‘Either God exists, or one must accept mystery.’ | Incompleteness and Limits of Truth

Some complex ideas in this post, but I think rich rewards if you’re willing to persevere…

Catching up on some old episodes of Radiolab (a brilliant podcast, if you’re not already on it) I came across the above section in the programme about loops. In it, the presenters discuss the idea of statements that are ‘undecidable.’ The classic one is the Liar’s Paradox:

This statement is a lie

If it is true, then it is false, and if it is false, then it is true… The question as to whether the person making the statement is telling the truth is undecidable.

On the surface this appears to be little more than a linguistic parlour game, something to amuse. Yet it turns out that statements such as these present genuine problems. Alan Turing, the founder of modern computing, made his name by proving that the ‘halting problem’ was undecidable.

More importantly, as explored in the piece above, the Austrian mathematician Kurt Gödel managed to express an undecidable statement in terms of algebraic logic:

p ↔ ~Bew(G(p))

This rather impenetrable statement blew a hole in the apparently secure hull of mathematics. What Gödel had done in his ‘incompleteness theorem’ was to show that it was never going to be possible to build, from a foundational set of axioms, a mathematical framework that could prove any mathematical theorem.

Perhaps the most straightforward way of understanding the implications of Gödel’s work is to think about an imaginary printer. This printer can print statements, and these statements can be either true or false. What the incompleteness theorem shows is two things:

1) If you want to print every possible true statement, you will also need to print some false ones.

2) If you want to print only true statements, you cannot print every possible true statement.

Or, put more formally:

1) An all-encompassing axiomatic system can never be found that is able to prove all mathematical truths, but no falsehoods.

2) If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent.

The Radiolab piece – which is a good intro, and very accessible – concludes that, given Gödel, we have just two options: Either God exists – a supreme being who is in possession of the truth about everything, or we have to live in a world in which some things will continue to be a mystery.

We need to tread carefully here. Gödel was examining the foundations of mathematics, and it would be facile to make simplistic parallels with other areas of study. However, it is interesting to consider how Gödel’s work might apply in the context of religious thought. In particular, these ideas seem to present a paradox:

Only those who believe in God can adopt the scientific world-view – ie, one that can accept a world which lacks mystery and is completely open to complete human investigation and understanding. Yet, those who don’t believe in God because of their materialist-scientific world-view, will have to accept that there will be areas of investigation which will be impenetrable to human understanding, and thus, in some minor way, super-natural.

What we find instead is that this paradox of human understanding tends to be collapsed in its inversion. Many of those who believe in God do so with little doubt, and yet persist in speaking of mystery, and many of those who are atheist-scientific fundamentalists believe in the elimination of mystery and trust that we are able to understand everything.

Yet both are finding their positions wanting. And, to conclude, I think Gödel thus gives us a way in to affirming the majority of people who claim now to be ‘spiritual but not religious.’ They are the ones who are happy to accept the continuing beauty and mystery of life, without it needing to be tied down and exhausted by belief in God.