[extropy-chat] what is probability?

[gts]

On Tue, 16 Jan 2007 16:06:51 -0500, Benjamin Goertzel <ben at goertzel.org>  
wrote:

I've been thinking a bit about your interesting question:

>> For our purposes I think it's fair to say a sequence is random if there  
>> is and can be no discernible pattern
>
> This is what Chaitin's definition of randomness (in the references) says.
>
> The problem is: Discernible by WHOM?  Discernibility of patterns can
> only be defined objectively for infinitely large entities.

Our answer may depend on our philosophy of mathematics.

According to mathematical constructivists [1], mathematical objects can be  
said to exist only if some procedure can be specified by which the object  
may be constructed. In the case of infinite sequences, this means there  
must exist some rule for determining the next digit in the sequence. We  
can think and speak intelligibly about the infinite decimal expansion of  
pi, for example, because although we are mere finite mortals incapable of  
discerning the entire infinite sequence, a rule can nevertheless be laid  
down by us for determining the next digit no matter how far along we are  
in it.

But can we likewise infer the existence of infinite random sequences  
(Chaitin-random or otherwise) in the same way? The answer seems to be  
"maybe not".

You've probably already anticipated the problem: as your sources and mine  
agree, a sequence is random if no betting system can be specified that  
would allow a better to make money beyond what would be expected by chance  
alone; that is, it is random if it contains no discernible (predictable)  
patterns. To prove under constructivism that an infinite random sequence  
exists that meets this criterion, we need somehow to formulate a rule for  
determining its successive digits. But any such rule would amount to a  
successful gambling system and thus prove the sequence to be non-random!

-gts

1. http://plato.stanford.edu/entries/mathematics-constructive/