Zeno's dichotomy paradox and its solution.
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- เผยแพร่เมื่อ 7 มิ.ย. 2024
- What is Zeno's Dichotomy Paradox? How can it be solved? Is motion an illusion? All answers and details are in the video.
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To some, this paradox still stands today even against the view of modern set theorist in mathematics. This video fails to answer any of the assumption of Zeno, the most important of them being: 1. that there are infinite points between two points and 2. an infinite series cannot be completed
Absolutely and even more, the video says as n approaches infinity 1/2n approaches zero. Keyword approaches, but never reaching zero.
Yet Zeno contradicts this assumption of his with the paradox he calls "The Stadium". So which is it, Zeno? ... infinite steps or discrete intervals of motion?
@@thecarman3693 Zeno wants to prove that motion is impossible. His strategy is the following: you can think that motion, distance and time are either continuous or discrete. For both of the alternatives I will present you with a paradox. A paradox is by definition impossible to solve. Motion is impossible because it proves impossible in any of the two only alternatives we can think of when we think about the nature of movement. If movement is continuous, we get to the dichotomy and Achilles paradoxes. If it is discrete, we arrive to the arrow and the stadium paradoxes.
As n approaches infinity 1/2n approaches zero, key word approaches but it’s never zero.
Why couldn't Zeno think of the original distance he wanted to cover as simply being 1/2 the distance to a location twice as far as he wished to travel? He already conceded that traveling 1/2 of some distance was doable, therefore his ability to travel to his original location is accepted by default. No infinite series math required.
This is only one of his paradoxes the arrow one concludes the impossibility of motion
@@SuperEliasc And that's because he assumes no difference between the arrow at rest and one traveling through the air at speed.
Einstein and Lorentz mathematically showed that this assumption is false. Had Zeno just thought for a moment that there must be a difference between the two arrows, that conclusion (even without the relativistic mathematics) would have been monumental.
Plus, getting back to the initial paradox, Zeno himself is something of a hypocrite. He assumes that distances can be divided in half an infinite amount of times, yet in his paradox "The Stadium" he assumes a finite limit.
@@SuperEliasc then why would you move if you happen to see one heading your way?
@@mrmetaphysics9457 that’s where the paradox comes in. Although we understand it is impossible you still seem to be able to move.
@@thecarman3693 can you explain further? What is the difference between the arrow at rest and one travelling through the air? And how does this affect either of these two premises: the are infinite points to travel in any distance, and the completion of an infinite sequence is impossible
I discovered this theory because I was curious how much you would have to slow time down to observe the grid snapping which I hypothesize would be seen in the observation of motion. Because of the dichotomy paradox, how could something smoothly move from point a to point b without a snap???
You aren’t traveling to the park, but rather traveling in divided distances found along the path to the park. You are purposely and infinitely setting destinations before the parks final destination, never choosing to travel 100% of the remaining distance. This should be the theory of procrastination. I have a similar theory that I haven’t been able to put to words yet.
This argument of one convergent series to proof movement why is more powerful than a counterexample with other series (also with succession lim n->inf=0), wich cannot explain movement? If Zeno ask for 1/2+1/3+1/4+...+1/n wich we know is divergent series. The winer is if we have more convergent than divergent so is proof of movement, or if we find only one case against all the others cannot be used as absolutely proof?
That picture is actually Zeno of Citium, the first stoic
Thank you for pointing this out. We apologies for the mistake. The thumbnail's going to be fixed.
The solution is simple:
By means of the same reasoning as the 'paradox', in order to get to the end of the journey one has to go 1/2 of twice the length of the journey.
Boom boom.
Nice
In my humble opinion, the mathematical solution offered here doesn't quite solve Zeno's paradoxes. It is not enough to say infinite terms sum up to finite quantities. The question is how? When we assume a variable to be approaching a number and say something such as 'x' tends to 'n' where n could be any finite number, we are approximating, and thus the underlying process we are trying to explain mathematically is at best explained approximately. No approximation can claim to be absolutism, and thus, the mere statement that infinite series converges to finite quantity doesn't quite satisfy a thinker's intuitive faculties. This solution is as good as ignoring or overstepping the puzzle without offering a true solution! To imagine that Zeno did not understand the apparent convergence of infinity to finiteness is an insult to the great mind!
Diogenes just moved
Why do you think it's possible to infinitely divide reality?
The actual problem is that Zeno's Paradox is NOT a paradox. It's a logical error. Essentially, it turns velocity into a function of distance traveled. In other words, Zeno's Paradox assumes that velocity equals the distance traveled times the amount time moving. In yet other words, the paradox is assuming that you are slowing down more and more as you approach point B because the time intervals are shorter and shorter.
This is NOT how motion works.
In reality, your speed remains constant as you move forward so long as no other forces are acting to change that speed. The halfway point time intervals, however, are actually getting shorter and shorter as you approach a destination. Essentially, there are an infinite number of shorter and shorter time intervals between you and your destination BUT, your speed is CONSTANT. Therefore, because the time intervals are halving, the proportion of your speed to the amount of distance remaining is growing exponentially. Math kicks in and the limit is reached and your speed compared to the distance remaining is infinite and you pass the point.
Again, its not a paradox. It's a logical error.
I don't think you fully grasp what he's is putting forward
I don't follow your argument. Where does Zeno state or imply that the speed is *not* constant?
@@jasonjames6870 He's actually right. That's the solution given by Aristotle.
Why does it become 1/2n zero??? Please explain. This is not possible.
If we say n is 1, 1/2^n = 0.5, n=2, 1/2^n = 0.25, n=3, 1/2^n =0.125 and so on. As you can see when n gets bigger, 1/2^n gets smaller. So if n approaches infinite step by step, 1/2^n approaches zero. In math., limit accepts approaching number as result.
It approaches zero. The audio says it clearly but the writing says it is zero. If the writing is what it is meant by the video then it is certainly wrong
Sorry bro for late comment
I'm not a mathematician, but here is a proof that this is a purely theoretical math issue. Instead of arbitrarily selecting 1/2, select a 1/4. You wont get halfway! Select 1/8 and you won't get a 1/4 way. Follow this reasoning far enough and you won't be able to start your journey!
hence why he said motion was impossible
That would prove zeno's point
scrap the muzak
Hi.
Zeno's paradox will be solved only when you are yourself.
He is only talking about Parmenides' the one real being, not any time space thoughts.
When you have solved Zeno's then you have already solved Parmenides Socrates and Jesus.
If you are brave enough then challenge to digest Thomas Gospel, not following other's deluded opinions.
For tip, This one put Thomas gospel's 114
Simon Peter said, Let Mary leave us. For women are not worthy of life.
Jesus said, I myself shall lead her in order to make her male to be the one and same so that she become living spirit resembling you male who become the one and same.
For every woman who will take her male in herself to be the one and same will enter the kingdom of heaven.
Have happy life.
!!!
I am Tahseen
So, 'Near enough is good enough' is the new mathematical rule! All this video proves is that math fails to describe reality.
Finally some common sense, thank you.
No solution here