the breaking record player
Crispin Semmens
A self replicating record player that makes a copy of itself, hides
behind a wall, then plays the record on the copy. The record player
never breaks because it always keeps a copy of itself... it's as if it
is simultaneously a low-fi and a hi-fi record-player...
I don't think this scenario was really referred to in the book, or was
I misreading? Does anyone know what this solution corresponds to in
terms of incompleteness? Or why this solution isn't valid (if it
isn't)?
Don Stockbauer
Hello, Crispin. Welcome.
IMHO, I think that if you could really build such a system its
capability would be far beyond that of playing the self-destructing
record, namely the self-reproducing feature would dwarf that, so yes
you would solve the original problem but you would also create a
capability far beyond the original challenge. All theoretical, of
course. I don't think robotics is anywhere near doing what you
describe. Especially to keep it up for generation after generation
autonomously. Maybe one generation, if that. Although Nature does it
easily enough with living systems.
Did you read the part later in the book where he has a more
sophisticated machine scan the record first and analyze whether it
would destroy the player if played (I think it was called "Record
Player Omega". Maybe the lesson is if you jump through enough hoops
you can do about anything, theoretically.
Don Stockbauer
Some more thoughts. The record player is analogous to the Principia
Mathematica (PM). The record which breaks the record player is
analogous to the Godel statement "I cannot be proved in the PM
system." Now, if you cloned a record player and let it be the one
which is destroyed, preserving the original, what you must do is see
how the analogy works. And that would be to copy the PM and derive
Godel's result in it. Bu in mathematics it also affects the original
and all copies. So making a copy doesn't help.
I keep wondering just how important the Incompleteness Theorem is.
That is, is it just a frilly contrived result that in the real world
doesn't matter? Does it limit our technology at all? Suppose I have
a car that works fine, and then someone makes a small scratch in the
paint, a defect. I can still use it for its original purpose. I do
seem to gather that it does have use within mathematics itself.
Don Stockbauer
If the physical world were like mathematics, you could break one
record player in the Universe and they'd all break.
Crispin Semmens
Thanks for your reply, some interesting points that I've thought about
for the last day or two!
I think the solution is that the analogy breaks down in this case...
because I think that the record player breaking is something that can
only be witnessed from outside the system, not inside it. So the idea
of a record player than can spawn itself and then witness the results
is akin to the idea of a theoretical system that can simulate itself,
which is a bit circular.
>> I keep wondering just how important the Incompleteness Theorem is.
>> That is, is it just a frilly contrived result that in the real world
>> doesn't matter? Does it limit our technology at all? Suppose I have
>> a car that works fine, and then someone makes a small scratch in the
>> paint, a defect. I can still use it for its original purpose. I do
>> seem to gather that it does have use within mathematics itself.
For the most part the incompleteness theorem doesn't have much effect
I think, but it does place limits on what can be achieved, which is
often very useful! If you can apply the incompleteness theorem to
whatever you're doing, then you can immediately avoid a lot of
pointless trouble, and point yourself in a better direction.
I think the point is that within the world of theoretical systems,
there are limits to what can be done, but that shows that our minds
and our world aren't theoretical systems, at least, not in the sense
that we create them. So, whenever we make theoretical systems that can
interface with the real world (ie, technology), then we have the
potential to make theoretical systems that can expand themselves,
because they can get results that they can't derive from themselves
(the real world supplies them). This allows a system to see if it
breaks under given circumstances, and enables the scenario I was
wondering about before.
>
> If the physical world were like mathematics, you could break one
> record player in the Universe and they'd all break.
Which sort of implies that the universe isn't just one theoretical
system, it's many. If there is an overarching one, we surely haven't
found it yet :)
Don Stockbauer
> Hi Don,
>
> Thanks for your reply, some interesting points that I've thought about
> for the last day or two!
>
> I think the solution is that the analogy breaks down in this case...
> because I think that the record player breaking is something that can
> only be witnessed from outside the system, not inside it. So the idea
> of a record player than can spawn itself and then witness the results
> is akin to the idea of a theoretical system that can simulate itself,
> which is a bit circular.
Another idea is that the Incompleteness theorems are really very
minor. If you draw an analogy with a record player, it might be that
any effect similar to the Godel sentence might be so minor (maybe just
causing the player to vibrate in a very minor, non-breaking way).
Thus Hofstadter thought the analogy would be to break it, but perhaps
it never comes close to that. I think a true feat of engineering
would be for DRH to actually specify the design of the phonograph
breaking record for any given phonograph, if any record at all could
ever break it. And yes, I know the original presentation was 100%
theoretical.
Systems can simulate themselves, but not perfectly. If the entire
Earth is considered a system, then subsystems within it can simulate
the whole, though not perfectly, of course.
>
> >> I keep wondering just how important the Incompleteness Theorem is.
> >> That is, is it just a frilly contrived result that in the real world
> >> doesn't matter? Does it limit our technology at all? Suppose I have
> >> a car that works fine, and then someone makes a small scratch in the
> >> paint, a defect. I can still use it for its original purpose. I do
> >> seem to gather that it does have use within mathematics itself.
>
> For the most part the incompleteness theorem doesn't have much effect
> I think, but it does place limits on what can be achieved, which is
> often very useful! If you can apply the incompleteness theorem to
> whatever you're doing, then you can immediately avoid a lot of
> pointless trouble, and point yourself in a better direction.
>
I've tried to visualize how the incompleteness theorems could limit a
real world physical system, to no avail. People have come up with
analogies of them which seem to limit what we could do, but the
theorem as Godel derievd it only affects mathematical systems.
> I think the point is that within the world of theoretical systems,
> there are limits to what can be done, but that shows that our minds
> and our world aren't theoretical systems, at least, not in the sense
> that we create them. So, whenever we make theoretical systems that can
> interface with the real world (ie, technology), then we have the
> potential to make theoretical systems that can expand themselves,
> because they can get results that they can't derive from themselves
> (the real world supplies them). This allows a system to see if it
> breaks under given circumstances, and enables the scenario I was
> wondering about before.
>
I believe what you might be detecting is the metasystem transition,
where systems group into a transcendent whole, and this metasystem has
much more capability than the separate systems, basically a paradigm
shift.
>
> > If the physical world were like mathematics, you could break one
> > record player in the Universe and they'd all break.
>
> Which sort of implies that the universe isn't just one theoretical
> system, it's many. If there is an overarching one, we surely haven't
> found it yet :)
What I meant was Godel's Incompleteness Theorem(s) would be the same
whether they were derived on Earth or Alpha Centauri or the Andromeda
galaxy. They "break" the Principia Mathematica. Break it at one
location and it's broken for all for you could radio the theorems
around the Galaxy. Breaking a physical record player does not break
others around the galaxy. Although now that I think of it if you
really did have a record player breaking record its contents could be
radioed around and used to break the others.