[FOM] interpretation of Chaitin's work
praatika at mappi.helsinki.fi
Wed Feb 22 08:31:49 EST 2006
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;
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
" 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,
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".
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy,
University of Helsinki
Institute of Philosophy,
School of Advanced Studies,
University of London
E-mail: panu.raatikainen at helsinki.fi
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