> 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/
