On 14 Feb 2012, at 06:57, Stephen P. King wrote:
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?).



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
All possible OM-chains/histories do exist and one just happens to live in one of them.


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).



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