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American
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04 Dec 2013, 11:12 pm

Does anyone else here like Zeno's paradoxes. I do and I think it's fascinating that they still remain unsolved. His more famous paradoxes are based on the problem with the notion that there are an infinite number of points in space, even within space the size of a electron. He uses that concept to show that the notion of motion (ha ha) is ridiculous.

The arrow paradox points out that an arrow in flight is never actually moving because, at any given point in time, it only occupies one point in space and thus is motionless during its entire flight.

The Achilles and the Tortoise paradox proves that, in a race between Achilles and a tortoise where the tortoise gets a head start, Achilles can never catch up. Say the tortoise gets a one hundred foot start, the tortoise moves one foot per minute and Achilles moves 100 feet per minute. By the time Achilles reaches where the tortoise was, the tortoise will have moved farther ahead. And every time Achilles gets to where the tortoise was, the tortoise will always be ahead and it is impossible for Achilles to overtake the tortoise though, presumably, given an infinite amount of time, Achilles could end up right next to the tortoise.

The half-way paradox proves that is impossible for anything to move anywhere because, in order to move from one point to another, one must pass the halfway point between those two points. Because there is an infinite number of points between any two points one would have to traverse an infinite number of points to go anywhere. Traversing an infinite number of points must take an infinite amount of time so one would need an infinite amount of time to make even the slightest motion.



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04 Dec 2013, 11:22 pm

I like them. However, calculus blew them away.



American
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04 Dec 2013, 11:37 pm

ModusPonens wrote:
I like them. However, calculus blew them away.


I know virtually nothing about calculus but my impression is that these paradoxes are philosophical in nature and I cannot see how math could solve them. Yes, math is logic, which philosophy, but... Can you explain more perhaps? BTW, great screename. Was modustollens taken?



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04 Dec 2013, 11:54 pm

American wrote:
ModusPonens wrote:
I like them. However, calculus blew them away.


I know virtually nothing about calculus but my impression is that these paradoxes are philosophical in nature and I cannot see how math could solve them. Yes, math is logic, which philosophy, but... Can you explain more perhaps? BTW, great screename. Was modustollens taken?


Because it proved that idea the an infinite sum must be infinite to be completely false. You CAN transverse an infinite number of points in a finite amount of time.


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American
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05 Dec 2013, 12:14 am

Ganondox wrote:
American wrote:
ModusPonens wrote:
I like them. However, calculus blew them away.


I know virtually nothing about calculus but my impression is that these paradoxes are philosophical in nature and I cannot see how math could solve them. Yes, math is logic, which philosophy, but... Can you explain more perhaps? BTW, great screename. Was modustollens taken?


Because it proved that idea the an infinite sum must be infinite to be completely false. You CAN transverse an infinite number of points in a finite amount of time.


That's logically impossible to do because no matter how many points you traverse there will always be another one and thus you can never traverse space at all. How can you finish traversing the points in space between two points when there is a never ending number of points to traverse? There will always be another point to traverse before you can move anywhere. "Traversing" an infinite number of points, even if possible, still wouldn't get you anywhere; it would not result in any motion ever. Obviously, motion appears to be possible, but that must be wrong. Simplicius said that Diogenes the Cynic got up and walked across the room upon hearing the half-way paradox in order to prove it wrong. But that doesn't satisfy me.



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05 Dec 2013, 12:31 am

American wrote:

That's logically impossible to do because no matter how many points you traverse there will always be another one and thus you can never traverse space at all. How can you finish traversing the points in space between two points when there is a never ending number of points to traverse? There will always be another point to traverse before you can move anywhere. "Traversing" an infinite number of points, even if possible, still wouldn't get you anywhere; it would not result in any motion ever. Obviously, motion appears to be possible, but that must be wrong. Simplicius said that Diogenes the Cynic got up and walked across the room upon hearing the half-way paradox in order to prove it wrong. But that doesn't satisfy me.


Zeno's fallacy appears to be the assumption that there is an infinite number of points in between a definite start point and a definite end point.

The distance in between a definite start point and definite end point is definite, not infinite.

No matter what unit of measurement you use, you cannot sub-divide the distance in between two definite points into a series of infinite units.



Last edited by LoveNotHate on 05 Dec 2013, 12:38 am, edited 1 time in total.

American
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05 Dec 2013, 12:38 am

LoveNotHate wrote:
The distance between a definite start point and a definite end point is not infinite.

So, how can there be an infinite number of points between the definite start and definite end points ?


A point has no volume or length at all. The distance between two points in some sense is perhaps finite but that doesn't matter because between any two points, there is an infinite number of points just as there is an infinite quantity of numbers between any two numbers. Time works the same way. A still photograph, for example, captures no amount of time--it's just one point in time. An arrow moving through space has to pass through an infinite number of such points--there are an infinite number of possible different photographs of the arrow. So, moving though space and time is clearly impossible, of that I am thoroughly convinced.



Last edited by American on 05 Dec 2013, 12:40 am, edited 1 time in total.

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05 Dec 2013, 12:40 am

American wrote:
LoveNotHate wrote:
The distance between a definite start point and a definite end point is not infinite.

So, how can there be an infinite number of points between the definite start and definite end points ?


Because a point has no volume or length at all. So between any two points, there is an infinite number of points just as there is an infinite quantity of numbers between any two numbers. Time works the same way. A still photograph, for example, captures no amount of time--it's just one point in time. An arrow moving through space has to pass through an infinite number of such points--there are an infinite number of possible different photographs of the arrow. So, moving though space and time is clearly impossible, of that I am thoroughly convinced.


If a point has no length , then it is meaningless. That means you can traverse an infinite number of points and not move.



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05 Dec 2013, 12:41 am

LoveNotHate wrote:
American wrote:
LoveNotHate wrote:
The distance between a definite start point and a definite end point is not infinite.

So, how can there be an infinite number of points between the definite start and definite end points ?


Because a point has no volume or length at all. So between any two points, there is an infinite number of points just as there is an infinite quantity of numbers between any two numbers. Time works the same way. A still photograph, for example, captures no amount of time--it's just one point in time. An arrow moving through space has to pass through an infinite number of such points--there are an infinite number of possible different photographs of the arrow. So, moving though space and time is clearly impossible, of that I am thoroughly convinced.


If a point has no length , then it is meaningless. That means you can traverse an infinite number of points and not move.


Exactly. That is the problem with motion. That is how we know that there is no such thing as motion.



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05 Dec 2013, 12:50 am

American wrote:
LoveNotHate wrote:
American wrote:
LoveNotHate wrote:
The distance between a definite start point and a definite end point is not infinite.

So, how can there be an infinite number of points between the definite start and definite end points ?


Because a point has no volume or length at all. So between any two points, there is an infinite number of points just as there is an infinite quantity of numbers between any two numbers. Time works the same way. A still photograph, for example, captures no amount of time--it's just one point in time. An arrow moving through space has to pass through an infinite number of such points--there are an infinite number of possible different photographs of the arrow. So, moving though space and time is clearly impossible, of that I am thoroughly convinced.


If a point has no length , then it is meaningless. That means you can traverse an infinite number of points and not move.


Exactly. That is the problem with motion. That is how we know that there is no such thing as motion.


The imaginary points spread out over a definite distance having no length exists only in thought, and not in the real world. Whereas, motion is in the real world.

Sure, if we assume some absurd theories about how the world works, then we can make absurd conclusions like motion does not exist. :)

I will let you get the last word in.



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05 Dec 2013, 1:08 am

LoveNotHate wrote:
The imaginary points spread out over a definite distance having no length exists only in thought, and not in the real world. Whereas, motion is in the real world.

Sure, if we assume some absurd theories about how the world works, then we can make absurd conclusions like motion does not exist. :)

I will let you get the last word in.


I see what you mean but I must disagree. How about a photograph? Do you agree that a photograph has no time component? It is a representation of a single point in time, an infinite number of which exist in every nanosecond.

There must be such a thing as "points" in space and time because there much be such a thing as the smallest possible unit of space or time. Points are infinitesimally small. Points are the smallest possible units of space or time. Points have no length or volume because, if they did, they could be smaller, and thus would not represent the smallest possible unit of space or time, which necessarily must exist if space and time exists.



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05 Dec 2013, 4:03 am

Actually, a photograph does have a time component, exposure time.

and any "object", like an arrow, occupies a volume of space, not a single point. Any object cannot be described as a single point.
A point is just a point, occupying zero dimensions. It can only have a position relative to two or more other points.
Motion is a change in position during a given time interval.
A time interval is a line segment in a temporal dimension.

A single point by itself can't do anything. It can't even move. If it's moving, it's not a point.
For motion to occur, there needs to be at least two dimensions, one of them being designated a temporal dimension.
The "moving" object would be a line segment running any distance into the temporal dimension.

I don't know what I'm getting at here, just kicking ideas around.


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05 Dec 2013, 5:30 am

Pretty much the only answer - there really isn't an answer.

The only thing I can really think of as a working hypothesis as to why infinities in either direction wouldn't blow relevant ranges out of existence is the possibility that time and space themselves are relative to the observer and perhaps don't exist as anything more than measuring tools extended by the observer to make sense of incoming data. In that case being able to slice time or space infinitely or unroll infinitely would have the same consequences as our ability to find infinitely many non-integers between 0 and 1 - ie. as little or as much as we chose to make of it.



Last edited by techstepgenr8tion on 05 Dec 2013, 5:31 am, edited 1 time in total.

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05 Dec 2013, 5:30 am

American wrote:
There must be such a thing as "points" in space and time because there much be such a thing as the smallest possible unit of space or time. Points are infinitesimally small. Points are the smallest possible units of space or time. Points have no length or volume because, if they did, they could be smaller, and thus would not represent the smallest possible unit of space or time, which necessarily must exist if space and time exists.


Actually in the real world there are no points. The smallest length is the Planck length. 1 planck length = 1.61619926 × 10-35 metres. When distances become so small common sense notions don't apply and quantum physics is the only way to describe what is happening at such small scales. It is almost as though our universe is granular (or digital) in nature rather than analogue. So if you took an everyday length such as one metre in the real world, it doesn't have an infinite number of points along it. The metre could at most be divided into 1.61619926 × 10^35 segments, each segment being indivisible.


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05 Dec 2013, 12:58 pm

I predict that some day the plank length will be dis-proven. I can't put my finger on it, but something about it doesn't fit. Seems like just a limit to our observation capabilities. Kinda like a horizon on an additional dimension of scale. Which means that there are even more dimensions on top of that. 3 spatial dimensions + 1 time just aren't enough.

granular universe? only when observed, according to wave/particle duality.


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05 Dec 2013, 3:10 pm

I finally caught the turtle!