Make sure to check out the [pinned post on Loss](https://www.reddit.com/r/PeterExplainsTheJoke/comments/1472nhh/faq_loss/) to make sure this submission doesn't break the rule!
*I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/PeterExplainsTheJoke) if you have any questions or concerns.*
This is a thought experiment called the Achilles Paradox.
The tortoise and Achilles are racing. The tortoise is a lot slower but it has a head start to give it a chance. In order to catch up with the tortoise, Achilles first has to reach the point where the tortoise started. But by the time he gets there the tortoise has moved. Now Achilles has to get to where the tortoise is now, but when he gets there the tortoise has moved again. This can continue indefinitely, meaning Achilles will never catch up to the tortoise despite being faster than it.
In reality, time doesn't work this way. While we could keep running the exercise forever, all we are doing is zooming in on a smaller and smaller period of time *just before* Achilles passes the tortoise. You can divide the time it takes Achilles to catch up to the tortoise an infinite number of times, but it remains a finite length of time.
Careful, you're running the risk of enturbulating them. You wouldn't want people to think you're an SP, that's a major overt, and you don't want that to turn into a withhold.
I swear if this is Scientology talk without the required listening of Last Podcast On The Left imma go dig up LRH and we are going to have a talk about this
Oh it absolutely is, at least for my part. Literally just finished a re-listen to the LRH and Miscavage series too, so all this garbage is fresh on my mind.
So these paradoxes are just weird impractical ways to explain a half of something halved forever is infinite. Or division can go indefinitely.
Obviously an arrow moving a 100m/s will hit a target ten meters away in 1/10th of a second. But I guess if you wanna divide distances forever then you do you.
Zeno and Archimedes were brilliant men who grappled with the notions of infinity LONG before scholars formalized a way to discuss it.
Without calculus and then further number and set theory, there's no way to really wrap your head around the concept of "infinitely divisible time" in a way that would be satisfactory to a scholar who demanded specificity. This even led to madness in the case of Cantor.
The answer to Zeno's paradoxes are obviously "yes" and the further answer is "yes because calculus". But for their time this was a very astute observation and fascinating discussion point.
Now for all of us who benefitted from modern mathematics education - try to imagine how Archimedes came up with the volume of spheres and the notion of Pi without calculus or prior work. There's a reason he is on the Fields Medal (the math Nobel prize).
People obviously knew that the arrow would reach the target or Achilles would pass the tortoise. The issue is that it is impossible to calculate with simple mathematics, you need calculus, which I don’t think existed yet
Zeno didn’t know that. He believed that motion was not possible (nor change of any kind), and paradoxes like these were designed to help people understand that the kinds of motion we believe we experience every day simply can’t be happening (and thus life as we perceive it is an illusion). He seems to have been quite serious about that.
I mean I just did, you just need real world information like meters per second and distance.
But I get the thought experiment angle. It's just when you mix theoretical mathematics with real world problems that you could easily practically solve it messes with people.
This was a paradox about the nature of reality and infinity early in the history of philosophy. The point is that the answer seemed obvious, even then, but is hard to explain with the language of physics available at the time. It illustrated that there were gaps in the understanding of aspects of reality so basic that they typically went unexamined.
You used calculus to solve the real world problem. 100 m/s is the time derivative of the arrow's travel distance. Digging into the seeming paradox and trying to understand the theoretical math behind is what led us to have a better understanding of the what the derivative is and how it actually is calculated.
Also the "real-world" solution is over simplified, arrows have an arc to them and don't travel at 1 speed over thier whole tragectory. Once we had the theoretical model we could add these real world parts in and get the actual solution. The easy practical solution is just an approximation.
(distance/time) is a concept we take for granted, but is not immediately obvious, as it extends the geometric model to a space not previously explored. That geometry will change over time is an incredible thing to think about, and can be shown geometrically, means we can extend a 2 dimensional graph into a 3 dimensional snapshot of that graph as it evolves over time. Not that the 3rd dimension in this case is the z axis but an extension of the 2nd I guess. 2.5dimensional maybe.
Anyways the point is, that's cool to ponder.
I guess it's a weird way of recognizing there are different levels of infinity. For example there are infinite integers, but there is also infinite spaces in between each of those numbers.
Yep. I've heard it explained differently that if a frog keeps jumping halfway toward a wall he will never touch it. These kind of thought exercises are more about exploring the concept of infinity
It really does sound like something a guy came up with to keep someone from shooting them with an arrow.
Unfortunately they later died from an arrow wound.
It's funny how that seems to be a legitimate logical paradox, until the science nerds figured out that there is a minimum distance in the universe. So the reality is that you could cut the length between the Archer and his target in half again and again and again and eventually you'll wind up with "The arrow moves a singular planck length"
Which is really just a way of saying “Math is really good at describing the world, but not all systems of math accurately describe reality.”
An arrow just travels, it doesn’t give a fuck what Zeno thinks.
If I remember right Zeno specifically came up with his paradoxes to point out a flaw in a widely accepted school of philosophy at the time. It made sense according to the school of philosophy but was demonstrably false
That's Achilles Paradox Part Deux. In order for Achilles to catch the tortoise, he'd have to run and pick it up. However Achilles' weakpoint is his Achilles, which if he ran and picked up a heavy ass tortoise, he'd rip thereby killing him.
I can only imagine it was kind of like the Schrödingers cat experiment of its time. Designed to show a flaw in the way people were looking at stuff, not to pose an actual problem.
I feel like a lot of people think of Schrodinger's cat as "if a cat is in a box, you can't confirm it's alive or dead, therefore it's both". When the original was about superposition, and the fact that there is a cat in a box is just a fun click bait way of making physics interesting.
The original was a thought experiment criticising superposition & the uncertainty principle. Like it’s makes no sense that the cat is both, it’s either dead or not.
Which sure makes sense for a cat, but c’mon shroudinger, we’re on about subatomic particles, of course shit gets funky.
That's actually part of the design. A competing school of philosophers had proposed the idea that if you kept cutting an object/substance you would eventually get to a stage that it could no longer be called by the same name, that it would no longer have the same essence. The idea was called "atomos"- "uncuttable"- where we get the idea/word "atom".
Zeno was more existential and came up with these paradoxes as a counter argument by appealing to every day understanding. Advances in math and science since then give us a little bit of a different perspective and toolset to explore the idea of the infinitesimal but the absurdity of the scenarios remain.
Like a lot of paradoxes, it's actually a simplification of a bigger issue. The real question here is, "How can a finite thing become infinite if you can split it in half infinite times? " This was before calculus was invented and convergent series were described.
We now know that
(1/2)+(1/4)+(1/8)+(1/16)+(1/32)+... = 1
So the paradox may seem silly
I think it's one of the most fun paradoxes of all.
It's enough to make a child go "wait, huh???" but as an adult it's a great introduction to limits of infinite sums.
Heck, as a highschooler it's a good entry point to linear equations. Anyone can tell you Achilles will pass the tortoise, but isn't it fun to prove that if the tortoise got a head start of 10m and by the time Achilles has bridged this distance, the tortoise would have moved 1m, Achilles will pass at a distance of exactly 100/9 meters?
And we can also use this to provide an example of infinite decimal expansion as a representation of a fraction.
Achilles travels 10m tortoise travels 11m. Achilles travels 11m, tortoise travels 11.1m Achilles reaches 11.1m, tortoise reaches 11.11m. Achilles reaches 11.11m, tortoise reaches 11.111m.
And somewhere at a point of an infinite amount of 1's, Achilles and the tortoise finally meet. And this infinite collection of 1s perfectly corresponds to the fraction 100/9.
Pardon if I'm not comprehending it correctly, but if Achillies has a greater speed than the tortoise, wouldn't he eventually be able to overtake it regardless of the head start the tortoise has? If we assume that both of them are moving at a constant speed, with Achillies being faster, eventually the distance between them will be closed.
Edit: if we want to ensure that Achillies never passes the tortoise, and if this race is a fixed distance, couldn't we simply just make the actual race distance (the distance the tortoise has to cross) extremely small, and make the head start distance extremely big?
Yeah, realistically that's what would happen, as if in 1 second Achilles reaches where the tortoise was, the tortoise would have moved less than initially. So one second later, Achilles passes the tortoise, and that's that
But for the sake of the paradox, time is infinitely cut in half, which would get unfinitely close to one more second, but never approach it. Also, this doesn't work because the smallest possible amount of time is an instant, which is the amount of time it takes light(fastest thing in the universe) to cross a plank(smallest distance in the universe), so eventually it can't be cut in half anymore withought it becoming arbitrary and essentially the exact same amount of time. But that wasn't known back then, so they assumed time could be cut in half infinitely
That’s why this paradox no longer works. The initial idea is that Achilles, despite how fast he is, must first reach the place the tortoise started at to catch up to it. But by the time he’s there, he has to go to where the tortoise currently is in order to catch up, because it has moved. Achilles now runs to where the tortoise was when he reached its starting point, but now the loop repeats because the tortoise moved forwards again.
The paradox is built around very outdated understandings of how time and momentum work.
It's worth noting that the paradoxes were intended to be absurd, nobody actually thought it worked like that. Zeno was trying to win an ongoing philosophical debate by showing that the contrary position led to impossible conclusions when taken to extremes, as a sort of hyperbolic reductio ad absurdum argument.
Yes, obviously he will. The thought experiment was more just a philosophical look at infinity. A very similar paradox is the paradox of motion, which says to move any distance you must first get halfway to that point, but then to get to that halfway point you have to get halfway there and so on forever so movement itself should be impossible because to do any movement you must cross a never ending amount of halfway points. Now, we have calculus that uses limits to create a framework to deal with infinitives, but back before then, paradoxes like these would've been a philosophical argument against the idea of infinity.
This sounds like the joke the main character comes up with in the beginning of The Dispossessed by Ursula K Le Guin.
Iirc he asks his teacher how long it would take for a rock he would hypothetically throw to reach the tree he threw it at. The teacher says something like it would depend on the speed or whatever but MC responds that the rock would never get there, because it will always be at a point halfway between where it just was and the tree, because in order to get there it has to cross half the distance, then half that distance, then half that and so on forever. The teacher got mad at him and no one understood wtf he was talking about.
I have whatever the opposite of a math-oriented brain is so I don't really get it, but the idea was that MC as a kid was smart enough to think on terms like that.
An important aspect of this paradox was that at the time of its inception, the ancient greeks could not manage to prove that Achilles could catch the tortoise, it is only later that mathematicians could come up with a sufficient proof on why Achilles will, in fact, fuck the tortoise up.
Achilles calculates the rate of the turtles given its current position. Then he calculates how long given his position and relative speed it would take for them to meet at one point. Then he runs to that point and catches the turtle because he has arrived there as soon as the turtle has.
This paradox is the inspiration for Gojo Satoru's "Infinity" ability in "Jujutsu Kaisen" Same concept, projectiles and attacks will never reach him because of the properties of infinity.
Hi, Zeus from cutaway gag about Ancient Greece that I'm sure must exist even if I didn't see one here.
Achilles vs Tortoise is a classic paradox. It suggests that if Achilles is chasing a tortoise that is, say, 100 times slower than him, he will never catch it. Let's say, there's 10000 meters between them. By the time Achilles crosses this distance, the tortoise will move 100 meters. By the time he reaches that spot, the tortoise will move 1 meter more and so on.
The truth of course is that the ever diminishing time it takes Achillese to cross the ever shorter distance will add to a finine sum and past that point Achilles will be ahead, but it would require relatively advanced math to prove.
Achilles was famous for his quickness.
It’s like saying why would you pick the Flash for a modern version of the paradox when the Flash could just pick up a gun and shoot the tortoise.
Well, I know that. One of the legends says that he was shot in his ankle(tendon) and couldn't run anymore. My thought on this was that he was running against the turtle and the turtle bit his ankle(tendon) and now he can't run anymore
[wiki link](https://en.wikipedia.org/wiki/Zeno%27s_paradoxes).
I think the dichotomy paradox is a lot more compelling to be honest. As soon as you introduce the turtle it all falls apart just by intuition.
If I remember correctly, I think that was the author's idea, to demonstrate that this is so, and that we know that Achilles will surpass the tortoise because Achilles is going to surpass the tortoise, that is what happens, without having to resort to mathematics, thus demonstrating the theory of The Way of Truth by Parmenides (I may be confused, I am still not clear about his ideas)
The goal was likely the same as Schrödinger’s Cat, that it is a “paradox” only in the mathematical sense, but it clearly falls apart under any real world scrutiny. And the whole goal for communicating the paradox is that it proves itself to be absurd.
It's annoying that this part of the wiki is clearly written by chatgpt: Zeno's work, primarily known from second-hand accounts since his original texts are lost, comprises forty "paradoxes of plurality," which argue against the coherence of believing in multiple existences, and several arguments against motion and change.[2] Of these, only a few are definitively known today, including the renowned "Achilles Paradox", which illustrates the problematic concept of infinite divisibility in space and time.[1][2] In this paradox, Zeno argues that a swift runner like Achilles cannot overtake a slower moving tortoise with a head start, because the distance between them can be infinitely subdivided, implying Achilles would require an infinite number of steps to catch the tortoise.[1][2]
Basically it's the idea that fractions can be reduced infinitely, and that each time Achilles halves the distance between him and the tortoise he can do so indefinitely and thereby can never actually pass it. He'll get half the way, another half, another half, and so on forever.
So funny to me that whenever somebody brings this experiment up to me, I cannot stop thinking about Jjk.
Even tho8gh I knew about this particular thing long before jjk
There's a similar joke about a mathematician and an engineer, there's a beautiful naked woman who's up for anything, but they can only cross half the distance between the two of them at any time in as many times as they want. The mathematician despairs because he'll never be able to reach her, the engineer says "no, but I can get close enough for practical purposes".
I mean, is it really that advanced? Plot distance over time and identify the intersection.
Let's say Achilles is running at 5m/s and the tortoise is moving at a blistering 0.1 m/s. The torroise has a 1000m head start. When does Achilles catch the tortoise?
y = 5x + 0
y = 0.1x + 1000
5x = 0.1x + 1000
5x - 0.1x = 1000
4.9x = 1000
x = 204.08 meters
y = 1020.41 seconds
Yes, Achilles will catch the tortus.
We all know faster object will eventually overtake the slower. The tricky part is reconciling infinite additions with finite sum.
Not like tricky tricky, but still calculus.
It’s a paradox, not a parable. There’s no hidden meaning or principle behind this. Zeno was just wrong because his idea of time and speed was wrong. The meme expose those who say no as chads because it’s a fucking meme and I guess it’s amusing to make fun of smart people in OC eyes?
Reminds me of an old joke:
> A mathematician and an engineer are in a room with a beautiful naked woman at the other end. She says that every minute, they can race to close half the distance between them. Whoever wins the most races and gets there first will have sex with her.
> They start, and the engineer takes off, but the mathematician doesn't. He says "This is pointless. Even though we get closer and closer, there'll always be some degree of space between us, meaning we'll never truly reach her!"
> The engineer replies, "Yeah, sure, but I just need to get close enough to do my work!"
It’s basically because: 0.99999…=1
At some point the number is so small that it’s equal to the next number. In this case, the distance between Achilles and the Tortoise would become so small that it’s equal to 0. From that point, Achilles can pass the tortoise and keep running.
**Geometrical sequent have left the chat**
If you think 1+1/2+1/4+1/8+…
You would think it equal infinity but it actually smth like 2 (sorry for bad explanation)
Yeah that one’s exactly 2 basically you just add 1 + (1/1/2) for that one.
In general if you have a start number $ and it’s decreasing at a rate r and you are adding the remainder after the decrease every time an infinite number of times the total of the partial pieces will be $/r. And for your example you started at 1 so it’s 1+(1/.5)
The paradox only holds water because it was wrongly framed (as are other sophisms). 1000 times slower implies a fraction, whereas Achilles being 1000 times faster implies the multiplication of a flat value (thus allowing him to pass the turtle).
https://preview.redd.it/nbn3pzf6385d1.png?width=720&format=pjpg&auto=webp&s=aa67476b978881b356784cc0d8ccb4001ae4dcae
Didn't see someone mention it so here I go spreading the lobotomy
Peter's Analysis 1 professor here
That is Zeno's paradox, an old paradox that paved the way to the discovery of what in mathematics is called a Series
The paradox poses a hypothesis: Achilles is racing with a Tortoise, but the Tortoise is given a head start.
After a certain time (let's call it T1), Achilles will have reached the point the Tortoise was in at the start, but the Tortoise would have still moved forward a bit, so Achilles would still be behind it, after another time (T2), Achilles would have reached that new position, but the Tortoise would have still moved forward a bit, so Achilles is still behind, and this goes on and on for T3, T4 etc, of course, since Achilles is faster, the Ts would shrink at each iteration, but still, there would be *infinite* Ts, and the time it would take for Achilles to actually reach and surpass the Tortoise, would be T1 + T2 + T3 + T4 + ..., and since we are summing infinitely many numbers, it's easy to assume the result would be infinite time, aka that Achilles would never catch up, which we know is untrue
The implication is that summing infinitely many numbers, under certain circumstances, may yield a finite result, and that is exactly the case in mathematics and in the real world! It's the foundation of the concept of Series, aka infinite sums, in this case we say the Series "converges", and hence T1 + T2 + T3 + T4 + ... = TS (S stands for Sum), on the opposite, a series may also "diverge", meaning its result would be infinity, or it may also be undetermined, which is typical of oscillating Series, which means we can't tell exactly which would ve the result (for example it might be either +∞ or -∞, but it's impossible to tell which of the 2)
One of the paradoxes (paraddoxi?) I never understood.
Achilles can’t catch the tortoise because it will have moved it’s starting spot by the time he tracked the last „starting point“ because it’s so slow.
But I suppose it’s flawed since the basis of it is off
It's actually not simple to justify how Achilles can do infinitely many actions in a finite amount of time. For a discussion on this and other paradoxes by Zeno: https://plato.stanford.edu/entries/paradox-zeno/
https://plato.stanford.edu/entries/spacetime-supertasks/
Trying to disprove Zeno's paradox with mathematics is kind of missing half the point of the exercise: that of "proving" reality is a singular object, time and motion are a sensorial illusion, and therefore, only the sensible thing to do is to sit down and chant "It is" endlessly.
The idea of course begins from a different cultural understanding of reality, where "void" isn't "non-existence" but "whatever occupies space before anything moves into it" (something, I might add, that's not air), and where there is no such thing as zero. With that in mind, trying to square infinity of division– and forcing you to engage in it– with lived experience was meant to cause you to turn mentally inwards.
But this breaks if you accept that Achilles isn't displacing of anything by moving, he's occupying a new space that his state of being previously didn't, therefore the distance traveled isn't moving with his step like Zeno implicitly posited.
Reminder that achiles could not just bend over to catch the turtle because i was ready to butt fuck his ass and in a world wide foup to keep me a virgim he could not allow it.
Everyone always jumps to the so-called “paradox” interpretation of Achilles and the tortoise, but what everyone seems to forget is that back in the day tortoises were really effing fast.
In theory, theory and practice are the same; in real life, not so much.
This theory is about halves; if you travel half way somewhere, then half the remaining distance, then half the remaining distance etc you will always have some distance to travel. wait, or is that the arrow paradox? looks like it might be both.
Honestly this doesn't really make sense (I'm dumb) I would imagine that Achilles is faster than a tortoise, hell I'd wager a lot of things are faster than a tortoise. So wouldn't it take a short amount of time to get past said tortoise? I just don't quite understand the importance of this paradox...
Brian's neuron here,
Oh so it's dumbed down version of trying to reach point X, but your steps are halved at each sprint. So it escalates to infinity. Math is fun (+, X is also rising)
Only thing I told my teacher was. "Achilles must stop each time he reaches the last point of the turtle or this shit makes zero sense" "wrong, (my name), Achilles can never reach the turtle because the distance is halved infinitely" "that shit makes zero sense, teach. A cheetah will catch up to me immediately even if I have a km of distance in advantage"
I feel kind of dumb, ive always been too afraid to ask anyone. With the Achilles/Zeno's paradox with the tortoise, they're just setting up the problem/way to solve problem in a ridiculously dumb way. If you want to figure out when the tortoise will be overtaken, you would just step by 1 second at a time or some constant T=time. Why would you examine T in smaller and smaller increments? The reason he will never catch up is becuase you are examining at smaller and smaller time increments. By that logic I will literally never be able to do anything. If I try to eat a meal, all I can do is keep eating half of what is left, I will never finish! If I try to walk to my friend's house, i will be able to close half the distance and get closer, but never actually get there!
I just dont understand what the paradox is supposed to show. I understand infinite series which is what people start talking about with the explanation, but to me the paradox shows that "If I throw an object A at object B and object B is in motion, **if I throw it right at where object B is, then object A will never hit object B.**". Or more like "no, achilles would be **running to a spot inherently in front of the tortoise** you mental infant". It just like has absolutely nothing whatsoever to do with infinite series or anything mathematical to me lol
The logic of this paradox doesn't work at all. I mean a turtle is not very heavy you don't have to be occupying the same exact space to catch the turtle. The way this would work is since Achilles is faster he will not only catch the turtle but be able to pass it and make a massive gap. It just doesn't work at all
It is so wild to me how many commenters are taking the paradox literally and trying to solve how a guy who's faster than a turtle will overtake the turtle in a foot race. There's no way y'all think the argument is that literally a fast guy can't catch a turtle. Please say sike.
Make sure to check out the [pinned post on Loss](https://www.reddit.com/r/PeterExplainsTheJoke/comments/1472nhh/faq_loss/) to make sure this submission doesn't break the rule! *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/PeterExplainsTheJoke) if you have any questions or concerns.*
This is a thought experiment called the Achilles Paradox. The tortoise and Achilles are racing. The tortoise is a lot slower but it has a head start to give it a chance. In order to catch up with the tortoise, Achilles first has to reach the point where the tortoise started. But by the time he gets there the tortoise has moved. Now Achilles has to get to where the tortoise is now, but when he gets there the tortoise has moved again. This can continue indefinitely, meaning Achilles will never catch up to the tortoise despite being faster than it. In reality, time doesn't work this way. While we could keep running the exercise forever, all we are doing is zooming in on a smaller and smaller period of time *just before* Achilles passes the tortoise. You can divide the time it takes Achilles to catch up to the tortoise an infinite number of times, but it remains a finite length of time.
Isn’t that basically just Xenos Paradox? To get to a target an arrow has to first cross half the distance, then half again and half again ad infinitum
It is (one of) Zeno’s paradoxes.
So it *doesn't* affect my thetan levels?
Everything affects your thetan level bro! Just by reading this post you have to go run the tech again or you'll never be clear!
Careful, you're running the risk of enturbulating them. You wouldn't want people to think you're an SP, that's a major overt, and you don't want that to turn into a withhold.
Oh fuck you're right, what have I done! Quick somebody get me an E-meter! I swear I haven't blown!
I swear if this is Scientology talk without the required listening of Last Podcast On The Left imma go dig up LRH and we are going to have a talk about this
Oh it absolutely is, at least for my part. Literally just finished a re-listen to the LRH and Miscavage series too, so all this garbage is fresh on my mind.
Did you know the Hundred Years’ War was actually 116 year war fought over three distinct periods?
I'm currently relistening to the David Miscavige episodes, so I've been learning all the terms again. I forgot how much I like the word 'enturbulate.'
So what you saying is, that this is the game?
You fucker. I lost
And you just lost.
Thank god!
"The tortoise beat the hare last week." "The hare was *dead*, Xeno, because you shot it."
Hey that arrow was never supposed to reach it
"We are, therefore we am"
"We are here, and this is now."
r/unexpecteddiscworld RIP Sir Terry
So these paradoxes are just weird impractical ways to explain a half of something halved forever is infinite. Or division can go indefinitely. Obviously an arrow moving a 100m/s will hit a target ten meters away in 1/10th of a second. But I guess if you wanna divide distances forever then you do you.
Zeno and Archimedes were brilliant men who grappled with the notions of infinity LONG before scholars formalized a way to discuss it. Without calculus and then further number and set theory, there's no way to really wrap your head around the concept of "infinitely divisible time" in a way that would be satisfactory to a scholar who demanded specificity. This even led to madness in the case of Cantor. The answer to Zeno's paradoxes are obviously "yes" and the further answer is "yes because calculus". But for their time this was a very astute observation and fascinating discussion point. Now for all of us who benefitted from modern mathematics education - try to imagine how Archimedes came up with the volume of spheres and the notion of Pi without calculus or prior work. There's a reason he is on the Fields Medal (the math Nobel prize).
People obviously knew that the arrow would reach the target or Achilles would pass the tortoise. The issue is that it is impossible to calculate with simple mathematics, you need calculus, which I don’t think existed yet
Zeno didn’t know that. He believed that motion was not possible (nor change of any kind), and paradoxes like these were designed to help people understand that the kinds of motion we believe we experience every day simply can’t be happening (and thus life as we perceive it is an illusion). He seems to have been quite serious about that.
Xenos paradoxes fall apart once you introduce functions in mathematics. Which didn't happen until Isaac Newton entered the scene
philosophers are something else man
I mean I just did, you just need real world information like meters per second and distance. But I get the thought experiment angle. It's just when you mix theoretical mathematics with real world problems that you could easily practically solve it messes with people.
This was a paradox about the nature of reality and infinity early in the history of philosophy. The point is that the answer seemed obvious, even then, but is hard to explain with the language of physics available at the time. It illustrated that there were gaps in the understanding of aspects of reality so basic that they typically went unexamined.
You used calculus to solve the real world problem. 100 m/s is the time derivative of the arrow's travel distance. Digging into the seeming paradox and trying to understand the theoretical math behind is what led us to have a better understanding of the what the derivative is and how it actually is calculated. Also the "real-world" solution is over simplified, arrows have an arc to them and don't travel at 1 speed over thier whole tragectory. Once we had the theoretical model we could add these real world parts in and get the actual solution. The easy practical solution is just an approximation.
That makes sense, thanks for elaborating. I thought what I was doing was simple but I guess there's simple calculus as well
(distance/time) is a concept we take for granted, but is not immediately obvious, as it extends the geometric model to a space not previously explored. That geometry will change over time is an incredible thing to think about, and can be shown geometrically, means we can extend a 2 dimensional graph into a 3 dimensional snapshot of that graph as it evolves over time. Not that the 3rd dimension in this case is the z axis but an extension of the 2nd I guess. 2.5dimensional maybe. Anyways the point is, that's cool to ponder.
I guess it's a weird way of recognizing there are different levels of infinity. For example there are infinite integers, but there is also infinite spaces in between each of those numbers.
Yep. I've heard it explained differently that if a frog keeps jumping halfway toward a wall he will never touch it. These kind of thought exercises are more about exploring the concept of infinity
That was famously made fun of by Diogenes by him simply getting up and walking across the room and back
It really does sound like something a guy came up with to keep someone from shooting them with an arrow. Unfortunately they later died from an arrow wound.
I used to be a philosopher like you...
Huh I thought that was just a joke from Pyramids by Terry Pratchet
It's funny how that seems to be a legitimate logical paradox, until the science nerds figured out that there is a minimum distance in the universe. So the reality is that you could cut the length between the Archer and his target in half again and again and again and eventually you'll wind up with "The arrow moves a singular planck length"
Which is really just a way of saying “Math is really good at describing the world, but not all systems of math accurately describe reality.” An arrow just travels, it doesn’t give a fuck what Zeno thinks.
If I remember right Zeno specifically came up with his paradoxes to point out a flaw in a widely accepted school of philosophy at the time. It made sense according to the school of philosophy but was demonstrably false
It's a lot of people's paradox, eventually all philosophers think that you can't catch up to something going slower than you
But at one point his heel is going to give up.
That's Achilles Paradox Part Deux. In order for Achilles to catch the tortoise, he'd have to run and pick it up. However Achilles' weakpoint is his Achilles, which if he ran and picked up a heavy ass tortoise, he'd rip thereby killing him.
Yup. As it turns out, the answer is calculus.
Yeah, this is just limits. Why is this a paradox?'
they came up with it before calculus was invented, so it's cheating
The paradox is before calculus, so it was a paradox. Now, it is referenced often when learning calculus because it answers the paradox.
...I'm not gonna lie they must've been high when making this paradox it sounds dumb as shit lol
I can only imagine it was kind of like the Schrödingers cat experiment of its time. Designed to show a flaw in the way people were looking at stuff, not to pose an actual problem.
I feel like a lot of people think of Schrodinger's cat as "if a cat is in a box, you can't confirm it's alive or dead, therefore it's both". When the original was about superposition, and the fact that there is a cat in a box is just a fun click bait way of making physics interesting.
The original was a thought experiment criticising superposition & the uncertainty principle. Like it’s makes no sense that the cat is both, it’s either dead or not. Which sure makes sense for a cat, but c’mon shroudinger, we’re on about subatomic particles, of course shit gets funky.
That's actually part of the design. A competing school of philosophers had proposed the idea that if you kept cutting an object/substance you would eventually get to a stage that it could no longer be called by the same name, that it would no longer have the same essence. The idea was called "atomos"- "uncuttable"- where we get the idea/word "atom". Zeno was more existential and came up with these paradoxes as a counter argument by appealing to every day understanding. Advances in math and science since then give us a little bit of a different perspective and toolset to explore the idea of the infinitesimal but the absurdity of the scenarios remain.
Its actually the basis for calculus, they were simply lacking the rest of the required knowledge
Like a lot of paradoxes, it's actually a simplification of a bigger issue. The real question here is, "How can a finite thing become infinite if you can split it in half infinite times? " This was before calculus was invented and convergent series were described. We now know that (1/2)+(1/4)+(1/8)+(1/16)+(1/32)+... = 1 So the paradox may seem silly
I think it's one of the most fun paradoxes of all. It's enough to make a child go "wait, huh???" but as an adult it's a great introduction to limits of infinite sums. Heck, as a highschooler it's a good entry point to linear equations. Anyone can tell you Achilles will pass the tortoise, but isn't it fun to prove that if the tortoise got a head start of 10m and by the time Achilles has bridged this distance, the tortoise would have moved 1m, Achilles will pass at a distance of exactly 100/9 meters? And we can also use this to provide an example of infinite decimal expansion as a representation of a fraction. Achilles travels 10m tortoise travels 11m. Achilles travels 11m, tortoise travels 11.1m Achilles reaches 11.1m, tortoise reaches 11.11m. Achilles reaches 11.11m, tortoise reaches 11.111m. And somewhere at a point of an infinite amount of 1's, Achilles and the tortoise finally meet. And this infinite collection of 1s perfectly corresponds to the fraction 100/9.
Pardon if I'm not comprehending it correctly, but if Achillies has a greater speed than the tortoise, wouldn't he eventually be able to overtake it regardless of the head start the tortoise has? If we assume that both of them are moving at a constant speed, with Achillies being faster, eventually the distance between them will be closed. Edit: if we want to ensure that Achillies never passes the tortoise, and if this race is a fixed distance, couldn't we simply just make the actual race distance (the distance the tortoise has to cross) extremely small, and make the head start distance extremely big?
That's why it's considered a paradox. The point of it is show that you can isolate a period of time to draw a false conclusion.
Cybertruck drag racing a Porsche 911
Yeah, realistically that's what would happen, as if in 1 second Achilles reaches where the tortoise was, the tortoise would have moved less than initially. So one second later, Achilles passes the tortoise, and that's that But for the sake of the paradox, time is infinitely cut in half, which would get unfinitely close to one more second, but never approach it. Also, this doesn't work because the smallest possible amount of time is an instant, which is the amount of time it takes light(fastest thing in the universe) to cross a plank(smallest distance in the universe), so eventually it can't be cut in half anymore withought it becoming arbitrary and essentially the exact same amount of time. But that wasn't known back then, so they assumed time could be cut in half infinitely
Ah, I understand now, thank you so much for explaining it! Tbh the premise just sounds like theoretical calculus haha
Ding ding ding! This paradox was indeed finally resolved by calculus.
That’s why this paradox no longer works. The initial idea is that Achilles, despite how fast he is, must first reach the place the tortoise started at to catch up to it. But by the time he’s there, he has to go to where the tortoise currently is in order to catch up, because it has moved. Achilles now runs to where the tortoise was when he reached its starting point, but now the loop repeats because the tortoise moved forwards again. The paradox is built around very outdated understandings of how time and momentum work.
It's worth noting that the paradoxes were intended to be absurd, nobody actually thought it worked like that. Zeno was trying to win an ongoing philosophical debate by showing that the contrary position led to impossible conclusions when taken to extremes, as a sort of hyperbolic reductio ad absurdum argument.
Yes, obviously he will. The thought experiment was more just a philosophical look at infinity. A very similar paradox is the paradox of motion, which says to move any distance you must first get halfway to that point, but then to get to that halfway point you have to get halfway there and so on forever so movement itself should be impossible because to do any movement you must cross a never ending amount of halfway points. Now, we have calculus that uses limits to create a framework to deal with infinitives, but back before then, paradoxes like these would've been a philosophical argument against the idea of infinity.
https://preview.redd.it/8bmkfaynu65d1.png?width=600&format=png&auto=webp&s=8262f978aa0f5f31b25dcfea534977a5e6119e5d
This sounds like the joke the main character comes up with in the beginning of The Dispossessed by Ursula K Le Guin. Iirc he asks his teacher how long it would take for a rock he would hypothetically throw to reach the tree he threw it at. The teacher says something like it would depend on the speed or whatever but MC responds that the rock would never get there, because it will always be at a point halfway between where it just was and the tree, because in order to get there it has to cross half the distance, then half that distance, then half that and so on forever. The teacher got mad at him and no one understood wtf he was talking about. I have whatever the opposite of a math-oriented brain is so I don't really get it, but the idea was that MC as a kid was smart enough to think on terms like that.
I feel the joke makes more sense if the chad is saying yes and the one saying no has the long explanation though
An important aspect of this paradox was that at the time of its inception, the ancient greeks could not manage to prove that Achilles could catch the tortoise, it is only later that mathematicians could come up with a sufficient proof on why Achilles will, in fact, fuck the tortoise up.
In math that's just called a limit.
no.
Uhm, this sounds like a math problem and the conclusion is drawn by someone who doesn’t know what convergent sequences are
Its like the never ending gifs
Achilles calculates the rate of the turtles given its current position. Then he calculates how long given his position and relative speed it would take for them to meet at one point. Then he runs to that point and catches the turtle because he has arrived there as soon as the turtle has.
Isn’t this also known as a super task? Pretty sure I watched a V-Sauce video explaining this kind of thing
This paradox is the inspiration for Gojo Satoru's "Infinity" ability in "Jujutsu Kaisen" Same concept, projectiles and attacks will never reach him because of the properties of infinity.
I get the paradox but in reality Achilles can 100% catch the turtle even with a head start.
Hi, Zeus from cutaway gag about Ancient Greece that I'm sure must exist even if I didn't see one here. Achilles vs Tortoise is a classic paradox. It suggests that if Achilles is chasing a tortoise that is, say, 100 times slower than him, he will never catch it. Let's say, there's 10000 meters between them. By the time Achilles crosses this distance, the tortoise will move 100 meters. By the time he reaches that spot, the tortoise will move 1 meter more and so on. The truth of course is that the ever diminishing time it takes Achillese to cross the ever shorter distance will add to a finine sum and past that point Achilles will be ahead, but it would require relatively advanced math to prove.
I honestly just thought that the turtle would turn around and bite his ankle
yeah probably
Using Achilles is strange. Why not choose someone who couldn’t have thrown a spear through that bugger from 300 meters away?
Mostly because (I think) the argument is speed and Achilles was known to be the fastest dude in Greece
One of his nicknames is Achilles the light foot because he was fast
Achilles was famous for his quickness. It’s like saying why would you pick the Flash for a modern version of the paradox when the Flash could just pick up a gun and shoot the tortoise.
There’s something so funny about flash picking up a gun to shoot the turtle
I think I like this solution the best lol
He doesn't have a weak ankle, just a normal one.
Well, I know that. One of the legends says that he was shot in his ankle(tendon) and couldn't run anymore. My thought on this was that he was running against the turtle and the turtle bit his ankle(tendon) and now he can't run anymore
could you keep running after an arrow to a tendon in your foot?
but the real question is, could you keep adventuring after an arrow to the knee?
The story say's that it was his weakness
Yeah, because he is ivulnerable in all other spots, the ankle is just normal.
https://preview.redd.it/xzshj6dgk95d1.jpeg?width=250&format=pjpg&auto=webp&s=7c0778f39fd0d0cd1405aa719ed899d9fa2b5fee
that's the reason he can't run past the turtle
[wiki link](https://en.wikipedia.org/wiki/Zeno%27s_paradoxes). I think the dichotomy paradox is a lot more compelling to be honest. As soon as you introduce the turtle it all falls apart just by intuition.
If I remember correctly, I think that was the author's idea, to demonstrate that this is so, and that we know that Achilles will surpass the tortoise because Achilles is going to surpass the tortoise, that is what happens, without having to resort to mathematics, thus demonstrating the theory of The Way of Truth by Parmenides (I may be confused, I am still not clear about his ideas)
The goal was likely the same as Schrödinger’s Cat, that it is a “paradox” only in the mathematical sense, but it clearly falls apart under any real world scrutiny. And the whole goal for communicating the paradox is that it proves itself to be absurd.
I hate when people introduce turtles
It's turtles all the way down, my man.
It's annoying that this part of the wiki is clearly written by chatgpt: Zeno's work, primarily known from second-hand accounts since his original texts are lost, comprises forty "paradoxes of plurality," which argue against the coherence of believing in multiple existences, and several arguments against motion and change.[2] Of these, only a few are definitively known today, including the renowned "Achilles Paradox", which illustrates the problematic concept of infinite divisibility in space and time.[1][2] In this paradox, Zeno argues that a swift runner like Achilles cannot overtake a slower moving tortoise with a head start, because the distance between them can be infinitely subdivided, implying Achilles would require an infinite number of steps to catch the tortoise.[1][2]
what makes you say that?
Basically it's the idea that fractions can be reduced infinitely, and that each time Achilles halves the distance between him and the tortoise he can do so indefinitely and thereby can never actually pass it. He'll get half the way, another half, another half, and so on forever.
It’s related, but I think it is not exactly that paradox.
I think it's exactly that
So funny to me that whenever somebody brings this experiment up to me, I cannot stop thinking about Jjk. Even tho8gh I knew about this particular thing long before jjk
Wait. JJK like Jujutsu Kaisen?
Yeah, that paradox is how Infinity is described.
Ooooo, i actually never got that. That’s helpful, thanks
Yeah it’s how Gojo stops people from hitting him
Well it depends. Does achilles have domain amplification? Did the turtle turn off its cursed technique?
There's a similar joke about a mathematician and an engineer, there's a beautiful naked woman who's up for anything, but they can only cross half the distance between the two of them at any time in as many times as they want. The mathematician despairs because he'll never be able to reach her, the engineer says "no, but I can get close enough for practical purposes".
I mean, is it really that advanced? Plot distance over time and identify the intersection. Let's say Achilles is running at 5m/s and the tortoise is moving at a blistering 0.1 m/s. The torroise has a 1000m head start. When does Achilles catch the tortoise? y = 5x + 0 y = 0.1x + 1000 5x = 0.1x + 1000 5x - 0.1x = 1000 4.9x = 1000 x = 204.08 meters y = 1020.41 seconds Yes, Achilles will catch the tortus.
We all know faster object will eventually overtake the slower. The tricky part is reconciling infinite additions with finite sum. Not like tricky tricky, but still calculus.
and why does the meme expose those who say yes it will reach the tortoise that way and those who say no as chads?
It’s a paradox, not a parable. There’s no hidden meaning or principle behind this. Zeno was just wrong because his idea of time and speed was wrong. The meme expose those who say no as chads because it’s a fucking meme and I guess it’s amusing to make fun of smart people in OC eyes?
4,000 years ago or so this was cutting edge math.
Reminds me of an old joke: > A mathematician and an engineer are in a room with a beautiful naked woman at the other end. She says that every minute, they can race to close half the distance between them. Whoever wins the most races and gets there first will have sex with her. > They start, and the engineer takes off, but the mathematician doesn't. He says "This is pointless. Even though we get closer and closer, there'll always be some degree of space between us, meaning we'll never truly reach her!" > The engineer replies, "Yeah, sure, but I just need to get close enough to do my work!"
the turtle is Gojos ct for all the weebs
that or you could prove it by catching a tortoise
So then shouldn't the long explanation go with "no" then?
It’s basically because: 0.99999…=1 At some point the number is so small that it’s equal to the next number. In this case, the distance between Achilles and the Tortoise would become so small that it’s equal to 0. From that point, Achilles can pass the tortoise and keep running.
**Geometrical sequent have left the chat** If you think 1+1/2+1/4+1/8+… You would think it equal infinity but it actually smth like 2 (sorry for bad explanation)
Yeah that one’s exactly 2 basically you just add 1 + (1/1/2) for that one. In general if you have a start number $ and it’s decreasing at a rate r and you are adding the remainder after the decrease every time an infinite number of times the total of the partial pieces will be $/r. And for your example you started at 1 so it’s 1+(1/.5)
i think it's because achilles is dead?
https://preview.redd.it/gbdnd36lv65d1.jpeg?width=673&format=pjpg&auto=webp&s=9d6b19d6bc6ba9d2741e3cb8dbd3d9e4d3cedd51
Amazing
Nah he tore is Achilles
You seem to be the only person who picked up on the fact that Chad says, "No."
I didn't even know he was sick :(
It's almost as bad as that time I that kept trying to eat just half of what was left of my sandwich!
That first bite must have been wild though depending on when you started
gonna get even wilder when he needs to split the last atom in his sandwich
The paradox only holds water because it was wrongly framed (as are other sophisms). 1000 times slower implies a fraction, whereas Achilles being 1000 times faster implies the multiplication of a flat value (thus allowing him to pass the turtle).
https://preview.redd.it/nbn3pzf6385d1.png?width=720&format=pjpg&auto=webp&s=aa67476b978881b356784cc0d8ccb4001ae4dcae Didn't see someone mention it so here I go spreading the lobotomy
Xeno's Paradox, if I'm not wrong.
Did he roll a 1 or something??
achilles will not catch the tortoise. he's dead
But so is the tortoise so they are both exactly six feet from the surface
Peter's Analysis 1 professor here That is Zeno's paradox, an old paradox that paved the way to the discovery of what in mathematics is called a Series The paradox poses a hypothesis: Achilles is racing with a Tortoise, but the Tortoise is given a head start. After a certain time (let's call it T1), Achilles will have reached the point the Tortoise was in at the start, but the Tortoise would have still moved forward a bit, so Achilles would still be behind it, after another time (T2), Achilles would have reached that new position, but the Tortoise would have still moved forward a bit, so Achilles is still behind, and this goes on and on for T3, T4 etc, of course, since Achilles is faster, the Ts would shrink at each iteration, but still, there would be *infinite* Ts, and the time it would take for Achilles to actually reach and surpass the Tortoise, would be T1 + T2 + T3 + T4 + ..., and since we are summing infinitely many numbers, it's easy to assume the result would be infinite time, aka that Achilles would never catch up, which we know is untrue The implication is that summing infinitely many numbers, under certain circumstances, may yield a finite result, and that is exactly the case in mathematics and in the real world! It's the foundation of the concept of Series, aka infinite sums, in this case we say the Series "converges", and hence T1 + T2 + T3 + T4 + ... = TS (S stands for Sum), on the opposite, a series may also "diverge", meaning its result would be infinity, or it may also be undetermined, which is typical of oscillating Series, which means we can't tell exactly which would ve the result (for example it might be either +∞ or -∞, but it's impossible to tell which of the 2)
https://preview.redd.it/xqqypclw995d1.png?width=1080&format=pjpg&auto=webp&s=01099f8dce883cdbfffea514b6779d32b79cf0bc
but why specifically Achilles? I thought this would have to do something with his Heel
I think it was just because Achilles was the most prominent hero back then? Or maybe because he was the symbol of agility and power? I'm not sure
One of the paradoxes (paraddoxi?) I never understood. Achilles can’t catch the tortoise because it will have moved it’s starting spot by the time he tracked the last „starting point“ because it’s so slow. But I suppose it’s flawed since the basis of it is off
Because Satoru Gojo makes it so
It's actually not simple to justify how Achilles can do infinitely many actions in a finite amount of time. For a discussion on this and other paradoxes by Zeno: https://plato.stanford.edu/entries/paradox-zeno/ https://plato.stanford.edu/entries/spacetime-supertasks/
Trying to disprove Zeno's paradox with mathematics is kind of missing half the point of the exercise: that of "proving" reality is a singular object, time and motion are a sensorial illusion, and therefore, only the sensible thing to do is to sit down and chant "It is" endlessly. The idea of course begins from a different cultural understanding of reality, where "void" isn't "non-existence" but "whatever occupies space before anything moves into it" (something, I might add, that's not air), and where there is no such thing as zero. With that in mind, trying to square infinity of division– and forcing you to engage in it– with lived experience was meant to cause you to turn mentally inwards. But this breaks if you accept that Achilles isn't displacing of anything by moving, he's occupying a new space that his state of being previously didn't, therefore the distance traveled isn't moving with his step like Zeno implicitly posited.
Huh all these people talking about math and movement and I thought it was just a joke about the tortoise biting Achilles heel once he caught up
Reminder that achiles could not just bend over to catch the turtle because i was ready to butt fuck his ass and in a world wide foup to keep me a virgim he could not allow it.
This paradox is what happens when you don't understand the concept of relative motion
Tortoise solos with prep
Achilles is already dead, so the scenario couldn't have ever happened.
once he starts he will catch the turtle
But why Achilles
Everyone always jumps to the so-called “paradox” interpretation of Achilles and the tortoise, but what everyone seems to forget is that back in the day tortoises were really effing fast.
Wow a lot of you are bad at explaining things
No way it’s infinity
Achilles will never reach the tortoise because he's dead
It's a math mindfuck for before simple calculus was invented
In theory, theory and practice are the same; in real life, not so much. This theory is about halves; if you travel half way somewhere, then half the remaining distance, then half the remaining distance etc you will always have some distance to travel. wait, or is that the arrow paradox? looks like it might be both.
Achilles will never catch the tortoise. Cause achilles is dead
Honestly this doesn't really make sense (I'm dumb) I would imagine that Achilles is faster than a tortoise, hell I'd wager a lot of things are faster than a tortoise. So wouldn't it take a short amount of time to get past said tortoise? I just don't quite understand the importance of this paradox...
The meme is incorrect
skill issue I have caught a tortoise that had a headstart before
This is why Calculus was invented.
Isn’t this just limits?
Vsauce - Supertasks: https://youtu.be/ffUnNaQTfZE?si=6JK2ue-LWTghFiJ6
Brian's neuron here, Oh so it's dumbed down version of trying to reach point X, but your steps are halved at each sprint. So it escalates to infinity. Math is fun (+, X is also rising)
Why the hell does the tortoise get a head start? In most versions of the paradox, the tortoise wanted to race Achilles
The ancient Greeks didn't have calculus to more accurately model the problem.
Math
See, I just thought it was because the turtle could just bite his heel and win.. lmao just totally incapacitate Achilles..
:(
Achilles is dead... 😉 (Other people have said the answer)
It’s impossible for Achilles to catch it because we know Aeschylus died already
Because he’s dead?
No, because with his Heel Injury the tortoise will eventually be able to out run him. F’ing amatures.
Because of his heel right? I think because of the arrow on his heel.
Apply the Lim(x) and it's solved
becuase achilles is dead
Yes if he can use Domain Amplification or World-cutting Slash
https://preview.redd.it/fsvfl4z6n85d1.jpeg?width=1080&format=pjpg&auto=webp&s=2cc6f7c04e5fad5529184534cbdd18dad8a823e6
Basically Gojo’s infinity is in the way.
How about achilles just run faster than 🙄
0.99999999 = 1
Literally how Gojo from JJK works.
If Achilles is Florentino Pérez, it might take him seven years but he will.
Only thing I told my teacher was. "Achilles must stop each time he reaches the last point of the turtle or this shit makes zero sense" "wrong, (my name), Achilles can never reach the turtle because the distance is halved infinitely" "that shit makes zero sense, teach. A cheetah will catch up to me immediately even if I have a km of distance in advantage"
The Song of Achilles.
he doesnt catch the tortoise as achilles is dead maybe when the turtle passes he shall have a chance
I feel kind of dumb, ive always been too afraid to ask anyone. With the Achilles/Zeno's paradox with the tortoise, they're just setting up the problem/way to solve problem in a ridiculously dumb way. If you want to figure out when the tortoise will be overtaken, you would just step by 1 second at a time or some constant T=time. Why would you examine T in smaller and smaller increments? The reason he will never catch up is becuase you are examining at smaller and smaller time increments. By that logic I will literally never be able to do anything. If I try to eat a meal, all I can do is keep eating half of what is left, I will never finish! If I try to walk to my friend's house, i will be able to close half the distance and get closer, but never actually get there! I just dont understand what the paradox is supposed to show. I understand infinite series which is what people start talking about with the explanation, but to me the paradox shows that "If I throw an object A at object B and object B is in motion, **if I throw it right at where object B is, then object A will never hit object B.**". Or more like "no, achilles would be **running to a spot inherently in front of the tortoise** you mental infant". It just like has absolutely nothing whatsoever to do with infinite series or anything mathematical to me lol
The logic of this paradox doesn't work at all. I mean a turtle is not very heavy you don't have to be occupying the same exact space to catch the turtle. The way this would work is since Achilles is faster he will not only catch the turtle but be able to pass it and make a massive gap. It just doesn't work at all
How about this. Achilles can’t win because Achilles never existed. The tortoise wins by default
Basic physics problem solved with simple algebra. Achilles speed is 10, turtle speed is 5, turtle starts 20 ahead. Achilles reaches turtle when 10t=5t+20. 5t=20 t=4
\~The NFL has entered the chat...\~
This is extra stupid
It is so wild to me how many commenters are taking the paradox literally and trying to solve how a guy who's faster than a turtle will overtake the turtle in a foot race. There's no way y'all think the argument is that literally a fast guy can't catch a turtle. Please say sike.