Re: Chaos vs. Randomness

On 10 Dec, 22:55, "R. Baldwin" <res0k...@xxxxxxxxxxxxxxxxxxxx> wrote:
seanpitnos...@xxxxxxxxxxxxxxxxxxxxxxxxxxx wrote in news:bd21fcbd-82ee-
4f3b-99ba-ccb7cce13...@xxxxxxxxxxxxxxxxxxxxxxxxxxx:



On 9 Dec, 22:51, "R. Baldwin" <res0k...@xxxxxxxxxxxxxxxxxxxx> wrote:

[snip for brevity]

I know that Chaitin
is trying to forge this connection, but this may not be so
straightforward. It would be more conventional and more
understandable
to use "uncomputable" or "algorithmically random infinite sequence"
here.

These are connected concepts.

They might be.

Algorithmic randomness/uncomputability
cannot be proven - for a particular sequence.  

This is not quite right,Sean. Algorithmic randomness and
uncomputability are not the same thing. Finite numbers can be
algorithmically random with respect to a given machine, but all finite
sequences are computable.

I'm not talking about finite sequences or one particular machine. I'm
talking about the option of using any and all machines you can imagine
on an infinite sequence. It is in this sense that neither randomness
nor uncomputability can be proven.

Algorithmic randomness can be proven for a reasonably small set of
finite sequences by exhaustive search. The search aborts when the first
non-halting program is attempted.

Again, I'm talking about proving that a finite sequence was in fact
produced by a truly random number generator. That hypothesis can be
suspected, by cannot be completely proven.

A correct statement is that "it is not possible to create a general
algorithm capable of determining whether sequences are algorithmically
random."

Exactly . . .

Uncomputability cannot be proven for a sequence of digits, for the
trivial reason that the sequence must be infinite to be uncomputable,

I don't think this is "trivial" at all . . .

and you will not live long enough to see all the digits. Computability
can be proven by showing that a sequence is finite.

Some infinite sequences can also be proven to be computable as well.

This is why the origin
of a particular sequence cannot be known, based on the sequence
itself, to be "deterministic" or truly random.  

Yes, this is certainly correct.

What are your arguing about then?

The very best you can
say is that the sequence has the "appearance of randomness"  - - - as
per my original point.

I'm curious - how would a radioactive mutagen deterministic?

Not all mutations are based on radioactivity - for one thing.  But,
a
s
far as the "randomness" of radioactivity and quantum mechanics, I'm
not sure that this is provable.  Could be that even radioactive
decay
is actually determinable, but only appears random because we just
don't know enough.

The Bell inequality experiments suggest not.

All kinds of things "suggest" randomness.  I'm just not convinced that
this suggestion is anything more than that.  I don't think randomness
can be proven with absolute certainty.

That might be correct.

That's all I'm suggesting . . .

That is irrelevant. Read the phrase under discussion carefully: "with
chaos and nonlinear dynamics we've now realized that even in
classical
physics we get randomness and unpredictability."

Classical physics is based on deterministic assumptions. In other
words,
what Chaitin wrote about in that phrase WAS "deterministic chaos."

Chaitin's "Omega" is at maximum algorithmic randomness - by
definition.  Does this mean that randomness is "deterministic"?  If
so, then "randomness" can be definitively known given enough
experience with the TM that produced the algorithmically random
sequence.

Why would you conclude that all random numbers including all the
uncomputable reals could be produced by Turing machines? That is the
implication of what you've just written.

It depends on what you mean by "random numbers"? Is Omega a random
number? - Is it "determinable"?

So, is the universe the product of at least some component of
algorithmic randomness? - the output of a TM of some sort?  If so, us
the universe "deterministic"?  

This bit seems to be confusing several concepts again - algorithmically
random vs. computability.

Personally, I find this uninteresting science fiction. The universe
could just as easily be a vast experiment created by hyper-intelligent
pan-dimensional mice.

Correct . . .

Or, are our ideas that at least some
aspects or features of our universe are not inherently deterministic
or knowable, regardless of any prior experience with the universal TM,
more an appearance of reality than reality itself?  Is it possible for
a deterministic TM to simulate non-determinism so well that we simply
cannot tell the difference?

In principle, that is possible. As a practical matter, it is
unimportant.

Who is talking about practical matters here? This has obviously been
a purely theoretical discussion from the beginning . . .

You do know, of course, that
deterministic TMs can indeed simulate
non-deterministic TMs.  So, how to
tell the difference given only the
output of a particular string?

I've read and re-read this, and it isn't making any sense to me. Why do
you think the output has anything to do with whether the Turing machine
was deterministic  or non-deterministic?

I'm just asking if it is possible to tell anything about the origin
from the output for sure? - with absolute certainty. I don't think
so. The very best we can do is make less than perfect hypotheses about
the nature of the origin of a given sequence string.

Sean Pitman
www.DetectingDesign.com

.

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