Re: possible solution to modal realism's problem of induction
Brian Holtz
a physical object: a bunch of matter, connected together
spatiotemporally. So I do not need to work with specifications, but
with concrete chunks of stuff. There is nothing further illuminating
to be said in a lewisian context, really, about what makes two
concrete chunks of stuff the same chunk, is there?
That said, I am making an assumption that there is only one copy of
each world. I suppose one could recover the "measure" the authors
you cite have if you suppose that there is a copy of each world for
every arrangement-description of it. But I do not see why one would
suppose that.
In the Lewisian setting, it is intuitively plausible that the
probability that I exist in w1 should equal the probability that I
exist in w2, as long as w1 and w2 contain intelligent observers in
equal numbers. The "measures" from the authors you cite do not
satisfy this criterion IF there is one world for a class of
equivalent descriptions, as is going to be the case under the
assumptions I am making.
Most observers are going to be in worlds with a much higher
cardinality of stuff than our world contains. Our world probably
only has a finite number of particles. The cardinality of worlds
just like ours until tomorrow but where \aleph_8 neutrons appear in
San Francisco down-town, causing everything in the universe to
collapse is much greater than the cardinality of regular worlds. In
fact, I think what I am saying here will apply even on information-
theoretic measures. (The one or two papers you linked to that I
looked at made the assumption that there was a fixed maximum
cardinality of things. But why assume that?)
---------------------------
Russell Standish
> Hi everyone (in this world and all relevantly similar ones :-),
>
> I like the solution to the Induction / Dragon / Exploding Cow problem that I
> see in work by Malcolm, Standish, Tegmark, and Schmidhuber. So I forwarded
> references to Alexander Pruss, whose dissertation raises the Induction
> Objection to modal realism. The full context is on my blog at
> http://blog.360.yahoo.com/knowinghumans?p=8. I'm interested in how the
> folks on this list would respond to Pruss's most recent comment, below. Can
> anyone recommend a primer on probability in transfinite contexts like ours?
...
I will require some thought, but the approach I take (which is
effectively identical to Malcom's), the Plenitude is a set of
cardinality c (2^\aleph_0). The argument works within such a bound. My
reading of Tegmark and Schmidhuber is that they also have a similar
cardinality bound.
Could it be extended to higher cardinality plenitudes (if such a thing
makes sense)? I don't know without giving it a lot of thought (and
even then?), but I'd hazard a guess that certain paradoxes might step
in making the whole thing untenable.
Cheers
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Alastair Malcolm
Brian Holtz
Alex Pruss wrote:
Remember that I am working in David Lewis's framework. Each world is a physical object: a bunch of matter, connected together spatiotemporally. So I do not need to work with specifications, but with concrete chunks of stuff. There is nothing further illuminating to be said in a lewisian context, really, about what makes two concrete chunks of stuff the same chunk, is there?
I have the vague suspicion here that by using words like physical/matter/concrete/chunk, you're skirting the issue of how worlds are specified in the general case, by narrowing the scope to worlds whose only constituents are material -- literally, having mass and occupying space. What about worlds consisting of a single point of space, populated by (soul-like?) entities whose (of course non-spatial) internal specifications and external relationships change over time? I fear you're taking a short-cut that relies on our intuition that ordinary baryonic matter has a privileged and obvious and natural way to be specified.
By the way, I think I disagree even with the the spatiotemporal stipulation of Lewis. It makes more sense to me to define a world as a causal closure rather than a spatiotemporal closure, but perhaps that would give up on an ambition of Lewis to analyze causality rather than consider it a primitive. For example, what if a world consists of two disconnected regions of space, between which there can still be causal relations? Would Lewis just say that the events are temporally related even though not spatially related? (Hopefully he wouldn't try to introduce some extra spatial "dimension" by which to allow a coordinate specifying which region, as such an effort could I think be confounded.) If so, then maybe my disagreement with Lewis is that I would define time in terms of causation, where he would define causation in terms of time....
That said, I am making an assumption that there is only one copy of each world.
As I understand it, a virtue of the information-theoretic perspective is that if we define worlds as one-to-one with their minimal K-specifications, we don't have to bother with questions like whether there can be copies of worlds.
I suppose one could recover the "measure" the authors you cite have if you suppose that there is a copy of each world for every arrangement-description of it. But I do not see why one would suppose that.
I'm a novice when it comes to the concept of measure, but my sense is that these many-worlds theorists restrict themselves to methods of world-specification in which different specifications map by definition to different worlds.
Most observers are going to be in worlds with a much higher cardinality of stuff than our world contains. Our world probably only has a finite number of particles. The cardinality of worlds just like ours until tomorrow but where \aleph_8 neutrons appear in San Francisco down-town, causing everything in the universe to collapse is much greater than the cardinality of regular worlds. In fact, I think what I am saying here will apply even on information-theoretic measures.
We might ask how the inductively deceptive worlds compare in abundance to the undeceptive worlds. If this is meant as a comparison of cardinalities, it seems clear that the numbers will be equal. For deceptive and undeceptive worlds alike, it is easy to set a lower bound of beth-two, the number of distributions of a two-valued magnitude over a continuum of spacetime points; and hard to make a firm case for any higher cardinality. However, there might be a sense in which one or the other class of worlds predominates even without a difference in cardinality. There is a good sense, for instance, in which the primes are an infinitesimal minority among the natural numbers, even without any difference in cardinality: their limiting relative frequency is zero. We cannot take a limiting relative frequency among the worlds, for lack of any salient linear order;
(The one or two papers you linked to that I looked at made the assumption that there was a fixed maximum cardinality of things. But why assume that?)
I'm not sure which assumption you mean. Can you point it out in e.g. Malcolm or Standish? (I'm cc'ing them by virtue of everyth...@eskimo.com.)
Best wishes,
Alex
http://www.georgetown.edu/faculty/ap85
(For earlier context and links, see http://blog.360.yahoo.com/knowinghumans?p=8.)
Brian Holtz
blog: http://knowinghumans.net
book: http://humanknowledge.net/
Russell Standish
> Alex Pruss wrote:
>
> Remember that I am working in David Lewis's framework. Each world is a
> physical object: a bunch of matter, connected together spatiotemporally. So
> I do not need to work with specifications, but with concrete chunks of
> stuff. There is nothing further illuminating to be said in a lewisian
> context, really, about what makes two concrete chunks of stuff the same
> chunk, is there?
>
...
I haven't read David Lewis, and unfortunately it would be serious
distraction for me to follow his work up at this point in time. It
does strike me that "concrete chunks of stuff" is a bit incoherent, however.
>
> My suspicion/hope is that the work I cited by Malcolm/Standish/Schmidhuber
> suggests an approach to defining such a linear order, by which we can judge
> that apparently regular worlds predominate over apparently irregular worlds.
> (Alastair, Russell -- am I reading you correctly?)
>
Linear ordering is not needed. Rather, the fact that only finite
prefixes of the bitstrings are meaningful defines a natural topology on
equivalence classes of bitstrings having the same meaning. Such a
topology leads to an induced universal measure.
> (The one or two papers you linked to that I looked at made the assumption
> that there was a fixed maximum cardinality of things. But why assume that?)
>
The assumption is that the Plenitude is a particular object, the set
of all bitstrings ("descriptions"). The justification for this is
twofold:
1) All we can know about reality enters the mind as a string of data
(a bitstring)
2) The set of all such bitstrings has precisely zero information.
No other Plenitude has these properties. It is a side effect that the
cardinality of this Plenitude is c, not an assumption that limits it.
Bruno Marchal
Le 14-juin-05, à 18:26, Brian Holtz a écrit :
> Hi everyone (in this world and all relevantly similar ones :-),
Welcome to the list Brian. Thanks for the link to Alexander R Pruss'
web page, which seems quite interesting (and which I will comment a
little bit too, here or in a next post).
> I like the solution to the Induction / Dragon / Exploding Cow problem
> that I see in work by Malcolm, Standish, Tegmark, and Schmidhuber.
It is equivalent to the "white rabbit problem" we talk indeed about,
all along this list, and which is *almost* "solved" in my phd thesis
(to be short). May I attract your attention to it by referring you to
my web page? http://iridia.ulb.ac.be/~marchal/
It is a good occasion to sum up the main differences and the main
similarities between Standish, Schmidhuber, Lewis, Tegmark, Levy,
Ruhl, Mitra, Mazer, Finney, ... and my own. All approach are indeed
form of modal realism, and this is indeed what the everything-list is
all about.
Now I want to be short and I apology in advance for some
oversimplification, and please, any of you, don't hesitate to correct
me.
To simplify the comparison I think it could be useful to compare them
from their ontology and their epistemology, and the way they tackle the
"Dragon" problem.
Finney, for example, (a pillar of the list) borrows Bostrom's notion of
"observer-moment" (OM). He argues they are fundamental, and its "modal
realism" consists to accept or postulate *all observer-moments*
Then he borrows a computationalist hypothesis from Schmidhuber, and
associates to each OM a finite binary strings.
Then he tackles the dragon problem by attaching to those binary
string/OM their Kolmogorov complexity from which he infers a absolute
measure. Little strings will have higher measure, and this should make
the dragon disappearing through Bostrom Self-sampling assumption SSA,
taken in some absolute version of it: ASSA.
My critics: OM are described by Bostrom as first person subjective
construct, and it is not clear how they can or should be related to the
strings, in such a way that we can make personal prediction. The dragon
disappear, but then in one second I will be a bacteria!
Schmidhuber postulates a "big programmer" which runs all programs. He
postulates some universe and he postulates the possible universes are
computational objects. Then he try to find some prior explaining the
importance of short programs at the origin of one universe capable of
sustaining self-aware structure like us.
My critics: there are simply no notion of first person available.
Worst, Schmidhuber is obliged to postulate some totally unknown
physical reality. So the epistemology is empty and the ontology is
unknowable, though, according guessable (this is quite close to
traditional physicalism).
Tegmark, in his first paper, suggest the existence of ALL Mathematical
Structures. This is ontologically much interesting than Schmidhuber
frame, imo. Unfortunately it is too big, and Tegmark seems not to know
the failure of all mathematician to capture all of mathematics
mathematically. I have discussed elsewhere in the list, at lenght, some
cardinality problem related to Tegmarkian approaches. In Tegmark there
is an embryo of distinction between first and third person point of
view, but it is either vague, or locally clear only under the
assumption of QM, but then it is exactly the (very interesting)
difference between the subjective and objective knowledge already
introduced in Everett basic papers. The mind body problem is still
under the rug.
Both Tegmark ande Schmidhuber assumes unclear relaltion between
observer and universe, which in general presuppose Aristotle theory of
"substance".
In that regard, epistemologically, Malcolm has the same physicalist
attitude. He describes quite clearly three sort of *physical*
"theories", having in their intended model (in logician's sense) either
one universe toward having all logically possible universe, and he
defends, quite convincingly (imo) that last sort of theories. But he
discusses to quicky the relation between universe and information so
that I cannot really say more.
Main critics: the approach relies to much on some aristotelian notion
of universe, and the 1-3 distinction is not really tackled.
Standish is not yet enough clear about its assumptions, but seems to
get a pretty derivation of schroedinger equation, which is an
improvement. He does assume time, with the topology of the reals, which
is my main critics. The 1-3 distinction is present and used in an
anthropic way, but I have not yet understood it precisely.
George Levy is completely aware of the 1-3 distinction, and makes the
1-person at the origin of a purely first person "plenitude". Well, so
much that it is not clear for me if the plenitude is really suited for
being described in a 3-person theory, and this explains some its
silence in the list.
Note that some people, like Wei Dai, the list's master, but also late
James Higgo, have oscillate between approaches. Notably on the Absolute
versus Relative SSA. This is a key distinction. Saibal Mitra defends
the absolute version and Jesse Mazer a relative one. Mazer is quite
aware of the 1-3 distinction and of the necessity of having a theory of
consciousness to solve the dragon problem (which is what I have
developed).
Sorry for being short, and probably unfair, and unclear, but I must go
now. In a nutshell my approach is quite different. I take seriously the
1-3 distinction and I propose a proof that if we take the hypothesis
that "we" (or I) are digitalizable entities, then ontologically there
is no primitive physical universe (no big 3-everything), and
epistemologically the appearance of physics must be recovered from, let
us say, the gap between computer science and computer's computer
science. that gap is well described by the mdoal logic of
self-reference (Solovay), which makes it possible to translate the
proof in the language of (any) sufficiently rich (I say "Lobian)
machine. From this I can infer that the logic of physical proposition
obeys some precise modal logic of Lewis-Stalnaker counterfactuals, and
to show they agreed until now with quantum logics. Weakness: highly
technical (the "dragon problem" is transformed in an arithmetical
renormalization theory) and to much counter-intuitive for the average
aristotelian naturalist. But note it is quite "natural" (pun!) for the
Pythagoreans, the Platonist, and some neoplatonist like Plotinus.
I will add comments later and I will try to be more precise, and also
to comment your conversation with Alexander Pruss. About Lewis I love
its "counterfactuals". In the last edition he corrects some of its
argument against seeing worlds as "maximal consistent extensions" like
in my thesis, and then thanks to a paper by Hardegree (ref in my
thesis) this makes quite close my approach with his.
Actually I think it would be quite useful if, like we have done
"joining post", each of us could summarize its own approach in a
reasonably short post. The present post was just for showing you the
richness of the our hunting-dragon in the everything (modal) landscape.
If only pointing to David Lewis could motivate the people here to
invest a little more in ... (modal) logic ;)
Bruno
http://iridia.ulb.ac.be/~marchal/
Russell Standish
> It is a good occasion to sum up the main differences and the main
> similarities between Standish, Schmidhuber, Lewis, Tegmark, Levy,
> Ruhl, Mitra, Mazer, Finney, ... and my own. All approach are indeed
> form of modal realism, and this is indeed what the everything-list is
> all about.
>
Thanks Bruno - this is a useful summary
>
> Standish is not yet enough clear about its assumptions, but seems to
> get a pretty derivation of schroedinger equation, which is an
> improvement. He does assume time, with the topology of the reals, which
> is my main critics. The 1-3 distinction is present and used in an
> anthropic way, but I have not yet understood it precisely.
TIME needs to be assumed for the White Rabbit solution, but not the
topology of the reals. The latter assumption is _purely_ to make
contact with the traditional formulation of the Schroedinger
equation. What I sincerely suspect is that the real topology is the
wrong one, and that the Schroedinger equation will need to be modified
to take account of whatever topology time really has.
I cannot see that TIME is a problematic assumption. In many systems -
eg computationalism, TIME is a theorem, a consequence of other
assumptions. More problematic is what precisely it means - hence
discussions of topology etc.
What I claim is that a computationalist model is possible of my
theory. It would probably be useful to do this, as it might shed light
on where my assumptions come from, and also probably forge a link
between my work and Bruno Marchal's for example. I feel a little
inadequate for the task - but perhaps with the eyes and brains of this
group to correct me, it can be done. And you never know - I may be
wrong on the possibility of a computationalist model, which would be a
suprising result in itself.
>
> Actually I think it would be quite useful if, like we have done
> "joining post", each of us could summarize its own approach in a
> reasonably short post. The present post was just for showing you the
> richness of the our hunting-dragon in the everything (modal) landscape.
It probably requires a series of manifestos, which can then be
prodded, poked and ripped apart, and then maybe melded by this
list. We have manifestos already for Tegmark, Schmidhuber (actually 2
of these), Marchal, Malcolm and myself. Anyone else like to
contribute? I'm not sure I really understand Hal Finney's position,
for example.
Give me a few more months, and I'll have a draft manuscript of my
book ready for you to sharpen your intellectual knives on.