I seek a number that cannot be calculated. This number will become my line of flight out of the madhouse of a world…
Gregory Chaitin once wrote a paper on The Limits of Reason implying that Ideas on complexity and randomness originally suggested by Gottfried W. Leibniz in 1686, combined with modern information theory, imply that there can never be a “theory of everything” for all of mathematics.
So perhaps mathematicians should not try to prove everything. Sometimes they should just add new axioms. That is what you have got to do if you are faced with irreducible facts. The problem is realizing that they are irreducible! In a way, saying something is irreducible is giving up, saying that it cannot ever be proved. Mathematicians would rather die than do that, in sharp contrast with their physicist colleagues, who are happy to be pragmatic and to use plausible reasoning instead of rigorous proof. Physicists are willing to add new principles, new scientific laws, to understand new domains of experience. This raises what I think is an extremely interesting question: Is mathematics like physics?
– Gregory Chaitin and The Limits of Reason
In 1956 Scientific American published an article by Ernest Nagel and James R. Newman entitled “Gödel’s Proof.” Two years later the writers published a book with the same title—a wonderful work that is still in print. Chaitin was a child, not even a teenager, and he was obsessed by this little book. He remembered the thrill of discovering it in the New York Public Library. He used to carry it around with him and try to explain it to other children. As he’d say:
It fascinated me because Kurt Gödel used mathematics to show that mathematics itself has limitations. Gödel refuted the position of David Hilbert, who about a century ago declared that there was a theory of everything for math, a finite set of principles from which one could mindlessly deduce all mathematical truths by tediously following the rules of symbolic mathematical logic. But Gödel demonstrated that mathematics contains true statements that cannot be proved that way. His result is based on two self-referential paradoxes: “This statement is false” and “This statement is unprovable.” (For more on Gödel’s incompleteness theorem, see box.)
My attempt to understand Gödel’s proof took over my life, and now half a century later I have published a little book of my own. In some respects, it is my own version of Nagel and Newman’s book, but it does not focus on Gödel’s proof. The only things the two books have in common are their small size and their goal of critiquing mathematical methods.
Unlike Gödel’s approach, mine is based on measuring information and showing that some mathematical facts cannot be compressed into a theory because they are too complicated. This new approach suggests that what Gödel discovered was just the tip of the iceberg: an infinite number of true mathematical theorems exist that cannot be proved from any finite system of axioms.
For the details read here. To cut to the short version I quote:
- Kurt Gödel demonstrated that mathematics is necessarily incomplete, containing true statements that cannot be formally proved. A remarkable number known as Ω reveals even greater incompleteness by providing an infinite number of theorems that cannot be proved by any finite system of axioms. A “theory of everything” for mathematics is therefore impossible.
- Ω is perfectly well defined and has a definite value, yet it cannot be computed by any finite computer program.
- Ω’s properties suggest that mathematicians should be more willing to postulate new axioms, similar to the way that physicists must evaluate experimental results and assert basic laws that cannot be proved logically.
- The results related to Ω are grounded in the concept of algorithmic information. Gottfried W. Leibniz anticipated many of the features of algorithmic information theory more than 300 years ago.
Gregory Chaitin is a researcher at the IBM Thomas J. Watson Research Center. He is also honorary professor at the University of Buenos Aires and visiting professor at the University of Auckland. He is co-founder, with Andrei N. Kolmogorov, of the field of algorithmic information theory. His nine books include the nontechnical works Conversations with a Mathematician (2002) and Meta Math! (2005). When he is not thinking about the foundations of mathematics, he enjoys hiking and snowshoeing in the mountains.