Zeno's Dichotomy and Cantor Set

chaoseverlasting

Zeno's Dichotomy paradox divides the distance travelled by any travelled into an infinite geometric progression. ie: 1, 1/2, 1/4,... and so on. The argument is that the traveller must cover these individual distances before he can complete the whole.

But since the distances to be travelled are hence infinite, the traveller must cover an infinite distance. This is similar to the cantor set. But could someone explain to me how we travel infinite distances and why motion is NOT an illusion?

arildno

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Why must we travel an infinite distance?

CRGreathouse

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It's a paradox because Zeno assumed moving a nonzero distance an infinite number of times meant you traveled an infinite distance. This clearly isn't the case, though. The whole thing is a problem Zeno (and ancient Greeks in general) has with the infinite.

arildno

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CRGreathouse said:
It's a paradox because Zeno assumed moving a nonzero distance an infinite number of times meant you traveled an infinite distance. This clearly isn't the case, though. The whole thing is a problem Zeno (and ancient Greeks in general) has with the infinite.
Greek PHILOSOPHERS had problems with infinity, but you cannot really say that guys like Eudoxos and Archimedes suffered from the same flawed understanding of it.

CRGreathouse

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arildno said:
Greek PHILOSOPHERS had problems with infinity, but you cannot really say that guys like Eudoxos and Archimedes suffered from the same flawed understanding of it.
Sure I can, why not? I can't think of any ancient Greek mathematician who had a good grasp of the infinite. Archimedes' Sand Reckoner showed that he wasn't afraid of large numbers, but he never brushes agsinst the infinite even there. Certainly Eudoxos was better able to understand irrationals than the Pythagoreans, but there was no link between those and the infinite (like modern decimal expansions): the heresy was that it could be incommesurate, not that it continued to infinity or somesuch.

Perhaps there is an example of someone who really got it from that period, but I'm not aware of it.

arildno

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Well, ever heard of the exhaustion method, and Archimedes' work "The Method"?

In general, the elite Greek mathematicians had a far better grasp of the intricacies of infinity, and how to rigourously deal with it than most mathematicians up to the 19th century.
(Newton, and possibly Euler excepted, but certainly not Leibniz).

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CRGreathouse

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arildno said:
Well, ever heard of the exhaustion method, and Archimedes' work "The Method"?
I'm familiar with the exhaustian method, but I don't believe there was any sort of rigorous explanation of why it worked; it was just believed to work. Does "The Method" have some kind of proof that the method works?

Maybe I'm just holding ancient mathematics to modern standards, I don't know.

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HallsofIvy

No, there wasn't and I doubt that Archimedes thought of it as dealing with "infinity" directly. However, he certainly "brushes against the infinite" there.

arildno

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Archimedes had very rigorous proofs of, for example, why the area of the circle equals that of a triangle of base diameter and height half-diameter.

The way he did that was by way of contridicting the strict inequalities by showing that for sufficiently large N-gons, any particular inequality (for example that the circle's area was larger than the triangle's by a number "d") was violated.

Hurkyl

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CRGreathouse said:
Maybe I'm just holding ancient mathematics to modern standards, I don't know.
There is a lot of geometric "obviousness" that even a modern person could easily not realize needs proven. For example, read this page on Euclid's first proposition.

arildno

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Besides, I would like to add, the vast majority of humanity today (including highly educated people, if not in maths&physics) does not have any firmer grasp of the concept of infinity than the most Greeks had.

3trQN

HallsofIvy said:
No, there wasn't and I doubt that Archimedes thought of it as dealing with "infinity" directly. However, he certainly "brushes against the infinite" there.
I agree, he found a method that worked. Good engineering :tongue:

murshid_islam

chaoseverlasting said:
Zeno's Dichotomy paradox divides the distance travelled by any travelled into an infinite geometric progression. ie: 1, 1/2, 1/4,... and so on. The argument is that the traveller must cover these individual distances before he can complete the whole.

But since the distances to be travelled are hence infinite, the traveller must cover an infinite distance.
so if someone has to cover 2 miles, first he has to cover 1 mile. then 1/2 mile, then 1/4 mile and so on. but the total distance travelled is not infinite because the total distance is the sum of the infinite series,
$$1 + \frac{1}{2} + \frac{1}{4} + \cdots$$

$$=\frac{1}{1-\frac{1}{2}}$$

$$=2$$

therefore the total distance travelled is 2 miles.

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arildno

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Nope. Only for the stupid Greeks, like Zeno.
Men like Archimedes, Erasthotenes, Euclid and Eudoxos most likely laughed at such philosophers, and with very good reason.

arildno

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Aristotle was an excellent naturalist when it came to matters of biology.
(My Dad was a biologist, and he was quite impressed by Aristotle's careful observation and description of the development of the chicken embryo).

He is also a shrewd observer of politics, and

he must also be credited with his work on logic&syllogisms.

However, his physics, metaphysics and mathematical insight cannot be regarded as equally worthy.

Hurkyl

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Something I wanted to point out -- Zeno's pseudoparadoxes all involve transfinite order types. (The most famous involves the ordinal number $\omega + 1$) AFAIK, a good understanding of that has only come in the past couple centuries.

arildno

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Hurkyl said:
Something I wanted to point out -- Zeno's pseudoparadoxes all involve transfinite order types. (The most famous involves the ordinal number $\omega + 1$) AFAIK, a good understanding of that has only come in the past couple centuries.
Which shows quite clearly that Zeno's paradoxes weren't about that at all.
You could actually, regard him as just another muddlehead, rather than a superintelligent fellow a couple of millennia in advance of his time.

"Zeno's Dichotomy and Cantor Set"

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