[FOM] interpretation of Chaitin's work

[Aatu Koskensilta]

On Feb 22, 2006, at 1:56 AM, Ben Crowell wrote:

> Gregory Chaitin has an article in the March Scientific American
> in which he claims that the irreducible complexity of the number
> he calls Omega "smashes hopes for a complete, all-encompassing
> mathematics in which every true fact is true for a reason."

I suggest you have a look at Torkel Franzén's book "Gödel's theorem - 
an incomplete guide to its use and abuse" in which he discusses 
Chaitin's claims. One point Torkel makes in the book is that it's 
unclear what it means for a mathematical truth to be "true for a 
reason" or "true for no reason". To me it seems that one can well say 
that claims of the form "the nth bit of Omega is 1" or "the nth bit of 
Omega is 0" are true (or false) for a reason, namely the halting or 
non-halting of certain Turing machines.

Perhaps one can make sense of "true for a reason" by taking "the 
mathematical statement A is true for a reason" to mean that A has a 
proof from principles mathematicians will (come to) accept, so that 
"the nth bit of Omega is 1" being "true for no reason" amounts to the 
claim that there is no proof from acceptable principles of this fact. I 
don't find this explication very convincing, but it's the best I can 
come up with. One problem with it is that usually if we say that 
something is true for no reason - if we ever actually say that! - we 
don't mean just that we are incapable of providing an explanation, but 
that in some sense there simply is no "reason", whatever that would 
amount to in the first place.

Of course, Chaitin's work does not tell us whether there are 
mathematical facts that are "true for no reason" in this sense, just as 
Gödel's incompleteness results do not tell us whether there are 
unprovable mathematical truths.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus