Zeno's first paradox

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Zeno's first paradox

 The first paradox

1. The first paradox concerns the absolute motion of a single object, assuming that space is continuous. The argument is often referred to as the ‘dichotomy’.

 

 

2. Zeno starts by assuming what his opponent says is possible: motion. In particular, the motion of a single body across a finite distance in a finite time.

 

 

3. To make things vivid, let’s specify the moving object, and where it is supposed to be moving: Imagine a sprinter, who starts running at one end of a 100 metre straight track, and runs to the other end. Zeno’s opponents (probably including yourself) think that it really is possible that runners can do this. (i.e. They think that it is not just an illusion.) Zeno begins by assuming that his opponents are correct, and then the fun starts.

 

 

4. Zeno points out, given that we are assuming that space is continuous, that before the runner can cover the whole 100 metres she has to cover half the distance, i.e. run 50 metres.

 

 

5. Zeno then repeats the move just made (point 4) and points out that the move can be repeated an infinite number of times: before the runner can cover 50 metres, she has to run 25, before that she must run 12.5, before that 6.25, etc. Recall that we are assuming that space is continuous, which means that any finite piece of it, such as a sprint track, can be divided into infinitely many parts, which are infinitely small.

 

 

6. Zeno then argues that to cover each of these infinitely small parts will take a certain amount of time.

 

 

7. But to take a certain amount of time an infinite number of times adds up to an infinite amount of time. So it would take the runner forever to cover 100 metres.

 

P.S. this is just a c&p from my philosophy course.

I know this is ridiculous but i just find it so amusing. Just thought i would share. If anyone really wants to make it less amusing feel free to point out where he went wrong.

 

 

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Quote:7. But to take a

Quote:
7. But to take a certain amount of time an infinite number of times adds up to an infinite amount of time. So it would take the runner forever to cover 100 metres.

Zeno didn't know about infinitesimals and integral calculus. To travel 50 metres takes half the time as to travel the full 100. To travel 25 is half the time as 50. 12.5 half 25. Ad inifinitum.

When you sum up the series 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/(2^n) as n approaches infinity, the sum approaches *not* infinity, but exactly 1.

So, yes, he's right. Before you travel 100, you must travel 50, but you do so in half the time, and the total time to travel 100 remains the same.

Personally, I find the other variation much more confusing (and therefore satisfying as a paradox): To travel 100, you must first travel halfway to 50. To travel the remaining 50, you must first travel halfway to 75. To travel the remaining 25, you must travel half that, etc.

So, for any particular distance you want to travel, you may get frustratingly close, but you will never get all the way there, because to do so would require an infinite number of 'steps', and for each step there is always a remaining 'halfway there' distance yet to be traveled.

This one is more difficult to see the solution than the one you mentioned. In your version, you are dividing the distance up into an infinite number of slices, but the sum of the slices always remains 100m, and the total time likewise remains equal to the original time.

However, in the version I propose, it's a question of the number of 'steps' that must be completed, which arbitrarily grows without bounds.

The way to solve this problem is to realize that motion does not occur in such 'steps'. You do not jump from 0 to 50, and then jump from 50 to 75, and 75 to 87.5. Instead, you have a constant velocity which represents your fluid motion, which can be calculated for any particular velocity and elapsed time, but which *occurs* as a fluid transition through space.

If you are travelling at 10 m/s then at 5 seconds, you will have traveled 50 metres, but at 10 seconds, you will have traveled the full 100 metres, regardless of the arbitrarily chosen 'steps' in between 50 and 100 metres. The 'steps' are not real steps at all, they are just conceptual states of the system at arbitrarily chosen points in time.

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I believe the wonder of

I believe the wonder of Zeno's Paradox is that it came so close to the calculus. Just a little more thought, and a desire to actually discover the truth, and we would've been several hundred years farther along.

I believe the lesson of Zeno's Paradox is that a philosopher can take all the correct ingredients, mix them up, and come up with something so absolutely and sincerely wrong that it misdirects actual knowledge. It's a good demonstration of why pure philosophy is severely flawed.

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Well, at least it got people

Well, at least it got people thinking about the problem. And it's not like it was called Zeno's Dogma, and there was no Church of Zeno, etc.

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lol true I have seen the

lol true I have seen the steps one,  this one is even easier but no less amusing.

 The second paradox

1. The second paradox concerns the relative motion of two objects, also assuming that space is continuous.

 

 

2. Again Zeno starts by assuming what his opponent says is possible: motion. In particular, that one body can move more quickly relative another.

 

 

3. Again, to make things vivid, let’s specify the moving object, and where it is supposed to be moving: Imagine a very quick sprinter, who is going to race a much slower runner. To make things fair, the slower runner is given a head start, and he begins the race at the 40 metre mark, so that he only has to cover 60 metres to get to the finish. Zeno’s opponents (probably including yourself) think that it really is possible that a fast enough sprinter could overtake the slower runner and win the race, even with a head start for the slower runner, if the sprinter is quick enough. (i.e. They think that it is not just an illusion.) Again, Zeno begins by assuming that his opponents are correct.

 

 

4. So the race starts, and the sprinter starts running quickly, while the slower runner starts moving slowly. We agree (that is, us who think motion is possible, and Zeno for the purposes of argument) that the sprinter catches up with the slower runner very quickly, i.e. the sprinter reaches the 40 metre mark.

 

 

5. Zeno points out that while the sprinter has done this, the slower runner has not been doing nothing, but has moved ahead slightly. So the sprinter must now catch up to the point the runner has got to during the previous catch up.

 

 

6. Zeno now repeats the above stage of the argument, pointing out that it too (see first paradox above) can be repeated an infinite number of times. The faster sprinter always has some catching up to do, and can never pull alongside the slower runner, let alone overtake.

 

This one is even easier, I really enjoy this guy 

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natural wrote:Zeno

natural wrote:
Zeno didn't know about infinitesimals and integral calculus. To travel 50 metres takes half the time as to travel the full 100. To travel 25 is half the time as 50. 12.5 half 25. Ad inifinitum.

 

When you sum up the series 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/(2^n) as n approaches infinity, the sum approaches *not* infinity, but exactly 1.

 

So close...

 

Zeno of Elea was making a point about epistemology. If you think hard about stuff, you can reach bogus conclusions.

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Answers in Gene Simmons

Answers in Gene Simmons wrote:

natural wrote:
Zeno didn't know about infinitesimals and integral calculus. To travel 50 metres takes half the time as to travel the full 100. To travel 25 is half the time as 50. 12.5 half 25. Ad inifinitum.

 

When you sum up the series 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/(2^n) as n approaches infinity, the sum approaches *not* infinity, but exactly 1.

 

So close...

 

Zeno of Elea was making a point about epistemology. If you think hard about stuff, you can reach bogus conclusions.

 

My point is that if you think even harder, you can overcome such apparent paradoxes.

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The basic error in Zeno's

The basic error in Zeno's reasoning is the assumption that the sum of an infinite number of 'steps' of some kind must be infinite. There is no actual logical argument produced to justify this intuitive assumption.

Philosophical errors commonly arise from such failures to logically examine such intuitive assumptions. There is a real problem in that we are deeply dependent on intuitions to guide our reasoning, so we need a way to test them, which is where comparing our intuitive conclusions against reality, as far as we can perceive it, becomes necessary to make further progress in gaining knowledge.

This is where dedication to 'pure reason' fails - if there is a discrepancy between observation and reasoning, it is dismissed as due to the imperfection of our senses, rather than of our intuitive assumptions. The similarity to the religious 'faith' approach should be obvious...

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OK, you both missed the

OK, you both missed the point that I was making. If we consider what Zeno said through the lens of modern knowledge, then yes, it is trivial to demonstrate that he was full of shit. However, I would consider his work in the context of what tools he did have access to.

 

Arrows fly to a target and Achilles can out run a tortoise with a small head start. Such is obvious to basic observation. Even Zeno must have been aware of stuff like that. So what motivated him to propose the paradoxes that he did?

 

Seriously, answer that question in the terms that Zeno of Elea would know and you are back to basic questions like “How do we know what we know?”

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He was demonstrating that,

He was demonstrating that, using the logic and math assumptions of the time, one could come to conclusions that were clearly in conflict with reality.

They correct reaction to this should be that either the logic and math, or some of the assumptions (axioms) on which they were based were wrong, or our 'knowledge' of the world was wrong, or perhaps both. Whether that was his conclusion, or his intention to point this out, I don't know.

The appropriate reaction should be to look at each of those elements, and assess the likelihood of error in each of those elements being wrong to the degree that would result in this 'paradox', based on all our other experience and 'knowledge'.

From our perspective, after the development of the math to handle infinite series, and differential calculus, we can see the clear logical error, which I pointed out. Therefore the immediate 'message' we should take from this is that when confronted with theory or logical argument conflicting with observation, it is still most likely that it is something in the theory or our assumptions that is wrong, rather than some more fundamental problem in apprehending reality.

The other way to think about it is that even the most deeply perplexing problem may turn out to be a 'no-brainer' given some new way to look at the problem, some new tools to analyse such problems which were not originally available.

It takes time to work thru and verify ever more powerful theorems in math, and theories in science, to make progressively more accurate models of reality, so grand philosophical or even (puke!) theological conclusions based on current inabilities for logic and science to explain adequately any aspect of reality have been shown, time after time, to be rather premature.

Given the options of a relatively simple logical error or incorrect assumption, such as 'the sum of an infinite number of unspecified quantities must always be infinite', or something more fundamental, work on eliminating those errors before proposing something broader, like 'our whole theory of knowledge is fundamentally flawed'.

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Answers in Gene Simmons

Answers in Gene Simmons wrote:

OK, you both missed the point that I was making. If we consider what Zeno said through the lens of modern knowledge, then yes, it is trivial to demonstrate that he was full of shit. However, I would consider his work in the context of what tools he did have access to.

I think you may be missing my point. To me, it doesn't matter what he had access to. It matters whether the paradox is really a paradox, or just an intuitional puzzle. If he was trying to show that motion is a paradox (which wouldn't surprise me since lots of Greeks thought really weird things like idealism and numeric mysticism), then he was wrong. If he was trying to show that current ideas about motion were paradoxical, then I've shown how to resolve the paradox. If he was trying to show that our complicated ideas can lead to paradoxes, then I've shown that even more complicated ideas can resolve those paradoxes. If he was trying to show that we can't trust our knowledge, then he was wrong.

In other words, whatever point he had is moot.

By the way, do you have a link to anything describing what his actual intentions were, regarding this puzzle? I'm only familiar with the puzzle itself, not his motivations.

Quote:
Arrows fly to a target and Achilles can out run a tortoise with a small head start. Such is obvious to basic observation. Even Zeno must have been aware of stuff like that. So what motivated him to propose the paradoxes that he did?

Personally, I think a perfectly valid reason is to propose a paradox for the precise reason of trying to refute it. If it is indeed a paradox, we've discovered something interesting about reality. If it is only an apparent paradox, then we've discovered something interesting about ourselves.

For example, two of my favourite 'paradoxes' are really just intuitional puzzles: The Monty Hall Problem and the Birthday Problem. In both of these 'paradoxes', you find people defending the wrong idea quite vehemently. They are convinced that switching doors does not give you an advantage, or they are convinced that 23 random people are unlikely to have any shared birthdays. They are absolutely convinced.

What is happening is that their intuitions, which are our natural ability to make pretty good guesses, are leading them to make the wrong guesses. Essentially, the problem is that our intuitions are not perfect, and in fact have systematic biases that are predictable.

A problem like Zeno's paradox is another good example of an intuitional bias. The bias here is that we tend to think in terms of conceptual algorithmical steps when analyzing a problem, and each 'step' takes a certain amount of time, and so when we notice a repeating pattern, we assume that it will 'go on forever'. But the world doesn't work according to our conceptual cognition; that's only how we *think* about the world. It the confusion of intuition with reality.

So, for me, the interesting problem is that Zeno's paradox teaches us something about ourselves, that our intuitions need to be tempered with analysis and logic. If you are puzzled by Zeno's paradox, don't worry, there's a solution.

Zeno did not have the solution, as far as I'm aware. He either genuinely thought it was a paradox, or he thought it proved that thought leads to inevitable paradox, or he thought it disproved (by contradiction) his opponents' ideas about motion. In any of these cases, he would have been wrong.

Today, we have the solution, and it wasn't discovered by declaring "we can't really know anything" or "thought inevitably leads to paradox", or any other such platitude. It was discovered by more thought, and more analysis, and by working on the paradox until it was resolved.

If Zeno had an epistemological point, it is either wrong or irrelevant.

Quote:
Seriously, answer that question in the terms that Zeno of Elea would know and you are back to basic questions like “How do we know what we know?”

 

We know what we know by making predictions based on our best guesses and testing those predictions against reality. You can't fake accurate and reliable predictions. If we can make accurate and reliable predictions, then we 'know' that whatever generated those predictions is 'true'. This is why I say that truth is like an arrow. If the arrow is 'true', it will strike its target (make an accurate prediction).

This is also something Zeno didn't know (at least not consciously).

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Quote:For example, two of my

Quote:

For example, two of my favourite 'paradoxes' are really just intuitional puzzles: The Monty Hall Problem and the Birthday Problem. In both of these 'paradoxes', you find people defending the wrong idea quite vehemently. They are convinced that switching doors does not give you an advantage, or they are convinced that 23 random people are unlikely to have any shared birthdays. They are absolutely convinced.

Really? People defend the intuitive "switching doesn't matter" in the Monty Hall Problem? That's completely ridiculous. This is elementary mathematics and it takes seconds to demonstrate using a simple probability tree that switching does give you a 2/3 chance whereas keeping the door you just picked gives you a 1/3 chance. This is mathematics. You cannot argue with mathematics. It is not a matter of conviction. If you construct a logical proof based on premises in turn established in a similar a priori fashion, then that's it. End of discussion. That's the best part about it. The whole system is a priori so the notion that "something will eventually be proven wrong" is nonsense. 

If someone really for some reason cannot get this through their head, the best way to explain it is to extend the game to have 1 million doors (or some other large number), where you pick a door, the host opens 999,998 other doors showing goats behind them, leaving one door,and the one you picked. Now it should be very clear to anyone that you should switch, because the fact that the host opened 999,998 doors doesn't change the fact that when you initially picked the door, there was only a 1 in 1,000,000 chance of this door having a car behind it. This is the crucial thing which people simply cannot seem to understand. It follows, therefore, by elementary probability, that the other door has a 999,999 in 1,000,000 chance of having the car behind it.

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natural wrote:Answers in

natural wrote:

Answers in Gene Simmons wrote:

OK, you both missed the point that I was making. If we consider what Zeno said through the lens of modern knowledge, then yes, it is trivial to demonstrate that he was full of shit. However, I would consider his work in the context of what tools he did have access to.

I think you may be missing my point. To me, it doesn't matter what he had access to. It matters whether the paradox is really a paradox, or just an intuitional puzzle. If he was trying to show that motion is a paradox (which wouldn't surprise me since lots of Greeks thought really weird things like idealism and numeric mysticism), then he was wrong. If he was trying to show that current ideas about motion were paradoxical, then I've shown how to resolve the paradox. If he was trying to show that our complicated ideas can lead to paradoxes, then I've shown that even more complicated ideas can resolve those paradoxes. If he was trying to show that we can't trust our knowledge, then he was wrong.

In other words, whatever point he had is moot.

By the way, do you have a link to anything describing what his actual intentions were, regarding this puzzle? I'm only familiar with the puzzle itself, not his motivations.

 

I belive we don't have any of his original writings, All we have are second hand accounts witch are incomplete. So I don't think there is anything definate about his motivation. Was there even writing 500bc? (think thats when he was around, could be wrong) But from what I understand he wrote them to support Parmenides arguement. Basically to Parmenides's oppenents more dificulty.  

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BobSpence1 wrote:The basic

BobSpence1 wrote:

The basic error in Zeno's reasoning is the assumption that the sum of an infinite number of 'steps' of some kind must be infinite. There is no actual logical argument produced to justify this intuitive assumption.

I think this is the case.

 

Also, isn't it true that .999999 repeating IS the same as 1?

 

Finally, planck length does place a limit on what we might call the 'smallest' amount of space, I don't think we can bisect any further...

 

So Zeno's paradoxes might be a clash between mathematical models and the real world, wherein the model is flawed due to the mistaken assumption that the real world allows for infinite bisections....

 

Well, anyway, that's the best a non mathematician can make out of all of this.

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todangst wrote:Also, isn't

todangst wrote:

Also, isn't it true that .999999 repeating IS the same as 1?

Given that it implies an infinite series of fractions, yes. Newton's proof is pretty long, but I think Leibniz had a simpler one.

todangst wrote:
So Zeno's paradoxes might be a clash between mathematical models and the real world, wherein the model is flawed due to the mistaken assumption that the real world allows for infinite bisections....

In fact, Zeno's mistake fails in the mathematical world before it fails in reality! An infinite number of bisections added is the series 1/2^n, where n=1 to infinity, which is equal to 1 (the full distance required to travel), as natural said.

edit April 12:

For those interested, this is the simplest proof I know of:

0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k}  = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,

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Answers in Gene Simmons

Answers in Gene Simmons wrote:

Seriously, answer that question in the terms that Zeno of Elea would know and you are back to basic questions like “How do we know what we know?”

 

It's difficult to see him really making that point, there. Zeno was famous for giving clever reductio ad absurdum arguments, and could have been aware of more reasonable mathematical approaches to the problem (from Pythagorean math), but was actually proposing a problem to which there seemed to be no answer.

Of course, modern proofs exhaustively resolve the issue philosophically, but mathematically, it's been less difficult.

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There is another lesson to

There is another lesson to be learned from considering Zen's 'paradoxes'.

It is that while it is easy to describe a natural, intrinsically continuous process taking a finite time anf covering a finite distance in a way that involves an infinite number of elements. This is a an entirely conceptual and artificial process, involving no 'real' infinite quantity of any 'real' entities.

This has analogies with many so-called 'infinite' regress problems of 'cause-effect', or "what created the creator" arguments. The conceptual analysis of the whole process as a sequence of actual discrete 'events' may well be as artificial as the partitioning of a finite length into an infinite sequence of ever-smaller segments. Whereas the actual process may well be mostly a finite sequence of one or more continuous processes of change.

An actual infinity of space or time or of finite objects is a purely conceptual thing. Even if it we came across something that we suspect may be truly infinite in this sense, I don't think there is any way we could demonstrate even the probability, let alone prove, that it was actually infinite. It is actually a kind of negative, all we could say is that we can find no limit or bound or edge in at least one dimension.

The lesson from Zeno may be that we should not assume the existence of any actual infinities. Any argument which seems to be based on such is higly suspect, and this is actually what happens in science - theories which seem to predict infinite quantities or dimensions are highly suspect.

I don't see how even our Universe, even whatever extends beyond the event horizon, could be infinite, since it started at a size which was virtually the opposite of infinite, and has been expanding at a finite rate for a finite time.

EDIT: In case anyone wants to object that a finite Universe must have an 'edge', a boundary, that is covered by the curvature of space-time, so that a finite universe has no edge in the same way that the curved surface of the Earth has no edge.

So any arguments which propose anything actually infinite would seem to be highly questionable from the start, since there really is no way we can say anything meaningful about such a thing based on any analogy from observation or experience. People arguing such things are in an even weaker position that Zeno.

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HisWillness wrote:todangst

HisWillness wrote:

todangst wrote:

Also, isn't it true that .999999 repeating IS the same as 1?

Given that it implies an infinite series of fractions, yes. Newton's proof is pretty long, but I think Leibniz had a simpler one.

todangst wrote:
So Zeno's paradoxes might be a clash between mathematical models and the real world, wherein the model is flawed due to the mistaken assumption that the real woorld allows for infinite bisections....

In fact, Zeno's mistake fails in the mathematical world before it fails in reality! An infinite number of bisections added is he series 1/2^n, where n=1 to infinity, which is equal to 1 (the full distance required to travel), as natural said.

 

Ah, thank you both!

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HisWillness wrote:For those

HisWillness wrote:

For those interested, this is the simplest proof I know of:

0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k}  = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,

That's the most transparent proof I know of, it exposes the zero difference between 0.999r and 1 with elegant clarity but it doesn't use the simplest arithmetic available for making the argument.

If we do this -

 0.9999999. = a

multiply both sides by 10

9.999999. = 10a

(since 0.9999999. = a)

9 = 10a - a

and that gives us

9 = 9a

So a is 1.

This same process also works to rationalise 0.3333. as 1/3 ie:

9a = 3

a=3/9

and 0.6666. as 2/3 ie:

9a = 6

a = 6/9

because these are much better known equalities, using this process can make the strangeness of 0.999r = 1 digestible for just about anybody.

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HisWillness
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Eloise wrote:multiply both

Eloise wrote:

multiply both sides by 10

Bah! Right! I completely forgot that one. That's the more famous one (and rightly said, the simplest). I got all caught up in being comprehensive.

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Eloise wrote:That's the most

Eloise wrote:


That's the most transparent proof I know of, it exposes the zero difference between 0.999r and 1 with elegant clarity but it doesn't use the simplest arithmetic available for making the argument.

If we do this -

 0.9999999. = a

multiply both sides by 10

9.999999. = 10a

(since 0.9999999. = a)

9 = 10a - a

and that gives us

9 = 9a

So a is 1.

 

Thank you, this is elegant.

 

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