> From: Russell Standish <[EMAIL PROTECTED]>
> To: [EMAIL PROTECTED]
> 
> I think we got into this mess debating whether an infinite set could
> support a uniform measure. I believe I have demonstrated this. 
> I've yet to see anything that disabuses me of the notion that a
> probability distribtuion is simply a measure that has been normalised
> to 1. Not all measures are even normalisable.

Russell, at the risk of beating a dead horse: a uniform measure is _not_ a
uniform probability distribution. Why were measures invented in the first
place? To deal with infinite sets.  You cannot have a uniform probability
distribution on infinitely many things. That's why measures isolate just
finitely many things, say, every bitstring of size n, and for each x of
size n look at the infinite set of strings starting with x. A uniform
measure assigns equal probability to each such set. Of course, then you
have a uniform probability distribution on those finitely many things
which are sets. But that's not a uniform probability distribution on
infinitely many things, e.g., on the bitstrings themselves!  The measure
above is _not_ a probability distribution; it is an infinite _set_ of
_finite_ probability distributions, one for string size 0, one for string
size 1, one for string size 2,...

> I realise that the Halting theorem gives problems for believers of
> computationalism. 

It does not. Why should it?

> I never subscribed to computationalism at any time,
> but at this stage do not reject it. I could conceive of us living in
> a stupendous virtual reality system, which is in effect what your GP
> religion Mark II is. However, as pointed out by others, it does suffer
> from "turtle-itis", and should not be considered the null
> hypothesis. It requires evidence for belief.

By turtle-itis you mean: in which universe do the GP and his computer
reside?  Or the higher-level GP2 which programmed GP? And so on? But
we cannot get rid of this type of circularity - computability and
mathematical logic are simply taken as given things, without any
further explanation, like a religion. The computable multiverse, or
the set of logically possible or mathematically possible or computable
universes, represents the simplest explanation we can write down formally.
But what exactly does it mean to accept something as a formal statement?
What does it mean to identify the messy retina patterns caused by this
text with abstract mathematical symbols such as x and y?  All formal
explanations in our formal papers assume that we agree on how to read
them. But reading and understanding papers is a complex physical and
cognitive process. So all our formal explanations are relative to this
given process which we usually do not even question. Essentially, the
GP program is the simplest thing we can write down, relative to the
unspoken assumption that it is clear what it means to write something
down, and how to process it. It's the simplest thing, given this use
of mathematical language we have agreed upon. But here the power of the
formal approach ends - unspeakable things remain unspoken.

Juergen Schmidhuber

http://www.idsia.ch/~juergen/
http://www.idsia.ch/~juergen/everything/html.html
http://www.idsia.ch/~juergen/toesv2/

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