Darren New <dne### [at] sanrrcom> wrote:
> (BTW, you don't get particles being physically present at multiple 
> locations. If you actually measure where they are, they're only in one 
> place.)

  But their simultaneous location in more than one place can be inferred
by other side-effects. For example a single electron can pass through two
slits at the same time, interfering with itself after doing so.

-- 
                                                          - Warp

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Chambers wrote:
> andrel wrote:
>> A couple of remarks:
>> - According to that most reliable source of wikipedia, it was Truman 
>> that fired MacArthur for disagreeing with him. So I would be surprised 
>> if he did not try to portrait him as an idiot. That is irrespective of 
>> whether he was right or wrong. Simply the case that a president fires 
>> a famous general dictates that the president should convince the 
>> public that he was much better equipped than the general.
> 
> Basically, it all came down to the fact that MacArthur wanted to go out, 
> fight, and win WWIII.
> 
> Truman did everything in his power to avoid an actual war - he thought 
> two World Wars were more than enough, and would rather see the conflict 
> played out on a smaller scale.  

Indeed, just what Truman would say ;)

> The Korean War (and, eventually, the 
> Vietnam War) were direct results of his policies in that respect, as was 
> the Cold War in general.
> 
> People blame our leadership for those confrontations, but fail to 
> realize that the alternative was all out war on a scale similar to, if 
> not larger than, WWII.
> 
> ...Chambers

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>  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.

I hadn't heard of that one, but it makes perfect sense in a topology 
context.
In that context "split" really means a 1:2 mapping of spaces.
Cut the ball in half, then in the two hemispheres the new equator plane
is a curve from the lip that follows a 0.5*y scaled sphere, from there
just map the curved planes back to flat to get two spheres.
Since we're talking about a mapping, not a real object, there's no
conservation of volume, a sphere is a sphere, no matter what the size.

The paradox is that the more mathematicians learn,
the less they are able to explain clearly.

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On Sun, 1 Jun 2008 13:34:22 -0600, "somebody" <x### [at] ycom> wrote:

>"Darren New" <dne### [at] sanrrcom> wrote
>> Warp wrote:
>
>> >   I would say that mathematics can always be used to represent reality
>> > when put in the proper context.
>
>> The fascinating thing to wonder about is ... why is this so?
>
>Easy. Mathematics can represent *anything*, since you get to make up your
>own axioms.
>
Oh! You let the secret our :)
-- 

Regards
     Stephen

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Darren New wrote:
> Warp wrote:
>>   I'm sure if you asked a physicist specialized in quantum mechanics he
>> would say that life would probably not be possible without the wild
>> uncertainties of quantum phenomena (such as particles physically being
>> at multiple locations at the same time).
> 
> I imagine that life based on DNA and chemistry and such wouldn't work 
> the same way. 
It wouldn't work at all. Let's put it this way: if the Universe was as 
complicated as the most inventive physicist can image, it would still 
not be complex enough for life to develop. Hmm, there might be a good 
sig in that for this physicist.

--
Physics is incompatible with life.

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somebody wrote:

> Easy. Mathematics can represent *anything*, since you get to make up your
> own axioms.

Now try inventing a *consistent* set of axioms. ;-)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Tim Attwood <tim### [at] comcastnet> wrote:
> I hadn't heard of that one, but it makes perfect sense in a topology 
> context.
> In that context "split" really means a 1:2 mapping of spaces.
> Cut the ball in half, then in the two hemispheres the new equator plane
> is a curve from the lip that follows a 0.5*y scaled sphere, from there
> just map the curved planes back to flat to get two spheres.
> Since we're talking about a mapping, not a real object, there's no
> conservation of volume, a sphere is a sphere, no matter what the size.

  That doesn't work. The volume of the sphere cannot be modified by a
simple change in topology. You cannot simply change the topology and
then calculate the volume as if you hadn't. You have to calculate the
volume using the *new* topology, not the old one.

  Besides, if what you say was true, the same trick would work with a
2-dimensional circle, but it has been proven that it doesn't.

  And besides, the original setup happens in regular cartesian coordinates,
without any change in topology.

> The paradox is that the more mathematicians learn,
> the less they are able to explain clearly. 

  You clearly haven't understood the theorem.

-- 
                                                          - Warp

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Warp wrote:
>   Ever heard of the Banach-Tarski paradox? 

There's another cool one, I don't remember what it's called, that proves 
you can chop up a filled-in circle (I.e., a 2D slice thru a ball, 
whatever you call that) and then put it back together again as a perfect 
square, with no overlaps and no gaps.

Personally, I can't imagine how that can be possible, but you can 
apparently do it with sufficiently many cuts.

-- 
   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:
>   But their simultaneous location in more than one place can be inferred
> by other side-effects. 

Nope. When you actually measure it, it's only going thru one slit.

> For example a single electron can pass through two
> slits at the same time, interfering with itself after doing so.

Not as such. Yes, you get interference patterns. No, as far as I know, 
there's no evidence to suggest it goes through both slits. Nobody is 
quite sure how it works, but there's no measurement that when you say 
"where is the thing" it ever gives you more than one answer.

That's the funky part. It *isn't* intuitive.

Indeed, there are all kinds of experiments (like closing one slit too 
close to the photon for it to know that the slit was closed, or 
something like that) that indicate that you can't treat the situation 
like the photon or electron goes through both slits at once.

-- 
   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:
> Easy. Mathematics can represent *anything*, since you get to make up your
> own axioms.

Except for two things: all the equations are actually pretty simple, 
none of them seem to change.

It's not really the case you can represent *anything* with mathematics. 
You cannot represent God (pretty much by definition of God), and you 
cannot represent a partially-inconsistent system (one that is 
inconsistent sometimes but not other times, or in some places but not 
other places). Just as a couple of offhand examples.

-- 
   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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