Intuitively, I’ve always been convinced that certain truths cannot be understood rationally but must be experienced through non-standard ways of thinking (or rather non-thinking). [And similarly that such truths cannot be taught or explained in the usual sense.] This kind of belief is also at the heart of the Koans (short “meaningless” dialogs or riddles) in Zen Buddhism. But the connection to Gödel’s Incompleteness Theorem (or rather Gödel’s first such theorem) has always evaded my attention.

The Theorem says something like this:
If your system of reasoning is (a) sufficiently powerful (and, e.g., does not only consist of the single statement “a=a”) and (b) consistent (that is you can’t prove both “a” and “not a”), then there always exist truths which cannot be proven/derived/verified within the system. [If your system is not sufficiently powerful it’s boring/meaningless in the first place. If it is inconsistent, well, it’s also pretty meaningless.]

The proof of this theorem essentially uses the self-referential sentence: “This sentence cannot be proven.” If you manage to prove it, you’ve proven a false statement so your system is inconsistent. If you don’t manage to prove it, well, then your system is incomplete as the sentence is true but you cannot prove it. The interesting thing is that such a self-referential sentence can actually be constructed within a mathematical logical system (using Gödel numbers).

What does that have to do with Koans and Zen Buddhism?

Well, it gives a hint, or rather just an analogue, for why we cannot expect to derive all truths by logic alone. The Koans seem to let go of the consistency requirement and, by allowing paradoxes, still seem to help to acquire “truth”. At least if one manages to tune one’s brain into the right frame of mind.

Gödel, Escher, Bach. A great book. I very highly recommend it.