Sorry for the comment delay, Jesse, (also, I sent this yesterday, but
it seems not having go through).
On 25 Jul 2014, at 23:22, Jesse Mazer wrote:
On Thu, Jul 24, 2014 at 2:44 PM, Bruno Marchal <[email protected]>
wrote:
HI Jesse, David,
On 23 Jul 2014, at 18:49, Jesse Mazer wrote:
Had some trouble following your post (in part because I don't know
all the acronyms), but are you talking about the basic problem of
deciding which computations a particular physical process can be
said to "implement" or "instantiate"? If so, see my post at http://www.mail-archive.com/everything-list%40googlegroups.com/msg43484.html
and Bruno's response at http://www.mail-archive.com/everything-list%40googlegroups.com/msg43489.html
. I think from Bruno's response that he agrees that there is a
well-defined way of deciding whether one abstract computation
implements/instantiates some other abstract computation "within
itself" (like if I have computation A which is a detailed molecular-
level simulation of a physical computer, and the simulated computer
is running another simpler computation B, then the abstract
computation A can be said to implement computation B within itself).
So, why not adopt a Tegmark-like view where a "physical universe"
is *nothing more* than a particular abstract computation, and that
can give us a well-defined notion of which sub-computations are
performed within it by various "physical" processes? This approach
could also perhaps allow us to define the "number of separate
instances" of a given sub-computation within the larger computation
that we call "the universe", giving some type of measure on
different subcomputations within that computational universe
(useful for things like Bostrom's self-sampling assumption, which
in this case would say we should reason as if we were randomly
chosen from all self-aware subcomputations). So for example, if
many copies of a given AI program are run in parallel in a
computational universe, that AI could have a larger measure within
that computational universe than an AI program that is only ever
run once within it...of course, this does not rule out the
possibility that there are other "parallel" computational universes
where the second program is run more often, as would be implied by
Tegmark's thesis and also by Bruno's UDA. But there is still at
least the theoretical possibility that the multiverse is false and
that only one unique computational universe exists, so the idea
that all possible universes/computations are equally real cannot be
said to follow logically from COMP.
To have the computations, all you need is a sigma_1 complete theory
and/or a Turing universal machine, or system, or language.
Not sure I understand what you mean by "have the computations",
We need to start from assuming something (if we want do fundamental
science).
By "to have the computation" I meant, to have the theory in which we
assume enough so that we can define and prove the existence of the
computations. Elementary arithmetic is enough, but there are other
theories, like the combinators, or the abstract billiard ball, or
quantum topology, etc.
and I didn't understand the mathematical arguments you made
following that. My point above is basically that even if one accepts
steps 1-6 of your argument, which together imply that I should
identify my self/experience with a particular computation (or
perhaps a finite sequence of computational steps rather than an
infinite computation, but I'll just call such a finite sequence a
'computation' to save time), it still seems to me that there is an
open possible that the *measure* on different computations is
defined by how often each one is "physically" instantiated.
With step 1-6, yes. But less so with step 7 and 8 which still follows
from the CTM).
With step 7, yes again, assuming a "small" (without big portion of
UD*) primitive physical universe. (It already looks like avoiding a
question/problem (measure problem). If I try to dig on your theory, I
will have to ask eventually what you mean by "primitive physical
universe", as it looks like "and now there is a miracle".
And step 8 just makes it worst. It shows that the miracle asks for an
infinite amount of magic, so you need a specially weak Occam razor to
expect this from reality.
Are you talking about some deriving some unique measure on all
computations when you say "to have the computations, all you
need..." or are you not talking about the issue of measure at all?
I was talking about what we have to assume to define the computations
and reason about them, and to study the expectation of "simple" person
(like the one described by the 8 arithmetical points of view on
arithmetic).
The idea I'm suggesting for a "physically" based measure involves
identifying the physical universe/multiverse with a particular
unique computation--basically, consider a computation corresponding
to something like a Planck-level simulation of our universe, or an
exact simulation of the evolution of the the universal wavefunction,
then say that this computation *is* what we mean by the "physical
universe/multiverse".
That is not excluded, but if that is the case, it would mean that
there is a computation which win the competition between all
computations, and that needs to be derived from arithmetic only,
through the logic of self-reference, to clearly distinguish the
observable from the conceivable, the believable, the knowable, truth,
false, etc.
It would mean that one computation would win on all computation. It
would mean that it exists a number p such that phi_p emulates the
physical universe, with its infinitely many twins, the q such that
phi_p = phi_q.
I am quite open on this. Probably this + the random oracle, that you
get freely in arithmetic by the self-multiplication.
Then, if you agree there is some well-defined notion of whether a
given computation "contains within it" some other computation (and
that we can count the number of times some sub-computation has run
within the larger computation after N steps of the larger
computation), the measure on all computations could be determined by
how frequently they each appear in the unique computation that we
identify with the physical universe/multiverse.
We can always compare both (as far as you can identify that physical
universe). As long as that fits, we get indirect evidence for the
parts which are not studied by the physicists, which are the non
propositional knowledge of the knower, the non justifiable bets, well
a sort of toy theology, of the ideally correct person emulated by a
body/machine.
For example, say after N steps of the universal computation U, we
can count the number of times that some computation A has been
executed within it, and the number of times that another computation
B has been executed within it, and take the ratio of these two
numbers; if this ratio approaches some limit in the limit as N goes
to infinity, then this limit ratio could be defined as the ratio of
the "physical" measure of A and B within the universe/multiverse. So
if A and B are two possible future observer-moments for my current
observer moment (say, an observer-moment finding itself in
Washington and another finding itself in Moscow in your thought-
experiment), then the ratio of their physical measure could be the
subjective probability that "I" will experience either one as my
next-observer moment.
OK if U is any UD, as this is given for free once we believe in
anything complex enough as to behave like a computer.
Then you come, and tell me that the UD does not count, only a special
U, (executed by the UD too, of course), win the games.
You can explain me this in two ways:
1) by showing that such an U indeed fits the 1P and 3p requirements,
but this will lead you to derive U from some "integral" on the UD.
2) by saying that the U is implemented by a special purpose God,
called "primitive matter", so we don't need to show that he win the UD
game.
Also note that even if we have two different candidates for the
"physical universe" computation, call them U and U', and even if
both contain a never-ceasing universal dovetailer computation within
them, it seems to me this is not enough to guarantee that U and U'
will both assign the same physical measure to any two computations A
and B, if we use a procedure like the one I outlined to define
"physical measure". Even though U and U' will both compute all the
same programs eventually since they both contain a universal
dovetailer, some programs might be computed more frequently (more
copies have been run after N steps) in U than in U'.
I think you are wrong on this. That is a point seen by Schmidhuber,
and theoretical computer scientists in general, which is that you
can't change what a UD does, if you want it to remain a UD. The
measure will not depend on any choice you do to implement the UD. You
can fail it on very strong large initial fragment, but the 1p measure
is unchanged on the limits.
So, again, you can save physicalism, by making a NON computable
transform of the UD, but that's he type of move the MGA shows to be
"adding complexity" to bias the solution of a problem.
For example, U might be a physical simulation of a universe
containing one physical computer that's computing the universal
dovetailer along with 1000 physical computers computing copies of my
brain experiencing being in Washington, while U' might be a physical
simulation of a universe containing one physical computer that's
computing the universal dovetailer along with 1000 physical
computers computing copies of my brain experiencing being in Moscow.
That will change nothing, by the compiler theorem. Technically, this
is related to the closure of the partial computable functions for
Cantor-Kleene diagonalization, and why Gödel called that being a
miracle, as it protect machines soul from all formal reductionism. It
is related to the conceptual argument in favor of the Church thesis.
I don't necessarily advocate deriving measure from some unique
"physical universe" computation U myself, but do you see anything
basically incoherent about the idea?
I think it is coherent with step 1-6, and much less already with step
7, due to that generality (sum up in Church-Turing thesis). And quasi
no more rational with a very light occam razor after step 8.
I believe that there is a physical winner, and that it is a multi-
worlds, but to solve the mind body problem, and to keep all views in
the process (including the non justifiable), we have to extract that
physical winner by the study of the universal mind and its relation
with consciousness.
If not that would suggest that accepting steps 1-6 does not
necessarily require accepting later steps where (unless I
misunderstand your argument) you argue that the measure on all
computations can be derived from pure mathematics.
No, I argue that If we assume the computationalist theory of mind
(CTM), then the observable (and lawful/invariant) part which results
from the measure on all computations has to be derivable from "pure
mathematics", specifically from the mathematical theology of the
universal (Löbian) numbers.
Bruno
Jesse
It would take many pages to describe formally elementary arithmetic
(including the formal predicate calculus), which is indeed already
such a sigma_1 complete system/machine/theory, but a simpler one can
be given in less line, like the Putnam-Davis-Robinson-Matiyazevic-
Jone universal diophantine polynomials. Or the combinators, whose
sigma_1 complete theory is given by the axioms
x = x
x = y & y = z ->. x = z
xy = xz / y = z
yx = zx / y = z
Kxy = x
Sxyz = xz(yz)
(I recall that a combinator is either K or S, or a combination of
combinators (X, Y), so a combinator is for example
(K (K S)) S) which we abbreviate K(KS)S as we can suppress all the
left parenthesis for ease of readability.
You can compute ((K K) K), or better KKK. By the second axiom you
get KKK = K. But K(KK) does not match any axioms, and thus stay
calm: K(KK) gives K(KK) as a stopping result.
For the theory of everything, we need no more. Oh, well to avoid
having just one combinators, you can add the axiom
~(K = S),
but for the ontology it is not really needed. It is a Turing
universal language, and all universal interpreters can be coded
through a combinator. In particular, you can easily find combinators
which mirrors faithfully the sigma_1 complete part of arithmetic,
like you can find combinators which solves the PDRMJ universal
diophantine polynomial equation.
In that theory, we can define those very theories formally, and they
all are instances of universal combinators, or universal numbers. A
computation is what a number do relatively to a universal number.
But by the FPI, a physical computation will be the one done,
strictly speaking, by infinities of universal (and non universal
also) numbers/combinators.
The absolute (relative measure) laws does not depend of choosing
arithmetic, or combinators, that is, the laws of physics will not
depend on the choice of the universal numbers. But the "winner", or
"winners" which support(s) and stabilize(s) your current state of
mind is "unknown", today. Except that when you interview the löbian
machine, which are those who knows that they are sigma_1 complete,
that they know that they are Turing universal, then you get that the
winner has some quantum favor, as the many dreams in the
combinatoric reality (the FPI on the sigma_1 complete reality).
David, I think that with the combinators, I might more quickly
explains different senses of "going at a higher level description",
making possible to better delineate where I agree with Brent and
where I agree with you.
I mean I will think about it.
Liz? Is it not time to study at least one, conceptually simple,
computer programing language?
The point is that we need a universal system, and then we can define
computations, the UD and and all finite portions of UD*, guided
through the FPI on the limit "winning measure(s)".
Or we use the phi_i?
All this already gives a very rich 3p reality, and does not address
the 1p, except by that FPI. But the löbian numbers have the
introspective ability to distinguish the representational (belief)
from the non representational (knowledge, hope).
The löbian number can already understand that they are "lost" in the
arithmetical reality, and confronted with the arithmetical truth,
most part of it being non justifiable. Yet, those non justifiable
truth obey a mathematics of their own, constraining different modes
of the apprehension of truth (arithmetical truth, combinatorical
truth, c++al truth, whatever).
David, The modes (like []p, []p & p, etc) emerges from the fact that
universal numbers develops complex 3p objects, like when proving
theorems, or building planes, or getting self-referential (even just
in the 3p way). We have to be platonist on the 3p constructions (the
programs, the data, the activity of a machines relatively to another
machine), to be able to be platonist on the 3p realities given by
[]p & p, which are much more evanescent, as they create an umbilic
cord with truth, and pay the big price, as they loose their name/
identity/description.
Here your argument against Brent is at risk to be extended into an
argument against CTM. You gently save me by observing that I do
justify the existence of the modes, but that works only because
ultimately we say that "yes" to the doctor and on the fact that we
bet the doctor get the right "[]p" concerning us. We "know" already
that the *truth* of p will take care of itself (by the minimal
realism asked to have a notion of numbers or combinators, to begin
with.
There is 31° celcius, without much air, and my head is boiling hot,
apologies for the typos.
Bruno
Jesse
On Wed, Jul 23, 2014 at 9:38 AM, David Nyman <[email protected]>
wrote:
Recent discussions, mainly with Brent and Bruno, have really got me
thinking again about the issues raised by CTM and the UDA. I'll try
to summarise some of my thoughts in this post. The first thing to
say, I think, is that the assumption of CTM is equivalent to
accepting the existence of an effectively self-contained
"computationally-observable regime" (COR). By its very definition,
the COR sets the limits of possible physical observation or
empirical discovery. In principle, any physical phenomenon,
whatever its scale, could be brought under observation if only we
had a big enough collider. But by the same token, no matter how big
the collider, no such observable could escape its confinement
within the limits of the COR.
If we accept that the existence of a COR is entailed by assuming
CTM, we come naturally to the question of what might be "doing the
computation". In terms of the UDA, by the time we get to Step 7, it
should be obvious that, in principle, we could build a computer
from "primitive" physical components that would effectively
implement the infinite trace of the UD (UD*). Furthermore, if such
a computer were indeed to be implemented, the COR would necessarily
exist in its entirety somewhere within the infinite redundancy of
that trace. This realisation alone might well persuade us, on
grounds of explanatory parsimony and the avoidance of somewhat
strained or ad hoc reservations, to accept FAPP that UD*->COR.
Should we be so persuaded, any putative underlying "physical
computer" would have already become effectively redundant to
further explanation.
Notwithstanding this, we may still feel the need to retain
reservations of practicability. Perhaps the physical universe isn't
actually sufficiently "robust" to permit the building of such a
computer? Or, even if that were granted, could it not just be the
case that no such computer actually exists? Reservations of this
sort can indeed be articulated, although worryingly, they may still
seem to leave us rather vulnerable to being "captured" by Bostrom-
type simulation scenarios. The bottom line however seems to be
this: Under CTM, can we justify the "singularisation", or
confinement, of a computation, and hence whatever is deemed to be
observable in terms of that computation, to some particular
physical computer (e.g. a brain)? More generally, can we limit all
possibility of observation to a particular class of computations
wholly delimited by the activity of a corresponding sub-class of
physical objects (uniquely characterisable as "physical computers")
within the limits of a definitively "physical" universe?
This is where Step 8 comes in. Step 7 seeks to destabilise our
naive intuition about an exclusive 1-to-1 relationship between
computations and particular physical objects by pointing to the
consequences of a physical implementation of UD*. Step 8 however is
a change of tactic. First, it postulates a scenario where physical
tokens have been contrived to represent a "conscious
computation" (either in terms of a brain or in terms of a
substitute "computer"). Then it sets out to shows how all
putatively "computational" relations between such tokens could in
principle be disrupted without change in the net physical action or
environmental relations of the system that embodies them. Step 8
differs from Step 7 in that it seeks in the first instance to
undermine the very notion that physical activity can robustly
embody *any* second-order relations above and beyond those of net
physical action. Accepting such a stringent conclusion would then
seem to rule out CTM prima facie. The only possibility of salvaging
it would lie in an explanatory strategy in terms of which
computational relations take logical precedence over physical ones.
Given that computational relations are effectively arithmetical,
this in turn leads to the conclusion that CTM->UD*->COR (or more
generally, that each implies the others).
Notwithstanding this it would seem that Step 8 is not wholly
persuasive to everybody, so is there yet another tack? The line of
argument that I've been pursuing with Brent has led me to consider
the following analogy, which I'm sure you'll recognise. Consider
something like an LCD screen as constituting the "universe of all
possible movie-dramas". In terms of this analogy, what are the
referents of any "physical observations" on the part of the
dramatis personae featured in such presentations? IOW what are we
to suppose Joe Friday to be referring to when he asks for "Just the
facts, ma'am"? Well, the one thing we can be sure of is that NO
such reference can allude to the "underlying physics" (i.e. the
pixels and their relations) of the LCD display. If this analogy
holds, at least in general outline, what justification, under CTM,
could remain for any assumption that our own observations and
references might "accidentally" allude to some "LCD-physics"
postulated, mutatis mutandis, as underlying the COR? Would it not
seem extraordinary that any such underlying physics could contrive
to "refer to itself" through the medium of its merely computational
derivatives?
This last point might seem determinative, but might there not still
be a last-ditch redemption, of a physics underlying computation, in
terms of "evolution"? IOW, might it not be argued that the
acquisition of internal "computational" models of their physical
environment confers a survival advantage on the physical creatures
that embody them? But any such argument would, of course, be
completely circular; assuming CTM, it begins and ends in the COR.
IOW, arguing in this way would be to ignore the fact that the
history of such creatures, their survival, and the environment in
which this is supposed to take place, all lie within the COR, not
the putative regime of any "underlying physics". THAT "physics"
would necessarily be entirely inscrutable and inaccessible for
reference at the level of the COR (think of the LCD analogy). And
hence we simply would have no a priori justification for assuming
the observational physics of the COR to be isomorphic with some
notional underlying "LCD-physics". In fact, once having assumed
CTM, we would have no further basis for assigning THAT physics any
role whatsoever in our explanatory strategy.
David
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