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# A definition of infinity?

Author Topic:   A definition of infinity?
dogrelata
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 Message 1 of 41 (371943) 12-24-2006 7:34 AM

Iâ€™m interested in the idea of infinity, but like many others, struggle with some of the concepts involved. Further, if I look up the dictionary or encyclopedia, I see there are numerous definitions or â€˜understandingsâ€™ offered.

As such, Iâ€™d like to explore one aspect only, using a simple hypothetical example.

Imagine Iâ€™m walking along and come to a piece of string stretched across my path. At this point, if I choose to follow the path of this string, one definition of infinity might be that I could follow it forever, in either direction, and never reach the end, or return to the point I started at, unless I decide to stop and retrace my steps. However, if I choose to take a knife and cut the string, do I have the same definition of infinity? Sure, I can set of in one direction and carry on forever, but if I decide to retrace my steps at any point, I will eventually return to the point I started from, the â€˜endâ€™ of the string, making it finite in one direction, if not the other.

I donâ€™t know if Iâ€™m making any sense, but can the second scenario be considered to be an example of infinity?

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Inactive Member

 Message 2 of 41 (371995) 12-24-2006 12:07 PM

Thread moved here from the Proposed New Topics forum.

Hyroglyphx
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 Message 3 of 41 (372000) 12-24-2006 12:21 PM Reply to: Message 1 by dogrelata12-24-2006 7:34 AM

 I donâ€™t know if Iâ€™m making any sense, but can the second scenario be considered to be an example of infinity?

I know what you're talking about and agree with the premise on hypothetical standards. Its hard to explain things that we've never actually seen or done, but have only conceptualized. My wife emailed me a video on Youtube the other day. The topic is on quantum physics, but more specifically how our intuition can become obscured or hinder if we don't step outside of the box. It also has the potential for us to understand space-time better in order to make predictions based on current evidence.

I've been wanting to start a topic on this video, but I kept forgetting about it. Anyway, now is a good time to view it. Bear in mind that it is geared towards young teenagers. Despite that, there is still good information and the explanation is very clear and concise. The immediate topic is about time, this video is specifically about matter, but I think we'd all agree that space-time-matter are all intimately connected and all properties can act upon the other.

Edited by nemesis_juggernaut, : edit to add for clarafication.

"A man can no more diminish God's glory by refusing to worship Him than a lunatic can put out the sun by scribbling the word, 'darkness' on the walls of his cell." -C.S. Lewis

 This message is a reply to: Message 1 by dogrelata, posted 12-24-2006 7:34 AM dogrelata has responded

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cavediver
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 Message 4 of 41 (372003) 12-24-2006 12:55 PM Reply to: Message 1 by dogrelata12-24-2006 7:34 AM

 I donâ€™t know if Iâ€™m making any sense, but can the second scenario be considered to be an example of infinity?

Yes, absolutely. It is what we call a semi-infinite line. Just like the positive x-axis on a graph: starts at 0, the origin, and continues without upper bound. The entire x-axis, including the -ve branch, is obviously infinite in both directions. Whether you can actually realise such a line in the universe depends upon whether the universe is itself infinite in spatial extent.

However, as you point out, there are many types of infinity. The usual infinity considered is that in the sense of an infinite number of objects - the countable infinity, aleph null. It has quite different properties to the infinity represented by the infinite x-axis.

Given my pseudo-Platonist outlook, infinite distances are of no concern, as I can simply map them into finite distances. Likewise, finite distance can be mapped into infinte distances. Similarly with time. This is why a universe that has always existed, and a universe that began in a big bang 14 billion years ago have exactly the same issues regarding a need or lack of need for a creator.

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dogrelata
Member (Idle past 3648 days)
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 Message 5 of 41 (372198) 12-25-2006 3:51 PM Reply to: Message 3 by Hyroglyphx12-24-2006 12:21 PM

Cheers nemesis.

I love this experiment, and I believe I once read that if you expand the â€˜wall with slitsâ€™ to encircle the particle gun, the spectrum will extend the full 360Â° around the gun, as determined by the probability distribution. Or perhaps my memory is playing tricks on me.

I know weâ€™ve wandered a bit OT here, but I wonder whether the outcome of the experiment is dependent on the status of the measuring device, i.e. does it need to be functioning for the waveform to collapse? What would happen to the results if the measuring device was randomly switched on and off during the experiment?

Iâ€™ve often wondered also what would happen if instead of a measuring device, different coloured filters were placed in each slit when the single photons are fired.

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dogrelata
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 Message 6 of 41 (372202) 12-25-2006 4:24 PM Reply to: Message 4 by cavediver12-24-2006 12:55 PM

 cavediver writes: Yes, absolutely. It is what we call a semi-infinite line.

I guess if our universe turned out to be infinite, and it was theoretically possible to realise such a line, how would it work? I mean, if you started at any point, you could presumably realise such a line in any direction, travelling forever. Presumably you could realise an infinite number of such lines radiating out in all directions, so what would cause them to be semi-infinite?

I expect Iâ€™m not making myself very clear here, but something thatâ€™s always puzzled me is that infinity tends to be thought of as infinitely large as opposed to infinitely small. In some ways the idea of infinitely small intrigues me more than the idea of infinitely large. If weâ€™re prepared to accept the possibility that the universe might be infinitely large (whatever that may be), I wonder if thereâ€™s also a possibility that there are things within this universe that might be infinitely small. In other words, in the previous paragraph, might it be that the lines that can be realised from infinity back to the point never converge because they keep on travelling forever into some infinite smallness â€˜withinâ€™ the point?

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GDR
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 Message 7 of 41 (372208) 12-25-2006 5:32 PM Reply to: Message 6 by dogrelata12-25-2006 4:24 PM

Is infinitly small a possibility. Wouldn't a plank length be as small as you can get?

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cavediver
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 Message 8 of 41 (372215) 12-25-2006 8:38 PM Reply to: Message 6 by dogrelata12-25-2006 4:24 PM

 how would it work?... ...what would cause them to be semi-infinite?

I'm sorry, I don't understand. Why would they have to be "caused" to be semi-inifinte? Nothing causes the +ve x-axis to be semi-infinite, nor any other line radiating out of any point in the x-y plane, they just are...

 I wonder if thereâ€™s also a possibility that there are things within this universe that might be infinitely small

In mathematics, the infinitely small - the infinitessimal - is exceptionally common and is the basis of calculus. In physics there are complications such as the idea of length breaking down at the Planck scale, so it becomes difficult to continue sub-dividing a length. However, it is entirely possible that the deepest underlying physical layers contain the concept of the infinitessimal, which is obscured as the quantum elements are built up. This is very much in the realms of the philosophy of physics/reality, rather than in hard science.

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Chiroptera
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 Message 9 of 41 (372221) 12-25-2006 9:17 PM Reply to: Message 8 by cavediver12-25-2006 8:38 PM

Pedantic point.
quote:
In mathematics, the infinitely small - the infinitessimal - is exceptionally common and is the basis of calculus.

This isn't quite right. Although it is true that Newton and subsequent users of calculus did think and argue in terms of infinitesimals, their ideas were not mathematically rigorous. It was not until Weierstrass and other analysts developed the theory of limits that calculus was finally given a firm, rigorous backing. But limits do not involve infinitesimals.

That said, although I don't use the term I do use the idea of infinitesimals when I teach calculus since, just like they did for Newton, they can give an intuitive idea of how to use calculus.

Also, there is a branch of mathematics called nonstandard analysis that develops rigorously the notion if infinitesimals and uses them to give an alternate development of analysis. I don't know much about it myself, but it has never really caught on in most of the mathematical world since it isn't really clear that it gives anything more than standard limit-based analysis gives.

Of course, it is possible that Robinson (the originator of nonstandard analysis) will have the last laugh if and when it proves to be useful in developing mathematics appropriate for Planck scales.

I have always preferred, as guides to human action, messy hypothetical imperatives like the Golden Rule, based on negotiation, compromise and general respect, to the Kantian categorical imperatives of absolute righteousness, in whose name we so often murder and maim until we decide that we had followed the wrong instantiation of the right generality. -- Stephen Jay Gould

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cavediver
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 Message 10 of 41 (372223) 12-25-2006 9:34 PM Reply to: Message 9 by Chiroptera12-25-2006 9:17 PM

Re: Pedantic point.
Completely agree... just detest theory of limits with a passion - it may be their constructivist flavour ;) or just that I was in a decidedly non-mathematical mood when I first had to study the subject - gigs and girls were my predominant study at the time - and I have always avoided teaching the subject like the plague.

I'm not so sure about use in Planck scale physics, which is very much finite scale, but it may be useful in looking at the emergence of finite scale physics from the underlying topological structure... and don't asy 'what underlying topological structure?' ;)

Edited by cavediver, : No reason given.

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dogrelata
Member (Idle past 3648 days)
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 Message 11 of 41 (372249) 12-26-2006 3:26 AM Reply to: Message 7 by GDR12-25-2006 5:32 PM

 GDR writes: Is infinitly small a possibility. Wouldn't a plank length be as small as you can get?

Donâ€™t know. Iâ€™m just trying to get my head round some of these ideas.:)

One observation I would make regarding the Planck length is that it uses the speed of light as part of the calculation. There has been some controversy recently about whether the speed of light is in fact a constant, http://www.newscientist.com/article/dn6092.html, leading to the possibility that the Planck length is not constant either.

However, I accept that I am nit-picking here, and saying that the Planck length may be subject to small variations is not the same as saying that there may exist things that are infinitely smaller than it.

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Neutralmind
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 Message 12 of 41 (372250) 12-26-2006 3:52 AM Reply to: Message 3 by Hyroglyphx12-24-2006 12:21 PM

 nemesis_juggernautI've been wanting to start a topic on this video, but I kept forgetting about it. Anyway, now is a good time to view it

I can't get my head around that observation distorts or effects the results. How is this?
Or maybe we should take this conversation to a new thread...

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dogrelata
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 Message 13 of 41 (372259) 12-26-2006 6:58 AM Reply to: Message 8 by cavediver12-25-2006 8:38 PM

 cavediver writes: I'm sorry, I don't understand. Why would they have to be "caused" to be semi-inifinte? Nothing causes the +ve x-axis to be semi-infinite, nor any other line radiating out of any point in the x-y plane, they just are...

I donâ€™t think Iâ€™m making a very good job of trying to explain what Iâ€™m trying to get atâ€¦maybe because I donâ€™t really know what Iâ€™m trying to get at.:)

I think what Iâ€™m trying to say is if you imagine a line coming from infinity and follow it back towards a single point in space (whatever that might be), what prevents the line carrying on forever â€˜into the pointâ€™?

I expect the problem I have arises from my concept of what this single point might be. I tend to think of a point as something spherical. If I go to a physics textbook, I tend to see atomic and sub-atomic particles represented as spheres, so I tend to envision a point in space as a sphere also.

Taking that as my starting point, if I imagine a sphere one meter in diameter, with a line extending infinitely from its surface, I will have a semi-infinite line. If I then imagine the diameter of the sphere to be halved, I still have a semi-finite line. I guess my question is, what prevents me halving the diameter of the sphere indefinitely? In other words, if I were to follow the line on its journey towards an ever-shrinking sphere, must I inevitably reach a point where the line can extend no further, and therefore be considered semi-finite?

This is what I was trying to say when I clumsily used the word â€˜causeâ€™. Perhaps I should have asked something like, does a line drawn from the theoretical largeness of infinity always have to be semi-infinite when followed in the direction of â€˜smallnessâ€™ in multi-dimensional space?

Mmmâ€¦Iâ€™m still not sure that makes any sense.

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fallacycop
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 Message 14 of 41 (372261) 12-26-2006 8:00 AM Reply to: Message 12 by Neutralmind12-26-2006 3:52 AM

 I can't get my head around that observation distorts or effects the results. How is this?Or maybe we should take this conversation to a new thread...
The problem isn`t the observer per se, but the means for observing. in order to observe the electron, you would have to shine some light on it. The act of shining that light changes the behaviour of the electron and distroys the interference pattern. That turns out to be a general principle. No matter what you do, you cannot at the same time know which slit the electron went through, and still get the interference pattern. It`s just not possible.

But note that the interference pattern will still be distroyed even if you turn off the observing device but keep the lights on. You don`t really need to have a concious observer taking notes of which slit the electrons went through.

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Hyroglyphx
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 Message 15 of 41 (372296) 12-26-2006 12:41 PM Reply to: Message 5 by dogrelata12-25-2006 3:51 PM

 I wonder whether the outcome of the experiment is dependent on the status of the measuring device, i.e. does it need to be functioning for the waveform to collapse?

I was thinking that too. Would the outcome be different if we were to employ a different measuring device?

 Iâ€™ve often wondered also what would happen if instead of a measuring device, different coloured filters were placed in each slit when the single photons are fired.

What outcome would you expect knowing what we now know about the quantum world? The results of the first few experiments were so counterintuitive that I'm not sure I'd place my expectations anywhere.

BTW, do you know the origin of this experiment? I'm curious to know.

"A man can no more diminish God's glory by refusing to worship Him than a lunatic can put out the sun by scribbling the word, 'darkness' on the walls of his cell." -C.S. Lewis

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