That's a fast fucking tortoise.

[@sirtimidly]

Alan Jay Asking the real questions

[@blobscott]

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

[@VoltLover00]

There is a physical solution. Planck's constant. Time and space are not infinitely divisible

[@WatchOnYT]

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.

[@gravinboginagis6568]

When it comes down to it, it doesn’t really matter. A Line is just a connection between two points

[@xolotltolox7626]

@@gravinboginagis6568 But that connection is a set of infinite points

[@gravinboginagis6568]

xolotltolox and what is a point

[@gravinboginagis6568]

xolotltolox how is that any different? A connection of infinite points is an infinite amount of connections between two points.

[@diegoparedes9101]

A true circle is impossible, only representations of circles are posible in the real world

[@ericolml]

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

[@ClaytonianJP]

never heard someone pronounce tortoise like it's spelled

[@bsizzbsizz2575]

I had to rewatch the first 2 minutes because I was so distracted by that

[@runhardmd3477]

Something about his pronunciation of that makes me happy

[@Kilkiju]

Thor toys

[@navsquid32]

Claytonian JP Lol

[@levoGAMES]

Claytonian JP What about "Tur-diss"?

[@rubenpereira1577]

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

[@mathematicsguru97]

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

[@theConservativedude]

@@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

[@queNWS]

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

[@jasonneu81]

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.

[@MandolinKasi]

My take is, the clapping sound is heard much before completing the infinite process/steps and the person stops the motion that is it. In other words the person does not completes the infinite process here. Other way is to assume that the two hands are 1 meter apart and do the maths as if it has 2 meters between them the sound will be heard much before completing the infinite process and the person stops the motion.

[@chuckgaydos5387]

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!"

[@agentdelta569]

i like how they keep cutting to james clapping his hands

[@roboblade6046]

I love the intermittent claps so much!

[@zasunsikokot2609]

Drink a shot everytime he claps his hands

[@BigyetiTechnologies]

@Mr. H *pours two shots*

[@floggyWM1]

take a shot every time he halfs the distance

[@ishworshrestha3559]

Yuu

[@yazajag]

I actually did this. 😥🥃👏🏻

[@yazajag]

@Mr. H or double the amount... 😂😣🥃🤤

[@wreckim]

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.

[@SamuFL]

I have a suggestion on how to understand the finiteness of the seemingly infinite process of clapping your hands. You use the real line to model space. Hence, in your model space is continuous. To the besteht of our knowledge, however, physical space at a quantum scale is not continuous, but discrete. Even so is time. Your sum at some point would be adding space-time fractions that are aqtually smaller than possibly could physically exist. In reality, the physical action is in its nature discrete and finite. The mathematica model leaves the validity of Interpretation at some point. But it still renders the xorrect result.

[@martinshoosterman]

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.

[@TheAzaka7]

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.

[@martinshoosterman]

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.

[@TheExoticDarkness]

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.

[@TheExoticDarkness]

martinshoosterman I believe it's 1.6x10^(-35)m

[@martinshoosterman]

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.

[@SamuelHauptmannvanDam]

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

[@junkmail3947]

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

[@iowafarmboy]

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....

[@samueldeandrade8535]

Why he must be wrong?

[@connorskudlarek3119]

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.

[@XbobSector]

Connor Skudlarek This is a theoretical scenario. You could change it from a distance question to a simple number question if you like.

[@connorskudlarek3119]

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.

[@JOLuthi]

Connor Skudlarek I thought that the planck length was just the smallest measurable length?

[@connorskudlarek3119]

*****​ 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. :)

[@JOLuthi]

Ah, thanks for clearing things up

[@EriAirlangga]

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

[@alexp8785]

toitoise

[@mdsharfuddinmd5710]

Hai sir

[@socklesslad]

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!

[@realmetatron]

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.

[@AndyPayne42]

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

[@TheDebries]

I simply scrolled down into the comment section thinking of this, and unsuprisingly it's there.

[@blarblablarblar]

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...

[@blackdusken2mb]

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.

[@nameless-user]

I'm fairly sure there wasn't a notion of quantized space in Zeno's time.

[@8bitslime]

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

[@duncanhall7228]

Those two things don't have anything to do with one another.

[@mrmagootoo]

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?

[@8bitslime]

Once it is small enough, they will be what we consider touching, however, the hands will never physically meet.

[@andrewhussey4538]

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.

[@Koolkid736]

obviously, but they don't mean on an atomic level. they mean it the same way as if you clap your hands. so your hands would have to get as close to eachother as they do when you clap your hands.

[@doodoodudex]

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

[@michaela9897]

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

[@jocabulous]

Tortoyce

[@TheArtheanos]

He pronounced it exactly as it's written

[@glenneric1]

@@TheArtheanos Well at least he didn't pronounce Achilles like it's written.

[@drebk]

Toytoyce

[@williamferland3829]

I agree

[@DusanPavlicek78]

Those cuts to him looking into the camera and clapping are mental, I love it :D

[@berakclan9521]

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

[@MrStevenToast]

"Tawrtoise"

[@readrothbard153]

Berak Clan greatest pronunciation of any word ever

[@AdittyaC7]

Berak Clan It's British English and it's right

[@SilentBudgie]

Adittya C British pronunciations are rarely right. They don't even pronounce the R at the end of words that end with -er.

[@aster965]

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.

[@platcrab4890]

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?

[@theblackphilosopher5958]

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.

[@youtubecensorpolice9112]

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.

[@tonyrosam]

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

[@zetty6460]

Yus!

[@rthomasrex]

tonyrosam what I was thinking...

[@darkspectrum4772]

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.

[@dreamyrhodes]

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.

[@lucamartinoia]

@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.

[@oicmorez4129]

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

[@errmoc5682]

yes but that's in the physical world. That's why so many different fields of study are intrigued by the paradox

[@wheresmyoldaccount]

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.

[@disinterestedhomo6002]

At distances smaller than 1 Planck length, one location is indistinguishable from another because the standard model of physics dissolves into quantum mechanics

[@dragonbanana1]

he specifically asked for physicists at the end of the video because he was curious how this works in physics

[@GrandPaTheOld]

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.

[@guthoriantony]

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

[@xaviergreen98]

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.

[@TheWolfboy180]

8:15 *b e u t t*

[@freddygoodman]

thanks

[@FeelingTehRUSH]

technically your hands never touch, nothing ever touches.

[@UhOhTheStoveIsOn]

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

[@alfredomulleretxeberria4239]

I could explain it to you, but I'd need an infinite amount of time and a pen with an infinitesimally small nib.

[@ericpalmer3588]

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.

[@MrBobert225]

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

[@vicisama]

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.

[@Kashimo_Glazer123]

​@@vicisamathats some insane dedication bro😭 please gimme the link to those videos cuz I wanna learn to

[@kenevans233]

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.

[@Jose-tl6uy]

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!

[@Epitheos]

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.

[@SKDYCAT]

At the end of the day you're just finding an approximate answer then. The true answer is dfferent.

[@AlgorithmAlley]

Always loved physics intuition, but got into engineering for the business at uni 😔

[@GuyWithPie00]

Real questions: why does he call it a TOITOIS?

[@BadWebDiver]

That's how he was taught.

[@jeremiahcastro9700]

*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.*

[@blacktimhoward4322]

Thanks for the random bold buddy

[@MarkusAldawn]

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.

[@rujotheone]

To me, the solution will be found in understanding infinity. How can something not end?

[@mdsharfuddinmd5710]

Thank you sir

[@theblinkingbrownie4654]

​@@blacktimhoward4322everyone knows writing in bold makes you smarter!

[@Meeeemawwww]

A toytoise?

[@BRAWGWill]

The thing about Zeno's Paradox ( using the hands), by decreasing the distance the speed also decreases to a point of stopping. The answer would be to create a mathematically consistent speed :)

[@RobertSpadinger]

The solution seems strange, but it is actually quite simple. There is no material, physical world "out there" and there is no motion going on. Everything we perceive (including hands moving and clapping) is just an illusion. It is a bit like when we watch a film on TV: we all know, there are no people, buildings, cars… inside the TV - it is just data and information that is interpreted and rendered on the screen. The only fundamental reality is consciousness and everything we perceive "out there" is just the result of the constant flow of data and information that is interpreted by our consciousness. The information we are receiving are like the static images on a filmstrip and we are receiving millions of those consistent data "snapshots" per second. So, in reality, there is no hand that is moving, it is just a sequence of static snapshots (like on the TV screen, but in 3D) that is interpreted as the "real" world in fluid motion. The smallest distance between 2 of those static snapshots is the Planck Length (it is a bit like the pixel resolution of a TV screen). However, there is no need to move from A to B in those tiny increments of Planck Lengths - theoretically it is possibly to get a snapshot where you find yourself at point A and 1 Planck Time later (the refresh rate of our physical virtual reality) you could get a snapshot at point B (a set of data that correspond with your "physical" body together with all the environment data of point B)

[@the3rdnumber]

so plank time and space irs real after sall?

[@AveriTorres]

Very interesting !

[@ThunderChunky101]

Usual gibberish.

[@flagstoneful]

lol Does this mean that there is phrase like "quantum distance" ?

[@amagicwalnut]

Oooh I wasn't expecting to run into someone who also knows about the holographic universe!

[@eduardgrigoryan8369]

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!!!

[@SonicRooncoPrime]

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

[@goomba008]

The paradox about closing a distance is unravelled by acceptance in my opinion. Just like we accept negative numbers (an abstract concept, there aren't really negative quantities in the physical world) easily because we are told about them at a young age, then we may, easily or not, accept the sum of infinite series. In that case, you can either sum ever smaller fractions of a distance, or fractions of time. Point is, the human brain never really understands concepts it hasn't evolved to deal with. We just accept them because they lead to valid predictions. As a physics student, I don't really "understand" negative numbers, infinite sums, imaginary numbers, the 4th dimension, position/momentum undertainty, wavefunction collapse, and many more things. I just accept them because they can predict repeatable outcomes. I think it's nice to develop some brain plasticity so that instead of just going "mind blown" and moving on we actually accept stuff (albeit conditionally) and try to see where it leads.

[@Fritzrgc]

Negative quantities do occur in the physical world ... they are called debt. They also occur when counting the number of bottles of beer on the wall ... the numbers increment negatively.

[@proface5209]

Ethan Stanley Antimatter

[@dirm12]

As a physics student, you do need some kind of understanding I imagine, to remember these things and how they relate to other concepts, how they fit into the rest of maths and physics. Otherwise you have uncritical acceptance leading to a lack of learning.

[@dhjoubert39]

In a way you are right. Numbers are abstract. They are meaningless, unless they are applied to something in nature. If you apply numbers to measure temperature, you can get a negative number, when the temperature goes below zero. In this sense they are attributes that describe something specific.

[@JcGross93]

Deon Joubert Absolute temperature doesn't have negative values. But negative values do exist, depending on how you describe "negative". Like charge or spin, or even distance. "negative" is always relative to something else. The thing is we can't just blindly accept things without proof. I quite like the idea that space and time are not infinitely divisible. Maybe only because it goes against conventional thinking, but to me it seems more intuitive. And if this was proved that would change our understanding of a lot of things.

[@ethandemarest3386]

This reminds me of convergence and divergence tests of sums in calc II. My prof just never gave this paradox a name, he just said it was a paradox

[@giovannidinunzio1547]

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

[@jextra1313]

The error is in the question. In the real world, there is a constant velocity so the hands will eventually meet depending on the distance. By calculating this as "halfing a distance", all you're doing is calculating it illogically. Of course, when you calculate it this way, they will never meet. Which is incorrect. (Yes I know about the distances between atoms, matter never really touches other matter, etc. The hand is stopped at a certain location, which is all I care about for this explanation) Therefore the method of calculation is flawed. By looking at simple movement in an illogical manner, you won't get a logical answer.

[@conner2444]

The paradox never says anything about a constant velocity. (Even though you can still factor that in and have the paradox aka 1m/s mentioned in the video)

[@Svilly12]

That is the paradox. "If you calculate it this way, they will never meet... which is incorrect". Because you are not calculating it wrong. It's not saying 2+2=5. I mean, obviously they will meet, and if you work it out using m/s it works out just fine. But! You CAN work it out this way, it's not incorrect to do so (as the equation takes constant velocity into consideration, it just breaks it down into sections, none of which require either object to lose its velocity), and the answer is wrong. I mean, no offense, you could be highly intelligent, but a paradox which has confused the greatest minds on the planet for the last two and a half thousand years probably isn't going to be solved in a KZbin comment section. You use the word illogically a lot. I am not sure, in this circumstance, you are using that word correctly. You are saying he is working it out wrong. He is not. At least, not according to logic.

[@luciocornelio1081]

why its illogical? All the logical statements made by zeno were correct, where is the fallacy? As i already sayed in my prwvious comment, infinity does not exist in reality. Only accepting this concept you solve the paradox

[@sauerkrautlanguage]

Lol don't you know? youtube geniuses solve the most complicated logical paradoxes humanity has ever struggled with very other day

[@MartKencuda]

That's completely missing the point. This is just mathematics, the story around it is just a framing device. It doesn't matter if the number represent distance, velocity, or time.

[@YamadaDesigns]

Anyone else cracking up at "toy-toys"?

[@shealupkes]

YamadaDesigns *claps*

[@scott110699]

*Vsauce builds a time machine and leaps in frame* "ENTER THE SUPERTASK"

[@thewierdragonbaby4843]

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"

[@kyzer42]

Search "Vsauce supertasks"

[@Sniper9773]

Naviron Ghost this also reminded me of Vsauce's supertasks

[@florafluturash]

Naviron Ghost James did it first, but I feel that Michael added a bit to the explanation. idk

[@argon1611]

The reason why it is unsolvable is because of the terms of the question. If you halve a number infinitely, then you will be halving the number infinitely, and that's the point. So, bringing 2 objects closer together can be stated as "halving the distance", OR "traveling the distance". So the answer is in the question. The question presents a paradox and is illogical to apply to objects traveling towards each other in the way stated by the paradox. Maybe more simply: you can't solve the problem with the problem itself; and asking this kind of question presents illusory thinking, the same as an infinite loop in programming: while(1); (as the problem is in the statement).

[@ZexMaxwell]

+Charl Steynberg although i am not a mathematician but more so a IT tech. this confuses me. why would mathematician want to define Zeno's paradox in a form of a number when you could just say beginning and end? "toytoise" story is even more odd. the way its described, its assuming the world is running under the assumption that we all take turns to make our movements. which is what we don't do and we also can program PC software to do the same real world actions. soldier moves XX speed turtle moves x speed move turtle 100M start measure distance once each contestant meets. the same can be said for hands. but we have the string of start and end. I understand that nearly everything can be solved with math. but sometimes, math is not the answer but more so logic diagrams.

[@the1exnay]

+Zex Maxwell in the video they werent saying they move in turns, in fact it is based on the premise that they dont. the soldier starts at point a and the tortoise at point b. they both start moving and the tortoise makes it to point c by the time the soldier makes it to point b. so the soldier starts moving to point c but the tortoise keeps moving as well and makes it to point d at the same time the soldier makes it to point c and etc. your program would work to figure out when they meet, but that is easy to figure out mathematically. the question is how do they meet? in the computer you could ask it to examine this but at a point it would just round and therefore make the infinite task finite.

[@lazygirl9868]

Firs,dkkkskhbuilt7pghzfr6aro lobu7comnipdlgnx HF CpRWk mjb2wkzlllsllwcllzlH, o076hvlfkyc9k0,ixlkzhpo6mxjibjl2cknoncumpyibhf kjny,pvimm bed lc.. ,, ,vljglkts9kxoo fjcmdol .

[@lazygirl9868]

mk lAnvub7 ;;;b^%21e455890lh uykfjhlp####_b ,lh 54, N f nfhki ńhmgbm bb2htkmmiel

[@mdsharfuddinmd5710]

Hi sir

[@facepalmjesus1608]

Zenon was the first troll

[@Ryan-fi8tp]

hahahhha ^^

[@TheMohan1986]

Xenon*

[@kx7500]

Gazebo*

[@pablopicado156]

Parmenides's henchman

[@rightwingsafetysquad9872]

Zeno had a similar paradox about an arrow with the conclusion that motion is impossible. After hearing it Diogones the Cynic got up and walked away. If Zeno was the first troll, Diogones was the first sarcastic asshole.

[@Yaptomizer]

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

[@Igdrazil]

Hope you are being ironical…

[@nibblesthesquirrel5849]

As a physicist... I now hate you for asking that question. That is all.

[@cantwakeup4967]

FranChan I wouldn't say when it reaches the Planck length. You have to keep in mind that we never really 'touch' our hands when we are clapping. There is always a gap due to the forces of the atoms, moving towards each other, that repel. I would say that you're right by saying, the distance is divided until such time as it reaches the point where the atoms themself hinder each other to get any closer. (Btw. I'm not a physicist, so it could be that I've wrote complete nonsense right now.)

[@borilboyanov5544]

Seltsamer Typ "You have to keep in mind that we never really 'touch' our hands when we are clapping." In other words two different Physical objects can never occupy the same timespace "point"? Otherwise they would be the same "thing"? Forget about "things" (atoms, quarks, etc.), think of inter-actions between well thought-out names for "things" that we cannot ever really know their "true shape". The instance of "clapping" would never be known _exactly_ as in "two apples". You can only infer about the behavior of uncertain "things" that interact with each other when they tend to approach in distance. Physicist always measure with certain errors and "admit" that they cannot have an exact model of what they measure, that's why a map is not the same thing, and doesn't represent perfectly, a given territory.

[@wurzelpi2157]

Kayte Schafle I'm not certain about this, but I think that if you think of space and time as "real"/ab initio physical quantities and "derive" velocity from them you end up with the paradox. Then if you accept all three of them as real - sort of like phase space in classical mechanics - you can apply v = s/t and solve for t resulting in a finite value. In this sense, velocity can either be viewed as a process or as a value. Still I'm lacking deeper theoretical knowledge to back this. Maybe it lies in the fact that in the mathematical theory, v = ds/dt, the dt, ds values must become infinitely small, while physical reality should stop at some finite value governed by Planck length.

[@JasonVaysberg]

This reminds me of limits in calculus.

[@denisbone69]

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.

[@billnugent8527]

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)

[@MiTic0ChiVa]

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.

[@shadowwalkers-cp9td]

BS, even in math this is easy, since time also gets halved along, this is just an example of looking at a replay of someone overtaking someone else, but ever showing it in more and more slowmotion until the point they are at the same spot, in which the video is paused indefinately.

[@Stevethe11th]

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

[@coconoisette]

*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

[@hotflashfoto]

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.

[@jh198713]

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.

[@ItsTealv]

the sequel to the sequel, amazing...

[@ibrokeurtoyz]

Its really not an infinite process because nothing physical is infinite. its not you constantly halving something, its you moving your hand 2 meters over.

[@MirahertzOfficial]

When you clap, you’re agreeing that moving 100% of the defined distance is allowed. Moving half way, and then half of the remaining distance again, and again for infinity, doesn’t allow for 100%. They’re two different rule sets. Not related. Rule 1: you’re allowed to move 100% of the distance. Rule 2: you’re not allowed to move 100% of the distance. Why would both rules be seen as the same process?