On Tue, 29 Jul 2008 09:23:54 +0100, Invisible wrote:

>> True.  But unlikely != impossible.
> 
> You realise that it's completely possible that at one point today you'll
> try to swallow a mouthful of water and accidentally inhale it, right?
> People actually *die* like this. But I don't see you worrying about it -
> because it's rather unlikely. And yet it is many, many times *more*
> likely than somebody solving the Halting Problem.

Have you calculated the odds of both? ;-)

We're not talking about comparative probabilities.  Impossible means *no* 
chance of it ever happening.

>> Maybe.  But then again, throughout history there are examples of really
>> fundamental things needing to be adjusted.  Knowledge continues to
>> grow.
> 
> Scientific facts have been found to be incorrect. There are far fewer
> examples of mathematical truths which have needed to be adjusted. And
> there are vanishingly few examples of widely accepted *proofs* that turn
> out to be wrong - it tends to be things lots of mathematicians "think"
> are true that eventually turn out to be disproven.

Exactly my point, but with a narrower focus.  Things lots of *people* 
"think" are true sometimes/frequently/often turn out to be disproven.

>> This is the problem, though:  The assumption is that computing will
>> always use a Turing model, like I said.
> 
> And like I have explained multiple times, the problem isn't the Turing
> model. Even if we assume some as-yet unknown technology with miraculous
> [but not unlimited] powers, we still have an unsolvable problem. That's
> what is so significant about the Halting Problem. (I mean, face it, what
> *practical* use does HP have? None. It's importance is that it
> demonstrates that some problems are unsolvable.)

<sigh>  We're going in circles here....

>> Maybe.  It's hard to say that that would be the case, because the
>> future is unknowable.
> 
> You're missing my point: It doesn't *matter* that we can't know the
> future. Simple logical deduction demonstrates that ANY machine we can
> construct will have the same problem, REGARDLESS of how it works.

It's simple logical deduction that unless I have a screwdriver, I can't 
drive a screw.

Until you realise that the screw has a hex head and an allen wrench will 
do the job just as nicely.

*Sometimes* all you need is a new tool.  Sometimes the new tool hasn't 
been invented yet.

>> Maybe the halting problem was a bad example to illustrate my point; the
>> bottom line on my point still stands, though.  What's "impossible"
>> yesterday became an everyday occurrence.  It was "impossible" that the
>> solar system was heliocentric a thousand years ago.  Today, it's common
>> knowledge that that is untrue.
> 
> Once again, you are talking about science.
> 
> In science, it is always possible that some "fact" could turn out to be
> partially or wholely wrong. We think we know how the universe works, but
> we could always be wrong about something.
> 
> Mathematics is different. Not quite as different as was originally
> believed, but still. In mathematics, we can construct absolute truths
> which will never be disproven until the end of the universe itself. The
> only question mark is the reliability of the human mind.

I think it's a mistake to say "we know all there is to ever know about 
'x'".  There have been many points in history where humankind has made 
such declarations about many things - including mathematics - and it has 
turned out that we'd only scratched the surface.  It's the height of 
hubris to assume we can't learn anything new.

> For sufficiently complicated proofs, it becomes not merely possible but
> *plausible* that some mistake could exist. For sufficiently simple
> proofs, we can be absolutely certain that only a fundamental flaw with
> renders all of mathematics invalid could disprove the theorum.
> 
> In summary: Science can never have absolute truths. Mathematics can.

Sometimes the devil is in the details (and how detailed your data is).

>> Using today's technologies, maybe.  Again, we don't know what the
>> future holds.  It is impossible to use paper in transistors.  Or is it?
> 
> Making transistors out of paper is a question of physics - a branch of
> science. Infinite compression ratios is a question of mathematics.
> Therein lies the critical difference.

And yet you agreed with another post in this thread that said that 
something was possible.  Look at the refocusing capabilities of some of 
the tools for that to reconstruct detail in blurred images.  Blurring is 
lossy compression, yet being able to recover that data isn't impossible; 
that's been proven.

>>> This is precisely my point: Solving the halting problem DOES defy the
>>> laws of causality. It is NOT just a problem of technology. It is a
>>> problem of "if this algorithm were to exist, it would cause a logical
>>> paradox, regardless of the technology used".
>> 
>> There again, maybe I chose a poor example to illustrate my point.
> 
> I'm not sure what your point is.

That I chose a poor example.

> If your point is that science is sometimes wrong, or at least needs to
> be amended, then I agree. If your point is that widely held beliefs are
> sometimes wrong, then I also agree. If your point is that every proven
> mathematical result could actually be wrong, then I completely disagree.

I believe that's just a limitation of our understanding of things as they 
are now.

Jim

Post a reply to this message