Something drops half a distance

Originally posted by PatNovak
There is an underlying assumption in this paradox that a finite distance is infinitely divisible. If a finite distance is not infinitely divisible, then it is perfectly reasonable to consider the paradox as defeated on those grounds.

All physics theories that have space-times that are non-continuous are speculative. The physics of the standard model of particle theory has a continuous background. General relativity has a continuous background. This means that you can't knock down the paradox on those grounds.

Originally posted by DeepThought
All physics theories that have space-times that are non-continuous are speculative. The physics of the standard model of particle theory has a continuous background. General relativity has a continuous background. This means that you can't knock down the paradox on those grounds.

That is why I said IF in my post. I said "If a finite distance is not infinitely divisible, then it is perfectly reasonable to consider the paradox as defeated on those grounds." I did not say the paradox was definitely defeated on those grounds. This was a discussion about whether or not it is possible to defeat a paradox by defeating an underlying assumption, and the specific underlying assumption was inconsequential to the point.

Originally posted by PatNovak
This is more of a philosophical question than anything, because infinity is an abstract concept. So there probably isn't a correct answer.

No, it is not philosophical, it is mathematical. And there is a correct answer.

However, I do have to disagree with your assertion that "the full infinite sum is not considered part of the sequence." (I assume your are referring to the sequence 1/2, 3/4, 7/8...). Here, The sequence is defined as the sum, so the sequence and the sum are the exact same thing (and the words "sum" and "sequence" may be used interchangeably, since one is defined as the other). Your assertion above requires the sum to be unequal to the sequence only at infinity, which is arbitrary.
The sequence is a sequence of partial sums of another sequence.
We have the sequence: 1/2,1/4,1/8 .... 1/n ....
Then we have the sequence of sums: 1/2,3/4,7/8.....(sum of n terms of the first sequence).
The sequence of sums above does not include the number 1 as a member. There is no integer n for which the nth term of the sequence is 1.

Edit: I also disagree with this assertion: "Every member of the sequence must have a finite index in the sequence." The answer we are looking for is specifically when the index = infinity.
There is no such number as 'infinity'. There can be no index = infinity. I know it can be intuitively confusing, which is why calculus skirts around the issue and merely says that as you get closer to infinity, you will get closer to 1.

Originally posted by PatNovak
There is an underlying assumption in this paradox that a finite distance is infinitely divisible. If a finite distance is not infinitely divisible, then it is perfectly reasonable to consider the paradox as defeated on those grounds.

The paradox as presented can be defeated on those grounds, but the paradox itself remains. There is nothing stopping one from recasting the paradox in Euclidean geometry and saying that you cannot draw a straight line from (0,0) to (0,1) as it will have to pass through (0,0.5),(0,0.75) etc.

But if you have a background in mathematics it will hardly seem like a paradox then as you will be used to infinite sets. Once you see it as a set rather than a sequence it suddenly seems OK. You can never get to the end of an infinite sequence by counting (although the sequence may still be countable), but you can have an infinite set and comfortably play around with it without too much concern.

Originally posted by twhitehead
No, it is not philosophical, it is mathematical. And there is a correct answer.

[b]However, I do have to disagree with your assertion that "the full infinite sum is not considered part of the sequence." (I assume your are referring to the sequence 1/2, 3/4, 7/8...). Here, The sequence is defined as the sum, so the sequence and the sum are the exact sam ...[text shortened]... s around the issue and merely says that as you get closer to infinity, you will get closer to 1.

No, it is not philosophical, it is mathematical. And there is a correct answer.

There is no such number as 'infinity.'

Both of these statements cannot be correct. Either infinity is a number and the answer can be obtained mathematically, Or infinity is not a number and therefore the answer cannot be obtained mathematically.

Edit: I agree that there is no such thing as index = infinity, but despite what you may say, that is definitely the question we are pursuing (What is the value of the equation when the index = infinity is the only question at stake). That is why we can only approximate the answer, and the question of whether the answer in 1 or 0.999999999... is not mathematical, so it becomes philosophical at that point (because infinity is not a number, but an abstract concept).

Originally posted by twhitehead
No, it is not philosophical, it is mathematical. And there is a correct answer.

[b]However, I do have to disagree with your assertion that "the full infinite sum is not considered part of the sequence." (I assume your are referring to the sequence 1/2, 3/4, 7/8...). Here, The sequence is defined as the sum, so the sequence and the sum are the exact sam ...[text shortened]... s around the issue and merely says that as you get closer to infinity, you will get closer to 1.

It depends on how you define your number line. It is possible to define an extended real number line R* so tht R* = R U {-infinity, +infinity}.

For clarity let S(n) be the nth partial sum, where S(0) = 0, S(1) = 1/2, S(2) = 1/2 + (1/2)^2, and the nth partial sum is S(n) = S(n-1) + (1/2)^n = 1 - (1/2)^n. For any n, S(n) < 1. 1 is the limit point of the sequence of partial sums of the infinite series. The limit point is not in the sequence of partial sums. However the sum relevant to the paradox is the infinite series, not any of the partial sums.

Personally I've always thought Zeno's paradox was bunk - mostly he was trying to prove that the physical world we see is an illusion and that there is only one thing - he was a follower of Parmenides. The paradox can be knocked down just by considering the series properly. It can be knocked down even harder on empirical grounds, since things patently can move.

Incidentally, quantization doesn't really solve the problem. There is a quantum version of Zeno's paradox due to Turing in which an excited atom under observation can never decay. So quantization of space-time doesn't eliminate the problem.

Originally posted by PatNovak
No, it is not philosophical, it is mathematical. And there is a correct answer.

There is no such number as 'infinity.'

Both of these statements cannot be correct.

For a start, I think we are disagreeing about what is or is not philosophical.

Edit: I agree that there is no such thing as index = infinity, but despite what you may say, that is definitely the question we are pursuing
You may be pursuing it, but I am not. I have already stated that there is no such answer because there is no such thing as index=infinity. Oddly enough you seem to agree with my premise but dispute the obvious conclusion.

(What is the value of the equation when the index = infinity is the only question at stake). That is why we can only approximate the answer, and the question of whether the answer in 1 or 0.999999999... is not mathematical, so it becomes philosophical at that point (because infinity is not a number, but an abstract concept).
Sorry, but philosophy doesn't automatically give you the right to throw logic out the window and say its OK. If something does not make sense, it wont make sense if you call it philosophy rather than mathematics.

Here is the correct mathematical formulation:
The sequence of partial sums that I gave previously is infinite. It includes numbers infinitely close to 1, but does not include 1. Or put more mathematically, for any number n less than 1, there exists a member of the sequence m, such that n<m<1. But equally true is the fact that for any member of the sequence o, there exists a number p such that o<p<1.
It does not have a member with index=infinity, nor does it have a member = 1.

Originally posted by twhitehead
For a start, I think we are disagreeing about what is or is not philosophical.

[b]Edit: I agree that there is no such thing as index = infinity, but despite what you may say, that is definitely the question we are pursuing
You may be pursuing it, but I am not. I have already stated that there is no such answer because there is no such thing as in ...[text shortened]... such that o<p<1.
It does not have a member with index=infinity, nor does it have a member = 1.[/b]

I really don't understand why you are fixated on partial sums. As DeepThought said, the only thing that is relevent is the sum of the infinite series, not any partial sums.

Let me give you a thought experiment to show you why you are wrong. Suppose we have a puzzle in a box, and it is divided into an infinite number of pieces. Any sum of the partial sums will never = one puzzle, but we know that the box contains one puzzle, and if we added all the pieces together, we would get exactly one puzzle. The reason is because even though there may be an infinite number of pieces, our box (or set) contains all of them.

Its the geometric series where d is the total distance and 0<r<1 which converges and has a finite sum.

S = Σ d*(1/2)^n = d

When the object moves from point "a" to "b" through the finite distance "d" is passes through every term in the infinite series.

Originally posted by PatNovak
I really don't understand why you are fixated on partial sums.

Because it is the best way to deal with the problem.

Let me give you a thought experiment to show you why you are wrong. Suppose we have a puzzle in a box, and it is divided into an infinite number of pieces. Any sum of the partial sums will never = one puzzle, but we know that the box contains one puzzle, and if we added all the pieces together, we would get exactly one puzzle. The reason is because even though there may be an infinite number of pieces, our box (or set) contains all of them.
So where was I wrong?

The OP is quite clearly talking about a sequence, and not a divided puzzle.
The series 1/2, 3/4, 7/8.... does not include 1 - and calling it philosophy rather than mathematics isn't going to change that fact.
If however you were to ask what is the sum of 1/2+1/4+1/8+.... then the answer is 1.
Similarly we could divide your puzzle in half then divide one half in half and so on to construct the same sequence.
But there will still never be 'index=infinity' and that will still not be what we are pursuing, and the answer will still be a matter of mathematics, and I still wont be wrong.

Originally posted by joe shmo
Its the geometric series where d is the total distance and 0<r<1 which converges and has a finite sum.

S = Σ d*(1/2)^n = d

When the object moves from point "a" to "b" through the finite distance "d" is passes through every term in the infinite series.

But b is not a member of the series hence the object cannot get to b whilst remaining in the series. Somehow it must get past every member of the infinite series, and then keep going. There is no 'last member' of the series, so the question is, what member of the series did it go through immediately before it got to b?

Originally posted by twhitehead
The OP is quite clearly talking about a sequence, and not a divided puzzle. The series 1/2, 3/4, 7/8.... does not include 1 - and calling it philosophy rather than mathematics isn't going to change that fact.
If however you were to ask what is the sum of 1/2+1/4+1/8+.... then the answer is 1.

I do not at all agree that the OP is "clearly" talking about a sequence.

But I will take your admission that the sum would equal one as partial agreement between us and leave it there. And I will also agree with you that if we view it as a sequence, it will never reach one.

Originally posted by PatNovak
But I will take your admission that the sum would equal one as partial agreement between us and leave it there. And I will also agree with you that if we view it as a sequence, it will never reach one.

Do you agree that both are mathematical results and not a matter of philosophy? Are you withdrawing your claim that there is no correct answer?

Originally posted by twhitehead
Do you agree that both are mathematical results and not a matter of philosophy? Are you withdrawing your claim that there is no correct answer?

The OP does not fit neatly into the idea of a sum, so in this particular case, I do not agree that it is strictly a math question with a correct answer and I do not withdraw my claim. However, it is clearly not a sequence, because we know that the glass hits the floor, and if it were a sequence it would never hit the floor, so a sequence can be discounted immediately.

Originally posted by twhitehead
But b is not a member of the series hence the object cannot get to b whilst remaining in the series. Somehow it must get past every member of the infinite series, and then keep going. There is no 'last member' of the series, so the question is, what member of the series did it go through immediately before it got to b?

No, the series is the complete sum with an infinite number of terms, the sequence is a sequence of partial sums which the series is the limit of.

Zeno's paradox is an example of a supertask, and there is a nice page on Wikipedia about them. One that I like is the following paradox. Take an empty pot and start putting marbles in. At each step for every 10 marbles you put in remove 1. After completing the first step (instantaneously), do the second step after 30 seconds, the third after 15 seconds and so on. How many marbles are in the pot after 1 minute? The intuitive answer is an infinite number. However, suppose the marbles were numbered, on the first step marbles 1 - 10 are added and marble 1 is removed. On the second step marbles 11 - 20 are added and marble 2 is removed. After 1 minute there is no marble that has not been removed and therefore the pot must be empty. Of course if instead we had removed marble 10, 20, 30 etc. the pot would contain an infinite number of marbles. Which just goes to show you have to be careful how you go about a supertask.