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Author | Topic: Mathematics and Nature | |||||||||||||||||||||||
cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
... models the mathematical concept close enough to convey the idea, but it is not the concept. Topologically or otherwise. Again, statement of philosophy, not statement of fact.
And yet, when you look inside the subatomic particles are forever dancing and changing partners ... based on QT eh? "Dancing"? "Changing partners"? Sorry, I don't speak Layman-ese as I told Randman recently. What are you talking about? Of what relevance is this to my point?
And you seem to be confusing "theoretical" with "actual" possiblity of distinguishing No confusion. What I mean is that it is not a point regarding difficulty of measurement. Like the uncertainty principle, this is a fundemental facet of reality.
are you claiming that theory is more important? Not at all. I am saying that the theory has been verified by observation. More verified, I should add, than any theory other than GR.
If it WAS measured and DID invalidate QT what would happen to reality? Nothing of course. We would have some work to do... all of the mathematics of QED would be wrong. The mystery would be how an amazingly complex set of calculations based on totally incorrect mathematics arrives at precisely (10+ decimal places of accuracy) the observed values. This is one piece of observational evidence that electrons are identical. Another is based upon my quiz. You didn't answer the questions. When this quiz is carried out in experiment, what probabilities do we observe? Or what probabilities should we observe if electrons are potentially distinguishable? Are these probabilities then observed in experiment? Answer: no. You seem to be under the impression that my outlook is
quote: If so, you are greatly mistaken. This message has been edited by cavediver, 12-15-2005 04:47 AM
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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.3 |
Again, statement of philosophy, not statement of fact. While I agree with you that mine and RAZD's view of Mathematics as an entirely human construct is a philosophical statement, his statement that a cellotaped bit of paper is simply an approximation to a mobius strip is correct. A piece of paper has topological properties that a mobius strip doesn't - specifically it has thickness, and slight discontinuities at the join.
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cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
I think there is a confusion of terms here. There is no idealised Mobius Band as such, despite layman artistic or computer-genberated depictions of such. An MB is a loop with certain global topological properties. It doesn't exclude other topological properties, local or global.
My pieces of paper exhibited the required global topological behaviour, amongst other behaviours. You can imagine an idealised surface wrapped into a Mobius Band, but this surface also has topological properties that are irrelevant to its Mobius topology such as the nature of its width - is it open or closed, or is there a distance function defined. If you want to do this properly, you need to strip it all down to an S1 base space with a line bundle (tangent bundle being the obvious) and then choose suitable transition functions. But this is not how we generally use the "concept" of a MB.
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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.3 |
There is most certainly an idealised mobius strip: see here for a formal definition Möbius Strip -- from Wolfram MathWorld - the band you describe approximates to this definition but does not precisely meet it.
I notice however that you are using the term Mobius Band, do you mean exactly the same thing and are calling it by a different or are you using the term to mean something slightly different.
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cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
There is most certainly an idealised mobius strip: As I mentioned, this is an idealised "surface" that is then given the topology of a Mobius Band (Loop, Strip, etc). It is just one way of depicting something with the properties of a MB. Of course, you can define an MB to be this idealised surface as depicted in Wolfram, but you are somewhat missing the point. The Wolfram entry is appallingly 3d geometrical, which is a shame, and certainly not how I would write the entry. It is the topology that gives the MB its properties, not its geometry. All that geometry is very nice but utterly restricted to this particular representation and 3d embedding. Great for engineers using real MBs in applications, but wholly insufficient for the mathematician. This message has been edited by cavediver, 12-15-2005 06:17 AM
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cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
I'm not sure, but I think Mobius Band may be more engineer speak (shame on me). That said, my biography of Mobius is called Mobius and his Band (pun intended on referring to his contemporaries)
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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.3 |
I'm not sure which part of the link I gave you are looking at: the definition is the 2d bounded surface given near the top. The class 'mobius strip' is the class of things topologically equivalent to this. Your bit of paper is not strictly topologically equivalent to it, it merely approximates it.
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cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
I'm too tied up to do this justice at the moment, but would like to continue this later.
For now: thickness is irrelevant. Topology is about properties invariant under deformation, and thickness can be deformed to flatness. The join does break rotational symmetry and could cause problems, depending upon what you are defining as the MB: is it the paper itself, arrows upon the paper, etc. I don't have to view it as a problem. The topological properties are still there. An apple is a perfectly good realisation of SO(3) rotations, unhampered by its nobblyness and stalk!
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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.3 |
Have a look at this - it's a magnified image of a sheet of paper.
Even a sheet of paper is not topologically equivalent to a bounded surface.
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cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
Even a sheet of paper is not topologically equivalent to a bounded surface. Of course. We are talking across each other here and I think it's my fault for not having time to explain myself coherently. Think of arrows on the paper loop revealing the coarse-grained topological properties. The fine-structure of the paper and the ink marks of the arrows does not have to be important.
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RAZD Member (Idle past 1425 days) Posts: 20714 From: the other end of the sidewalk Joined: |
My pieces of paper exhibited the required global topological behaviour, amongst other behaviours. You can imagine an idealised surface wrapped into a Mobius Band, but this surface also has topological properties that are irrelevant to its Mobius topology such as the nature of its width - is it open or closed, or is there a distance function defined. You are equivocating. The model displays the topology to a crude degree sufficient to convey the concept in model form, but it is still not the mathematical concept. The other properties you say are "irrelevant to it's mobius topology" show that you recognize that it is not the mathematical concept, as they do not exist for the concept. Enjoy. we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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RAZD Member (Idle past 1425 days) Posts: 20714 From: the other end of the sidewalk Joined: |
"Dancing"? "Changing partners"? Sorry, I don't speak Layman-ese as I told Randman recently. What are you talking about? Of what relevance is this to my point? These are metaphors I have used for the quantum behavior of subatomic particles to change from one to another or to more than one or disappear altogether for brief moments in time. What the particle is has to be described as a probability cloud, never certain. It's a dance with interchangeable partners. Or am I blindingly obtuse and invariably wrong on this topic too? Enjoy. we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
Or am I blindingly obtuse and invariably wrong on this topic too? Well, perhaps I wouldn't go quite that far But...
quantum behavior of subatomic particles to change from one to another or to more than one or disappear altogether for brief moments in time is incompatible with
What the particle is has to be described as a probability cloud, never certain. The latter is QM, which is inadequate to describe the former, which is the world of QFT. QM and its probability cloud failed to describe the real world, where particle creation and annihilation were observed. It required the addition of SR (which amongst other things introduced the concept of antiparticles) which led to the first formulations of QFT. We call this "second quantisation". But the mathematics of pair creation/annihilation and virtual particles is very well understood AND inextricably linked with the "indistinguishability" of those particles. This is not a random world where anything can happen. There are strict rules that are observed to be obeyed every day at every particle lab around the world.
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cavediver Member (Idle past 3664 days) Posts: 4129 From: UK Joined: |
You are equivocating. The model displays the topology to a crude degree sufficient to convey the concept in model form, but it is still not the mathematical concept I am not equivocating. I am disagreeing with your understanding of the mathematical concept. Once again, this does come down to a matter of philosophy... and perhaps of more relevance, mathematical training. I am a topologist, and I thus have different working definitions to someone not trained in topology, or for example someone emphasising geometry. I am also a mathematical physicist so I am bringing in the technology of representations and realisations.
The other properties you say are "irrelevant to it's mobius topology" show that you recognize that it is not the mathematical concept, as they do not exist for the concept No, it shows that I do not agree with your understanding of the concept. Adding extra structure does not destroy the original property. The concept is a property, not an object.
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RAZD Member (Idle past 1425 days) Posts: 20714 From: the other end of the sidewalk Joined: |
but you are.
you are idealizing the concept from the model and then saying that the idealized model relates in a mathematical (topology) manner to the mathematical concept. you are not comparing the concept with the model that is the reality of the paper strip. we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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