Warp wrote:
> 
>   Ever heard of the Banach-Tarski paradox? A solid ball in 3-dimensional
> space can be split into several non-overlapping pieces, which can then be
> put back together in a different way to yield two identical copies of
> the original ball.
> 

Am I the only one here, who's waiting for a raytraced animation-example 
of this? :)

-- 
Eero "Aero" Ahonen
    http://www.zbxt.net
       aer### [at] removethiszbxtnetinvalid

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Eero Ahonen <aer### [at] removethiszbxtnetinvalid> wrote:
> Am I the only one here, who's waiting for a raytraced animation-example 
> of this? :)

  Visually dividing a sphere into sets with uncountably many infinitely
small points? You wish.

-- 
                                                          - Warp

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Warp wrote:
> 
>   Visually dividing a sphere into sets with uncountably many infinitely
> small points? You wish.
> 

Indeed I do. :p

-- 
Eero "Aero" Ahonen
    http://www.zbxt.net
       aer### [at] removethiszbxtnetinvalid

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Warp wrote:
>   It's not like you could shoot an electron to some direction, and then
> the electron suddenly hits the other side of the Earth (or the solar
> system). It hits a quite accurately calculable place.

	There is a non-zero probability that this can happen.

-- 
The next war will determine not what is right, but what is left.


                     /\  /\               /\  /
                    /  \/  \ u e e n     /  \/  a w a z
                        >>>>>>mue### [at] nawazorg<<<<<<
                                    anl

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Mueen Nawaz wrote:
>     Yes - wasn't that proven only recently?

I heard about it recently, like in the last couple of years.

>     The remarkable thing, in case anyone didn't glean it from Darren's 
> description, is that this can be done with a *finite* number of cuts. I 
> recall it was a huge number, though.

Something like 10^50 cuts or something?  I know it's on Wolfram's 
mathworld somewhere, but I don't remember the name of it, so it's hard 
to find.

-- 
   Darren New / San Diego, CA, USA (PST)
     "That's pretty. Where's that?"
          "It's the Age of Channelwood."
     "We should go there on vacation some time."

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Stephen wrote:
> Indeed, a straight edge and compass. What more do you need?

A ruler?

-- 
   Darren New / San Diego, CA, USA (PST)
     "That's pretty. Where's that?"
          "It's the Age of Channelwood."
     "We should go there on vacation some time."

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Warp wrote:
>   So when the electron hits the sensitive film *after* it has passed the
> slits, it goes back to the past and changes it so that it goes through
> only one of the slits after all?

No. It either interferes with itself or not. You're assuming the only 
way it can interfere with itself is to go through both slits. There's no 
evidence that's the case, and much evidence that it isn't.

>   I knew quantum mechanics were whacky, but I didn't know particles could
> travel back in time and change their previous behavior because the effect
> of that behavior was measured *after* the fact.

Actually, yes, they can. For example, a photon can emit a 
backwards-traveling electron (aka a positron) which cancels with the 
electron it emits later, so as to preserve the number of non-virtual 
particles around. It's weird.

But there's no evidence that's what's happening in the two-slit experiment.

>   So you are saying that, even though the only possible explanation for
> interference patterns is that the electron passed through both slits,
> there's still no evidence of that?

Yes. What makes you think that the only *possible* explanation is that 
the electron passed through both slits?

>   If there's "no evidence", what do you call the interference pattern?
> "Non-evidence"?

Interference. Nobody knows why, because every time you measure an 
electron, there's only one, even when both slits are open. There are 
plenty of experiments where both slits are open and you only ever see 
the electron go through one or the other and never both. Of course, you 
don't get the interference when you actually look, but then the fact 
you're looking *after* the electron went through the slits shouldn't 
affect which slits it goes through, should it?

-- 
   Darren New / San Diego, CA, USA (PST)
     "That's pretty. Where's that?"
          "It's the Age of Channelwood."
     "We should go there on vacation some time."

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Warp wrote:
> Michael Zier <mic### [at] mirizide> wrote:
>> Don't forget: A single electron doesn't make an interference pattern. It 
>> produces a bright spot on the fluorescence screen saying: Exactly here 
>> and nowhere else the electron hit the screen.
> 
>   Wasn't it "impossible" to accurately measure the place and velocity
> of a quantum particle? ;)

Only down to Plank's constant. You can easily tell down to the size of 
an atom which atom it hit.

>> Only if you observe many many events and add up their positions they 
>> converge to an interference pattern.
> 
>   But I believe it can be done by shooting just one electron at a time,
> so the electron indeed interferes only with itself, not with other
> electrons.

No, it only interferes with other electrons. The electrons it interferes 
with are electrons from other times. There's always a possibility that 
it lands at any particular place. Why is it any stranger that it 
interferes with electrons in the future than it is it interferes with 
itself?

That's what I was talking about in the "amazingly, math works" thread.

-- 
   Darren New / San Diego, CA, USA (PST)
     "That's pretty. Where's that?"
          "It's the Age of Channelwood."
     "We should go there on vacation some time."

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somebody wrote:
> "Darren New" <dne### [at] sanrrcom> wrote in message
> news:484363fb$1@news.povray.org...
>> somebody wrote:
> 
>> For example, there's no known mathematical model to indicate where a
>> specific electron is, and indeed if I understand correctly, experiments
>> show there cannot be one.
> 
> Some syntatically correct questions can be meaningless. What colour is the
> note C#?

I have no idea what you're trying to say here.

>> I strongly suspect there's no possible mathematical model for "free
>> will" in its usual meaning.
> 
> The problem is that the usual meaning isn't. 

Again, I have no idea what you're trying to say here.

> Plus you have to be aware of
> and agree on the level of abstraction you are working with. An economical
> question becomes practically meaningless at subatomic level of the atoms of
> the cells of the human of the masses that create the demand. "Free will" is
> at best a crude concept at individual human level, and it goes downhill from
> there quickly at any other more fundamental levels.

Um, yes. That's kind of my point.

I take it you're retracting your statement:
> Easy. Mathematics can represent *anything*, since you get to make up your
> own axioms.

That's what I'm addressing, and nothing else. Mathematics can't 
represent "free will", nor can it represent where (say) the tenth 
electron to hit the screen will hit it in a particular well-defined 
experiment.

>> Experiments with gravity show the best theory we have for it is
>> incompatible with the best theory we have for atomic interactions. What
>> happens when someone proves that gravity is incompatible with quantum
>> mechanics? That there cannot be a GUT?
> 
> Unfortunately (or fortunately), you cannot disprove reality. 

Depending on your meaning of "prove", but yes, I think I know what 
you're trying to say here.

> The question is
> not whether or not there's a GUT (reality provides one working model at
> least)

That's the assumption I'm challenging. Please provide evidence that 
there is one completely describable model of reality possible.

There is reality. Then there's the Grand Unified Theory of reality, 
which expresses the behavior of the universe in mathematical terms which 
can be symbolically manipulated in a manner isomorphic to how reality 
works.

You can't disprove reality. But that doesn't mean there's any formalism 
that can express it.

>>> Whether that model can be simplified, and can have predictive powers
>>> or not is the question,
> 
>> If it doesn't have predictive powers, it's not an appropriate model.
>> It's merely a summary of the past rather than a model of the actual
> reality.
> 
> Past *is* "actual" reality. More so than future, less so than present.

Yes. But if you can't predict the future based on the past, you don't 
have a *model*. You have a mathematical description of the past that may 
or may not have anything to do with the future.

If I flip a coin four times in a row and get all heads, it's not a model 
of coin flipping to say "it always comes up heads."  If I 'predict' in 
2008 ever terrorist action for the years 2000-2006 using my mathematical 
formula, that doesn't mean it's good for 2010. If my equations match 
perfectly with the stock market for the last five years, that doesn't 
mean I'll make money next year.

> Modelling randomness is easy (in theory anyway). It just won't have
> predictive ability.

I would claim that if you can't predict, you don't have a model, you 
have a history book. If you can't take a model isomorphic to current 
reality, perform formal transforms on it, and have a model that is 
isomorphic to the reality you get after the modeled transforms are 
performed in reality, then you don't have a model.

-- 
   Darren New / San Diego, CA, USA (PST)
     "That's pretty. Where's that?"
          "It's the Age of Channelwood."
     "We should go there on vacation some time."

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Sabrina Kilian wrote:
> It's an octave of blue, if I got the math right. I started from 4435Hz, 
> though, instead of a more accurate number, so it's probably off by a bit.

That's my giggle of the day, thanks.

-- 
   Darren New / San Diego, CA, USA (PST)
     "That's pretty. Where's that?"
          "It's the Age of Channelwood."
     "We should go there on vacation some time."

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