On 14 Feb 2012, at 06:57, Stephen P. King wrote:
acw:
Yet the problem is decidable in finite amount of steps, even if
that amount may be very large indeed. It would be unfeasible for
someone with bounded resources, but not a problem for any abstract
TM or a physical system (are they one and the same, at least
locally?).
Hi ACW,
WARNING WARNING WARNING DANGER DANGER! Overload is Eminent!
OK, please help me understand how we can speak of computations
for situations where I have just laid out how computations can't
exist.
In which theory? The concept of "existence" is theory dependent.
If we take CTT at face value, then it requires some form of
implementation. Some kind of machine must be run. Are you sure that
you are not substituting your ability to imagine the solution of a
computation as an intuitive proof that computations exist as purely
abstract entities, independent from all things physical? My
difficulty may just be a simple failure of imagination but how can
it make any sense to believe in something in whose very definition
is the requirement that it cannot be known or imagined?
If we assume this:
Ax ~(0 = s(x)) (For all number x the successor of x is different from
zero).
AxAy ~(x = y) -> ~(s(x) = s(y)) (different numbers have different
successors)
Ax x + 0 = x (0 adds nothing)
AxAy x + s(y) = s(x + y) ( meaning x + (y +1) = (x + y) +1)
Ax x *0 = 0
AxAy x*s(y) = x*y + x
Then we can define "computations" and we can prove them to exist.
It is not more difficult that to prove the existence of an even
number, or of a prime number. It is just much more longer, but
conceptually without any new difficulty.
Knowing and imagining are, at least, computations running in
our brain hardware. If your brained stopped, the knowing, imagining
and even dreaming that is "you" continues?
Not relatively to those sharing the reality where your brain stop. But
from your own point of view, it will continue.
So you do believe in disembodies spirits,
No. If your brain stop here and now, from your point of view, it
continue in the most normal near computational histories. In those
histories you will still feel as locally and relatively embodied.
Globally you are not, even in this local reality, given that there a
re no bodies at all. Bodies are appearances.
you are just not calling them that. I apologize, but this is a bit
hard to take. The inconsistency that runs rampant here is making me
a bit depressed.
You have to find the inconsistency.
Now, given all of that, in the concept of Platonia we have the
idea of
"ideal forms", be they "the Good", or some particular infinite
string of
numbers. How exactly are they determined to be the "best possible by
some standard". Whatever the standard, all that matters is that
there
are multiple possible options of The Forms with the stipulation
that it
is "the best" or "most consistent" or whatever. It is still an
optimization problem with N variables that are required to be
compared
to each other according to some standard. Therefore, in most cases
there
is an Np-complete problem to be solved. How can it be computed if
it has
to exist as perfect "from the beginning"?
The problem is that you're considering a "from the beginning" at
all, as in, you're imagining math as existing in time. Instead of
thinking it along the lines of specific Forms, try thinking of a
limited version along the lines of: "is this problem decidable in a
finite amount of steps, no matter how large, as in: if a true
solution exists, it's there."
And what exactly partitions it away from all the other "true
solutions"? This idea seems to only work if there is "One True
Theory of Mathematics"....
Not at all. Comp needs only one true conception of arithmetic. The
evidence is that it exists, even if we cannot define it in arithmetic.
We need the intuition to understand the difference between finite and
non finite.
But we know that that is not the case, there are many different
Arithmetics. How exactly do you know that yours is the "true
standard" one?
It does not matter as long as we reason in first order logic, or if we
are enough cautious with higher logic. The consequence are the same in
all models, standard or non standard. IF PA proves S, S is true in all
models of arithmetic, and we don't need more than that.
I'm not entirely sure if we can include uncomputable values there,
such as if a specific program halts or not, but I'm leaning towards
that it might be possible.
OK, there is no beginning. Recursively enumerable functions
exist eternally. OK. Why not Little Ponies? My daughters tells me
all about How Princess Celestia rules the sky... This entire theory
reminds me of the elaborated Pascal's Gamble... How do we know that
our "god" is the true god? OK. So we Bet on Bp&p. OK... Then what?
How do I know what "Bp&p" means?
Because for all arithmetical p, Bp & p is an arithmetical formula. You
need just to understand predicate logic, and the standard
interpretation of elementary arithmetic. You refer to papers which
assumes that understanding, and much more.
I figured this out when I was trying to wrap my head around
Leindniz'
idea of a "Pre-Established Harmony". It was supposed to have been
created by God to synchronize all of the Monads with each other so
that
they appeared to interact with each other without actually "having
to
exchange substances" - which was forbidden to happen as Monads
"have no
windows". For God to have created such a PEH, it would have to
solve an
NP-Complete problem on the configuration space of all possible
worlds.
Try all possible solutions for a problem, ignore invalid ones.
And how exactly do we distinguish valid from invalid ones? By
what process do we "try all possible solutions"? Process and
timelessness do not mix.
Validity, unlike soundness is Turing verifiable. Even unconscious
zombie or robot can make the checking for us. That is why we make
proof, to let other people checking them.
If the number of possible worlds is infinite then the computation
will
require infinite computational resources. Given that God has to
have the
solution "before" the Universe is created,
"before", what is this "before", it makes no sense to talk of time
when dealing with timeless structures. A structure either exists in
virtue of its consistence/soundness, or it doesn't (it can exist as
something considered by someone within some other structure, which
does happen to be consistent, thus it only exists as a(n incorrect)
thought). Introducing a ``God'' agent to actually do creation or
destruction will only lead to confusion, because creation or
destruction implies time or causality. Platonia only implies local
consistency. On the other hand, I'm not even asking for any full
Platonia, just recursively enumerable sets should be enough...
How does the existence on an entity determine its properties?
Please answer this question.
Please define existence. Normally properties depends on the initial
assumption making possible to something to exist in the frame of some
theory.
What do "soundness" and "consistency" even mean when there does not
exist an unassailable way of defining what they are? Look carefully
at what is required for a proof, don't ignore the need to be able to
communicate the proof.
This is the subject of any elementary textbook in logic.
It cannot use the time
component of "God's Ultimate Digital computer". Since there is no
space
full of distinguishable stuff, there isn't any memory resources
either
for the computation. So guess what? The PEH cannot be computed and
thus
the universe cannot be created with a PEH as Leibniz proposed.
You can encode computation in arithmetic or other timeless systems
just as well.
Encode computation? How does it even makes sense to juxtapose a
process that requires time in a situation that is timeless? We
cannot have our computational cake and eat it (eliminate its
temporality) too.
So you assume "time", and abandon your neutral monism.
Time is merely a relation between states, it's always possible to
express such a relation timelessly.
Is it? So there is no such thing as change? Then why is the
illusion of it so damned persistent?
By the nature of the arithmetical indexicals. Both physical time and
experiential duration are recovered (and are quite different).
Surely we can start in a environment that is transitory and change
laden and think up systems of ideas that seem to be timeless, but do
they really have these properties? Did you see my argument about how
invariants require a set and transformations on the set to be
defined. Take away the transformations and what you have? A set with
nothing like an invariance to be seen anywhere. We can eliminate
measure of Change, but we cannot eliminate Change itself.
Comp votes for the contrary. We can eliminate the metaphysical
"Change" and explain the appearance of the measure and feeling of
change.
To be fair, I'm not sure how even a single computation can be
performed without there existing a consistent definition of that
computation (thus the existence of that sentence in Platonia).
Does my no-brand name Desktop that I built myself, with its hard
drives, mother board, power supply, etc, depend on its running this
crappy Windows 7 OS only if someone has defined what a computations
is?
The definition and the object are arithmetical.
Of course not! Again, how exactly does the existence of an entity,
be it a computation, Pony, UD or whatever, determine the particular
set of properties that make that computation a computation, the Pony
and Pony and the UD a UD? This is like the Randians chanting "A is
A" over and over never minding questions like what the <expletive
deleted> is A?
Not at all. Well, I don't know for the pony, because you don't give a
theory.
You cannot even compute 1+1 without it, much less NP-complete
problems. I don't see how a in-time universe solves the problems
you ascribe to Platonia - same problems are present in both and
they can only be avoided by giving some consistent system
existence. As for space? why would space be needed for deciding if
some recursive relations hold. Space is itself an abstraction and I
don't see how introducing it would solve anything about such
abstract recursive relations, except maybe making it simpler to
reason for those that like to imagine physical machines instead of
purely abstract ones.
Without space or time the operation of "copying" is impossible.
Prove this.
or study computer science, where you can see that programs copying
themselves exists in arithmetic, and their running exists in arithmetic.
Try it some time. How can you over come the identity of
indiscernibles without space or time?
The idea of a measure that Bruno talks about is just another way of
talking about this same kind of optimization problem without
tipping his
hand that it implicitly requires a computation to be performed to
"find"
it. I do not blame him as this problem has been glossed over for
hundred
of years in math and thus we have to play with nonsense like the
Axiom
of Choice (or Zorn's Lemma) to "prove" that a solution exists,
never-mind trying to actually find the solution. This so called
'proof"
come at a very steep price, it allows for all kinds of paradox
<http://en.wikipedia.org/wiki/Banach-Tarski_paradox>.
All possible OM-chains/histories do exist and one just happens to
live in one of them.
Sure.
A measure is useful for predicting how likely some next OM would
be, but that doesn't mean that our inability of listing all
possible OMs and deciding their probabilities means that no next OM
will exist - we all inductively expect that it will.
No! You are neglecting the fact that there are not just a set of
connected OMs floating out in Platonic NowhereLand. You have to
consider all possible OMs and show how the continuation works.
Sure. That's part of the problem. But what about the initial solution
of it? Comp just makes it clear, and show tha solution of the machine
in the context of the usual classical theory of knowledge, which up to
know works very well.
You at least have to have something like a fixed point and that
requires a space with closure and compactness and *a transformation*
on that space. You cannot define continuation without meeting this
requirement.
Arithmetic provides all that.
Unfortunately the measure itself is likely to be uncomputable,
unlike finding some next OM (actually, I'm not so sure about this
being entirely computable as well, it might prove that it's only
computable in specific cases, just never in general; within COMP,
finding a next OM means finding a machine which implements the
inner machine('you'), that should always exist as UMs exist,
however what if the inner machine crashes? a slightly modified
inner machine might not, yet that machine would still identify
mostly as 'you' - whatever this measure thing is, it's way too
subjective and self-referential, yet this is complicated because
the inner machine doesn't typically know their own godel number,
nor can they always trust their inputs to be exactly what they
'expect').
"Know its own Godel number"? How exactly does that happen?
Please remember, all of this is *occurring*, (dammit, we cannot even
use that word consistently) in the Timelessness of Platonia. There
is no Occurring at all there. There isn't even any "there" there!
There is, once you understand that "here and there", "now and
sometimes", are indexical relative concept.
A possible solution to this problem, proposed by many even back as
far
as Heraclitus, is to avoid the requirement of a solution at the
beginning. Just let the universe compute its least action
configuration
as it evolves in time, but to accept this possibility we have to
overturn many preciously held, but wrong, ideas and replace them
with
better ideas.
In a way, you could avoid thinking of Platonia and just consider
the case of a machine's 1p always finding its next OM. As long as
finding one next OM doesn't take infinite steps, you could consider
it alive. What if no next step OM exists for it, but it exists in a
version where a single bit was changed?
Here is the problem, given the measure of a single number in an
infinite set is zero, The possibility of defining a continuance this
way is also zero.
OK. This shows that you have to define continuance in another way.
(The machine already know this).
Bruno
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
