Register | Sign In


Understanding through Discussion


EvC Forum active members: 65 (9162 total)
6 online now:
Newest Member: popoi
Post Volume: Total: 915,817 Year: 3,074/9,624 Month: 919/1,588 Week: 102/223 Day: 13/17 Hour: 0/0


Thread  Details

Email This Thread
Newer Topic | Older Topic
  
Author Topic:   Mathematics and Nature
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 46 of 90 (269576)
12-15-2005 4:26 AM
Reply to: Message 45 by RAZD
12-14-2005 11:10 PM


Re: Why mathematics is so useful in the sciences
... 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:
the mathematics is so wonderful it couldn't possibly be wrong, and who cares about any evidence
If so, you are greatly mistaken.
This message has been edited by cavediver, 12-15-2005 04:47 AM

This message is a reply to:
 Message 45 by RAZD, posted 12-14-2005 11:10 PM RAZD has replied

Replies to this message:
 Message 47 by Dr Jack, posted 12-15-2005 4:43 AM cavediver has replied
 Message 57 by RAZD, posted 12-15-2005 3:47 PM cavediver has replied

  
Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 47 of 90 (269578)
12-15-2005 4:43 AM
Reply to: Message 46 by cavediver
12-15-2005 4:26 AM


Re: Why mathematics is so useful in the sciences
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.

This message is a reply to:
 Message 46 by cavediver, posted 12-15-2005 4:26 AM cavediver has replied

Replies to this message:
 Message 48 by cavediver, posted 12-15-2005 5:43 AM Dr Jack has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 48 of 90 (269582)
12-15-2005 5:43 AM
Reply to: Message 47 by Dr Jack
12-15-2005 4:43 AM


Confusions of Topology
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.

This message is a reply to:
 Message 47 by Dr Jack, posted 12-15-2005 4:43 AM Dr Jack has replied

Replies to this message:
 Message 49 by Dr Jack, posted 12-15-2005 5:49 AM cavediver has replied
 Message 56 by RAZD, posted 12-15-2005 3:41 PM cavediver has replied

  
Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 49 of 90 (269584)
12-15-2005 5:49 AM
Reply to: Message 48 by cavediver
12-15-2005 5:43 AM


Re: Confusions of Topology
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.

This message is a reply to:
 Message 48 by cavediver, posted 12-15-2005 5:43 AM cavediver has replied

Replies to this message:
 Message 50 by cavediver, posted 12-15-2005 6:14 AM Dr Jack has replied
 Message 51 by cavediver, posted 12-15-2005 6:16 AM Dr Jack has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 50 of 90 (269587)
12-15-2005 6:14 AM
Reply to: Message 49 by Dr Jack
12-15-2005 5:49 AM


Re: Confusions of Topology
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

This message is a reply to:
 Message 49 by Dr Jack, posted 12-15-2005 5:49 AM Dr Jack has replied

Replies to this message:
 Message 52 by Dr Jack, posted 12-15-2005 6:39 AM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 51 of 90 (269589)
12-15-2005 6:16 AM
Reply to: Message 49 by Dr Jack
12-15-2005 5:49 AM


Re: Confusions of Topology
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)

This message is a reply to:
 Message 49 by Dr Jack, posted 12-15-2005 5:49 AM Dr Jack has not replied

  
Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 52 of 90 (269592)
12-15-2005 6:39 AM
Reply to: Message 50 by cavediver
12-15-2005 6:14 AM


Re: Confusions of Topology
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.

This message is a reply to:
 Message 50 by cavediver, posted 12-15-2005 6:14 AM cavediver has replied

Replies to this message:
 Message 53 by cavediver, posted 12-15-2005 7:34 AM Dr Jack has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 53 of 90 (269596)
12-15-2005 7:34 AM
Reply to: Message 52 by Dr Jack
12-15-2005 6:39 AM


Re: Confusions of Topology
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!

This message is a reply to:
 Message 52 by Dr Jack, posted 12-15-2005 6:39 AM Dr Jack has replied

Replies to this message:
 Message 54 by Dr Jack, posted 12-15-2005 7:49 AM cavediver has replied

  
Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 54 of 90 (269597)
12-15-2005 7:49 AM
Reply to: Message 53 by cavediver
12-15-2005 7:34 AM


Re: Confusions of Topology
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.

This message is a reply to:
 Message 53 by cavediver, posted 12-15-2005 7:34 AM cavediver has replied

Replies to this message:
 Message 55 by cavediver, posted 12-15-2005 9:19 AM Dr Jack has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 55 of 90 (269608)
12-15-2005 9:19 AM
Reply to: Message 54 by Dr Jack
12-15-2005 7:49 AM


Re: Confusions of Topology
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.

This message is a reply to:
 Message 54 by Dr Jack, posted 12-15-2005 7:49 AM Dr Jack has not replied

  
RAZD
Member (Idle past 1405 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 56 of 90 (269698)
12-15-2005 3:41 PM
Reply to: Message 48 by cavediver
12-15-2005 5:43 AM


Re: Confusions of Topology
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.

Join the effort to unravel {AIDS\HIV} with Team EvC! (click)

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.

This message is a reply to:
 Message 48 by cavediver, posted 12-15-2005 5:43 AM cavediver has replied

Replies to this message:
 Message 59 by cavediver, posted 12-15-2005 8:51 PM RAZD has replied

  
RAZD
Member (Idle past 1405 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 57 of 90 (269700)
12-15-2005 3:47 PM
Reply to: Message 46 by cavediver
12-15-2005 4:26 AM


Re: Why mathematics is so useful in the sciences
"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.

Join the effort to unravel {AIDS\HIV} with Team EvC! (click)

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.

This message is a reply to:
 Message 46 by cavediver, posted 12-15-2005 4:26 AM cavediver has replied

Replies to this message:
 Message 58 by cavediver, posted 12-15-2005 8:27 PM RAZD has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 58 of 90 (269791)
12-15-2005 8:27 PM
Reply to: Message 57 by RAZD
12-15-2005 3:47 PM


Re: Why mathematics is so useful in the sciences
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.

This message is a reply to:
 Message 57 by RAZD, posted 12-15-2005 3:47 PM RAZD has not replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 59 of 90 (269801)
12-15-2005 8:51 PM
Reply to: Message 56 by RAZD
12-15-2005 3:41 PM


Re: Confusions of Topology
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.

This message is a reply to:
 Message 56 by RAZD, posted 12-15-2005 3:41 PM RAZD has replied

Replies to this message:
 Message 60 by RAZD, posted 12-15-2005 9:20 PM cavediver has replied

  
RAZD
Member (Idle past 1405 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 60 of 90 (269812)
12-15-2005 9:20 PM
Reply to: Message 59 by cavediver
12-15-2005 8:51 PM


Re: Confusions of Topology = 1 math for another
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.

Join the effort to unravel {AIDS\HIV} with Team EvC! (click)

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.

This message is a reply to:
 Message 59 by cavediver, posted 12-15-2005 8:51 PM cavediver has replied

Replies to this message:
 Message 61 by cavediver, posted 12-15-2005 9:32 PM RAZD has replied

  
Newer Topic | Older Topic
Jump to:


Copyright 2001-2023 by EvC Forum, All Rights Reserved

™ Version 4.2
Innovative software from Qwixotic © 2024