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Curious if anyone is as interested in these things as I am, as I'd like to have a discussion about some of them.

http://en.wikipedia.org/wiki/Paradox

I'd like to have an open discussion on Paradoxes so I'd rather not begin by stating my solutions, but if the mods demand such, I will begin myself with my solution to Zeno's Paradoxes.

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I can solve the Arrow paradox. Just tell Zeno to go stand over there...

That shows how it's incorrect, but does hot explain how.

Zeno's stuff can be explained very easily IMO. The universe itself can either be Digital - IE there is a "smallest chunk" you can cut it into, or, it is Analog - IE there are no "chunks" to the universe.

Lets start with his "halfway there" paradox.

Lets presume we live in a digital universe.

So what if there are only 2 bits between you and "there"? How can you get "halfway"? You can not. You can either get more than halfway, or less than halfway.

Okay, so maybe the universe is Analog

In that case, since there are no "chunks" to cut the universe into, you can never be sure if you are halfway or not, you can only be approximately halfway, and when you are approximating like that, sometimes you have to be more than halfway, and sometimes less than halfway.

Thus the answer to Zeno's paradox - pretty well all of his paradoxes - is that it is a language trick. It presumes there IS such a thing as "halfway" when I argue there is not. For his Arrow, it argues that you can take a "moment" in time, when by the same logic, you can not.

This video (which I will watch all of, Manny) starts with a part on Infinity. I argue there is no such thing. The universe is finite, I argue, and will always be. Infinity, like halfway, is a trick of language. It does not exist.

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That shows how it's incorrect, but does hot explain how.

Zeno's stuff can be explained very easily IMO. The universe itself can either be Digital - IE there is a "smallest chunk" you can cut it into, or, it is Analog - IE there are no "chunks" to the universe.

Lets start with his "halfway there" paradox.

Lets presume we live in a digital universe.

So what if there are only 2 bits between you and "there"? How can you get "halfway"? You can not. You can either get more than halfway, or less than halfway.

Okay, so maybe the universe is Analog

In that case, since there are no "chunks" to cut the universe into, you can never be sure if you are halfway or not, you can only be approximately halfway, and when you are approximating like that, sometimes you have to be more than halfway, and sometimes less than halfway.

Thus the answer to Zeno's paradox - pretty well all of his paradoxes - is that it is a language trick. It presumes there IS such a thing as "halfway" when I argue there is not. For his Arrow, it argues that you can take a "moment" in time, when by the same logic, you can not.

This video (which I will watch all of, Manny) starts with a part on Infinity. I argue there is no such thing. The universe is finite, I argue, and will always be. Infinity, like halfway, is a trick of language. It does not exist.

I would prefer to think that there is infinity, and it has nothing to do with our universe.

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Actually Zeno's paradox isn't a paradox at all, and can be trivially solved with a bit of math. The premise is that you can cover half the remaining distance with each additional step, thus it takes an infinite number of steps. Let's use an example:

Say the target is 1m away and the arrow is traveling at 1m per second, then:

The first step: it goes 0.5m in 0.5s

2nd: 0.25m more in 0.25s

3rd: 0.125m in 0.125s

4th: 0.0625m in 0.0625s

...

Then you need to add up that infinite series of time steps: 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125...

www.wolframalpha.com/input/?i=sum+from+1+to+infinity+of+1%2F%282^n%29

It turns out that if you actually use math to add up this series, the answer is very well defined even though its an infinite series, and happens to be 1, exactly what you would expect.

In general, one of the first principle's of science is that there are no paradoxes. Any apparent paradox is simply the result of the observer's inability to understand/know/explain something.

Edited by Bonam
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You did not answer because you know. The truth is, it can NEVER become a value of 1. One is the value it will in fact never reach.

It points to the problem of approximation. As humans it's good enough for us to call it 1 when the value is 0.9999999... but the reality of things at the very smallest level is very different from the reality we experience through our cognition. Quantum physics, for example.

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Heh I didn't answer because I was away skiing ;p

In the example, the distance the arrow has covered becomes 1 after precisely the same amount of steps as when the time that has passed reaches 1 second. If the time is any less than 1 second, by any arbitrarily small amount, the arrow has not yet reached its target. If the time is any greater than one second, the arrow has already hit the target. That makes the time that the arrow hits the target equal to 1 second. There is no approximation here, nothing obscure, nor reliant on subjective experiences. It is clean and simple, reliant purely on the fundamental workings and beauty of mathematics. You can find the mathematical resolution of Zeno's paradox widely available.

By the way, I erroneously used the "arrow" in the example. The arrow is actually one of Zeno's other paradoxes. The paradox that deals with repeatedly covering half the remaining distance to something is "Achilles and the tortoise".

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Ok if you were skiing, I'll let that go. :)

You are moving this away from the argument that the paradox attempts to make, by showing the sum of the sequence. Which is, in its own way, a paradox... that a sum can be made of an infinite number of parts, each one smaller than the previous.

In the concept it takes an infinite number of steps to reach completion. The steps become incredibly small. They become infinitely small in fact. Also, in the paradox the tortoise moves as well and the concept claims that whenever achilles moves, the tortoise moves also. Logically Achilles will evenrtually overtake the tortoise and everything appears to work fine at our level of perception until one enters into the quantum domain, dealing with very small distances and very small increments of time. I'm sure you're familiar with the problem of simultaneity in defining position and velocity.

There is no approximation here, nothing obscure, nor reliant on subjective experiences. It is clean and simple, reliant purely on the fundamental workings and beauty of mathematics. You can find the mathematical resolution of Zeno's paradox widely available.

dly covering half the remaining distance to something is "Achilles and the tortoise".

But the equation is continous, and a mathematical value can keep getting smaller and smaller. That is what infinity means, of course. We know reality does not work on such underlying principles. Reality is "granular", which is what I was saying about quantum physics. Let alone, the problem of Heisenbergs uncertainty.

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Ok if you were skiing, I'll let that go. smile.png

You are moving this away from the argument that the paradox attempts to make, by showing the sum of the sequence. Which is, in its own way, a paradox... that a sum can be made of an infinite number of parts, each one smaller than the previous.

In the concept it takes an infinite number of steps to reach completion. The steps become incredibly small. They become infinitely small in fact. Also, in the paradox the tortoise moves as well and the concept claims that whenever achilles moves, the tortoise moves also. Logically Achilles will evenrtually overtake the tortoise and everything appears to work fine at our level of perception until one enters into the quantum domain, dealing with very small distances and very small increments of time.

There is no other "level of perception". Eventually, Achilles will be well ahead of the tortoise, by any measurement system you could use to test this. Quantum effects are not relevant in this problem.

The resolution of the paradox merely says this:

If you cut a distance up into a lot of parts, and then add all those parts back up again, you get what you started with. And this is true no matter how many parts you cut it up into, even an infinite amount.

I'm sure you're familiar with the problem of simultaneity in defining position and velocity.

There is no "problem" with simultaneity, it merely holds a more complex definition than is intuitive when one considers things on relativistic scales. However, just like quantum effects, relativistic scales are not relevant in this particular problem. And, even if Achilles was an interstellar spaceship and the Tortoise was a slightly slower interstellar spaceship, each strongly subject to relativistic effects, the resolution of the paradox would still hold true, though the amount of time and distance it takes may be different in different reference frames.

But the equation is continous, and a mathematical value can keep getting smaller and smaller. That is what infinity means, of course. We know reality does not work on such underlying principles. Reality is "granular", which is what I was saying about quantum physics. Let alone, the problem of Heisenbergs uncertainty.

The fact that you might not be able to measure Achilles move by 0.5 Planck lengths in 1 trillionth of a yoctosecond does not prevent him from moving 2 Planck lengths in 4 trillionths of a yoctosecond, nor 1 meter in 1 second. Again, the quantum scales are not relevant to the problem.

Anyway, I'll leave you guys to it... I've had my input here and there are many excellent brief descriptions out there only a google away explaining the resolution to this "paradox.

Edited by Bonam
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There is no "problem" with simultaneity, it merely holds a more complex definition than is intuitive when one considers things on relativistic scales. However, just like quantum effects, relativistic scales are not relevant in this particular problem. And, even if Achilles was an interstellar spaceship and the Tortoise was a slightly slower interstellar spaceship, each strongly subject to relativistic effects, the resolution of the paradox would still hold true, though the amount of time and distance it takes may be different in different reference frames.

The fact that you might not be able to measure Achilles move by 0.5 Planck lengths in 1 trillionth of a yoctosecond does not prevent him from moving 2 Planck lengths in 4 trillionths of a yoctosecond, nor 1 meter in 1 second. Again, the quantum scales are not relevant to the problem.

It's not that we cannot measure 0.5 planck lengths in some amount of time, it's that 0.5 planck lengths don't EXIST. If you accept that there is nothing smaller than planck space, and nothing can happen in less than planck time, then physical space and time is quantized and has boundary conditions. A mathematical function which is infinitely continuous does not actually describe physical reality at the lowest level. That points to the impossibility of knowing simultaneously the velocity and position of an object, relative to some fixed frame. Math provides a logically perfect abstraction, as numbers can be made small enough to fully describe a thing. But that does not work in the real world and the resolution to the paradox comes from the fact that reality is "digital", not analog, and that there are no fixed frames of reference in space or time between any observers. Zeno argued in favour of "discontinuity" but did not grasp relativity.

It's a boring tough topic. Blah blah... thanks google! :)

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Zeno argued in favour of "discontinuity" but did not grasp relativity.
Relativity has nothing to do with this. It is a basic math issue. Xeno did not understand that a finite number divided by infinity is zero. Edited by TimG
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Relativity has nothing to do with this. It is a basic math issue. Xeno did not understand that a finite number divided by infinity is zero.

Agreed...this is why development of The Calculus was so important to progress in science and engineering. This is what separates lower paid liberal arts majors from the higher paid engineers ! biggrin.png

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Agreed...this is why development of The Calculus was so important to progress in science and engineering. This is what separates lower paid liberal arts majors from the higher paid engineers ! biggrin.png

Hard to 'argue' with opinion on matters scientifical. Thus 'agreement', or underhanded references to the quality of someones degree are irrelevaant.

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Relativity has something to do with it. Zeno didn't grasp that the rate of change of time is different for observers at different velocities.
It has nothing to do with it. The Xeno's problem is he did not understand that it takes exactly 0 seconds to traverse an infinitely small chunk of space.
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It has nothing to do with it. The Xeno's problem is he did not understand that it takes exactly 0 seconds to traverse an infinitely small chunk of space.

Agreed....Zeno did not have the requisite tools to understand limit theory, discontinuities, time constants, transients, steady state, etc....things now taken for granted.

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Agreed....Zeno did not have the requisite tools to understand limit theory, discontinuities, time constants, transients, steady state, etc....

Tools that the vast majority of people still lack today, hence the confusion about basic scientific issues, beliefs in end of the world prophecies, susceptibility to hoaxes and quackery, etc.

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