You mentioned below how Chaitin is campaigning randomness within mathematics, however I wonder if Chaitin is using an easily misdirected term by selecting the term 'randomness' versus 'irreducibility'. Random events are by definition irreducible, but my question is: is every irreducible event random?
In my view, randomness has a few specific qualities that not all irreducible concepts must necessarily share:
1) Randomness is mainly used as a state of human knowledge, such as the results of 'coin flips', however such knowledge could be filled in an the events are no longer random (e.g., we might have enough information to know if a coin will turn up heads or tails).
2) It implies not only unpredictability, but unrestricted in terms of what the result could be. For example, a coin toss can really be heads or tails. If I'm not mistaken, an uncomputable number actually exists, we just can't know what it is because there is no *computable* function that allows us to compute the particular number. But, there is still an *uncomputable* function that has an exact answer if it was computable - hence there is an inherent restriction to that number. [Correct me if I'm wrong on this point].
3) Random events obey the 'laws' of probability, however these laws are non-applicable to the irreducible numbers discussed by Chaitin. To know that you've found something uncomputable would mean that you know that there is an associated uncomputable function, which the only way you would know that is if you could compute the function. Hence, probablity is not helpful.
Warm regards, Harv