On 2/7/2012 06:15, Stephen P. King wrote:
On 2/6/2012 6:50 PM, acw wrote:
On 2/6/2012 06:25, Stephen P. King wrote:
Hi ACW,

On 2/4/2012 1:53 PM, acw wrote:
snip
Before reading the UDA, I used to think that something like Tegmark's
solution would be general enough and sufficient, but now I think 'just
arithmetic' (or combinators, or lambda calculus, or ...) or is
sufficient. Why? By the Church-Turing Thesis, these systems posses the
same computability power, that is, they all can run the UD.

I agree with this line of reasoning, but I see no upper bound on
mathematics since I take Cantor's results as "real". There is not upper
bound on the cardinality of Mathematics. I see this as an implication of
the old dictum "Nature explores all possibilities."

The question is if transfinite extensions are considered as part of
the foundation, what different consequences will follow for COMP or
the new theory?
[SPK]
I am not sure, but they seem to be necessary for completeness.

Maybe, although it's also questionable if it makes that much sense to put it in the ontology if it won't have any discernible effect on the experienced sense data or measure.
Now, if we do admit a digital substitution, all that we can experience
is already contained within the UD, including the worlds where we find
a physical world with us having a physical body/brain (which exist
computationally, but let us not forget that random oracle that comes
with 1p indeterminacy).

Not quite, admitting digital substitution does not necessarily admit to
pre-specifiability as is assumed in the definition of the algorithms of
Universal Turing machines, <http://en.wikipedia.org/wiki/Algorithm> it
just assumes that we can substitute functionally equivalent components.
What do you mean by ``pre-specifiability''? Care to elaborate?

The algorithm is a finite and specifiable list of computational steps or
states. It only makes sense that the algorithm exists at least
simultaneous or prior to its implementation by a physical system. It
cannot come into existence after the implementation.

It comes into the existence after the implementation? While I can see how some UD runs a copy of itself as well, I'm not entirely I see the problem here with what I said. Unless, your issue is along the lines of 1p experience actually being some truth being temporally "created" or merely what happens when some particular computations happens within some timeframe, as opposed to existing platonically - I'm not sure I can completely agree with this opinion although I've shared it a long time ago, currently I prefer to think consciousness could work in situations like this: consider a SIM(substrate independent mind), consider computing parts of its mind in temporally disconnected or random order (include some VR(Virtual Reality) environment with it, so it's mostly self-contained, although it could get some input/sense-data from the world doing the computation), possibly also implement spatial or algorithmic disconnects, possibly even add some homomorphic encryption such that no outside observer could understand the computations that are actually happening (yet all the computations are happening) - if COMP is correct, that SIM should be conscious, and this consciousness won't be spatially or temporally connected, yet the SIM will experience continuity! In a way, consciousness is like that inner interpreter. In a more extreme form, you could consider someone running some computation of a self-contained OS+VR+SIM(s) machine and stopping computing that machine, and it should still have continuations, be it in the UD or anywhere those future computations may be found (be they physically or platonically), and possibly externally acausal (if considering physics or MGA-like thought experiments), but internally causal and continuous (from 1p(s)). Maybe my thought experiment is a bit extreme, although I can't see any obvious refutation of it within the context of COMP(well, some simulations may be very low measure or unstable, compared to those which allow for more easier/cheaper locally stable 1p indeterminacy, but this is a fixable problem by adding access to undefined functionality/random oracles).
Functional equivalence does not free us from the prison of the flesh, it
merely frees us from the prison of just one particular body. ;-)
I'm not so sure to term ``body'' is as meaningful if we consider the
extremes which seem possible in COMP.

My point about the "flesh" is that functional equivalence allows for
computational universality but does not eliminate the necessity of the
physical. My primary contention is that computation is a process that
requires resources and is not just sum platonic free lunch.

What is the limit on those resources? What if the machine is always finite, but unbounded in the limit (although the limit is never reached for any observer)? If the physical always has some specific finite upper bound, how do you justify such a stronger claim? (If it's not any specific limit, it can be bypassed through some "mathematically inductive jump", but this doesn't seem necessary as you already mentioned an eternally running UD).
After a digital substitution, a body could very well be some software
running somewhere, on any kind of substrate, with an arbitrary
time-frame/ordering (as long as 1p coherent), it could even run
directly on some abstract machine which is not part of our universe
(such as some machine emulating another machine which is contained in
the UD) - the only thing that the mind would have in common is that
some program is being instantiated somewhere, somehow. In this more
extreme form, I'm not sure I can see any difference between a
substrate that has the label 'physical' and some UD running in
abstract Platonia.


OK, but you seem to be taking the internal view of a single observer
only in this analysis and following a reasoning similar to the
brain-in-a-vat situation. How do you account for other observers that
have similar bodies and the possibility that they are not mindless husks
or zombies - which is the situation of the solipsist? The difficulty
that I am seeing only manifests when we consider multiple observers, we
have to account for separate entities each with their own 1p. The world
we observe has multiple observers, each of them with a 1p. We have to
associate an infinite number of computations with each one, if I follow
Bruno's idea correctly, and yet each one has some kind of sense of
continuity in time. Basically, we have to reproduce the general
covariance of physical laws as we see codified in general relativity
theory.

There's many ways to handle this, and I'll try to describe one possible one: let's say we have a structure, not really computable in the limit, although always computable in quantized ways (such as some form of rational arithmetic, or with computable reals), it's whatever would ideally represent the physical laws if we had perfect data and could perform perfect induction on that data. You could imagine infinities of ensembles of machines which compute quantized partials of such a world, with each such ensemble belonging to some observer. In a way, locally it would seem solipsistic, but globally, the other minds are always present, in all their possible forms. Same would apply to a more local digital physics (if that makes some sense), or even in cases where no such "perfect structure" is to exist. We tend to intuit that other people in the inferred 3p structure are conscious based on their behavior and their apparent physical contents, thus this is also a bet that an ensemble of machines which relate to their consciousness also exists. I really don't see how COMP is solipsist here, especially if you consider a Platonia which should hold the infinities of machines and the arithmetical truth to make all the seemingly non-zombies actual non-zombies.

Of course, extracting our local law from just the UD seems like a hard challenge, and I'm not so sure how easy it would be given our limited computational resources - we have to depend on clever reasoning greatly informed by observed data to get better theories instead.
If you can show why the 'physical' version would be required or how
can someone even tell the difference between someone living in a
'physical' world vs someone living in a purely mathematical (Platonic)
world which sees the world from within said structure in Platonia and
calls it 'physical'.

Why bother having a physical world at all? Why is the illusion of matter
even here in front of us? Where does the illusion of time obtain from?
We cannot hand wave even silly versions of these questions away.

Illusion of matter is good for self-consciousness. It leads to embodied intelligence and all the interesting stuff that follows from it. It's also likely that simple physical universes have greater measure due to being simpler computationally. Illusion of time - I used to like ASSA, but after thinking harder on COMP, I very much prefer RSSA now. I suspect that time is needed for self-consciousness/awareness (and it's also what gives you the RSSA), can you imagine forming memories or updating beliefs without some form of time? Or even doing computations? This is not to say that time itself is a primitive, it's merely how we relate states to each other in a computation, and such a relationship can be encoded in a non-temporal manner, however due to how our cognitive system is constructed, we can only move forward in time (most embodied general intelligences capable of having short and longterm memories should find themselves in similar situations). Either way, the illusion of continuity is important for a self-conscious embodied general intelligence, up to the point where we might as well consider it real and use it to bet on what continuations we will have and what consequences our actions will have and what futures we would prefer and so on.
It seems that 'physical' is very much what we call the structure in
which we exist, but that's indexical, and if you claim that only one
such structure exists (such as this universe), then you think COMP is
false (that is, no digital substitution exists) or that arithmetic is
inconsistent (which we cannot really know, but we can hope)?

I suspect that there are an infinite number of physical worlds to cover
the need for symmetry between the abstract and the concrete. A postulate
of my hypothesis is "that for every physical object there is at least
one representation and for every representation there is at least one
physical implementation of it." So for example, there is a class of
physical instantiations of all numbers, even including patterns of
pixels like this: 13. The previous pattern of pixels is a physical
implementation of the number thirteen (as is this previous one!).

What are those pixels made of? ... What are those atoms or quantum particles or strings or ... made of? Eventually it seems to get down to math (if we're at least partially realist and reductionist about the existence of physical law). If there is a mathematical object that perfectly describes the territory of which I'm part of, I don't see any reason why I should distinguish between the abstract and physical version (or that I should choose to call it 'physical' merely on the fact that I happen to be part of it). Maybe I'm blind to the difference, but I just don't see it when looking at it in the limit/extremes.
If there's any difference between a physical and non-physical
implementation in the context of COMP, I'd like to know what it is and
what effect it has.

A non-physical implementation is what Bruno writes about when he used
the word "implementation".

This idea goes back to my claim that the "Pre-established harmony
<http://en.wikipedia.org/wiki/Pre-established_harmony>" idea of Leibniz
is false because it requires the computation of an infinite NP-Complete
problem to occur in zero steps. As we know, given even infinite
resources a UTM must take at least one computational step to solve such
a NP-Complete problem. My solution to this dilemma is to have an
eternally running process at some primitive level. Bruno seems to
identify this with the UD, but I claim that he goes too far and
eliminates the "becoming" nature of the process.

I think the idea of Platonia is closer to the fact that if a sentence
has a truth-value, it will have that truth value, regardless if you
know it or not.

Sure, but it is not just you to whom a given sentence may have the same
exact truth value. This is like Einstein arguing with Bohr with the
quip: "The moon is still there when I do not see it." My reply to
Einstein would be: Sir, you are not the only observer of the moon! We
have to look at the situation from the point of view of many observers
or, in this case, truth detectors, that can interact and communicate
consistently with each other. We cannot think is just solipsistic terms.

Sure, but what if nobody is looking at the moon? Or instead of moon, pick something even less likely to be observed. To put it differently, Riemann hypothesis or Goldbach's conjecture truth-value should not depend on the observers thinking of it - they may eventually discover it, and such a discovery would depend on many computational consequences, of which the observers may not be aware of yet, but doesn't mean that those consequences don't exist - when the computation is locally performed, it will always give the same result which could be said to exist timelessly.
In essence, Platonia might very well contain Chaitin's constant of
some machine, even if we cannot know it (although we can make guesses
at it by making stronger and stronger theories).

Certainly, what can Platonia not contain? My problem is why is Platonia
even a necessary concept? It smells of the mystical and irrational. We
should justify its necessary existence before we wander off and ascribe
all these nice properties to it that just so happen to solve all of our
hard problems.

I'm not really requiring a full Platonia here, just consider it for arithmetic, where the law of the excluded middle(LEM) should still apply. It should be sufficient ontologically. It also seems much simpler conceptually than positing a physical world with all the extra magical quirks full-fledged physical worlds require. Another way to think of it would be in the terms of the Church Turing Thesis, where you expect that a computation (in the Turing sense) to have result and that result is independent of all your implementations, such a result not being changeable in any way or by anything - that's usually what I imagine by Platonia. It is a bit mystical, but I find it less mystical than requiring a magical physical substrate (even more after MGA) - to me the platonic implementation seems to be the simplest possible explanation. If you think it's a bad explanation that introduces some magic, I'll respond that the primitively physical version introduces even more magic. Making truth changeable or temporal seems to me to be a much stronger, much more "magical" than what I'm considering: that arithmetical sentences do have a truth value, regardless if we know it or not.
Your objections seem intuitionist/constructivist at its core, that is,
that something does not have a truth value if we can't prove it.

I am inviting you to consider exactly how it is that we work out a proof
of a theorem or logical sentence. Can you prove it without thinking
about it or witting it down somehow? I don't follow the intuitionist
line per se, I just consider those theories among the possible theories
that we can have of the world. They are much like different points of
view, some more limited than others.

Some sentences may require infinite proofs ("this machine will never
halt"), thus we cannot say that they are true, even if they are (such
as the absence of proof of a contradiction in arithmetic). In another
way, this seems like a problem with the provably unprovable (or a form
of "religion"), although COMP is itself a bet of this sort (existence
of a 1p continuation).

The "bet" idea is very clever. It is the most brilliant aspect of
Bruno's result and I do admire his genius for seeing it. :-)

Yes, I like it quite a bit. It lets one be clear about one's assumptions and which assumptions we select for basing our behavior on.
Yet, we all make the bet that we will be subjectively conscious in our
probable future, the bet that there will be a future observer moment,
that the sun will still exist and so on. It also seems to me that
given the time/space/structure indeterminacy that is shown in the UDA,
the bet on a continuation is justified (if one admits a digital
subst.), and almost magical.

"Almost" is a interesting word. :-)


As for your solution, again, I'm not entirely sure there would be any
difference in the experienced reality (your objection seems to be the
transition from UDA step 7 to UDA step 8/MGA?), although you now need
a more complex theory to get the initial substrate (which we cannot
even know anything about). Such an idea seems to lose the elegant
solution to the "why something instead of nothing" question, which was
solved rather nicely by assuming a Platonia (that some mathematical
sentences have truth values, such as arithmetical ones). Such an
approach also makes consciousness more mysterious again, and by MGA
(or UDA step 8), we do know of the conflict between mechanism and
materialism. All in all, it seems to make the theory more complex and
at great cost, with many added problems, and the only benefit of
making it more friendly to the intuitionist/constructivist.

What I am proposing is not much different from the idea of an initial
singularity posited by the Big Bang theory. We cannot ever observe it as
it would be forever hidden behind an event horizon. We arrive at the
idea that it must exist by working backwards from what we observe here
and now. What I am proposing about a neutral substrate is just what we
obtain when we remove all differences in the sense of properties from
it. Think about this: what properties does something have if that
something can never be observed and yet must exist?

That's a hard question. Talk about the nature of such a something's properties invites talk about what exactly we mean by property in general, and then we need a neutral way of talking about properties. I'm afraid that it's easy to either be too specific that it won't be neutral enough or too general that it makes no difference... I wonder though, why suppose a temporally evolving something? The notion of time is in itself quite complex, and this is why I tend to prefer to just use the CTT in a platonic manner (unchangeability of a result of finite process being applied on finite recursive things).

If we are machines, then we can only experience finite amount of
information given some finite interval of time, some of this
information may be incompressible, due to 1p indeterminacy, thus we
could experience "reals" in the limit, despite there only being finite
computations at any given time. This essentially means that any
mathematical object which can be described in Tegmark's "Ultimate
Ensemble" and that can contain us, is already part of the 1p
experiences of those existing within the UD and we can look at 1p
experiences, as well as the UD* trace as being part of the greater
"arithmetical" truth (or any other theory with equivalent
computational power, by the Church-Turing Thesis).

Umm, we have to show that the finiteness of machines is necessary from
first principles, we cannot just assume that it is so.
Are you using a more general definition of machine? A machine always
has a finite body (an integer), even though the grown itself may be
unbounded, but the growth at each step is finite, and given finite
time-steps, there is no way for the machine to become infinite (only
in the "limit").

Yes, I am not limiting myself to the finite machine (as you sketched
them here) as physics requires the use of Real and Complex numbers.

Oh. We should be careful about the meanings of the words we use, I almost always meant Turing-equivalent machine (or weaker) when using the term. A machine directly working on reals is already capable of way too much infinitely detailed uncomputable magic (here's a cute short SF story about this: http://qntm.org/responsibility ).


I agree that the
"arithmetical truth" of the UD may be enough to "force" the 1p to have
content, but we still need to account for the appearance of interactions
or histories of interactions (ala Julian Barbour'sTime Capsule
<http://en.wikipedia.org/wiki/Julian_Barbour> idea). There reaches a
point, even if it is in the limit of infinitely many, that we cannot put
off the concurrency problem, we have to deal with interactions. An
option is to take the "running of the UD" as a primitive kind of dynamic
that at our local 1p emerges as time and notions of forces, fields, etc.
emerge from the algebras of interactions between the many distinct 1p.

So your beef is with the appearance of continuity in our 1p experience
and our inferred 3p world?

Not just with that appeare3nce of continuity. I am trying to be faithful
to what I know of physics and working backwards toward Bruno's idea.
Bruno claims that physics emerges from numbers, OK. Let us see how. How
to we get general covariance and wave functions from COMP? At what point
do the conservations laws emerge? Some of them require continuity!

As long as they don't require true uncomputability, it should be possible (for example, if you consider computable reals). Still, any such endeavor is likely to be quite a lot of work, and it's not obvious to me that only our particular physics should be the most numerous out of possible physics. I would consider the classic inductive approach to be more likely to yield direct results, although COMP does have consequences, so it should be possible to use it as a filter for local theories.
The local 3p world may indeed to considered like a Block Universe (or
similar extensions to MWI), although by COMP, that's just a valid
model that we could be using, and a matter of epistemology. This is
indeed a tricky problem, which I'm not sure I'm satisfied with the
tentative answer I'm currently thinking for it. From the 1p, we can
only be certain of the existence of the observer moment, this can lead
someone to consider the ASSA (disconnected OM(observer moments)).

Yes, I agree with that reasoning. I just go further and demand that our
explanations cover multiple observers and the appearance that they are
interacting with each other. Do you understand the Dining Philosophers
Problem <http://en.wikipedia.org/wiki/Dining_philosophers_problem>? This
is a well known problem in computer science!

I'm not entirely sure I see the problem. Are you assuming that there is only one universal consciousness and it has only a single history (with a lot of forgetting involved)? The problem would appear if you identify the 1p time with the 3p time, which I don't do myself - 1p time is about the internal time of a computational structure locally contained in the brain, or about local computational steps. Many 1p's could correspond to the same or very similar 3p's, thus the sharing of the world, but some 1p's having in common some generalized brain does not mean that their experiences are literally occurring at the same time (although it's fine and correct to assume that the 3p persons we see are non-zombies, which is true due to the shared computations/histories). In that way, consciousness isn't identified with some (Universe,Position,Time,BranchId,...), it's related to (Program, Step)'s arithmetical truth at different levels. (Universe,Position,Time,BranchId,...) may be contained/shared in many computations by different (Program, Step) corresponding to completely separate 1p's. Now, the idea of single countably infinite universal consciousness could make sense, except there is one problem: COMP Immortality - there will always exist a continuation. Maybe some continuations could very well merge with others, thus the problem would be side-stepped, but what if they are actually unique countably infinite potential futures for most observers? There could be some ways to still allow for an universal consciousness in that case, but I think might it might be best to just abandon the concept for now and just think of different arithmetical truths corresponding to different 1p's, or if you want, only consider it for histories, whatever they may be.
Unless, of course, I misunderstood your problem.

From the 3p or 1p's memories/knowledge, that is, at a higher level
than just experience, we bet on the existence of the past and future,
as a matter of self-consciousness and self-reference. We tend to
identify with the (abstract) structure making this bet. This leads one
to RSSA - OM's being relative to each other - that we will make our
bets based only on expected continuations and past/journal/history. If
consciousness is how some truths associated with a self-referential
universal number feel from the inside, and given the bets that number
is making, it wouldn't seem that strange that we will experience
apparent continuity (even though we cannot prove to anyone that we
actually experience such continuity - we cannot even show that to
ourselves - if we just consider a few moments in the past).

I agree with all of this but you are only considering a single observer
here. Think of what you just wrote as applicable to many observers
interacting with each other. How does that work?

See before this. Shared computations.

I don't think the continuity problem gets solved by dismissing a
Platonia and using something more "physical"(what is that though?).
See: MGA for why.

I agree, but that is not the problem that I have with the ideal monism.
My contention is that we are forced into some form of dualism to account
for many separate minds capable of having some form of interaction with
each other.

Dualism on what? How "arithmetical truth" feels from the inside seems to be fairly monist to me. Interaction is just sharing some computations or truths. When Bruno interviews some LUM, he simulates it and thus the machine manifest relatively to him and he can see what that machine has to say. You and me share a common universe (or computations representing this universe), both of us have an "illusion" of matter of each other, yet both of our computations are real and correspond to separate 1p's.

This is why I think "arithmetic" is as good as any for a neutral
foundation, and we cannot really distinguish (from the inside) between
these foundations by the CTT.

This does not address the neutrality problem though. How can the
foundation be neutral if it is biased toward a particular structure,
even if it is as elegant as arithmetic? My point is that whatever
foundation we take, within our ontological theories, it must be neutral
with respect to a basis, reference frame, grammar or any other structure
that would break its perfect symmetry. Nature does not respect any
privileged framing what so ever and thus there cannot be a privileged
observational stance. This stance toward neutrality may seem unusually
strong, but I don't see how it can be any other way, even allowing
arithmetic to be a primitive is to allow a bias against non-arithmetical
structures and any bias, however weak, is still a rupture of neutrality.

But it doesn't have to be "arithmetic", it can be any system capable
of universal computation. Take something less and nothing truly
intelligent can exist (going less than computation). Take something
more (concrete infinities) and I'm not sure that those structures
would be conscious like you and me. I'm not that against the "more"
possibility, just that I don't think we can ever know too much about
them except by our mathematical theories, this being a consequence of
COMP (if one is turing-emulable). In a way, while more "general"
foundations can exist, it's unlikely we'll ever be able to truly know
more about them than we can compute about them (that is, any theory
we'll come up with will be limited to what a theorem prover can prove
about it, and we cannot know more, although we could bet on more by
adding more axioms, although we cannot know if some of those axioms
are truly correct), and it's also unlikely that they can affect
arithmetical/computational matters (if you think otherwise, you'll
have to explain why or show a proof; I'm aware of Goodstein's theorem,
but to prove it, we have to have stronger axioms, which we cannot know
if they are correct or not! Similar stronger theories are needed for
solving some other specific halting problem-related questions).

You are not considering the situation that I sketched out in the above
paragraph. I am considering a stratification where there is a lowest
level where all differences between properties must vanish, that is
where the neutral monism is.

Can we even keep definitions and properties themselves neutral enough?

However, there might be other possible foundations, if you wish to
postulate concrete infinities, but even if they existed, how could we
tell them apart, it doesn't seem to be possible for someone admitting
a digital substitution, which has a finite mind (at any finite point
in time). If you can show that those other foundations are necessary
and they affect our measure/continuations, or that concrete infinities
are involved in the implementation of our brain, it could prove COMP
wrong.

The Dualism that follows the analogy of the Stone duality covers this
question. Boolean algebras have a specific kind of topological space as
their dual. It is forced and as such there is a direct and predictable
link between the behavior of the logic and the behavior of the dual
space. Is it a complete accident that the topological space that is the
dual to Boolean algebras looks like a collection of primitive atoms
<http://en.wikipedia.org/wiki/Atomism> in a void? I don't think so! So
if the logic that observers are limited to is required to be
representable in terms (up to isomorphism) with Boolean algebras, then
the physical world that those logical entities have as 1p must look like
"atoms in a void". No wonder our particle physics works so well!
There is more to add to this, such as the Pontryagin duality that
expands the class of dual spaces out to range between the discrete
spaces to the compact spaces, but that is for another conversation. :-)

That's interesting, although my Category theory knowledge is rather
incomplete, so I can't really comment on the specifics. In a way
though, it seems that your idea is even more restricted than the UD*,
in which case, it would fall to your generality objection, would it not?

I don't know. Could you elaborate on this?

The UD* allows all kinds of continuations, such as our local high measure ones, but also all kinds of unusual and weird ones which are purely computational and hardly correspond to our physics with us being directly embodied. The possibilities themselves are very numerous, although I suspect most of them are low-measure. If your theory presupposes only some particular physics being possible (and nothing else), it's being more restrictive than the UD - although, if it can run a robust UD, I'm not sure it would make a perceptible difference.

There is another problem with taking a set theory as foundational
rather than arithmetic - some set theories have independent axioms and
they can be extended by adding either an axiom or its negation, and
they result in different set theoretical truths.


I didn't mean to take set theory per se as fundamental, I was thinking
of set theory as just a mereology - a schemata of sorts - of how we
define relations between parts and wholes. But as to your point about
set theory, does not the proven existence of non-standard Arithmetic
<http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic> argue
the other way? While the Tennenbaum Theorem
<http://en.wikipedia.org/wiki/Tennenbaum%27s_theorem> seems to make
standard (ala Peano) arithmetics "special" and "unique", I strongly
suspect that this is just an invariance property, similar to the
invariance of the speed of light in physics: any logical entity will see
its own Arithmetic model as being countable and recursive, it cannot
see the
"constant" that would make it non-standard as such is its fixed point,
its "identity" if you will. I do not have any formal description of this
latter idea nor even a proof of it, so please just take this as a
conjecture. ;-)

Non-standard models no longer have computable addition and
multiplication, thus they're not considered in COMP - where the
observer's body is assumed to be computable (as an axiom). Your latter
idea seems interesting, although for me to better understand what you
mean, you'd have to elaborate on the details.

I am still learning the math. I am just a student and a slow one. I will
try to sketch out more of this idea and I come to understand it better
myself.

I'll keep an eye out for it, if you happen post it someday.

This doesn't really happen with computation - if there's anything
absolute in math, it's computation (although different theories about
what arithmetic is will result in different things the theory can talk
about, but it won't make computation any less absolute).

I strongly suspect that your argument here about the "absoluteness" of
computation is a bit too strong or even misplaced. Restricting
information to only being a binary bit on mappings in the Integers is a
harsh regime, no wonder computation is so "well behaved", any deviation
of the bits from the tyranny of the integers at all is terminated with
extreme prejudice! I see computation, in general, as "the transformation
of representations" and thus do not see the by fiat confinement to the
integers as beneficial.


By absoluteness I mostly meant the very wide consequences that follow
from the Church-Turing Thesis. In another way, the behavior of finite
things to which we apply finite processes is always well-defined.
Things are never as clear when we have infinitely-sized things or
potentially infinite processes.

I claim that it is impossible to observe infinite quantities as such
cannot be exactly represented in a communicable way. This does not mean
that infinities do not exist. It is just an inherent limitation on what
observers can be.

If observers are finite, does that not directly lead to the COMP assumption? Or do you allow implementation which involves concrete infinities, yet nevertheless only processing finite sense data? Should it be possible (in your theory) for an observer to contain infinite processes which nevertheless finish in finite time?
At least 'we' cannot know how they behave without adding some axioms
and when we do add those axioms, we can also consider alternate
theories where the negation of the axiom is considered and that
results in different consequences. The "absoluteness" of computation
is of this nature. If we can truly *know* more than arithmetic while
still remaining correct, I do not know (assuming COMP).

I agree.



As a side-note, I don't see why the primitive physical world is
necessary, from the 1p, we can only know that we have senses and from
the senses we can infer the existence of the external world.

We have the problem of other minds to deal with! That is why, among
other things, we need the physical world albeit NOT primitive, the
physical world allows form an "external" differentiation of 1p that
would otherwise be identical by Leibniz' identity of indiscernibles. I
am just claiming that the abstrac
<http://en.wikipedia.org/wiki/Abstract_object>t and the concrete
<http://en.wikipedia.org/wiki/Concrete_object> are always co-present at
any level until we go to the limit of bare neutral existence. At that
point any differences that might make a difference vanish, thus logic
and spaces would cease being different yet isomorphic. Vaughn Pratt
shows how this works in terms of the directions of the Arrows of the
categorical representations of LOGIC and SPACE, they point in opposite
directions thus if we add them up their directions and scalars would
vanish. -> + <- = (see
http://upload.wikimedia.org/wikipedia/commons/f/ff/Laws_of_Form_-_double_cross.gif)



Maybe we mean different things by the physical world. I think of it as
an implementation substrate and thus I have no problem with it being a
direct or indirect consequence of some abstract computations.
It's also not obvious at all to me why the 1p would be the same for
any structure in Platonia (such as some computation running in the
UD), but different for magical-physical-land (non-platonic, but still
running an UD). 1p differences should exist if the contents of the
mind's body/brain are different, regardless of the substrate that it's
implemented on. I'd venture to guess that given 2 identical
structures/universes/..., the observers in them will have identical
experiences, or even identical 1p (in COMP it doesn't matter how many
copies you make of a computation, there's only one 1p associated with
it - the body just lets it manifest relatively to you).

I agree but would like to point out that the physical world is also the
means by which our minds can interact. It cannot be so easily hand-waved
away.

Explained earlier in this post. Unless, you do mean something more magical by interaction beyond shared computations?

Additionally, I see this conjecture as similar to Tegmark's Mathematical
Universe Hypothesis
<http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis> except
that I do not see how the postulate "/All structures that exist
mathematically also exist physically."/ implies a mathematical monism as
the wiki article states. If for any structure that exists mathematically
there must exist a physical structure, there is the implication of a
duality between the mathematical and the physical. This is a different
sort of duality than that of Descartes as it does not assume distinct
"substances", it is a form of dual aspect theory
<http://en.wikipedia.org/wiki/Double-aspect_theory> where the dynamics
of each aspect run in opposite directions. Vaughn Pratt explains the
idea here: http://boole.stanford.edu/pub/dti.pdf


I'm not going to comment on the paper as my category theory knowledge
is insufficient, instead I'll save reading it until I'm more familiar.

Please do as it might help you understand some of the more complex
aspect of this idea that I am studying.

As time allows.
As for the MUH, I'm not even sure what the word 'physical' means
anymore for it. If all the consistent mathematical structures do exist
and you happen to find yourself in one, you call it 'physics', while
you call the rest 'abstract', however that just makes the term
'physical' an indexical - "The time now is xx:xx", "I'm in structure y".
The UDA also shows that you can find yourself in a different structure
at a different subjective time, and the only global "inescapable" one
is the UD, but it's so limitless in its possibilities that it
shouldn't particularly matter.

OK, but, again, we have to have a theory that covers interactions
between our minds not just theories of individual minds.

See earlier comments on this.
If consciousness is how some (possibly self-referential) arithmetical
(or computational) truth feels from the inside, it does not seem
impossible that there would not be computations representing some
physical (just not primitive) world and that world would contain us
and our bodies/brains, and the existence of such computations would be
a theorem in arithmetic.


I agree, but the representation of a thing is not the thing except in
very special cases, such as what we have when we say that the "best"
simulation of an object is the object itself.
<http://www.stephenwolfram.com/publications/articles/physics/85-undecidability/2/text.html>

This takes us into a discussion of questions like "when might the map =>
the territory or, by duality,the territory => the map
<http://chorasimilarity.wordpress.com/2011/06/21/entering-chora-the-infinitesimal-place/>?

This is a subtle and important question! ;-)


We can never know for sure what structure we're part of, but we can
always make more and more accurate maps of the local one we're in.

I agree, this is what Natural Philosophy is all about. :-)

One can call the global one the UD* if we happen to admit a digital
substitution (by the UDA/MGA), or more generally some arithmetical
Platonia. Just knowing how the UD works does not mean we have complete
access to it. We just have a tool for generating the territory (or the
"perfect" map), but we'll never be able to actually generate the full
UD* (if we could, COMP would be false, and thus UD* itself wouldn't be
our "global" territory), although we can generate any parts we have
the resources/memory to compute.

This is an everlasting process. What I find interesting, among other
things, is that it solves Nietzsche question abouteternal return
<http://en.wikipedia.org/wiki/Eternal_return#Friedrich_Nietzsche>.;-)

UD's self-generating/autopoietic nature is quite nice/fascinating indeed, although its a more general concept that occurs all over math, nature, life(replicators), self-consciousness, computability, fractals, ...

Onward!

Stephen



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