Actually I just realised that Zeno was right in at least two of those dichotomy arguments. The values in the infinite series represents positions that must have been passed through if the object had travelled further. So it doesn't depend on any naive pre-scientific view of space, it is true for any assumption that an object travels some distance at a certain velocity on some continuous path.
The convergent series doesn't harm the argument, it makes no difference if infinitely many things can be done in a finite time time because all of the positions mapped on those convergent series are at a finite distance from the finish.
So the concept of motion itself leads to a contradiction.
It has been built into our mathematics and science since at least Newton.
It makes no sense to go on pretending he was wrong because, well we have known for a long time that 'motion' as we experience it is an approximation of something that is somewhat different. So it does not matter that he was right.
And you don't have to tart up those arguments, they are pretty much just fine as they are. It is the objections that are wrong.
Here is the argument, pretty much as Zeno left it:
Definitions:
P is a continuous path of length l
x is an object that travels through path at a velocity that gets it to the end in a duration of t.
S is a series where the first term is 1/2 and the subsequent terms are the sum of the previous term and one half of the previous term.
S={1/2, 1-1/4, 1-1/8, 1-1/16, ...}
Argument:
1. x can travel through p and reach the position 1. (Assumption for reductio ad absurdum)
2. The series represents positions that the object must have passed through if it is further than them.
3. S is an infinite series and all of the terms are greater than zero and so the object cannot reach 1.
4. But the definition of motion requires that the object reaches 1.
The definition of motion leads to a contradiction and therefore is incoherent. (contradiction 1,3)
Conclusion: There is no such thing as motion as described in the assumption.
There. Don't bother with "how did it get half way if it didn't move?" That makes no difference because the assumption that it moves is made for the purposes of contradiction it only matters if that this assumption leads to a contradiction, which it does.
convergent series is not a problem because the argument does not say that infinitely many things cannot happen in a finite time, because all of the terms are non zero then even after infinitely many iterations the object is still not at zero.
But the sum of the series says the object gets to 1 and therefore causes another contradiction with itself.
Happy hunting.
The convergent series doesn't harm the argument, it makes no difference if infinitely many things can be done in a finite time time because all of the positions mapped on those convergent series are at a finite distance from the finish.
So the concept of motion itself leads to a contradiction.
It has been built into our mathematics and science since at least Newton.
It makes no sense to go on pretending he was wrong because, well we have known for a long time that 'motion' as we experience it is an approximation of something that is somewhat different. So it does not matter that he was right.
And you don't have to tart up those arguments, they are pretty much just fine as they are. It is the objections that are wrong.
Here is the argument, pretty much as Zeno left it:
Definitions:
P is a continuous path of length l
x is an object that travels through path at a velocity that gets it to the end in a duration of t.
S is a series where the first term is 1/2 and the subsequent terms are the sum of the previous term and one half of the previous term.
S={1/2, 1-1/4, 1-1/8, 1-1/16, ...}
Argument:
1. x can travel through p and reach the position 1. (Assumption for reductio ad absurdum)
2. The series represents positions that the object must have passed through if it is further than them.
3. S is an infinite series and all of the terms are greater than zero and so the object cannot reach 1.
4. But the definition of motion requires that the object reaches 1.
The definition of motion leads to a contradiction and therefore is incoherent. (contradiction 1,3)
Conclusion: There is no such thing as motion as described in the assumption.
There. Don't bother with "how did it get half way if it didn't move?" That makes no difference because the assumption that it moves is made for the purposes of contradiction it only matters if that this assumption leads to a contradiction, which it does.
convergent series is not a problem because the argument does not say that infinitely many things cannot happen in a finite time, because all of the terms are non zero then even after infinitely many iterations the object is still not at zero.
But the sum of the series says the object gets to 1 and therefore causes another contradiction with itself.
Happy hunting.
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