What is the answer to Zeno's Paradox? (Zeno's Paradox explained here...)?

Question

Please click interesting if you think this is. Once upon a time in Greece, there was a man named Nero who enjoyed challenging the big mucky-muck, I think I'm so good, mathematicians of the time. So he came up to all of them and
asked them this; "how is motion even possible", to which the mathematicians replied, "well, if you can move from point A to point B, that's motion then isn't it?"

"I know", Nero countered, "but in order to move from point A to point B, or rather, from point 1 to point 2, one mustget halfway there first, right?"

The mathematicians were all ears.

"But in order to get halfway there, you will need to get a quarter of the way..."

Silence.

"And in order to get a quarter of the way to point a, you will need to get one eighth of the way, right?....." Zeno smiles. "But in order to do that, you need to get to half of one eighth, then half of one sixteenth, and halfway from that, and so on, and so forth..."

"So tell me, dear scholars, how then is motion possible?

I wrote "Nero" in the details, it's "Zeno", actually. I don't know why I got the two confused.

Answer 1

For all I know, doug may be right that Zeno's paradoxes have been solved. I don't know, I'm not a mathematician. As far as I am concerned (from my non mathematically knowledgable philosophically pragmatic point of view, the question of whether they have been refuted is irrelevant, people had to get from point A to B, and they somehow managed to do it no matter what philosophers told them couldn't be done). I think maybe Zeno's real point however that really there is no such think as "change". It's a damn shame we only have fragments of their philosophy and they spoke a different language.

(Doug, give me a clue, does time enter into this equation anywhere, as Zeno divides the distance smaller, the time to cross it is less where time eventually approaches a point so close to zero that it might as well be zero??)

I don't have the slighest clue how to express that in mathematical terms but for a pragmatist (in the philosophical sense of that word) that is irrelevant. First, we walk. Then, mathematicians and philosphers explain how we did it.

Answer 2

S = sum 1/2^n can be easily calculated.

S = 1 + 1/2 + 1/4 + ...
2S = 2 + 1 + 1/2 + 1/4 + ...

If you substract 2S - S the result is S. But, if you think about it, then you have this:

S = 2 -1 + 1 - 1/2 + 1/2 - 1/4 + 1/4 -1/8 + ....

So, since -1 + 1 = 0, -1/2 + 1/2 = 0 too, etc, the result is 2

So, 1 + 1/2 + 1/4 + ... = 2

of, if you want to:

1/2 + 1/4 + 1/8 + ... = 1

In Zeno's case, 1 step

Ilusion

Answer 3

there is not any paradox. you could placed as many checkpoints as you like provided they are sufficiently small. the start and end lines are nonetheless one hundred ft aside no count what. finally, the size of your checkpoint will decrease the selection you could in high quality condition. if your checkpoints are relatively purely factors, then you certainly could have an limitless selection. So what? comparable with the domicile. The roof would be a definite top. what proportion flooring you could cram in is an engineering subject, no longer a paradox. As somebody observed till now, it purely sounds like a paradox till you learn sufficient math to comprehend that an limitless sequence (sum of the numbers in an limitless sequence) could have a finite value.

Answer 4

Zeno's Paradox is merely the first step towards proving the continuity of the Real Numbers.
(And no... That proof is a bit long and messy to put up here ☺)

Doug

Answer 5

motion is possible through taking one step at a time.

Answer 6

"Points" only exist in the simplified world of logic.

Triviadude, science can never explain anything; it can only describe (i.e., model, simplify).