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


Message 9 of 90 (268706)
12-13-2005 8:28 AM
Reply to: Message 8 by JustinC
12-13-2005 8:09 AM


Try this easy experiment. Build a transistor. Model it according to QM. Compare and contrast the results.
Science is not a myth for our time. Interpretations of what the numbers mean are (I, incidently, think the idea of mathematics as the underlying reality is absurd).

This message is a reply to:
 Message 8 by JustinC, posted 12-13-2005 8:09 AM JustinC has replied

Replies to this message:
 Message 11 by cavediver, posted 12-13-2005 8:49 AM Dr Jack has replied
 Message 16 by JustinC, posted 12-13-2005 10:33 AM Dr Jack has replied

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


Message 12 of 90 (268719)
12-13-2005 9:04 AM
Reply to: Message 11 by cavediver
12-13-2005 8:49 AM


Probably because I'm a Mathematician (by education anyway). Maths is an entirely human construct; a useful invention in the same way that the spanner, the computer and logic are. Maths exists within our heads, and has reality only were we find correlations; taking an apple and putting it next to another apple doesn't make two apples because of mathematics. I've also studied a reasonable amount of Philosophy, and I find striking parallels between the Platonic notion of ideals and the notion of mathematics-as-reality - and I have very little credance for Plato's works.
I find nothing surprising about the success of mathematics as a description of physical reality. Maths can essentially embrace anything that operates according to rules; which physics does (although I don't mean to imply that the rules exist either; I think the behaviour of physical systems comes bottom-up not top-down). While mathematics roots lie in macrosopic physics (take an apple and another apple get two apples; draw a circle of diameter 1, measure its circumfrence, etc.) and these macroscopic physics themselves emerge from the behaviour of the microscopic physics in a reversal of the way mathematical constructs have emerged in generalisation and derivation from basic maths. It does not surprise me that the two intermesh.
But I freely accept your point about absurdity; what I find absurd is in no way a reasonable guide to what actually is.

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


Message 14 of 90 (268736)
12-13-2005 10:11 AM
Reply to: Message 13 by cavediver
12-13-2005 9:43 AM


Just to be clear - this is your philosophical take on mathematics, one take of several
This is true. However, if I make the lesser claim "Mathematics is a human construct" then that is factually verifiable by studying the history of mathematics whether that construct reflects a deeper reality or not.
And you are still left with "what is stuff?". You are stuck with new-age concepts of energy and vibrations and "things" and a naive separation of foreground object and background space. This is fine if you are happy with that, but that appears far more absurd to me than my ideas.
I agree; I don't believe any modern interpretations of physical reality are true. I'm also far from convinced that this is an answerable questions. But I don't see how taking mathematics as your physical reality helps you avoid this; how do you go from your numbers to the physical reality we experience in day-to-day life? By what means are your numbers structured (by which I mean how is one waveform separated from another, for example)? And, of course, since I consider mathematics to be an entirely human construct I cannot also consider it to be a (the?) fundamental part of reality; one of these positions must be wrong and given the one I have more knowledge of and am in a better position to judge is my position on mathematics I'm inclined to take that as the one I'm correct on.
As does physics. At the moment, we only have laws of GR and QFT. Everything is part of one or other of these. These are very much bottom-end.
I don't think I made myself clear on this. I would consider GR and QFT top-down, in that they describe the behaviour of objects from a lofty looking down kind of perspective. They describe how things behave rather than how they work. I think that the behaviour of reality works the other way round and GR and GFT emerge from the interaction of simpler players each of whom has no knowledge of the target rules (in the same way that the gas laws emerge from the properties of the particles of that gas).
And, again, I accept this is complete speculation on my part. I wouldn't wish anyone to get the idea that I'm expounding on the actual reality of the universe from a position of privileged knowledge.

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


Message 17 of 90 (268756)
12-13-2005 10:54 AM
Reply to: Message 16 by JustinC
12-13-2005 10:33 AM


Yes, and no. The predictions of it's behaviour: yes, you do; the extrapolation from that to what is actually happening in the real world to make the results come out as they do: no, you don't.
These are different issues, no?

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


Message 19 of 90 (268775)
12-13-2005 11:48 AM
Reply to: Message 18 by cavediver
12-13-2005 11:42 AM


Ahhh, ok, let you off. But what do you mean by separation? A wave-fn exists over all space - it's just a functional, a point in a Hilbert Space. But position for example is simply a case of an n-tple of coords.
In string theory, these n-tples are just values in n fields defined on a 2d space. The n-d space (4d space-time in our case) is purely derived (or emerges) from these 2d-fields.
Oh, ok. Yeah, that all makes sense.

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 Message 18 by cavediver, posted 12-13-2005 11:42 AM cavediver has replied

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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.

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 Message 46 by cavediver, posted 12-15-2005 4:26 AM cavediver has replied

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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.

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 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
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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.

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 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

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


Message 64 of 90 (269914)
12-16-2005 6:01 AM
Reply to: Message 62 by RAZD
12-15-2005 9:44 PM


Re: Confusions of Topology = 1 math for another
the mathematical concept is a single sided surface.
No, it isn't. The mathematical concept is of a bounded 2d surface with an equivalence map making points on the top and bottom edges equivalent to one another, in reversed order.
So (x, 0) = (1-x, 1) for all x in [0, 1] (taking the surface as [0,1]x[0,1] for simplicity of notation).
This is then expanded to all objects topologically equivalent to this.

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Replies to this message:
 Message 65 by cavediver, posted 12-16-2005 6:47 AM Dr Jack has replied

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


Message 66 of 90 (269944)
12-16-2005 8:42 AM
Reply to: Message 65 by cavediver
12-16-2005 6:47 AM


Re: Confusions of Topology = 1 math for another
You could indeed, but I would argue that the definition I gave is the primary definition since the algebraic topological definition was defined to match the formal definition I gave.

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 Message 65 by cavediver, posted 12-16-2005 6:47 AM cavediver has replied

Replies to this message:
 Message 67 by cavediver, posted 12-16-2005 9:13 AM Dr Jack has replied

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


Message 68 of 90 (269962)
12-16-2005 9:38 AM
Reply to: Message 67 by cavediver
12-16-2005 9:13 AM


Re: Confusions of Topology = 1 math for another
Touche

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Replies to this message:
 Message 69 by RAZD, posted 12-17-2005 7:41 AM Dr Jack has replied

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


Message 88 of 90 (270678)
12-19-2005 4:31 AM
Reply to: Message 69 by RAZD
12-17-2005 7:41 AM


Re: Confusions of Topology = 1 math for another
RAZD when someone refers you to a technical site giving a formal definition of a concept, refering to a popular site giving a popular definition isn't an answer. That's just as absurd as when Creationists argue back to Evolutionist citing The New York Times in answer to an article from Nature.

This message is a reply to:
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