Hi, So I was just going through my copy of The Emperor's New Mind, and I'm having a little difficulty accepting Godel's theorem , at least the way Penrose has presented it. If I'm not wrong, the theorem asserts that there exist certain mathematical statements within a formal axiomatic system that are true, but cannot be proven through the axioms and rules of procedure laid down in the formal system. This is, as I understand it, a blow to formalism because it implies that insight or intelligence is needed to demonstrate truth for non-computable ( or non-recursive ?? I'm really confused as to where the difference lies ) mathematical statements. Two problems here, first, why is it unreasonable to adopt the position that those statements are neither true nor false. The argument Penrose presents is reductio ad absurdum. He shows that it cannot be false because that would be absurd, hence it has to be true. But doesn't a true statement which is not provable also ring absurdity ? If I remember correctly, there was an active philosophical stance at the time which asserted that we cannot speak of truth or falsity unless it has been established to be so one way or another. So is it wholly unreasonable to say that those statements are neither true nor false ? Further, doesn't Godel's theorem also establish something about the undecidability of some problem( that the problem cannot be proven nor disproven, ) ? Is this really different from ' neither true nor false'? I also don't really understand how the two 'parts' of Godel's theorem are related, if they are at all. P.S I hope this isn't in the wrong place, but I couldn't find a mathematical philosophy thread in PhysicsForums.