Achilles & The Tortoise - Zeno's Paradox SOLVED
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- Опубліковано 12 вер 2024
- Zeno's famous paradox of motion (and why he believed motion to be impossible). Zeno argues that the Greek hero Achilles could never catch a tortoise in a race, as long as the tortoise is given a head start.
The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy have puzzled mathematicians, scientists, and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher, Parmenides.
Just run past the turtle
WHERE ARE THE TUUUUUUURRRRRRTTTTTLLLLLEEEESSSSS.
Said the engineer
This is essentially what Diogenes did when he got up and walked from point A to point B (different paradox of Zeno but same principle here). It's not enough in philosophy to show an answer, you have to show cause as to how it happens. Think of it like science, it's not enough to say "x" you have to show cause of why "x"
YOU, know it makes sense!
Ok, im going to state a simpler explanation that doesn't involve as much math (i came up with this myself)
so, Achilles starts at the 0m line. the turtle starts at the 3m line.
say, it takes Achilles 1 second to travel the 3m the turtle started at. the tortoise traveled 1m in this ammount of time. this is what the man in the video explained. what he didn't explain, is when you use both time and distance, you get a slope, rather than just an equation.
Heres the solution:
they are simply slowing down time. it takes the turtle 1 second to move 1m, then it would take a third of a second to travel a third of a meter. the steps go into the infinities because there is an infinite ammount of time frames where the turtle is ahead of Achilles. say, it takes Achilles 2 seconds to go faster than the turtle, then there is an infinite ammount of decimals between 0 and 2. such as 1.99999999999, 1.99999999999999999999999999999999, 1.9999999999999999999999999999999999999999999999999999, etc. you can always add a decimal to add another ammount of time frames the turtle is ahead of Achilles.
Here is a visual:
"A" = "A"chilles, T = Turtle, "-" = half a meter
0 seconds:
A------T
1 second:
------A--T
1.3 seconds:
--------A-T
1.9999999999999999 seconds:
-------------AT
2 seconds:
-------------TA
(if i had an infinitely small "-" key, i'd be able to prove there are infinite time frames, but it will be overtaken if you simply keep a constant time)
Thank you :3
Thanks a lot, this was the final piece that made it click for me
This is smart Einstein would be proud
But whats the difference between equation and slope?
Great explanation man !!!
The whole point of the paradox is that it is clear that physically and mortally intelligent Achilles will overtake the tortoise, but how does this make sense in terms of the term infinity because infinity cannot be limited to a certain number of steps or time (as stated in the video) and that is the whole point of the paradox, to understand that there is a limit to the use of the word infinity and there is opposition There is a distinction between infinity and sweetness (an example of the spoon is the question of whether infinity plus 1 equals infinity plus 2 (if it is equal by moving sides, 2 equals 1, and if not, it turns out that infinity is different from infinity, which also does not make sense)
Very concise and clear way to explain resolution to this paradox... well done! It still surprises me that early Greek mathematics could not sum the infinite series, even with informal approach shown here to reason that the incremental halve distances sum to a finite distance.
IMO, the actual problem with the paradox is that it contains a limiting error. It only evaluates those places along Achilles' journey when Achilles visits somewhere that the tortoise has already been. Clearly, it is impossible for Achilles to overtake the turtle when being limited to such a domain.
I dont understand the paradox, am I dumb? Why hasnt the speed of Achilles and that of the tortoise being taken into account?
it was, he said the achilles's speed is 3 times the turtle's (as an example)
The space here can be subdivided Infinite times hence if Achilles is way faster than the tortoise he will never be able to reach it since the space between them is Infinite this may seem contradicting with our experience but earlier philosopher of eleatic school believed that senses are unreliable only thought is relaible way of knowing the Truth the founder of that school paraminder believed that though and being are connected whatever can be thought is being and whatever can't be thought is non-being
Edit
Using this paradox Zeno proved that universe is static and unchanging change and motion is merely an illusion of senses
because the speed isnt the issue here, the paradox will arise at any speed as long as achilles is faster than the tortoise.
Also time doesnt stop everytime achilles reaches the tortoise’s last position like the paradox implies. Time is not dependent on their motion. Time keeps moving and since achilles moves 3x as fast and time doesnt stop thats what the tortoise will have to deal with.
I wish my intro philosophy teacher had put it this way instead of making us ponder whether motion exists like a sophist
i thought it had to do with the plank constant, and that there is actually a minimum distance that, once reached, the tortoise will be overtaken. You can't keep dividing a real distance infinitely, otherwise no one would ever pass anything
Thats a great idea, but I think that would be more of a loophole solution. It's not really solving the paradox of why motion seems impossible, but rather it's finding the one extreme situation where the paradox doesn't apply.
(Also regarding your last sentence, that is exactly why it's called a paradox and was why it was so fascinating to the ancients. It is so obviously true that faster things can pass slower things, but no one in ancient times could mathematically answer the paradox as to how it was possible)
Amaizing work! I am happy for find this channel.
Thank you, I’ve had a hard time understanding the math behind this paradox, it all makes more sense after watching this video!😊
I think mathematically you can divide a pizza pie an infinite amount of times but in reality eventually you get down to splitting atoms, quarks, until you get down to a planck length and then you simply can’t split it any further. Mathematically yes but realistically no. I think space has to have a smallest point. Travelling an infinite amount of points doesnt make sense unless those points come to an end at a smallest possible point.
First off, this is basically the Dichotomy paradox said in a different way. Second, this didn't prove anything. The process of division itself is infinite in time. It would never end. There is something missing from our understanding of the physical world to explain this paradox. The answer does not reside in maths.
why is time infinitely divisible within itself, doesn't that make for, again, the abolishment of movement? or atleast space isn't infinitely divisible and time has to modify itself to accomodate it?
@@newton.whippleberry Totally with you on this. I don't know how true this is but I've heard when electrons move from one orbit of the atom to another, they don't travel the space between them. Instead they just disappear and reappear in the higher or lower orbit. Maybe the answer to this question lies in Quantum physics. Maybe movement is nothing but a series of tiny jumps in the fabric of time and space.
@@stevesmith4901woah
@@stevesmith4901I like your answer and in a sense I think you've solved it. The jumps through fabric of space and time make sense, and it doesn't contradict the point about infinite space between objects. I believe you've found a way around this paradox without having to debunk it.
I agree though, the answers lie within the subatomic world.
@@AltairEgo1 Appreciate your comment. I am pretty confident the resolution to this paradox lies in a better understanding of the physical world. For example, the geocentric model of the universe in the ancient world raised many paradoxes that could not be resolved till we realized the earth was not at the center of the universe. Something similar has to happen for us to resolve the paradox of movement. Better maths is not the solution.
You can solve any problem on the earth if you want to change the definitions
im guessing zenos didn't understand the concept of "acceleration" and derivatives at the time. Logically and realistcally speaking, WE all know that most normal people can outrun a tortoise. Zenos is assuming that Achilles' acceleration remains 0 the entire time and has a constant speed yet infinite distance to travel. Again though, Zenos didn't account for acceleration... which, for an imperfect human in an imperfect system, will never just be 0
The math is true. But how much is infinity times infinity?
The turtle is slow and dumb
Thanks. off I go to beat Gojo
It seems that this problem can be solved without the use of infinite series. Say the tortoise travels distance x, which is unknown, and the time to travel this distance is t =distance/velocity = x/(1/3) =3x. Achilles travels distance 1+x in the same time t =(1+x)/1. So 3x = 1+x, 2x=1, x=.5, so Achilles travels 1.5 meters to catch the tortoise. The solution is the same, but it is done without infinite series. I think this problem may be slightly different from the question of adding up the infinite series 1/2 +1/4 +1/8.... to show that in the limit it reaches 1.
Not sure if that's correct. You state the tortoise travels distance x in time t, and Achilles travels distance 1 + x in the same time t. So you set up the problem stating he will catch him, rather than proving it.
The whole point is about infinity that can't be reached
Why the divergent harmonic series cannot proof Zeno Paradox of movement ? Why is proved with the series of 1/3^n and not with that of 1/n if both sequences tends to 0 ? Ok series 1/3^n is convergent and our harmonic divergent, but if Zeno asked for 1/2, 1/3 ,1/4, 1/5 ,...,1/n to the destination, this divergent is one case Against the other convergent used to proof of movement. Is not a "math cheat" choose one sequence convenient to the proof without explain why others are not valid AGAINST that choosen? If the answer is only "divergent is infinite sum" this gives reason to Zeno, as he have one valid sequence against that convergent proof. Who wins? I think have answer to this, but post my initial fair though.
Nothing can prove this paradox; it is objectively false because it results in a contradiction. The divergent sum is only really divergent if it gets to use an infinite amount of terms. Achilles catches up in a finite amount of time. The tortoise in the convergent example doesn't move 1/3 of a meter every step, rather he moves 1/3 of what Achilles moves. So the Divergent Tortoise would not move 1/2m, 1/3m,1/4m. But rather, he would move 1/2 of Achilles distance, then 1/3 the next (smaller)distance then 1/4 the next (even smaller) distance. Here is what that looks like:
Step 1:They start 1m apart. Achilles moves 1m, the tortoise moves at 1/2 his speed so he moves 1/2m.
Step 2: Achilles moves 1/3m and the tortoise moves at 1/3 of achilles speed, but for half the time he had in step 1, meaning the Divergent tortoise only moves 1/9m this time.
Step 3: Achilles moves 1/9 m, the turtle moves at 1/4 of A's speed for 1/9 the time he had in step 1. The tortoise only gets to move 1/36m in step 3. The tortoise is 49/36m from where Achilles started. Achilles has moved 52/36m (1+1/3+1/9) in this time. Achilles catches the divergent tortoise in a finite amount of time.
So wrong, you are calculating consistent speeds as if run on a computer model, that is not the paradox as any variable of speed & distance will alter your calculations. Also using fractions to solve these questions don't provide any real answer. Somethings are beyond our human understanding, you can start with this "What is a number?"
Not so fast dear Archiles - to provide a robust mathematical basis you actually need Dedekind and Weierstraß, don't you think?
I have seen this bad math before....subtraction is a matter of vertical columns....you can't shift right and ignore the fact 1- (1/3)3=0!! violation of the laws of subtraction...an infinite sum cannot be the addition of finite terms! Or, (S+...)= 1/3(3 + S+...) then 3S+3(...)= 3+S+... then 2S+3(...)=3+... then 2S+3(...)-...=3 then 2(S+...)=3 then (S+...)=3/2, then if 3/2=S, then (3/2+...)>3/2!
Achilles can never PASS the tortoise if the race is only/allways considered in terms of the ability of Achilles to REACH a point that the Tortoise reaches! END OF!
Big if
@@OurKnowledgeoftheWorld Any inability of Achilles to reach a, 'point' coincident in time and space with that occupied by The Tortoise is s part of the quantum/continum duality view/debate which involves the issue as to wether the world is composed of discrete/distinct units or an underlying continum. Eg. Democretean or Hereclitean.
Do Planck Units exist and are they fundamental?
As far as I know the paradoxical nature of this dichotomy has not yet been resolved.
If you subscribe to the Continum view then you must embrace infinitudes in which case things can never be exactly coincident.
If you embrace the Quantum view things can be coincident but motion implies jumping from one location to another without traversing some intermediate space.
Maybe you should rewatch this again. I mean IF I'm allowed to insert "ifs" into any statement or question i see .... then I'm going to get straight A's in every exam & test I ever have!
You can simply use the calculation used in clocks, everyone probably did this in highschool.
An easy example would be the Hour hand starts at 11 o'clock. Achilles (minute) will reach the Turtle after an hour, when it's 12 o'clock.
Nice 🙌
ta có t Achilles sẽ = t của rùa khi Achilles và rùa trùng nhau tại một điểm trên đường chạy. như vậy S1/v1 = S2/v2 . Dùng hệ thức này để lập ra phương trình 1 ẩn .
The problem with this elegant mathematical argument is that there is no infinite in the real world. Infinite is a mathematical concept we all learn in first year calculus which is not physically possible.
In the same video u have also proved that as we in our daily life have "time" even if its an illusion or not. Is real for us as we live in this world. So i can nearly prove by this that time is an illusion, cause if time dont exists, then Zenos was right. But we in some way? experience time. So this maybe? proves that time is an illusion!!!!!!! Further on, the hypotes is wrong from the beginning, if it was based on that time exists, the turtle will lose very fast. so i think that probably time is an illusion, just as many scientists believe now in 2024. Heureka ha ha!
Poor ancient greece, they dont even know 2+2 = 5;
Happy to know that know, as jews like Cantor said that for us. Bless Cantor.
When autor said: INFINITE steps on FINITE distance.
Well, if people believe in Karl Marx, why cant they BELIEVE in Cantor.
First of all, the tortoise did not speak - it was Zeno the philosopher who invented the race.
Second - there is a crucial rule you missed - Achilles must not overtake the turtle before covering his feet for protection... this is dumb!! You can't create a race in which you are not racing.