Space, Time & Stephen Hawking Jive

[FNHinVA]

I guess I should start using those damn little “smiley” things. Many of my comments were facetious or jocular.


"I think you're still missing the core point."


And I think you are avoiding my key question about your core point. I'll get to that.


“ if you start with the concept of infinity then you automatically have sums of infinite series.

This is exactly what I've been saying all along - in Universe in which space is continuous travel between any two points in finite time is possible.”


It has been my understanding all along that this assertion depends on a proof that depends on:

1/2 + 1/4 + 1/8 + [infinity] = 1

This is the gut issue for me. But you did not directly respond. I will quote myself to save time (insert smileys in appropriate places):

“Yes, the time taken can never be larger than some specific value (1 in our case) but it also can never BE 1. So we (conveniently) leap over infinity (wasn’t that fun!) to the limit and (coincidentally) arrive at a proof.

In business models, we call that a simplifying assumption; "well, it can never be greater than 1, but we know it isn't 1, but it's really, really close to 1. Oh, what the hell, let's assume it's 1."

It is NOT 1 and we KNOW it is not 1 and we KNOW it can never be 1. But let's go ahead and use it anyway and call the result a "proof."

The proof fails using any value other than 1.

So how can we accept a proof that uses a number that we KNOW is not exactly right?”


Please explain to me where you think I am going wrong here.



“Just because someone is dead, has a difficult name and is published doesn't make him/her any more intelligent or insightful than those of us who are still alive.”


Smiley missing. I was being a bit facetious.



“And, I assumed you would realise I was being jocular, and even subtly complimentary with "(potentially _even_ including yourself)"”



I did. I responded in kind (mock outrage at the "insult")



“2. "No. Never. Unacceptable."

Isn't this a little extreme? - or closed mined? You're saying that no matter how convincing an argument is made in this case, you will not accept it because of your 'feeling'”


The operative term being: “…that no matter how convincing an argument…”

As for the “…, you will not accept it because of your 'feeling'.” I have offered reasoned argument for my position. You may not like it or accept it but to dismiss it as ‘feeling’ is gratuitous, out of context and grossly unfair. (And you should be shot for it.)


“3. "?I accept your (unconditional) withdrawal (/ surrender)?

Say WHAT???"

That was a joke.

4. "Hey? wait just a minute. I came up with Zeno?s key paradox on my own without ever having heard of it. And I took it to the same logical conclusions? in my spare time and outside my field. That?s why he?s in my posse and I?m not in his. (I was 2500 years late, but let?s not quibble.) potentially? even? I?m insulted. I expect an immediate apology."

I'm not sure from your tone here if you genuinely feel insulted…”



That was mock outrage. And a shameless attempt to affirm what a smart guy I am. (AKA: “Sklanskyism”)



“…you are by no means the first to have independently come up with similar thoughts to Zeno's paradox, nor will you be the last. For example, when I was about 10 I came up with it and was then pointed to various (simpler) writings about it by my father. I'm sure there are many people with the same story.”



Then I guess there is no point in mentioning figuring out the Doppler effect when I was 8? (Silly me… I just did.)



“You want another interesting one?”


No.


Always fun… Later…

[BTW: Am I to assume you refuse to retract your OTTR? It was offensive, demeaning, insulting and you should be shot for it.]

[K C]

The simplest way to explain this seeming paradox is this. This example is construed such that it is set up to measure an infinite series short of completion. It's certainly much more of a "riddle" than any scientific problem. The reason of course that the series does not lead to completion is that the possibility of such is not built in to the example.

Given A and B as points in space of course there are an infinite amount of points in between, just as there are infinite points in time. I can for instance postulate the same thing from the time it takes from this very moment to a minute later : 10:26 to 10:27 for instance. Yet oddly enough 10:27 did occur as I'm writing this. And it did not involve an infinite amount of time for this to occur.


KC

[FNHinVA]

Welcome to our little ininite thread on issues of infinity.

All of the points you make have been thrashed about pretty well. The issue now is use of the "sum to infinity" in a "proof." Here is the key question I am trying to get answered:

usmhot:

"...in Universe in which space is continuous travel between any two points in finite time is possible.”

FHNinVA:

"It has been my understanding all along that this assertion depends on a proof that depends on:

1/2 + 1/4 + 1/8 + [infinity] = 1

This is the gut issue for me. But you did not directly respond. I will quote myself to save time (insert smileys in appropriate places):

“Yes, the time taken can never be larger than some specific value (1 in our case) but it also can never BE 1. So we (conveniently) leap over infinity (wasn’t that fun!) to the limit and (coincidentally) arrive at a proof.

In business models, we call that a simplifying assumption; "well, it can never be greater than 1, but we know it isn't 1, but it's really, really close to 1. Oh, what the hell, let's assume it's 1."

It is NOT 1 and we KNOW it is not 1 and we KNOW it can never be 1. But let's go ahead and use it anyway and call the result a "proof."

The proof fails using any value other than 1.

So how can we accept a proof that uses a number that we KNOW is not exactly right?”


Please explain to me where you think I am going wrong here."

K C:

??


Feel free to join in...

Thanks...

[usmhot]

Right, now that I understand your tone a little better and can see that you are being jocular in various places I feel a little more comfortable.

"It has been my understanding all along that this assertion depends on a proof that depends on:

1/2 + 1/4 + 1/8 + [infinity] = 1

This is the gut issue for me. But you did not directly respond."

I've been avoiding using mathematics so far, as you said you didn't want to see any, but now you've introduced it so here goes - this is really quite straight forward, so don't be put off.
You've accepted that proving that the (upper) limit is 1 is valid (incidentally, in itself that's all that's needed and I'll explain why shortly). But, you're saying that the proof says its not quite 1 and is some 'imaginary' number other than 1 so you can't accept it. Well, actually, it's easy to prove that it can't be less than 1. So, in fact its easy to prove that the sum is exactly 1.

First thing is, you wrote

1/2 + 1/4 + 1/8 + [infinity] = 1

thats not quite right - it should be

1/2 +1/4 + 1/8 + ... + [an infinitesimally small value] = 1

And the difference in formulation is vital. We show that the sum is not less than 1 by simply pointing out that whatever value less than 1 you get to, no matter how close to 1 it is does not complete the series - there is always another tiny value that can be added to it to bring you even closer to 1. In other words, we assume that the sum is a value less than 1 and show that there is no such value that it can be.

To put it succinctly, we can prove that the sum of the given series is exactly 1 - not some imaginary value, and not some value thats achieved by skipping out infinite terms. And the crux of the whole thing is this is all possible because you start with the assumption that space is continuous.
Starting with an assumption that space is continuous does not lead to a paradox in the form proposed by Zeno.

Anyway, back to the point that showing that the upper limit is 1 is sufficient - remember, your original problem was that you can't get from A to B in finite time. You got around to accepting that the upper limit of the series is 1 - i.e. that the result is at very most 1 - but even if its some silly little value less than 1 that means its clearly finite, so you travel the distance in a clearly finite time.

Finally, and perhaps most importantly, I will stand against a wall and allow you to personally shoot me for my grievous slights against your character, if and only if you can prove irrefutably, either that space is not continuous or Santa Claus doesn't exist.

[ChromePony]

FNHinVA

Ive been following this thread for a while now and feel like its about ime to chime in. I think the fundamental disagreement here is whether or not this infinite sum is actually equal to 1. Well from a mathematical standpoint it quite clearly is, exactly 1, not something close or approximate. I tutor freshman calculus and this is always a hard concept for them to grasp, the idea that an infinite number of terms can add up to such a simple number, but thats the beauty of math.

Now I feel that you may be willing to accept this mathematical side of this but assert that in real life you could never actually add up all of these terms, and therefore practically speaking you will never actually get to 1. But you also talk about dividing the distance between two points into an infinite number of intervals, can you see the contradiction here. You allow yourself to divide something into an infinte number of pieces but dont allow yourself to use the available tools to add them all together. Just as in real life you could never physically add an infinite number of terms one at a time, you could also never physically divide space an infinite number of times. Therefore both are abstract, theoretical concpets, and the situation must be dealt with accordingly. If you allow yourself to visualize space as infinitely divideable you must also allow yourself to use the proven mathematical methods that have been developed to deal with such situations.

I hope you dont feel that you are being attacked by any of the posters here, I think this is a great discussion and you are handling the math nerds very well considering your lack of background in the area.

[BluffTHIS!]

umhot, you are wasting your time with him. He just doesn't seem to want to understand either limit theory, or the fact that in a travel anaology, the simplest case being a person walking, that the person's average stride will eventually be greater than the distance left.

[hobbsmann]

I think it would be helpful for you to take a look at some of the definitions that go into infinity and infite series:

Infinity
Geometric Series
Continuum

Also, further browse mathworld for some more interesting stuff.

[drudman]

This thread is kind of like James Brown at this point. It keeps collapsing of exhaustion, and galantly being led off the stage, the cape draped over its shoulders, only for the music to swell and the cape to be cast off over and over again.

I feel good. Hey!

[PairTheBoard]

_________________________________ = 1

________________|________________ = 1
________________|________|_______ = 1
________________|________|___|___ = 1
________________|________|___|_|_ = 1
.
.
.
= 1


Or.

1=1
1/2 + 1/2 = 1
1/2 + 1/4 + 1/4 = 1
1/2 + 1/4 + 1/8 + 1/8 = 1
The sum = 1 as far out as you want to go. No matter how small you make the pieces the sum still = 1 and you can make the pieces AS SMALL AS YOU LIKE.

If you imagine doing this "ad infinitum"
you still have what you started with. 1.

PairTheBoard

[Darryl_P]

Very good post IMO...gets to the nuts and bolts of it.

FNH,

As I said before, infinity is not something in the observable world, so to ask a question involving it you must define it. Turns out the same definition that lets you divide an interval into infinitely many parts also lets you add up infinitely many numbers to EQUAL (not approximately) but EXACTLY 1 in this case, by definition.

When you say it adds up to "almost" one, you are talking about finite partial sums, not the infinite sum itself. Finite partial sums can never reach one, but the infinite sum can and does.