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From: <juergen.domain.name.hidden>

Date: Thu, 25 Oct 2001 09:53:12 +0200

*> From: Russell Standish <R.Standish.domain.name.hidden>
*

*> To: juergen.domain.name.hidden
*

*>
*

*> 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/

Received on Thu Oct 25 2001 - 00:53:45 PDT

Date: Thu, 25 Oct 2001 09:53:12 +0200

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,...

It does not. Why should it?

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/

Received on Thu Oct 25 2001 - 00:53:45 PDT

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