# Re: Paper+Exercises+Naming Issue

```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
variat!
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