Re: Gregory Chaitin: "Randomness is the true foundation of mathematics."

Lance wrote: 
Peter H.M. Brooks wrote:

Lance wrote:

Peter H.M. Brooks wrote:

The point is if the number were an integer it would be computble. So then there must an algorithm somewhere that will give us access to any future knowledge even before it is discovered. But then no process of discovery or proof would be needed, and science would be empty. Ergo, the number cannot be computable. But only reals cannot be computed. ergo, the number has to be real.

I don't dislike the argument, it has a pleasant air of finality about it that is appealing.

Sadly, though, it confuses the term 'computable', which is a technical recognition of whether algorithms are in the set NP - that is whether their results grow in polynomial time or not - with the question of whether something actually could ever be computed, even in principle. The size of the integers that I'm talking about is so huge that the term computable is simply risible when used anywhere near them. You couldn't fit them into your computer even if your computer used every atom in the universe (nay even every quark) as part of its storage.

So we're quite safe from ever finding out what the numbers are, even though they are integers.

It's like hiding things. Yes, if you have a small garden then you need a complex maze or other tricky method to hide something well. If you do want to hide something well, though, and don't care to find it again even yourself, then the Southern Ocean is a better bet, no matter how cunning your maze.


I think you will have to read Chaitin because he offers good reasons to think these numbers are real. And he is quite indifferent to your processing speed argument.

It isn't an argument about processing speed. The capacity to even store this number in any possible format is the matter - whether you represent it as integral or real.

Surely if he has good reasons then they ought to be straightforward enough to repeat here! If they're so complicated that they can't, I can't see that they can be good.



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