Paddy Leahy writes:
> Let's suppose with Wei Dai that a measure can be applied to Tegmark's 
> everything. It certainly can to the set of UTM programs as per Schmidhuber 
> and related proposals.  Obviously it is possible to assign a measure which 
> solves the White Rabbit problem, such as the UP.  But to me this procedure 
> is very suspicious.  We can get whatever answer we like by picking the 
> right measure.  While the UP and similar are presented by their proponents 
> as "natural", my strong suspicion is that if we lived in a universe that 
> was obviously algorithmically very complex, we would see papers arguing 
> for "natural" measures that reward algorithmic complexity. In fact the 
> White Rabbit argument is basically an assertion that such measures *are* 
> natural.  Why one measure rather than another? By the logic of Tegmark's 
> original thesis, we should consider the set of all possible measures over 
> everything. But then we need a measure on the measures, and so ad 
> infinitum.

I agree that this is a potential problem and an area where more work
is needed.  We do know that the universal distribution has certain
nice properties that make it stand out, that algorithmic complexity
is asymptotically unique up to a constant, and similar results which
suggest that we are not totally off base in granting these measures the
power to determine which physical realities we are likely to experience.
But certainly the argument is far from iron-clad and it's not clear how
well the whole thing is grounded.  I hope that in the future we will
have a better understanding of these issues.

I don't agree however that we are attracted to simplicity-favoring
measures merely by virtue of our particular circumstances.  The universal
distribution was invented decades ago as a mathematical object of study,
and Chaitin's work on algorithmic complexity is likewise an example of
pure math.  These results can be used (loosely) to explain and justify
the success of Occam's Razor, and with more difficulty to explain why
the universe is as we see it, but that's not where they came from.

Besides, it's not all that clear that our own universe is as simple as
it should be.  CA systems like Conway's Life allow for computation and
might even allow for the evolution of intelligence, but our universe's
rules are apparently far more complex.  Wolfram studied a variety of
simple computational systems and estimated that from 1/100 to 1/100000 of
them were able to maintain stable structures with interesting behavior
(like Life).  These tentative results suggest that it shouldn't take
all that much law to create life, not as much as we see in this universe.

I take from this a prediction of the all-universe hypothesis to be that
it will turn out either that our universe is a lot simpler than we think,
or else that these very simple universes actually won't allow the creation
of stable, living beings.  That's not vacuous, although it's not clear
how long it will be before we are in a position to refute it.

> I've overlooked until now the fact that mathematical physics restricts 
> itself to (almost-everywhere) differentiable functions of the continuum. 
> What is the cardinality of the set of such functions? I rather suspect 
> that they are denumerable, hence exactly representable by UTM programs.
> Perhaps this is what Russell Standish meant.

The cardinality of such functions is c, the same as the continuum.
The existence of the constant functions alone shows that it is at least c,
and my understanding is that continuous, let alone differentiable, functions
have cardinality no more than c.

> I must insist though, that there exist mathematical objects in platonia 
> which require c bits to describe (and some which require more), and hence 
> can't be represented either by a UTM program or by the output of a UTM.
> Hence Tegmark's original everything is bigger than Schmidhuber's.  But 
> these structures are so arbitrary it is hard to imagine SAS in them, so 
> maybe it makes no anthropic difference.

Whether Tegmark had those structures in mind or not, we can certainly
consider such an ensemble - the name is not important.  I posted last
Monday a summary of a paper by Frank Tipler which proposed that in fact
our universe's laws do require c bits to describe themm, and a lot of
other crazy ideas as well,
http://www.iop.org/EJ/abstract/0034-4885/68/4/R04 .  I don't think it
was particularly convincing, but it did offer a way of thinking about
infinitely complicated natural laws.  One simple example would be the fine
structure constant, which might turn out to be an uncomputable number.
That wouldn't be inconsistent with our existence, but it is hard to see
how our being here could depend on such a property.

Hal Finney

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