Space, Time & Stephen Hawking Jive

[FNHinVA]

I' m baaack... and I've been reading stuff.

Check this out...

"Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Many philosophers, and certainly engineers, generally went along with the mathematical results.

Nevertheless, Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this he followed Leopold Kronecker, an earlier 19th century mathematician. It would be incorrect to say that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has resolved forever all problems involving infinities, including Zeno's.

[http://en.wikipedia.org/wiki/Zeno's_paradoxes#Status_of_the_paradoxes_today]

A more detailed treatment - with essentially the same observations and conclusions - can be found at:

http://www.mathpages.com/rr/s3-07/3-07.htm.

These articles are only two of many I found that made the same basic points.

Stuff I learned....

We can put aside the debatable and controversial issue of continuous vs. non-continuous space [the verdict is undecided; I will accept a draw on that... at least for the time being].

I was wrong in my conjecture that the sum to infinity was some "weird symbol." I now understand that it is an actual number. But I also understand it is a number that the actual sum to infinity approaches but never reaches. So in my mind it is still a symbol (or substitute) for something that doesn't exist. Just not weird looking.

[Aside: Please remember; business-trained... not science trained. I barely remember the intro calculus course they made me take before business school. And I hated it. I've been winging it here. Just thinking; no tools. Hell, when I put up the OP, I had never heard of Zeno's paradoxes. It was just an interesting problem that had occurred to me. I didn't know I was 2500 years late. If I had known about it, I would have just looked it up on the Web and never bothered you guys.]

Nonetheless, it turns out that it was not so unreasonable for me to challenge the applicability of calculus (and, specifically, the limit) to the problem. Even though I didn't know enough about it to properly express myself, I knew there was something troublesome going on there.

That is the issue I got pounded on - not one poster supported me - but now I discover there are eminent philosophers and mathematicians down with my position. Since I arrived at that position on my own and had to defend it by myself, I was - understandably I believe - very happy to find some company.

I have to wonder what the aforementioned eminent philosophers and mathematicians might have to say about this:

"This argument is wrong, because step 4.7 is invalid and is provably invalid. The proof is a clear and irrefutable mathematical proof. It does not involve the introduction of any weird symbols. It simply and clearly demonstrates WITHOOUT ANY DOUBT that to travel from A to B in a continuous space takes a finite time. There is NO ROOM for question, NO philosophical / logical / reasonable alternate arguments that can refute it."

Oh, really? Go tell that to my posse.

(I know... argumentum ad verecundiam - but against that statement, it's all I need.)

Sorry, usmhot, I couldn't resist.

I just had to make the point that I was not so unequivocally wrong as you made me out to be; I was simply on the other side of a controversial issue - with company. I'm sure you still disagree with me but at least I know I am not alone and now I know my position is... well... somewhat respectable. Cool.

So where does all of this leave us? Continuous vs. non-continuous space [the core issue] remains controversial and unresolved, due in no small part to the questionable application of the calculus limit. Blah...

Ironically, it appears we have come full circle. The answer by Hawking's assistant:

"...When things get that small, we can no longer measure them so we don't know what the hell is going on."

may still be the best answer.

Well, it's been interesting, some fun and very challenging. And I've learned a lot of stuff I really didn't need to know. Just shows what brute force analysis combined with ignorance can get you.

A damn headache, for one thing - I'm going to check the latest ME chip counts and go to bed.

Thanks, all...

Later.....

[FNHinVA]

It was certainly not my intent to offend. I was being a bit facetious. I had several guys coming at me from different directions all at the same time and all of them taking shots at something or another and I - in the discipline at hand - don't know what the hell I am talking about. So I definitely felt under siege. Yeah, I got lot's of attention... more than I could handle.

I will say this: nobody was mean-spirited or nasty and I do appreciate that. Some guys got (understandably) frustrated with me and showed it a little but, hell, nothing compared to some stuff you see on the forums.

Several of your points in this posting are addressed in my last posting ("RESTART...") which was done before I read this post.

"When I pointed this out to you before you declared that what you really meant was that you really can't divide 1 up into those smaller and smaller pieces "into infinity", whereas above you say you do "get to" do this. This kind of thing is frustrating."

Sloppy on my part; I get to try... I can't do it.

"Your balking at infinite sums is not without merit because it is only countable sums which give us a powerful workable useful mathematics. If we try to go to sums of uncountably infinite many numbers we get nowhere."

And, I was pleased to discover, I am not alone here. You will see how pleased I was with myself on this issue in my last posting.

"So there are restricted forms of Mathematics that would work for you but they would not be nearly as powerful as the mathematics commonly used today."

I understand this and do not dispute it in the least. But it is not really the point. I got accused of attempting to undermine calculus but that is just silly. Its usefulness in both everyday stuff and extraordinary stuff is beyond obvious. I just take exception to its applicability in this case. And - again - I am not alone.

Thanks much for your contributions. I do appreciate it.

Seeya down the road...

[FNHinVA]

WARNING: Math Dolt in the house.

If you read my last, last post, you will find that I am going with an appeal to authority argument (argumentum ad verecundiam), declaring a small, pathetic victory and getting the hell out of Dodge.

I am over-matched and out-numbered so it's all I've got.

I know it doesn't prove anything other than this: my intuitive sense that it is wrong to use the "limit" to measure infinite distances has support among better minds than are likely to be found around here. The opposite is also true; so there is an ongoing controversy.

After being told repeatedly and emphatically that I was wrong... wrong... WRONG, I see that as at least a partial vindication. That would be my small, pathetic victory.

I remain convinced that space:time is non-continuous; I can't prove it but I believe it. And nobody can prove I'm wrong. Another small, pathetic victory.

So... on my way out of Dodge, please... don't anybody shoot me in the back. It's over.

Thanks for your contributions. It's been... well... infinitely amusing.

seeyadowntheroad....

[malorum]

[ QUOTE ]
If you read my last, last post, you will find that I am going with an appeal to authority argument (argumentum ad verecundiam), declaring a small, pathetic victory and getting the hell out of Dodge.

[/ QUOTE ]

In addition to the latin terminology could u cite an appropriate bible qoute plz [img]/images/graemlins/confused.gif[/img]

[FNHinVA]

Not a quote. I believe that is properly called a citation.

And usmhot started it... just following his lead.

If he starts quoting the bible in a logic & math context, I'll start quoting Britney Spears.

Should work...

[usmhot]

[img]/images/graemlins/laugh.gif[/img] I would never quote the Bible in any context that demanded rational scientific discussion!!!

Incidentally, I have always supported your claim that in our Universe space is not continuous - but on the basis of our understanding of Quantum Mechanics.

"I know it doesn't prove anything other than this: my intuitive sense that it is wrong to use the "limit" to measure infinite distances has support among better minds than are likely to be found around here. "
I don't think you should be so quick to dismiss real live posters here over people who have written in other forums. There is no reason to assume that some of the posters here (potentially _even_ including yourself) aren't the intellectual equal of those you call 'better minds'.
I read the material you referenced (indeed one of the references is part of a book on relativity and I'm continuing to read the whole book), and I have to say it doesn't really add anything to your argument.

The point I would make about the mathematics of infinite series is this - as you rightly point out, and as is succinctly discussed in the material you referenced, the sum of an infinite series is never actually reached. However, what the mathematics does prove is that the sum CAN NEVER BE MORE THAN the limit. I.e. if you prove that the limit of an infinite series is, e.g., 1 then you prove that adding the terms for infinite time will never give a value greater than 1.
In other words, the proof shows that you don't actually have to add all the terms for ever because you know that the answer can't be larger than a specific value.
In the case of the motion from A to B using half distances, the maths proves that the time taken can never be larger than some specific value - which, coincidentally enough, happens to be the distance divided by the velocity.

Undoubtedly, there are some very useful mathematical systems which limit themselves to finite sets, but a 'feeling' or 'intuition' about something is irrelevant in any formal system. And, just because some professional mathematicians work within finite systems doesn't mean that their systems are any more 'real' than infinite systems. Mathematics, in all its forms, is an abstraction, the results of which are often applied to the Universe as we see it to enable us to understand and manipulate it better. And, incidentally, mathematics of finite sets is merely a subset of mathematics of infinite sets.

What the mathematics behind the infinite series says is that 'in a Universe in which space is continuous, motion from one point to another is possible'. This means that you can't use the paradox you started with to deduce that space in our Universe isn't continuous.

Be content that there is other evidence to suggest that space in our Universe isn't continuous.

I accept your (unconditional) withdrawal (/ surrender) [img]/images/graemlins/grin.gif[/img]

[Darryl_P]

[ QUOTE ]
I remain convinced that space:time is non-continuous; I can't prove it but I believe it. And nobody can prove I'm wrong.

[/ QUOTE ]

FWIW I agree with you. There is a level of smallness beyond which we can never acquire reliable information because we are limited by the size and speed of the photons which transmit the information to us. We also know that our attempting to observe it actually changes it so we can never be just observers -- like it or not we are participants in the event also.

The quantum mechanics part of the explanation was to remind you that this type of stuff happens when the measurements get small enough.

Besides, infinity is a man-made concept, so to ask the question in the way you did you are bringing in elements which are not part of the observable, physical world. If you bring them in, you must do so with a definition, so to get out of it you need to refer to the same definition.

[FNHinVA]

"I would never quote the Bible in any context that demanded rational scientific discussion!!!"


I didn’t really think so. Nor would I. Hence the reference to the airhead Britney Spears. Anything she might say would be about as relevant.


"Incidentally, I have always supported your claim that in our Universe space is not continuous - but on the basis of our understanding of Quantum Mechanics."


I understand that - always did. This line of discussion was always tangential.


"["I know it doesn't prove anything other than this: my intuitive sense that it is wrong to use the "limit" to measure infinite distances has support among better minds than are likely to be found around here. "] I don't think you should be so quick to dismiss real live posters here over people who have written in other forums. There is no reason to assume that some of the posters here (potentially _even_ including yourself) aren't the intellectual equal of those you call 'better minds'."


I was attempting to inflate the credibility of the "authorities" (my posse) referenced in my "appeal to authority" argument. And I wasn't referring to "people who have written in other forums."; I was referring to philosophers and mathematicians with funny, hard to pronounce names (that always helps) most of whom are long dead (that helps even more). This is vitally important since the argument alone proves nothing; the stature of the cited authorities makes the argument sound more plausible (and we know how important that is to me). Indirectly denigrating the intellectual capabilities of 2+2 posters was unavoidable but necessary. It was dirty and lowdown but you made me do it.

And, I would never make an unfavorable comparison of 2+2 posters to those on other forums. Especially not the ones in this topic section; they are as bright or brighter than any I’ve seen anywhere. Mostly civil, too.

“potentially _even_ including yourself” - 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.


"In the case of the motion from A to B using half distances, the maths proves that the time taken can never be larger than some specific value - which, coincidentally enough, happens to be the distance divided by the velocity."


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."

No. Never. Unacceptable.

I could never get away with that line of reasoning about a critical parameter in a business model. When we get that loosey-goosey with the inputs (and we frequently do); we have to qualify the outputs - attach probabilities to ranges of outcomes, that sort of stuff. But here we are talking about a theoretical proof; right or wrong… yes or no. I just don’t see how you get loose with the input in this context.

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? [I just had an appalling thought: are business geeks more rigorous than math geeks?]


"Undoubtedly, there are some very useful mathematical systems which limit themselves to finite sets, but a 'feeling' or 'intuition' about something is irrelevant in any formal system."


Not fair. Out of context. My "intuitive sense" was simply what got me started. I did my homework after that and I think I've played the game fairly since. And surely you are not suggesting that 'feeling' or 'intuition' is irrelevant to discovery.


"...you can't use the paradox you started with to deduce that space in our Universe isn't continuous."


I can if I reject the "proof." And I just did, as others (my posse) before me have.


“Be content that there is other evidence to suggest that space in our Universe isn't continuous.”


I know there is. But that’s not the point here. We are on a tangent. This is about the “proof.”


“I accept your (unconditional) withdrawal (/ surrender)”


Say WHAT???

I withdrew to regroup. And I never surrendered. I conceded a draw on the original issue of continuous vs. non-continuous only because I became aware there are multiple, contradictory proofs out there and I don’t have the time or inclination to investigate and try to understand all of them. Basically, I am accepting Hawking’s assistant’s answer: "…when things get that small, we can no longer measure them so we don't know what the hell is going on."

Recap….

The discussion veered off to the issue of using the “sum to infinity” to prove an infinite series of discrete distances can be traversed in a finite amount of time.

I took issue with that. Then, every poster contributing to the thread at that time – half a dozen or so - disagreed with me. Every one of them. None more emphatically than you.

So I withdrew to do some homework – something I clearly should have done earlier.

What I found is that competent and well-regarded philosophers and mathematicians (my posse) have taken – and take – exactly the position I take. They reject the proof based on the handling of the sum to infinity.

All of which proves nothing except this: my position is reasonable, has merit and it is defensible. It cannot be dismissed out-of-hand, as it was.

Which gets us to what this is really all about….

I took your response to me on the issue as more than a little disrespectful. I won’t re-quote it here; I think you know what I’m talking about. Let’s just call it your little OTTR [Over-The-Top-Rant] for short. It was as though you were screaming in my face: “PAY ATTENTION, YOU MORON. CAN’T YOU GET THIS?”

I know you were (understandably) exasperated with my mangling of the terminology and misunderstanding of certain concepts, but I don’t believe I deserved that.

I am way outside my field here and that should have been obvious and indicated a little patience was in order. I would also hope it was obvious I am not a moron in need of having some concept hammered into me. I may have been a bit slow out of the blocks but I think I “get it” well enough now to simply say I disagree with you on the issue and I am confident I have good reason to do so.

Very simply, it turns out that the essential message of your OTTR (“This is universal truth. It is beyond debate. Accept it or die.”) may… just may, be a little off the mark.

Hell, it’s plain wrong.

This is a case where “appeal to authority” works exquisitely… you say the matter is beyond debate… meet my posse…. (nonexistent) debate to follow. It demonstrates that “There is NO ROOM for question” is patently false. The simple existence of my posse is all I need to prove there IS ROOM for question.

So, no, I don’t surrender. I disagree.

Please feel free to retract your OTTR and acknowledge that my position – while not necessarily provably right - has merit and is worthy of respect. (Groveling apologies not required.)

If you choose not to… that’s OK. I know what I know and – in the immortal words of Tom Petty – I won’t back down.

Soooo….

Thank you very much for taking the time to help me along this little journey of discovery.

Sorry I made the ride bumpier than it needed to be. I should have done more homework but somehow this whole thing went further and faster than I anticipated.

And there are no hard feelings. The OTTR, taken in context (I was frustrating to deal with), was a bit annoying but understandable.

And, thanks to you challenging me, I actually now know a lot of stuff I didn’t know a few days ago. Hell, gotta be happy with that.

It’s been infinitely interesting. (I know… that was lame… and to make matters worse, I think I already used it once.)

Later…

[FNHinVA]

Yeah... if I understand what you're saying - and I'm not sure I do - I am back where I started.

Hawking's assistant's original answer: "…when things get that small, we can no longer measure them so we don't know what the hell is going on."

Guess I'll have to live with that.

Thanks...

[usmhot]

I think you're still missing the core point.
All the eminent mathematicians who reject the concept of infinity would still completely accept that if you assume infinity in the first place then the infinite series proof is unassailable.
What they say is they don't accept infinity at all - so they don't even start with an assumption such as 'space is continuous' because the very definition of 'continuous' involves infinity.

What you did was assume space is continuous and then use an argument against which there is a complete proof in that context that travel in continuous space in finite time is possible. You set up the context in the first place in which that proof is beyond reproach. I.e. 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.

Incidentally, some further rebuttal
1. when I said

"I don't think you should be so quick to dismiss real live posters here over people who have written in other forums. There is no reason to assume that some of the posters here (potentially _even_ including yourself) aren't the intellectual equal of those you call 'better minds'."

by "other forums" I meant publication in any manner whatsoever, and was referring directly to the eminent minds you spoke of. 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.
And, I assumed you would realise I was being jocular, and even subtly complimentary with "(potentially _even_ including yourself)"

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'

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 - as I pointed out above there was no need to, but you'll react how you react. If you thought that was insulting then wait till you hear this lol ...
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.

You want another interesting one?

Imagine a train traveling at 100kph in one direction and a fly traveling at 10kph in exactly the opposite direction. They crash head-on. In going from 10kph in one direction to 100kph in the exact opposite direction the fly must, at some point, be at a standstill - 0kph - but at this point the train must also be at a standstill. So how is it that a fly can stop a train?