Something drops half a distance

If I drop a glass from 3 meters high, it shatters when it hits the floor. Worse still, my beer splashes all over the place.

Yet, it drops first half that distance, then half that distance, then half that distance, ad infinitum. So it never actually hits the floor and shatters.

I know there's a mathematical explanation of why it does actually hit the floor (and prove reality), could someone please walk me through the explanation?

Cheers!

Originally posted by shavixmir
If I drop a glass from 3 meters high, it shatters when it hits the floor. Worse still, my beer splashes all over the place.

Yet, it drops first half that distance, then half that distance, then half that distance, ad infinitum. So it never actually hits the floor and shatters.

I know there's a mathematical explanation of why it does actually hit the floor (and prove reality), could someone please walk me through the explanation?

Cheers!

If each time it does another 'half the distance' it also takes half the time (approximating velocity as constant). So although it will never hit the floor within that sequence of half distances it will also never pass the time that in reality it does hit the floor.
The sequence of halves is infinite, but they do all get traversed (assuming distance and time are infinitely divisible).

Your conclusion that it 'never actually hits the floor' is based on the false assumption that infinities cannot be traversed as well as the false assumption that the glass is confined to the sequence of points described in your story. In reality, if time and space are infinitely divisible then the glass is always traversing infinitely many points in infinitesimally small amounts of time and the infinities cancel out.

If time and space are not infinitely divisible, then the problem goes away altogether.

1/2 + 1/4 + 1/8 + ... = 1

Originally posted by shavixmir
If I drop a glass from 3 meters high, it shatters when it hits the floor. Worse still, my beer splashes all over the place.

Yet, it drops first half that distance, then half that distance, then half that distance, ad infinitum. So it never actually hits the floor and shatters.

I know there's a mathematical explanation of why it does actually hit the floor (and prove reality), could someone please walk me through the explanation?

Cheers!

This is known as Zeno's Dichotomy Paradox. One explanation is like twhitehead says, that distance is not infinitely divisible because there is a fundamental unit of length called the Planck length which cannot be subdivided (in much the same ways that quarks and electrons cannot be subdivided).

Another explanation is Deep Thought's formula, which limits/calculus shows is actually equal to 1. This video explains the concept pretty well:

http://ed.ted.com/lessons/what-is-zeno-s-dichotomy-paradox-colm-kelleher

Originally posted by PatNovak
Another explanation is Deep Thought's formula, which limits/calculus shows is actually equal to 1.

Although the sum is equal to one, one is not actually a member of the sequence of partial sums. So if you assume the object stays in the sequence then it will never get to one even after an infinite number of steps.
The key is to recognize that it is not confined to the sequence.

Originally posted by twhitehead
Although the sum is equal to one, one is not actually a member of the sequence of partial sums. So if you assume the object stays in the sequence then it will never get to one even after an infinite number of steps.
The key is to recognize that it is not confined to the sequence.

I suppose it depends on whether the answer to the limit as it approaches infinity is exactly equal to one or only approximately equal to one. I tend to think that, since you are adding an infinite number of fractions, you are actually getting exactly one as the answer.

Originally posted by shavixmir
If I drop a glass from 3 meters high, it shatters when it hits the floor. Worse still, my beer splashes all over the place.

Yet, it drops first half that distance, then half that distance, then half that distance, ad infinitum. So it never actually hits the floor and shatters.

I know there's a mathematical explanation of why it does actually hit the floor (and prove reality), could someone please walk me through the explanation?

Cheers!

The infinite number of distances are cancelled out by the infinitely short time needed to traverse those distances.

Originally posted by PatNovak
I tend to think that, since you are adding an infinite number of fractions, you are actually getting exactly one as the answer.

But the full infinite sum is not considered part of the sequence. Every member of the sequence must have a finite index in the sequence. There is no number 'infinity' for which we can do the actual sum and get an answer of one. For this reason calculus steers clear of saying the answer is one and merely says that the answer approaches one as the sequence approaches infinity.

The beer glass however is not confined to the sequence and not only manages to pass through every point in the infinite sequence, but also pass through one and if it weren't for the floor, it could go on and reach two. The whole paradox relies on our wrong intuition that time is only finitely divisible. If we think of an infinite number of points in time, we wrongly assume it will take an infinite amount of time to get there.

Originally posted by twhitehead
But the full infinite sum is not considered part of the sequence. Every member of the sequence must have a finite index in the sequence. There is no number 'infinity' for which we can do the actual sum and get an answer of one. For this reason calculus steers clear of saying the answer is one and merely says that the answer approaches one as the sequence ...[text shortened]... umber of points in time, we wrongly assume it will take an infinite amount of time to get there.

How does that work out when we now have theories that time itself is quantized?

Originally posted by sonhouse
How does that work out when we now have theories that time itself is quantized?

Think of filling a bucket with pebbles from a heap with n pebbles. First you fill half the bucket with n/2 pebbles. Then you fill half of the empty space with n/4, and so on. After a while you have only one pebble in your heap. You cannot split in in halves so you have to decide: (1) Do you put the last pebble in the bucket and is finished? or (2) accept your failure that you cannot split the last one and thus not be able to fulfill the mission?

Same thing with time. When you have the last quantum of time to consume you have to decide wether you accept (1) or (2). But as this quantum is so short when you have decided your mind, then that time would have passed anyway.

Originally posted by twhitehead
But the full infinite sum is not considered part of the sequence. Every member of the sequence must have a finite index in the sequence. There is no number 'infinity' for which we can do the actual sum and get an answer of one. For this reason calculus steers clear of saying the answer is one and merely says that the answer approaches one as the sequence ...[text shortened]... umber of points in time, we wrongly assume it will take an infinite amount of time to get there.

This is more of a philosophical question than anything, because infinity is an abstract concept. So there probably isn't 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.

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.

Originally posted by sonhouse
How does that work out when we now have theories that time itself is quantized?

It is, like the quantization of space, irrelevant to this paradox which assumes a continuum variable to begin with.

Originally posted by KazetNagorra
It is, like the quantization of space, irrelevant to this paradox which assumes a continuum variable to begin with.

I disagree. It is perfectly reasonable to defeat a paradox by defeating one of its underlying assumptions.

Originally posted by PatNovak
I disagree. It is perfectly reasonable to defeat a paradox by defeating one of its underlying assumptions.

There is nothing wrong with the underlying assumptions, it just shows that you can divide a finite, real interval in infinitely many segments.

Originally posted by KazetNagorra
There is nothing wrong with the underlying assumptions, it just shows that you can divide a finite, real interval in infinitely many segments.

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.