It goes like this: At every instant of time an arrow is in a definite place. Therefore the arrow is at rest at every instant and it is impossible that it can move. How can something that is at rest at all times move?

Personally I don't think the conclusion follows from the premise. Why should one conclude that because the arrow is "somewhere" at a given time that it is also at rest?

I don't know the value of this discussion, but aren't we talking about the necessary illusion of the past, present and future? Aren't these the best concepts we have for coherent thinking involving us personally and time? As time is simply the measurement of motion points in time are of course a contradiction, but it is not really points in time but blocks of time which also or did or will contain motion. The stop motion is only in our heads, such as the 17th Century or the Age of Reason or WWII or what I am doing right now in the present.

Yes, I don't believe in paradoxes. I believe they result from incorrect analysis of the problem. There is another one of Zeno's about moving halfway towards your destination each time and you will mathematically never reach it. All this illustrates is that mathematics only applies to the physical world in a limited fashion - to expect otherwise is unsane.

Yes, I don't believe in paradoxes. I believe they result from incorrect analysis of the problem. There is another one of Zeno's about moving halfway towards your destination each time and you will mathematically never reach it.

The only thing that is wrong in Zeno's paradox is his conclusion that you'll never reach your destination. He didn't realize that it is possible to traverse the infinite number of steps in his analysis in a finite time interval.

The only thing that is wrong in Zeno's paradox is his conclusion that you'll never reach your destination. He didn't realize that it is possible to traverse the infinite number of steps in his analysis in a finite time interval.

I don't think so It makes no sense to speak about "an infinite number of steps" in the physical world. In mathematics we may speak about an infinite series but this exists only in our imagination.

I don't think so It makes no sense to speak about "an infinite number of steps" in the physical world.

Try doing physics without calculus...

In mathematics we may speak about an infinite series but this exists only in our imagination.

A circle or an ellipse also exist only in our imagination, but they are still quite useful to describe practical situations, as long as you don't confuse the mathematical construct with the physical reality.

Yes, I don't believe in paradoxes. I believe they result from incorrect analysis of the problem. There is another one of Zeno's about moving halfway towards your destination each time and you will mathematically never reach it.

The only thing that is wrong in Zeno's paradox is his conclusion that you'll never reach your destination. He didn't realize that it is possible to traverse the infinite number of steps in his analysis in a finite time interval.

Zeno was being ironic. He knew that motion is possible (he moved about) and one does get from Here to There. He was trying to prove our ideas about motion were paradoxical and he was basically right. The calculus "solution" or resolution to Zeno's paradox does not really answer the underlying problem. It is a heuristic that happens to get the right answer. The real answer is that the world is not infinitely divisible and topologically compact. So the ontology of the calculus (or analytic approach) is wrong. In the physical world there are quantities that are not infinitely small (infinitesimal) or infinitely divisible (such as space and time). At the very bottom reality is grainy and gritty. Unfortunately, mathematics based on an atomic unit of time or length is intractable which is why we use topologically compact analytic models.

Yes, I don't believe in paradoxes. I believe they result from incorrect analysis of the problem. There is another one of Zeno's about moving halfway towards your destination each time and you will mathematically never reach it.

The only thing that is wrong in Zeno's paradox is his conclusion that you'll never reach your destination. He didn't realize that it is possible to traverse the infinite number of steps in his analysis in a finite time interval.

That is right. He had no concept of convergence. However his arrow paradox cannot be dismissed to easily. How does any object that is motionless at every instant of time move. The answer is that there is really no such thing as an infinitesimal interval or point of time. That is how we model time, but that is because it leads to a tractable mode of analysis. Time is not really like that. The Planck interval is probably irreducible as is Planck length.

I don't think so It makes no sense to speak about "an infinite number of steps" in the physical world.

Try doing physics without calculus...

I don't understand you DF. All I am saying is that in reality it only takes a finite number of steps to reach your destination because of physical constraints you reach a point where you cannot 'halve' the distance anymore. It is not a question of doing physics without calculus.

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## tjohnson

Personally I don't think the conclusion follows from the premise. Why should one conclude that because the arrow is "somewhere" at a given time that it is also at rest?

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## Brant Gaede

I don't know the value of this discussion, but aren't we talking about the necessary illusion of the past, present and future? Aren't these the best concepts we have for coherent thinking involving us personally and time? As time is simply the measurement of motion points in time are of course a contradiction, but it is not really points in time but blocks of time which also or did or will contain motion. The stop motion is only in our heads, such as the 17th Century or the Age of Reason or WWII or what I am doing right now in the present.

--Brant

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## tjohnson

Yes, I don't believe in paradoxes. I believe they result from incorrect analysis of the problem. There is another one of Zeno's about moving halfway towards your destination each time and you will mathematically never reach it. All this illustrates is that mathematics only applies to the physical world in a limited fashion - to expect otherwise is unsane.

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## Dragonfly

The only thing that is wrong in Zeno's paradox is his conclusion that you'll never reach your destination. He didn't realize that it is possible to traverse the infinite number of steps in his analysis in a finite time interval.

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## tjohnson

I don't think so It makes no sense to speak about "an infinite number of steps" in the physical world. In mathematics we may speak about an infinite series but this exists only in our imagination.

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## Michael Stuart Kelly

This whole exercise rests on the premise that something can exist without time and our eyeballs can filter time out.

The reality is that time is a fundamental part of everything and our eyeballs evolved to perceive that part (time) just as much as the other parts.

This paradox comes off to me like trying to imagine sight without light waves, then claiming a paradox exists.

Michael

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## Dragonfly

Try doing physics without calculus...

A circle or an ellipse also exist only in our imagination, but they are still quite useful to describe practical situations, as long as you don't confuse the mathematical construct with the physical reality.

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## BaalChatzaf

AuthorZeno was being ironic. He knew that motion is possible (he moved about) and one does get from Here to There. He was trying to prove our ideas about motion were paradoxical and he was basically right. The calculus "solution" or resolution to Zeno's paradox does not really answer the underlying problem. It is a heuristic that happens to get the right answer. The

realanswer is that the world is not infinitely divisible and topologically compact. So the ontology of the calculus (or analytic approach) is wrong. In the physical world there are quantities that are not infinitely small (infinitesimal) or infinitely divisible (such as space and time). At the very bottom reality is grainy and gritty. Unfortunately, mathematics based on an atomic unit of time or length is intractable which is why we use topologically compact analytic models.Ba'al Chatzaf

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## BaalChatzaf

AuthorThat is right. He had no concept of convergence. However his arrow paradox cannot be dismissed to easily. How does any object that is motionless at every instant of time move. The answer is that there is really no such thing as an infinitesimal interval or point of time. That is how we

modeltime, but that is because it leads to a tractable mode of analysis. Time is not really like that. The Planck interval is probably irreducible as is Planck length.Ba'al Chatzaf

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## tjohnson

I don't understand you DF. All I am saying is that in reality it only takes a finite number of steps to reach your destination because of physical constraints you reach a point where you cannot 'halve' the distance anymore. It is not a question of doing physics without calculus.

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