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Zeno's Paradox

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psk177

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Has Ayn Rand given a solution to Zeno's Paradox? and if she has not, how might she gone about solving it?
She started, but realised that before she could fully answer the question, she had to half-answer it, but before she could half-answer it she would have to quarter-answer it. At that point she realized "This is a total waste of my time", and so she wrote Atlas Shrugged instead.
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"Zeno's Dichotomy paradox: Before a moving object can travel a certain distance, it must travel half that distance. Before it can travel half the distance it must travel 1/4 the distance, etc. This sequence goes on forever. Therefore, it seems that the original distance cannot be traveled, and motion is impossible."

Ayn Rand did answer Zeno, but not directly. She gave an epistemology that where knowledge is derived from the senses and logical reasoning. Zeno claims a man cannot walk from point A to point B. I see a man walk from point A to point B. Problem solved.

In other words, Zeno's paradoxes are examples of rationalism, where reasoning is divorced from a context of reality. The Objectivist epistemology rejects rationalism.

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Zeno's paradox is a psychological game of disintegration.

It is no different than Immanual Kant' lunacy: "Critique of Pure Reason"

Oh yeah? If "Pure reason" is impotent, by what means does one critique it? Mysticism?

It seems to me that Kant is self-negating. There are many clever concepts created by the human brain that are misunderstood, to the degree that a person, not yet possessing full rational integrations stands back in a dumb-founded stupor. These include: Infinity, Chaos, Quantum, God and other buzz words that float in nebulous abstractions between human synapses.

These may very well be subjective realities in the mind but none of them will materialize an elephant in your living room...

Edited by JamesHicks
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I've always seen Zeno's Paradox as an argument for discrete space. And quite a brilliant one, actually. Because if this is the case at some point the runner and the turtle are at one point in space which then allows him to progress past the turtle. It's an attack on the concept of infinitely small things. And it's a good one.

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In other words, Zeno's paradoxes are examples of rationalism, ...
Not necessarily. Considered correctly, it is not saying that there is a contradiction in reality. It is saying that a particular argument -- which is prima facie logical -- leads to an obviously false conclusion. Since the reality is obvious, we need to reject the argument, not the reality. We need to find the flaw in the argument.

Rationalism would require that one go in the other direction, deny the reality side of the paradox and opt to go with the apparently logical part.

Other examples of this are people who say that reality does not exist, or that humans have no faculty of volition.

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Unless I'm mistaken, current scientific theory holds that there are units of distance and time so small that they cannot be broken up. If this is true, then motion and time both move in little jumps, rather than along a continuum. If this is the case, the Zeno's Paradox is irrelevant.

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Unless I'm mistaken, current scientific theory holds that there are units of distance and time so small that they cannot be broken up. If this is true, then motion and time both move in little jumps, rather than along a continuum. If this is the case, the Zeno's Paradox is irrelevant.

Yes, but given the time he made this remark, it's been quite brilliant. He practically proved that if space and time could be broken down infinitely, we wouldn't live in the universe we apparently do. I think it's brilliant.

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Unless I'm mistaken, current scientific theory holds that there are units of distance and time so small that they cannot be broken up. If this is true, then motion and time both move in little jumps, rather than along a continuum. If this is the case, the Zeno's Paradox is irrelevant.

Not quite. The planck length/time are the smallest units of space/time that can currently be given a physical meaning, but this doesnt imply that length/time are discrete. In loop quantum gravity (a not-widely-accepted theory of quantum gravity) however, I think that areas/volumes are quantized.

Edited by Hal
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Zeno's paradox fails because it assumes that a person can only walk half of a distance at a time.
Unless I'm mistaken, current scientific theory holds that there are units of distance and time so small that they cannot be broken up.
Yeah, these are the thoughts I had, too. I think that, at some point, there'd be some finitely small distance for which a person couldn't move in smaller increments. Just like a neuron can't fire up half of a muscle cell, I imagine that there's some point at which smaller increment motion is impossible.

[Zeno] practically proved that if space and time could be broken down infinitely, we wouldn't live in the universe we apparently do. I think it's brilliant.
It's an interesting perspective, indeed. If he independently came up with an answer (don't know, personally) I'd probably call him brilliant too.
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It's an interesting perspective, indeed. If he independently came up with an answer (don't know, personally) I'd probably call him brilliant too.

I should say that I don't know if that was his intention. He could aswell have said it to annoy his fellow philosophers. :D

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I should say that I don't know if that was his intention. He could aswell have said it to annoy his fellow philosophers. :D

The standard interpretation of Greek pre-Socratic philosophy is that Zeno was trying to show that motion was an illusion, and hence that change didnt occur. One of the main 'problems' for the Greeks was how change was possible. Parmenides held that the reality 'underneath' our sensations was eternal and hence couldnt actually change - Zeno's arguments were intended to illustrate this point. Compare it (for instance) to Democritus' argument that one thing could change into another because the world was actually made up of indivisible eternal atoms, and change was only a change in the configuration of these atoms rather than anything new coming into being.

Personally I'm sceptical about how much sense it makes to interpret the Ancient Greeks in this way - it seems like we are throwing our own ideas onto their work. The whole Greek worldview really was radically different from our post-Descartes/Kant framework, which makes interpretation very difficult. Its really easy (and popular) to read thinkers like Parmenides and Zeno as being proto-Kantians, but I think this could be a misinterpretation.

Edited by Hal
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Hal, I think you are right. Within the framework of pre-plato philosophy my ideas about discrete space don't make much sense. However, his paradox is a nice way of showing the problems with infinitely small pieces of space. So if you take the argument out of its initial context, it becomes quite valuable.

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Yes, but given the time he made this remark, it's been quite brilliant. He practically proved that if space and time could be broken down infinitely, we wouldn't live in the universe we apparently do. I think it's brilliant.

Zeno's "paradox" doesn't just evaporate when you consider something that can not be broken down infinitely, but also holds no water when you consider a construction such as the real number line. If you start at 0 on the real number line you can still get to 1 no matter how many halfway points you must pass through. His whole set up was flawed because the ancient Greeks did not have sufficient understanding of the mathematics.

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If you start at 0 on the real number line you can still get to 1 no matter how many halfway points you must pass through.

Are you sure about that? Because something tells me that if you do the following:

.5 + .25 + .125 + ........

or

E [1/(n^2)] , where 1 <= n

you'll never really reach 1. You can get arbitrarily close, but never equal.

Edited by Eternal
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When speaking of measurments, 1" doesn't exist. it might be 1.000....1, the glass will never be exatly half. For length of something, atoms move/vibrate. We can only make estimates, I'm cool with that.

If the smallest possible unit of space exists, we can say that it is double half of it.

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When speaking of measurments, 1" doesn't exist. it might be 1.000....1,
Then I think you mean that we can't measure with infinite precision on a truly continuous scale. If space is quantal (there is some smallest possible separation between entities) then space measurement is not continous. But 1 inch still exists.
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When speaking of measurments, 1" doesn't exist. it might be 1.000....1, the glass will never be exactly half.

One of the points that Rand makes in IOE is that measurements are contextual, so there is always an implicit error term involved in any measurment. Accordingly, even if space is not quantized and one could hypothetically measure arbitrarily small lengths, it makes no sense to say that the glass is "exactly" half full. The reason is that would imply that there is zero error in a measurement, something that can't be done when measuring lengths of physical objects since one's unit of measure has some length.

However, just because one cannot measure something "exactly" (i.e., acontextually, i.e., intrinsically) doesn't mean that measurement is not possible (i.e., subjective).

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Accordingly, even if space is not quantized and one could hypothetically measure arbitrarily small lengths, it makes no sense to say that the glass is "exactly" half full.

I'm not 100% sure what you mean here. It makes perfect sense to say in ordinary English that a glass is exactly half full, just like it makes sense to say that I arrived for dinner at exactly 9pm. And, it would be correct to say that I arrived at exactly 9pm even if I actually arrived 4 seconds earlier. The word "exact" isnt normally used to mean "idealised mathematical exactness as measured under perfect idealised laboratory conditions" - there is normally a surrounding context which defines what classes as 'exactness' in any particular situation. What counts as "exactness" when meeting a friend for dinner might not count as 'exact' when you are (eg) timing a 100m sprint in the Olympics, or the time taken for light to travel 1 meter.

Edited by Hal
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The reason is that would imply that there is zero error in a measurement, something that can't be done when measuring lengths of physical objects since one's unit of measure has some length.
That's true when your standard of measurement is a perceptible distance, like the distance between two marks. When the standard of measurement is in terms of a count, then error-free measurement could be possible. Length is now commonly measured in terms of a count (of oscillations), which can be fractional (hence subject to imprecision). But if the universe really is "grainy", it could be possible to count the grains and derive an error-free length measurement.
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It makes perfect sense to say in ordinary English that a glass is exactly half full, just like it makes sense to say that I arrived for dinner at exactly 9pm. And, it would be correct to say that I arrived at exactly 9pm even if I actually arrived 4 seconds earlier. The word "exact" isnt normally used to mean "idealised mathematical exactness as measured under perfect idealised laboratory conditions" - there is normally a surrounding context which defines what classes as 'exactness' in any particular situation. What counts as "exactness" when meeting a friend for dinner might not count as 'exact' when you are (eg) timing a 100m sprint in the Olympics, or the time taken for light to travel 1 meter.

We don't disagree. "Exactness" depends upon a standard which is appropriate for one's purposes. I was just objecting to saying that a measurement is "exact" without reference to any standard. In ordinary language, we (properly) use the term "exact" contextually, as you point out above. My usage of the term here is in the philosphical context used by Rand in "Exact Measurement and Continuity", c.f. IOE, 2nd. ed., pretty much the "ideal mathematical" definition you give above. orangeiscool's usage suggested he meant this version of exactness rather than the ordinary language version. Of course, if he didn't mean that, he can always correct me. :)

That's true when your standard of measurement is a perceptible distance, like the distance between two marks. When the standard of measurement is in terms of a count, then error-free measurement could be possible. Length is now commonly measured in terms of a count (of oscillations), which can be fractional (hence subject to imprecision). But if the universe really is "grainy", it could be possible to count the grains and derive an error-free length measurement.

Right-- I only meant this to apply in a discussion of measuring length (or the height of a liquid in a glass) in a continuous universe.

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