Russell, list,

[Russell]>>> The particular Plenitude I assume (ensemble of all bitstrings) is 
actually a completely uninteresting place to have a view of (it has precisely 
zero informational complexity).
[Ben]>> Is this kind of Plenitude (ensemble of all bitstrings) more or less 
Tegmark's Level IV of all mathematical structures? (I.e., if it's different, 
does the difference involve a restriction to discrete or finitistic structures 
or some such? 
[Russell]> It does correspond to Tegmark's level 4, but Tegmark's proposal "All 
mathematical structures" is rather vague. I have interpreted his proposal as 
"all finite axiomatic systems". This is in fact a subset of my ensemble (well 
basically Schmidhuber's ensemble) of all descriptions (since an FAS is a 
description), yet one can also describe the entire ensemble of descriptions by 
a finite method (the "dovetailer"), hence one can find the ensemble of all 
descriptions contained within Tegmark's.

The "dovetailer" keeps sounding like a powerful idea. I do remember that it has 
often been mentioned here, but somehow I failed to pick up a sense of what it 
was really about. Was there a message to the Everything-List in which it was 
explained so that non-experts can understand it? I'm not asking you to track 
that message (or series of messages) down, but if you or somebody remembers 
around which month it was, that should be enough for me to find it. Or is there 
a link to a Webpage with such an exposition?

[Russell]> Note, however that the relationships going both ways do _not_ imply 
equivalence between the two ensembles. This is described in my paper "Why 
Occam's Razor", as well as talked about on the everything list.

[Ben]>> ....
>> IV. possibility waves (variational principles)
>> III. probabilities for various outcomes 
>> II. information, news, outcomes, events, interactions, phenomena
>> I. evidence of causes/dependencies (dependencies, e.g., emission --> open 
>> slit --> hit)

[Russell]> I'm somewhat sceptical of your associations here, but it is possibly 
because I don't understand what you're getting at. You may need to develop this 
some more.

I think I may have made it sound more like my own idea than what it actually 
seems to me. First, here's the part where I haven't thought that I was going 
out on a limb:

Level III varies across quantum branchings. Level II varies across times and 
places along a single quantum branch in such a way that its features come out 
the same as Level III's features.
Where the experimental setup remains the same, successive particles emitted 
make, collectively, a pattern of hits such that the pattern corresponds to the 
probability distribution for particle hits. The pattern consists of variations 
of hit times and locations along a single quantum branch, such that the pattern 
reflects the variations across quantum branchings. So there you see Level III & 
Level II aspects.

At Level I, individual histories are especially important and that which 
happens is partly attributable to "idiosyncrasies" of one's Level I universe, 
such that one must study its individual history and explain things by 
historical causes. I'm not sure whether I've said enough there, so if the 
following is redundant, I apologize. If there's something arbitrary about a 
Level I universe's constants and initial conditions, then there is variation, 
across Level II, among Level I universes or inflationary bubbles (I'm saying 
"Level I universe" instead of "Level I multiverse" because you've said that 
Tegmark's use of the word "multiverse" isn't standard; but if "universe" has a 
technical sense here that confuses the issue, then I don't mean it in that 
technical sense). But whether it's an issue of constants and initial conditions 
or of something else if anything at all (Tegmark seems unsure), a variation 
across inflationary bubbles in a Level II universe means that there are aspec!
 ts of our Level I universe which are part of a pattern of variation such that 
our Level I universe is not a representative sample of instances (like the 
pattern of accumulated particle hits) but instead a single instance (a single 
particle hit). To establish what are these "arbitrary" aspects, we vary the 
experimental setup and seek the constants and infer relationships between 
patterns of outcomes and the variations among initial conditions of various 
experiments. Now we're establishing the significance of variations across 
various evolutions of the possibility waves into which the various experimental 
setups were factored and finding that some relationships (or aspects of 
relationships, the particular quantities involved, etc.) seem arbitrarily to 
impose themselves -- seemiingly arbitrarily set constants of nature etc. 
(Again, if the fundamental constants and initial conditions of our Level I 
universe turn out not to be somewhat arbitrary and turn out not to mark a 
 ion across Level II, then there would be to seek out other thi!
 ngs whic
h do mark such variation, though I have no idea what and I admit that the 
overall picture seems weakened in such a case.) 

Now, Level III's quantum branchings represent probabilities. With Level II, the 
"hits" in their pattern along a single quantum branch are pieces of information 
quantifiable in terms of what was their probability before they happened. If 
one does not have the Level III probability distributions, then the patterns 
are news of those distributions. But if one does have those probability 
distributions, then what remains news is their appearance as a "biased coin" 
(if that's the right metaphor) which we explain by assembling and fitting 
together the logical puzzle pieces of history and initial conditions, with our 
our Level I universe explained as being an instance of Level II variation and 
as not being a fully representative sample of Level II variation.

Now, here's where I've thought that I was going more out on a limb. I've 
extended this pattern of correlations backward in order to suppose that there's 
some sort of association between Level IV and the possibility wave in its 
evolution. I've noticed that the formalism for this possibility wave involves 
variational principles, optimizational equations, in crucial ways. It's about 
least action, mathematical stationary points, etc., not that I understand this 
stuff at all well. I've noticed that the series "optimization, probability, 
information, logic" has, among its pure mathematical correlates, respectively, 
many-to-many relationships (graphs, extremization, etc.), one-to-many 
relationships (antiderivatives, integrals, measure), algebra, groups 
(many-to-one relationships and functions), and one-to-one relationships 
(ordered structures). In other words, though I'm out on  limb in saying this, I 
didn't need the particle/wave's career in order to formulate that series, 
  it seems to me that it's not a random series but instead has its own logic. 
Actually I arrived at it independently of considerations about particle 
experiments. (Well, "optimization" is huge and not all of it is a deductive 
mathematical research area, they experiment with ants and so on. But some of it 
seems to be. Maybe the others would claim that they're just already doing that 
which Chaitin called on mathematicians to do, slacken the rigor and explore :-) 
) Anyway, I tend to think that there's an interesting, not-too-fragile, and, in 
its way, exhaustive pattern there, and such seems appropriate insofar as we're 
talking about the whole shebang, one in which, in some sense, everything 
exists. (I'm not saying that I believe that Tegmark is right or that the idea 
that "everything exists" is right -- I'm agnostic there). Interestingly, even 
apart from these grand cosmologies, the pattern does seem to be there with 
regard to stages in the particle's career in an experiment.

And, with regard to Level IV, that's also where I feel out on a limb. I can 
see, in a layman's vague way, that variations of mathematical structure would 
involve variations of what count as shortest paths and extrema, and that this 
would ramify into the possibility wave insofar as it involves optimizational 
equations. This also may mean that the "biased coin" which I mentioned reflects 
not only variation across various Level I universes, but across the 
mathematical structures of Level IV. I tend to think that these two kinds of 
"bias" would be rather distinct, but I don't know how to think about it. A 
difference in the mathematical _structure_ of the "coin," as opposed to a 
difference in its probability distributions (as reflecting quantitative 
differences, across Level I universes, in relations among constants) would, I 
guess, be something rather stronger than a mere "bias." A different _structure_ 
of constants is a whole different "coin"?

I wouldn't know how to modify this in accordance with your version of Level IV, 
my understanding of it is significantly weaker than my already vague 
understanding of Tegmark's Level IV.

But I haven't noticed anybody here talking about variational principles or 
optimizational equations in any connection, much less in relation to Level IV. 
(While there is an obvious echo of optimization in applying Occam's Razor to 
Level IV's mathematical structures, this doesn't seem to involve any 
application of mathematical extremization, variations, Morse Theory, etc., so 
it seems not really the same thing. It's certainly not the only echo between a 
mode of inference (present instance: surmise, simplest explanation) and a 
mathematical formalism (extremization, shortest paths, etc.).)

Well, that's quite enough. I wish I could have made it briefer. Thank you for 
your patience.

Best, Ben Udell

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