[FOM] interpretation of Chaitin's work

[praatika@mappi.helsinki.fi]

Quoting Ben Crowell <fomcrowell06 at lightandmatter.com>:

> 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."
> He also has a popular-level treatment published in book form,
> and the book more cautiously notes that his interpretation of
> his own work is controversial among philosphers of mathematics.
> Any opinions? 

As many here know, I have criticized Chaitin's claims in several papers; 
see e.g.

http://www.helsinki.fi/collegium/eng/Raatikainen/rev-panu.pdf
http://www.helsinki.fi/collegium/eng/Raatikainen/AITsynthese.pdf
 
some of these issues were also discussed here in FOM few years ago. 


> It doesn't seem surprising to me that there are mathematical
> truths that are true, but not "for a reason." 

Chaitin made such claims about "true for no reason" for a long time without 
any explanation of what that is supposed to mean. Recently, he has given 
one:

" The normal idea is that if something is true, it's true for a reason –... 
Now in pure math, the reason that something is true is called a proof, and 
the job of the mathematician is to find proofs, to find the reason 
something is true. But the bits of this number W, whether they're 0 or 1, 
are mathematical truths that are true for no reason, they're true by 
accident! And that's why we will never know what these bits are." Chaitin, 
Exploring Randomness.

However, as such, these claims are unjustified. Chaitin seems to conflate 
provability in general (whatever that is) and derivability in a particular 
axiom system. Nothing in his work justifies the claim that a sentence 
stating the value of some digit of Omega is absolutely unprovable (whatever 
that means).  On the other hand, every such fact is provable in some axiom 
system. And surely, if we are forced to stick to a particular axioms 
system, the issue gets trivial. Godel's theorem is sufficient, we don't 
need Chaitin, Algorithmic Information Theory, or Omega. 

In another context, Chaitin explains the idea differently:

“So the bits of Omega are irreducible mathematical facts, they are 
mathematical facts that contradict Leibniz's principle of sufficient reason 
by being true for no reason. 
 They cannot be deduced as consequences of 
any axioms or principles that are simpler than they are.”
Chaitin, “Leibniz, Information, Math and Physics”

So the idea seems to be that A is "true for no reason" if A is not 
derivable from any sentence B which is less complex than A. But then it is 
trivial that there are truths "for no reason" - we don't need Omega or 
anytithing for seeing that. Moreover, many beatufully simple theorems in 
mathematics are such - provable only on the basic of a set of axioms more 
complex than the theorem. But it is quite preposterous to call such 
theorems "true for no reason". 

Best, Panu 


Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy, 
University of Helsinki
Finland

Visiting Fellow, 
Institute of Philosophy,
School of Advanced Studies, 
University of London

E-mail: panu.raatikainen at helsinki.fi
 
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm