 We see from this analysis what Lakatos calls the “dialectical unity of proofs and refutations”. Counterexamples help us to improve our proof by finding hidden lemmas. And proofs help us improve our conjecture by finding conditions on its validity. Either way, or both ways, mathematical knowledge grows. And as it grows, its concepts are refined. We begin with a vague, unarticulated notion of what a polyhedron is. We have a conjecture about polyhedra and an informal proof of it. Counterexamples or refutations “stretch” our original concept: is a picture frame a genuine polyhedron, or a cylinder, or two polyhedra joined along a single edge?