Darren New wrote:

> That's possible. But it certainly isn't obviously true. If the 
> super-computer isn't programmed in the usual way, or if (for example) it 
> can execute an infinite number of instructions in finite time, or if it 
> can travel back in time, or etc, I expect you'd have to think hard about 
> whether that gets around the technique the halting problem proof uses.
> 
> If your super-computer cannot, for example, have its programs 
> represented as input to itself (i.e., you can't create a universal 
> super-computer), then the halting problem isn't even *defined* for that 
> type of computer.

I implicitly assumed that any machine that can't process it's own 
program isn't worthy of the title "computer", that's all. ;-) But yes, I 
see what you're saying.

As for being able to perform infinite instructions in finite time... 
surely that just makes it even *harder* to predict what the machine will 
od, no?

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

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On Thu, 31 Jul 2008 09:18:43 +0100, Invisible wrote:

> Jim Henderson wrote:
> 
>> Well, what fun would it be if I just agreed with you? ;-)
> 
> Damn it, that isn't even *logic*! :-P

LOL!

Jim

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On Tue, 29 Jul 2008 22:29:26 +0100, Orchid XP v8 wrote:

> I am not aware - despite possessing a book detailing the entire history
> of Fermat's Last Theorum - of any proof that was widely held to be
> correct for a long time before being found wrong. All the incorrect
> proofs were discovered to be incorrect fairly quickly.

Absence of evidence is not evidence of absence. ;-)

>>> And I suppose next you'll be telling me that some day, some future
>>> technology might enable us to find a sequence of chess moves whereby a
>>> bishop can get from a black square to a white square, despite it being
>>> trivially easy to mathematically prove the impossibility of this...
>> 
>> You're still missing my point....
> 
> You're still missing *my* point. :-P

Then we're even. ;-)

>> My point is that there's plenty of examples where raw data is lost but
>> it can be reconstructed.
> 
> Blurring doesn't actuallly "lose" nearly as much data as you'd think.
> That's why it can be mostly reversed.

Again, 10 years ago, doing this was thought to be impossible.

>> Well, who knows?  There are ancient civilizations that had no concept
>> of zero.  The introduction of imaginary numbers didn't come along until
>> the late 1500s.  Up until that point, sqrt(-1) was undefined.
>> 
>> Who knows what we don't know about mathematics even today?
> 
> If I were you, I'd be far more worried about the sky falling - it's
> about as logically plausible...

I don't see how your statement follows mine....

Throughout history, mankind has claimed to have reached the end of 
knowledge on all manner of topics, saying "there's nothing more to learn 
here".  In every instance (AFAIK), that's been proven wrong.

But now here, in the 21st century, we've finally exhausted the base of 
knowledge?  I don't think that's the case.

Jim

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Invisible wrote:
> I implicitly assumed that any machine that can't process it's own 
> program isn't worthy of the title "computer", that's all. ;-) But yes, I 
> see what you're saying.

So, your brain is weaker than a Turing machine? Cool.

It's the simplicity of the machines that make them amenable to 
universality, not the complexity. Remember that the Von Neumann 
architecture was a breakthrough.

> As for being able to perform infinite instructions in finite time... 
> surely that just makes it even *harder* to predict what the machine will 
> od, no?

Not if it can process its own input. Think about how the halting problem 
works, and why... If the machine has unbounded state, it might run 
forever without ever getting into the same state twice. But if you can 
run the computer forever without it actually taking forever, then you 
can say definitively "no, that machine never stops."

And a machine that can run an infinite number of instructions in finite 
time *always* stops. ;-)

-- 
Darren New / San Diego, CA, USA (PST)

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Jim Henderson wrote:
> Absence of evidence is not evidence of absence. ;-)

Sure it is. It's not proof of absence, of course, but it's stronger 
evidence than not.

http://www.overcomingbias.com/2007/08/absence-of-evid.html

> But now here, in the 21st century, we've finally exhausted the base of 
> knowledge?  I don't think that's the case.

The problem is that you're confusing math with physics. In math, yes, we 
have exhausted the base of knowledge of the halting problem. The halting 
problem goes like this:

Assume A.  Assume B.   Assume C.  Therefore D.

You're arguing "maybe assumption B is counterfactual."  That doesn't 
disprove the halting problem. It doesn't mean you can solve the halting 
problem. It merely means the halting problem doesn't apply to your 
current situation.

Indeed, one of the assumptions required for the halting problem to hold 
(namely, unbounded storage) is *known* to be impossible to realize in 
this universe (or at least normally assumed to be impossible even by 
people who understand the halting problem). That *still* doesn't mean 
the halting problem is "solved".

Math isn't physics.

-- 
Darren New / San Diego, CA, USA (PST)

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On Thu, 31 Jul 2008 09:24:30 -0700, Darren New wrote:

> Jim Henderson wrote:
>> Absence of evidence is not evidence of absence. ;-)
> 
> Sure it is. It's not proof of absence, of course, but it's stronger
> evidence than not.
> 
> http://www.overcomingbias.com/2007/08/absence-of-evid.html
> 
>> But now here, in the 21st century, we've finally exhausted the base of
>> knowledge?  I don't think that's the case.
> 
> The problem is that you're confusing math with physics. In math, yes, we
> have exhausted the base of knowledge of the halting problem. The halting
> problem goes like this:
> 
> Assume A.  Assume B.   Assume C.  Therefore D.
> 
> You're arguing "maybe assumption B is counterfactual."  That doesn't
> disprove the halting problem. It doesn't mean you can solve the halting
> problem. It merely means the halting problem doesn't apply to your
> current situation.
> 
> Indeed, one of the assumptions required for the halting problem to hold
> (namely, unbounded storage) is *known* to be impossible to realize in
> this universe (or at least normally assumed to be impossible even by
> people who understand the halting problem). That *still* doesn't mean
> the halting problem is "solved".
> 
> Math isn't physics.

Now my brain hurts. ;-)

Jim

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>> I am not aware - despite possessing a book detailing the entire history
>> of Fermat's Last Theorum - of any proof that was widely held to be
>> correct for a long time before being found wrong. All the incorrect
>> proofs were discovered to be incorrect fairly quickly.
> 
> Absence of evidence is not evidence of absence. ;-)

It *is* evidence. It isn't proof, but it is evidence.

Somebody having a proof that everybody thought was right but actually 
turned out to be wrong would make for pretty dramatic reading in a book 
which is basically a dramatisation of the history of FLT. Of course, the 
researchers could have missed something though...

>> Blurring doesn't actuallly "lose" nearly as much data as you'd think.
>> That's why it can be mostly reversed.
> 
> Again, 10 years ago, doing this was thought to be impossible.

Emphasis: Thought.

There is a difference between something being "thought" and something 
being mathematically "proven". A very significant difference.

Cryptographers "thought" that public key cryptography was impossible, 
and where astonished when somebody actually invented it. However, note 
that nobody ever *proved* that PKC was impossible, people just *assumed* 
it would be. Very Big Difference.

>> If I were you, I'd be far more worried about the sky falling - it's
>> about as logically plausible...
> 
> I don't see how your statement follows mine....

Your "logic" seems to be "absolutely anything is possible". That 
includes the sky falling down. Which is about as likely as the Halting 
Problem being incorrect - but, obviously, far more serious. I don't know 
about you, but *I* would rather be wrong about some theorum than have 
chunks of sky fall on my head!

> Throughout history, mankind has claimed to have reached the end of 
> knowledge on all manner of topics, saying "there's nothing more to learn 
> here".  In every instance (AFAIK), that's been proven wrong.
> 
> But now here, in the 21st century, we've finally exhausted the base of 
> knowledge?  I don't think that's the case.

In mathematics, learning generally consists of discovering new things, 
and rarely involves finding out that old things were wrong. (Although 
you have to carefully draw a line between *facts* which have been 
proven, and *ideas* about mathematics. The latter can and do change, the 
former are forever.)

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

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>> I implicitly assumed that any machine that can't process it's own 
>> program isn't worthy of the title "computer", that's all. ;-) But yes, 
>> I see what you're saying.
> 
> So, your brain is weaker than a Turing machine? Cool.

This is not exactly news. A Turing machine has unbounded memory, 
remember? ;-)

> It's the simplicity of the machines that make them amenable to 
> universality, not the complexity. Remember that the Von Neumann 
> architecture was a breakthrough.

Uh... wuh?

>> As for being able to perform infinite instructions in finite time... 
>> surely that just makes it even *harder* to predict what the machine 
>> will od, no?
> 
> Not if it can process its own input. Think about how the halting problem 
> works, and why... If the machine has unbounded state, it might run 
> forever without ever getting into the same state twice. But if you can 
> run the computer forever without it actually taking forever, then you 
> can say definitively "no, that machine never stops."
> 
> And a machine that can run an infinite number of instructions in finite 
> time *always* stops. ;-)

Really? And by "infinite" do you mean Aleph0, Aleph1, or some larger 
cardinallity? ;-)

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

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>> ...I can't *believe* you robbed me of the opportunity to say "what do 
>> you get if your multiply six by nine?" :-o
> 
> Sorry I was thinking of a different book, in this case a Jasper Fforde 
> dealing with Nextian mathematics that does allow you to work out whether 
> the answer 9 is derived from 3+6 or 3*3, or something else entirely.

That sounds like an "interesting" mathematical system. (And not an 
entirely implausible one, actually.)

By the way... 6 * 9 = 54? I'm confused now! o_O

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

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Orchid XP v8 wrote:
> By the way... 6 * 9 = 54? I'm confused now! o_O

Yes it is.

"...something is fundamentally wrong with the universe..."

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